n8 TEE POPULAR SCIENCE MONTHLY 



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 mag- 

 nificent 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 con- 

 ceivable, 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 to- 

 gether 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 obser- 

 vation 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 applica- 



