## 7-2-63. Felix Klein to H. Poincaré, English translation

Düsseldorf, 3 April 1882

Addr. Bahnstraße 15

Dear Sir,

When I received your letter yesterday via Leipzig, I was just about to
write you a few words to accompany the page proofs of my new note in
the *Annalen*, the page proofs of which should already be in your
hands.^{1}^{1}endnote:
^{1}
Klein (1882a), reedited in
Fricke et al. (1923, 627–629). In the meantime I have obtained a copy
of Prof. Fuchs’s note published in the *Göttinger
Nachrichten*.^{2}^{2}endnote:
^{2}
Fuchs (1882), reedited in
Fuchs and Schlesinger (1906, 285–287). If I could say a few words about
the latter, they would be to the effect that I consider it to be
completely mistaken. I have claimed only that Fuchs has never
published anything on “fonctions fuchsiennes”. It follows that the
second article that he cites (which, by the way, I have procured in order to
study more closely) is pointless. The first may, however, be
considered to be concerned with “fonctions fuchsiennes” insofar as it
deals with modular functions, but, lacking geometric intuition, Fuchs
has not fully grasped the proper character of the latter, which
lies in the nature of singular lines, as Dedekind showed already in
Volume 83 of * Borchardt*.^{3}^{3}endnote:
^{3}
Dedekind (1877). Finally, as for the
insinuation at the end of his note to the effect that my own work has
been essentially stimulated by his, historically this is simply
incorrect. My research began in 1874 with the determination of all finite
groups of linear transformations in one
variable.^{4}^{4}endnote:
^{4}
Klein (1876), reed. Fricke and Vermeil (1922, 275–301). Then in
1876 I showed that the problem raised at that time by Fuchs of
determining all integrable second order linear differential equations
was *eo ipso* solved.^{5}^{5}endnote:
^{5}
Klein (1877). The situation is precisely the
reverse of what Fuchs claims. It was not that I took his ideas, but
rather that I showed that his topic should be treated using my ideas.

In addition, and as you may imagine, I do not agree with your
analysis. If it were a matter of a general appreciation of Fuchs’s
works, I would willingly have his name bestowed on some *new*
class of functions that no one had yet studied, or even, for instance,
on the functions of several variables that he has put forward.*
However, the functions you have named after Fuchs already belonged to
others before you suggested the name. I am also quite sure that you
would have not proposed this name had you then (at the beginning) been
familiar with the literature. You then offer me, in compensation, one
might say, “fonctions keinéennes”. To the extent that I sense your
friendly intention here, to just that extent is it impossible for me
to accept the offer, as again perpetrating an historical untruth. If
my memoir in Volume XIX might give the impression that I am now
especially preoccupied with the “kleinéennes”, my more recent work
in Volume XX shows that as before I continue to regard the
“fuchsiennes” as my domain.^{6}^{6}endnote:
^{6}
Klein (1882b, a).

Enough on this topic. I immediately dispatched your note to the
printer, appending only a remark to the effect that for my part I adhere
to my previous opinion (and on this occasion drawing the public’s
attention to Mr. Fuchs’s note). You will receive the page proofs very soon
and I ask you to return them quickly to me here (where I am spending the
Easter holidays), and I will then do what is necessary as far as
publication is concerned. (Your note will appear directly after mine!) As
far as the passage about Schottky is concerned, I would like to point out
to you a posthumous article in Riemann’s *Works*, p. 413, where
exactly the same ideas are developed.^{7}^{7}endnote:
^{7}
The reference is to chapter 15, entitled
“Gleichgewicht der Electricität auf Cylindern mit kreisförmigem
Querschnitt und parallelen Axen” (Weber and Dedekind, 1876, 413–416). I should say, however, that it would
be difficult to determine the extent of a possible contribution by the
editor, Prof. Weber. Riemann’s *Works* appeared in 1876, and Schottky’s
dissertation was completed in 1875 and published in 1877 as a memoir in *Borchardt’s Journal*.^{8}^{8}endnote:
^{8}
Schottky (1875, 1877). However, the 1875 dissertation
constitutes only a part of the 1877 memoir, and I cannot recall if the
figure in question had already appeared in the 1875 text.

