A quaternion is best defined as a symbol of the type q = Za s e s = aoeo + ales = ale, + a3e3, where eo, ...
Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.
The equations q'+x = q and y+q' = q are satisfied by the same quaternion, which is denoted by q - q'.
In the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.
The outer and inner products of two extensive quantities A, B, are in many ways analogous to the quaternion symbols Vab and Sab respectively.
These may be compared and contrasted with such quaternion formulae as S(VabVcd) =SadSbc-SacSbd dSabc = aSbcd - bScda+cSadb where a, b, c, d denote arbitrary vectors.
The word "quaternion " properly means " a set of four."
From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space - or, what comes to the same thing, as the factor, or operator, which changes one directed line into another.
We may state, in passing, that every quaternion can be represented as a (cos 0+ 7 sin 9), - where a is a real number, 6 a real angle, and it a directed unit line whose square is - 1.
He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: - I T = 12 '=' 2 = _ 1, ij= - ji=k, jk= - kj=i, ki= - ik =j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'+jc'+kd') = A+iB+jC+kD, where A=aa' - bb' - cc' - dd', B = ab'+ba'+cd' - dc', C = ac'+ca'+db' - bd', D =ad' +da'+bc' - cb'.
And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion - + yy ' +2z') +i (yz ' - zy') +j (zx' - xz') +k (x y ' - yx').
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.
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.
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.
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.
The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product.
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.
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.
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+a, or as the product, fly, or the quotient, be-', 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.
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.
Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ kz, effectively expressed by a single letter.
No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like ClerkMaxwell (in his Electricity and Magnetism).
Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol;' these applications may be said to require no addition to the quaternion algebra; (b) W.
Shaw, in America, independently of Joly, has interpreted the quaternion as a point-symbol.
0) 2 = i suitable for non-Euclidean space, and w 2 = o suitable for Euclidean space; we confine ourselves to the second, and will call the indicated bi-quaternion p+wq an octonion.
This is the basis of a method parallel throughout to the quaternion method; in the specification of rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires.
He arrives at the tri-quaternion as the suitable fundamental concept.
We believe that this tri-quaternion solution of the very interesting problem proposed by Combebiac is the best one.
Hamilton, of quaternion fame.
The plane is of vector magnitude ZVq, its equation is ZSpq=Sr, and its expression is the bi-quaternion nVq+wSr; the point is of scalar magnitude 4Sq, and its position vector is [3, where 1Vf3q=Vr (or what is the same, fi = [Vr+q.
(Note that the z here occurring is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq i is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq i with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position vector of the point relative to the centre is Srg i.
Putting 1 - n _ E we get Combebiac's tri-quaternion under the form Q= Ep+nq+wr.
Combebiac's tri-quaternion may be regarded from many simplifying points of view.
Thus, in place of his general tri-quaternion we might deal with products of an odd number of point-plane-scalars (of form, uq+wr) which are themselves point-plane-scalars; and products of an even number which are octonions; the quotient of two point-plane-scalars would be an octonion, of two octonions an octonion, of an octonion by a point-plane-scalar or the inverse a point-plane-scalar.
If Q= Ep+nq+wr and we put Q= (I +Zwt)(Ep-i-nq)X (1 Zwt) -1 we find that the quaternion t must be 2f (r) /f (q - p), where f(r)=rq - Kpr.
When Su = o, (I + 2wu) () (1 + zwu) -1 is an operator which shifts (without further change) the tri - quaternion operand an amount given by u in direction and distance.