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

Chapter 19
The Paris memoir
N. H. ABEL’S (1802–1829) most famous result was first communicated in a paper
which he delivered to the Parisian Academy of ScienceAcadémie des Sciences in 1826.
The result which ABEL obtained in the so-called Paris memoir was a rather technical
one which dealt with the integration of algebraic differentials. 1 In its original, it was
formulated in the typical style of ABEL, his predecessors, and many of his contem-
poraries but during the century ABEL’S result was recast in a quite different language
and in another mathematical structure. 2 In modern mathematics, ABEL’S result is typ-
ically considered a part of algebraic geometry; readers who wish to see a presentation
of the result from such a modern perspective can consult e.g. (Shafarevich, 1974).
Besides presenting the main results, the present rendering of ABEL’S Paris memoir
aims at describing the central tools which ABEL employed in his reasoning. The fo-
cus on tools facilitates a continuation of the comparison with the methods involved in
ABEL’S purely algebraic works and also serves to support the discussion of the imme-
diate reception of ABEL’S results taken up in section 19.5. ABEL’S arguments in the
Paris memoir were conducted in a style heavily dependent on explicit manipulations
of formulae. In section 19.5.1, his subsequent announcements of the main results and
the much clearer sketches of proof contained therein are described and discussed.
19.1 ABEL’s approach to the Paris memoir
ABEL’S Paris memoir represents a pivotal point in his mathematical production. Many
investigations which ABEL had previously undertaken for their own sake and brought
to interesting conclusions were surpassed by the main result of the Paris memoir. The
Paris memoir was three-fold monumental: it was the culmination of a line of research
which ABEL had undertaken for years, it contained the result which brought him
widespread fame in the nineteenth century, and yet it provided this result with an
incredibly long and cumbersome proof.
1 ABEL’S result is also discussed in e.g. (Cooke, 1989; J. Gray, 1992).
2 For a brief discussion of styles, see chapter 21.
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