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

DIFFERENTIAL CALCULUS 223
the moment) is the same as the rate of increase of y' on the tangent at P. Hence, if
we have a general method of determining
the rate of increase of a function f(x) of a
variable x, we can determine the slope of the
tangent at any point (x, y), on a curve, and
thence can draw it. Thus the problems of
drawing tangents to a curve, and of determining the rates of increase of a function
are really identical.
It will be noticed that, as in the cases of
Conic Sections and Trigonometry, the more
artificial of the two points of view is the one
in which the subject took its rise. The really
fundamental aspect of the science only rose
into prominence comparatively late in the
day. It is a well-founded historical generalization, that the last thing to be discovered
in any science is what the science is really
about. Men go on groping for centuries,
guided merely by a dim instinct and a
puzzled curiosity, till at last "some great
truth is loosened."
Let us take some special cases in order to
familiarize ourselves with the sort of ideas
which we want to make precise. A train is in
motion-how shall we determine its velocity
at some instant, let us say, at noon? We can
take an interval of five minutes which includes
noon, and measure how far the train has gone
in that period. Suppose we find it to be five