NUMERICAL LAWS AND MATHEMATICS 149 



by taking half the distance travelled in the two seconds 

 of which " 3 " is the middle (from 2 to 4) which is again 6. 

 We get different values for the velocity according to 

 which of these alternatives we adopt. There are doubt- 

 less good reasons in this example for choosing the alterna- 

 tive 6, for two ways (and really many more than two ways, 

 all of them plausible) lead to the same result. But if 

 we took a more complicated relation between time and 

 distance than that of Table II, we should find that these 

 two ways gave different results, and that neither of them 

 were obviously more plausible than any alternative. Do 

 then we mean anything by velocity in such cases and, 

 if so, what do we mean ? 



It is here that mathematics can help us. By simply 

 thinking about the matter Newton, the greatest of 

 mathematicians, devised a rule by which he suggested 

 that velocity might be measured in all such cases. 1 It is 

 a rule applicable to every kind of relation between time 

 and distance that actually occurs ; and it gives the 

 " plausible " result whenever that relation is so simple 

 that one rule is more plausible than another. Moreover 

 it is a very pretty and ingenious rule ; it is based on ideas 

 which are themselves attractive and in every way it 

 appeals to the aesthetic sense of the mathematician. 

 It enables us, when we know the relation between time 

 and distance, to measure uniquely and certainly the 

 velocity at every instant, in however complicated a way 

 the velocity may be changing. It is therefore strongly 

 suggested that we take as the velocity the value obtained 

 according to this rule. 



But can there be any question whether we are right or 

 wrong to take that value ; can experiment show that we 

 ought to take one value rather than another ? Yes, it 



1 This is part of the great mathematical achievement mentioned on 

 p. 100. I purposely refrain from giving the rule, not because it is 

 really hard to explain, but because I want to make clear that what 

 is important is to have some rule, not any particular rule, 



