NUMERICAL LAWS AND MATHEMATICS 139 



but only by relations between numerals. Let me take 

 an example. Consider the pairs of numerals (i, i), (2, 4), 

 (3, 9), (4, 16) ... Our present familiarity with 

 numerals enables us to see at once what is the relation 

 between each pair ; it is that the second numeral of the 

 pair is arrived at by multiplying the first numeral by 

 itself ; i is equal to i x i, 4 to 2 x 2, 9 to 3 x 3 ; and 

 so on. But, if the reader will consider the matter, he will 

 see that the multiplication of a number (the physical 

 property of an object) by itself does not correspond to 

 any simple relation between the things counted ; by the 

 mere examination of counted objects, we should never 

 be led to consider such an operation at all. It is suggested 

 to us only because we have drawn up our multiplication 

 table and have reached the idea of multiplying one 

 numeral by another, irrespective of what is represented 

 by that numeral. We know what is the result of 

 multiplying 3x3, when the two 3's represent different 

 numbers and the multiplication corresponds to a physical 

 operation on things counted ; it occurs to us that the multi- 

 plication of 3 by itself, when the two 3's represent the 

 same thing, although it does not correspond to a physical 

 relation, may yet correspond to the numerical relation 

 in a numerical law. And we find once more that this 

 suggestion turns out to be true ; there are numerical 

 laws in which this numerical relation is found. Thus if 

 we measure (i) the time during which a body starting 

 from rest has been falling (2) the distance through which 

 it has fallen during that time, we should get in our 

 notebook parallel columns like this : 



TABLE II 



TIME DISTANCE 



1 .. I 



2 .. 4 



3-9 



TIME DISTANCE 



4 . . 16 



5 25 



6 .. 36 



