366 INDUCTION. 



to produce that particular aggregate. If the aggregate be of peb- 

 bles, and we call it tivo, the name implies that to compose the 

 aggregate, one pebble must be joined to one pebble. If we call it 

 three, we mean that one and one and one pebble must be brought to- 

 gether to produce it, or else that one pebble must be joined to an 

 aggi'egate of the kind called two, already existing. The aggregate 

 which we call_/o«?" has a still greater number of characteristic modes 

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

 together; or two aggi'egates 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 aggre- 

 gate present themselves, not by the union of smaller, but by the dis- 

 memberment of larger aggregates. Thus, three pehhles may be formed 

 by taking away one pebble from an .aggregate of four ; tioo 2>ehbles, 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 the 

 formation of a given number. It affinns that a certain aggregate 

 might have been formed by putting together certain other aggregates, 



by vvithdrawing certain j^ortions of some aggregate; and that, by 

 consequence, we might reproduce those aggregates from it, by revers- 

 ing 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 twelve ; and put together these twelves again into 

 similar collections ; and, finally, make up twelve of these largest par- 

 cels; the aggregate thus formed will be such a one as we call 1728; 

 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 decom- 

 posed into the twelve twelves of twelves of pebbles which it consists of. 



The modes of formation of any number are innumerable; but when 

 we know one mode of formation of each, all the rest may be deter- 

 mined deductively. If we know that a is formed from h and c, h from 

 d and c, c from d and f, and so forth, until we have included all the 

 numbers of any scale we choose to select, (taking care that for each 

 number the mode of formation is really a distinct one, not bring- 

 ing us round again to the former numbers, but introducing a new 

 number,) we have a set of propositions fi'om which we may reason 

 to all the other modes of formation of those numbers from one 

 another. Having established a chain of inductive truths connecting 

 together all the numbers of the scale, we can ascertain the formation 

 of any one of those numbers from any other by merely travelling 

 from the one to the other along the chain. Suppose that we knew 

 only the following raode.sof foraiation : 6 = 4-|-2, 4 = 7 — 3, 7 = 5 + 2, 



