Editing Hyperbolic quaternion (section)

==Discussion==

The hyperbolic quaternions form a [[nonassociative ring]]; the failure of [[associativity]] in this algebra curtails the facility of this algebra in transformation theory. 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'' <sup>2</sup> = +1. The plane <math>D_r = \lbrace t + x r : t, x \in R \rbrace </math> with hyperbolic quaternion multiplication is a commutative and associative subalgebra isomorphic to the split-complex number plane.
The [[versor#Hyperbolic versor|hyperbolic versor]] <math>\exp(a r) = \cosh(a)  + r \sinh(a) </math> transforms D<sub>r</sub> by
:<math>\begin{align}
t + x r && \mapsto \quad & \exp(a r) (t + x r)\\
&&=\quad&  (\cosh(a) t + x \sinh(a)) + (\sinh(a) t + x \cosh(a)) r .
\end{align}</math>
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
:''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.''<ref name=Crowe>{{cite book |first=M.J. |last=Crowe |title=A History of Vector Analysis |publisher=University of Notre Dame |year=1967 |page=191 |title-link=A History of Vector Analysis }}</ref>