Question

It's a common observation in Lie theory that Cartan matrices and the Killing form are named after the wrong people; they were discovered by Killing and Cartan, respectively. I remember learning about many other examples of this phenomenon, but can't think of too many at the moment. Wikipedia has some examples here and here, but I'm curious about more obscure examples.

Bonus points for an interesting story behind why the concept was incorrectly named. Concepts that were deliberately named in honor of another mathematician don't count.

Answer 1

Zorn's lemma is neither due to zorn, nor is it a lemma. It is a theorem due to Kuratowski.

Answer 2

Pythagoras' Theorem apparently predates Pythagoras.

Answer 3

Euler's nine point circle was never discussed by Euler. This is an error of the "argument by authority" type: Catalan propagated that incorrect attribution made by another scholar, the "learned Terquem", without checking it himself.

Answer 4

Banach algebras should probably be called Gelfand Algebras, or something similar. I'm not sure of the history here, but presumably the "Banach" is attached because this is the study of "complete" normed algebras. I don't believe that Banach actually did much work on algebras (as opposed to Banach spaces).

Answer 5

It seems to me that Lagrange's theorem may well be one of the most prominent examples of the phenomenon under discussion.

According to J. J. Rotman,

the theorem was inspired by work of Lagrange (1770), but it was probably first proved by Galois.

Curiously enough, the Wikipedia article adscribes the first complete proof of the theorem to Pietro Abbati Marescotti.

Answer 6

The Banach-Steinhaus theorem was first proved by Hahn, the Hahn-Banach theorem was first proved by Helly.

Answer 7

Jones' Conjecture. Jones does not even exist; it's a Western pseudonym of Chuan-Min Lee.

Answer 8

Perhaps counterexamples to Stigler's/Arnol'ds law are actually the rare items. The most significant one that I know is the Cartesian coordinate system which, strangely, seems to have actually been invented by Descartes!

Answer 9

In reference to exactly this phenomenon (and in particular to the case of Pell's equation), Andre Weil once observed that "This has happened many times in mathematics. For example, I live on von Neumann Circle. I live there. Yet still it is called von Neumann Circle".

Answer 10

The "Lichnerowicz formula" relating the square of the Dirac operator to the Laplacian has been proved more than 30 years earlier by Schrödinger.

See: E. Schrödinger, Dirac'sches Elektron im Schwerefeld, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1932, 105-128 (1932).

Answer 11

Farey fractions were introduced by C. Haros. John Farey asked a question about them that reached Cauchy, and Caucy then attributed the question and result to Farey, and the rest is history.

Answer 12

The algebraic numbers that are now commonly called "Gauss sums" were studied in more general form than that introduced in Gauss's Disquisitiones by Lagrange [1]. In that same work, Lagrange shows how to generate an abelian extension of degree n by adjoining an nth root after, if necessary, adjoining the nth roots of unity. These generators were later called "Kummer generators". Jacobi sums, which are closely related to Gauss sums, were studied earlier than Jacobi by Gauss and Cauchy.

Finally, a story best recounted by Weil [2]: "For reference, we recall that the Gauss sums appear among the local constant factors in the functional equations of the $L$ functions; these factors are also called "nombres radiciels" ("root-numbers", "Wurzelzahlen"), undoubtedly because of Hilbert, who a had a sort of genius for bad terminology, where it would have been sensible to name "Wurzelzahl" that which before him had been named a "Lagrange resolvent" , and "Lagrangian Wurzelzahl" that which here has been called a Gauss sum".

[1] Lagrange, Reflexions sur la resolution algebrique des equations, Nouveaux Mem. de l'Acad. R. des Sc. et B.-L. de Berlin, 1770-1771 = Oeuvres, vol. III, p. 332;

[2] Weil, La Cyclotomie Jadis et Naguere.

Answer 13

Farey series, attributed to Farey, were actually first studied by Haros.

Answer 14

The Shimura-Taniyama conjecture was originally known as the Weil conjecture see http://www.ams.org/notices/199511/forum.pdf, also see the comment of Weil on page 7 (with other examples) in his response to Lang on the same issue as in the question posed here.

Additionally, the Frey curve was actually first considered by Yves Hellegouarch.

Answer 15

Some people have tried to give examples with as high a multiplicity as possible. I want to try to break the record for the smallest non-zero example: Martin's axiom was introduced by Martin and Solovay. (I judge that to have multiplicity 1/2.)

