In the operations of multiplication and division the rule of signs is thus shown :— It is said that two negative quantities multiplied together produce a positive quantity, which means that a mistake of direct for inverse, or vice versd, made in both the terms of a pro duct, produces no mistake in the product, when the latter is formed by the usual rules. Thus, if a, which should be x—y, has been taken to be y—x, and if b, whidh should be v — w, has been taken to be w — v, the algebraical product (w—v) x) or vs at which we arrive in the mistaken process, is precisely the same as (v — w) (x—y) or ax—vy -MX wy at which we should have arrived in the correct process.
The first step then from arithmetic to algebra is made by the follow ing definitions : 1. Quantities are distinguished into positive and negative, which are to be considered as of diametrically opposite kinds ; and common arith metical quantities ( abstract numbers without signs ) are to be con sidered as positive. 2. The rules of arithmetical algebra are to be applied to the extended algebra, and in all cases in which the latter presents a case unknown to the former, the rule of signs already known in the former must be applied. The extension which takes place in the terms less than nothing, &c., will be considered under the word
NOTHING.
The preceding extension gives an extended meaning to all the terms of operation ; thus addition is no longer the simple arithmetical pro cess, ,but a compound operation, first reducing a multiplicity of signs to one alone, and then following the direction of that sign ; and the same of subtraction. Thus a — ( — 6) is a +b. It may be asked then how we are to trace our steps through any problem so as to form its equation out of symbols which seem to have various meanings ; for it might appear as if the + of algebra were either the + or —of arithmetic, as the case may be. The answer is very simple : since the extended algebra is no more than arithmetic iu its actual operations, how ever the meaning of those operations may be extended, we may be sure that if we assign a particular case of a problem, and treat it entirely as in arithmetic, we are, though with one case only in view, performing upon limited symbols (limited because we think at the time only of a limited meaning) the same steps which we should have to follow if we could, by one act of the mind, grasp the symbols in their utmost generality.