Introduction
Alexander was a mathematical adept and physics professor who emigrated from Scotland to Texas. Later he taught Electrical Engineering
at Lehigh University and gave an important series of lectures about personalities in Mathematics and Physics.
He commuted over the Atlantic to several International Congresses of Mathematicians.
In his day, he was a leading link in the international community of Physics and Mathematics. This biography highlights the "space analysis" techniques he developed that presaged the velocity geometry of modern spacetime theory.
Edinburgh and London
Macfarlane's first mathematical publication appeared in 1878 as follows:"Boole's symbolic logic was brought to my notice by Professor Tait, when I was a student in the physical laboratory of Edinburgh University. I studied the Laws of Thought and I found that those who had written on it regarded the method as highly mysterious; the result wonderful, but the processes obscure. I reduced everything to diagram and model, and I ventured to publish my views on the subject in a small volume called Principles of the Algebra of Logic;...".This revelation came 19 April 1901 while he was reflecting on George Boole at Lehigh University.
His association with the famous mathematican Arthur Cayley was recounted the next day: "of his kindliness to young investigators I can speak from personal experience. Several papers which I read before the Royal Society of Edinburgh on the Analysis of Relationships were referred to him, and he recommended their publication. Soon after I was invited by the Anthropological Society of London to address them on the subject, and while there, I attended a meeting of the Mathematical Society of London. The room was small and some twelve mathematicians were assembled round a table, among whom was Arthur Cayley, as became evident from the proceedings. At the close of the meeting Cayley gave me a cordial handshake and referred in the kindest terms to my papers which he had read."
But when Cayley died in 1895, Macfarlane, knowing the importance of algebraic motors, had the following to say about Cayley: "He regarded the complex number a + b i as the fundamental quantity of mathematical analysis, and considered that with such a basis, algebra was a complete and bounded science, in which no further imaginary symbols could spring up."
The New World
Let us consider the draw of America on the young professor: America was the land of Nathaniel Bowditch,
Benjamin Franklin, and Joseph Henry. Each of these luminaries had focused on natural philosophy early in life, contributing later through more sophisticated social involvements:
Though offered the presidency of Harvard, Bowditch worked and prospered as an actuary. Franklin's role in establishing
the American nation is well known and celebrated on the $100 dollar bill. Joseph Henry deftly steered the Smithsonian
Institution of Washington D.C. through its first decades, befriending several Presidents of the Republic.
Such was the pattern in Macfarlane's career: He was a studentteacher from a young age in Scotland; he taught at the University of Texas and at Lehigh University, and later lead
an international society of linear algebraists and participated in early International Congresses of Mathematicians, and in particular the primordial meeting in Chicago, 1893. The international society has a history written by Eduardo L. Ortiz, and linked from the references section below. It was called The International Society for the Study of Quaternions and Allied Systems of Mathematics
George Bruce Halsted
Alexander Macfarlane began correspondence with Halsted when Principles of the Algebra of Logic was published.
G.B. Halsted was teaching at the University of Texas in 1885 when Alexander came out to Texas to teach there. While Alexander served the University of Texas for nine years, Halsted gave 19 years without a pension. He took three more posts before laying down the chalk. While Halsted taught math and Macfarlane taught physics, together they projected science as a study in integrity. Halsted was a leader in axiomatic pragmatism for teaching, and passionate about Bolyai and Loabachevski contributions to modern thought in geometry.
After UT, in 1895, they were instrumental in bringing about the New York Mathematical Society, the American Mathematical Society, and the American Mathematical Monthly.
Helen Swearingen
In 1886 George Bruce Halsted wed Margaret Swearingen in Austin.
She was the daughter of Patrick Sweringen from one of the founding families of New Amsterdam (later renamed New York City}. Patrick Sweringen had another daughter Helen Martha, who became Alexander Macfarlane's wife in 1895. Thus the university colleagues were in fact brothersinlaw. The wedding was announced in the American Mathematical Monthly (2:135). Alexander and Helen had three sons: Alexander S., Robert H.K., and Henry S. It was Helen that arranged for Lectures on Ten British Mathematicians to be published in 1916 by John Wiley and Sons, after Alexander had passed.
George Washington Pierce
Macfarlane was the first physics teacher of George Washington Pierce who later became professor of physics at Harvard. G.W. Pierce is featured in the Dictionary of Scientific Biography, in the Biographical Memoirs of the National Academy of Sciences (v.33, 1959), and now online. As a teacher is known by his students, Macfarlane the educator is confirmed.
Lehigh University
Lehigh University was an outgrowth of the Society for the Promotion of Useful Knowledge and the benevolence of
Asa Packer. "An advanced course in Electricity was founded in 1884, and this was expanded in 1888 to meet the needs of the new profession of electrical engineers, and a regular course with an appropriate degree was established." (see Hyde reference) C.D. Bowen writes "[H. Wilson] Harding started the course in electrical engineering, first a one year course, then four years of electrical engineering and physics; then in 1887 established a full course  the course guided from '95 to '97 by the stong hand of Alexander MacFarlane M.A., D.Sc. LL.D." (see references).
In 1901 Lehigh University was the site of his memorable lecture series "Ten British Mathematicians of the Nineteenth Century". In the period 1902 1904 he continued to lecture on "Ten British Physicists of the Nineteenth Century" at Lehigh.
Bibliography
As secretary of the (international) Quaternion Society, MacFarlane was charged with editing a Bibliography of Quaternions and Allied Systems of Mathematics. He explains in the preface of this work that he benefited from help from the Library of the University of Michigan. The single column volume provided ample margins for annotation. It was published in Dublin, 1904. The inclusion of James Cockle's articles, then 50 years old, on tessarines and coquaternions, shows some of the scope of the phrase "allied systems of mathematics". He writes that "almost all" living authors had replied to requests to complete the bibliography, a positive note to buoy up the spirits of volunteers like MacFarlane working on a daunting project.
