176 A History ofMathematics
Leibniz’s manuscript notes). However, none of this matters in terms of the impact such a solution
would have had on any supposed audience. Rapidly solving a problem which, in contemporary
terms, was interesting and difficult, it must have been, of all the sections of the paper, the one
which most invited the reader to make an effort, and it is unfortunate that it came at the end. In our
terms, we would say that XC (the subtangent) isw/(dw/dx), and we require this to be a constanta.
We derive the equation
dw
dx=a.w (3)which has the solutionw=Aeax, or (as Leibniz says)xis proportional to the logarithm ofw.
Leibniz’ solution is, in our terms, ‘from first principles’. First, he has to assume thatdxis constant.
In modern notation this isalwaysan unspoken assumption, based on the choice of xas the
‘independent variable’ in terms of whichwis expressed. (Bos 1991, chapter 5 is an interesting
essay on the nature of the ‘early calculus’ which brings this point out.) Next, he notices that as you
adddxtox, you changewtow+dw=w+(b/a)w=w( 1 +(b/a)). (Note that ifdxis infinitely
small, which it should be, then so isb, but this is not stated.) Hence, as Leibniz states, arithmetic
progression in thexs corresponds to geometric progression in thews. We would then say thatw
was a power function (e.g.ex)ofx, but in the 1680s they were not used to such functions. Hence
Leibniz’s statement goes the other way round:xis a logarithmic function ofw.Exercise 5.Deduce the formula for d(x^2 )from that for d(xv). Next, see if you can generalize to d(xn),
where n=3, 4,...Exercise 6.Solve the differential equation (3). What properties of differentials and integrals have you
had to use?7. ThePrincipiaand its problems
After they had been some time together, the Dr asked him what he thought the Curve would be that would be described
by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from
it. Sr Isaac replied immediately that it would be an ellipsis, the Doctor struck with joy and amazement asked him how
he knew it, Why saith he I have calculated it, Whereupon Dr Halley asked him for his calculations without any further
delay, Sr Isaac looked among his papers but could not find it, but he promised to renew it, & then to send it him.
(De Moivre quoted in I. B. Cohen 1971, pp. 297–8)‘The Dr’ is Newton’s friend Halley, and the above breathless quote (all those commas) is the classic
account of how Halley’s question on the paths of planets plunged Newton into three years’ intense
mathematical work which issued in thePrincipia(1687). If its three massive volumes may have
been as strange and new as Leibniz’s work, they were certainly more impressive, and backed
up by a formidable apparatus of proof. As the quote above indicates, the work deduces (using
Newton’s three laws as starting point) that the observed paths of the planets are compatible with
an inverse-square law of attraction; and—more uncertainly—that such a law is the only one
which can account for the observations. Whatever he may have said—as in the opening quote of
the chapter—about his calculus having been used to deduce his results, the book presents itself as
new physics demonstrated by means of old (i.e. of course, Greek) mathematics. The reasons for this