An introduction to mathematics, by A. N. Whitehead.

222 INTRODUCTION TO MATHEMATICS
fication of curves and the quadrature of
curves as they are named, belong to the Integral Calculus, which is however involved in
the same general subject as the Differential
Calculus.
The introduction of coordinate geometry
makes the two points of view coalesce. For
(cf. fig. 32) let AQP be any curved line and let
PT be the tangent at the point P on it. Let
the axes of coordinates be OX and OY and
let y=f(x) be the equation to the curve,
so that OM =x, and PM= y. Now let Q be any
moving point on the curve, with coordinates
x1, yi; then y1 =f(xi). And let Q' be the point
on the tangent with the same abscissa xl;
suppose that the coordinates of Q' are xi and
y'. Now suppose that N moves along the
axis OX from left to right with a uniform
velocity; then it is easy to see that the ordinate y' of the point Q' on the tangent TP
also increases uniformly as Q' moves along
the tangent in a corresponding way. In fact
it is easy to see that the ratio of the rate of
increase of Q'N to the rate of increase of ON
is in the ratio of Q'N to TN, which is the same
at all points of the straight line. But the
rate of increase of QN, which is the rate of
increase of f(xi), varies from point to point of
the curve so long as it is not straight. As Q
passes through the point P, the rate of increase of f(xi) (where xi coincides with x for