A History of Mathematics From Mesopotamia to Modernity

170 A History ofMathematics

separately. This is how Newton’s first private draft of his ideas begins, if with no explanation of

where he is heading:

As if the bodyAwith the velocitypdescribe the infinitely little line (cd=)poin one moment, in that moment the

bodyBwith the velocityqwill describe the line (gh=)qo.Forp:q::po:qo. So that if the described lines bee

(ac=)x,&(bg=)yin one moment, they will bee (ad=)x+poand (bh=)y+qoin the next. (Newton 1967–81, 1 ,

p. 414)

There is no record that this draft was shown to anyone; and if it had been, it might well have

been found strange. Newton was confidently using Descartes’s notation ofxs andys for varying

quantities, but he had added, as a new feature, that these quantities varied in time. He had, it

appears, read Galileo (at least theDialogues), but this bold introduction shows how far he was

prepared to go beyond Galileo and Kepler in thinking of infinitely short distances and times. ‘It

is possible that Dr Barrows Lectures might put me on considering the generation of curves by

motion, tho I not now remember it’, he reflected many years afterwards (Westfall 1980, p. 131);

but as Westfall points out, the idea was quite widespread (it went back to the Greeks, like so much

else, in the case of Archimedes’s spirals), and the cycloid (path of a fly glued to a rotating wheel)

which was currently being bitterly argued about, was a prime example. In any case, if the idea of

generating curves by motion was old, the idea of motion in an infinitely small instant was relatively

new, and the use of coordinates to describe it even newer. Where Galileo, as we saw, had to go

through a substantial argument to explain how one could make sense of ‘velocity at an instant’,

and where Kepler avoided defining it at all, Newton took it for granted; and, moreover, by using an

infinitely small timeo, he worked the argument backwards, to deduce from a (supposed known)

velocitypan infinitely small changepoin thex-coordinate. The infinitely small quantities are not

introduced or defended—they are simply there. [The idea of a ‘moment of time’opresupposes

some measurement of time which remains unclear, as Westfall points out (1980, p. 134), but the

advance is still substantial.]

What Newton did next in the 1665 tract was naturally to bring in a curve, in its Cartesian form.

He supposed thatxandywere related by an equation:

x^3 −abx+a^3 −dyy= 0 (2)

wherea,b,dare constants, and he showed how to find its tangent using his idea of change in time.

Rather than Newton’s complicated equation (whose curve he did not draw), let us look at a

simpler one, say (in seventeenth-century notation)y+xx=ax, which is the curve represented in

Fig. 3. If the velocities—or, as Newton was later to call them, ‘fluxions’—ofx,yarep,q, then in the

momento,xbecomesx+poandybecomesy+qo. The point A′whose coordinates arex+poand

y+qois (a) still on the curve and (b) infinitely near to the original one A.

Since the new point is on the curve, we havey+qo+(x+po)(x+po)=a(x+po).Becausethe

two points are infinitely near, the line AA′which joins them is the tangent. Subtracting the original

equation, and expanding the brackets, we get:

qo+ 2 xpo+ppoo=apo

We now divide this byoto getq+ 2 xp+ppo =ap. We want a relation betweenpandq(see

Exercise 2). Discard the termppo, which is infinitely small (this is the important part), and you end