[Note: This is the same question on MSE, but it has only five answers, one of which doesn't give an exact answer (it tells that a conjecture of Euler was wrong, but a conjecture isn't a mistake) and the question is unlikely to get answers since it was active 2 years ago.]
Euler was one of the greatest and smartest mathematician in the history. He contributed to many fields of mathematics. But no one is perfect, and this holds for Euler also. For example, this tells that Euler thought that $$\sqrt{ab}=\sqrt{a}\sqrt{b}$$ whether or not $a$ and $b$ are positive or negative. So this is one of Euler's mistake. I want to see his some more mistakes. So my question is:

What are some of Euler's mistakes?

Like most big list questions, please give only one mistake per answer.

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    $\begingroup$ His proof of the existence of God was a bit ropy, though Catherine the Great didn't seem to mind. $\endgroup$ – Nick Oct 29 '20 at 15:17
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    $\begingroup$ $\sqrt{ab}=\sqrt{a}\sqrt{b}$ is only a "mistake" on the modern complex analysis interpretation of $\sqrt{\ }$, which did not exist at the Euler's time. Another "mistake" is manipulation of divergent series in a way inconsistent with Weierstrass's interpretation of real analysis, which did not exist either. They fare better on non-standard analysis interpretations, but... those did not exist as well. And this illustrates the problem with the whole conception of this question, it is completely ahistorical. $\endgroup$ – Conifold Oct 29 '20 at 20:27
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    $\begingroup$ @Conifold You're right; it's like saying Newton made a "mistake" inventing classical mechanics. $\endgroup$ – Spencer Oct 29 '20 at 22:24

There is Betteridge's law of headlines: "Any headline that ends in a question mark can be answered by the word no." The article you quote (Euler's “Mistake”?) is not an exception. $\sqrt{ab}=\sqrt{a}\sqrt{b}$ is not a mistake. It is true if you understand square root as a multivalued function. Another common understanding of √ is positive square root, but it is defined only for positive numbers. The author of the paper gave two weird interpretations of square root, under which this equality indeed is not true. However there is no evidence that these interpretations were common or even used in Euler time. So, this is not a mistake at all, at least until someone proves that Euler meant one of these interpretations.


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