TheCalculus 179
Leibniz was in Italy and did not reply to this crawling letter (there is more in the same vein).
However, the brothers’ persistence bore fruit; in the end, as Jakob had prophesied, they clarified,
adapted, and extended Leibniz’s method so that it could solve a vast range of problems; and they
proved effective and aggressive propagandists. As Roero characterizes it, the Bernoullis can be
credited with the ‘rediscovery’ of the Leibnizian algorithm (Roero 1989, p. 142).
Instructed by Johann, in 1696 the Marquis de l’Hôpital produced what might have seemed
impossible 10 years earlier, an ‘elementary’ introduction to the theory, theAnalyse des infiniment
petits, pour l’intelligence des lignes courbes.^10 In this admirable book, the student can with relative
ease learn both how the calculus works and why it works. A certain amount of what, in terms of
all previous mathematical practice, would be considered nonsense must be accepted along the way
(the problems are identical to those I have already indicated in Newton’s theory); but within its own
boundaries the theory works, and you will not make mistakes. The serious limitation of theAnalyse
is that it deals only with differential problems and does not touch integration. (Information on that
was to come from various sources, notably Newton’sQuadratura curvarum, in 1704.) However, you
can for the first time find a published proof of the ‘Leibniz rule’ for the differential of a product.Proposition II. To find the differentials of the products of several quantities multiplied, or drawn into each other.(Fauvel and
Gray 11.B.6.)
The differential ofxyisydx+xdy: forybecomesy+dy, whenxbecomesx+dx; and thereforexythen becomes
xy+ydx+xdy+dxdy. Which is the product ofx+dxintoy+dy, and the differential thereof will beydx+xdy+dxdy,
that is,ydx+xdy: becausedx dyis a quantity infinitely small in respect of the other termsydxandxdy: For if, for
example, you divideydxanddx dybydx, we shall have the quotientsyanddy, the latter of which is infinitely less than
the former.
Whence it follows, that the differential of the product of two quantities is equal to the product of the differential of the
first of these quantities into the second plus the product of the differential of the second into the first.
The core ‘problem’ of early calculus is neatly set out here; and you cannot really derive the product
rule for differentials in any other way. The differentialsdx,dyare serious quantities which cannot be
neglected, and which enter into the formula ford(xy). And yet their productdxdycan be neglected.
You can get used to doing things this way, but its justification is still shaky. However, at least what
was obscure in Leibniz has become transparently clear. If you suspend your worries about the
infinitely small, it is easy to follow the instructions, and (for example) to find tangents to any curve.
From the moment that the new methods became at all understood, the Leibnizians had to defend
them against those who held (with some justice) that they broke the rules of correct practice in
mathematics. In the 1690s, Leibniz was already defending his ‘new analysis’ against Nieuwentijt
(a Dutch philosopher), and the continued success of the calculus against such attacks is a good
demonstration of how much importance its defenders attached to its value in producing results,
as opposed to mere logical coherence. The most damning and serious attack was to come, some
years after the founders were dead, from Bishop George Berkeley. Berkeley’sThe Analyst(1734) is
logically (and in terms of classical mathematics) hard to fault, as well as being fine rhetoric; which
perhaps goes to show that logic is not always what determines the progress of mathematics. He
made the well-founded point that it is bad logic to claim that a theory is correct because it leads
to correct conclusions (it is often possible to deduce the truth from false assumptions). And by
a careful analysis of the version of the ‘product rule’ for differentiation given in thePrincipiahe
- One feels that L’Hôpital deserves some credit, although it is now recognized that his text is essentially due to Johann Bernoulli
and his introduction to Fontenelle. Such is the destructive power of scholarship.