## 7-2-59. Felix Klein to H. Poincaré, English translation

Leipzig, 13 January 1882

Dear Sir!

I have not yet thanked you personally for transmitting your article, for which I am very much obliged. As things stand, it will go to press in a few days. You will be getting page proofs, which I ask you to return to Teubner Publishing in Leipzig. Would you, in particular, examine the short commentary that I have, along the lines mentioned earlier, appended to your article, and in which I protest, as strongly as I can, against the two names Fuchsian and Kleinian, citing Schottky with respect to the the latter, and, incidentally, pointing to Riemann as the one who initiated all these investigations? I endeavored to preserve as moderate a tone as possible in this commentary, but I ask you to write me immediately if you wish to make a change. My comments avoid any diminishment of the merit of your investigations. Furthermore, I have written another short article which will appear right after yours.11endnote: 1 Klein (1882), reed. Klein (1923, 622–626). It contains, also without proof, some results relating to the area in question, primarily the following one: Every algebraic equation $f(w,z)=0$ can be solved in one and only one way by $w=\phi(\eta)$, $z=\psi(\eta)$ in terms of $p$ independent reentrant cuts on the corresponding Riemann surface, where $\eta$ is a discontinuous group of the kind you spoke of regarding my letter. This theorem is all the more beautiful in that this group has exactly $3p-3$ essential parameters, that is, the same as the number of moduli possessed by the equations of the given $p$. In this connection further considerations arise which seem to me to be of interest. In order to share this with you fully I have ordered the publisher to send you the page proofs of my article for you to have at your disposal.

As far as the proofs are concerned, this is a troublesome affair. I always operate with Riemann’s ideas respecting “geometria situs”. This is difficult to express quite explicitly. I shall make every effort to achieve this in due time. Meanwhile I would like very much to correspond with you on this subject and also on your proofs. You can be sure that I will study with the greatest interest any letters on this subject that you may send me, and that I will answer them accordingly in detail. If you should wish to publish them in one form or other, the Annalen are naturally at your disposition.

Very respectfully, your devoted,

F. Klein

PTrL. Translated by S.A. Walter from the original German (§ 4-47-14), after R. Burns in Saint-Gervais (2016). See also the French translation (§ 7-2-38).

Time-stamp: " 1.05.2021 00:07"

## References

• F. Klein (1882) Ueber eindeutige Functionen mit linearen Transformationen in sich. Mathematische Annalen 19, pp. 565–568. Cited by: endnote 1.
• 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.