NUMERICAL LAWS AND MATHEMATICS 151 



ment the distance moved than to measure the velocity 

 and calculate the distance. But there are closely 

 analogous cases one of which we shall notice immedi- 

 ately in which the position will be reversed. Let us 

 therefore ask what is the assumption which, in accord- 

 ance with the conclusion reached on p. 146, must be 

 introduced, if the solution of the problem is to give new 

 experimental knowledge. 



We have seen that the problem could be solved easily 

 if the velocity were constant ; what we are asking, is 

 how it is to be solved if the velocity does not remain 

 constant. If we examined the rule by which the solution 

 is obtained, we should find that it involves the assump- 

 tion that the effect upon the distance travelled of a certain 

 velocity at a given instant of time is the same as it would 

 be if the body had at that instant the same constant 

 velocity. We know how far the body would travel at 

 that instant if the velocity were constant, and the assump- 

 tion tells us that it will travel at that instant the same 

 distance although the velocity is not constant. To obtain 

 the whole distance travelled in any given time, we have to 

 add up the distances travelled at the instants of which 

 that time is made up ; the reversed Newtonian rule gives 

 a simple and direct method for adding up these distances, 

 and thus solves the problem. It should be noted that 

 the assumption is one that cannot possibly be proved by 

 experiment ; we are assuming that something would 

 happen if things were not what they actually are ; and 

 experiment can only tell us about things as they are. 

 Accordingly calculation of this kind must, in all strictness, 

 always be confirmed by experiment before it is certain. 

 But as a matter of fact, the assumption is one of which 

 we are almost more certain than we are of any experiment. 

 It is characteristic, not only of the particular example 

 that we have been considering, but of the whole structure 

 of modern mathematical physics which has arisen out 



