116 WHAT IS SCIENCE? 



The two rules that have been mentioned are necessary 

 to explain what we mean by " the number " of a collec- 

 tion and how we ascertain that number. There is a 

 third rule which is of great importance in the use of 

 numbers. We often want to know what is the number 

 of a collection which is formed by combining two other 

 collections of which the numbers are known, or, as it is 

 usually called, adding the two collections. For instance 

 we may ask what is the number of the collection made 

 by adding a collection of 2 objects to a collection of 3 

 objects. We all know the answer, 5. It can be found by 

 arguing thus : The first collection can be counted against 

 the numerals i, 2 ; the second against the numerals 

 i, 2, 3. But the numerals i, 2, 3, i, 2, a collection formed 

 by adding the two first collections, can be counted against 

 i, 2, 3, 4, 5. Therefore the number of the combined 

 collection is 5. However, a little examination will show 

 that in reaching this conclusion we have made use of 

 another rule, namely that if two collections A and a, 

 have the same number, and two other collections 

 B and b, have the same number, then the collection 

 formed by adding A to B has the same number as that 

 formed by adding a to b ; in other words, equals added to 

 equals produce equal sums. This is a third rule about 

 numbers and counting ; it is quite as important as the 

 other two rules ; all three are so obvious to us to-day 

 that we nevei think about them, but they must have 

 been definitely discovered at some time in the history of 

 mankind, and without them all, our habitual use of 

 numbers would be impossible. 



we must add to our standard series are 131680, 131681, and so on. 

 This is a triumph of conventional nomenclature, much more satis- 

 factory than the old convention that when we have exhausted our 

 fingers we must begin on our toes, but it is not essentially different. 



