148 INDUCTION. 



What, then, is that which is connoted by a name of 

 number ? Of course some property belonging to the agglo 

 meration of things which we call by the name; and that 

 property is, the characteristic manner in which the agglo 

 meration is made up of, and may be separated into, parts. 

 I will endeavour to make this more intelligible by a few ex 

 planations. 



When we call a collection of objects two, three,- or four, 

 they are not two, three, or four in the abstract ; they are two, 

 three, or four things of some particular kind ; pebbles, horses, 

 inches, pounds weight. What the name of number connotes is, 

 the manner in which single objects of the given kind must 

 be put together, in order to produce that particular aggregate. 

 If the aggregate be of pebbles, and we call it two, the name 

 implies that, to compose the aggregate, one pebble must be 

 joined to one pebble. If we call it three, one and one and 

 one pebble must be brought together to produce it, or 

 else one pebble must be joined to an aggregate of the kind 

 called two, already existing. The aggregate which we call 

 four, has a still greater number of characteristic modes of 

 formation. One and one and one and one pebble may be 

 brought together ; or two aggregates of the kind called two 

 may be united ; or one pebble may be added to an aggregate 

 of the kind called three. Every succeeding number in the 

 ascending series, may be formed by the junction of smaller 

 numbers in a progressively greater variety of ways. Even 

 limiting the parts to two, the number may be formed, and 

 consequently may be divided, in as many different ways as 

 there are numbers smaller than itself; and, if we admit of 

 threes, fours, &c., in a still greater variety. Other modes of 

 arriving at the same aggregate present themselves, not by the 

 union of smaller, but by the dismemberment of larger aggre 

 gates. Thus, three pebbles may be formed by taking away one 

 pebble from an aggregate of four ; two pebbles, by an equal 

 division of a similar aggregate ; and so on. 



Every arithmetical proposition; every statement of the 

 result of an arithmetical operation; is a statement of one 

 of the modes of formation of a given number. It affirms 



