## 7-2-46. Felix Klein to H. Poincaré, English translation

Leipzig, 12 June [1881]

Address: Leipzig, Sophienstraße 10/II

Dear Sir,

Your three Notes in the *Comptes rendus*: “On fuchsian functions”,
which I first became acquainted with yesterday and then only
fleetingly, are so closely related to the reflections and endeavors that
have occupied me these last years, that I feel obliged to write to you.^{1}^{1}endnote:
^{1}
Poincaré
1881a; 1881b; 1881c;
reed. Nörlund and Lebon (1916, 1–10).
First of all I would like to tell you of various works of mine on elliptic
functions published in Volumes XIV,
XV,
and XVII
of the *Mathematische
Annalen*. As far as modular elliptic functions are concerned, I dealt with
only a special case of the independence relation that you consider; but a
closer examination will show you that I did indeed have a general point of
view. In this regard, I draw your attention to certain particular points:

Volume XIV, p. 128 deals with general functions representable by modular functions, independently of being connected with doubly periodic functions. Then follows, first in a special case, the important theorem on the fundamental polygon.

Volume XIV, pp. 159–160 is where I expound the result that every hypergeometric series can be represented by single-valued functions of suitable modular functions.

Volume XIV, p. 428ff. contains a table
illustrating the mutual disposition of triangles with sides of circular arcs
and angles
$\frac{\pi}{7},\frac{\pi}{3},\frac{\pi}{2}$
(which is also an example from the classes of special functions studied by Halphen), apropos
of which I should mention by the way that Mr. Schwarz has elaborated the case
$\frac{\pi}{2},\frac{\pi}{4},\frac{\pi}{4}$
in Volume LXXV of Crelle’s *Journal*.^{2}^{2}endnote:
^{2}
Schwarz (1873).

Starting from p. 62 of Volume XVII, I present a rapid overview of the more mature conceptions of the theory of elliptic modular functions which in the meantime had occurred to me. I have not published anything on these; however I did present them during the summer of 1879 in a course at the Munich Polytechnic. My line of thought, which comes very close to your exposition at many points, was as follows:

1. Periodic and doubly periodic functions are just examples of single-valued functions admitting linear transformations. It is the task of modern analysis to determine all these functions.

2. The number of such transformations may be finite; these give the
equations of the icosahedron, octahedron, …, that I investigated
earlier (*Math. Annalen* IX, XII)
and served
as the point of departure for all of these sets of ideas.^{3}^{3}endnote:
^{3}
Klein (1876, 1877a, 1877b).

3. Groups of infinitely many linear transformations, giving rise to useful
groups (discontinuous groups, in your terminology), are obtained *for
example* when one starts with a polygon with circular arcs as sides, such
that these circles cut a fixed circle orthogonally and have angles
equal to exact
simple fractions of $\pi$.

4. One should investigate all such functions (as indeed you have already begun doing); however, in order to attain concrete goals, let us confine ourselves to triangles of circular arcs, and, in particular, to elliptic modular functions.

Since then I have been much occupied with these questions, also in
discussions with other mathematicians, but except to say that I have not
yet obtained any definitive result, this is really not the place for them.
I would like to limit myself to what I have published or lectured on.
Perhaps I should have made contact with you sooner, or with one of your
friends, such as Mr. Picard. (When the occasion presents itself,
would you draw Mr. Picard’s attention to *Annalen*, XIV,
p. 122, § 8?)
For, the line of development of the set of
ideas that have engaged you for the last 2 or 3 years, is, in actuality,
very close to mine. I would also be very happy if this first letter led to
a continuing correspondence. It is true that at the moment other business
takes me from these questions, but I am the more encouraged to take them
up again in that next Winter I am to give a course on differential
equations.

Please convey my compliments to Mr. Hermite. I have often thought of starting a correspondence with him, and would have done so long ago — doubtless to my great profit — if it were not for the language problem. As you may perhaps know, I was in Paris long enough to learn to speak and write in French; however, in the meantime this ability has faded through disuse.

With the greatest respect,

Prof. Dr. F. Klein

PTrL. Translated by Robert Burns from the original German in Nörlund (1923, 97–99) (§ 4-47-1), and published in Saint-Gervais (2016).

Time-stamp: "20.07.2021 13:18"

## References

- Ueber binäre Formen mit linearen Transformationen in sich selbst. Mathematische Annalen 9, pp. 183–208. Cited by: endnote 3.
- Ueber lineare Differentialgleichungen. Mathematische Annalen 12, pp. 167–179. Cited by: endnote 3.
- Weitere Untersuchungen über das Ikosaeder. Mathematische Annalen 12, pp. 503–560. Cited by: endnote 3.
- Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. link1 Cited by: endnote 1.
- Correspondance d’Henri Poincaré et de Felix Klein. Acta mathematica 39, pp. 94–132. link1 Cited by: 7-2-46. Felix Klein to H. Poincaré, English translation.
- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92 (7), pp. 333–335. link1 Cited by: endnote 1.
- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92 (8), pp. 395–398. link1 Cited by: endnote 1.
- Sur une nouvelle application et quelques applications importantes des fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92, pp. 859–861. link1 Cited by: endnote 1.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-46. Felix Klein to H. Poincaré, English translation.
- Ueber diejenigen Fälle, in welchen die Gaussiche hypergeometrische Reihe eine algebraische Function ihres vierten Elements darstellt. Journal für die reine und angewandte Mathematik 75 (4), pp. 292–335. link1 Cited by: endnote 2.