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 are parallel, or =0, the product, which is now the square of any (unit) line is −1. And when the two factor lines are at right angles to one another, or =/2, the product is simply ix″+jy″+kz″, the unit line perpendicular to both. Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k. In other words, if the system (considered as rigid) be made to turn about till the first factor coincides with i and the second with j, the product will coincide with k. This fundamental system, therefore, becomes unnecessary; and the quaternion method, in every case, takes its reference lines solely from the problem to which it is applied. It has therefore, as it were, a unique internal character of its own.

Hamilton, having gone thus far, proceeded to evolve these results from a characteristic train of a priori or metaphysical reasoning. Let it be supposed that the product of two directed lines is something which has quantity; i.e. it may be halved, or doubled, for instance. Also let us assume (a) space to have the same properties in all directions, and make the convention (b) that to change the sign of any one factor changes the sign of a product. Then the product of two lines which have the same direction cannot be, even in part, a directed quantity. For, if the directed gart have the same direction as the factors, (b) shows that it will e reversed by reversing either, and therefore will recover its original direction when both are reversed. But this would obviously be inconsistent with (a). If it be perpendicular to the factor lines, (a) shows that it must have simultaneously every such direction. Hence it must be a mere number.

Again, the product of two lines at right angles to one another cannot, even in part, be a number. For the reversal of either factor must, by (b), change its sign. But, if we look at the two factors in their new position by the light of (a), we see that the sign must not change. But there is nothing to prevent its being represented by a directed line if, as further applications of (a) and (b) show we must do, we take it perpendicular to each of the factor lines. Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a numerical and a directed part. He indulged in a great deal of speculation as to the existence of an extra-spatial unit, which was to furnish the raison d’étre of the numerical part, and render the quaternion homogeneous as well as linear. But for this we must refer to his own works.

Hamilton was not the only worker at the theory of sets. The year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann in which results closely analogous to some of those of Hamilton were given. In particular, two species of multiplication (“inner” and “outer ”) of directed lines in one plane were given. The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton’s quaternion product. But Grassmann distinctly states in his preface that he had not had leisure to extend his method to angles in space. Hamilton and Grassmann, while their earlier work had much in common, had very different objects in view. Hamilton had geometrical application as his main object; when he realized the quaternion system, he felt that his object was gained, and thenceforth confined himself to the development of his method. Grassmann’s object seems to have been, all along, of a much more ambitious character, viz. to discover, if possible, a system or systems in which every conceivable mode of dealing with sets should be included. That he made very great advances towards the attainment of this object all will allow; that his method, even as completed in 1862, fully attains it is not so certain. But his claims, however great they may be, can in no way conflict with those of Hamilton, whose mode of multiplying couples (in which the “inner” and “outer” multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had not even been attacked. Grassmann made in 1854 a somewhat savage onslaught on Cauchy and De St Venant, the former of whom had invented, while the latter had exemplified in application, the system of “clefs algébriques,” which is almost precisely that of Grassmann. But it is to be observed that Grassmann, though he virtually accused Cauchy of plagiarism, does not appear to have preferred any such charge against Hamilton. He does not allude to Hamilton in the second edition of his work. But in 1877, in the Mathematische Annalen, xii., he gave a paper “ On the Place of Quaternions in the Ausdehnungslehre,” in which he condemns, as far as he can, the nomenclature and methods of Hamilton.

Relations to other Branches of Science.-The above narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry. Were this all, the gain by their introduction would consist mainly in a clearer insight into the mechanism of co-ordinate systems, rectangular or not—a very important addition to theory, but little advance so far as practical application is concerned. But, as yet, we have not taken advantage of the perfect symmetry of the method. When that is done, the full value of Hamilton’s grand step becomes evident, and the gain is quite as extensive from the practical as from the theoretical point of view. Hamilton, in fact, remarks, “I regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to have recourse to the resources of ordinary algebra, for the solution of equations in quaternions.” This refers to the use of the x, y, z co-ordinates,—associated, of course, with i, j, k. But when, instead of the highly artificial expression ix+jy+kz, to denote a finite directed line, we employ a single letter,. (Hamilton uses the Greek alphabet for this purpose), and find that we are permitted to deal with it exactly as we should have dealt with the more complex expression, the immense gain is at least in part obvious. Any quaternion may now be expressed in numerous simple forms. Thus we may regard it as the sum of a number and a line, a+, or as the product,, or the quotient, −1, of two directed lines, &c., while, in many cases, we may represent it, so far as it is required, by a single letter such as q, r, &c.

Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical trigonometry. Also, everything relating to change of systems of axes, as for instance in the kinematics of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with regard to axes fixed in the system, is a matter of troublesome complexity by the usual methods. But every quaternion formula is a proposition in spherical (sometimes degrading to plane) trigonometry, and has the full advantage of the symmetry of the method. And one of Hamilton’s earliest advances in the study of his system (an advance independently made, only a few months later, by Arthur Cayley) was the interpretation of the singular operator qq−1, where q is a quaternion. Applied to any directed line, this operator at once turns it, conically, through a definite angle, about a definite axis. Thus rotation is now expressed in symbols at least as simply as it can be exhibited by means of a model. Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms.

The physical properties of a heterogeneous body (provided they vary continuously from point to point) are known to, depend, in the neighbourhood of any one point of the body, on a quadric function of the co-ordinates with reference to that point. The