ON A THEORY OF QUATERNIONS

By William Rowan Hamilton

(British Association Report, 1844, Part II, p. 2.)

Edited by David R. Wilkins On a Theory of Quaternions. By Sir William R. Hamilton, M.R.I.A.

[Report of the Fourteenth Meeting of the British Association for the Advancement of Science; held at York in September 1844. (John Murray, London, 1845), Part II, p. 2.]

It has been shown, by Mr. Warren and others, that the results obtained by the ordinary processes of algebra, involving the imaginary symbol \sqrt{−1}, admit of real interpretations, such as those which relate to compositions of linear motions and rotations in one plane. Sir W. Hamilton has adopted a system of three such imaginary symbols, i, j, k, and assumes or defines that they satisfy the nine equations

i^{2} = j^{2} = k^{2} = −1, ij = k = −ji, jk = i = −kj, ki = j = −ik,

which however are not purely arbitrary, and for the adoption of which the paper assigns reasons. He then combines these symbols in a quaternion, or imaginary quadrinomial, of the form

q = w + ix + jy + kz,

in which w, x, y, z, are four real quantities; and states that he has established rules for algebraical operations on such expressions, and has assigned geometrical interpretations corresponding; so as to form a sort of Calculus of Quaternions, which serves as an instrument to prove old theorems, and to discover new ones, in the geometry of three dimensions, and especially respecting the composition of motions of translation and rotation in space.