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

2.3. The European tour 31
A.-L. CAUCHY’S (1789–1857) new program of founding analysis on the notion of
limits as expressed by inequalities had not reached Norway before ABEL left. CAUCHY
had expressed his thoughts most influentially in the textbook Cours d’analyse intended
(but never used) for instruction at the École Polytechnique which was printed in 1821. In
a review of CAUCHY’S Exercises de mathématiques, CRELLE spoke highly of CAUCHY’S
insights and his other works, and there can be little doubt that during ABEL’S time in
Berlin, CAUCHY’S new analysis was discussed by a circle of mathematicians around
CRELLE. One member of the circle, the mathematician M. OHM (1792–1872) 45 reacted
to CAUCHY’S new rigorization by devising his own approach to algebraic analysis
which was kept the formal aspect which CAUCHY had rejected. 46 In a letter, ABEL
records how the circle discontinued its meetings because of G. S. OHM’S (1789–1854)
arrogant mentality. 47 It is possible, that mathematical topics may also have played a
role.
“The work [Exercises de mathématiques] is full of deep analytical investigations
as it would be expected from the acute and inventive author of Cours d’analyse,
Leçons sur le calcul infinitésimal, Leçons sur l’application du calcul infinitésimal à la
géométrie, etc.” 48
ABEL discovered the works of CAUCHY in 1826. CAUCHY’S new approach to the
theory of infinite series took ABEL by storm, and soon ABEL became one of CAUCHY’S
most devoted missionaries (see part III). In print, ABEL first disclosed his sympathies
in his work on the binomial theorem, which was printed in the early spring of 1827
(see table 2.1), but ABEL’S letters allow us to date his encounter with the new Cauchian
rigor in analysis more precisely. In a famous letter dated 16 January 1826, i.e. while in
Berlin, ABEL wrote to HOLMBOE:
“Taylor’s theorem, the foundation for the entire higher mathematics, is equally
ill founded. I have found only one rigorous proof which is by Cauchy in his Resumé
des leçons sur le calcul infinitesimal.” 49
It is likely, that it was also in CRELLE’S library that ABEL came across CAUCHY’S
famous textbook Cours d’analyse, a work which had tremendous consequences for
ABEL’S attitude toward rigor. 50
45 For the years of birth and death, see (Jahnke, 1987, 103). OHM was the younger brother of the famous
physicist OHM.
46 (Jahnke, 1987; Jahnke, 1993).
47 (Abel→Hansteen, Berlin, 1825/12/05. N. H. Abel, 1902a, 11).
48 “Das Werk ist voller tiefer analytischer Untersuchungen, wie sie sich von dem scharfsinningen und
an neuen Ideen reichen Verfasser des “Cours d’analyse”, “Leçons sur le calcul infinitésimal,” der “Leçons
sur l’application du calcul infinitésimal à la géométrie, etc.” erwarten lassen.” (A. L. Crelle, 1827, 400).
49 “Det Taylorske Theorem, Grundlaget for hele den høiere Mathematik er ligesaa slet begrundet. Kun
eet eneste strængt Beviis har jeg fundet og det er af Cauchy i hans Resumé des leçons sur le calcul
infinitesimal.” (Abel→Holmboe, 1826/01/16. N. H. Abel, 1902a, 16).
50 (I. Grattan-Guinness, 1970b, 79). Again, this influence will be documented in part III.