Hyperbolic Quaternion - Historical Review

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 of the Royal Society A where he says his vector system is

simply the elements of Quaternions without quaternions, with the notation simplified to the uttermost, and with the very inconvenient minus sign before scalar product done away with.

So the appearance of Macfarlane's hyperbolic quaternions had some motivation, but the disagreeable non-associativity precipitated a reaction. Cargill Gilston Knott was moved to offer the following:

Theorem (Knott 1893)

If a 4-algebra on basis is associative and off-diagonal products are given by Hamilton's rules, then .

Proof:

, so . Cycle the letters, to obtain . QED.

This theorem needed statement to justify resistance to the call of the physicists and the Electrician. 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:

Macfarlane constructed a new system of vector analysis more in harmony with Gibbs-Heaviside system than with the quaternion system. ...he...defined a full product of two vectors which was comparable to the full quaternion product except that the scalar part was positive, not negative as in the older system.

In 1899 Charles Jasper Joly noted the hyperbolic quaternion and the non-associativity property while ascribing its origin to Oliver Heaviside.

The hyperbolic quaternions, as the Algebra of Physics, undercut the claim that ordinary quaternions made on physics. As for mathematics, the hyperbolic quaternion is another hypercomplex number, as such structures were called at the time. By the 1890s Richard Dedekind had introduced the ring concept into commutative algebra, and the vector space concept was being abstracted by Peano. 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. More generally, abstract algebra developed in the twentieth century with the sorting of mathematical structures as well as delving into their contents. Referring to the abstract structures as categories, the science of category theory describes the system of structures.