Alexander McAulay

“ 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 a system there will be no line in space specially distinguished as the real unit line: all will be alike imaginary, or rather alike real. We may state, in passing, that every quaternion can be represented as a (cos θ+ π sin θ) , where a is a real number, θ a real angle, and π a directed unit line whose square is −1. ”

“ But one most important connexion with modern physics must be pointed out. In the theory of surfaces, in hydro kinetics, heat-conduction, potentials, &c., we constantly meet with what is called “Laplace’s operator,” viz. d2/dx2+ d2/dy2+d2/dz2. We know that this is an invariant; i.e. it is independent of the particular directions chosen for the rectangular co-ordinate axes. Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression d/dx+ d/dy+d/dz could be, like ix+jy+kz, effectively expressed by a single letter. ”

“ 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. ”

“ We believe that this tri-quaternion solution of the very interesting problem proposed by Combebiac is the best one. But the first thing that strikes one is that it seems unduly complicated. A point and a plane fix a line or axis, viz. that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also. To fix a weighted point and a weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion. We should expect some species of bi quaternion to suffice. ”

“ 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. ”

“ Combebiac is not content with getting rid of the origin in these limited circumstances. The fundamental geometrical conceptions are the point, line and plane. Lines and complexes thereof are sufficiently treated as rotors and motors, but points and planes cannot be so treated. He glances at Grassmann's methods, but is repelled because he is seeking a unifying principle, and he finds that Grassmann offers him not one but many principles. He arrives at the tri-quaternion as the suitable fundamental concept. ”

“ 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 [6] 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. ”

“ In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q−1 is meaningless.
Putting 1−η=ξ we get Combebiac's tri-quaternion under the form Q=ξp+ηq+ωr. ”

“ That we have here a perfectly real and intelligible interpretation of the ordinary algebraic imaginary is easily seen by an illustration, even if it be a somewhat extravagant one. Some Eastern potentate, possessed of absolute power, covets the vast possessions of his vizier and of his barber. ”

“ 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. ”

“ He considers two cases, viz. ω2=1 suitable for non-Euclidean space, and ω2=0 suitable for Euclidean space; we confine ourselves to the second, and will call the indicated bi-quaternion p+ωq an octonion. In octonions the analogue of Hamilton's vector is localized to the extent of being confined to an indehnitely long axis parallel to itself, and is called a rotor; if p is a rotor then wp is parallel and equal to ρ, and, like Hamilton's vector, ωρ is not localized ”

“ 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. ”

“ C. J. Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol; [8] these applications may be said to require no addition to the quaternion algebra; (b) W. K. Clifford's bi-quaternions and G. Combebiac's tri-quaternions, which require the addition of quasi-scalars, independent of one another and of true scalars, and analogous to true scalars. As an algebraic method quaternions have from the beginning received much attention from mathematicians. An attempt has recently been made under the name of multenions to systematize this algebra. ”