A History of Mathematics From Mesopotamia to Modernity

TheCalculus 171

T

qo

po

AA'

BCB' x

y

Fig. 3A and A′are infinitely close on the curve (a parabola, represented as a polygon with infinitely small sides). AA′Cis

the tangent at A; it meets thex-axis at C and BC is the ‘subtangent’. BB′=TA′is dx, and AT (negative in the picture) is dy.

up with:

q

p

=a− 2 x

(If you are familiar with the calculus, you will notice that the right-hand side is obtained by

differentiatingax−xx, according to the usual rules; and Newton began to formulate such rules for

dealing with the velocities of curves.) In the seventeenth century, they found the tangent by finding

the ‘subtangent’—the length BC in Fig. 3.—which is equal toyp/q; in our case,y/(a− 2 x).[We

can also say thatyis a maximum whenx=a/2, since thenq=0 so the tangent is horizontal.]

As Newton summed up his method:

Hence may bee observed: First, that those termes ever vanish in whichois not because they are the propounded

equation [in our example,y+xx=ax]. Secondly the remaining Equation being divided byothose termes also vanish

in whichostill remaines because they are infinitely little. Thirdly that the still remaining termes will ever have that

forme which by the first preceding rule [the rule for differentiating] they should have. (Newton 1967–81, p. 387)

Newton realized that he had made a major discovery; there are references in his texts on

calculus from the 1660s to the possibility of solving with ease problems which had been diffi-

cult or impossible before. As we have seen, he took no serious steps to make it public. Under

pressure (he said, from Dr Barrow), he finally in 1670 produced a serious Latin exposition, untitled,

unfinished, generally known as theMethodof FluxionsandInfiniteSeries. Where the 1665 notes had

been essentially for his own use, this was (in intention) addressed to a circle of practitioners—‘for

the satisfaction of learners’, as he puts it in the quotation which opens this section. Unfortunately,

it did not reach even that narrow circle, although it was passed around among friends to the point

of becoming dog-eared^7 ; and its existence was only generally made known 30 years later in the

priority dispute. However, it remains the clearest guide we have to what Newton’s early calculus

was like at the time of its discovery. The variable quantitiesxandyare in this text called ‘fluents’;

and their rates of change have the name ‘fluxions’ (which was to become a fixture in English math-

ematical language for a century). The relationship of differentiation—finding the fluxion—and

7. See Gjertsen (1986), p. 157. The exact date when theMethod of Fluxionswas observed to be dog-eared is a typical point of
   contention in Newton studies.