# Alexander Macfarlane

Alexander Macfarlane
Alexander Macfarlane (1851 – 1913)
Born21 April 1851
Blairgowrie, Scotland
Died28 August 1913 (aged 62)
ResidenceChatham, Ontario
NationalityScottish
Alma materUniversity of Edinburgh
Known forScientific biographies
Algebra of Physics
Scientific career
FieldsLogic
Physics
Mathematics
InstitutionsUniversity of Texas
Lehigh University
InfluencesWilliam Rowan Hamilton
William Kingdon Clifford
Arthur Cayley
InfluencedG. W. Pierce
Quaternion Society

Prof Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.

## Life

Macfarlane was born in Blairgowrie, Scotland and studied at the University of Edinburgh. His doctoral thesis "The disruptive discharge of electricity"[1] reported on experimental results from the laboratory of Peter Guthrie Tait.

In 1878 Macfarlane was elected a Fellow of the Royal Society of Edinburgh. His proposers were Peter Guthrie Tait, Philip Kelland, Alexander Crum Brown, and John Hutton Balfour.[2]

During his life, Macfarlane played a prominent role in research and education. He taught at the universities of Edinburgh and St Andrews, was physics professor at the University of Texas (1885 – 1894),[3] professor of Advanced Electricity, and later of mathematical physics, at Lehigh University. In 1896 Macfarlane encouraged the association of quaternion students to promote the algebra.[4] He became the Secretary of the Quaternion Society, and in 1909 its President. He edited the Bibliography of Quaternions that the Society published in 1904.

Macfarlane was also the author of a popular 1916 collection of mathematical biographies (Ten British Mathematicians), a similar work on physicists (Lectures on Ten British Physicists of the Nineteenth Century, 1919). Macfarlane was caught up in the revolution in geometry during his lifetime,[5] in particular through the influence of G. B. Halsted who was mathematics professor at the University of Texas. Macfarlane originated an Algebra of Physics, which was his adaptation of quaternions to physical science. His first publication on Space Analysis preceded the presentation of Minkowski Space by seventeen years.[6]

Macfarlane actively participated in several International Congresses of Mathematicians including the primordial meeting in Chicago, 1893, and the Paris meeting of 1900 where he spoke on "Application of space analysis to curvilinear coordinates".

Macfarlane retired to Chatham, Ontario, where he died in 1913.

## Space analysis

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. This collection is now available on-line.[7] 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),[8][9] Macfarlane uses the hyperbolic versor as the hyperbolic quaternion corresponding to the versor of Hamilton. The presentation is encumbered by the notation

${\displaystyle h\alpha ^{A}=\cosh A+\sinh A\ \alpha ^{\pi /2}.}$

Later he conformed to the notation exp(A α) used by Euler and Sophus Lie. The expression ${\displaystyle \alpha ^{\pi /2}}$ 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.

Papers three and four are "Fundamental Theorems of Analysis Generalized for Space" and "On the definition of the Trigonometric Functions", which he had presented the previous year in Chicago at the Congress of Mathematicians held in connection with the World's Columbian Exhibition. He follows George Salmon in exhibiting the hyperbolic angle, argument of hyperbolic functions. 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"[10] 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.

## Notes and references

1. ^ A Marfarlane (1878) The disruptive discharge of electricity from Nature 19:184,5
2. ^
3. ^
4. ^ A. Macfarlane (1896) Quaternions Science (2) 3:99–100, link from Jstor early content
5. ^ 1830–1930: A Century of Geometry, L Boi, D. Flament, JM Salanskis editors, Lecture Notes in Physics No. 402, Springer-Verlag ISBN 3-540-55408-4
6. ^ A. Macfarlane (1891) "Principles of the Algebra of Physics", Proceedings of the American Association for the Advancement of Science 40:65–117. It was 1908 when Hermann Minkowski proposed his spacetime.
7. ^ A. Macfarlane (1894) Papers on Space Analysis, B. Westerman, New York, weblink from archive.org
8. ^ Cox, H. (1883) [1882]. "On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space". Trans. Camb. Phil. Soc. 13: 69–143.
9. ^ Cox, H. (1883) [1882]. "On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space". Proc. Camb. Phil. Soc. 4: 194–196.
10. ^ A. Macfarlane (1900) "Hyperbolic Quaternions" Proceedings of the Royal Society at Edinburgh, vol. 23, Nov. 1899 to July 1901 sessions, pp. 169–180+figures plate. Online at Cambridge Journals (paid access), Internet Archive (free), or Google Books (free). (Note: P. 177 and figures plate incompletely scanned in free versions.)
11. ^ Mason, Thomas E. (1917). "Review: Alexander Macfarlane, Ten British Mathematicians". Bull. Amer. Math. Soc. 23 (4): 191–192. doi:10.1090/s0002-9904-1917-02913-8.
12. ^ G. B. Mathews (1917) Review:Ten British Mathematicians from Nature 99:221,2 (#2481)
13. ^ N.R.C. (1920) Review:Ten British Physicists from Nature 104:561,2 (#2622)