XV

for $x^2$ 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 ${\displaystyle \frac{f(x+h)-f(x)}{h}}$ of the argument $h$ at the value $0$ of its argument.

How have we, by this definition and the subsidiary definition of a limit, 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,

at the end of our exposition of the essential notions of mathematical analysis, we are led back to the ideas with which in II. we commenced our enquiry–-that in mathematics the fundamentally important ideas are those of "some things" and "any things."