69

The other answer is correct. In addition, there is significant evidence that Fermat did not have a proof of the theorem now known as Fermat's Last Theorem. First, we should note that Fermat was not a professional mathematician, only an amateur. He never published any mathematics himself. With just that, it would not seem strange that he did not publish his ...


54

It may seem strange but logarithms were invented much earlier. Napier used the base $(1-10^{-7})^{10^7}$ which is very close to 1/$e$ (within 0.00000002 of 1/$e$). Number $e$ (as a limit) was formally defined by Euler approximately 100 years after Napier. Napier's MIRIFICI LOGARITHMORUM CANONIS CONSTRUCTIO (English translation by Ian Bruce) contains tables ...


40

An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to ...


40

It's not exactly the war, but the Nazi regime more generally that caused the decline of Göttingen. When the Nazi's came to power in 1933, they started implementing antisemitic measures quite quickly. An important step in the Nazi policy was what is now dubbed 'the great purge of 1933', which (basically) aimed to expel all Jews from positions in government or ...


40

Just prior to Grothendieck's entry to the subject, Weil had gotten terrific results in number theory by algebro-geometric arguments, and pointed the way to far more, but some of his methods went beyond existing rigorous foundations. He aimed to supply new foundations adequate to his ideas. Around the same time Zariski and van der Waerden were also ...


36

Documents like the one you linked to were not typed on a typewriter. When writing on typewriters, it was common to leave some space in the document in which the formulas could be inserted by hand. For professional publications, this was then given to a printer (the person, not the machine on your desk ;-)) who hot metal typeset the document. Printers had ...


32

According to the paper of Davies cited by MJD, Archimedes actually gives a double inequality $$\frac{265}{153}<\sqrt{3}<\frac{1351}{780}.$$ As both of these fractions are not just random approximations, but are the 9-th and 12-th convergents for the continued fraction expansion of $\sqrt{3}$, there is no doubt that Archimedes used the continued ...


30

The idea was to simplify multiplicaton of numbers. If you ever tried to multiply $10$-digit numbers by hand you will see what I am talking about. The idea is this. We have $a^{m+n}=a^m a^n.$ On the right hand side we have a product of $a^m$ and $a^n$, while on the left hand side a sum $m+n$. So if you write two progressions, one arithmeric and one geometric ...


29

We must locate this passage into a double context : the general historical context the specific context of the "rivalry" between Hooke and Newton. See Robert Purrington, The First Professional Scientist : Robert Hooke and the Royal Society of London (2009), Ch.8 : And All Was Light : Hooke and Newton on Light and Color, page 135-on. For the general ...


29

The truth is that we do not know. We do know of the person who is credited with the discovery, Menaechmus (c. 350 BC), a student of Eudoxus of Cnidus and a friend of Plato's, one of the most prominent mathematicians of his time. The names ellipse, parabola and hyperbola were given to them by Apollonius of Perga over a century later however. Menaechmus called ...


29

First hand testimony and insightful thoughts on Ramanujan's background and way of doing mathematics can be found in Hardy's lecture Indian Mathematician Ramanujan. Hardy is the British mathematician who first appreciated the full extent of Ramanujan's talent, knew him well personally, and had a fruitful mathematical collaboration with him. Here is Hardy's ...


28

It is really funny to read that in the beginning of 21-st century, some young people may think that journals and books printing had something to do with typewriters:-) If you look attentively at the page you scanned you will easily see that this is not a TeX font and not a typewriter. It is more beautiful. Before the middle 1990s we lived in the "...


27

There is no way that Fermat could have had anything approaching the now commonly-accepted proof. Almost none of the concepts in that proof were known in any form in Fermat's time. Further, Fermat is known for publishing very few of his proofs; almost none survive today, and even in the 1800s there was significant doubt in the mathematical community that he ...


27

I will start by answering why matrix algebra became important, and then discuss approximately when. "Matrices" underpin what is often called operations research. That is, the theory of decision making. They are particularly useful in computer science, which features strings, arrays, etc., with machines substituting for human beings in (mechanical) decision ...


