TheCalculus 169
the Indian work as, precisely,nota version of the (later) European ‘method of infinite series’, let
alone that broader field which we call the calculus.
Exercise 1.Derive equation (1) from Jyesthadeva’s formulation.5. Newton, an unknown work
Observing that the majority of geometers, with an almost complete neglect of the ancients’ synthetical method,
now for the most part apply themselves to the cultivation of analysis and with it have overcome so many formidable
difficulties that they seem to have exhausted virtually everything apart from the squaring of curves and certain topics
of like nature not yet fully elucidated: I found not amiss, for the satisfaction of learners, to draw up the following
short tract in which I might at once widen the boundaries of the field of analysis and advance the doctrine of series.
(Newton 1967–81, 3 ,p.33)So what was it that Newton, and later Leibniz, invented (assuming with the Royal Society that
they were essentially the same)? To clarify, consider the ‘fundamental problem’ of finding tangents.
A variety of ad hoc methods were around by the 1660s, following on from the rather complicated
one which Descartes had proposed (see Fauvel and Gray 11.A.9). For us today, (and this is how
Newton and Leibniz saw it), the question can be posed:Problem.Given a curve specified by an equation betweenxandy, to find the tangent at the point
whose coordinates are(x,y)(see Fig. 2).It is easy to see that we know the tangent if we know its gradient (its inclination to thex-axis), and
this is now usually given—once you know calculus—by the expression ‘dy/dx’. You will probably
have been told that this isnota fraction, and thatdyanddxdo not mean anything on their own.
Many students naturally find this idea confusing, if few question it, and indeed this is not how it
started out. Instead, a number of writers in the mid-seventeenth century had arrived at the idea
that the tangent was the line which joined two infinitely near points on the curve; and the challenge
was to find a simple way of working out what that line was.
In one respect, Newton was more inventive. His fundamental idea was that the curve was
described by the motion of a point in time, and that the tangent was the direction of its velocity at
an instant. In itself, the idea could be thought of as a geometrical one; but if the curve was described
in Cartesian form by coordinatesx,y, then one could also think of thex-velocity andy-velocityTOP
(x,y)Fig. 2PT is the tangent at the point P on the curve.