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{{For|other persons of the same name|Alexander Macfarlane (disambiguation)}}
{{Infobox scientist
|name = Alexander Macfarlane
|image = Macfarlane_Alexander_math.jpg
|image_size = 300px
|caption = Alexander Macfarlane (1851 — 1913)
|birth_date = {{birth-date| April 21, 1851}}
|birth_place = [[Blairgowrie and Rattray|Blairgowrie]], [[Scotland]]
|death_date = {{death-date|August 28, 1913 }}
|death_place = [[Chatham, Ontario]], [[Canada]]
|residence = Chatham, Ontario
|citizenship =
|nationality = [[Scotland|Scottish]]
|ethnicity =
|field = [[Logic]] [[Physics]] [[Mathematics]]
|work_institutions = [[University of Texas]] [[Lehigh University]]
|alma_mater = [[University of Edinburgh]]
|doctoral_advisor = [[Peter Guthrie Tait]]
|doctoral_students =
|known_for = Scientific biographies Algebra of Physics
|author_abbrev_bot =
|author_abbrev_zoo =
|influences = [[William Rowan Hamilton]] [[William Kingdon Clifford]] [[Arthur Cayley]]
|influenced = [[G. W. Pierce]] [[Quaternion Society]]
|prizes =
|religion =
|footnotes =
|signature =
}}

'''Alexander Macfarlane''' [[Fellow of the Royal Society of Edinburgh|FRSE]] (April 21, 1851 – August 28, 1913) was a [[Scotland|Scottish]] [[logic]]ian, [[physicist]], and [[mathematician]].

Macfarlane was born in [[Blairgowrie and Rattray|Blairgowrie]], [[Scotland]] and studied at the [[University of Edinburgh]]. His [[doctoral thesis]], on the conditions governing the [[electric spark]], was subsequently published in the ''[[Transactions of the Royal Society of Edinburgh]]''. It brought him to the notice of [[James Clerk Maxwell]], and in 1878 Macfarlane was elected a [[Fellow of the Royal Society of Edinburgh]]

During his life, Macfarlane played a prominent role in research and education. He taught at the universities of Edinburgh and [[University of St Andrews|St Andrews]], was physics [[professor]] at the [[University of Texas at Austin|University of Texas]] (1885 – 1894),<ref>See the [http://www.lib.utexas.edu/taro/utcah/00515/cah-00515.html Macfarlane papers at the University of Texas].</ref> professor of Advanced Electricity, and later of [[mathematical physics]], at [[Lehigh University]]. MacFarlane was the secretary of the [[Quaternion Society]] and compiler of its publications.

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), and he compiled a [[bibliography]] on [[quaternion]]s in 1904.
Macfarlane was caught up in the revolution in [[geometry]] during his lifetime,<ref>''1830–1930: A Century of Geometry'', L Boi, D. Flament, JM Salanskis editors, Lecture Notes in Physics #402, Springer-Verlag ISBN 3-540-55408-4</ref> 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.<ref>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.</ref>

Macfarlane actively participated in several [[International Congress of Mathematicians|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 MacAulay]]’s ''Utility of Quaternions in Physics''. This collection is now available on-line.<ref>A. Macfarlane (1894) ''[http://www.archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from [[archive.org]]</ref>
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". Here he introduces the [[versor#Hyperbolic versor|hyperbolic versor]] as the hyperbolic quaternion corresponding to the [[versor]] of Hamilton. The presentation is encumbered by the notation
:<math>h \alpha ^ A = \cosh A + \sinh A \ \alpha ^{\pi/2}.</math>
Later he conformed to the notation exp(A α) used by Euler and Sophus Lie. The expression <math>\alpha ^{\pi/2}</math> is meant to emphasize that α is a ''right versor'', where π/2 is the measure of a [[right angle]] in [[radian]]s. 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 function]]s. 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 [[hyperboloid]]s of one and two sheets, where he provides the [[hyperbolic law of cosines]].

