Alexander Macfarlane

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

Life

Macfarlane was born in 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" 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. The next year he published ''Principles of the Algebra of Logic'' which interpreted Boolean variable expressions with algebraic manipulation.

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), 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. 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, 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.

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''. 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), Macfarlane uses the hyperbolic versor as the hyperbolic quaternion corresponding to the versor of Hamilton. The presentation is encumbered by the notation
:$h \alpha ^ A = \cosh A + \sinh A \ \alpha ^.$
Later he conformed to the notation exp(A α) used by Euler and Sophus Lie. The expression $\alpha ^$ 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.

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." 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.

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" 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.

Works

* 1879
Principles of the Algebra of Logic
from Internet Archive.
* 1885
Physical Arithmetic
from Internet Archive.
* 1887
The Logical Form of Geometrical Theorems
from Annals of Mathematics 3: 154,5.
* 1894
Papers on Space Analysis

* 1898
Book Review: “La Mathematique; philosophie et enseignement” by C.A. Laissant
in Science 8: 51–3.
* 189
The Pythagorean Theorem
from ''Science'' 34: 181,2.
* 1899
The Fundamental Principles of Algebra
from ''Science'' 10: 345–364.
* 1906
Vector Analysis and Quaternions

* 1910
Unification and Development of the Principles of the Algebra of Space
from Bulletin of the Quaternion Society.
* 1911
Book Review: ''Life and Scientific Work of P.G. Tait'' by C.G. Knott
from ''Science'' 34: 565,6.
* 1912
A System of Notation for Vector-Analysis; with a Discussion of the Underlying Principles
from ''Bulletin of the Quaternion Society''.
* 1913
On Vector-Analysis as Generalized Algebra
address to 5th International Congress of Mathematicians, Cambridge, via Internet Archive
*
* N.R.C. (1920
Review:''Ten British Physicists''
from ''Nature'' 104:561,2 (#2622)

Publications of Alexander Macfarlane
from ''Bulletin of the Quaternion Society'', 1913

References

*
* Robert de Boer (2009
Biography of Alexander Macfarlane
from WebCite.
* Robert de Boer (2009
Alexander Macfarlane in Chicago, 1893
from WebCite


* Knott, Cargill Gilston (1913
Alexander Macfarlane
Nature.

Macfarlane papers at the University of Texas

External links

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{{DEFAULTSORT:Macfarlane, Alexander
1851 births
1913 deaths
People from Blairgowrie and Rattray
Scottish logicians
19th-century Scottish philosophers
20th-century Scottish philosophers
Scottish physicists
19th-century Scottish mathematicians
20th-century Scottish mathematicians
Academics of the University of Edinburgh
Academics of the University of St Andrews
Alumni of the University of Edinburgh
Fellows of the Royal Society of Edinburgh
Lehigh University faculty
People from Chatham-Kent
Relativity theorists
Scottish expatriates in the United States
Scottish emigrants to Canada
University of Texas at Austin faculty
British geometers