His earliest work dealt mainly with mathematical subjects, and especially with quaternions (q.v.), of which he may be regarded as the leading exponent after their originator, Hamilton.
He was the author of two text-books on them - one an Elementary Treatise on Quaternions (1867), written with the advice of Hamilton, though not published till after his death, and the other an Introduction to Quaternions (1873), in which he was aided by Professor Philip Kelland (1808-1879), who had been one of his teachers at Edinburgh.
In addition, quaternions was one of the themes of his address as president of the mathematical section of the British Association in 1871.
Among his articles may be mentioned those which he wrote for the ninth edition of this Encyclopaedia on Light, Mechanics, Quaternions, Radiation and Thermodynamics, besides the biographical notices of Hamilton and Clerk Maxwell.
Under the general heading "Fundamental Notions" occur the subheadings "Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; "Universal Algebra," with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and "Theory of Groups," with the topics finite and continuous groups.
For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.
The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.
Thus e 1 e 2 = - e2ei, and if q, q are any two quaternions, qq is generally different from q'q.
Clifford's biquaternions are quantities Eq+nr, where q, r are quaternions, and E, n are symbols (commutative with quaternions) obeying the laws E 2 = E, n 2 =,g, = 1 j E=0 (cf.
All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.
As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.
Quaternions afford an example of a quadruple algebra of this kind; ordinary algebra is a special case of a duplex linear algebra.
Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices.
This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general.
Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866); H.
QUATERNIONS, in mathematics.
Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space.
The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.
Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730).
The course of his investigations is minutely described in the preface to his first great work (Lectures on Quaternions, 1833) on the subject.
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'.
But in 1877, in the M athematische 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.
More general systems, having close analogies to quaternions, have been given since Hamilton's discovery was published.
- The above narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry.
Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms.
The 2 Lectures on Quaternions, § 513.
Here the symmetry points at once to the selection of the three principal axes as the directions for i, j, k; and it would appear at first sight as if quaternions could not simplify, though they might improve in elegance, the solution of questions of this kind.
Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol.
Sufficient has already been said to show the close connexion between quaternions and the theory of numbers.
- There are three fairly wellmarked stages of development in quaternions as a geometrical method.
Combebiac's tri-quaternions, which require the addition of quasi-scalars, independent of one another and of true scalars, and analogous to true scalars.
Method quaternions have from the beginning received much attention from mathematicians.
We select for description stage (3) above, as the most characteristic development of quaternions in recent years.
For (3) (a) we are constrained to refer the reader to Joly's own Manual of Quaternions (1905).
Clifford in his paper of 1873 (" Preliminary Sketch of Bi-Quaternions," Mathematical Papers, p. 181) seems to have come from Sir R.
Clifford makes use of a quasi-scalar w, commutative with quaternions, and such that if p, q, &c., are quaternions, when p-I-wq= p'+wq', then necessarily p= p', q = 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.
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.
In 1904 Alexander Macfarlane published a Bibliography of Quaternions and allied systems of Mathematics for the International Association for promoting the study of Quaternions and allied systems of Mathematics (Dublin University Press); the pamphlet contains 86 pages.
Hamilton's classical Elements of Quaternions of 1866 was republished under C. J.
Tait's Elementary Treatise on Quaternions appeared (Cambridge).
Joly published his Manual of Quaternions (London); the valuable contents of this are doubled by copious so-called examples; every earnest student should take these as part of the main treatise.
McAulay, Octonions, a development of Clifford's Bi-quaternions (Cambridge, 1898); G.
The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamiltons Quaternions,the tensor expressing the velocity ratio, and the versor the directional relation.
He was much interested, too, in universal algebra, non-Euclidean geometry and elliptic functions, his papers "Preliminary Sketch of Bi-quaternions" (1873) and "On the Canonical Form and Dissection of a Riemann's Surface" (1877) ranking as classics.