Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

ian,

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

, and mathematician.

Life

Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowire) and Ann Small. He studied at the

University of Edinburgh The University of Edinburgh ( sco, University o Edinburgh, gd, Oilthigh Dhùn Èideann; abbreviated as ''Edin.'' in post-nominals) is a public research university based in Edinburgh, Scotland. Granted a royal charter by King James VI in 15 ...

. 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: ...

"The disruptive discharge of electricity" reported on experimental results from the laboratory of

Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he co-wrote wi ...

. In 1878 Macfarlane spoke at the Royal Society of Edinburgh on algebraic logic 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 ...

. 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 letters, judged to be "eminently distinguished in their subject". This socie ...

. His proposers were

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 Alexander Crum Brown FRSE FRS (26 March 1838 – 28 October 1922) was a Scottish organic chemist. Alexander Crum Brown Road in Edinburgh's King's Buildings complex is named after him. Early life and education Crum Brown was born at 4 Belle ...

, and

John Hutton Balfour John Hutton Balfour (15 September 1808 – 11 February 1884) was a Scottish botanist. Balfour became a Professor of Botany, first at the University of Glasgow in 1841, moving to the University of Edinburgh and also becoming the 7th Regius Kee ...

. 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 The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 ...

(1885–1894), professor of Advanced Electricity, and later of

mathematical physics Mathematical physics refers to the development of 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 developme ...

, at

Lehigh University Lehigh University (LU) is a private research university in Bethlehem, Pennsylvania in the Lehigh Valley region of eastern Pennsylvania. The university was established in 1865 by businessman Asa Packer and was originally affiliated with the Epi ...

. In 1896 Macfarlane encouraged the association of quaternion students to promote the algebra. He became the Secretary of the

Quaternion Society The Quaternion Society was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the ac ...

, 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, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

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 mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...

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 Alexander McAulay (9 December 1863 – 6 July 1931) was the first professor of mathematics and physics at the University of Tasmania, Hobart, Tasmania. He was also a proponent of dual quaternions, which he termed "octonions" or "Clifford biqua ...

'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 In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form :q = a + bi + cj + dk, \quad a,b,c,d \in \mathbb \! where the squares of i, j, and k are +1 and distinct eleme ...

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 In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Wil ...

as the hyperbolic quaternion corresponding to the

versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...

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

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. The unit was formerly an SI supplementary unit (before tha ...

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 George Salmon FBA FRS FRSE (25 September 1819 – 22 January 1904) was a distinguished and influential Irish mathematician and Anglican theologian. After working in algebraic geometry for two decades, Salmon devoted the last forty years of his ...

in exhibiting the

hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...

, argument of

hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...

s. The fifth paper is "Elliptic and Hyperbolic Analysis" which considers the

spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "sph ...

as the fundamental theorem of the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

, 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 hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigono ...

. 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 (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth 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, ca ...

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 co ...

s, an algebra used by students of Hamilton since 1853.

Works

* 1879
Principles of the Algebra of Logic
from

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...

. * 1885
Physical Arithmetic
from Internet Archive. * 1887
The Logical Form of Geometrical Theorems
from

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 the ...

3: 154,5. * 1894
Papers on Space Analysis
* 1898
Book Review: “La Mathematique; philosophie et enseignement” by C.A. Laissant
in

Science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...

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
from ''Bulletin of the Quaternion Society''. * * 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 WebCite was an on-demand archive site, designed to digitally preserve scientific and educationally important material on the web by taking snapshots of Internet contents as they existed at the time when a blogger or a scholar cited or quoted ...

.
Electric Scotland historical biography
* Knott, Cargill Gilston (1913
Alexander Macfarlane

Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...

.
Macfarlane papers at the University of Texas

External links

* * * * * {{DEFAULTSORT:Macfarlane, Alexander 1851 births 1913 deaths People from Blairgowrie and Rattray Scottish mathematicians Scottish logicians Scottish philosophers Scottish physicists 19th-century British mathematicians 19th-century Scottish people 20th-century British 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