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What mathematical errors is Leonhard Euler known to have made?

PS: As I wrote in a comment below: "However, I would not consider proof to be an error merely because it's not a proof by present-day standards." Everybody knows Euler wrote about infinitely large integers and about infinitesimals in ways differing from what today is considered logically rigorous. I had in mind actually erroneous conclusions or arguments that we cannot today replace with any we consider rigorous.

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Either, but maybe the latter are more interesting. However, I would not consider proof to be an error merely because it's not a proof by present-day standards. –  Michael Hardy Apr 22 at 2:48
Dying. Actually, this is not an error he made, since –  Euler....IS_ALIVE Apr 22 at 2:55
@anorton Sure, that's fine. It seems a deliberately obtuse reading of an interesting request, but go ahead. Perhaps there are some errors on his computation of a tip when dining out in St. Petersburg one night on July, 1727, and it may do for good gossip. –  Andres Caicedo Apr 22 at 3:02
Euler is more likely to be KNOWN (in the present day) to have made a particular mistake if he published it than if he whispered it to his psychiatrist, but if it is somehow known that he made a mistake that he never published, I see no reason why that shouldn't be included here. –  Michael Hardy Apr 22 at 3:27
You might like to look at some of William Dunham's books on Euler's work. My impression is that Euler was pretty careful about what he wrote up and published, so "errors" tend to run more toward the limit of how reliable his intuition was or how well certain concepts (such as convergence of infinite series) were understood in the 18th Century. –  RecklessReckoner Apr 22 at 3:30
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5 Answers 5

Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals.

He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he described as "obviously wrong and even meaningless."

He then got $$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}\right)\,\mathrm{d}t\,\mathrm{d}v$$ which was not symmetric in the variables, and therefore would not do.

Finally, he derived the correct $$\mathrm{d}x\,\mathrm{d}y=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}-\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and lamented that simply multiplying out $$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial x}{\partial t}\,\mathrm{d}t+\frac{\partial x}{\partial v}\,\mathrm{d}v\right)\left(\frac{\partial y}{\partial t}\,\mathrm{d}t+\frac{\partial y}{\partial v}\,\mathrm{d}v\right)=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}+\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and shredding the squared differentials yielded an incorrect but annoyingly close answer.

But let us remember, if Euler committed errors it was only because of the unrivaled breadth of his work. If I could finish with a quote from the article cited below: "As a developer of algorithms to solve problems of various sorts, Euler has never been surpassed."

Source: For an excellent review of the history of the Jacobian, and to learn more about the details of what I have written, I highly recommend reading this article by Prof. Victor J. Katz.

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To be fair, I just taught an entire chapter's worth of material on what amounts to this very error. –  Ryan Reich Apr 22 at 14:40
@RyanReich Really? Was it for a history of maths class or something? –  NotNotLogical May 3 at 21:02
Not at all! This computation is the basis for the modem theory of integration on manifolds using differential forms. –  Ryan Reich May 4 at 1:31
@RyanReich I see. I really need to learn about that :) –  NotNotLogical May 4 at 1:32
This calculation of Euler's is of course the best proof that what one is really integrating is a differential form whose alternating sign property makes a minus sign appear so that one gets the determinant as expected (rather than the permanent). This also allows one to get rid of absolute values around the determinant when doing multiple integrals. The main point is that change of variables should be viewed as an operation on the signed area 2-form. –  user72694 May 5 at 12:54
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Euler conjectured that for $n=2\pmod 4$ there are no mutually orthogonal Latin squares of size $n\times n$. Bose and Shrikande disproved it by construction and earned the name Euler's Spoilers. See http://en.wikipedia.org/wiki/Graeco-Latin_square

