## 7-2-53. H. Poincaré to Felix Klein, English translation

Caen, 5 July 1881

Sir,

I received your letter and read it with the utmost interest. A thousand pardons I ask for my question about Geschlecht im Sinne der Analysis Situs. I could have saved you the trouble of responding since I found the explanation on the subsequent page of your memoir.

You will undoubtedly recall that in one of my recent letters I asked your permission to quote a phrase in a communication in which I was seeking to generalize your results. You did not respond, and I took your silence for acquiescence. The communication was made in two parts, during the meetings of 27 June and 4 July.

You will find that we have overlapped on some points. However, I think you will find the citation of your phrase to be a sufficient guarantee.

Allow me, sir, another question: where will I find the works you mentioned by Messrs. Schwarz and Weierstrass, first concerning the theorem that:

*Man kann immer die Halbebene so auf ein Kreisbogenpolygon abbilden:*

*dass die Punkte I, II, III, IV, V welche den 1, 2, 3, 4, 5 auf der Begränzung der Halbebene entsprechen, beliebige lage haben*.^{1}^{1}endnote:
^{1}
“One can always map the half-plane onto a polygon of
circular arcs in such a way that the points I, II, III, IV, V
corresponding to the points 1, 2, 3, 4, 5 of the boundary of the
half-plane are positioned arbitrarily”.

This theorem was not unknown
to me, since I proved it in my communication of 23 May.^{2}^{2}endnote:
^{2}
Poincaré (1881), reedited in
Nörlund and Lebon (1916, 12–15).
But where will I find the works of my anticipators? In Volume
70 of *Crelle*? Where also will I find the developments you
mention in the following phrase: “Demgegenüber haben Weierstrass und Schwarz beider von mir berührten Frage der Abbildung von Keisbogenpolygon wirkliche Bestimmungen der in Betracht kommenden Constanten durch convergente prozesse gegeben”?^{3}^{3}endnote:
^{3}
“On the other hand, in connection
with the above-mentioned problem of mapping a polygon of circular arcs,
Weierstrass and Schwarz have effectively determined the relevant constants
by means of convergent processes.”

The theorem you tell me you have discovered has interested me a great deal. It is
clear that, as you say, your result includes as special cases “alle
meine Existenzbeweise”.^{4}^{4}endnote:
^{4}
“All my existence proofs.” However, it comes later.

I now come to your remark about Abelian functions. When I spoke of $4p+2$
constants, it wasn’t a question of the number of moduli. What I
said was this: “An algebraic equation of genus $p$ can always
be reduced to degree $p+1$. An equation of degree $p+1$ depends on $4p+2$
parameters; for, a *general* equation of degree $p+1$ depends on

$\frac{(p+1)(p+4)}{2}$ |

parameters. But there are

$\frac{p(p-1)}{2}-p$ |

double points. Therefore there remain $4p+2$ independent parameters. I thus obtain, not the number of moduli, but an upper bound for that number, and this was sufficient for my aim.

Please be assured, sir, of my respect,

Poincaré

PTrL. Slightly emended from the translation by R. Burns of the original French (§ 4-47-8) in Saint-Gervais (2016).

Time-stamp: "20.09.2023 18:42"

### Notes

- 1 “One can always map the half-plane onto a polygon of circular arcs in such a way that the points I, II, III, IV, V corresponding to the points 1, 2, 3, 4, 5 of the boundary of the half-plane are positioned arbitrarily”.
- 2 Poincaré (1881), reedited in Nörlund and Lebon (1916, 12–15).
- 3 “On the other hand, in connection with the above-mentioned problem of mapping a polygon of circular arcs, Weierstrass and Schwarz have effectively determined the relevant constants by means of convergent processes.”
- 4 “All my existence proofs.”

## References

- Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. link1 Cited by: endnote 2.
- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92, pp. 1198–1200. link1 Cited by: endnote 2.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-53. H. Poincaré to Felix Klein, English translation.