MEASUREMENT 123 



Suppose I have a collection of bodies, each of .which has 

 the same weight 3, the number of bodies in the collection 

 being 4. I may ask what is the weight of the whole 

 collection. The answer is given of course by multiplying 

 3 by 4, and we all know now that the result of that opera- 

 tion is 12. That fact, and all the other facts summed up 

 in the multiplication table which we learn at school, can 

 be proved from the rules on which weighing depend 

 together with facts determined by counting numerals. 

 But the point I want to make is that multiplication repre- 

 sents a definite experimental operation, namely the com- 

 bination into a single collection, placed on one pan of the 

 balance, of a set of bodies, all of the same weight, the 

 number of those bodies being known. Division arises 

 directly out of multiplication. In place of asking what 

 will be the weight of a collection formed of a given number 

 of bodies all of the same weight, we ask what must be the 

 weight of each of a collection of bodies, having a given 

 number, when the whole collection has a given weight. 

 E.g. what must each body weigh in order that the whole 

 collection of 4 bodies weighs 12 ? The answer is obtained 

 by dividing 12 by 4. That answer is obtained, partly 

 from the multiplication table, partly by inventing new 

 numerals which we call fractions ; but once again division 

 corresponds to a definite experimental operation and has 

 its primary significance because it corresponds to that 

 operation. This is this conclusion that we shall use in 

 the sequel. But it te worth while noting that the fractions 

 which we obtain by this method of addition overcome 

 the difficulty from which this paragraph started. If 

 we make all possible fractions of our original weight (i.e. 

 all possible bodies, such that some number of them formed 

 into a single collection have the same weight as the original 

 body), then, by adding together suitable collections of 

 these fractions, we can make up a collection which will 

 have the same weight as any body whatever that we 



