Page 6 - Foundations Of Differential Calculus

Preface

What differential calculus, and, in general, analysis of the infinite, might be
can hardly be explained to those innocent of any knowledge of it. Nor can we
here offer a definition at the beginning of this dissertation as is sometimes
done in other disciplines. It is not that there is no clear definition of this
calculus; rather, the fact is that in order to understand the definition there
are concepts that must first be understood. Besides those ideas in common
usage, there are also others from finite analysis that are much less common
and are usually explained in the course of the development of the differential
calculus. For this reason, it is not possible to understand a definition before
its principles are sufficiently clearly seen.
In the first place, this calculus is concerned with variable quantities.
Although every quantity can naturally be increased or decreased without
limit, still, since calculus is directed to a certain purpose, we think of some
quantities as being constantly the same magnitude, while others change
through all the stages of increasing and decreasing. We note this distinc-
tion and call the former constant quantities and the latter variables. This
characteristic difference is not required by the nature of things, but rather
because of the special question addressed by the calculus.
In order that this difference between constant quantities and variables
might be clearly illustrated, let us consider a shot fired from a cannon with
a charge of gunpowder. This example seems to be especially appropriate to
clarify this matter. There are many quantities involved here: First, there is
the quantity of gunpowder; then, the angle of elevation of the cannon above
the horizon; third, the distance traveled by the shot; and, fourth, the length