Coworkers
The acknowledgements in the bibliography provide us with a view of the peers of the man. His life path crossed those of Halsted and Pierce as mentioned above, but neither of these men had the taste for linear algebra and differential geometry, in relation to physics, that Macfarlane had. Of the six people he acknowledges in the bibliography, four are wellknown persons and two are harder to trace:
* Charles Jasper Joly (18641906) was the Astronomer Royal of Ireland from 1897. He is the author of ?A Manual of Quaternions? (1905), now available online through Cornell University.
* Samuel Dickstein (1851 ? 1939) of the Warsaw Scientific Society. (see Kuratowski, A HalfCentury of Polish Mathematics)
* Charles Gaston Combebiac, author of ?Les actions a distance? (1910).
* Victor Schlegel, a leading exponent of Grassmann's extension theory with publications beginning in 1872.
* (someone named) Grassman. (Herman Grassmann died in 1877.)
* (someone named) von Elfrinkhof.
(MacFarlane decided to omit the first names of his actual collaborators, so there is some obscurity.)
3D Hyperbolic Model and Protorelativity
Significantly, Alexander Macfarlane sowed the seed concepts of the theory of
relativity of space and time with his hyperbolic quaternion work since its structure foreshadowed Minkowski's space of 1908.
Before him, W.R. Hamilton had called algebra the "science of pure time". Hamilton's invention of the quaternion
number ring was a precursor to general linear algebra of ndimensional space. He also promoted the abstract "space of velocities" with his work
in Mechanics.
Alexander first widened the scope of linear algebra with his lecture "Principles of the Algebra of Physics".He was speaking, in August 1891, as secretary of the physics section of the American Association for the Advancement of Science. M.J. Crowe(see references) details an intense international dialogue of the period. One can confirm that his
ideas suggested a foundation for relativity by review of his "Hyperbolic Quaternions" paper (1900) in the
proceedings of the Royal Society at Edinburgh. This essay uses the hyperboloid
t^{2}  x^{2}  y^{2}  z^{2 }= 1 and a type
of quaternion algebra to discuss the trigonometry typical of 3D LobachevskiBolyai space.
Recall that Macfarlane had been at the University of Texas with George Bruce Halsted, explicator of Lobachevski, so Macfarlane had some time to consider his hyperboloid model. He also brought up the model in his biography of William Kingdon Clifford given at Lehigh on 23 April 1901: "there is another surface, complementary to the sphere, such that [the sum of] the angles of any triangle on it [is] less than two right angles. The complementary surface to which I refer is not the pseudosphere, but the equilateral hyperboloid. As the plane is the transition surface between the sphere and the equilateral hyperboloid, and a triangle on it is the transition triangle between the spherical triangle and the equilateral hyperboloidal triangle, the sum of the angles of the plane triangle must be exactly equal to two right angles."
We find vindication of Macfarlane's model as a velocity space in Emile Borel's 1913 contribution to Comptes
Rendues Acad. Sci. Paris. He says, "It is natural to call the space of velocity points the kinematic space. In classical kinematics, the kinematic space is Euclidean. The principle of relativity corresponds to the hypothesis that the kinematic space is a space of constant negative curvature, the space of Lobachevski and Bolyai."
In 1849, James Cockle noted a second alternative quaternion system, the coquaternions. Without citation, Macfarlane exhibited the vector profile of this second system in the 1900 meeting of the International Congress of Mathematicians at Paris. More than a century later, these "Paris quaternions" which have multiplicative associativity (a property missing in his "hyperbolic quaternions") turn out to also speak to relativity. The "space of velocities" has a peculiar "azimuth structure" here, breaking the isotropic presumption usually held in mechanics for velocity.
Space Analysis
W.K. Clifford died too early (age 34) to clarify all his insights; his Mathematical Papers were introduced by H.J.S. Smith who worked to make a clarification. He writes (p.xl): ...the unknown, or at least the unforeseen, seems to be excluded from geometry, because whatever may be found out hereafter must be latent in what is already known. But upon the view put forward by Riemann and adopted by Clifford, the essential properties of space have to be regarded as things still unknown, which we may one day hope to find out by closer observation and more patient reflection, and not as axioms to be accepted on the authority of universal experience, or of the inner consciousness.
Thus MacFarlane referred to his original geometric analysis as ?space analysis?. He knew some algebras had nonreal elements squaring to +1 (for example, jj = +1), and used these to form "hyperbolic versors": exp(a j) = cosh a + j sinh a. For a fixed j and variable a taken from a real line, these form a oneparameter transformation group as found in the theory of Sophus Lie. An example of MacFarlane?s confidence in his contribution to science shows when he compares his view of quaternion theory and the view of Felix Klein:
The existence of the ... expression r e^{b β}, and the application of these expressions to develop the trigonometry of surfaces of the second order show that his [Klein?s] theory of quaternions is inadequate, and that the sphere of applicability which he assigns them too narrow. According to his idea, quaternions will be in place when we wish to have a convenient algorithm for the combinations of rotations and dilations; the true idea is that the quaternions contain the elements of the algebra of space.
Here we see MacFarlane practicing a branch of differential geometry, trigonometry on surfaces of the second order.
Likewise in Paris, 1900, his title to the International Congress of Mathematicians was "Applications of Space Analysis to Curvilinear Coordinates", highlighting the utility of the tool he advances.
The phrase "space analysis" is an extension of "complex analysis", the study of functions of a complex variable, especially the differentiable (analytic) functions. Through study of functions of a quaternion variable there is suitable basis for space analysis, Macfarlane's term.
