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

11.2. CAUCHY’s concepts of limits and infinitesimals 209
11.2 CAUCHY’s concepts of limits and infinitesimals
In the preliminaries, CAUCHY defined what he understood under the terms limit and
infinitely small. The two concepts are closely related in CAUCHY’S book although their
interrelation is not trivial. The proper interpretation of the concepts has sparked some
controversy in the historical literature which has sometimes chosen to focus on one
of the concepts and neglecting the other. To be fair to the source and to facilitate a
discussion of N. H. ABEL’S (1802–1829) reading of it, both CAUCHY’S definitions are
reproduced and translated here. 5
First, CAUCHY defined his concept of limit:
“Whenever the values successively attributed to one and the same variable
approach a fixed value indefinitely in such a way that it eventually differs from it
by as little as one desires, the latter is called the limit of all the others.” 6
A few sentences later, CAUCHY gave his definition of infinitely small quantities which
he based on the notion of limits:
“Whenever the successive numerical values of one and the same variable decrease
indefinitely in such a way that they become less than any given number,
this variable becomes what is called infinitely small or an infinitely small quantity. A
variable of this kind has zero for limit.” 7
Various conceptions of limits and infinitesimals had previously been suggested as
the foundations for the calculus and EULER called the calculus the “algebra of zeros”
because of his prolific use of infinitesimals. However, CAUCHY gave a process-based
definition of limits and — more importantly — showed how to work with it. Corre-
spondingly, to CAUCHY, infinitesimals were variable quantities which were involved
in limit processes and could be made as small as desired by particular choices of the
variable of the limit process.
It has puzzled certain historians of mathematics why CAUCHY simultaneously
employed limits and retained the older concept of (completed) infinitesimals. 8 The
fact remains that CAUCHY employed both concepts in different proofs and probably
thought of them as equivalent but suited for different purposes. 9
Importantly, CAUCHY sometimes used symbols to denote infinitely small quanti-
ties which were really (according to the definition) variables which tended toward the
5 A good interpretation is provided in (Grabiner, 1981b, 80–81).
6 “Lorsque les valeurs successivement attribuées à une même variable s’approchent indéfiniment
d’une valeur fixe, de manière à finir par en différer aussi peu que l’on voudra, cette dernière est
appelée la limite de toutes les autres.” (A.-L. Cauchy, 1821a, 4).
7 “Lorsque les valeurs numériques successives d’une même variable décroissent indéfiniment, de
manière à s’abaisser au-dessous de tout nombre donné, cette variable devient ce qu’on nomme
un infiniment petit ou une quantité infiniment petite. Une variable de cette espèce a zéro pour
limite.” (ibid., 4).
8 See discussion in (Lützen, 1999, 198–211).
9 See e.g. (Grabiner, 1981b, 87).