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 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 deductively. If we know that a is formed from b and c, b from a and e, 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 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 traveling 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 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 retained 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 characteristic 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
