The introduction of coordinate geometry

makes the two points of view coalesce. For (cf. [fig.]32) let $A Q P$ be any curved line and let $P T$ be the tangent at the point $P$ on it. Let the axes of coordinates be $O X$ and $O Y$ ; and let $y = f (x)$ be the equation to the curve, so that $O M = x$ , and $P M = y$ . Now let $Q$ be any moving point on the curve, with coordinates $x_{1}$ , $y_{1}$ ; then $y_{1} = f (x_{1})$ . And let $Q^{'}$ be the point on the tangent with the same abscissa $x_{1}$ ; suppose that the coordinates of $Q^{'}$ are $x_{1}$ and $y^{'}$ . Now suppose that $N$ moves along the axis $O X$ from left to right with a uniform velocity; then it is easy to see that the ordinate $y^{'}$ of the point $Q^{'}$ on the tangent $T P$ also increases uniformly as $Q^{'}$ moves along the tangent in a corresponding way. In fact it is easy to see that the ratio of the rate of increase of $Q^{'} N$ to the rate of increase of $O N$ is in the ratio of $Q^{'} N$ to $T N$ , which is the same at all points of the straight line. But the rate of increase of $Q N$ , which is the rate of increase of $f (x_{1})$ , varies from point to point of the curve so long as it is not straight. As $Q$ passes through the point $P$ , the rate of increase of $f (x_{1})$ (where $x_{1}$ coincides with $x$ for 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

miles, we may then conclude that the train was running at the rate of $60$ miles per hour. But five miles is a long distance, and we cannot be sure that just at noon the train was moving at this pace. At noon it may have been running $70$ miles per hour, and afterwards the brake may have been put on. It will be safer to work with a smaller interval, say one minute, which includes noon, and to measure the space traversed during that period. But for some purposes greater accuracy may be required, and one minute may be too long. In practice, the necessary inaccuracy of our measurements makes it useless to take too small a period for measurement. But in theory the smaller the period the better, and we are tempted to say that for ideal accuracy an infinitely small period is required. The older mathematicians, in particular Leibniz, were not only tempted, but yielded to the temptation, and did say it. Even now it is a useful fashion of speech, provided that we know how to interpret it into the language of common sense. It is curious that, in his exposition of the foundations of the calculus, Newton, the natural scientist, is much more philosophical than Leibniz, the philosopher, and on the other hand, Leibniz provided the admirable notation which has been so essential for the progress of the subject.

Now take another example within the region of pure mathematics. Let us proceed to find the rate of increase of the function x 2 for any value x of its argument. We have not yet really defined what we mean by rate of