REMAINING LAWS OF NATURE. 149 



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 reproduce 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 into 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 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 decomposed 

 into the twelve twelves of twelves of pebbles which it con- 

 sists 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 determined deductively. If we know that a is formed 

 from b and c, b from a and e, c from d and /, 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 forma- 

 tion be really a distinct one, not bringing us round again to 

 the former numbers/ but introducing a new number,) we have 

 a set of propositions from 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 one to the other along the chain. 

 Suppose that we know only the following modes of formation : 

 6 -- 4 + 2, 4 -- 7 3, 7 - 5 + 2, 5 = 9 4. We could determine 

 how 6 may be formed from 9. For 6 = 4 + 2 = 7 3+2=5+ 

 2 3 + 2 = 9 4 + 2-3 + 2. It may therefore be formed by 



