therefore confine ourselves, so far as his predecessors are concerned,
to attempts at interpretation which had geometrical
applications in view.

One geometrical interpretation of the negative sign of algebra was early seen to be mere reversal of direction along a line. Thus, when an image is formed by a plane mirror, the distance of any point in it from the mirror is simply the negative of that of the corresponding point of the object. Or if motion in one direction along a line be treated as positive, motion in the opposite direction along the same line is negative. In the case of time, measured from the Christian era, this distinction is at once given by the letters A.D. or B.C., prefixed to the date. And to find the position, in time, of one event relatively to another, we have only to subtract the date of the second (taking account of its sign) from that of the first. Thusl to find the interval between the battles of Marathon (400 B.C.) and Waterloo (Ap. 1815) we have 1

-i-1815-(-490) =2305 years.

And it is obvious that the same process applies in all cases in which we deal with quantities which may be regarded as of one directed dimension only, such as distances along a line, rotations about an axis, &c. But it 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.

Tolin 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 Abbé Buée (Phil. Trans., 1806), apparently without any knowledge 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 com planar directed lines. His constructions are based on the idea that the imaginaries #= ~j - 1 represent a unit line, and its reverse, perpendicular to the line on which the real units 1 1 are measured. In this sense the imaginary expression a + b xl - 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 kinernatical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

In 1806 (the year of publication of Buée's paper) Jean Robert Argand published a -pamphlet* in which precisely the same ideas are developed, but to af 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 1 Strictly speaking, this illustration of Tait's is in error by unity because in our calendar there is no year denominated zero. Thus the interval between June the first of 1 B.C. and June the first of I A.D. is one year, and not two years as the text implies. (A.l/ICA.) 2Exsai sur une manrére de représenter les Qnanlités Imaginaires dans les Conslruclions G€0méff10ll€$. A second edition was published by j. 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 he was born at Geneva in 1768.

term is the real (positive) unit-line, and the other twoare the factor-lines. Argand's work remained unnoticed until the question was again raised in Gergonne's Annales, 1813, by J. F. F rancais. This writer stated that he had found the germ of his remarks among the papers of his 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 communications with Legendre, and gave a résumé 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 existence of n roots, and no more, in every rational algebraic equation of the nth order with real coefficients. About 1828 John Warren (1796-1852) in England, and C. V. Mourey in France, independently of one another and of Argand, reinvented these modes of interpretation; and still later, in the writings of Cauchy, Gauss and others, the properties of the expression a -l- b / - 1 were developed into the immense and most important subject now called the lheory of complex numbers (see NUMBER). From the more purely symbolical view it was developed by Peacock, De Morgan, &c., 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 an angle 0 with the real(positive) unit line, is expressed by a(cos 0-{~i sin 0), where i is regarded as -l- / -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 I:a(cos0~l-isin 0): 2 a'(cos6'+isin 0'):P, or P=ao”{cos (0~{-0')-l-isin (04-0')}.

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-l-Bi, A”+B'i, the product is AA'-BB'-l-'i(AB'+BA'); and the fact that the length of the product line is the product of those of the factors is seen in the form

(AZ-l-Bl) (A'2-l-Bm) = (AA' -BB')2+(AB'-l-BA')“. 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 generally were fruitless. The reasons will be obvious later; but we mention them just now because they called forth from F. T. Servois (Gergonne's Annales, 1813) a very remarkable 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 if 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-1r/2. Servois says, with reference to the general representation of a directed line in spacezf

“ ljanalogie semblerait exiger que le trinome fiit de la forme p cos a-I-q cos B-l-r cos 11: a, (3, 7 étant les angles d'une droite avec trois axes rectangulaires; et qu'on eut

(pcosa-l-qcos{3+rcos'y)(, b' cos a ~|-g'cos5-l-r'cos 7) =cos2a-l-cos”/3-l-cos"y= I. Les valeurs de p, g, r, p', g, ' r', qui satisferaient at cette condition seraient absnrzles; mais seraient-elles imaginaires, deductibles at la forme générale A-l-B / -1? Voila une question d'analyse fort singuliere que je soumets a vos lumiéres. La simple proposition que jevous en fais suffit pour vous faire voir que je ne crois point que toute fonction analytique non réelle soit vraiment deductible a la forme A+B x/ - I.”

As will be seen later, the fundamental i, j, k of quaternions, 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 ordinary imaginaries. Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730). Instead of regarding, as Buée and Argand had done, the expression a(cos 0-I-i sin 0) as a directed line, let us suppose it to represent the operator Which, when applied to any line in the plane in which 0 is measured, turns it in that plane through the angle 0, and at the same time increases its length in the ratio az 1. From the new point of view we see at once, as it were, why it is true that (cos 0-{~'i sin 8)"' =cos m6~~ -i sin m0.