NEGATIVE QUANTITIES are generally defined as quantities the opposite of "posi tive" or " numerical " quantities, and form the first and great point of difference between algebra as a separate science and arithmetic. In the oldest treatises on algebra they are ree041fized as distinct modifications of quantity, and existing apart from, and inde pendent of, positive quantity. In later times, this opinion was vigorously combated by Many nuithematiciims, among whom Vieta occupied a prominent place; but the more eminent analysts retained the old opinion. Newton and Euler distinctly assert the existence of negative quantities as quantities less than zero, and the latter supports his opinion by the well-anown illustration of a man who has no property, and is 1'50 in debt, to whom £50 requires to be given in order that he may have nothing. Alter all, this discussion is little inora than a verbal quibble, though interesting from the prominent position it for a long time held. It had its rise in the difficulty of satisfying the requirements of a constantly progressing science by the use of signs and forms retaining their original limited,signilleation. It was soon felt that the limited interpre tation must be given up; and accordingly an extension of signification was allowed to signs and modes of operation. + and —, which were formerly considered as merely symbols of the arithmetical operations of addition and subtraction, were now considered "general cumulative symbols, the reverse of each other," and could signify gain and loss, upward and downward, right and left, same and opposite, to and front, etc. Apply ing this extended interpretation of signs to as quantity such as —4, we obtain at once a true idea of a negative quantity; for it 4 signifies 4 in. above a certain level, —4 signifies 4 in. below that level, and therefore, though a positive quantity in itself (a negative being,
strictly speaking, an impossible existence), it may be fairly considered to be less than zero, as it expresses a quantity less by 4 than 0 inches above the level. Keeping this i,lea in view it has been conventionally agreed to admit the existence of negative quan tities as existing per se. The only errors which can flow front this arise from misinter pretation of results, for the four fundamental operations of addition. subtraction, multi plication, and division arc unaffected by the extended interpretation of signs. The following is an illustration of the value of an extended interpretation of the negative sign, snowing at the same time how much more general are the ideas conveyed by alge braic expressions than by ordinary language: If at the present time a father is 50 years, and Ills son 20 years old, when will the father be three times as old as his son, This probfem when solved. gives —5 as the number of years which must elapse before the father's age is three times the son's. Now, at first sight.. this result appxirs to be absurd, but when we consider the terms of the problem, its explanation is easy. The question asked pointed to a number of years to cow, and had the result turned cut to be positive, such would have been the case, and the fact of its being negative directs us to look in a " contrary" direction, or backward to time past; and this is found to satisfy the problem as five years " ago" the father was 45 and his son 15.
Negative quantities arise out of the use of general symbols in subtraction, as in the formula a — b, where we may afterward find that b is greater than a. See SUBTRACTION.