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Alexander Macfarlane
(1851 - 1913)

Alexander MacFarlane


Born: 21 April 1851, in Blairgowrie, Scotland
University of Edinburgh: 1871 - 1884
Awarded D.Sc.: 1878
University of Texas, Austin: 1885 - 1894
Awarded Honorary LL.D. (U. Michigan): 1887
Chicago ICM 1893
Weds Helen Swearingen 8 April 1895
Lehigh University, Pennsylvania: 1895 - 1897
Chatham, Ontario: 1898 - 28 August 1913
Paris ICM 1900
Rome ICM 1908
Cambridge ICM 1912

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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 student-teacher 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 brothers-in-law. 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 on-line. 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.


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.


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 well-known persons and two are harder to trace:
* Charles Jasper Joly (1864-1906) was the Astronomer Royal of Ireland from 1897. He is the author of ?A Manual of Quaternions? (1905), now available on-line through Cornell University.
* Samuel Dickstein (1851 ? 1939) of the Warsaw Scientific Society. (see Kuratowski, A Half-Century 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 Proto-relativity

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 n-dimensional 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

t2 - x2 - y2 - z2 = 1
and a type of quaternion algebra to discuss the trigonometry typical of 3D Lobachevski-Bolyai 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 non-real 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 one-parameter 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 eb 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.

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Contextual Reading

One can glean an impression of the age and context in which Macfarlane worked by reading from relevant books:

The precociousness of Alexander Macfarlane’s contributions to mathematical electromagnetism can be seen by contrast with Charles Steinmetz of General Electric. Macfarlane and Steinmetz crossed paths in 1893 at the same session of the International Electrical Congress (see RR Kline, p. 70). In Steinmetz’ case, his paper "Complex Quantities and their use in Electrical Engineering" opened the door to applied science: "[the complex number] method became associated with Steinmetz because he developed it more fully than his predecessors and spent the better part of his career applying it to the entire range of AC circuits and machines" (Kline p. 77). In contrast, Macfarlane’s application involved hyperbolic angles and functions, a topic more deftly explored by Arthur Kennelly two decades later. Since Steinmetz was born in 1865, midway between the births of Macfarlane and Einstein, he represents an intermediary generation of electrical philosophers. As the biographer Kline explains, the rapidly-growing marketplace for electrical arts, and the support of General Electric, made Steinmetz famous as the "Wizard of Schenectady". Further, in 1922 he contributed the book Relativity and Space to the growing literature on spacetime.

The Maxwellians, by Bruce J. Hunt, Cornell University Press, 1991, recounts the mathematical physics context that ultimately lead to a successful radio technology. It includes references to Oliver Heaviside's contributions, which anticipated the need to use algebraic motors.

Joseph Henry: His Life and Work, by Thomas Coulson, Princeton University Press, 1950, details America's contender in the electromagnet contests inspired by Michael Faraday's magnets.

While these texts serve to illustrate the human and social context of electric science in the time of MacFarlane, they are recommended here for that purpose only. Mathematical physics anticipated by Macfarlane has only been assimilated in modern spacetime study.

A contrasting picture of another mathematical immigrant can be read in "Charlotte Angas Scott (1858 - 1931)" by Patricia Clark Kenschaft and found in A Century of Mathematics in America, Part III, pp.241-252. When Bryn Mawr College of Pennsylvania opened in 1885, she became its first department head in mathematics. Kenschaft says, "American mathematicians joked about her leaving for Europe every spring as soon as exams were marked, but this was not literally true. Still, she crossed the Atlantic Ocean often, at a time when each voyage involved at least a week of discomfort and danger."

One should not get the impression that the flow of professors of natural philosophy (physics) was uniformly from Europe to North America. One can note the contrary direction evidenced by James Gordon MacGregor ( 1852 - 1913), a perfect contemporary of Macfarlane. MacGregor was born in Halifax and took his first diploma from Dalhousie in Halifax. Before gaining his doctorate from the University of London in 1876, he studied with P. G. Tait in Edinburgh and Wiedeman at Leipzig. Dalhousie then invited him to take up the newly endowed ?Munro Professor of Physics? chair in the Nova Scotia capital, which he did until 1901 when he returned to Europe. The call came from Edinburgh for MacGregor of Nova Scotia. The terms were attractive and the honour immense, so the Maritimer went to Scotland to lead young students to experiment as the way to knowledge. See The Lives of Dalhousie University (1994) by P.B. Waite (pp.188-90).

