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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....


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, ...


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

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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