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

26 Chapter 2. Biography of NIELS HENRIK ABEL
Norway, the means of publication were limited and advanced knowledge of the sci-
ences was confined to a few men. Therefore, foreign experts were invited to judge —
and possibly publish — ABEL’S first papers. DEGEN made reservations concerning the
completeness of ABEL’S research on the solution of the quintic and refused to present
ABEL’S paper to the Danish Academy until he had seen the method applied to a nu-
merical example. The numerical example was explicitly requested as a lapis lydius —
a test of correctness — and it is not improbable that the confrontation with an explicit,
difficult problem was what later led ABEL to realize his being in error.
The HANSTEEN-DEGEN period. In his historical introduction to the centennial memo-
rial volume, 27 E. B. HOLST (1849–1915) has emphasized the influence of HANSTEEN
and DEGEN on ABEL’S research 1821–24. P. L. M. SYLOW’S (1832–1918) analysis seems
applicable on at least two levels: topics and methods. On the topical level, the two pro-
tagonists exerted contrary influences. Responding to HANSTEEN’S suggestion, ABEL
had briefly worked on a physical problem; either because of this failed encounter or
because of a personal inclination, he subsequently focused exclusively on working
within pure mathematics. Following an advice of DEGEN, ABEL ventured into the
theory of integration in the tradition of EULER and LEGENDRE. 28 Being an impor-
tant theory of the eighteenth century left open for further developments, DEGEN had,
himself, spent some time studying elliptic integrals. But there is no real evidence to
suggest that DEGEN could have foreseen what his new disciple would do for the disci-
pline. Although their interests differed, both HANSTEEN and DEGEN were trained in
the typical eighteenth century mathematical literature including the men whom ABEL
considered his masters at the time. Formal manipulations and physical applicability
were considered positive aspects of the approaches of EULER and his mid-eighteenth
century contemporaries. The DEGEN-HANSTEEN period marks the end of ABEL’S
youthful encounters with the formal approach to analysis and at the same time marks
the beginning of a period of intense study of the theory of higher transcendentals
which would be ABEL’S masterpiece when judged by his contemporaries.
2.3 The European tour
After ABEL returned to Christiania from his first trip to Copenhagen, he soon real-
ized that there was little more for him to gain while isolated in the limited Norwegian
mathematical community. In 1824, ABEL applied with the support of the professors
HANSTEEN and RASMUSSEN for a travel grant from the university. ABEL’S primary
aim was to visit to the mathematical capital of his time, Paris. There, in Paris, math-
ematics had been institutionalized and cultivated to the highest level in the wake of
27 (Holst, 1902, 22).
28 In part IV, the influence on ABEL of these mathematicians will be traced, documented, and analyzed.