32

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, ...


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)....


11

The original motivation, and an inkling of the integral formula, came from Cauchy's attempts to compute improper real integrals. Here is from A Brief History of Complex Analysis in the 19th Century: "Cauchy’s first work on complex integration appeared in an 1814 paper on definite integrals (improper real integrals) that was presented to the Institute but ...


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

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 ...


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 ...


9

The systematic modern terminology and presentation of the Galois theory is due to Artin, a part of his joint project with Emmy Noether to reformulate the "concrete" older algebra in abstract terms, inspired by Dedekind. It was Artin who finally detached the Galois theory from the problem of solving equations algebraically, and gave a presentation that freely ...


9

"Corollary" is similar to the word "bonus": a little extra (i.e. an extra proposition coming from a demonstration). The term Euclid uses is πόρισμα "porism," which Liddell-Scott-Jones cite as akin to πορίζω in the sense of "to find (money)." For instance, after I.15: Πόρισμα ἐκ δὴ τούτου φανερὸν ὅτι, ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς πρὸς τῇ ...


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 ...


8

The following is a slightly edited version of my 31 January 2003 sci.math post archived at Math Forum. Riemann [6] introduced his integral in his December 1853 Habilitationsschrift thesis. In his thesis he also gave an example, correctly verified, of a Riemann integrable function whose discontinuities form a dense set. Riemann's thesis wasn't widely known ...


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 "...


8

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,...


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

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. ...


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

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


6

I found no trace of factoring polynomials being done before Descartes, in his La Géométrie, written in 1637. In it, he wrote: It is evident from the above that the sum of an equation having several roots [that is, the polynomial itself] is always divisible by a binomial consisting of the unknown quantity diminished by the false roots. In this way, the ...


6

It was known even before Newton and Leibniz that areas under curves can be found by inverting the "computation of derivatives" (drawing tangents). In explicit geometric form this "fundamental theorem of calculus" was derived by Newton's teacher Barrow, see Barrow's Fundamental Theorem by Wagner. Newton and Leibniz developed explicit symbolic methods for ...


6

Cauchy made integral rigorous, and proved that integral (in the sense of Cauchy) exists for continuous functions. Riemann proposed a more general definition, (integral in the sense of Riemann) and introduced the new class of functions, which are called now Riemann-integrable. This class is strictly larger than continuous (or piecewise-continuous) functions. ...


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