7-2-62. H. Poincaré to Felix Klein, English translation

[28.03.1882]

Sir,

Recently you were good enough to publish in the Mathematische Annalen my article on single-valued functions that replicate themselves under linear substitutions, to which you appended a note outlining the reasons why you find poorly suited the names I bestowed on these transcendental functions. Allow me to address you a few lines in defense of my terminology, which I did not choose haphazardly.

If I believed I should bestow on these new functions the name of Mr. Fuchs, it was not out of disregard for the value of your work and that of Mr. Schwarz. I am the first, on the contrary, to to appreciate its great significance. However, it was impossible for me to ignore the remarkable discoveries published by the Heidelberg professor in Crelle’s Journal. They form the basis of the theory of linear equations and without them I would not have been able to begin my investigations of my transcendental functions, so directly linked to that theory. In his first articles, it is true, Mr. Fuchs takes a point of view a little different from mine and concerns himself neither with the discontinuity of the groups nor with the single-valuedness of the functions. However, Mr. Schwarz, in his memoirs in Volumes 70 and 74 of Crelle’s Journal, is not preoccupied by these matters, either; he has a few words to say in a very special case in his memoir in Volume 75, which I cite in my note. It is only there that he finds himself “Auf dem Gebiete des fonctions fuchsiennes”. In your beautiful investigations of modular functions, your approach differs little from mine, but you had more in mind the study of elliptic functions than that of linear equations. As for Mr. Fuchs, in his memoirs in Volumes 83 and 89 of Crelle’s Journal he ascended to a new point of view and illuminated the close connection between the theory of differential equations and that of certain single-valued functions. It was the reading of these memoirs that became the point of departure for my investigations.

As far as Kleinian functions are concerned, were I to have bestowed any other name than yours to them, I would have felt guilty of injustice. It was Mr. Schottky who discovered the figure you discuss in your letter, but it was you who had “ihre principielle Wichtigkeit betont”, as you say at the close of your learned article: “Ueber eindeutige Funktionen mit linearen Transformationen in sich”.

Concerning your remarks about Riemann, I can do nothing but subscribe to them completely. He was one of those geniuses who so change the face of science that they leave their imprint, not only on the works of their current crop of students, but on those of all their successors over the years. Riemann created a new theory of functions, and it will always be possible to find there the seed of everything that has been done and will be done after him in mathematical analysis.

PTrL. Translated by S.A. Walter from the original French (§ 4-47-17). Previously translated by R. Burns in Saint-Gervais (2016).

Time-stamp: " 2.05.2021 09:11"

Notes

• 1 The manuscript, destined for publication in Mathematische Annalen 20 (Poincaré, 1882b), accompanied Poincaré’s letter to Klein of 28 March 1882 (§ 4-47-16). It bears several annotations by Klein, all of which are markup for publication; these are ignored here.
• 2 Poincaré (1882a).
• 3 “[I]n the domain of fuchsian functions.”
• 4 [I]t was you who had “underlined their fundamental importance”.
• 5 Klein (1882), reedited in Fricke et al. (1923, 622–626).