The Quaternion Society was a

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

, 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. Hamilton defined a quatern ...

s and other

hypercomplex number In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...

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, Blairgowire) 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 i ...

for topics in

vector analysis Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...

and

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

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

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

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 matrices. ...

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, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, 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. ...

Science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...

, 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. I ...

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 notation. The stand ...

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 Chie ...

met in

Toronto Toronto ( ; or ) is the capital city of the Canadian province of Ontario. With a recorded population of 2,794,356 in 2021, it is the most populous city in Canada and the fourth most populous city in North America. The city is the ancho ...

where vector products were discussed: :Professor

Henrici The surname Henrici is the possessive form of the Latin language, Latin name "Henricus", i.e. "Henry (given name), Henry". Henrici may refer to: * Christian Friedrich Henrici, known as Picander (1700–1764), German poet and librettist * Cornelius ...

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 related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

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. 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 centur ...

for Germany, Joly for Great Britain and Ireland, Giuseppe Peano for Italy, Kimura for Japan,

Aleksandr Kotelnikov Aleksandr Petrovich Kotelnikov (russian: Алекса́ндр Петро́вич Коте́льников; October 20, 1865 – March 6, 1944) was a Russian and Soviet mathematician specializing in geometry and kinematics. Biography Aleksandr w ...

for Russia,

F. Kraft F is the sixth letter of the Latin alphabet. F may also refer to: Science and technology Mathematics * F or f, the number 15 in hexadecimal and higher positional systems * ''p'F'q'', the hypergeometric function * F-distribution, a cont ...

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 Charles-Ange Laisant (1 November 1841 – 5 May 1920), French politician and mathematician, was born at Indre, near Nantes on 1 November 1841, and was educated at the École Polytechnique as a military engineer. He was a Freemason and a liberta ...

, author of ''Methode des Quaterniones'' (1881).

reported on the new institution in the

Monatshefte für Mathematik '' Monatshefte für Mathematik'' is a peer-reviewed mathematics journal established in 1890. Among its well-known papers is " Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" by Kurt Gödel, published in 193 ...

Officers

When the society was organized in 1899,

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 ...

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

Robert Stawell Ball Sir Robert Stawell Ball (1 July 1840 – 25 November 1913) was an Irish astronomer who founded the screw theory. He was Royal Astronomer of Ireland at Dunsink Observatory. Life He was the son of naturalist Robert Ball, and Amelia Gresley Hel ...

, and

served as secretary and treasurer. In 1905

Charles Jasper Joly Charles Jasper Joly (27 June 1864 – 4 January 1906) was an Irish mathematician and astronomer who became Royal Astronomer of Ireland.Obituary, New York Times, 5 January 1906 Life He was born at St Catherine's Rectory, Hop Hill, Tullamore, ...

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 the person responsible for running the treasury of an organization. The significant core functions of a corporate treasurer include cash and liquidity management, risk management, and corporate finance. Government The treasury o ...

. Today

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

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 Louis Ferdinand Ferber (8 February 1862 – 22 September 1909) was a French Army officer who played an important role in the development of aviation during the early 1900s. Although his aircraft experiments were belatedly successful, his early ...

. "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 M.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 Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch-born American (since 1934) mathematician, historian of mathematics and Marxian theoretician who spent most of his life in the U.S. Life Dirk Jan Struik was born in 1 ...

, 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 (abbreviation: AAA&S) 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 ...

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

Ludwik Silberstein Ludwik Silberstein (1872 – 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 Macmillan in 1914 with a sec ...

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

, 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 co ...

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 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 mathematics, is mostly conce ...

and expanded

mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...

, and expressing

mathematical physics Mathematical physics refers to 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 t ...

Notes and references

The Quaternion Association
at

MacTutor History of Mathematics archive The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathemati ...