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

Chapter 11
CAUCHY’s new foundation for analysis
Against the background of J. L. LAGRANGE’S (1736–1813) algebraic foundation for the
calculus, another and radically different program of rigorization emerged when A.-L.
CAUCHY (1789–1857) set out to write a textbook suitable for his courses at the École
Polytechnique. In a sense, CAUCHY’S famous textbook Cours d’analyse continues the
Lagrangian program as its subtitle Analyse algébrique testifies, 1 but its contents consti-
tuted a remarkable break with the Lagrangian system. In the Cours d’analyse, CAUCHY
reformulated and revised the theory of infinite series from his novel viewpoint based
on a shift in the conception of equality (see below). 2 Later, CAUCHY continued the
reworking of the foundations of the calculus in two further textbooks dealing with the
differential and integral calculus (see table 11.1).
The Lagrangian foundation for the calculus relied on a notion of equality between
functions (expressions) which was largely formal and had been inherited from L. EU-
LER (1707–1783) (see chapter 10). In CAUCHY’S hands, the concept of equality shifted
toward focusing on numerical or arithmetical equality: to CAUCHY, two functions
were only equal if they produced equal numerical results for equal numerical values
of the arguments. By way of a few central examples, I will document how this change
of approach was implemented and what its consequences were.
11.1 Programmatic focus on arithmetical equality
Generality of algebra. Describing the methods used in the Cours d’analyse, CAUCHY
stressed the way in which he had fought to obtain the standard of rigor which is char-
acteristic of geometry by denouncing arguments relying on the “generality of alge-
bra”. Such arguments could be, CAUCHY admitted, suitable inductions for obtaining
1 (A.-L. Cauchy, 1821b).
2 CAUCHY’S Cours d’analyse marks a turning point in the history of the calculus and has been given
due attention by historians of mathematics. The most comprehensive presentation is probably BOT-
TAZZINI’S introduction to a photographic reproduction of the Cours d’analyse (Bottazzini, 1990). In
particular, the section entitled The “Generality of Algebra” (ibid., xliv–xcvii) is of direct relevance to
the present discussion.
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