114 WHAT IS SCIENCE? 



in the sack I propose to buy is the same as that in a 

 sack I have at home, I need not bring my sack to the 

 shop ; I can count the potatoes at the shop against 

 some third collection, take this collection home, and 

 count it against my potatoes. Accordingly the discovery 

 of this first rule immediately suggests the use of portable 

 collections which can be counted, first against one collec- 

 tion and then against another, in order to ascertain 

 whether these two have the same number. 



The value of this suggestion is increased greatly by 

 the discovery of a second rule. It is that by starting with 

 a single object and continually adding to it another single 

 object, we can build up a series of collections of which 

 one v/ill have the same number as any other collection 

 whatsoever. This rule helps us in two ways. First, 

 since it states that it is possible to make a standard 

 series of collections one of which will have the same 

 number as any other collection, it suggests that it might 

 be well to count collections, not against each other, but 

 against a standard series of collections. If we could 

 carry this standard series about with us, we could always 

 ascertain whether any one collection had the same number 

 as any other by observing whether the member of the 

 standard series which had the same number as the first 

 had also the same number as the second. Next, it shows 

 us how to make such a standard series with the least 

 possible cumbrousness. If we had to have a totally 

 different collection for each member of the standard 

 series, the whole series would be impossibly cumbrous ; 

 but our rule shows that the earlier members of the series 

 (that is those with the smaller number) may be all 

 parts of the later members. Suppose we have a collec- 

 tion of objects, each distinguishable from each other, 

 and agree to take one of these objects as the first member 

 of the series ; this object together with some other as 

 the next member ; these objects with yet another as 



