RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

15.5. Chronology of ABEL’s work on elliptic transcendentals 297
terminatio attractionis. 20 Thus, when GAUSS spoke of the third of his research which
ABEL had anticipated, he probably referred to the theory of lemniscate functions. Con-
cerning the lemniscate function, GAUSS had performed the inversion, extended to
complex variables, found the resulting function to be doubly periodic, expressed its
addition formulae, and obtained infinite representations of it. The division problem of
the lemniscate had played an important role (together with the arithmetic-geometric
means) in motivating his research. Concerning all these results and aspects, GAUSS
was certainly correct in observing the similarity with ABEL’S approach. It is possible,
but not a necessary assumption, that rumors of GAUSS’ investigations and their meth-
ods had spread to ABEL through H. C. SCHUMACHER (1784–1873) and C. F. DEGEN
(1766–1825); certain of GAUSS’ letters to SCHUMACHER suggest that GAUSS for a short
while considered it a possibility. 21
15.5 Chronology of ABEL’s work on elliptic
transcendentals
Little is known of ABEL’S first encounters with elliptic functions. Presumably, ABEL
took up DEGEN’S suggestion in the letter to C. HANSTEEN (1784–1873) and began
studying the higher transcendentals possibly through the works of EULER and LEG-
ENDRE. A letter from his stay in Copenhagen 1823 indicates that he had shown DEGEN
a small paper in which “inverse functions of elliptic transcendentals” played a role. 22
In both editions of ABEL’S Œuvres, a number of manuscripts are included which were
among the papers destroyed in a fire in B. M. HOLMBOE’S (1795–1850) house in 1849. 23
According to HOLMBOE, the manuscripts date from before ABEL’S European tour, i.e.
they were written before 1825. 24 Apparently based on these manuscripts and the let-
ter from Copenhagen (as there are no other primary sources), 25 some historians have
credited ABEL with possessing the key results and methods around 1823.
It was, however, not until during and after the European tour that ABEL developed
and published his research on elliptic functions and higher transcendentals which
would merit so much attention. ABEL’S mature research on the topics can be seper-
ated into three categories. His first publication on the subject was the Recherches, which
introduced the crucial idea of inverting elliptic integrals of the first kind into elliptic
functions and established the latter as doubly periodic functions of a complex variable.
Simultanously with the publication of ABEL’S Recherches, the German mathemati-
cian CARL GUSTAV JACOB JACOBI announced some results on the transformation of
20 (C. F. Gauss, 1818).
21 (Schlesinger, 1922–1933, 167). I hope to have more to say on this at a later stage in connection with
future research on the mathematical milieu in Copenhagen in the early nineteenth century.
22 (Abel→Holmboe, Kjøbenhavn, 1823/08/04. N. H. Abel, 1902a, 5).
23 (N. H. Abel, 1881, II, 324) and (Stubhaug, 1996, 560).
24 (Holmboe in N. H. Abel, 1839, i)
25 (Abel→Holmboe, Kjøbenhavn, 1823/08/04. In N. H. Abel, 1902a, 4–8).