To the same head may be referred the meaning of =, as connecting an infinite series with its finite source of development (or its invelop meut). Arithmetical equality may not exist, for the series may be divergent ; but between the development and its invelopmeut exists the relation of sameness of properties, and the relation of sameness of source. The infinite series 1-1 +1-1+ , eec., is equated to half a unit ; that is, the sign = is put between and 1-1 +1-1+ , fie., ail inf. The relation of sameness of magnitude has no existence, for 1-1 + 1—, &c., ad infinitum, fmniahcs no definite idea of magnitude; but in properties, the two are the same.
2. The sign = means the relation of sameness of magnitude, without reference to form; and in this sense its use generally imposes conditions on one or more of the symbols employed, and always does so unless when the sign might also truly have the meaning described under the first head. Thus 2x+ 3=x + 4+x-1 imposes no condition on the value of x, because the first side is only a more simple performance of the operations indicated on the second side ; but 2 x +3=21—x is the assertion cf a relation existing which is not true of the forms, and is not generally true as to the magnitudes. The condition is neces sary to the truth of the relation asserted to exist. Relations of this sort, under the name of equations, are the first which meet the student at his entrance into algebra ; and he frequently has a subse quent difficulty in extending the use of the symbol =. Being accus tomed to see it impose conditions of magnitude, he cannot easily cease to imagine that it always does so ; and he Rieke upon the two equations e= = I + + + to., and x + 1 = 2, as things of the same kind, differing only in complexity. To prevent this, the distinction between idratica/ equations (so called),—namely, assertions of the relation described under the first head,—and equations of condition, should be strongly marked at the outset of his course. It would even be wise to use somewhat different symbols for the two relations: thus* might denote the first described relation, and = the second. The learner might drop the slight distinction which exists between the two symbols when he finds himself able to do with out it ; but we are satisfied that those who had once learned to use it would never think the time was come when they might safely drop it.
3. The sign = means the relation of algebraical identity between the results of different operations, when the symbols are net symbols of magnitude, but of OPERATION : that is, it asserts the relation of sameness of effect between the two operations which are written on one side and the other of it. And here it is in truth used in the first
sense described, the difference being in the meaning of the symbols, not in that of the relation. And here again there is the distinction between the case in which the relation is explicable from definitions, and that in which it requires interpretation. Thus, in the relation 1 +2n we can prove and verify that the operation 1+ is of that sort which, if performed twice following, will yield the same result as the sum of the results of the operations 1, 2n, and ns. But when, having established, as in the article cited, a right to the use of all the ordinary transformations of algebra, we come to 1 + = ED and n = log (1 +n), we have results of which the first side only is explicable, and the second requires interpretation. It might be satis factory to consider such symbols as log (1 + a), &c., in no other light than as abbreviations of the aeries into which they might be developed in common algebra ; but as such a use of interpretation seems to a beginner to be more arbitrary than it really is, we may point out how to make the passage in a somewhat more guarded manner, presuming the reader to be perfectly well acquainted with the results of the article IJ PERATION.
It A, IS, &C., stand for symbols of operation, then A + B, A Bp A ÷ B, are compound results of operation, which are capable OT and actually receive a distinct definition. Similarly, A" is also deducible in meaning from the definition when a is any number, whole or fractional, positive or negative ; but A', where s is also a symbol of operation, cannot be immediately explained from definition. But it is to he remembered that an algebraic quantity may be susceptible of different definitions, though really amounting to the same definition. Sometimes nothing more than a mere change of the form of words will render a notion capable of being rationally extended further than it could have been before the change was made. For instance, in FRACTIONS, we under stand the division of 7 into 3 equal parts, and into 4 equal parts ; but a division into 3,1 equal parts is a set of words without meaning. But if we only speak of taking parts of which three make 7, and other parts of which four make 7, it is perfectly easy to imagine parts such that three parts and half a part make 7. Can we not, then, take such a method of defining A' as, without in any way altering its common meaning, shall present that common meaning in a form which will be intelligible when A and Is are Symbols of operation ? In BINOMIAL THEOREM it is proved that the equation sIix x oz= can only be satisfied for all values of x and; by tax= e' where c is independent of x.