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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.

Michael Hardy
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    Do you mean in proofs, or in theorem statements? – Asaf Karagila Apr 22 '14 at 02:47
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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 '14 at 02:48
  • Define "mathematical error." This is fairly vague. If I'm writing out a problem (not to be published, but just for fun), and I say $2^3 = 6$, is that an error? ;) – apnorton Apr 22 '14 at 02:48
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    @anorton That seems off topic. Euler is one of the most prolific mathematicians of all time. The question refers to his body of published works. – Andrés E. Caicedo Apr 22 '14 at 02:49
  • @AndresCaicedo Unless the question has an edit which is not showing up on my screen, it does not explicitly specify that. I figure that it would be reasonable to assume so, given the questioner, but I'm simply holding to the same standard I would if a 1 rep user walked in and asked that question. In other words, Euler being prolific does not imply that the question refers to published works. – apnorton Apr 22 '14 at 02:52
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    Dying. Actually, this is not an error he made, since – Euler....IS_ALIVE Apr 22 '14 at 02:55
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    @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. – Andrés E. Caicedo Apr 22 '14 at 03:02
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    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 '14 at 03:27
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    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. – colormegone Apr 22 '14 at 03:30
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    @Euler....IS_ALIVE: That is a result of his previous error. He challenged Chuck Norris to a contest which included counting all the digits of $\pi$; finding the least number which cannot be described in less than ten words; and a roundhouse kicking tourney. – Asaf Karagila Apr 22 '14 at 04:08
  • BTW, I have been assured by a Fully Licensed Poet that the subject line I gave this question is "extremely euphonious". – Michael Hardy Apr 23 '14 at 02:57
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    See http://math.stackexchange.com/questions/113162/why-is-sqrt-2-sqrt-3-neq-sqrt6. – lhf Apr 25 '14 at 03:16

6 Answers6

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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 (Internet Archive, jstor.

Martin Sleziak
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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 '14 at 14:40
  • @RyanReich Really? Was it for a history of maths class or something? –  May 03 '14 at 21:02
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    Not at all! This computation is the basis for the modem theory of integration on manifolds using differential forms. – Ryan Reich May 04 '14 at 01:31
  • @RyanReich I see. I really need to learn about that :) –  May 04 '14 at 01:32
  • 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. – Mikhail Katz May 05 '14 at 12:54
  • The link is broken. – Ian Mateus Sep 11 '15 at 00:05
  • And the error is . . .? –  Aug 06 '17 at 20:43
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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

P Vanchinathan
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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. – Andrés E. Caicedo Apr 22 '14 at 03:04
  • @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: 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. – P Vanchinathan Apr 22 '14 at 03:11
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    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. – Sachin Valera Apr 22 '14 at 03:16
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    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. – Andrés E. Caicedo Apr 22 '14 at 03:17
  • I admit, I have not read the original work of Euler where this conjecture is stated. If gaps/unverified assumptions in an argument of a proof should be considered errors then this may not count as error by such a definition. – P Vanchinathan Apr 22 '14 at 03:33
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    Every time mathematicians attempt to axiomatise spoken language ... we get very long paragraphs. – Evgeni Sergeev Apr 22 '14 at 08:37
  • @Evgeni Sergev. That is an amusing and valid observation. – P Vanchinathan Apr 22 '14 at 10:52
  • Hi sir! May I know where you are teaching? – N.S.JOHN Aug 05 '16 at 15:27
  • @N.S.JOHN When your comment (question?) is addressed to a person make it explicit who you are addressing. If anyone wants to reveal personal info about themselves it will be available when you click on their names. – P Vanchinathan Aug 05 '16 at 17:41
  • @P Vanchinathan Yes I saw your profile sir. I'm from India too. I wanted to know where you are teaching. I'm a sutdent of 10th grade – N.S.JOHN Aug 05 '16 at 17:42
  • See the profile in the linked site "mathoverflow" – P Vanchinathan Aug 05 '16 at 17:47
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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

Euler....IS_ALIVE
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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? – Andrés E. Caicedo Apr 22 '14 at 03:18
  • 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". – colormegone Apr 22 '14 at 03:39
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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.

