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

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

Sir,

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.^{2}^{2}endnote:
^{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.^{3}^{3}endnote:
^{3}
Theory of the fundamental
polygon.

I would also ask you to clarify the following points: Have you found
all the Kreisbogenpolygone^{4}^{4}endnote:
^{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

Time-stamp: "27.04.2021 13:44"

## References

- Correspondance d’Henri Poincaré et de Felix Klein. Acta mathematica 39, pp. 94–132. Link Cited by: 7-2-47. H. Poincaré to Felix Klein, English translation.