# Tag Info

33

Lagrange enrolled to the university at the age of 14, to study law. But he quickly switched to mathematics. According to his biography, his mother was surprised when the French ambassador presented himself to congratulate her and give her the prize of the Paris Academy which Lagrange won in a public competition (by correspondence). (His mother was surprised ...

22

Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $1$-$1$, but $2$-$1$. Klein in Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree (1888) replaced the unit quaternions by $2 × 2$ unitary ...

19

This question was actually discussed on this site several times, for example here: When was the vector notation in physics and other sciences first introduced? It indeed looks strange to modern people that this simple idea came so late. Maxwell never uses vectors in his Treatise on electricity and magnetism, which makes his notation somewhat clumsy. In fact ...

19

It is not random. These names are of Greek origin, and -ic or -ics are Anglicizations of the Greek suffix -ikos, which meant "pertaining to". In other languages it can be rendered as -ika or -ica, Wolfram's "Mathematica" uses such a version. From the Online Etymology Dictionary: "-ics in the names of sciences or disciplines (acoustics, aerobics, ...

16

The letter Ш (sha) of the Cyrillic alphabet is widely accepted in theoretical computer science as the symbol for the shuffle product, which gives the shuffle algebra. The same letter is also used to denote the Tate-Shafarevich group, but I'm not sure if it's really a standard (the letter was introduced by Cassels only in 1990 in 1962 instead of TS, see below ...

15

No, it is not widely accepted. The language of "crises" is rather obsolete and mostly reflects the attitudes of the early 20th century projected backwards. At that time, the contemporaries did indeed characterize the situation in mathematics (and physics) as a crisis. For example, Weyl's 1920 address was titled “The new foundational crisis in mathematics”, ...

14

Pretty much every mathematics textbook (school or college) before the early 1980s (and many even up to the late 1980s), at the algebra level or above, as well as many (most?) chemistry and physics and engineering textbooks, had such a table at the back of the book (as an appendix or something, where "selected answers" and "index" and "glossary" would appear)....

14

The same person who introduced it, Cayley. Sylvester first used the term "matrix" (womb in Latin) for an array of numbers in 1848, but did not do much with it. Cayley started developing matrix algebra in 1855 and summarized his theory in A Memoir on the Theory of Matrices (1858). In the opening paragraphs he writes: "It will be, seen that matrices (...

14

Archimedes' books, Stomachion and The Method of Mechanical Theorems were lost until rediscovered in 2006. The only known copy is the Archimedes Palimpsest. These two texts comprise many theorems. The Method describes Archimedes' very early use of Riemann sums to compute areas and a variation of Dedekind cuts (via a pair, one a strictly monotonically ...

12

Georg Cantor was a German mathematician who created set theory that has become a fundamental theory in mathematics. However, his original work on infinity and ordinal number was considered unconventional or even unorthodox, which was under heavy attacks from some of the famous contemporary mathematicians for a long time. Another example is Charles Sanders ...

12

The Grothendieck Circle site suggests a more innocent explanation for the loss these letters. Having left Montpellier in 1984.... In May of that year [1985] a secretary informed him that his office on the fourth floor of the institute had been cleared out. Seeing this incident as an egregious example of the general decline of mores, an outraged ...

11

The idea, yes, Aryabhata speculated about something like that as early as c. 500 AD, Brahmagupta called it gurutvākarṣaṇ. So did Kepler, at about the same time as Ahmad Baba al Massufi (late 1500-s), and much less vaguely. Russo even ascribes the idea to Hipparchus (c.150 BC), although this is far fetched. Even the inverse square law for gravity predates ...

10

Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite ...

10

It is sometimes asserted that $\varnothing$ for the empty set was introduced by Bourbaki using a Danish and Norwegian letter. EDIT: The source is the Weil autobiography, cited in Jeff Miller's collection of the origins of mathematical expressions: André Weil (1906-1998) says in his autobiography that he was responsible for the symbol: Wisely, ...

9

The formula for $\Gamma$ function which is valid for all complex values of the argument was introduced by Euler, $$\Gamma(z)=\frac{1}{z}\prod_{n=1}^\infty\left\{\left(1+\frac{1}{n}\right)^z\left(1+\frac{z}{n}\right)^{-1}\right\}.$$ This is how he originally defined it, and then he showed that it is equal to the "Euler's integral". Of course at the time of ...

