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”.

Sir,

Your letter proves to me that you noticed before me a few of the results I obtained in the theory of fuchsian functions. I am not at all surprised by this, 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 procure volumes 14, 15 and 16 of Mathematische Annalen, which are not in the University of Caen’s library.

As for the talk you gave at Munich Polytechnic, would you provide a few 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 a few explanations of a few 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 period ratio, 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 will also ask you to clarify for me 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,

Poincaré

PTrL. Translated by S.A. Walter from the original manuscript in French (§ 4-47-2).

Time-stamp: "28.10.2026 10:37"

Notes

  • 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.