The Quaternion Society was a

scientific society A learned society ( ; also scholarly, intellectual, or academic society) is an organization that exists to promote an academic discipline, profession, or a group of related disciplines such as the arts and sciences. Membership may be open to a ...

, 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 academic world that were experimenting with

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 ...

s and other

hypercomplex number In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...

systems. The group's guiding light was

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 s ...

who served as its secretary initially, and became president in 1909. The association published a ''Bibliography'' in 1904 and a ''Bulletin'' (annual report) from 1900 to 1913. The ''Bulletin'' became a

review journal A review article is an article that summarizes the current state of understanding on a topic within a certain discipline. A review article is generally considered a secondary source since it may analyze and discuss the method and conclusions ...

for topics in

vector analysis Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

and

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

such as the theory of equipollence. The mathematical work reviewed pertained largely to

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

and

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

as the methods were in rapid development at the time.

Genesis

In 1895, Professor P. Molenbroek of The Hague, Holland, and Shinkichi Kimura studying at Yale put out a call for scholars to form the society in widely circulated journals:

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 ...

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 ...

, and the

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notati ...

also announced the society formation in his ''Rivista di Matematica''. The call to form an Association was encouraged by Macfarlane in 1896: :The logical harmony and unification of the whole of mathematical analysis ought to be kept in view. The algebra of space ought to include the algebra of the plane as a special case, just as the algebra of the plane includes the algebra of the line…When vector analysis is developed and presented...we may expect to see many zealous cultivators, many fruitful applications, and, finally, universal diffusion ... May the movement initiated by Messrs. Molenbroek and Kimura hasten the realization of this happy result. In 1897 the

British Association The British Science Association (BSA) is a charity and learned society founded in 1831 to aid in the promotion and development of science. Until 2009 it was known as the British Association for the Advancement of Science (BA). The current Chief ...

met in

Toronto Toronto ( , locally pronounced or ) is the List of the largest municipalities in Canada by population, most populous city in Canada. It is the capital city of the Provinces and territories of Canada, Canadian province of Ontario. With a p ...

where vector products were discussed: :Professor Henrici proposed a new notation to denote the different products of vectors, which consists in using square brackets for vector products and round brackets for scalar products. He likewise advocated adoption of Heaviside’s term "ort" for vector, the

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

of which is the number 1. Prof. A. Macfarlane read a communication on the solution of the cubic equation in which he explained how the two binomials in Cardano’s formula may be treated as complex quantities, either circular or hyperbolic, all the roots of the cubic can then be deduced by a general method. Also that year, L. van Elfrinkhof described rotations in 4-dimensional Euclidean space. A system of national secretaries was announced in the AMS Bulletin in 1899:

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 biquate ...

for Australasia,

Victor Schlegel Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams. In the nineteenth ce ...

for Germany, Joly for Great Britain and Ireland, Giuseppe Peano for Italy, Kimura for Japan, Aleksandr Kotelnikov for Russia, F. Kraft for Switzerland, and Arthur Stafford Hathaway for the USA. For France the national secretary was Paul Genty, an engineer with the division of Ponts et Chaussees, and a quaternion collaborator with Charles-Ange Laisant, author of ''Methode des Quaterniones'' (1881).

reported on the new institution in the Monatshefte für Mathematik.

Officers

When the society was organized in 1899,

Peter Guthrie Tait Peter Guthrie Tait (28 April 18314 July 1901) was a Scottish Mathematical physics, mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he ...

was chosen as president, but he declined for reasons of poor health. The first president was Robert Stawell Ball, and

served as secretary and treasurer. In 1905 Charles Jasper Joly took over as president and L. van Elfrinkhof as treasurer, while Macfarlane continued as secretary. In 1909 Macfarlane became president, James Byrnie Shaw became secretary, and van Elfrinkhof continued as treasurer. The next year Macfarlane and Shaw continued in their posts, while Macfarlane also absorbed the office of treasurer. When Macfarlane died in 1913 after nearly completing the issue of the ''Bulletin'', Shaw completed it and wound up the association. The rules state that the president had the power of veto.

