increase. We will try and grasp its meaning in relation to this particular case. When x increases to x + h , the function x 2 increases to ( x + h ) 2 ; so that the total increase has been ( x + h ) 2 − x 2 , due to an increase h in the argument. Hence throughout the interval x to ( x + h ) the average increase of the function per unit increase of the argument is ( x + h ) 2 − x 2 h . But ( x + h ) 2 = x 2 + 2 h x + h 2 , and therefore ( x + h ) 2 − x 2 h = 2 h x + h 2 h = 2 x + h . Thus 2 x + h is the average increase of the function x 2 per unit increase in the argument, the average being taken over by the interval x to x + h . But 2 x + h depends on h , the size of the interval. We shall evidently get what we want, namely the rate of increase at the value x of the argument, by diminishing h more and more. Hence in the limit when h has

decreased indefinitely, we say that $2 x$ is the rate of increase of $x^{2}$ 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 so 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 mathematical

procedure was substantially right. In fact, the subject was right, though the explanations were wrong. It is this possibility of being right, albeit with entirely wrong explanations as to what is being done, that so often makes external criticism–-that is so far as it is meant to stop the pursuit of a method–-singularly barren and futile in the progress of science. The instinct of trained observers, and their sense of curiosity, due to the fact that they are obviously getting at something, are far safer guides. Anyhow the general effect of the success of the Differential Calculus was to generate a large amount of bad philosophy, centring round the idea of the infinitely small. The relics of this verbiage may still be found in the explanations of many elementary mathematical text-books on the Differential Calculus. It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.

Newton would have phrased the question

by saying that, as $h$ approaches zero, in the limit $2 x + h$ becomes $2 x$ . It is our task so to explain this statement as to show that it does not in reality covertly assume the existence of Leibniz's infinitely small quantities. In reading over the Newtonian method of statement, it is tempting to seek simplicity by

saying that $2 x + h$ is $2 x$ , when $h$ is zero. But this will not do; for it thereby abolishes the interval from $x$ to $x + h$ , over which the average increase was calculated. The problem is, how to keep an interval of length $h$ over which to calculate the average increase, and at the same time to treat $h$ as if it were zero. Newton did this by the conception of a limit, and we now