Page:Popular Science Monthly Volume 76.djvu/122

118 of no particular intellectual eminence. Presumably he had never tried it. To the often-repeated charge that mathematics will turn out only what is put in we may reply that while from incorrect assumptions it can not get correct results it has the power of so transforming the data as to reveal to us totally unexpected truths. Witness the magnificent generalizations of Adams and Leverrier, of Hamilton, and of Maxwell already quoted. There is no doubt that the invention of the infinitesimal calculus has furnished man with the most powerful and elegant instrument of thought ever devised. Allow me to try in a few words to tell why this is so. Natural phenomena are not, as a rule, discrete, like integral numbers, but continuous, like points on a line, so that there is no least difference between one and another. We say that they are continuous, and that they vary continuously. The examination of continuous change is the function of the differential calculus. When we undertake to define so simple a matter as the speed of a point, we can not say that the velocity is the distance traversed in a given time, unless during the whole of that time the speed is the same. If it is continually changing we must divide the time into less and less intervals, and find the ratio of the distance to the time required when both become smaller than any quantity conceivable, in other words we must find the limit approached by this ratio. Thus all questions relating to rates of change, to slopes of curves, to curvature, and the like, require the method of limits, as applied in the differential calculus. On the other hand, consider the case of two bodies attracting each other according to any law of the distance. Since the body is more than a point, from what point of the body shall the distance be measured. Obviously each small portion of the body contributes its part in the attraction, with a different amount according to where it is, all these amounts requiring to be added together to make the whole. But how many parts shall there be, and how large. Obviously there is no bound to the number, nor to the size, one increasing as the other decreases. We must accordingly take the limit which this sum of all the actions approaches as we increase the number of parts while diminishing their size below any limit whatever. This is the method of the integral calculus, Now as observation enables us to deal with bodies of finite size only, the inference to the laws of the ultimate parts can be made only deductively by the calculus. In practise, however, the inverse process is more frequently employed, that is, the actions of points infinitely near each other in space, time or other circumstances are assumed to follow some simple law, thus giving us what are called differential equations, the integration of which gives us conclusions as to what happens on the large scale, which conclusions can be compared with experiment. It is on account of the logical importance of the method, the universality of its