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 1892) 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 Giuseppe 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. 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 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