I should add that on my part I have no intention of prolonging our
*terminological* disagreement (once I have added the above-mentioned
footnote to your explanation). However, if I should be led to intervene in
the matter anew then I would, it is true, give a very complete and frank
account of it. Let us rather compete to see which of us is best equipped
to advance the entire theory in question! On my side, I believe my new note
represents a certain advance. A whole series of theorems on algebraic
functions can be proved using the new $\eta$-function – for instance, the
theorem that, in my book on Riemann, I initially indicated as only
probably true, namely that a surface of genus $p>0$ never admits
infinitely many single-valued *discrete* transformations (since
otherwise it would decompose into an infinite number of “equivalent
fundamental polygons”). And also the theorem that various of Picard’s
results for the case $p=0$ extend to general $p$, etc.

As for the methods I use to prove my theorems, I will inform you as
soon as I have further clarified them. In the meantime, could you not
describe for me the ideas you are currently pursuing? I scarcely need
add that we would be pleased to publish in the *Mathematische
Annalen* any article you wished to communicate. It will be crucial
to remain in active contact with you. For me, lively contact with
mathematicians aspiring to similar ends has always been a precondition
for mathematical production.

Very respectfully, your devoted

F. Klein

* Are these really *single-valued*? All I understand is
that over the whole of the range of values taken by them there is no
branching. However, I may be mistaken in this.

Dr. Hurwitz’s address, until further notice, is: *Hildesheim*, Langer
Hagen.

PTrL. Translated by S.A. Walter from the original German (§ 4-47-18), after R. Burns in Saint-Gervais (2016). See also the French translation (§ 7-2-39).

Time-stamp: "16.02.2023 15:38"

### Notes

- 1 Klein (1882a), reedited in Fricke et al. (1923, 627–629).
- 2 Fuchs (1882), reedited in Fuchs and Schlesinger (1906, 285–287).
- 3 Dedekind (1877).
- 4 Klein (1876), reed. Fricke and Vermeil (1922, 275–301).
- 5 Klein (1877).
- 6 Klein (1882b, a).
- 7 The reference is to chapter 15, entitled “Gleichgewicht der Electricität auf Cylindern mit kreisförmigem Querschnitt und parallelen Axen” (Weber and Dedekind, 1876, 413–416).
- 8 Schottky (1875, 1877).

## References

- Schreiben an Herrn Borchardt über die Theorie der elliptischen Modul-Functionen. Journal für die reine und angewandte Mathematik 83, pp. 265–292. link1 Cited by: endnote 3.
- Felix Klein Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. link1 Cited by: endnote 1.
- Felix Klein Gesammelte mathematische Abhandlungen, Volume 2. Springer, Berlin. link1 Cited by: endnote 4.
- Über Functionen, welche durch lineare Substitutionen unverändert bleiben. Nachrichten von der Königliche Gesellschaft der Wissenschaften und der Georg-August-Universität zu Göttingen, pp. 81–84. Cited by: endnote 2.
- Gesammelte mathematische Werke von L. Fuchs, Volume 2. Mayer & Müller, Berlin. link1 Cited by: endnote 2.
- Ueber binäre Formen mit linearen Transformationen in sich selbst. Mathematische Annalen 9, pp. 183–208. Cited by: endnote 4.
- Ueber lineare Differentialgleichungen. Mathematische Annalen 12, pp. 167–179. Cited by: endnote 5.
- Ueber eindeutige Functionen mit linearen Transformationen in sich (zweite Mittheilung). Mathematische Annalen 20, pp. 49–51. link1 Cited by: endnote 1, endnote 6.
- Ueber eindeutige Functionen mit linearen Transformationen in sich. Mathematische Annalen 19, pp. 565–568. link1 Cited by: endnote 6.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-63. Felix Klein to H. Poincaré, English translation.
- Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen. Ph.D. Thesis, University of Berlin, Berlin. Cited by: endnote 8.
- Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen. Journal für die reine und angewandte Mathematik 83, pp. 300–351. Cited by: endnote 8.
- Bernhard Riemann’s gesammelte mathematische Werke und wissentschaftlicher Nachlaß. Teubner, Leipzig. link1 Cited by: endnote 7.