NEGATIVE

, in Algebra, something marked with the sign -, or minus, as being contrary to such as are positive, or marked with the sign plus +. As Negative powers and roots, Negative quantities, &c. See Power, Root, Quantity, &c.

Negative Sign, the sign of subtraction -, or that which denotes something in defect. Stifel is the first author I find who used this mark - for subtraction, or negation, before his time, the word minus itself was used, or else its initial m.

The use of the Negative sign in algebra, is attended with several consequences that at first sight are admitted with some difficulty, and has sometimes given occasion to notions that seem 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, which is of the greatest use in this art.

Hence it serves to express a quantity of an opposite quality to a positive; such as a line in a contrary position, a motion with opposite direction, or a centrifugal force in opposition to gravity; and thus it often saves the trouble of distinguishing, and demonstrating separately, the various cases of proportions, and preserves their analogy in view. 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 quantities relates to their magnitude only, without determining whether they are to be considered as increments or decrements; fo there is no ground to imagine any other proportion of + a and - b, than that of the real magnitudes of the quantities represented by a and b, whether these quantities are, in any particular case, to be added or subtracted.

As to the usual arithmetical operations of addition, subtraction, &c, the case is different, as the effect of the Negative sign is here to be carefully attended to, and is to be considered always as producing, in those operations, an effect just opposite to the positive sign. Thus, it is the same thing to subtract a decrement as to add an equal increment, or to subtract - b from a - b, is to add + b to it: 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 - b to nb, according to the rule, that unit is to one of the factors as the other factor is to the product, there is not ground to imagine that there is any mystery in this, or any other meaning than that the real quantities represented by 1, n, b, and nb are proportional. For that rule relates only to the magnitude of the factors and product, without determining whether any factor, or the product, is additive or subtractive. But this likewise must be determined in algebraic computations; and this is the proper use concerning the signs, without which the operation could not pro- ceed. 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 a root. Hence the √- 1, or the square root of a Negative, implies an imaginary quantity, and in resolution is a mark or character of the impossible cases of a problem, unless it is compensated by another imaginary symbol or supposition, for then the whole expression may have a real signification. Thus 1 + √- 1, and 1 - √- 1, taken separately, are both imaginary, but yet their sum is the number 2: as the conditions that separately would render the solution of a problem impossible, in some cases destroy each others effect when conjoined. In the pursuit of general conclusions, and of simple forms for 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, wherein each particular term may have a real signification, as well as the whole expression.

The theorems that are sometimes briefly discovered by the use of this symbol, may be demonstrated without it by the inverse operation, or some other way; and though such symbols are of some use in the computations in the method of fluxions, &c, its evidence cannot be said to depend upon any arts of this kind. See Maclaurin's Fluxions, book 2, chap. 1.

Mr. Baron Maseres published a pretty large book in quarto, on the use of the Negative Sign in algebra.

For the rules or ways of using the Negative sign in the several rules of Algebra, see those rules severally, viz, Addition, Subtraction, Multiplication, &c. And for the method of managing the roots of Negative quantities, see Impossibles.

NEPER. See Napier.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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NAVIGATOR
NEAP
NEBULOUS
NEEDHAM (John Tuberville)
NEEDLE
* NEGATIVE
NEWEL
NEWTON (Dr. John)
NEWTON (Sir Isaac)
NICHE
NICOLE (Francis)