7-2-64. H. Poincaré to Felix Klein, English translation

Paris, 4 April 1882

Sir,

I have just received your letter and hasten to reply. You tell me you wish to close a debate sterile for science and I can only congratulate you on your resolution. This cannot have cost you dearly, I know, since you have the last word, in the note appended to my last letter, but I am grateful to you nonetheless. As for me, if I opened and engaged the debate, it was only to voice once and once only an opinion that was impossible for me to silence. It is not I who will prolong the debate, and I would speak further only if forced to do so; besides, I doubt anything could so force me.

If I named Kleinian functions after you, it was for the reasons given and not, as you insinuate, ‘‘zur Entschädigung’’,11endnote: 1 “By way of compensation”. since there is nothing for me to compensate; I will only recognize a property right prior to mine when you will have shown me that someone had earlier investigated the discontinuity of the groups and the single-valuedness of the functions in a minimally-general case, with the series development of these functions. I respond to a query found in a footnote to your letter. Speaking of the functions defined by Mr. Fuchs in Volume 89 of Crelle, you ask:22endnote: 2 Fuchs (1880a); Fuchs and Schlesinger (1906, 191–212). Klein asked Poincaré this question in his letter of the previous day (§ 4-47-18). Sind diese Funktionen wirklich eindeutig? Ich verstehe nur dass sie in jedem Werthsysteme welches sie erreichen, unverzweigt sind.33endnote: 3 “Are these really single-valued? All I understand is that there is no branching over the entire range of values taken by them.” Here is my response. The functions investigated by Mr. Fuchs subdivide into three large classes. Those of the first two are indeed single-valued; those of the third are in general only unverzweigt;44endnote: 4 “Unbranched”. they are only single-valued if a condition is added to those given by Mr. Fuchs. These distinctions are not made in the first of Mr. Fuchs’s works; they can be found in two additional notes, unfortunately too concise, one included in Borchardt’s Journal 90, the other in Göttinger Nachrichten 1880.55endnote: 5 Fuchs (1881); (1880b).

I thank you very much for the last note that you had the kindness to send me.66endnote: 6 Klein (1882); (1923, 627–629). The results you state interest me greatly, here is why: I found them quite a while ago, but withheld publication since I wished to clarify the proof a bit; for this reason I would like to see your proof when you will have clarified it on your side.

I hope that the struggle we’ve just engaged in over a name – as it happens, with courteous arms – will not alter our good relationship. In any case, for your offensive I harbor no ill will at all, and for my defense I hope you harbor none towards me either. Besides, it would be ridiculous for us to debate further over a name. “Name ist Schall und Rauch’’ and after all, it’s all the same to me, do as you please, and I, for my part, will do as I please.77endnote: 7 The line is from Goethe’s Faust, Part 1, Scene 16, where Gretchen asks Faust if he believes in God. Faust, having sold his soul to the Devil, replies evasively that he has no name for what he believes in, and that the name itself is not essential: Name ist Schall und Rauch / Umnebelend Himmelsglut (The name is noise and smoke / beclouding the heavens’ blaze); see Gray (2000, 300 n14).

Please be assured, sir, of my greatest regard.

Poincaré

PTrL. Translated by S.A. Walter from the original French (§ 4-47-19). Previously translated by R. Burns in Saint-Gervais (2016).

Time-stamp: "15.05.2021 20:36"

Notes

  • 1 “By way of compensation”.
  • 2 Fuchs (1880a); Fuchs and Schlesinger (1906, 191–212). Klein asked Poincaré this question in his letter of the previous day (§ 4-47-18).
  • 3 “Are these really single-valued? All I understand is that there is no branching over the entire range of values taken by them.”
  • 4 “Unbranched”.
  • 5 Fuchs (1881); (1880b).
  • 6 Klein (1882); (1923, 627–629).
  • 7 The line is from Goethe’s Faust, Part 1, Scene 16, where Gretchen asks Faust if he believes in God. Faust, having sold his soul to the Devil, replies evasively that he has no name for what he believes in, and that the name itself is not essential: Name ist Schall und Rauch / Umnebelend Himmelsglut (The name is noise and smoke / beclouding the heavens’ blaze); see Gray (2000, 300 n14).

References

  • L. Fuchs (1880a) Über eine Classe von Functionen mehrerer Variabeln, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Koeffizienten entstehen. Journal für die reine und angewandte Mathematik 89, pp. 151–169. Link Cited by: endnote 2.
  • L. Fuchs (1880b) Ueber eine Classe von Funktionen mehrerer Variabeln, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Coefficienten entstehen. Nachrichten von der Königliche Gesellschaft der Wissenschaften und der Georg-August-Universität zu Göttingen, pp. 170–176. Link Cited by: endnote 5.
  • L. Fuchs (1881) Auszug aus einem Schreiben des Herrn L. Fuchs an C. W. Borchardt. Journal für die reine und angewandte Mathematik 90, pp. 71–73. Link Cited by: endnote 5.
  • R. Fuchs and L. Schlesinger (Eds.) (1906) Gesammelte mathematische Werke von L. Fuchs, Volume 2. Mayer & Müller, Berlin. Link Cited by: endnote 2.
  • J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. Birkhäuser, Boston. Cited by: endnote 7.
  • F. Klein (1882) Ueber eindeutige Functionen mit linearen Transformationen in sich (zweite Mittheilung). Mathematische Annalen 20, pp. 49–51. Link Cited by: endnote 6.
  • F. Klein (1923) Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. Link Cited by: endnote 6.
  • H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. Link Cited by: 7-2-64. H. Poincaré to Felix Klein, English translation.