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

298 Chapter 15. Elliptic integrals and functions: Chronology and topics
VII Propriétés remarquables de la fonction
y = φx déterminée par l’équation f y.dx −
f x � (a − y) (a1 − y) (a2 − y) . . . (am − y) =
0, f étant une fonction quelconque de y
qui ne devient pas nulle ou infinie lorsque
y = a, a1, a2, . . . , am
VIII+IX Sur une propriété remarquable d’une classe très
étendue de fonctions transcendantes
X Sur la comparaison des fonctions transcendantes
XIII Théorie des transcendantes elliptiques
Table 15.3: ABEL’S early unpublished works on elliptic integrals and related topics.
The manuscripts are no longer extant but HOLMBOE dated them all to the period before
ABEL’S European tour. The roman numerals indicate the position of the manuscript
in (N. H. Abel, 1881, II).
elliptic integrals which were astounding to ABEL. JACOBI had obtained results which
were special cases of ABEL’S own findings, and ABEL was surprised by the sudden
element of competition. For a period of time, ABEL devoted himself to explaining and
elaborating the results of JACOBI within his own framework and this constituted a
second topic in his research on elliptic functions.
ABEL’S last approach to elliptic functions was the most general. Applying the
theory which he developed in the Paris memoir concerning integration of algebraic
differentials (see chapter 19) to the special case of elliptic functions, ABEL could sketch
a very general approach to elliptic functions which — based on functions of the first
kind — introduced all kinds of elliptic functions.
These aspects — which far from exhaust the discipline of elliptic and higher tran-
scendentals in the early nineteenth century — will be addressed in the subsequent
chapters. A complete description of the history of these transcendental objects is way
outside the scope of the present work as their study was one of the most important
and widely studied mathematical topics in the period. Instead, selections have been
made to illustrate how new objects were being introduced and how new tools — and
primarily algebraic tools — were being put to use in the investigation of these new
objects.