7-2-47. H. Poincaré to Felix Klein, English translation

15 June [1881]11endnote: 1 The manuscript is annotated in an unknown hand: “Caen 1881”.


Your letter proves to me that you noticed earlier than myself some of the results I obtained in the theory of fuchsian functions. This does not surprse me at all, as I know how well versed you are in knowledge of non-Euclidean geometry, which is the real key to the problem occupying us.

When I publish my results I will acknowledge the fact; I hope to be able to get Volumes 14, 15, and 17 of Mathematische Annalen, which are not in the University of Caen’ library.

As for the talk you gave at Munich Polytechnic, would you provide some details on the topic, so that I may append to my memoir a note delivering full acknowledgment? Otherwise it is doubtful that I will be able to procure your work directly.

As undoubtedly I won’t immediately be able to get hold of Mathematischen Annalen, I would also ask you to agree to provide some explanation of certain points in your letter. You speak of

Die elliptische Modulfunktionen.22endnote: 2 Elliptic modular functions.

Why the plural? If the modular function is the square of the modulus expressed as a function of the ratio of the periods, then there is only one such function; thus the expression Modulfunktionen must mean something else.

What do you mean by these algebraic functions capable of being represented by modular functions? Also, what is the

Theorie der Fundamentalpolygone.33endnote: 3 Theory of the fundamental polygon.

I would also ask you to clarify the following points: Have you found all the Kreisbogenpolygone44endnote: 4 Polygons of circular arcs. giving rise to a discontinuous group? Have you proved the existence of functions corresponding to all discontinuous groups?

I have written to Mr. Picard to share your remark.

I congratulate myself, sir, on having the occasion to exchange with you. I have taken the liberty of writing to you in French as you tell me you know this language.

Please be assured, sir, of my respect.


PTrL. Translated by S.A. Walter from the original French in Nörlund (1923, 100) (§ 4-47-2).

Time-stamp: "27.04.2021 13:44"


  • 1 The manuscript is annotated in an unknown hand: “Caen 1881”.
  • 2 Elliptic modular functions.
  • 3 Theory of the fundamental polygon.
  • 4 Polygons of circular arcs.