Answer 16

If the Russians are to believed, Kuratowski's theorem in graph theory was proved earlier by Pontryagin, but he hadn't published his notes.

Answer 17

I had a companion observation that almost noone attributes the well-known sum-of-roots, product-of-roots etc. polynomial formulas as Vieta's formulas as I posted on Yahoo!Answers.

Because as user absird pointed out, it makes that sort of topic Google-proof; at least a bad name is better than no name for purposes of searching or discussion.

('Yes it's very hard to refer to something when noone knows it by its proper name or uses that name. I tried many Google searches on "sum-of-roots product-of-roots" and it was almost impossible to get a coherent lead.')

MathWorld notes: The theorem was proved by Viète (also known as Vieta, 1579) for positive roots only, and the general theorem was proved by Girard.

Answer 18

Apparently the Robin boundary condition was never studied by Robin. (Gustafson, K. and T. Abe, (1998b). The third boundary condition – was it Robin's?, The Mathematical Intelligencer, 20, 63–71.)

Answer 19

A favorite of mine is l'Hôpital's rule. l'Hôpital paid Johann Bernoulli a retainer to keep him updated on developments in calculus and to solve problems he had. Correspondence shows that Bernoulli stated and proved the rule, which l'Hôpital then published.

Heine-Borel was first published by Borel, not Heine. In fact, Heine's name was attached because he was using similar methods to solve related problems. Too bad for both of them that it was actually Dirichlet who was the first recorded to have proved it, but his notes were published posthumously and after Borel's proof.

Cramer's Rule was published first by MacLaurin, and some believe MacLaurin knew the proof some 20 years before Cramer's publication.

Answer 20

I don't know if this is a real example, but it led to a nice gem in a recent abstract on the arXiv: "Glaisher's correspondence goes back to Euler."

(As far as I know Glaisher generalized Euler's bijection, which is why he gets the eponym -- in addition some people say "Euler-Glaisher" instead.)

Answer 21

Cartan discovered the Killing form, and Killing discovered the Cartan matrix.

Answer 22

Morse theory is named after Marston Morse; it was widely used at least 50 years earlier. Wikipedia mentions Cayley and Maxwell, in the context of topography. Maxwell also used it in his work on electromagnetism, as detailed (complete with extensive passages from Maxwell's treatise) in the appendix of Mystery of point charges (A. Gabrielov, D. Novikov and B. Shapiro) available here (subscription)

Answer 23

I think the Kazhdan-Lusztig Conjectures are due to Vogan.

EDIT.

True or false, the claim is mainly based on the very first two paragraphs of

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[II] Irreducible characters of semisimple Lie groups II. The Kazhdan-Lusztig conjectures. David A. Vogan, Jr. Duke Math. J. Volume 46, Number 4 (1979), 805-859. --- The link

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077313724

gives a universal access to the first page, which contains the two paragraphs in question. In case you don't have access to the full paper, here is a scan of the references (to completely understand the two paragraphs):

http://www.iecn.u-nancy.fr/~gaillard/vogan_ref.pdf

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Here are two more references:

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[I] Irreducible characters of semisimple Lie groups I, David A. Vogan, Jr., Duke Math. J. Volume 46, Number 1 (1979), 61-108.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077313255

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[KL] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae, Volume 53, Number 2, 165-184.

http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN356556735_0053&DMDID=dmdlog14

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I would summarize things as follows.

Step 1. In [I] Vogan made a certain conjecture.

Step 2. [II] and [KL] were written simultaneously. Each paper cites the other. In [KL] Kazhdan and Lusztig also made a certain conjecture. When he learned this, Vogan immediately (or at least very fast) proved that the "Step 1 conjecture" implies that of Kazhdan and Lusztig. (He even showed that the "Step 1 conjecture" generalizes that of Kazhdan and Lusztig.)

But, again, the best is to read carefully the first two paragraphs of [II]. Vogan explains this much more clearly than I, and it's always better to hear things from the horse's mouth.

Answer 24

Liouville talked about the Legendre function when he studied the so-called Euler Gamma function. It made me doubt about who defined the Gamma function first.

Answer 25

According to this link Steiner Systems were mentioned by by W Woolhouse in 1844 before the famous Kirkman Schoolgirl problem (P Kirkman 1847) - Steiner's work was more systematic and did advance the theory, but it came in 1853.

Answer 26

The notion of Frobenius manifold is due to Dubrovin