In 1900 Alexander published "Hyperbolic Quaternions"<ref>Macfarlane (1900) "Hyperbolic Quaternions" ''Proceedings of the Royal Society at Edinburgh'', 1899-1900 session, pp. 169–181</ref> with the Royal Society in Edinburgh, and included a sheet of nine figures, two of which display conjugate [[hyperbola]]s. Having been stung in the ''Great Vector Debate'' over the non-associativity of his Algebra of Physics, he restored associativity by reverting to [[biquaternion]]s, an algebra used by students of Hamilton since 1853.

==Works==
* {{cite book | last = Macfarlane | first = Alexander | title = Lectures on Ten British Mathematicians of the Nineteenth Century | publisher = John Wiley and Sons | location = New York | year = 1916 | url = http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=05070001}}<ref>{{cite journal|author=Mason, Thomas E.|title=Review: Alexander Macfarlane, ''Ten British Mathematicians''|journal=Bull. Amer. Math. Soc.|year=1917|volume=23|issue=4|pages=191–192|url=http://www.ams.org/journals/bull/1917-23-04/S0002-9904-1917-02913-8/}}</ref>
* {{cite book | last = Macfarlane | first = Alexander | title = Lectures on Ten British Physicists of the Nineteenth Century | publisher = John Wiley and Sons | location = New York | year = 1919 | url = http://www.archive.org/details/lecturesontenbri00macfrich}}
* {{gutenberg author|id=Alexander_MacFarlane|name=Alexander Macfarlane}}

==Notes and references==
{{reflist}}

* {{cite journal | author = Colaw, J. M. | title = Alexander Macfarlane, M.A., D. Sc., LL.D | journal = The American Mathematical Monthly | year = 1895 | volume = 2 | pages = 1–4 | doi = 10.2307/2971573 }}
* Robert de Boer (2009) [http://www.webcitation.org/5kmWeo7j0 Biography of Alexander Macfarlane] from [[WebCite]].
* [http://www.electricscotland.com/history/canada/kent/macfarlane_alexander.htm Electric Scotland historical biography]
* [[Cargill Gilston Knott|Knott, Cargill Gilston]], ''Alexander Macfarlane'' (obituary), [[Nature (journal)|Nature]], 25 September 1913.
* [http://www.lib.utexas.edu/taro/utcah/00515/cah-00515.html Macfarlane papers at the University of Texas]

{{Authority control|VIAF=50005071}}

{{Persondata 
| NAME = Macfarlane, Alexander
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = British mathematician and physicist
| DATE OF BIRTH = April 21, 1851
| PLACE OF BIRTH = [[Blairgowrie and Rattray|Blairgowrie]], [[Scotland]]
| DATE OF DEATH = August 28, 1913
| PLACE OF DEATH = [[Chatham, Ontario]], [[Canada]]
}}
{{DEFAULTSORT:Macfarlane, Alexander}}
[[Category:1851 births]]
[[Category:1913 deaths]]
[[Category:People from Blairgowrie and Rattray]]
[[Category:Scottish mathematicians]]
[[Category:Scottish logicians]]
[[Category:Scottish physicists]]
[[Category:19th-century British mathematicians]]
[[Category:19th-century Scottish people]]
[[Category:20th-century mathematicians]]
[[Category:Academics of the University of Edinburgh]]
[[Category:Academics of the University of St Andrews]]
[[Category:Alumni of the University of Edinburgh]]
[[Category:Fellows of the Royal Society of Edinburgh]]
[[Category:Lehigh University faculty]]
[[Category:People from Chatham-Kent]]
[[Category:Relativists]]
[[Category:Scottish expatriates in the United States]]
[[Category:Scottish emigrants to Canada]]
[[Category:University of Texas at Austin faculty]]
[[Category:ICM 1900 Plenary and Invited Speakers]]
[[Category:Geometers]]
[[Category:Scottish philosophers]]

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en>Mild Bill Hiccup: Cleaned up using AutoEd

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