486 16. THE CALCULUSdistance traveled in an instant of time by an object moving with finite velocity.
This debate was carried on in private in the Paris Academy during the first decade
of the eighteenth century, and members were at first instructed not to discuss it in
public, as if it were a criminal case! Rolle's criticism could be answered, but it was
not answered at the time. According to Mancosu (1989), the matter was settled
in a most unacademic manner, by making l'Hospital into an icon after his death in- His eulogy by Bernard Lebouyer de Fontenelle (1657-1757) simply declared
the anti-infinitesimalists wrong, as if the Academy could decide metaphysical ques-
tions by fiat, just as it can define what is proper usage in French:
[T]hose who knew nothing of the mysteries of this new infinitesimal
geometry were shocked to hear that there are infinities of infinities,
and some infinities larger or smaller than others; for they saw only
the top of the building without knowing its foundation. [Quoted
by Mancosu, 1989, 241]In the eighteenth century, however, better expositions of the calculus were pro-
duced by d'Alembert. In his article on the differential for the famous Encyclopedic
he wrote that 0/0 could be equal to anything, and that the derivative ^ was not
actually 0 divided by 0, but the limit of finite quotients as numerator and denomi-
nator tended to zero.
Lagrange's algebraic analysis. The attempt to be clear about infinitesimals or to
banish them entirely took many forms during the eighteenth and nineteenth cen-
turies. One of the most prominent (see Fraser, 1987) was Lagrange's exposition
of analytic functions. Lagrange understood the term function to mean a formula
composed of symbols representing variables and arithmetic operations. He argued
that "in general" (with certain obvious exceptions) every function f(x) could be
expanded as a power series, based on Taylor's theorem, for which he provided his
own form of the remainder term. Using an argument that resembles the one given
by Ruffini and Abel to prove the insolvability of the quintic, he claimed that the
hypothetical expansion
s/x + h = ,/x~ + ph + qh^2 + ••• + hm/ncould not occur, since the left-hand side has only two values, while the right-hand
side has ç values. In this way, he ruled out fractional exponents. Negative exponents
were ruled out by the mere fact that the function was defined at h = 0. The
determinacy property of analytic functions was used implicitly by Lagrange when
he assumed that any zero of a function must have finite order, as we would say
(Fraser, 1987, p. 42).
The advantage of confining attention to functions defined by power series is
that the derivative and integral of such a function have a perfectly definite meaning.
Lagrange advocated it on the grounds that it pointed up the qualitative difference
between the new functions produced by infinitesimal analysis: dx was a completely
different function from x.
Cauchy's calculus. The modern presentation of calculus owes a great deal to the
textbooks of Cauchy, written for his lectures at the Ecole Polytechnique during
the 1820s.^13 Cauchy recognized that calculus could not get by without something
(^13) Although we have mentioned particular results of Cauchy in connection with the solution of
algebraic and differential equations, his treatises on analysis are the contributions for which he is