On the nature of negative and imaginary quantities

Abstract

The object of this paper, the author observes, is one that has given rise to much controversy, and has been involved in much unnecessary mystery. Paradoxes and apparent solecisms, when involved with facts and indubitable truths, will always be found, upon accurate examination, to be near the surface, and to owe their existence either to ambiguities of expression, or to the unperceived adoption of some extraneous additions or limitations into the compound terms employed for definition, and which are subsequently taken as constituent parts of their essence. The first misapprehension pointed out, is that of considering any quantity whatever as negative per se , and without reference to another opposed to it, which has previously been established as positive. In order to avoid previously formed associations of ideas, the author prefers employing in his reasonings on this subject, the symbols ( a ) and ( b ) to express this quality of opposition, rather than the usual ones of plus and minus . By the aid of this notation he is enabled to present, in its full generalization, the law of the signs in multiplication, —a process which, it is well known, is founded solely upon the principle of ratios ; and to show that like signs invariably give the sign belonging to the assumed unity, or universal antecedent of the ratios ; and unlike signs, the contrary.