Editing Hyperbolic quaternion (section)

==Macfarlane's hyperbolic quaternion paper of 1900==
The ''Proceedings of the Royal Society of Edinburgh'' published "Hyperbolic Quaternions" 
in 1900, a paper in which Macfarlane regains associativity for multiplication by reverting 
to [[biquaternion|complexified quaternions]]. While there he used some expressions later 
made famous by [[Wolfgang Pauli]]: where Macfarlane wrote
:<math>ij=k\sqrt{-1}</math> 
:<math>jk=i\sqrt{-1}</math>
:<math>ki=j\sqrt{-1},</math>
the [[Pauli matrices]] satisfy
:<math>\sigma_1\sigma_2=\sigma_3\sqrt{-1}</math>
:<math>\sigma_2\sigma_3=\sigma_1\sqrt{-1}</math>
:<math>\sigma_3\sigma_1=\sigma_2\sqrt{-1}</math>
while referring to the same complexified quaternions.

The opening sentence of the paper is "It is well known that quaternions are intimately connected with [[spherical trigonometry]] and in fact they reduce the subject to a branch of algebra." This statement may be verified by reference to the contemporary work ''[[Vector Analysis]]'' which works with a reduced quaternion system based on [[dot product]] and [[cross product]]. In Macfarlane's paper there is an effort to produce "trigonometry on the surface of the equilateral hyperboloids" through the algebra of hyperbolic quaternions, now re-identified in an associative ring of eight real dimensions. The effort is reinforced by a plate of nine figures on page 181.  They illustrate the descriptive power of his "space analysis" method. For example, figure 7 is the 
common [[Minkowski diagram]] used today in [[special relativity]] to discuss change of velocity of a frame of reference and [[relativity of simultaneity]].

On page 173 Macfarlane expands on his greater theory of quaternion variables. By way of contrast he notes that [[Felix Klein]] appears not to look beyond the theory of [[Quaternions and spatial rotation]].