Proceedings of the Edinburgh Mathematical Society Volume 11-12 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 Excerpt: ... are attained, it is necessary to consider the laws of multiplication of quaternions: and, in doing so, it is not necessary to consider the stretching part (or Tensor) of the quaternion--for that part is a mere number and so obeys all the laws of ordinary algebra. We may represent quaternions by plane angles or by arcs of great circles on a unit sphere. Thus, if PQR be a spherical triangle whose sides p, q, r are portions of great circles on the unit sphere, the quantities p, q, r may represent the corresponding quaternions. Let a be the vector from the origin to the point Q. Then pa is the vector to the point R, and q-pa is the vector to P. But this is also ra, if r is measured from Q to P while p and q are measured from Q to R, and from R to P, respectively. And we are at liberty to define r = qp, so that q-pa = qpa. This makes the associative law hold when a, pa, and qpa are vectors--a fact which is pointed out by Hamilton, Lectures, 310, and by Tait, Elements, 54. It defines quaternion multiplication. Various proofs that the associative law holds in the multiplication of quaternions have been given. Of these, Hamilton's proof (Lectures, 296; Elements, 270; and Tait's Elements, 57-60) by spherical arcs and elementary properties of spherical conics involves, by definition, the particular assumption of association just alluded to. His alternative proof, by more elementary geometry (Lectures, 298-301), makes use of the same definition; and the same remark applies to the proof given in 358, 359 of the Lectures. On the other hand, the geometrical proof given in Hamilton's Elements, 266, 267, 272, is based upon the definition of the reciprocal of a quaternion, which makes the produ...