a mathematical model: When one selects a unit vector r in the hyperbolic quaternions, then r 2 = +1. The plane with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane. The hyperbolic versor 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, a historian commented The introduction of another system of vector analysis, even a sort of compromise system such as Macfarlane's, could scarcely be well received by the advocates of the already existing systems and moreover probably acted to broaden the question beyond the comprehension of the as-yet uninitiated reader. Geometry Later, Macfarlane published an article in the Proceedings of the Royal Society of Edinburgh in 1900. In it he treats a model for hyperbolic space H3 on the hyperboloid This isotropic model is called the hyperboloid model and consists of all the hyperbolic versors in the ring of hyperbolic quaternions. Historical review The 1890s felt the influence of the posthumous publications of W. K. Clifford and the continuous groups of Sophus Lie. An example of a one-parameter group is the hyperbolic versor with the hyperbolic angle parameter. This parameter is part of the polar decomposition of a split-complex number. 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, . Nevertheless, the set is closed under multiplication. It satisfies all the properties of an abstract group except the associativity property; being finite, it is a Latin square or quasigroup, a peripheral mathematical structure. Loss of the associativity property of multiplication as found in quasigroup theory is not consistent with linear algebra since all linear transformations compose in an associative manner. Yet physical scientists were calling in the 1890s for mutation of the squares of ,, and to be instead of : 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