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

14.1. Reception of ABEL’s rigorization 279
ABEL’S “exception” met with little response in the 1820s and 1830s. Actually, it
does not seem to have been quoted as influential in the first half of the 19 th century. In
the 1840s, however, other and probably independent events put Cauchy’s Theorem (see
page 217) back on the agenda. Independently and simultaneously in 1847, the math-
ematicians G. G. STOKES (1819–1903) and P. L. VON SEIDEL (1821–1896) published
investigations of the conditions under which a convergent sum of continuous func-
tions would not result in a continuous function. 8 In both cases, their research led to
the realization that a particular mode of convergence was involved and SEIDEL gave
it the name of “arbitrarily slow convergence”; 9 STOKES developed a refined hierarchy
of modes of convergence on intervals. 10 Of the two publications, I find SEIDEL’S par-
ticularly interesting because it was set up in the form of a proof analysis and stressed
the importance of keeping focus on the relations between limit processes. SEIDEL
even proposed notational advances which would help clarify the interdependence of
nested limit processes. Such thoughts were important in completely separating limit
processes from infinitesimals (see below).
CAUCHY’S eventual reaction. Apparently, even these researches of British and Ger-
man mathematicians did not directly prompt any reaction from the French mathemati-
cians, in particular CAUCHY. Eventually, CAUCHY did address the Cauchy Theorem
again in an address to the Paris Academy of 1853. 11 In a paper, prompted by remarks
made by French colleagues earlier that year, 12 CAUCHY described how the theorem of
the Cours d’analyse could be amended so that it no longer suffered any exceptions. The
fix which he proposed was the uniform convergence of the series in a form similar to
the modern requirement. CAUCHY refined the assumptions of the theorem by requir-
ing that a number N existed such that the difference |sm (x) − sn (x)| was less than ε
for all values of x in the interval I under consideration when m, n ≥ N. CAUCHY’S
requirement can easily be read as the modern definition of uniform convergence on
the interval I,
∀ε > 0 ∃N > 0 ∀m, n ≥ N ∀x ∈ I : |sm (x) − sn (x)| < ε.
With this stricter assumption, the original proof of the theorem carried through even
without a more elaborate notation to handle the two limit processes. In the paper, 13
CAUCHY considered the series
∞
sin nx
∑ n n=1
8 (Seidel, 1847; Stokes, 1847). GRATTAN-GUINNESS has pointed to the works of BJØRLING and considered
him the “fourth man” in this development besides STOKES, SEIDEL, and CAUCHY (who was
also involved, see below). See (I. Grattan-Guinness, 1986).
9 (Seidel, 1847, 37).
10 See (Stokes, 1847).
11 (A.-L. Cauchy, 1853).
12 (Briot and Bouquet, 1853a; Briot and Bouquet, 1853b).
13 (A.-L. Cauchy, 1853, 31).