If, then, we propose the equation axx az=qi(x+s) , (I) the only solution must be It is easy enough to show that eD cp= c'+'; the proof referred to shows that c' is the only solution of this equation. If z and x be symbols of operation, and if by tpx we mean a combination of operations performed with x, and by ax . az the result of succes sively performing the operations az and we may denote by yD an operation which is such, that calling it as, the successive performance of ax and az is equivalent to that of a (x+z); and that, calling it ay, the successive performance of if,y and 4,,z is equivalent to that of tp(yz). If we want to define the particular operation A', we must add to the equation (1) the following: Thus, let it be the definition of ED, D being a symbol of operation, that we have here an operation such that if it and eb were successively per formed, the result would be the same as if eD+"' were performed at once; this last symbol implying that the operation n + a' is used in the same way as D in the first. Moreover, let it be understood that if D were 1—that is, if the operation D produced no alteration in the func tion operated on—the result of s. would be simple multiplication by e. There is nothing in this definition which is unintelligible, though there is something unknown. An operation is defined by means of itself; the definition must then be developed before its object can be under stood, but it is not the less a definition—that is, a description of some one operation, and a distinction between it and every other. Thus, in common algebra, the magnitude of x may be defined by an equation, say x=12—x. Here x is only given in terms of its unknown self, but it is not the less defined to be 6, and nothing but 6. When the step above described has been made, it ie (owing to the demonstrated con nection of the rules of common algebra with those of the calculus of operations) the same process to prove that— e° 1 D D22 + • • • .
when n signifies an operation, as when it signifies a quantity.
The definition of log D is that this operation is the inverse of eD with respect to D; so that log means D. Those functions which in common algebra are trigonometrical [SINE] cannot be defined in the subject of which we are speaking otherwise than by reference to the well-known exponential forms. Thus, D denoting an operation— Cos 2 Cos D means -{ en Vc—i) Sin D means 2 ED e—n 441-1} It might perhaps be said that though we have constantly used the word relation, yet we have considered nothing but identity, that is, either identity of magnitude, form, process, or properties ; but that the term in common life refers to something short of complete identity, frequently meaning mere connection, and sometimes only analogy, or even nothing more than resemblance. We answer, that relation always refers to identity of some sort. For example, there is a relation be tween the position of the sun and moon and the state of the ocean. Here the word means merely a connection; but this connection in volves an absolute identity ; having given the position of those heavenly bodies with respect to any place, together with the direction and quantity of their motions, the height of the water at that place is connected with the quantities which express those positions and mo tions by an equation or a mathematical identity. Resemblance again
means identity in some respect, or near approach to identity : analogy, a term generally applied to relations of similarity, will be found to admit of the word sameness being used instead of similarity. Thus, when we say that substance is formed from substore in a manner similar to that in which distance is formed from distare, the analogy assertacl is one of absolute identity (of mode of derivation), Reasoning by analogy is either the same thing as common reasoning; or else analogy is but another word for induction. If A give and c have something in common with A, it may be a necessary consequence that c gives D ; n being connected with c in the same manner as e with A. But this happens only when the following of a from A is a news airy consequence of that which A and c have in common, and of that only : in which case the deduction of n from c by analogy with the deduction of D from A, is only an assertion of the possibility of applying the same mode of proof to that part or property of c which was pre viously applied to the same part or property of A. But when we conclude by analogy of a horned animal that it is not carnivorous, as it is raid; that is, when we conclude that the horned animal of which we speak will resemble all other horned animals which we know, in every point in which they resemble each other, we apply uo other process than the establishment of a highly probable result by in duction.
Reasoning by pure analogy, is then, not absolutely demonstrative reasoning, except in the case above described. in which we want no new name for the process. But attention to analogy in the structure of definitions, and in the route of investigation, is necessary to the success of many inquiries, and gives clearness and saves time in all. Indeed it may be taken as a maxim that whenever there is any species of resemblance pervading the results of two branches of inquiry, there ought to be a reason for that resemblance in the nature of the two subjects, expressed by a resemblance of the notations used ; and this reason ought to be made prominent and insisted on.
For instance, we have two distinct algebras (ALCEDRA], which, for temporary distinction, we may call arithmetical and geometrical, using the same symbols in the same manner, but proceeding upon meanings given to those symbols which appear altogether different. The only reason given to the student in the article cited, to justify the definitions of the latter, or geometrical algebra, was that they would be found to answer a certain purpose, namely, to make all theorems in the earlier algebra true, when no other alteration was made than that of the meanings of the symbols. It is now to be asked, why have the new definitions that property ? what relation have they to the old ones which gives the results of the two a perfect community of form ? The answer to this question is not very difficult ; but it will require us first to consider what are the operations of common arithmetic, and how they are to be described in terms of the simplest notions of the science.