Quaternion Society

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A scientific society, the Quaternion Society was 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 quaternions and other hypercomplex number systems. The guiding light was Alexander Macfarlane who served as its secretary throughout the life of the society. The society, through Macfarlane, published a Bibliography in 1904 and a Bulletin (annual report) from 1900 to 1913. By exercising mathematics and the imagination in four dimensions, the Society helped prepare for the theory of spacetime relativity.


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, Science, and the Bulletin of the American Mathematical Society (see references). Giuseppe Peano also announced the society formation in his Rivista di Matematica.

In 1897 the British Association met in Toronto 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 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.[1]

When the Society was organized in 1899, Peter Guthrie Tait was chosen as president, but he declined for reasons of poor health. Robert Stawell Ball then served for a year. Charles Jasper Joly, who had just completed editing the second edition of William Rowan Hamilton's Elements of Quaternions, then became president.

A system of national secretaries was announced in the AMS Bulletin in 1899: Alexander MacAulay for Australasia, Paul Genty for France, Victor Schlegel 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.

Victor Schlegel reported[2] on the new institution in the Monatshefte für Mathematik.


The annual report included articles, reviews, and society notes. It did not serve as a professionally reviewed scholarly journal, so there has been little reference to this publication in later work. However, in historical work the bulletin is of great value. One instance can be seen in M. J. Crowe's A History of Vector Analysis. P.R. Girard also referred to the Bulletin in his survey "The Quaternion Group and Modern Physics".[3] Girard notes that Macfarlane supplemented his 1904 Bibliography with lists in the bulletin for 1905, 08, 09, 10, 12, and 1913. Ortiz[4] notes also that the bulletin indicated when courses on quaternions were offered at various universities.


Main article: Alexander Macfarlane

As the society existed only with each individual, their writings and correspondence, the secretary was the motive force. By the time he steered the society Macfarlane had left the faculties of the University of Texas and Lehigh University. From his home base at Chatham, Ontario he ventured to campuses such as University of Michigan and Lehigh for research and lectures, and to three meetings of the International Congress of Mathematicians. Having been educated in Tait’s Lab in Edinburgh, he was grounded in physical science and, like many others at the time, thought quaternion methods could clarify physical theory. Furthermore, he developed the hyperbolic angle concept which arises naturally in the biquaternions. Macfarlane also fashioned an Algebra of Physics in the form of hyperbolic quaternions. Nevertheless, as Society secretary, he did not use the Bulletin to advance his particular system, but rather let it express the broad spectrum of linear algebra and multilinear algebra coming into existence at that time. In 1913 Macfarlane died, and as related by Dirk Struik, the Society "became a victim of the first World War".[5]


Published in 1904 at Dublin, cradle of quaternions, the 86 page Bibliography of Quaternions and Allied Systems of Mathematics[6] 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, 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 ...

Now available on-line in the Cornell University Historical Math Monographs collection, Macfarlane's work continues to be an important tool in research.


In 1912 the International Congress of Mathematicians met in Cambridge England. Macfarlane attended, as did Ludwik Silberstein, who used biquaternions in his presentation to the congress and in his book Theory of Relativity. Silberstein made no reference to the society or Macfarlane, viewing them as the common background of the time. Silberstein’s flowing prose and concrete application of relativity to electromagnetic theory shifted the attention of the scientific public from the abstract algebra with which the society had been concerned to physical science.

Notes and references[edit]

  1. ^ "Physics at the British Association" Nature 56:461,2 (# 1454)
  2. ^ Victor Schlegel (1899) "Internationaler Verein zur Beförderung des Studiums der Quaternionen und verwandter Systeme der Mathematik", Monatshefte für Mathematik 10(1):376
  3. ^ P.R. Girard (1984) "The Quaternion Group and Modern Physics", European Journal of Physics 5:25–32
  4. ^ Eduardo L. Ortiz International Association for Promoting the Calculus of Quaternions
  5. ^ Dirk Struik (1967) A Concise History of Mathematics, 3rd edition, page 172, Dover Books
  6. ^ Alexander Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, weblink from Cornell University Historical Math Monographs.