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

278 Chapter 14. Reception of ABEL’s contribution to rigorization
ties and were only partially within the Cauchyian approach. CRELLE only considered
real arguments and exponents and divided his research into two parts correspond-
ing to the two papers. First, he showed by formal arguments from the trivial identity
1 = 1 that the binomial and its series had to be identical. The argument involved finite
differences which had been such a key component of his research within the German
combinatorial school. Second, CRELLE investigated the convergence of the binomial
series dividing into separate cases corresponding to various assumptions on a and k
(he wrote his binomial as (1 + a) k ). In each case, CRELLE considered the remainder
terms of the series and established conditions of convergence or divergence.
Concerning CRELLE’S publications on the binomial theorem, two remarks can be
made. First, the fact that CRELLE published on a particular case of the binomial the-
orem (real arguments and exponents) after ABEL’S more general result testifies to the
debate between the Cauchyian program and the German algebraic school. CRELLE
wrote in his introduction that he considered his proof to fulfill all requirements in-
cluding being truly rigorous and general and simultaneously clear and elementary.
These positive attributes were obtained through the use of algebraic manipulations. 4
Second, CRELLE did not initially consider or even mention the necessity of conver-
gence of the binomial series. ABEL’S critical attitude may have provoked CRELLE to
take up the issue in the second paper. Thus, at least in Germany, A.-L. CAUCHY’S
(1789–1857) new program of numerical equality was not immediately accepted — not
even in CRELLE’S Journal after ABEL’S publication and the translation of the Cours
d’analyse. 5
Later in the nineteenth and the twentieth century, when the German combinato-
rial school eventually lost ground, ABEL’S proof of the binomial theorem was recog-
nized as the first rigorous and general proof. 6 The local criticism and scrutiny did not
severely impair the evaluation of ABEL’S proof — primarily because the fundamen-
tal notions and knowledge of power series developed immensely over the nineteenth
century.
14.1.2 From ABEL’s “exception” to uniform convergence
As already indicated and cited, one of the major historical interests in ABEL’S contri-
bution to the rigorization movement was the “exception” which he presented against
Cauchy’s Theorem (see section 12.6). 7 The Fourier series representation of the function
f (x) = x 2 on an interval such as ]−π, π[ provided an example that not every convergent
sum of continuous functions was itself a continuous function as the Fourier series
was periodically discontinuous at the end-points of the interval.
4 (A. L. Crelle, 1829a, 305).
5 (A. L. Cauchy, 1828).
6 See e.g. (Stolz, 1904).
7 This history is particularly well described in (Bottazzini, 1986). LAKATOS’ reconstruction (Lakatos,
1976) is also extremely interesting.