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

18.2. Integration in logarithmic terms 339
18.2 Integration in logarithmic terms
Another problem which figures significantly in ABEL’S approach to and research on
higher transcendentals was the question of integration in more elementary forms. In
chapter 15, it was described how mathematicians attacked the study of elliptic inte-
grals although these were non-elementary. One of the approaches adopted was to
relate a number of elliptic integrals by elementary functions or to investigate situa-
tions in which the integration could indeed be effected in elementary (or finite) terms.
A similar idea was pursued by ABEL in his investigations on what he called “theory
of integration”. Thus, ABEL’S understanding of this notion differed from the present
one in the sense that it was highly formal or algebraic and did not concern a numeri-
cal interpretation of the integral. Such an interpretation was, of course, part of A.-L.
CAUCHY’S (1789–1857) complete program of rigorization and became very important
in the 19 th century mainly in the efforts to answer the challenges raised by Fourier
series.
Reminiscences of the Collegium mémoire. The first evidence of ABEL’S interest in
the theory of integration (in finite terms) originates from descriptions of a paper which
is no longer extant. As was already described in section 2.3, ABEL had hoped to em-
bark on his European Tour shortly after the application was sent to the Collegium of
the University in 1824. Before that time, in March 1823, ABEL presented a manuscript
to the Collegium academicum through C. HANSTEEN (1784–1873). It concerned “a gen-
eral presentation of the possibility of integrating all possible differential formulae” 17 .
The manuscript was given to professors HANSTEEN and S. RASMUSSEN (1768–1850)
for their professional evaluation. Their review was positive but no means of publish-
ing the paper were at hand and it was subsequently lost. However, from ABEL’S pub-
lished research, we may get an impression of what it could have contained. ABEL’S
notebooks contain a number of entries related to the question of integration in finite
terms; in particular, a manuscript for a large memoir on the theory of elliptic transcen-
dentals from this perspective has been included in the Œuvres. 18 Nevertheless, the
present description focuses on his main publication on the subject which occurred in
the Journal in 1826. 19
The local context of ABEL’S work on integration in finite terms was mainly related
to the same theme as his research in the Paris memoir (see chapter 19, below). How-
ever, it also included such issues as the reduction of all elliptic integrals to four basic
kinds which ABEL undertook in his manuscripts and which had been a corner stone
of LEGENDRE’S theory of elliptic integrals. 20 In the 1840s, mainly through the works
17 “en almindelig Fremstilling af Muligheden at integrere alle mulige Differential-Formler” (N. H.
Abel, 1902d, 4).
18 (N. H. Abel, [1825] 1839b).
19 (N. H. Abel, 1826d).
20 (N. H. Abel, [1825] 1839b, 101); for LEGENDRE’S theory, see section 15.3.