A History of Mathematics From Mesopotamia to Modernity

TheCalculus 173

was in no hurry to publish—it took nine years from his discovery of the method in 1675 to his first

paper, which revealed very little; and he was only forced into publication by an apparent claim to

priority by his friend Tschirnhaus. Rushing into print, he produced what must surely be one of the

worst of all path-breaking papers, with no proofs, few and contradictory explanations and a long

list of misprints.

About the 1684 paper a great deal has been said. It had almost no impact at the time, because

it was so obscure; probably only Leibniz himself believed that he had revealed a revolutionary

discovery to the world. And yet, today, in spite of the quite marked differences between seventeenth-

century calculus and our own, a reader who knows what calculus is about can form a fair idea

of his aims and method; as with Descartes’sGéométrie, his language, which must have seemed so

strange, has passed into common currency, and his rules for differentiation which his readers had

to take on trust are (roughly) the ones we learn in school. A historical take on the paper, then,

might properly contain two components:

1. What was Leibniz trying to communicate?


2. How might his communication have been received by a reader?

It would have been easier for readers who had been softened up by Newton’s version of the calculus,

but there were none of them, and the new notation (∫ dx,dyfor infinitesimals or ‘differentials’ and

for integrals) was quite unconnected with anything which had gone before. There are two

extracts from the 1684 paper (Fauvel and Gray 13.A.3., pp. 428–34) in Appendix B to illustrate

the difficulties.

What is noteworthy about Leibniz’s exposition (and is often noted) is that at the outset the

differentialsdx,dy, and so on are not infinitely small. They are ‘quantities’ whose only property

is that (for example)dy/dxis the gradient of the tangent to the curve specified by some equation

betweenxandy. Theruleswhich Leibniz then gives for working out differentials are introduced,

to say the least, abruptly: ‘Now, addition...’. However, they do make it possible to work out the

relations betweendxanddy. So, for example, usingd(xv)=xdv+vdx, we can easily deduce that

d(x^2 )= 2 xdx

Equally, ify=x^2 ,x=

√

y; and, using the above equation,

d(

√

y)=dx=

1

2 x

d(x^2 )=

1

2 √y

dy

And many other formulae can be obtained by more or less ingenious applications of the ‘Leibniz

rule’ for multiplication. The problem is that the rule is stated without proof; and its proof depends

on some sort of limiting argument—in the language of 1684, you need to use the fact thatdx

anddyare infinitesimal. Nor is the proof to be found later in the paper; the simple version due to

L’Hôpital is given in Section 8.

What Leibniz does next must have seemed still more perplexing; as shown in the second extract,

he uses thedprocedure a second time to arrive at something called, for example,ddv. Since we have

a rather vague idea of howdvhas been defined, we may well ask how the preceding methods could

possibly be applied to it to get a second differential. We shall have to wait, although the descriptions

of its properties (in distinguishing maxima from minima, for example) show that it is useful.