8

I assume this refers to Lagrange's 1768 proof of the Diophantine approximation theorem. The proof was simplified by Dirichlet in 1842, using the idea twice. He named it Schubfachprinzip (drawer principle), and it is with Dirichlet that the principle came to be most commonly associated. Many authors date Dirichlet's use back to 1834, but without any reference....


6

“Mathemata mathematicis scribuntur.” is the original Latin of Copernic which is easily translated as “Mathematics is written for mathematicians.” but Edward Rosen chose to translate this famous passage as “Astronomy is written for astronomers.” Obviously "astronomy" is not the author's word and also it is generally agreed that there was no ...


5

Good sources on the history of fixed point theorems are Park, Ninety Years of the Brouwer Fixed Point Theorem and Kumar, A Short Survey of the Development of Fixed Point Theory. According to both, early versions of fixed point theorem concerned self-maps, from a ball or some other set to itself. The first version for non-self maps is in Rothe's Zur Theorie ...


4

According to Fowler's Ratio in Early Greek Mathematics, palindromic patterns in continued fractions of $\sqrt{p}:\sqrt{q}$ with primes $p>q$ were likely known already to Pythagoreans. In the modern times the interest in continued fractions was revived largely due to Euler's work (from 1731), although Wallis and Huygens worked on them earlier, see ...


4

For Ramanujan's background see How did Ramanujan learn to do mathematics? According to Hardy himself, he did not teach him any topics, only the idea and perhaps some methods of proof, see his lecture Indian Mathematician Ramanujan. Ramanujan did pick up sporadic bits and pieces of modern mathematics from various sources that Hardy is not too sure about, ...


3

According to Wikipedia, it is the Austrian mathematician Gerhard Wanner.


3

The cyrillic letter Ш (sha)is -- for obvious reasons when looking at the graph) also used to denote the "function" (well, it is a distribution if you want to be picky) given by the sum of integral displacements of the Dirac-delta function, see https://en.wikipedia.org/wiki/Dirac_comb


3

Lucky you! (or me :-) ). This question was answered a while back on Math.SE I found his original thoughts in the translated version of "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1", chapter 7. The translation is called "Foundations of Differential Calculus" and a link is found here https://...


3

The letter $\upsilon$ or $\Upsilon$ is the second letter of δύναμις, κύβος and δυναμοδύναμις (square, cube and bisquare) of the words the notation abbreviates. Being the same letter, it plays an additional role of identifying the power symbols as being of a kind, marking Diophantus's "numerical species" (squares, cubes, etc., multiplied by coefficients). ...


2

My vote goes to three out of {Michał Dziewicki, Stanisław Zaremba, Jan Śleszyński, Witold Wilkosz, Otto Nikodym, Leon Chwistek, Stanisław Bilski}; most probably the three whose names I put in bold. On May 10, 1915, Russell wrote to Ludwig Wittgenstein who was doing military service in Krakow: "If you have time, you should visit in Krakow a lonely old ...


1

Internet archive is one of the best resources for historical mathematical books from the 18th/19th century. Another professional level historical resource is Hathi Trust. This is not open access.


1

It does not seem like there was much of a contribution, if any Lagrange is sometimes mentioned, without reference, e.g. by Borell, along with Taylor and de Moivre, as one who "worked" on it. But while Euler's work is called "serious" and "influential" not much else is said about the other three. A detailed history in Rediscovery of the Knight's Problem by ...


1

Yes, he was, at least in a particular example. Near the end of his life, in 1808-09, Lagrange studied perturbative dynamics of a planet on an elliptic orbit, and derived what came to be called Hamiltonian equations for it in Second mémoire sur la théorie de la variation des constantes arbitraires dans les problèmes de mécanique, dans lequel on simplifie l'...


1

The earliest reference I can find to "banana brackets" is in: G. Malcolm. Data structures and program transformation. Science of Computer Programming, 14(2-3):255-280, October 1990. Where they are clearly crescent-moon/banana shaped symbols: ⦅...⦆. The later style using $($ and $|$ seems to be a typographic practicality, and is used by ...


1

The writings of Descartes and Viète do indeed suggest a certain "inferiority complex", believing that their own work was simply rediscovering Greek analytic methods which had been lost by the "barbarians". From Descartes' Rules for the Direction of the Mind: But when I afterwards bethought myself how it could be that the earliest pioneers of Philosophy ...


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