We have [Aternue] pointed out what is meant by symbolical algebra, as distinguished from explained or interpreted algebra. Granting a certain number of fundamental relations, which are to be true, the logical consequences of combining those relations must be true ;..lso thus, if it be universally true that a+ b=b+ a, and that xy= yx, it follows, even before a, b,+, x, xy, &c., have any meaning assigned, that (a+ b)z= z(a + b)= z(b + a). If, as in the article cited, we select all the primary relations on which algebraical transformations depend, and then bear in mind that the truth of all their consequences depends on the truth of those relations only, not on the relations being true for one meaning or another meaning of the symbols, but on the truth only of the relations, come how it may—we shall then see that all formulm of algebra may he used as expressions of truths, whatever may be the meaning of the symbols employed, provided only that, such meanings given, the fundamental relations are true. We have already seen that this may be carried the length of extending the meanings of all the symbols of algebra, in such manner that a science is created with definitions wide enough to include among its rational objects not only the negative quantity, but also its square root. This was extension only ; we shall now show a process which, though it be still extension, is of another character.
In our present inquiry, we need not trouble ourselves to make any particular consideration of the signs + and —. They retain their algebraical meaning, so that whatever A and is may stand for, + (+ s) = A, —( A)= —.A, &c.
If we now ask, What are the fundamental symbolical relations of algebra which remain, after those which depend on + and — are taken away, we shall find them to be as follows :-1. The distributive character, as it is called, of the operation ab, with respect to + and —, as shown in a(b + c—e)=ab +ac—ae. 2. The commutative or character of the same operation with respect to others of the same kind and itself, as shown in abc=acb=bea, &c., and ab=ba. 3. The depressible character of operations of the species co., when performed upon other operations of the same kind, as shown in ci=a" =a"+*, a."4. These laws of operation being granted, no matter what the nature of the interpretation under which it is found possible to grant them, all that is necessary to the mechanism of algebra will be found to have been granted. It will be remembered that we speak of I÷a under the symbol in arithmetic, as already seen, we may, if we please, consider the symbols 2, 3, etc., as indicative of operations performed upon the unit. Let us extend this notion, and, instead of the unit, make 0x, any function of a variable x, the subject of operation ; this function being always understood, if not expressed. Thus any symbol E has an operative meaning in itself, but when written in an equation stands for the result obtained by performing that operation upon which may also be signified by E (0x). Also let E F and E—F stand for the algebraical sum and difference of the results of the operations E and F performed upon cps. Let us now appropriate E to stand for the simple operation of changing x into x +It, or x + any quantity independent of x in value : and to distinguish the different increments, let E, &c., denote the operations of changing x into x + h, x + k, &e. It is then very easily shown that E possesses the distributive, convertible, and depressible characters essential to its being logically the object of algebraical transformation. Take two functions, 0x and tpx, either
assumed independently or resulting from preceding operations : it follows then that Es (0x 4x) is 0 (x + + 4' + h), which is cpx ± ; or the distributive character Ti established. Again, consider ster) and (r, 0x) : first, E, 0x means 0 (x + k), on which perform or change x into x + h, giving 0 (x + It + k); next, 0x is 0 (x h), on which r, being performed, gives (x + k + h), the same as cp (x + h + L). Consequently ; Os E, Os, or the convertible character of z is established. Thirdly, consider rs:, meaning that the operation having been twice formed, is to be three times performed upon the result : we have evidently 0 (x + 5h), or El 0x; and if n were to be performed four times running, we should have E Hence the depressible character of the successive operations is established : and though it be a wide step or the beginner to make, the applicability of all the formulae of algebra is now proved, subject, as in common algebra, to difficulties of interpretation oemrring In results.
A simplification of the preceding notation may be made as follows : if it or ei be simply a direction to increaser by unity, and Ee a direction to let it remain unaltered, it is clear that re or must mean (x+ 1 + 1), or rr.. or; so that r.' or and E, or are the Larne. Simi ar ramming applies to e. whenever A is a whole nnmIx•r ; and shows that it is to and nothing else. Similar reasoning also applies to r-e where A is a whole number : for must be so interpreted that t' performed upon it may give or e; that is, E--e or with x changed A times into x + 1 must be r° or, or or ; or or must be q5 (z - h). In like manner it may easily be shown that one of the meanings of st.o.r, where A is fractional, is + A): but, as in common algebra, of which all the conclusions, as shown, here apply, when A is a fraction, r.' may be any one out of operations as many in number as there are units in the denominator of A. To take a very simple case, required Es (x), meaning an operation which, twice repeated, gives EA 4,x, or 41$ (x + 1). This condition is evidently satisfied by ¢ (x + ii), but it is also satisfied by - + it), for if part of the operation consist in change of sign, two repetitions of the operation reproduce the original ? It must not be forgotten that, in finding new objects of algebraical reasoning, we have not lost our rights over the old ones • thus any letter may stand for a multiplier or divisor of the universal subject of calculation, or. Bret these independent multipliers must not contain x : for if they did contain x, the convertibility of the multipliers with E would not any longer be practicable. lf, for instance, we consider E+x, which is a op (x+1), we find it to be tho same as E (04 for a, being independent of x, is not affected by E. But if we consider .rEcp.r, and E (xor), we find the first to be rep (.r +1), and the second to be (x + 1) cp (x + I). A wide branch of the calculus of operations exists, in which the convertible character of the operations is not made a postu late. Our limits will not allow us to enter upon it, except to a very small extent, as presently given : but there is one elementary work on the subject, Carmichael, 'Calculus of Operations,' 1855 ; and there is a good deal on the subject in Boole, ` On the Calculus of Finite Differences, 1860.