Bulletin

The ''Bulletin of the Association Promoting the Study of Quaternions and Allied Systems of Mathematics'' was issued nine times under the editorship of Alexander Macfarlane. Every issue listed the officers of the Association, governing council, rules, members, and a

financial statement Financial statements (or financial reports) are formal records of the financial activities and position of a business, person, or other entity. Relevant financial information is presented in a structured manner and in a form which is easy to un ...

from the

treasurer A treasurer is a person responsible for the financial operations of a government, business, or other organization. Government The treasury of a country is the department responsible for the country's economy, finance and revenue. The treasure ...

. Today

HathiTrust HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries. Its holdings include content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digit ...

provides access to these publications that are mainly of historical interest:
March 1900
Published in Toronto by Roswell-Hutchinson Press.
March 1901
Published in Dublin at the University Press. President Charles J. Joly address.
March 1903
Dublin. Macfarlane announces Bibliography.
April 1905
Dublin. President C.J. Joly address.
March 1908
Published in Lancaster, Pennsylvania, by New Era Printing. J.B. Shaw reports on bibliographic supplement.
June 1909
Lancaster. President Macfarlane address on notation.
October 1910
Lancaster. J.B. Shaw challenged by "inclusion or exclusion of certain papers which are only remotely connected with the theory of operations in the abstract."
June 1912
Lancaster. Obituary: Ferdinand Ferber. "Comparative Notation for Vector Expressions" by J.B. Shaw. President Macfarlane address citing

Duncan Sommerville Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...

's comments.
June 1913
Lancaster. Secretary Shaw reports the death of A. Macfarlane and G. Combebiac.

Bibliography

Published in 1904 at Dublin, cradle of quaternions, the 86 page ''Bibliography of Quaternions and Allied Systems of Mathematics'' cited some one thousand references. The publication set a professional standard; for instance the ''Manual of Quaternions'' (1905) of Joly has no bibliography beyond citation of Macfarlane. Furthermore, in 1967 when Michael J. Crowe published ''

A History of Vector Analysis ''A History of Vector Analysis'' (1967) is a book on the history of vector analysis by Michael J. Crowe, originally published by the University of Notre Dame Press. As a scholarly treatment of a reformation in technical communication, the text i ...

'', he wrote in the preface (page ix) : :''Concerning bibliography''. No formal bibliographical section has been included with this book. ... the need for a bibliography is greatly diminished by the existence of a book that lists nearly all relevant primary documents published to about 1912, this is Alexander Macfarlane’s ''Bibliography'' ... Every year more papers and books appeared that were of interest to Association members so it was necessary to update the ''Bibliography'' with supplements in the ''Bulletin''. The categories used to group the items in the supplements give a sense of the changing focus of the Association:
1905 Supplement

1908 Supplement
Matrices, Linear substitutions, Quadratic forms, Bilinear forms, Complex numbers, Equipollences, Vector analysis, Commutative algebras, Quaternions, Biquaternions, Linear associative algebras, General algebra and operations, Additional.
1909 Supplement

1910 Supplement
Matrices, Linear groups, Complex numbers & equipollences, Vector analysis, Ausdehnungslehre, Quaternions, Linear associative algebras.
1912 Supplement
Equipollences, Commutative systems, Space-analysis, Dyadic systems, Vector analysis, Quaternions.
1913 Supplement
Commutative systems, Space analysis, Dyadic systems, Vector analysis, Other, Quaternions, Hypercomplex numbers, General algebra.

Aftermath

In 1913 Macfarlane died, and as related by Dirk Struik, the Society "became a victim of the first World War". James Byrnie Shaw, the surviving officer, wrote 50 book notices for American mathematical publications. The final article review in the ''Bulletin'' wa
The Wilson and Lewis Algebra of Four-Dimensional Space
written by J. B. Shaw. He summarizes, :This algebra is applied to the representation of the Minkowski time-space world. It enables all analytical work to be with reals, although the geometry becomes non-Euclidean. The article reviewed was "The space-time manifold of relativity, the non-Euclidean geometry of mechanics, and electromagnetics".E. B. Wilson & G. N. Lewis (1912) Proceedings of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

48: 389–507 However, when the textbook ''The Theory of Relativity'' by

Ludwik Silberstein Ludwik Silberstein (May 17, 1872 – January 17, 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmill ...

in 1914 was made available as an English understanding 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 ...

, the algebra of

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 was applied, but without references to the British background or Macfarlane or other quaternionists of the Society. The language of quaternions had become international, providing content to

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

and expanded

mathematical notation Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling ...

, and expressing

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 ...

Notes and references