It has always appeared a paradox in mathematics, that by employing what are called imaginary or impossible quantities, and subjecting them to the same algebraic operations as those which are performed on quantities that are real and possible, the results obtained should always prove perfectly correct. The author inferring from this fact, that the operations of algebra are of a more comprehensive nature than its definitions and fundamental principles, was led to inquire what extension might be given to these definitions and principles, so as to render them strictly applicable to quantities of every description, whether real or imaginary. This deficiency, he conceives, may be supplied by having recourse to certain geometrical considerations. By taking into account the directions as well as the lengths of lines drawn in a given plane, from a given point, the addition of such lines may admit of being performed in the same manner as the composition of motions in dynamics; and four such lines may be regarded as proportional, both in length and direction, when they are proportionals in length, and, when also the fourth is inclined to the third at the same angle that the second is to the first. From this principle he deduces, that if a line drawn in any given direction be assumed as a positive quantity, and consequently its opposite a negative quantity, a line drawn at right angles to the positive or negative direction will be represented by the square root of a negative quantity ; and a line drawn in an oblique direction will be represented by the sum of two quantities, the one either positive or negative, and the other the square root of a negative quantity. On this subject, the author published a treatise in April 1828; since which period several objections have been made to this hypothesis. The purpose of the present paper is to answer these objections. The first of these is, that impossible roots should be considered merely as the indications of some impossible condition, which the proposition that has given rise to them involves; and that they have in fact no real or absolute existence. To this it is replied by the author, that although such a statement may be true in some cases, it is by no means necessarily so in all; and that these quantities resemble in this respect fractional and negative roots, which, whenever they are excluded by the nature of the question, are indeed signs of impossibility, but yet in other cases are admitted to be real and significant quantities. We have therefore no stronger reasons,
à priori
, for denying the real existence of what are called impossible roots, because they are in some cases the signs of impossibility, than we should have for refusing that character to fractional or negative roots on similar grounds.
The Interaction between International and Domestic Legal Orders: Framing the Debate according to the Post-Modern Condition of International Law
