TheCalculus 163
on extremely short intervals of time, so that the question of whether Leibniz received three
communications about Newton’s work in June 1676 or two in July (see Hofmann 1974, p. 231)
comes to have capital importance.
Such infinitesimals are not properly the concern of this chapter, which will rather try to focus on
the larger questions. How did the change take place? Why was such a dubious way of proceeding so
quickly accepted? What end did it serve, and who profited? And what were the positive and negative
effects of all the quarrels? This chapter will try to give a broad outline of the history, with reference
to such questions.Note.The material which is dealt with in this chapter is necessarily mathematically harder than
that of the previous chapters. The reader who is unfamiliar with the ideas of the calculus (and they
do still form a major part of mathematical culture) will therefore have to skip some of the detail;
it is not possible to omit it from the text without losing what the history is about.2. Literature
As has already been indicated, the literature is large and many-sided. To begin with the primary
sources, the whole of Newton’s mathematical work is available in an excellent modern edition,
edited by D. T. Whiteside (Newton 1967–81). Substantial extracts of thePrincipia(from books I and
III) are available on the Internet at http://www.members.tripod.com/gravitee—it is not clear whether
it is intended to extend this selection. Fauvel and Gray’s sourcebook is good on the period also,
giving some essential Newton material and the bulk of Leibniz’s key 1684 paper, plus the opening
of L’Hôpital’s book. The other works of Leibniz and the Bernoulli brothers are less accessible, and
usually in Latin; and while L’Hôpital’s book was quite well translated in the eighteenth century (see
quote above), there is no modern edition.
With the secondary sources, the question is really that of differentiating between ways of
thinking about history. There are, for example, numerous biographies of Newton, of which the
definitive one is Richard Westfall’s massive (1980), supplemented by Gjertsen’s handbook (1986);
and there is a comparable, if less comprehensive biography of Leibniz by Aiton (1985). Close to the
biographical are studies of the small community of late seventeenth-century scientists who were
able to appreciate and develop the calculus; for example, Hall (1980) and Hofmann (1974). And
third, there are more technical, one could say ‘internal’, studies of what was involved in the early
calculus, its techniques, and how practitioners saw what they were doing: these include particu-
larly Guicciardini (1999), various essays of Henk Bos, with summaries and further thoughts in
(1991), and Dupont and Roero (1991).^3
However, all of these works, even the most ‘external’, are dealing with a tiny community, in
comparison with the wider society in which the calculus was born and flourished. The obvious
reason for this is that, at the outset and for quite a time afterwards, it was found incomprehensible
outside a small circle. It is significant that Bishop Berkeley’s damning critique inThe Analystdates
from 1734, but still gives the impression that to a well-informed bishop the methods of the calculus
were relatively new. Although L’Hôpital’s 1696 text was an attempt at a popularization, intended
to have the same impact as Descartes’sGéométrie, written in French in a very similar style and- This book is often referenced. Apart from being in Italian, it seems almost impossible to find, but as a detailed study of Leibniz’s
paper, it is very interesting. Perhaps one can hope for a translation.