Alexander Macfarlane stylized his work as "Space Analysis". In 1894 he published his five earlier papers and a book review of
Alexander McAulay's
Utility of Quaternions in Physics. Page numbers are carried from previous publications, and the reader is presumed familiar with quaternions. The first paper is "Principles of the Algebra of Physics" where he first proposes the
hyperbolic quaternion algebra, since "a student of physics finds a difficulty in principle of quaternions which makes the square of a vector negative." The second paper is "The Imaginary of the Algebra". Similar to
Homersham Cox (1882/83), Macfarlane uses the
hyperbolic versor as the hyperbolic quaternion corresponding to the
versor of Hamilton. The presentation is encumbered by the notation :h \alpha ^ A = \cosh A + \sinh A \ \alpha ^{\pi/2}. Later he conformed to the notation exp(A α) used by Euler and Sophus Lie. The expression \alpha ^{\pi/2} is meant to emphasize that α is a
right versor, where π/2 is the measure of a
right angle in
radians. The π/2 in the exponent is, in fact, superfluous. Paper three is "Fundamental Theorems of Analysis Generalized for Space". At the 1893 mathematical congress Macfarlane read his paper "On the definition of the trigonometric functions" where he proposed that the
radian be defined as a ratio of areas rather than of lengths: "the true analytical argument for the circular ratios is not the ratio of the arc to the radius, but the ratio of twice the area of a sector to the square on the radius." The paper was withdrawn from the published proceedings of mathematical congress (acknowledged at page 167), and privately published in his
Papers on Space Analysis (1894). Macfarlane reached this idea or ratios of areas while considering the basis for
hyperbolic angle which is analogously defined. The fifth paper is "Elliptic and Hyperbolic Analysis" which considers the
spherical law of cosines as the fundamental theorem of the
sphere, and proceeds to analogues for the ellipsoid of revolution, general
ellipsoid, and equilateral
hyperboloids of one and two sheets, where he provides the
hyperbolic law of cosines. In 1900 Alexander published "Hyperbolic Quaternions" with the Royal Society in Edinburgh, and included a sheet of nine figures, two of which display conjugate
hyperbolas. Having been stung in the
Great Vector Debate over the non-associativity of his Algebra of Physics, he restored associativity by reverting to
biquaternions, an algebra used by students of Hamilton since 1853. ==Works==