It was Alexander Macfarlane who promoted this concept in the 1890s as his Algebra of Physics, first through the American Association for the Advancement of Science in 1891, then through his 1894 book of five Papers in Space Analysis, and in a series of lectures at Lehigh University in 1900.
has these products: Using the distributive property, these relations can be used to multiply any two hyperbolic quaternions.
One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers.
In fact, for events p and q, the bilinear form arises as the negative of the real part of the hyperbolic quaternion product pq*, and is used in Minkowski space.
Nevertheless, this algebra put a focus on analytical kinematics by suggesting a mathematical model: When one selects a unit vector r in the hyperbolic quaternions, then r 2 = +1.
with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane.
transforms Dr by Since the direction r in space is arbitrary, this hyperbolic quaternion multiplication can express any Lorentz boost using the parameter a called rapidity.
However, the hyperbolic quaternion algebra is deficient for representing the full Lorentz group (see biquaternion instead).
Writing in 1967 about the dialogue on vector methods in the 1890s, historian Michael J. Crowe commented Later, Macfarlane published an article in the Proceedings of the Royal Society of Edinburgh in 1900.
The 1890s felt the influence of the posthumous publications of W. K. Clifford and the continuous groups of Sophus Lie.
But it is a startling aspect of finite mathematics that makes the hyperbolic quaternion ring different: The basis
of the vector space of hyperbolic quaternions is not closed under multiplication: for example,
: The Yale University physicist Willard Gibbs had pamphlets with the plus one square in his three-dimensional vector system.
Oliver Heaviside in England wrote columns in the Electrician, a trade paper, advocating the positive square.
In 1892 he brought his work together in Transactions of the Royal Society A[2] where he says his vector system is So the appearance of Macfarlane's hyperbolic quaternions had some motivation, but the disagreeable non-associativity precipitated a reaction.
The quasigroup stimulated a considerable stir in the 1890s: the journal Nature was especially conducive to an exhibit of what was known by giving two digests of Knott's work as well as those of several other vector theorists.
Michael J. Crowe devotes chapter six of his book A History of Vector Analysis to the various published views, and notes the hyperbolic quaternion: In 1899 Charles Jasper Joly noted the hyperbolic quaternion and the non-associativity property[4] while ascribing its origin to Oliver Heaviside.
As for mathematics, the hyperbolic quaternion is another hypercomplex number, as such structures were called at the time.
In 1899 Alfred North Whitehead promoted Universal algebra, advocating for inclusivity.
The concepts of quasigroup and algebra over a field are examples of mathematical structures describing hyperbolic 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."
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.
By way of contrast he notes that Felix Klein appears not to look beyond the theory of Quaternions and spatial rotation.