Editing Alexander Macfarlane

{{Short description|Scottish physicist and mathematician (1851–1913)}}
{{For|other persons of the same name|Alexander Macfarlane (disambiguation)}}
{{Use dmy dates|date=October 2020}}
{{Infobox scientist
| name              = Alexander Macfarlane
| image             = Macfarlane_Alexander_math.jpg
| image_size        =
| caption           = 
| birth_date        = {{birth date text|21 April 1851}}
| birth_place       = [[Blairgowrie and Rattray|Blairgowrie]], Scotland
| death_date        = {{death-date and age|28 August 1913|21 April 1851}}
| death_place       = [[Chatham, Ontario]], Canada
| field             = [[Logic]]<br />[[Physics]]<br />Mathematics
| work_institutions = [[University of Texas]]<br />[[Lehigh University]]
| alma_mater        = [[University of Edinburgh]]
| doctoral_advisor  = [[Peter Guthrie Tait]]
| doctoral_students = 
| known_for         = Scientific biographies<br />Algebra of Physics
| spouse            = Helen Swearingen
| prizes            = 
| footnotes         = 
| signature         = 
}}

'''Alexander Macfarlane''' [[Fellow of the Royal Society of Edinburgh|FRSE]] LLD (21 April 1851&nbsp;– 28 August 1913) was a Scottish logician, physicist, and mathematician.

==Life==

Macfarlane was born in [[Blairgowrie and Rattray|Blairgowrie]], Scotland, to Daniel MacFarlane (Shoemaker, Blairgowrie) and Ann Small. He studied at the [[University of Edinburgh]].  His [[doctoral thesis]] "The disruptive discharge of electricity"<ref>A Marfarlane (1878) [http://www.nature.com/nature/journal/v19/n478/pdf/019184a0.pdf "The disruptive discharge of electricity"] from ''[[Nature (journal)|Nature]]'' 19:184,5</ref> reported on experimental results from the laboratory of [[Peter Guthrie Tait]].

In 1878 Macfarlane spoke at the [[Royal Society of Edinburgh]] on [[algebraic logic]] as introduced by [[George Boole]]. He 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]].<ref>{{cite book|title=Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002|date=July 2006|publisher=[[The Royal Society of Edinburgh]]|isbn=0-902-198-84-X|url=https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp2.pdf|access-date=25 June 2017|archive-url=https://web.archive.org/web/20160304074135/https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp2.pdf|archive-date=4 March 2016|url-status=dead}}</ref> The next year he published ''Principles of the Algebra of Logic'' which interpreted Boolean variable expressions with algebraic manipulation.<ref>Stanley Burris (2015), "[http://plato.stanford.edu/entries/algebra-logic-tradition/ The Algebra of Logic Tradition]", Stanford Encyclopedia of Philosophy</ref>

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]]. In 1896 Macfarlane encouraged the association of [[quaternion]] students to promote the algebra.<ref>A. Macfarlane (1896) [https://www.jstor.org/stable/1624707 Quaternions] [[Science (journal)|Science]] (2) 3:99–100, link from [[Jstor]] early content</ref> 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,<ref>''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}}</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.<ref>{{Cite book |url=https://books.google.com/books?id=RxPiAAAAMAAJ&pg=PA50 |title=The Michigan Alumnus, Volume 22 |publisher=[[University of Michigan Library]] |page=50 |year=1916 |access-date=2020-04-02 |via=Google Books}}</ref>

==Space analysis==
Alexander Macfarlane stylized his work as "Space Analysis". In 1894 he published his five earlier papers<ref>A. Macfarlane (1894) ''[https://archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from [[archive.org]]</ref> 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 (mathematician)|Homersham Cox]] (1882/83),<ref>{{Cite journal|author=Cox, H.|year=1883|orig-year=1882|title=On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space|journal=Trans. Camb. Philos. Soc.|volume=13|pages=69–143|url=https://archive.org/details/transactions13camb}}</ref><ref>{{Cite journal|author=Cox, H.|year=1883|orig-year=1882|title=On the Application of Quaternions and Grassmann's Ausdehnungslehre to different kinds of Uniform Space|journal=Proc. Camb. Philos. Soc.|volume=4|pages=194–196|url=https://archive.org/details/proceedingsofcam4188083camb}}</ref> Macfarlane uses 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.

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."<ref>A. Macfarlane (1893) "On the definitions of the trigonometric functions", page 9, [https://archive.org/details/principlesalgeb01macfgoog/page/n138/mode/2up link at Internet Archive]</ref> 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.<ref>{{wikibooks inline|Geometry/Unified Angles}}</ref>

