7. The calculus
1. Introduction
By the help of this new Analysis Mr Newton found out most of the propositions in thePrincipia Philosophiae. (Newton
1967–1981 8 , 598–9)The ‘new Analysis’ was what we now call the calculus; a point in learning mathematics where
many students give up, and which many others never reach. Intended to make everything easy, it is
still found a stumbling-block. What, then, is it? A simple web encyclopaedia describes it as follows:
calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the
use of infinite processes, involving passage to a limit: the notion of tending toward, or approaching, an ultimate value.
(http://reference.allrefer.com/encyclopedia/C/calcul.html)
While not very clear unless you know what it is already, this definition would already have been
acceptable at the end of the seventeenth century. As our opening quote suggests, the calculus
begins with Newton; and in the extract he was, typically, discussing his own work (and somewhat
bending the truth) anonymously in the third person many years after the event. He had already
made himself into a monument, as his one-time colleague and later rival Leibniz never managed
to do, and even today the work continues. The British Library, whose courtyard is incongruously
dominated by a massive statue of the man,^1 lists in its catalogue 10 new books on Newton for
the year 2001 alone; and one must suppose that many lesser books have appeared, together with
articles learned and otherwise, student dissertations and entries on the many Newton websites.
Experience shows that ‘Newton-and-Leibniz’ is easily the most popular subject for student essays in
the history of mathematics—despite the difficulty of many of the early calculus texts. Is it not time
to call a moratorium, a temporary halt to all this industry? Given that most scholars have arrived at
a reasonable conclusion about the once burning question of priority in the discovery (to which this
chapter will return later), must they still be concerned with the date when Newton discovered the
inverse square law of gravitation, the reasons for his conversion from Cartesian geometry to the
methods of ‘the Ancients’, or the extent to which his religion, the pursuit of alchemy, or political
beliefs influenced his mathematics?
The answer is that at least some of the current research is important and necessary, even if much
of it seems to be lacking in the ‘social element’ which we have had occasion to single out for praise
in earlier chapters. This is partly because the story is still subject to myths and misunderstandings
(e.g. of when the calculus became widely known and available, and to whom); and partly because,
like the First World War (which has also had much too much written about its origins), the calculus
is an important founding event in European history. And while Newton and Leibniz get more
- Modelled, ironically one must suppose, on the drawing by Blake, for whom Newton represented the blindness and alienation of
rationalism.