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As far as I know, he published a paper containing serious mathematical mistakes about the three-body problem, but he soon spent a lot of money to retrieve the paper and prove that it was impossible to obtain a general solution to the three-body problem.

Aside from the three-body problem, has he ever made any mathematical mistakes in his published books or papers?

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    $\begingroup$ Poincare frequently wrote some non-rigorous arguments, and some of them were later seen to be incorrect. But he probably understood that they are non-rigorous, so it is hard to qualify them as formal mathematical mistakes. $\endgroup$ Commented May 13 at 11:46
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    $\begingroup$ Just out of curiosity (I know nothing about Poincare) - is he famous for being particularly error-free, to spawn the question? $\endgroup$
    – AnoE
    Commented May 14 at 7:25

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The book Lecat, Maurice, Erreurs des Mathématiciens, Castaigne, 1935 reports four mistakes of Poincaré (pp. 90-91).

I report below, in English translation, part of the fourth one, about calculus of variations, a mistake that was pointed out by Radon, Carathéodory and others:

Calculus of variations. Existence and analytical continuity of a closed geodesic on a convex surface. Establishing the necessary condition to minimize: ” that there is no pair of conjugate points […]", P. starts from the principal ellipses [...]. Not valid reasoning.

Poincaré, H. Les Methodes Nouvelles de la Mécanique Celeste, t. iii, P.,, G,-V, p. 285. – Sur les lignes géodesiques des surfaces convexes. Trans. Am. M. S. 6 (1905) p. 237/74, partic. P. 254 [17-IV-0

[…]Carathéodory (1873-)shows (*)that the condition of P. is not valid ($\alpha$).

(*)Über geshlossene Extremalen und periodische Variationsprobleme in der Ebene und im Raume, Ann. Di Mat. (4) 2 (1924/5) p. 297/320, partic. P. 298, 305.

($\alpha$) Radon [Zur Behandlung geshlossener Extremales, Abh. Math. Semin. Hamburger U., t.I. p. 195] had already pointed out that thing with regard to the space.(my transl.)

This question on Mathoverflow about mathematics mistakes has two answers about Poincaré's mistakes:

https://mathoverflow.net/a/1688

https://mathoverflow.net/a/202640

In addition, here is a lecture of the Fields medalist Cédric Villani about Poincaré and his mistakes, Les meilleurs and le pire des erreurs de Poincaré, (En. Tr. The best and the Worst of Poincaré’s Mistakes).

Unfortunately, I didn’t find an English translation.

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