REMAINING LAWS OF NATURE. 431 



Tluis, 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 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 determined de- 

 ductively. If we know that a is formed from h and c, h from a and e, c 

 from f?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 

 formation be really a distinct one, not bringing us round again to the for- 

 mer 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, Ave can ascertain the forma- 

 tion of any one of those numbers from any other by merely traveling from 

 one to the other along the chain. Suppose that we know only the follow- 

 ing modes of formation : 6=:44-2,4 = 7 — 3, 7 = 5 + 2, 5 = 9— 4. "VVe could 

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

 3 + 2 = 9—4 + 2 — 3 + 2. It may therefore be formed by taking away 4 and 

 3, and adding 2 and 2. If we know besides that 2 + 2 = 4, we obtain 6 from 

 9 in a simpler mode, by merely taking away 3. 



It is sufficient, therefore, to select one of the various modes of formation 

 of each number, as a means of ascertaining all the rest. And since things 

 which are uniform, and therefore simple, are most easily received and re- 

 tained by the understanding, there is an obvious advantage in selecting a 

 mode of formation which shall be alike for all ; in fixing the connotation 

 of names of number on one uniform principle. The mode in which our 

 existing numerical nomenclature is contrived possesses this advantage, with 

 the additional one, that it happily conveys to the mind two of the modes 

 of formation of every number. Each number is considered as formed by 

 the addition of a unit to the number next below it in magnitude, and this 

 mode of formation is conveyed by the place which it occupies in the series. 

 And each is also considered as formed by the addition of a number of 

 units less than ten, and a number of aggregates each equal to one of the 

 successive powers of ten; and this mode of its formation is expressed by 

 its spoken name, and by its numerical character. 



What renders arithmetic the type of a deductive science, is the fortunate 

 applicability to it of a law so comprehensive as " The sums of equals are 

 equals :" or (to express the same principle in less familiar but more charac- 

 teristic language), Whatever is made up of parts, is made up of the parts 

 of those parts. This truth, obvious to the senses in all cases which can be 

 fairly referred to their decision, and so general as to be co-extensive with 

 nature itself, being true of all sorts of phenomena (for all admit of being 

 numbered), must be considered an inductive truth, or law of nature, of the 

 highest order. And every arithmetical operation is an application of this 



