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

22 Chapter 2. Biography of NIELS HENRIK ABEL
FRANCOEUR’S Cours complet de mathematiques pures [Complete course on pure mathe-
matics], 13 which in two volumes dedicated to the emperor of Russia, introduced arith-
metic, geometry, algebra, and differential and integral calculus. The textbook had been
translated into German by one of ABEL’S mentors, C. F. DEGEN (1766–1825), 14 but
since ABEL only came to master German during his tour 1825–27, he probably studied
the French original.
Besides writing textbooks on algebraic analysis (LAGRANGE’S approach to the cal-
culus), in 1807, GARNIER translated EULER’S Vollständige Einleitung zur Algebra [Com-
plete introduction to algebra] of 1770 into French. 15 With his limited knowledge
of German, it is doubtful whether ABEL read EULER in the original language, but
through the translations by LAGRANGE or GARNIER or even through FRANCOEUR’S
complete course on pure mathematics, ABEL became acquainted with the elementary
parts of contemporary mathematical knowledge, in particular the solution of cubic
and bi-quadratic equations.
2.2.1 An alleged solution formula
While still in grammar school, ABEL approached one of the most prestigious prob-
lems of contemporary mathematics: the search for an algebraic formula expressing
the solution of the quintic equation. Since the Western Renaissance, similar formulae
for equations of the first four degrees had been known. In 1821, ABEL believed to
have found a closed algebraic expression solving the next case: the general quintic.
He wrote down his result and showed it to his teacher HOLMBOE, who took it to C.
HANSTEEN (1784–1873), one of the two professors in science at the Christiania Univer-
sity. 16 HANSTEEN, who together with HOLMBOE were among the few people in Nor-
way competent enough to have a chance of understanding ABEL’S tedious argument,
took it to the University’s collegium academicum. The collegium took note of ABEL’S ar-
gument and wanted to make it public to a broader mathematical audience. However,
as the young Norwegian state was itself without means of such a publication with a
wide circulation, ABEL’S paper was sent to professor DEGEN in Copenhagen with the
hope that it be published in the transactions of Royal Danish Academy of Sciences and
Letters. DEGEN’S assessment proved to have a profound influence on ABEL’S career.
Upon reception of the paper, DEGEN scrutinized ABEL’S solution, and DEGEN’S re-
sponse to HANSTEEN is the only existing written source of ABEL’S adventure. There,
DEGEN requested an elaboration, a numerical example, and a rewriting of the manu-
script for the other members of Videnskabernes Selskab to be able to read it. To ABEL,
the refusal to immediately publish his result must have been disappointing. However,
13 (L.-B. Francœur, 1809).
14 (L.-B. Francœur, 1815). In 1839, it was again translated into German by KÜLP.
15 It has been translated into English as (L. Euler, 1972).
16 Very unfortunately, ABEL’S supposed solution has not survived and only speculative reconstructions
can be suggested.