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(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...
|Country of origin:
Edinburgh Mathematical Society
||246 x 189 x 4mm (L x W x T)
||Paperback - Trade
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