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

Chapter 20
General approaches to elliptic
functions
The present chapter serves to present the ideas which matured the theory of elliptic
functions in the nineteenth century. In particular, attention is called to the variations
in the ways of introducing elliptic functions because these variations are indicators of
the changing conceptions of rigorous foundations. Moreover, it is interesting to follow,
how results are turned into definitions to meet the changing standards of rigorous
definitions.
20.1 ABEL’s version of a general theory of elliptic
functions
In his ultimate publication, the Précis d’une théorie des fonctions elliptiques, 1 N. H. ABEL
(1802–1829) addressed the theory of elliptic functions from a more general perspective
than the Recherches. Since the Recherches which dealt exclusively with elliptic func-
tions of the first kind, ABEL had published a number of smaller papers on the theory
of elliptic functions, some investigations on integration in finite form, and various
announcements of his Paris memoir. ABEL had also been working on a continuation
of the Recherches which he postponed to devote his energy to completing the Précis.
The second Recherches mémoire was eventually published by M. G. MITTAG-LEFFLER
(1846–1927) in 1902. 2 However, to focus on the most general of ABEL’S approaches, we
shall limit the discussion to the Précis. In the Précis, these threads were woven together
to present an exposition of the entire theory of elliptic functions which simultaneously
addressed all three kinds.
1 (N. H. Abel, 1829d).
2 (N. H. Abel, 1902c).
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