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>A. Macfarlane (1900) "Hyperbolic Quaternions" ''Proceedings of the Royal Society at Edinburgh'', vol. 23, November 1899 to July 1901 sessions, pp.&nbsp;169&ndash;180+figures plate. Online at [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8376797&fulltextType=RA&fileId=S0370164600010385 Cambridge Journals] (paid access), [https://archive.org/details/proceedingsroya37edingoog Internet Archive] (free), or [https://books.google.com/books?id=-DhrXa1ZX38C&oe=UTF-8 Google Books] (free). (Note: P. 177 and figures plate incompletely scanned in free versions.)</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==
* 1879: [https://archive.org/details/principlesalgeb03macfgoog Principles of the Algebra of Logic] from [[Internet Archive]].
* 1885: [https://archive.org/details/117770339 Physical Arithmetic] from Internet Archive.
* 1887: [https://archive.org/details/jstor-1967561 The Logical Form of Geometrical Theorems] from [[Annals of Mathematics]] 3: 154,5.
* 1894: [https://archive.org/details/cu31924001506769 Papers on Space Analysis].
* 1898: [https://archive.org/details/jstor-1626808 Book Review: “La Mathematique; philosophie et enseignement” by C.A. Laissant] in [[Science (journal)|Science]] 8: 51–3.
* 1899 [https://archive.org/details/jstor-1638030 The Pythagorean Theorem] from ''Science'' 34: 181,2.
* 1899: [https://archive.org/details/jstor-1627482 The Fundamental Principles of Algebra] from ''Science'' 10: 345–364.
* 1906: [https://archive.org/details/vectoranalysisan13609gut Vector Analysis and Quaternions].
* 1910: [http://babel.hathitrust.org/cgi/pt?id=pst.000052363498;view=1up;seq=337 Unification and Development of the Principles of the Algebra of Space] from [[Quaternion Society#Bulletin|Bulletin of the Quaternion Society]].
* 1911: [https://archive.org/details/jstor-1638016 Book Review: ''Life and Scientific Work of P.G. Tait'' by C.G. Knott] from ''Science'' 34: 565,6.
* 1912: [http://babel.hathitrust.org/cgi/pt?id=pst.000052363498;view=1up;seq=426 A System of Notation for Vector-Analysis; with a Discussion of the Underlying Principles] from ''Bulletin of the Quaternion Society''.
* 1913: [https://archive.org/details/proceedingsfifth00hobs/page/266/mode/2up On Vector-Analysis as Generalized Algebra], address to 5th [[International Congress of Mathematicians]], Cambridge, via Internet Archive
* {{cite book | last = Macfarlane | first = Alexander | title = Lectures on Ten British Mathematicians of the Nineteenth Century | series = Mathematical monographs, no. 17 | 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=https://www.ams.org/journals/bull/1917-23-04/S0002-9904-1917-02913-8/|doi=10.1090/s0002-9904-1917-02913-8|doi-access=free}}</ref><ref>[[G. B. Mathews]] (1917) [http://www.nature.com/nature/journal/v99/n2481/pdf/099221a0.pdf Review:''Ten British Mathematicians''] from ''Nature'' 99:221,2 (#2481)</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 = https://archive.org/details/lecturesontenbri00macfrich}}<ref>N.R.C. (1920) [http://www.nature.com/nature/journal/v104/n2622/pdf/104561a0.pdf Review:''Ten British Physicists''] from ''Nature'' 104:561,2 (#2622)</ref>
* [http://babel.hathitrust.org/cgi/pt?id=pst.000052363498;view=1up;seq=481 Publications of Alexander Macfarlane] from ''Bulletin of the Quaternion Society'', 1913

==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 | issue = 1 | pages = 1–4 | doi = 10.2307/2971573 | jstor = 2971573 }}
* Robert de Boer (2009) [https://web.archive.org/web/20091026233425/http://ca.geocities.com/macfarlanebio/ Biography of Alexander Macfarlane] from [[WebCite]].
* Robert de Boer (2009) [https://web.archive.org/web/20091027012441/http://ca.geocities.com/macfarlanebio/Chicago1893.html Alexander Macfarlane in Chicago, 1893] from WebCite
* [http://www.electricscotland.com/history/canada/kent/macfarlane_alexander.htm Electric Scotland historical biography]
* [[Cargill Gilston Knott|Knott, Cargill Gilston]] (1913) [http://www.nature.com/nature/journal/v92/n2291/pdf/092103a0.pdf Alexander Macfarlane], [[Nature (journal)|Nature]].
* [http://www.lib.utexas.edu/taro/utcah/00515/cah-00515.html Macfarlane papers at the University of Texas]

==External links==
*{{Commons category-inline}}
*{{wikisource author-inline}}
* {{gutenberg author|id=4911|name=Alexander Macfarlane}}
* {{Internet Archive author |sname=Alexander Macfarlane}}
* {{Librivox author |id=14893}}

{{Authority control}}

{{DEFAULTSORT:Macfarlane, Alexander}}
[[Category:1851 births]]
[[Category:1913 deaths]]
[[Category:People from Blairgowrie and Rattray]]
[[Category:Scottish logicians]]
[[Category:19th-century Scottish philosophers]]
[[Category:20th-century Scottish philosophers]]
[[Category:Scottish physicists]]
[[Category:19th-century Scottish mathematicians]]
[[Category:20th-century Scottish 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:Relativity theorists]]
[[Category:Scottish expatriates in the United States]]
[[Category:Scottish emigrants to Canada]]
[[Category:University of Texas at Austin faculty]]
[[Category:British geometers]]