9

Hermann Cäsar Hannibal Schubert invented what is called "Schubert calculus" (publ. in 1879). His highly original method was not sufficiently justified. One of the Hilbert problems (1900) was to justify Schubert Calculus. This was achieved in principle in the 1920s and the interest to the Schubert Calculus declined, and it was never very strong before that. ...

9

Before Hamilton (1847) one should cite Euler (1771), Gauss (1819), Rodrigues (1840), and Cayley (1845). Detailed references in e.g. Pujol, J., Hamilton, Rodrigues, Gauss, quaternions, and rotations: a historical reassessment, Commun. Math. Anal. 13, No. 2, 1-14 (2012). ZBL1268.01010. Specifically, to four numbers $p,q,r,s$ with $pp+qq+rr+ss=u$, Euler (1771,...

9

This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations. Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

9

There are several non-alphabetic symbols, the best known is the integral sign $\int$ and the Weierstrass $P$-function $\wp$. To be sure their origins are letters of Latin alphabet, but they are special stylized symbols, and as far as I know there is no computer code for them in the standard sets of computer characters. Strictly speaking they do not belong to ...

8

It would seem to be Euler. Dickson, in Ch. XVI of History of the Theory of Numbers, writes the following: "Euler discussed the numbers $a$ for which $a^2+1$ is divisible by a prime $4n+1=r^2+s^2$. Let $p/q$ be the convergent preceding $r/s$ in the continued fraction for $r/s$; then $ps-qr=1$. Thus every $a$ is of the form $(4n+1)m\pm k$, where $k=pr+qs$. ...

8

The idea of coding ruler and compass constructions algebraically was more or less known to Vieta, as described in Supplementum Geometriae (1593), see Viète's Relevance and his Connection to Euler. It was developed further by Descartes in La Geometrie (1637), where the idea of constructing segment arithmetic (including square roots) geometrically also appears....

8

Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt: "Invariant integration on one or another special class of groups has long been known and used. A detailed computation of the invariant integral on $\mathfrak{SD}(n)$ was given in 1897 by HURWITZ [1]. ...

8

It is a fun method but it appears to be very recent. It is characterized as Chinese, Japanese, Korean, Indian, or even Mayan method in various internet posts, all of them recent, and without attribution, naturally. The "ancient origin" story is most likely made up, it is reminiscent of the Lucas's towers of Hanoi hoax, still credited on the internet to the "...

7

Alexis Clairaut designed and led the detailed calculations of planetary perturbations to estimate the return-date in 1759 of what then became known as Halley's comet. The most detailed account of the work (in which Clairaut was assisted in the calculations by the young Jerome de Lalande and by Mme Lepaute) seems to be that given by Curtis Wilson in the ...

7

Poincare's influence is acknowledged by Fatou in his dynamics papers, but not the work on 3 body problem. Fatou was inspired by another work of Poincare, namely on Kleinian groups. The analogy between iteration of rational functions (Fatou) and Kleinian groups (Poincare) also plays an important role in the modern development. By the way, Fatou wrote the ...

7

According to Klein, the first mathematician who considered space-time as a 4-dimensional manifold was Lagrange, but these ideas were not immediately developed by others. Then he mentions Cauchy and Cayley (Cayley published in 1844 "Chapters on the analytic geometry in $n$ dimensions"), and credits Grassmann (1844) with the first systematic exposition. While ...

7

What motivated Euler was not any problem in particular, but rather the general need to solve differential equations approximately when an analytic solution could not be found. He explains the method in a general form in Section 2, chapter VII of volume I of Institutionum Calculi Integralis (Foundations of Integral Calculus, 1768), his textbook on integral ...

7

Yes, he did, multiple times. Singular double integrals (1814) In Mémoire sur les intégrales définies (1814) Cauchy studied why switching the order of integration in a double integral can sometimes lead to different results. This led him to introduce the notion of "singular integral". For $K = \frac{z}{x^2 + z^2}$ he showed that \$\int_0^1\int_0^1\frac{\...

7

Feynman is being... liberally creative. What he says is his own interpolation that "makes sense" from the perch of today. "Must have been psychologically wonderful", perhaps, but "freeing of man from the intimidation of the ancients" is not how the men of Renaissance generally felt. The intimidation they sought the freeing from was not of the ancients, but ...

7

I did once write an email to Robert Geroch, because an unpublished paper of his was listed as a bibliographical reference in a paper of Hajicek, which contained a few theorems regarding non-Hausdorff spacetimes. He apparently had no recollection of this paper, and it was never published. I assume there are many other such papers, the myriad of unpublished ...

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