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Yes, indeed when trying to obtain the law of falling bodies, Galileo's first conjecture was that the speed is proportional to the distance traveled. After some contemplation, Galileo understood that this cannot be the case and eventually came with the correct law. Good source on Galileo: S. Drake, Galileo at work. (There are many editions).


Contrary to what the name suggests, Fourier series were not invented/discovered by Fourier. They were considered by Euler and Bernoullis, in relation to the one dimensional wave equation, not the heat equation. This early story is described for example in the papers by Luzin in Amer. Math. Monthly: Luzin, N. Function. I. Amer. Math. Monthly 105 (1998), no. ...


§1.1 (+ supplement) of Bressoud’s A radical approach to real analysis, recommended here just recently, does pretty much exactly what you want.


The book Introduction to the Theory of Fourier's Series and Integrals by H. S. Carslaw answers your questions in the first chapter on the History of this subject. Many commonly held false beliefs are debunked in his first chapter, including the idea that Fourier failed to give a rigorous proof of convergence. Another common false belief is that Fourier ...


Yes, Galileo made that error (and so did Descartes). Only later did he realise that the speed is proportional to the time ellapsed, not to the distance already covered. I suggest that you read The new science of motion: A study of Galileo's De motu locali, by Winifred L. Wisan (Archive for History of Exact Sciences, June 1974, 13, Issue 2–3, pp 103–306).


You can download the journal here: https://www.biodiversitylibrary.org/bibliography/5919#/summary There are also Collected works of Schwarz: https://archive.org/details/gesammeltemathem02schwuoft


Here are plots using Jahrbuch for 1870–1940 and Zentralblatt after that. A glance at their detailed output suggests some unreliability (e.g. Zentralblatt counts a paper twice when both reviewed it), but hopefully not too much overall. Papers and books in other languages (Danish, Dutch, Hungarian, Japanese, Latin, Norwegian, Polish, Portuguese, Spanish, ...


MR0106313 Krasovskiĭ, N. N. {\cyr Nekotorye zadachi teorii ustoĭchivosti dvizheniya.} (Russian) [Certain problems in the theory of stability of motion] Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow 1959 211 pp. MR0052616 Barbašin, E. A.; Krasovskiĭ, N. N. On stability of motion in the large. (Russian) Doklady Akad. Nauk SSSR (N.S.) 86, (1952). 453-456.

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