NEGATIVE QUANTITIES. This subject is considered, as a part of the most complete algebra, in the article ALGETITIA. In the present article we confine ourselves to such a view as may be sufficient for ordinary algebra.
In the oldest treatises on algebra which exist, there is mention of a modification of quantity unknown in arithmetic, called negative quan tity, as distinguished from positive. In the Viga Ganita [Vice GANITA, Bloc. Div.] we find this distinction and the rules for its use precisely as in modern treatises. One of the commentators says that negation is contrariety ; and the Liliwati' contains the geometrical interpretation of a negative line—namely, a line measured in the direction contrary to that of a positive line. The commentator says that Patna is fifteen yojanas east, and Allahabad eight yojanas west, of a place called Varanasi ; " the interval or difference is twenty-three yojanas, and is not obtained but by addition of the numbers. Therefore, if the difference between two contrary quantities be required, their sum must be taken." Surely it will be said that algebra began in a strange confusion of ideas; but yet the fault is rather in expression than in conception. Au art was in existence presenting undoubted means of discovering truth, commencing with a generalisation of which the use was obvious, but not the meaning. In Diophantus we find the common rule announced as a definition (without even a previous notice of the distinction of quantities) in terms as broad as the following :—" Ainfas ?al M4iv roVtarAacriacrOgzera roeiz " "Defect upon defect repeated, makes In Mohammed Ben Musa [ALGEBRA] the rules are announced in the same way, though the separate existence of positive and negative quantities does not seem to be assumed : it must he remembered that this work was written for popular use. The European promoters of algebra, with the exception only of Vieta, adopted the use of two species of quantities, positive and negative, with the explanation above noticed. Vieta not only avoided the
negative quantity, but, so far as he could, dispensed with subtractive terms and subtraction itself. He discards the double nature of quan tities in the words "Plus autem vel minus non constituunt genera diverse." It is not our intention to follow the earlier algebraists through their different uses of negative quantities. These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterised every attempt at their theory. Newton and Euler distinctly admit the existence of the quantity less than nothing : the latter asserts that a mau who has no property, and is in debt 50 crowns, would only have nothing if any one else made him richer by a gift of 50 crowns, and therefore begins with 50 crowns less than nothing. Elementary treatises for the most part try to append an explanation of nsga`ive quantities to an algetta which is 3v nothing more than arithmetic, instead of introducing those new abstractions which are the basis of the separate science : so that algebra, instead of being systematically learnt, is collected by slow and often dubious steps from arithmetical examples, in which the rules of operation of the former science are employed, preceded by the prin ciples of the latter. Few, therefore, acquire a real perception of the meaning of the subject, except those who study mathematics to great extent. It is matter of notoriety that difficulties attend the beginner in algebra of a nature totally different from those which are found in geometry ; so that shills a person who has read a few books of Euclid may be imagined capable of writing an intelligent commentary on what he knows, another who has mastered a common elementary treatise on algebra is conscious only of a great increase of working power, with a glimmering of principles which owe their reception more to the never failing accuracy of their results than to native evidence or logical deduction from easily admitted truths.