166 INDUCTION. 



dismemberment of larger aggregates. 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 state- 

 ment of one of the modes of the formation of a given 

 number. It affirms that a certain aggregate might 

 have been formed by putting together certain other 

 aggregates, or by withdrawing certain portions of some 

 aggregate ; and that, by consequence, we might re- 

 produce those aggregates from it, by reversing the 

 process. 



Thus, when we say that the cube of 12 is 1728, 

 what we affirm is this : That if, having a sufficient 

 number of pebbles or of any other objects, we put 

 them together in the particular sort of parcels or 

 aggregates called twelves ; and put together these 

 twelves again into similar collections; and, finally, 

 make up twelve of these largest parcels ; the aggre- 

 gate thus formed will be such a one as we call 1 728 : 

 namely, that which (to take the most familiar of its 

 modes of formation) may be made by joining the 

 parcel called a thousand pebbles, the parcel called 

 seven hundred pebbles, the parcel called twenty 

 pebbles, and the parcel called eight pebbles. The 

 converse proposition, that the cube root of 1728 is 

 12, asserts that this large aggregate may again be 

 decomposed into the twelve twelves of twelves of 

 pebbles which it consists of. 



The modes of formation of any number are innu- 

 merable ; but when we know one mode of formation 

 of each, all the rest may be determined deductively. 

 If we know that a is formed from b and c, b from d 

 and e, c from d and /, and so forth, until we have 



