## 7-2-63. Felix Klein to H. Poincaré, English translation

Düsseldorf, 3 April 1882

Dear Sir,

When I received your letter yesterday via Leipzig, I was just about to write you a few words to accompany the page proofs of my new note in the Annalen, the page proofs of which should already be in your hands.11endnote: 1 Klein (1882a), reedited in Klein (1923, 627–629). In the meantime I have obtained a copy of Prof. Fuchs’s note published in the Göttinger Nachrichten.22endnote: 2 Fuchs (1882), reedited in Fuchs and Schlesinger (1906, 285–287). If I could say a few words about the latter, they would be to the effect that I consider it to be completely mistaken. I have claimed only that Fuchs has never published anything on “fonctions fuchsiennes”. It follows that the second article that he cites (which, by the way, I have procured in order to study more closely) is pointless. The first may, however, be considered to be concerned with “fonctions fuchsiennes” insofar as it deals with modular functions, but, lacking geometric intuition, Fuchs has not fully grasped the proper character of the latter, which lies in the nature of singular lines, as Dedekind showed already in Volume 83 of Borchardt.33endnote: 3 Dedekind (1877). Finally, as for the insinuation at the end of his note to the effect that my own work has been essentially stimulated by his, historically this is simply incorrect. My research began in 1874 with the determination of all finite groups of linear transformations in one variable.44endnote: 4 Klein (1876), reed. Klein (1922, 275–301). Then in 1876 I showed that the problem raised at that time by Fuchs of determining all integrable second order linear differential equations was eo ipso solved.55endnote: 5 Klein (1877). The situation is precisely the reverse of what Fuchs claims. It was not that I took his ideas, but rather that I showed that his topic should be treated using my ideas.

In addition, and as you may imagine, I do not agree with your analysis. If it were a matter of a general appreciation of Fuchs’s works, I would willingly have his name bestowed on some new class of functions that no one had yet studied, or even, for instance, on the functions of several variables that he has put forward.* However, the functions you have named after Fuchs already belonged to others before you suggested the name. I am also quite sure that you would have not proposed this name had you then (at the beginning) been familiar with the literature. You then offer me, in compensation, one might say, “fonctions keinéennes”. To the extent that I sense your friendly intention here, to just that extent is it impossible for me to accept the offer, as again perpetrating an historical untruth. If my memoir in Volume XIX might give the impression that I am now especially preoccupied with the ‘‘kleinéennes’’, my more recent work in Volume XX shows that as before I continue to regard the ‘‘fuchsiennes’’ as my domain.66endnote: 6 Klein (1882b, a).

Enough on this topic. I immediately dispatched your note to the printer, appending only a remark to the effect that for my part I adhere to my previous opinion (and on this occasion drawing the public’s attention to Mr. Fuchs’s note). You will receive the page proofs very soon and I ask you to return them quickly to me here (where I am spending the Easter holidays), and I will then do what is necessary as far as publication is concerned. (Your note will appear directly after mine!) As far as the passage about Schottky is concerned, I would like to point out to you a posthumous article in Riemann’s Works, p. 413, where exactly the same ideas are developed.77endnote: 7 The reference is to chapter 15, entitled “Gleichgewicht der Electricität auf Cylindern mit kreisförmigem Querschnitt und parallelen Axen” (Weber and Dedekind, 1876, 413–416). I should say, however, that it would be difficult to determine the extent of a possible contribution by the editor, Prof. Weber. Riemann’s Works appeared in 1876, and Schottky’s dissertation was completed in 1875 and published in 1877 as a memoir in Borchardt’s Journal.88endnote: 8 Schottky (1875, 1877). However, the 1875 dissertation constitutes only a part of the 1877 memoir, and I cannot recall if the figure in question had already appeared in the 1875 text.

