## 7-2-48. Felix Klein to H. Poincaré, English translation

Leipzig 19 June 1881

Dear Sir,

Upon receiving your welcomed letter yesterday, I immediately sent you those offprints I still have of works relating to our topic. Allow me to add today a few lines of explanation pertaining to them. Of course, a single letter will not suffice; rather must we correspond more extensively in order to establish mutually close contact. Today I would like to put forward the following points:

1. Of the papers sent to you the three most important from the 14th Volume of
the *Math. Annalen*
are missing, as are also my investigations of the icosahedron in Volumes
9 and 12,
and my second memoir on linear differential equations (which also
appears unknown to Mr. Picard) in
Volume 12.^{1}^{1}endnote:
^{1}
Klein (1879b, a, c, 1876, 1877c, 1877a, 1877b).
I entreat you to procure them somehow. I have been sending various
offprints to Paris – to Hermite, for instance.

2. The work of my students Dyck and Gierster complements my own. I am asking both of them to send you their offprints. Mr. Hurwitz’s doctoral dissertation, relating to these same theories, is in press and will be delivered to you in a few weeks.

3. Since last Autumn one of your countrymen, whose name you must surely know since he studied under Picard and Appell, namely Mr. Brunel (address Liebigstraße 38/II), has been here. Perhaps you would be interested in starting a correspondence also with him; he could recount – better than myself – the organization of our seminar and the role played therein by single-valued functions invariant under linear transformations.

4. I let Mr. Gierster write up a set of notes from the course I gave in the summer of 1879. At the moment it is on loan, but I should get it back in a few days and will go through it with Mr. Brunel before getting back to you with an account of it.

5. I reject the name “Fuchsian functions” although I understand that
you were led to these ideas via Fuchs’s work. In essence, all these
investigations are based on those of Riemann. My own evolution in this
regard was strongly influenced by Schwarz’s deliberations, closely linked
to Riemann’s and of great significance, appearing in Volume 75 of *Borchardt’s Journal*
(and which I can strongly recommend if you
are unaware of them).^{2}^{2}endnote:
^{2}
Schwarz (1873).
Mr. Dedekind’s memoir on elliptic modular functions
appeared only in Volume 83 of *Borchardt’s Journal*,
when the geometric representation of modular functions was already clear to me
(by the autumn of 1877).^{3}^{3}endnote:
^{3}
Dedekind (1877).
By their ungeometric form, Fuchs’s memoirs stand
in deliberate opposition to these works. I don’t deny the great
service that Mr. Fuchs has rendered other parts of the theory of
differential equations, but his work here leaves so much to be desired
that on the only occasion when, in a letter to Hermite, he expatiated on
elliptic modular functions, he made a fundamental mistake, which Dedekind
only gently criticizes in the above-mentioned
memoir.^{4}^{4}endnote:
^{4}
Fuchs (1877),
reed. Fuchs and Schlesinger (1906, 85–111).

6. One may, in particular, define a function invariant under linear
transformations by the property that it maps the *half-plane* onto a
given polygon with sides circular arcs. This in fact represents only a
special case of the general situation (I don’t know yet if you have been
limiting yourself to just this particular case). The corresponding group
of linear transformations is then characterized by the fact that it is
contained in a group of operations twice as large, which, in addition to
linear transformations, contains reflections (transformations by
reciprocal radii). In this case
the existence of the function has been rigorously established in
earlier work of Schwarz and Weierstrass, inasmuch as one prefers not to
appeal to general Riemannian principles. See Volume 70 of
*Borchardt*, Abbildung
der Halbebene auf Kreisbogenpolygone.^{5}^{5}endnote:
^{5}
Mapping
the half-plane on polygons of circular
arcs, by which title Klein presumably means to refer to Schwarz (1869);
reed. Schwarz (1890, 65–83).

7. Even in this special case I haven’t yet been able to find all the
“groupes discontinus”; I have only established that there are many for
which there is no determined fundamental circle, so that the analogy
with non-Euclidean geometry (with which, by the way, I am very familiar) does
not hold. If you take, for example, any polygon, the sides of which are any
tangentially incident circles *whatsoever*, then generation via symmetry
will always yield a *groupe discontinu*.

8. You will doubtless find answers to the other questions you pose in your letter in the articles I am sending you, in particular to those concerning the plural form of “Modulfunktionen” and, especially, “Fundamentalpolygonen”.

Hoping to hear from you again soon,

your very devoted

F. Klein

PTrL. Translated by S.A. Walter from the transcription in German by Nörlund (1923) (§ 4-47-3), in view of the translation by R. Burns in Saint-Gervais (2016). See also the French translation (§ 7-2-34).

Time-stamp: "27.04.2021 19:44"

### Notes

## 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.
- Sur quelques propriétés des intégrales des équations différentielles, auxquelles satisfont les modules de périodicité des intégrales elliptiques des deux premières espèces. Journal für die reine und angewandte Mathematik 83, pp. 13–37. link1 Cited by: endnote 4.
- Gesammelte mathematische Werke von L. Fuchs, Volume 2. Mayer & Müller, Berlin. link1 Cited by: endnote 4.
- Ueber binäre Formen mit linearen Transformationen in sich selbst. Mathematische Annalen 9, pp. 183–208. Cited by: endnote 1.
- Ueber lineare Differentialgleichungen. Mathematische Annalen 11 (1), pp. 115–118. link1 Cited by: endnote 1.
- Ueber lineare Differentialgleichungen. Mathematische Annalen 12, pp. 167–179. Cited by: endnote 1.
- Weitere Untersuchungen über das Ikosaeder. Mathematische Annalen 12, pp. 503–560. Cited by: endnote 1.
- Ueber die Erniedrigung der Modulargleichungen. Mathematische Annalen 14, pp. 417–427. Cited by: endnote 1.
- Ueber die Transformaiton der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades. Mathematische Annalen 14, pp. 111–172. Cited by: endnote 1.
- Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Mathematische Annalen 14, pp. 428–471. Cited by: endnote 1.
- Correspondance d’Henri Poincaré et de Felix Klein. Acta mathematica 39, pp. 94–132. link1 Cited by: 7-2-48. Felix Klein to H. Poincaré, English translation.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-48. Felix Klein to H. Poincaré, English translation.
- Ueber einige Abbildungsaufgaben. Journal für die reine und angewandte Mathematik 70, pp. 105–120. link1 Cited by: endnote 5.
- 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.
- Gesammelte mathematische Abhandlungen von H. A. Schwarz, Volume 2. Springer, Berlin. link1 Cited by: endnote 5.