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Lectures given at International Congresses of Mathematicians

Chicago 1893: He spoke on the definitions of trigonometric functions and on elliptic and hyperbolic analysis.
Paris 1900: "Space Analysis Applied to Curvilinear Coordinates".
Rome 1908: "On the Square of Hamilton's Delta".
Cambridge 1912: "Vector Analysis as Generalized Algebra".

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Original and Secondary Sources

  1. Michael J. Crowe (1967) History of Vector Analysis includes the dramatic story of how vector analysis was separated from its origins in the abstract algebra of quaternions (see Chapter 6). Macfarlane's contribution was through reaction to his proposal to take the square of unit vectors to be +1 and carry on in four dimensions.
  2. P.R. Girard (1984) "The Quaternion Group and Modern Physics" European Journal of Physics 5:25-32. Girard wrote a Ph.D. thesis "Conceptual development of Einstein's General Theory of Relativity";the EJP paper credits Macfarlane's bibliography and Bulletin of the International Quaternion Society for early quaternion references.
  3. G. Bruce Halsted: American Mathematical Monthly (1894) pp. 286-9. ?The Algebra of Physics, presented by Alexander MacFarlane to Brooklyn audience, this original algebra being a contender with Hamilton?s quaternions and with Grassmann?s extension theory."
  4. A.Macfarlane?s obituary of A. Cayley :American Mathematical Monthly 2; 99-106, has mixed admiration for the fallen (1895).
  5. Macfarlane(1892) "On the Imaginary of Algebra", Proceedings of the American Association for the Advancement of Science, v. XLI pp.33-55.The paper develops the expressions r eb β used in his method of space analysis.
  6. Founder of Edinburgh Maths Society Alexander Macfarlane was one of the 55 founding members of the EMS in February 1883.
  7. Macfarlane(1899) "A Treatise on Universal Algebra" by Alfred North Whitehead (book review), Science 9:324-8, cited in Whitehead's Early Philosophy of Mathematics by Granville C. Henry and Robert J. Valenza. For book review comment see Utility of Universal Algebra.
  8. Nature, 25 Sept 1913. Macfarlane?s death and life recounted by C.G. Knott.
  9. Ethel Ward-McLemor lists Alexander Macfarlane, engineer, as one of 14 founders of the Texas Academy of Science (1892-1913) .
  10. "Hyperbolic Quaternions" Proc. Royal Society of Edinburgh 23:169-181. This construct bears comparison with W.G. Reynolds (1993) American Mathematical Monthly 100:442.
  11. Eduardo L. Ortiz Remarks on the International Association for Promoting the Calculus of Quaternions
  12. Electric Scotland biographical sketch with genealogy.
  13. Giuseppe Peano, "Saggio di calcolo geometrico" (1896) cites Macfarlane as "seeking to simplify the theory of quaternions". See Hubert C. Kennedy (1973), Selected Works of Giuseppe Peano, p.170.
  14. Martin Gardiner, Logic Machines and Diagrams (1982) cites Macfarlane's "Logical Spectrum" as a contribution to the evolution of logical diagrams (see pp.44,5).
  15. Smith, J.D.H. and Anna B. Romanowska: Post Modern Algebra (1999) Wiley - Interscience. Exploits the algebraic category of quasigroups, a corner of algebra into which the loop {1, i, j, k} in hyperbolic quaternions falls.
  16. Edmund M. Hyde (1896) The Lehigh University - A Historical Sketch pp. 17,18.
  17. Catherine Drinker Bowen (1924) A History of Lehigh University p.34.

  18. Ronald R. Kline (1992) Steinmetz: Engineer and Socialist, John Hopkins University Press.

    These next four Monographs can be found at the
    Cornell University Historical Mathematical Digital Library
  19. A. Macfarlane (1916) "Lectures on Ten British Mathematicians": Marvelous revelations of people and ideas, delivered in 1901 at Lehigh University, Bethlehem, Pennsylvania.
  20. A. Macfarlane (1906) "Vector Analysis and Quaternions".
  21. A. Macfarlane (1904) "Bibliography of Quaternions and Allied Systems of Mathematics"
  22. E. Borel (1914) "Introduction Geometrique a Quelques Theories Physiques"
    The Physicists Lectures are available from
  23. A. Macfarlane (1919) Lectures on Ten British Physicists of the Nineteenth Century.

    First posted: 2002 October 16
    Last modified: 2009 October 17