I should add that on my part I have no intention of prolonging our terminological disagreement (once I have added the above-mentioned footnote to your explanation). However, if I should be led to intervene in the matter anew then I would, it is true, give a very complete and frank account of it. Let us rather compete to see which of us is best equipped to advance the entire theory in question! On my side, I believe my new note represents a certain advance. A whole series of theorems on algebraic functions can be proved using the new $\eta$-function – for instance, the theorem that, in my book on Riemann, I initially indicated as only probably true, namely that a surface of genus $p>0$ never admits infinitely many single-valued discrete transformations (since otherwise it would decompose into an infinite number of “equivalent fundamental polygons”). And also the theorem that various of Picard’s results for the case $p=0$ extend to general $p$, etc.

As for the methods I use to prove my theorems, I will inform you as soon as I have further clarified them. In the meantime, could you not describe for me the ideas you are currently pursuing? I scarcely need add that we would be pleased to publish in the Mathematische Annalen any article you wished to communicate. It will be crucial to remain in active contact with you. For me, lively contact with mathematicians aspiring to similar ends has always been a precondition for mathematical production.

F. Klein

* Are these really single-valued? All I understand is that over the whole of the range of values taken by them there is no branching. However, I may be mistaken in this.

Dr. Hurwitz’s address, until further notice, is: Hildesheim, Langer Hagen.

PTrL. Translated by S.A. Walter from the original German (§ 4-47-18), after R. Burns in Saint-Gervais (2016). See also the French translation (§ 7-2-39).

Time-stamp: " 3.05.2021 10:05"

### Notes

• 1 Klein (1882a), reedited in Klein (1923, 627–629).
• 2 Fuchs (1882), reedited in Fuchs and Schlesinger (1906, 285–287).
• 3 Dedekind (1877).
• 4 Klein (1876), reed. Klein (1922, 275–301).
• 5 Klein (1877).
• 6 Klein (1882b, a).
• 7 The reference is to chapter 15, entitled “Gleichgewicht der Electricität auf Cylindern mit kreisförmigem Querschnitt und parallelen Axen” (Weber and Dedekind, 1876, 413–416).
• 8 Schottky (1875, 1877).

## References

• R. Dedekind (1877) Schreiben an Herrn Borchardt über die Theorie der elliptischen Modul-Functionen. Journal für die reine und angewandte Mathematik 83, pp. 265–292. Cited by: endnote 3.
• L. Fuchs (1882) Über Functionen, welche durch lineare Substitutionen unverändert bleiben. Nachrichten von der Königliche Gesellschaft der Wissenschaften und der Georg-August-Universität zu Göttingen, pp. 81–84. Cited by: endnote 2.
• R. Fuchs and L. Schlesinger (Eds.) (1906) Gesammelte mathematische Werke von L. Fuchs, Volume 2. Mayer & Müller, Berlin. Cited by: endnote 2.
• F. Klein (1876) Ueber binäre Formen mit linearen Transformationen in sich selbst. Mathematische Annalen 9, pp. 183–208. Cited by: endnote 4.
• F. Klein (1877) Ueber lineare Differentialgleichungen. Mathematische Annalen 12, pp. 167–179. Cited by: endnote 5.
• F. Klein (1882a) Ueber eindeutige Functionen mit linearen Transformationen in sich (zweite Mittheilung). Mathematische Annalen 20, pp. 49–51. Cited by: endnote 1, endnote 6.
• F. Klein (1882b) Ueber eindeutige Functionen mit linearen Transformationen in sich. Mathematische Annalen 19, pp. 565–568. Cited by: endnote 6.
• F. Klein (1922) Gesammelte mathematische Abhandlungen, Volume 2. Springer, Berlin. Cited by: endnote 4.
• F. Klein (1923) Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. Cited by: endnote 1.
• H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich.
• F. Schottky (1875) Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen. Ph.D. Thesis, University of Berlin, Berlin. Cited by: endnote 8.
• F. Schottky (1877) Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen. Journal für die reine und angewandte Mathematik 83, pp. 300–351. Cited by: endnote 8.
• H. Weber and R. Dedekind (Eds.) (1876) Bernhard Riemann’s gesammelte mathematische Werke und wissentschaftlicher Nachlaß. Teubner, Leipzig. Cited by: endnote 7.