NUMERICAL LAWS AND MATHEMATICS 143 



represented a property of the system investigated and 

 permitted a derived measurement of that system. But 

 in Table II, there is no such constant numeral ; the 

 rule for obtaining the second from the first column is 

 simply that the numerals in the first column are to be 

 multiplied by themselves ; no other numeral is involved. 

 But this simplicity is really misleading ; we should not, 

 except by a mere " fluke/' ever get such a table as Table 

 II as a result of our measurements. The reason is this. 

 Suppose that, in obtaining Table II, we have measured the 

 time in seconds and the distance fallen in feet ; and 

 that we now propose to write down the result of exactly 

 the same measurements, measuring the time in minutes 

 and the distances in yards. Then the numerals in the 

 first column, representing exactly the same observations, 

 would all be divided by 60 and those in the second would 

 all be divided by 3 ; the observation which was repre- 

 sented before by 60 in the first column would now be 

 represented by i ; and the number in the second column 

 represented before by 3 would now be represented by i. 

 If I now apply the rule to the two columns I shall find it 

 will not work ; the second is not the first multiplied by 

 itself. But there will be a new rule, as the reader may 

 see for himself ; it will be that the second column is the 

 same as the first, when the first is (i) multiplied by itself, 

 and (2) the result multiplied by 1,200. And if we 

 measured the time and the distance in some other units 

 (say hours and miles), we should again have to amend 

 our rule, but it would only differ from the former rule in 

 the substitution for 1,200 of some other numeral. If we 

 choose our units in yet a third way, we should get a third 

 rule, and this time the constant 'numeral might be i. 

 We should have exactly Table II ; but we should get that 

 table exactly only because we had chosen our units of 

 time and distance in a particular way. 

 These considerations arejquite general. Whatever the 