Alex R.
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  • This wiki page: http://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). –  Apr 22 '14 at 05:02
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    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 – Mikhail Katz Apr 22 '14 at 09:03
  • In other words, it is not at all a "pitfall" that Euler failed to toe the line on Weierstrass. – Mikhail Katz Apr 22 '14 at 09:33
  • @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 '14 at 12:14
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    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 '14 at 16:59
  • @DanielFischer, obviously Euler knew nothing of modern foundational frameworks so his proofs cannot be claimed to be literally true without lapsing into presentism. However, I stand by my point that an implied assumption of Alex's answer is that the only way of justifying Euler's argument is by Weierstrassian means. This was shown not to be the case by Kanovei already in the 1980s, and discussed in detail here. So overall I would maintain that the "Weierstrass vs. infinitesimals" detector sounds red alarm here. – Mikhail Katz Apr 22 '14 at 18:07
  • @user72694 My point is that I'm convinced you're reading the answer wrongly. Alex's point, as I understand it, is not at all that one cannot give a rigorous proof of the product representation in whatever flavour of non-standard analysis one prefers. The point, as I understand it, is that Alex says that Euler had no rigorous (let's understand that as "rigorous by the standards of Euler's times") proof of the fact that $g(z)$ in the product representation $$\sin z = e^{g(z)}\cdot z\prod_{n=1}^\infty\left(1-\frac{z^2}{n^2\pi^2}\right)$$ vanishes identically. (I don't know whether Euler had.) – Daniel Fischer Apr 22 '14 at 18:20
  • @DanielFischer, that's just my point: no mathematician before the 1870s can be claimed to have given a rigorous proof of anything at all, because the standard of rigor we use today depends on the foundational frameworks that have only been developed since the 1870s. One can only talk about this or that formalisation of the arguments used by the great classical pioneers. Asserting that they were not "rigorous" is in a way a vacuous statement because they could not possibly be so in the sense in which we understand the term "rigor". Here Benacerraf's dichotomy of mathematical practice versus – Mikhail Katz Apr 22 '14 at 18:49
  • ... mathematical ontology can be very helpful in sorting out these issues. If you like I can send you a text that explores these issues specifically in the context of Euler. – Mikhail Katz Apr 22 '14 at 18:50
  • @user72694 I think you're still missing the point. The point is not that today's standards of rigour are different. Even at Euler's time, it was plain that $$"p(z) = z\prod_{n=1}^\infty \left(1 - \frac{z^2}{n^2\pi^2}\right)$$ is an entire function that has the same zeros as $\sin z$" is not a rigorous proof of $p(z) = \sin z$. Even without going to a library to look at Euler's original work, I'm willing to bet that Euler gave more arguments than that to conclude that $\frac{\sin z}{p(z)} \equiv 1$, but I don't find it impossible to believe that his arguments were lacking by then-standards. – Daniel Fischer Apr 22 '14 at 19:01
  • I withhold judgment on that point, because I haven't read what Euler actually wrote when he got his product representation. – Daniel Fischer Apr 22 '14 at 19:02
  • The "library" you can "go to" nowadays (provided you know which documents you're looking for, and can read the Latin) is here: http://eulerarchive.maa.org/ – colormegone Apr 23 '14 at 04:58
  • @DanielFischer, if either the OP or you are implying that Euler merely noticed that sine has the same zeros as the infinite product and from this jumped to the conclusion that they must be equal implicitly exploiting Liouville's theorem, then I can assure you that Euler did nothing of the sort. Rather, he provided an elaborate argument of about 7 distinct steps, as discussed in the reference I provided above. I think this discussion supports my suggestion that objectivity needs to be restored to Euler scholarship... – Mikhail Katz Apr 23 '14 at 15:39
  • ... and therefore I invite you to show open-mindedness and vote to reopen this thread: http://math.stackexchange.com/questions/764242/errors-of-euler-interpretation – Mikhail Katz Apr 23 '14 at 15:41
  • @user72694 As I said, without even looking I'm willing to bet that Euler gave more arguments for the identity. What I don't know is whether the arguments he gave were sufficient. But the question whether the arguments he gave were sufficient or not does most probably not depend on a use of infinitesimals in any way. I think you're reading things into this answer that the author did not in any way mean. I may be wrong in that interpretation, of course, but I see nothing in the answer that links the alleged lack of a rigorous proof to Euler's use of infinitesimals. – Daniel Fischer Apr 23 '14 at 15:57
  • @user72694: Your reference is enlightening. One thing that it corroborates is that there were indeed "hidden" lemmas at work in the steps of his derivation which needed more justification. Just briefly looking at these hidden lemmas, they seem awfully similar to the proof of the Weierstrass factorization theorem, in particular the rate of decay of the series involved. I don't think anyone here is disputing the usefulness of his methods. Perhaps a good classification would be "physics methods in math" that have some handwaviness in the process. – Alex R. Apr 23 '14 at 16:02
  • @DanielFischer, I certainly did not mean to imply that the OP is prejudiced against infinitesimals; on the contrary. I even left a comment on one of his recent answers thanking him for presenting a balanced view on these very infinitesimals. However there does exist a general perception that there was something logically unsound about early work in infinitesimal calculus, and this affects our views of Euler as well. And I certainly disagree with your comment above that Euler's proof of the infinite product formula has a good chance of being unrigorous even by the standards of his period. – Mikhail Katz Apr 23 '14 at 16:02
  • Also, when I say "pitfall" I really mean that perhaps if similar techniques were applied elsewhere then one would not get so lucky. I'm short of thinking of an example at the moment but it seems like one should try to find a function whose Taylor series decays slower than the requirement of the hidden lemma which would hopefully give a strange answer for the factorization. – Alex R. Apr 23 '14 at 16:03
  • @AlexR., I certainly appreciate your interest. Some of the infinitesimal arguments are similar to traditional ones, others are not. The "hidden lemma" viewpoint on Euler's work was initiated by Laugwitz in a series of serious studies published in top-notch history and philosophy journals, and has been fruitfully pursued by several authors in the past two decades. And again, I don't think Euler's infinitesimal methods involved any more "handwaving" than any other field in math at the time. – Mikhail Katz Apr 23 '14 at 16:05
  • @user72694 "I don't find it impossible to believe" is quite something different than "has a good chance". – Daniel Fischer Apr 23 '14 at 17:33
  • @DanielFischer, perhaps this should be moved to SE English at this point :-) – Mikhail Katz Apr 23 '14 at 17:40
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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

Grass M.
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In the Introduction of Rational Points on Elliptic Curves by Silverman and Tate, it is claimed that Euler, in the 1730s, provided an incorrect solution to a question posed by Fermat in the 1650s, which was to show that the equation $$ y^2 - x^3 = -2 $$ has only two solutions in integers, namely $(3,±5).$ There's no citation given, though.

Michael Hardy
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I. J. Kennedy
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