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

218 Chapter 11. CAUCHY’s new foundation for analysis
For now, we have to stress the importance this theorem played in CAUCHY’S proof
of the binomial theorem. The argument was, that since both the binomial (1 + x) m and
the infinite power series (11.4) satisfied the same functional equation for all rational m,
and since they were both continuous functions of m, they would coincide for all real
values of m. Thus, CAUCHY proved the binomial theorem for all real exponents (here
designated m).
In CAUCHY’S theory of rigorously restructuring calculus, the binomial theorem
played an extremely important role. 25 It provided one of the basic, theoretical bricks
which could be used to rebuild the existing theory of real analysis. Historians have
argued that CAUCHY’S rigorization and his proof of the binomial theorem constitute
a veiled attack on Fourier series. 26 However, although CAUCHY was critical toward
J. B. J. FOURIER’S (1768–1830) reasoning, I find such a hypothesis largely unnecessary
as CAUCHY’S rigorization program makes good sense from its own premises. 27
11.6 Early reception of CAUCHY’s new rigor
CAUCHY’S Cours d’analyse dealt exclusively with the theory functions from the per-
spective of infinite series. Later in the 1820s, he also published lectures pertaining to
rigorously founding the theory of differentiation and integration. 28
As a textbook for the École Polytechnique, the Cours d’analyse was not successful.
Because of internal animosities among the teachers, CAUCHY’S textbook was never
used as a textbook but it may have served as inspiration for students preparing for
the entrance exams of the school. Among his fellow mathematicians, CAUCHY’S pro-
gram also received mixed reactions. In Germany, A. L. CRELLE (1780–1855) men-
tioned CAUCHY’S textbooks in very positive terms, 29 and a German translation of the
Cours d’analyse appeared in 1828. 30 However, a distinct German reaction also existed
which sought to continue the formal, algebraic approach to foundations of analysis in
the so-called combinatorial school initiated by C. F. HINDENBURG (1741–1808), and M.
OHM (1792–1872) pursued his own rigorization program. 31
Thus, in the 1820s, the mathematical community could be divided into three camps
reflecting their attitudes toward rigor:
1. Some had picked up CAUCHY’S vision of a rigorization of analysis; both its
theme and its tools. They joined in the restriction to arithmetical equality and
adopted CAUCHY’S redefinition of central concepts in terms of limits.
25 (Grabiner, 1981b, 111).
26 See e.g. (Bottazzini, 1986, 110).
27 (Grabiner, 1981b, 111).
28 (A.-L. Cauchy, 1823; A.-L. Cauchy, 1829).
29 (Crelle, 1827; Crelle, 1828).
30 (A. L. Cauchy, 1828).
31 See (Jahnke, 1992).