Hyperbolic Quaternion - MacFarlane's Hyperbolic Quaternion Paper of 1900

MacFarlane's Hyperbolic Quaternion Paper of 1900

The Proceedings of the Royal Society at Edinburgh published "Hyperbolic Quaternions" in 1900, a paper in which Macfarlane regains associativity for multiplication by reverting to complexified quaternions. While there he used some expressions later made famous by Wolfgang Pauli: where Macfarlane wrote

,

the Pauli matrices satisfy

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.