172 A History ofMathematics
integration—finding the fluent—is established as the central problem; so is the fact that you can
find areas by finding fluents. In Appendix A, I give his method of finding tangents—the easiest
application—with an example. It is worth comparing with Leibniz’s text for its greater clarity and
(fairly) good explanations. The work, as far as it went, was well set out, and would have been more
than a ‘useful guide for beginners’ if they had been allowed to see it.
There has been naturally considerable speculation (a) on why Newton did not publish it and
(b) on what would have been the effect if he had. It might well, to begin with, have mystified
its readers as Leibniz’s later publication did, but the fact of publication would inevitably have
brought clarification and improvements. Unfortunately, by 1670, his interests had already turned
awayfrom mathematics, and in Westfall’s opinion ‘[n]early all of Newton’s burst of mathematical
activity in the period 1669–71 can be traced to external stimuli’ (1980, p. 232). As we have seen,
reasons for publication in the late seventeenth century were varied, and correspond badly with the
image which is often presented of a new open scientific society. While some were free with their
ideas, anxiety about theft and arguments about priority were widespread, and it was common for
publication, whether by book or in a letter, to take place to forestall a potential rival and stake
a claim for a discovery, rather than to reach an interested audience. ‘Huygens, for example, had
no intention of revealing his great discovery at once, but he did want to safeguard his priority by
allusions in letters to his friends’, is a typical comment (Hofmann 1974, p. 107); and Huygens
was one of the more open publishers. The nascent ‘community of savants’ of the mid-seventeenth
century had created a climate in which reputation and national rivalry rather than actual financial
reward often encouraged scientists to be secretive; and Newton, always isolated and increasingly
suspicious, needed little encouraging. For the next 15 years his main attention was focused on
the pursuits of alchemy and biblical study, whose importance in his own estimation of his work
equalled that of mathematics.Exercise 2.Suppose given infinitely near points A=(x,y)and A′=(x+po,y+qo)as above. Show
that the line AA′has gradient q/p.Exercise 3.Follow through the argument which leads to the gradient of the tangent to y+xx=ax,
above.What are its strengths and weaknesses?Exercise 4.Try to do the analogous calculation for the curve given by equation (2).What problems arise
in finding the ratio q/p?6. Leibniz, a confusing publication
For what I love most about my calculus is that it gives us the same advantages over the Ancients in the geometry of
Archimedes, that Viète and Descartes have given us in the geometry of Euclid or Apollonius, in freeing us from having
to work with the imagination. (Leibniz letter to Huygens, 29th Dec. 1691, in Gerhardt 1962, 2 , p. 123)If no one knew what was going on in 1670, by 1690 it is fair to say that the few who did have
an idea were either confused or misinformed. It would be quite inaccurate to suppose that there
was a recognized object called ‘the calculus’ which was clearly destined to betheway ahead for
mathematics. Leibniz’s discovery, as is always pointed out, was different in many repects from
Newton’s; but they shared a common language to the extent that they could communicate, even if
(as in the case of Newton’s letters) partly for the purpose of concealment. Also like Newton, Leibniz