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

19.2. The contents of ABEL’s Paris result and its proof 351
19.2 The contents of ABEL’s Paris result and its proof
ABEL’S objective in the memoir was to study integrals of the form
�
f (x, y) dx
in which x and y were related by some algebraic equation (19.1) and f was a rational
function. Such integrals provided a way of generalizing elliptic integrals; any elliptic in-
tegral could be written in the form above. However, it was not through a direct study
of one such integral that something new was to be learned, but by studying relations —
arising from the additional equation (19.3) — among a number of such integrals.
The contents of the Paris memoir can be structured into several results and their
primary applications:
1. The establishment of Main Theorem I on integration of certain sums of algebraic
differentials by elementary functions,
2. The establishment of Main Theorem II on the number of independent integrals of
algebraic differentials, and
3. Application of Main Theorem II to the simplest case, the case of hyperelliptic
integrals.
In the following, the results and methods of first two of these three parts will be
described; as will the other instances where ABEL presented his findings on related
issues. ABEL’S reasoning is rather cumbersome and not completely flawless. Some
of the subsequent objections and comments — primarily by P. L. M. SYLOW (1832–
1918) — are referred to in the course of the presentation. However, despite the reser-
vations, ABEL’S original argument is presented to illustrate how he ingeniously used
the tools at his disposal. Even if the contents and purpose of ABEL’S arguments can
seem to evade attention, his various tools and the contents of the Paris memoir are
subsequently summarized.
To various degrees of authenticity, ABEL’S argument has been described from the
viewpoint of the application to hyperelliptic integrals, see e.g. (Brill and Noether, 1894;
Cooke, 1989). However, as will be discussed in section 19.5.1, the chronology and in-
ternal logical structure suggests that the results of the Paris memoir were indeed prior
to and to some extent independent of the applications to this (afterwards) immensely
important special case.