NEGATIVE, in general, something which implies a negation. Thus we say negative quantities, negative powers, negative signs, and the like.

Negative Sign. The use of the negative sign, in algebra, is attended with several consequences which at first are admitted with difficulty, and has sometimes given occasion to notions which appear to have no real foundation. This sign implies, that the real value of the quantity represented by the letter to which it is prefixed is to be subtracted; and it serves, with the positive sign, to keep in view what elements or parts enter into the composition of quantities, and in what manner, whether as increments or decrements, that is, whether by addition or subtraction. It serves therefore to express a quantity of an opposite quality to the positive, as a line in a contrary position, a motion with an opposite direction, or a centrifugal force in opposition to gravity; and thus often saves the trouble of distinguishing and demonstrating separately the various cases of proportions, and preserves in view their analogy. But as the proportions of lines depend on their magnitude only, without regard to their position, and motions and forces are said to be equal or unequal, in any given ratio, without regard to their directions; and, in general, the proportion of quantity relates to their magnitude only, without determining whether they are to be considered as increments or decrements; so there is no ground to imagine any other proportion of — b and + a (or of — 1 and 1) than that of the real magnitude of the quantities represented by b and a, whether these quantities are, in any particular case, to be added or subtracted. It is the same thing to subtract the decrement as to add an equal increment, or to subtract — b from a — b, as to add to it + b. And because multiplying a quantity by a negative number implies only a repeated subtraction of it, the multiplying — b by — n is subtracting — b as often as there are units in n ; and is therefore equivalent to adding + b so many times, or the same as adding + nb. But if we infer from this, that 1 is to — n as — & to nb, according to the rule, that unit is to one of the factors as the other factor is to the product, there is no ground to imagine that there is any mystery in this, or any other meaning than that the real magnitudes represented by 1, n, b, and ηb are proportional. For that rule relates only to the magnitude of the factors and product, without determining whether any factor, or the product, is to be addpd or subtracted. But this must likewise be determined in algebraic computations; and such is the proper use of the rules concerning the signs, without which the operation could not proceed. Because a quantity to be subtracted is never produced in composition by any repeated addition of a positive, or repeated subtraction of a negative, a negative square number is never produced by composition from the root. Hence √ — 1, or the square root of a negative, implies an imaginary quantity; and, in resolution, it is a mark or character of the impossible cases of a problem, unless it is compensated by another imaginary symbol or supposition, when the whole expression may have a real signification. Thus 1 + √ — 1, and 1 — √ — 1, taken separately, are imaginary, buttheir sum is 2; as the conditions that would separately render the solution of a problem impossible, in some cases destroy each other’s effect when conjoined. In the pursuit of general conclusions, and of simple forms representing them, expressions of this kind must sometimes arise where the imaginary symbol is compensated in a manner that is not always so obvious.

By proper substitutions, however, the expression may be transformed into another, in which each particular term may have a real signification as well as the whole expression. The theorems which are sometimes briefly discovered by the use of this symbol may be demonstrated without it by the inverse operation, or in some other way; and though such symbols are of some use in the computations by the method of fluxions, their evidence cannot be said to depend upon arts of this kind.