140 WHAT IS SCIENCE? 



The numerals in the second column are arrived at by multi- 

 plying those in the first by themselves ; in technical 

 language, the second column is the " square " of the first. 

 Another example. In place of dividing one column 

 by some fixed number in order to get the other, we may 

 use the multiplication table to divide some fixed number 

 (e.g. i) by that column. Then we should ge^the table 



1 . . i-oo 



2 . . 0-50 



3 0'33 



4 0-25 



5 .. 0-20 



and so on. Here, again, is a pure numerical operation 

 which does not correspond to any simple physical relation 

 upon numbers ; there is no collection simply related 

 to another collection in such a way that the number of 

 the first is equal to that obtained by dividing i by the 

 number of the second. (Indeed, as we have seen that 

 fractions have no application to number, and since this 

 rule must lead to fractions, there cannot be such a rela- 

 tion.) And yet once more we find that this numerical 

 relation does occur in a numerical law. If the first 

 column represented the pressure on a given amount of 

 gas, the second would represent the volume of that gas. 



So far, all the relations we have considered were 

 derived directly from the multiplication table. But 

 an extension of the process that we are tracing leads to 

 relations which cannot be derived directly and thus 

 carries us further from the original suggestions indicated 

 by mere counting. Let us return to Table II, and 

 consider what would happen if we found for the numerals 

 in the second column values intermediate between those 

 given. Suppose we measured the distance first and 

 found 2, 3, 5, 6, 7, 8, 10, n, 12, 13, 14, 15 . . . ; what 

 does the rule lead us to expect for the corresponding 

 entries in the first column, the values of the time. The 

 answer will be given if in the multiplication table we 



