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

226 INTRODUCTION TO MATHEMATICS
has decreased indefinitely, we say that 2x is the
rate of increase of x2 at the value x of the
argument.
Here again we are apparently driven up
against the idea of infinitely small quantities
in the use of the words "in the limit when h
has decreased indefinitely." Leibniz held that,
mysterious as it may sound, there were actually existing such things as infinitely small
quantities, and of course infinitely small numbers corresponding to them. Newton's language and ideas were more on the modern
lines; but he did not succeed in explaining
the matter with such explicitness as to be
evidently doing more than explain Leibniz's
ideas in rather indirect language. The real
explanation of the subject was first given by
Weierstrass and the Berlin School of mathematicians about the middle of the nineteenth
century. But between Leibniz and Weierstrass a copious literature, both mathematical
and philosophical, had grown up round these
mysterious infinitely small quantities which
mathematics had discovered and philosophy
proceeded to explain. Some philosophers,
Bishop Berkeley, for instance, correctly denied
the validity of the whole idea, though for
reasons other than those indicated here. But
the curious fact remained that, despite all
criticisms of the foundations of the subject,
there could be no doubt but that the mathe