The fundamental operations of arithmetic arc addition, subtraction, multiplication, and division. Of these we may make the definitions of subtraction and division follow from those of addition and multipli cation : thus subtraction is the process which destroys the effect of addition, and division that which destroys the effect of multiplication.
The fundamental ideas of arithmetic are, first, that absence of all magnitude which must precede the consideration of any particular number; secondly. the particular magnitude which we choose for repetition, and to which we refer other magnitudes. Nothing and unity are the names of these ideas ; and 0 and 1 are their well-known symbols. The first, 0, reminds him who uses it, of the state in which ho is antecedently to thinking of any number ; the second, 1, of the succes sive accessions by which he passes from one object of consideration to another. If 0 do not present itself before we can think of any number, it is that we avoid it by an act of memory ; but if, for instance, a person had forgotten what seven was, as a young child might do in learning arithmetic, he would be obliged, beginning from 0, to con struct 7 by repeated accessions of a unit each time.
Now addition of one number to another is a process which merely puts a number in the place of nothing, and proceeds to count from that number in the same manner as when we form the number to be added from 0. Thus to add b to a we do with a what we should have done with 0 to form b : to add 4 to 3, we do with 3 what we should have done with 0 to form 4. If this last operation were performed on the fingers, we should first complete three, and then count the fingers which make four from and after the completion of the three; thus— This definition of addition, namely, that "a + 1) is a direction to do that with a, which would give is if a were nothing," will new be put by for a moment, until we are readyto apply it in the construction of the new algebra.
Multiplication of one number by another is a process which puts a number in the place of unity, and proceeds to use that number in the same manner As we use unity when wo make another number. Thus, to multiply a by 1), we do with a what we should have done with unity to make b ; to multiply 3 by 4, we do with 3 what we should have done with unity to make 4. Thus, The definitions of subtraetion and division arc then obtained, as before described, by the supposition of inverse operations, or operations destructive of the effects of addition and multiplication.
In consequence of the preceding considerations, we shall pares front the limited to the more extended algebra without anything of any arbitrary character, except only the choice of a meaning for the fundamental symbols. In arithmetic, the symbols a, b, e, &c., mean simply numbers; let their meaning in the geometrical algebra be not numbers but lengths, or if numbers, let them be numbers of lengths, a given length being taken as the unit. And let each symbol he ex pressive not only of a length, but of a length in some particular direction ; by which we mean that two lines are not to be denoted by the seine symbol, unless they have not only the same lengths but the same directions.
This one fundamental change in the meaning of the things signified by letters is now all that need be made ; for all the rest is absolutely the same as in arithmetic. For example, what is a + b t Let o A and o n be the lines represented in length and direction by a and 6 ; having completed o A, that is, having passed from o to A through the proper length and in the proper direction, do that which would have given o 11 if 0 A had been nothing, or if A had been at o ; that is. draw A c equal and parallel to on, and in the same direction. The point c thus attained terminates a lino o c, the length and direction of which is therefore to be denoted by a+ 6,, since every process is to resemble that of arithmetic in everything but the meaning of the objects of calculation.
Again, required the meaning of We must now choose a length" and direction which is to be represented by 1 ; let this be o II. We are now to do with o A or a what we should have dope with o u to make o n. Suppose for simplicity that 013 is double of o u. To turn o u into 0 A, we must double its length, and let it revolve through a certain angle u o n. Do this with o n ; that is, double its length, and make it, thus doubled, revolve to the position o c, so that the angles o n and A o c are equal. Then o c must be that which is repre sented by b a ; and the angle u o c is the stun of the angles IS o n and tro A.
If we examine the fundamental definitions of the geometrically defined system of Ataenun, we shall find that we have here described enough to deduce them all, and that we have done it by pure analogy. But also remark that analogy here means nothing but an identity of process ; we have described the processes of addition and multiplication in terms which connect them so closely with the objects to which they are applied, and at the same time make the process so distinct from the subject-matter of the process, that when we change the subject matter, we can still preserve the process. If then any other subject matter could be found. such that, with reference to meanings of a and is derived from it, a + b and a b could be consistently defined, or rather deduced, from 0 and 1, a new application of the rules of algebra would follow.
In the calculus of operations, the same steps might be made; and when this branch of algebra consisted of nothing but the separation of the symbols of operation and quantity in an arbitrary manner [OrenAatoe], analogy was the species of relation by which the deduc tion of results from this separation was connected with the results of common algebra. But this analogy was only the guide to the results, and not the proof of them : it became a proof when it was shown that the validity of the common algebra itself depended, not upon the whole meaning of the symbols, but upon that part of it which was preserved in the meanings of the symbols of operation.