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

12.4. Continuity 235
Actually, if taken literally, ABEL’S p would be the maximum of the sums in (12.9) — not
the supremum — but this distinction is beyond the point because for ABEL, the entire
discussion on the definition of p was a non-issue. The quantity p was simply (and
un-problematically) defined to be the greatest one among a sequence of numbers. The
nature of the number p will be pursued in section 12.6, where Lehrsatz V will shed even
more light on this non-issue. In the present case, the sequence of numbers is bounded
since the series is assumed to converge at δ, but this was not explicitly remarked by
ABEL.
In order to follow ideas of ABEL’S proof, it suffices to take either ABEL’S naïve
definition of p or the modernized one expressed in (12.9). From the equality (12.8),
ABEL then concluded, that
“for any value of α which is less than or equal to δ, m can be taken sufficiently
large that
ψ (α) = ω.” 30
Next, ABEL observed that φ (α) was an entire function of α, i.e. a polynomial, and
thus β could be taken small enough that
φ (α) − φ (α − β) = ω.
By combining these two results, ABEL concluded
f (α) − f (α − β) = ω.
Here we encounter ABEL’S way of operating with infinitesimals. The internal de-
pendencies among ω, m, and α have been completely obscured by the notation and
the argumentative style.
A simple observation, inspired by comparing ABEL’S proof with modern exposi-
tions of the calculus, concerns the use of infinitesimals. Today, infinitesimals have been
completely abandoned from “rigorous” presentations of the calculus, and to a person
trained within this program, ABEL’S usage of infinitesimals and even CAUCHY’S dual
definitions involving both limits and infinitesimals can be repulsive. But to ABEL they
were legitimate means of proving theorems.
ABEL’S Lehrsatz IV and the paradoxes of analysis. In the binomial paper, the fourth
theorem is given without further comments, but in his letters, ABEL had related it to
one of the strongest ongoing discussions among analysts. As already described in
30 “Mithin kann man für jeden Werth von α, der gleich oder kleiner ist, als δ, m groß genug annehmen,
daß
ist.” (ibid., 315).
ψ (α) = ω