136 WHAT IS SCIENCE? 



I have only to divide the numbers in the second column 

 by 7 in order to arrive at those in the first, or multiply 

 those in the first by 7 in order to arrive at those in the 

 second. That is a definite rule which I can always apply 

 whatever the numbers are ; it is a rule which might always 

 be true, but need not always be true ; whether or no it 

 is true is a matter for experiment to decide. So much is 

 obvious ; but now I want to ask a further and important 

 question. How did we ever come to discover this rule ; 

 what suggested to us to try division or multiplication 

 by 7 : and what is the precise significance of division and 

 multiplication in this connexion ? 



THE SOURCE OF NUMERICAL RELATIONS 



The answer to the first part of this question is given by 

 the discussion on p. 123. Division and multiplication 

 are operations of importance in the counting of objects ; 

 in such counting the relation between 21, 7, 3 (the third 

 of which results from the division of the first by the 

 second) corresponds to a definite relation between the 

 things counted ; it implies that if I divide the 21 objects 

 into 7 groups, each containing the same number of objects, 

 then the number of objects in each of the 7 groups is 3. 

 By examining such relations through the experimental 

 process of counting we arrive at the multiplication (or 

 division) table. This table, when it is completed, states 

 a long series of relations between numerals, each of which 

 corresponds to an experimental fact ; the numerals 

 represent physical properties (numbers) and in any given 

 relation (e.g. 7 x 3 = 21) each numeral represents a 

 different property. But when we have got the multipli- 

 cation table, a statement of relations between numerals, 

 we can regard it, and do usually regard it, simply as a 

 statement of relations between numerals ; we can think 

 about it without any regard to what those numerals 