27

The main question is why the Pythagorean theorem for right triangles: $$ a^2+b^2=c^2$$ is such a central tool of Euclidean geometry. There are many different approaches one can take to this; I'll give it a shot. One of the key observations is that the triangle is the most basic 'non-trivial' shape in plane (two-dimensional) geometry. Any three points - one ...


26

According to an American Scientist article (Gauss' day of reckoning by Brian Hayes, Volume 94 p. 200) mentioned in the comments, the original source for this story, or at least a story very similar to it, was Gauss zum Gedächtnis, a memorial written very soon after Gauss' death by Wolfgang Sartorius, a colleague of his at Göttingen (however, I am not sure if ...


26

This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to history, is that only Euler presented volume formulas in this form in his textbooks after 1737. The principal step was no doubt made by Archimedes in On Sphere and ...


24

Newton's notation, Leibniz's notation and Lagrange's notation are all in use today to some extent they are respectively: $$\dot{f} = \frac{df}{dt}=f'(t)$$ $$\ddot{f} = \frac{d^2f}{dt^2}=f''(t)$$ You can find more notation examples on Wikipedia. The standard integral($\displaystyle\int_0^\infty f dt$) notation was developed by Leibniz as well. Newton did ...


24

TL;DR: The modern zero was born in India, in the latter half of the first millennium. I'll start with Wikipedia, and then work with some better sources. But first, from the Wikipedia article, Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1740 BCE, the Egyptians had a symbol for zero in accounting ...


24

By about 1640, the solution to the "area problem" for curves with equation $Y^n = aX^m$ was known by Fermat for all integer cases except when $n = 1, m = -1$. I.e., the only unsolved area problem was for $Y = \frac 1X$ - the standard equation for the graph of a hyperbola. In 1647, Gregoire de St. Vincent showed the following special property ...


23

How To Solve It was originally published in English in 1945 by Princeton University Press in English, after being rejected by three other U.S. publishers. However, the original text at least started out in German as a draft. Pólya began writing the draft prior to 1940, while he was living in Zürich, and presumably initially intended for the text to be ...


23

This happens all the time, mathematicians are fond of publishing papers titled Elementary Proof of Such and Such [hard] Theorem. Original proofs are often tour de force (feats of strength), they introduce new concepts along the way, make convoluted constructions, involve heavy calculations, just to get to the final result somehow. Once a new framework is ...


23

Leibniz did use this notation for instance in his paper Supplementum geometriae practicae, Acta Eruditorum, April 1693, p. 179 (Google Books link):


23

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 ...


22

I do not agree on some details of the interpretation regarding the discovery of the irrationality of $\sqrt{2}$ as a confutation of the Pythagoreans [...] belief that all numbers could be constructed as the ratio of 2 numbers. My undestanding is that all "archaic" Greek mathematics shared the (implicit) assumption that, given two magnitudes, e.g. two ...


22

"Counterexamples" to Cauchy's theorem were "discovered" as soon as he proved it. Of course Cauchy knew all these "counterexamples" but he insisted that his theorem and its proof is correct until his death. In particular, Fourier's book on heat is full of these counterexamples. Fourier tries to clarify the matter by defining a continuous function as one "...


21

The issue is thorny ... According to Morris Kline, Mathematical Thought from Ancient to Modern Time. Volume I (1972), page 272 [only entry of the Subject Index regarding : mathematical Induction] : The method was recognized explicitly by Maurolycus in his Arithmetica of 1575 and was used by him to prove, for example, that $1+3+5+ \ldots + (2n+1)=n^2$. ...


21

According to Florian Cajori, A History of Mathematical Notations (1928 - Dover reprint), Vol II, page 128 : 498. It was Euler who first used the letter $i$ for $\sqrt -1$. He gave it in a memoir presented in 1777 to the Academy at St. Petersburg, and entitled "De formulis differentialibus etc.," but it was not published until 1794 after the death of Euler....


21

An excellent source for the early history of smooth non-analytic functions is the paper Gerald G. Bilodeau. The origin and early development of non-analytic infinitely differentiable functions, Arch. Hist. Exact Sci., 27 (1982), 115-135. MR84g:26017. Also, Dave Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two ...


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