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

228 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
12.2 Infinitesimals
ABEL’S reading and rendering of CAUCHY’S fundamental concepts came to influence
the development of analysis in the nineteenth century. At certain points, his interpre-
tations were clearer and more specific and some of them would eventually coincide
with the standardized interpretations laid down by men such as G. P. L. DIRICHLET
(1805–1859) and WEIERSTRASS.
One of the basic notions which ABEL introduced in his binomial paper was that of
infinitesimals. In a footnote, ABEL explained,
“For brevity, in this paper ω denotes a quantity which can be less than any
given arbitrarily small quantity.” 22
Despite the awe which ABEL felt for CAUCHY’S work, this definition is not truly in
accord with CAUCHY’S notion of infinitesimals. As discussed above, CAUCHY had
interpreted infinitesimals as variables with limit zero but in ABEL’S definition, the
infinitesimals seem to reenter as completed quantities less than any finite quantity
but different from zero. The limit process has seemingly faded into the background.
To illustrate this way of designating infinitesimals by symbols, we may reconsider
CAUCHY’S proof of the Cauchy Theorem (see page 217) interpreted in ABEL’S notation.
ABEL did not undertake this proof, but the arguments are directly to those which he
employed in proving the Lehrsatz IV (see below). By continuity of the finite polyno-
mial sn,
and by the convergence of s,
Therefore
sn (x + α) − sn (x) = ω,
rn (x + α) = ω, (12.5)
rn (x) = ω.
s (x + α) − s (x) = ω
and the continuity has been “proved”. This way of designating infinitesimals by the
same symbols regardless of the way in which they depend on other variables and
infinitesimals hid and obscured the basic problems of the above argument. In the ar-
gument, the n which appears in (12.5) must depend on α and ω and can be unbounded
as α and ω vanish. However, it took quite some time and a detailed analysis of these
dependencies to clear out the proof (see section 14.1.2, below).
22 “Die Kürze wegen soll in dieser Abhandlung unter ω eine Größe verstanden werden, die kleiner
sein kann, als jede gegebene, noch so kleine Größe.” (N. H. Abel, 1826f, 313, footnote).