## 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.^{1}^{1}endnote:
^{1}
Klein (1882), reed.
Fricke et al. (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

- Felix Klein Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. link1 Cited by: endnote 1.
- Ueber eindeutige Functionen mit linearen Transformationen in sich. Mathematische Annalen 19, pp. 565–568. link1 Cited by: endnote 1.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-59. Felix Klein to H. Poincaré, English translation.