MEASUREMENT 115 



the next member ; and so on. Then we shall obtain, 

 according to our rule, a series, some member of which 

 has the same number as any other collection we want to 

 count, and yet the number of objects, in all the members 

 of the standard series taken together, will not be greater 

 than that of the largest collection we want to count. 



And, of course, this is the process actually adopted. 

 For the successive members of the standard series com- 

 pounded in this way, primitive man chose, as portable, 

 distinguishable objects, his fingers and toes. Civilized 

 man invented numerals for the same purpose. Numerals 

 are simply distinguishable objects out of which we build 

 our standard series of collections by adding them in turn 

 to previous members of the series. The first member of 

 our standard series is i, the next i, 2, the next i, 2, 3 

 and so on. We count other collections against these 

 members of the standard series and so ascertain whether 

 or no two collections so counted have the same number. 

 By an ingenious convention we describe which member 

 of the series has the same number as a collection counted 

 against it by quoting simply the last numeral in that 

 member ; we describe the fact that the collection of the 

 days of the week has the same number as the collection 

 i, 2, 3, 4, 5, 6, 7, by saying " tha v . the number " of the 

 days of the week is 7. But when we say that what we 

 really mean, and what is really important, is that this 

 collection has the same number as the collection of 

 numerals (taken in the standard order) which ends in 

 7 and the same number as any other collection which 

 also has the same number as the collection of numerals 

 which ends in 7. 1 



1 Numerals have also an immense advantage over fingers and toes 

 as objects of which the standard series may be formed, in that the 

 series can be extended indefinitely by a simple rule which automatically 

 gives names to any new numerals that may be required. Even if we have 

 never hitherto had reason to carry the series beyond (say) 131679 

 in order to count all the collections we have met with, when we do 

 meet at last with a larger collection, we know at once that the objects 