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But this is not an error, surely. Unless he claimed the result, and only now we read his claim as a (false) conjecture. The link you provide does not specify whether this was indeed the case. –  Andres Caicedo Apr 22 at 3:04
@Andres Caicedo: I think it is fair to say someone was in error when the conjecture made by them is disproved. Let me copy-paste relevant portions from that Wikipedia article: <quote> In April 1959, Parker, Bose, and Shrikhande presented their paper showing Euler's conjecture to be false for all n ≥ 10. Thus, Graeco-Latin squares exist for all orders n ≥ 3 except n = 6. <end-quote> –  P Vanchinathan Apr 22 at 3:11
Yes but I believe that this answer coheres with the spirit of the question. While it isn't really an error as such, this conjecture was obviously something Euler thought to be true. Seeing as Euler's conjecture turned out to be false, it in an indicator for the existence of this mathemagician's coefficient of humanity - he did not have some completely infallible looking glass into the mathematical universe - but still had a darn good one. –  S Valera Apr 22 at 3:16
Again, it may just be a matter of definitions, but I do not consider a conjecture to be an error. It is a conjecture. When I conjecture something, even if in writing, I do not know whether it is true regardless of my intuition telling me it is. I do not claim that it is true, only that I believe it to be true. Now, if I claim that it is true, then it may be different, but it really depends on the customs for treating conjectural statements at the time. Which is why the reference to Wikipedia is not very satisfying. It would be best to examine what Euler actually wrote about it. –  Andres Caicedo Apr 22 at 3:17
Every time mathematicians attempt to axiomatise spoken language ... we get very long paragraphs. –  Evgeni Sergeev Apr 22 at 8:37
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Euler liked to play fast and loose with divergent series. Mathematicians of that era did not seem to be concerned with convergence issues.

For a more concrete example, Euler made a large mistake in trying to prove Fermat's Last Theorem for $n=3$. For details, check out http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_last_theorem.html

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Do you know whether he actually consider his computations with series valid, or just a heuristic means of arriving at conclusions? –  Andres Caicedo Apr 22 at 3:18
He also sank some time into looking for a method of solution of the general quintic equation. But it is unclear whether failing to solve a problem where a later broader understanding (and new "tools") would reveal that such effort would be unavailing is an "error". –  RecklessReckoner Apr 22 at 3:39
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This isn't a bona fide mistake but it's certainly a pitfall. Hopefully someone can verify the following. In Euler's original proof of the Basel Problem $(\zeta(2)=\pi^2/6$), he used the fact that

$$\sin(z)=z\prod_{n\geq 0}\left(1-\frac{z^2}{n^2\pi^2}\right).$$

This was well before Weierstrass's factorization theorem, which allows for a prefactor of $e^{g(z)}$ and in the case of sine, this prefactor is just 1. Rigorously showing that the above factorization holds and that the prefactor is 1 is nontrivial and as far as I know Euler had no solid proof of this fact.

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This wiki page: en.wikipedia.org/wiki/Basel_problem#Euler.27s_approach agrees with what you have claimed. Given that he solved a famous problem though, and correctly, I don't know if I would call this a mistake (or even a pitfall). –  NotNotLogical Apr 22 at 5:02
The idea that one needs Weierstrass to formalize Euler is not only tententious; it is actually wrong. A modern formalisation of Euler's infinite product decomposition of sine closely following Euler himself (rather than Weierstrass) is explained in this article –  user72694 Apr 22 at 9:03
In other words, it is not at all a "pitfall" that Euler failed to toe the line on Weierstrass. –  user72694 Apr 22 at 9:33
@user72694 Methinks your "Weierstraß vs. infinitesimals" detector gave a false alarm here. Weierstraß enters here only as the one who proved the general factorisation theorem, which would make proving the product formula for the sine more easy, since the structure of the parts is known with it. Whether Euler's derivation of the product was rigorous or not is a different question (which I don't know the answer to, I haven't read that of the man himself). –  Daniel Fischer Apr 22 at 12:14
I wouldn't really consider things like this mistakes in the sense I intended. Everybody knows Euler did lots of stuff like this and that's what I had in mind when I wrote in a comment above, "However, I would not consider proof to be an error merely because it's not a proof by present-day standards." But you've got $x$ as the independent variable on the left and $z$ on the right. –  Michael Hardy Apr 22 at 16:59
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It can be read on Peter Schumer's book "introduction to number theory" page 80, that Euler gave a defective proof that all primes have primitive roots

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