Editing Hyperbolic quaternion (section)

==Historical review==<!-- This section is linked from [[Quasigroup]] -->
The 1890s felt the influence of the posthumous publications of [[William Kingdon Clifford|W. K. Clifford]] and the ''continuous groups'' of [[Sophus Lie]]. An example of a [[one-parameter group]] is the [[versor#Hyperbolic versor|hyperbolic versor]] with the [[hyperbolic angle]] parameter. This parameter is part of the [[polar decomposition#Alternative planar decompositions|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 <math>\{1,\,i,\,j,\,k\}</math> of the vector space of hyperbolic quaternions is not [[closure (mathematics)|closed]] under multiplication: for example, <math>ji=-\!k</math>. Nevertheless, the set <math>\{1,\,i,\,j,\,k,\,-\!1,\,-\!i,\,-\!j,\,-\!k\}</math> 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 <math>i</math>,<math>j</math>, and <math>k</math> to be <math>+1</math> instead of <math>-1</math> :
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''<ref>{{harvnb|Heaviside|1892|pp=427–430}}</ref> 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<ref>{{cite journal |first=C.G. |last=Knott |title=Recent Innovations in Vector Theory |journal=Nature |volume=47 |issue=1225 |pages=590–3 |year=1893 |doi=10.1038/047590b0 |bibcode=1893Natur..47R.590. |doi-access=free }} read before the [[Royal Society of Edinburgh]] 19 December 1892 and published in ''Proceedings''</ref> 1892)
:If a 4-algebra on basis <math>\{1,\,i,\,j,\,k\}</math> is associative and off-diagonal products are given by Hamilton's rules, then <math>i^2=-\!1=j^2=k^2</math>.

'''Proof:'''
:<math>j = ki = (-ji)i = -j(ii)</math>, so <math>i^2 = -1</math>. Cycle the letters <math>i</math>, <math>j</math>, <math>k</math> to obtain <math>i^2=-1=j^2=k^2</math>. ''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 (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.''<ref name=Crowe/>

In 1899 [[Charles Jasper Joly]] noted the hyperbolic quaternion and the non-associativity property<ref>{{cite book |last=Hamilton |title=Elements of Quaternions |year=1899 |page=163 |url= https://archive.org/details/elementsquatern01hamigoog/page/n200 |edition=2nd |editor-first=C.J. |editor-last=Joly |place=London |publisher=Longmans, Green, and Co. }}</ref> 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 (mathematics)|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 structure]]s describing hyperbolic quaternions.