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

234 INTRODUCTION TO MATHEMATICS
crease for x2 as did the Leibnizian way of
making h grow "infinitely small."
The more abstract terms "differential coefficient," or "derived function," are generally used for what we have hitherto called the
"rate of increase" of a function. The
general definition is as follows: the differential coefficient of the function f(x) is the
limit, if it exist, of the function (x +h)fx)
h
of the argument h at the value O of its argument.
How have we, by this definition and the
subsidiary definition of alimit, really managed
to avoid the notion of "infinitely small numbers" which so worried our mathematical
forefathers? For them the difficulty arose
because on the one hand they had to use an
interval x to x+h over which to calculate
the average increase, and, on the other hand,
they finally wanted to put h =0. The result
was they seemed to be landed into the notion
of an existent interval of zero size. Now
how do we avoid this difficulty? In this
way-we use the notion that corresponding
to any standard of approximation, some interval with such and such properties can
be found. The difference is that we have
grasped the importance of the notion of "the
variable," and they had not done so. Thus,