MATHEMATICS AND METAPHYSICIANS 83 



which, however appropriate to monarchs, seems, un- 

 fortunately, to interest them even less than the infinitely 

 little interested the monarchs to whom Leibniz discoursed. 



The banishment of the infinitesimal has all sorts of odd 

 consequences, to which one has to become gradually 

 accustomed. For example, there is no such thing as the 

 next moment. The interval between one moment and the 

 next would have to be infinitesimal, since, if we take two 

 moments with a finite interval between them, there are 

 always other moments in the interval. Thus if there are 

 to be no infinitesimals, no two moments are quite con- 

 secutive, but there are always other moments between any 

 two. Hence there must be an infinite number of moments 

 between any two ; because if there were a finite number 

 one would be nearest the first of the two moments, and 

 therefore next to it. This might be thought to be a difn 

 culty ; but, as a matter of fact, it is here that the philo- 

 sophy of the infinite comes in, and makes all straight. 



The same sort of thing happens in space. If any piece 

 of matter be cut in two, and then each part be halved, 

 and so on, the bits will become smaller and smaller, and 

 can theoretically be made as small as we please. However 

 small they may be, they can still be cut up and made 

 smaller still. But they will always have some finite size, 

 however small they may be. We never reach the in- 

 finitesimal in this way, and no finite number of divisions 

 will bring us to points. Nevertheless there are points, 

 only these are not to be reached by successive divisions. 

 Here again, the philosophy of the infinite shows us how 

 this is possible, and why points are not infinitesimal 

 lengths. 



As regards motion and change, we get similarly curious 

 results. People used to think that when a thing changes, 

 it must be in a state of change, and that when a thing 



