Page:Encyclopædia Britannica, Ninth Edition, v. 20.djvu/173

QUATEBNIONS 161 is essential to notice that this is by no means necessarily true of operators. To turn a line through a certain angle in a given plane, a certain operator is required ; but when we wish to turn it through an equal negative angle we must not, in general, employ the negative of the former operator. For the negative of the operator which turns a line through a given angle in a given plane will in all cases produce the negative of the original result, which is not the result of the reverse operator, unless the angle involved be an odd multiple of a right angle. This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, which merely means that - 1 is commutative with all other quantities. The celebrated Wallis seems to have been the first to push this idea further. In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed. In 1804 the Abbe" Buee, 1 apparently without any know- ledge of Wallis's work, developed this idea so far as to make it useful in geometrical applications. He gave, in fact, the theory of what in Hamilton's system is called Composition of Vectors in one plane i.e., the combination, by + and - , of complanar directed lines. His construc- tions are based on the idea that the imaginaries */ - 1 represent a unit line, and its reverse, perpendicular to the line on which the real units 1 are measured. In this sense the imaginary expression a + b *J 1 is constructed by measuring a length a along the fundamental line (for real quantities), and from its extremity a line of length b in some direction perpendicular to the fundamental line. But he did not attack the question of the representation of products or quotients of directed lines. The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary. In 1806 (the year of publication of Bute's paper) Argand published a pamphlet 2 in which precisely the same ideas are developed, but to a considerably greater extent. For an interpretation is assigned to the product of two directed lines in one plane, when each is expressed as the sum of a real and an imaginary part. This product is interpreted as another directed line, forming the fourth term of a proportion, of which the first term is the real (positive) unit-line, and the other two are the factor-lines. Argand's work remained unnoticed until the question was again raised in Gergonne's Annales, 1813, by Fran^ais. This writer stated that he had found the germ of his remarks among the papers of Mr: deceased brother, and that they had come from Legendre, who had himself received them from some one unnamed. This led to a letter from Argand, in which he stated his communica- tions with Legendre, and gave c, resume of the contents of his pamphlet. In a further communication to the Annales, Argand pushed on the applications of his theory. He has given by means of it a simple proof of the exist- ence of n roots, and no more, in every rational algebraic equation of the nth order with real coefficients. About 1828 Warren in England, and Mourey in France, inde- pendently of one another and of Argand, reinvented these modes of interpretation ; and still later, in the writings of 1 Phil. Trans., 1806. 2 Essai sur une maniere de representer les Quantites Imaginaires dans les Constructions Geometriques. A second edition was published by Hoiiel (Paris, 1874). There is added an important Appendix, consisting of the papers from Gergonne's Annales which are referred to in the text above. Almost nothing can, it seems, be learned of Argand's private life, except that in all probability lie was born at Geneva in 1768. Cauchy, Gauss, and others, the properties of the expression a + 1 j _ 1 W ere developed into the immense and most important subject now called the theory of complex numbers (see NUMBERS, THEORY OP). From the more purely sym- bolical view it was developed by Peacock, De Morgan, <fec., as double algebra. Argand's method may be put, for reference, in the following form. The directed line whose length is a, and which makes au angle with the real (positive) unit line, is expressed by where i is regarded as + V - 1. The sum of two such lines (formed by adding together the real and the imaginary parts of two such expressions) can, of course, be expressed as a third directed line the diagonal of the parallelogram of which they are conterminous sides. The product, P, of two such lines is, as we have seen, given by 1 ' ' or Its length is, therefore, the product of the lengths of the factors, and its inclination to the real unit is the sum of those of the factors. If we write the expressions for the two lines in the form A + Bi, A' + B'i, the product is A A' - BB' + i( AB' + B A') ; and the fact that the length of the product line is the product of those of the factors is seen in the form (A 2 + B 2 )(A' 2 + B' 2 ) = (AA' - BB') 2 + (AB' + B A') 3 . In the modern theory of complex numbers this is expressed by saying that the Norm of a product is equal to the product of the norms of the factors. Argand's attempts to extend his method to space gene- rally were fruitless. The reasons will be obvious later ; but we mention them just now because they called forth from Servois (Gergonne's Annales, 1813) a very remark- able comment, in which was contained the only yet discovered trace of an anticipation of the method of Hamilton. Argand had been led to deny that such an expression as i l could be expressed in the form A + Bi, although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of - ir/2 . Servois says, with reference to the general representation of a directed line in space : "L'analogie semblerait exiger que le trinome fut de la forme jpcosa + geos/J + rcosy; a, &, 7 etaut les angles d'une droite avec trois axes rectangulaires ; et qu'on cut = l. Les valeurs de p, q, r, p', q', r' qiii satisferaieut & cette condition seraient absurdcs ; mais scraient-ellcs imaginaires, reductibles a la forme generale A + B/ - 1 ? Voila une question d'analyse fort singuliere que je soumets a vos lumieres. La simple proposition que je vous en fais suffit pour vous faire voir que je ne crois point que toute fonctionjinalytique non reelle soit vraiinent reductible a la forme A + B/- 1." As will be seen later, the fundamental *, j, Tc of quater- nions, with their reciprocals, furnish a set of six quantities which satisfy the conditions imposed by Servois. And it is quite certain that they cannot be represented by ordin- ary imaginaries. Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem J1730). Instead of regarding, as Buee and Argand had done, the expression a (cos + i sin 6) as a directed line, let us suppose it to represent the operator which, when applied to any line in the plane in which is measured, turns it in that plane through the angle 6, and at the same time increases its length in the ratio a : 1. From the new point of view we see at once, as it were, why it is true that (cos + isin 0) m =cosmO + isinmd. For this equation merely states that m turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle. To make this process applicable to any plane in space, it is clear that we must have a special value of i for each such plane. In other words, a unit line, drawn in any direction whatever, must have - 1 for its square. In such XX. 21