Alexander Macfarlane
FRSE
Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently distinguished in their subject". ...
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
The University of Edinburgh (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...
. His
doctoral thesis
A thesis (: theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: D ...
"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
The Royal Society of Edinburgh (RSE) is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was establis ...
on
algebraic logic
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
as introduced by
George Boole
George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...
. He was elected a
Fellow of the Royal Society of Edinburgh
Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently distinguished in their subject". ...
. His proposers were
Peter Guthrie Tait,
Philip Kelland
Philip Kelland PRSE FRS (17 October 1808 – 8 May 1879) was an English mathematician. He was known mainly for his great influence on the development of education in Scotland.
Life
Kelland was born in 1808 the son of Philip Kelland (d.1847), ...
,
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
St Andrews (; ; , pronounced ʰʲɪʎˈrˠiː.ɪɲ is a town on the east coast of Fife in Scotland, southeast of Dundee and northeast of Edinburgh. St Andrews had a recorded population of 16,800 , making it Fife's fourth-largest settleme ...
, was physics professor at the
University of Texas
The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas, United States. Founded in 1883, it is the flagship institution of the University of Texas System. With 53,082 students as of fall 2 ...
(1885–1894), professor of Advanced Electricity, and later of
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, at
Lehigh University
Lehigh University (LU), in Bethlehem, Pennsylvania, United States, is a private university, private research university. The university was established in 1865 by businessman Asa Packer. Lehigh University's undergraduate programs have been mixed ...
. In 1896 Macfarlane encouraged the association of
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
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
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
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
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
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
In mathematics, a versor is a quaternion of Quaternion#Norm, norm one, also known as a unit quaternion. Each versor has the form
:u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi
where the r2 = −1 conditi ...
of Hamilton. The presentation is encumbered by the notation
:
Later he conformed to the notation exp(A α) used by Euler and Sophus Lie. The expression
is meant to emphasize that α is a ''right versor'', where π/2 is the measure of a
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
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
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
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
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, and proceeds to analogues for the ellipsoid of revolution, general
ellipsoid
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a Surface (mathemat ...
, and equilateral
hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
s 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
hyperbola
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
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
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions cor ...
s, an algebra used by students of Hamilton since 1853.
Works
* 1879
Principles of the Algebra of Logicfrom
Internet Archive
The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
.
* 1885
Physical Arithmeticfrom Internet Archive.
* 1887
The Logical Form of Geometrical Theoremsfrom
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
3: 154,5.
* 1894
Papers on Space Analysis
* 1898
Book Review: “La Mathematique; philosophie et enseignement” by C.A. Laissantin
Science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
8: 51–3.
* 189
The Pythagorean Theoremfrom ''Science'' 34: 181,2.
* 1899
The Fundamental Principles of Algebrafrom ''Science'' 10: 345–364.
* 1906
Vector Analysis and Quaternions
* 1910
Unification and Development of the Principles of the Algebra of Spacefrom
Bulletin of the Quaternion Society.
* 1911
Book Review: ''Life and Scientific Work of P.G. Tait'' by C.G. Knottfrom ''Science'' 34: 565,6.
* 1912
A System of Notation for Vector-Analysis; with a Discussion of the Underlying Principlesfrom ''Bulletin of the Quaternion Society''.
* 1913
On Vector-Analysis as Generalized Algebra address to 5th
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
, Cambridge, via Internet Archive
*
*
[N.R.C. (1920]
Review:''Ten British Physicists''
from ''Nature'' 104:561,2 (#2622)
Publications of Alexander Macfarlanefrom ''Bulletin of the Quaternion Society'', 1913
References
*
* Robert de Boer (2009
Biography of Alexander Macfarlanefrom
WebCite.
* Robert de Boer (2009
Alexander Macfarlane in Chicago, 1893from WebCite
*
Knott, Cargill Gilston (1913
Alexander Macfarlane Nature
Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
.
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