The Quaternion Association

Founded in 1895

The Quaternion Association was officially named the 'International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics'. It was founded in 1895 by Pieter Molenbroek and Shunkichi Kimura. However, the quaternions had been discovered by William Rowan Hamilton in 1843 and his book Elements of Quaternions, published posthumously in 1866, was at that time the main textbook on the topic. Although many worked on quaternions in the second half of the 19th century, the main champion for their cause was Peter Guthrie Tait. The other input to the vector calculus related to this was from Hermann Grassmann who published Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik in 1844.

Pieter Molenbroek (born 1861) was professor at The Hague, Holland, and the author of Theorie der Quaternionen and Anwendung der Quaternionen auf der Geometrie (1893). Shunkichi Kimura was Japanese but was in Europe visiting Pieter Molenbroek when he published Note on Quaternions in 1895 giving his address as Japanese Legation, The Hague. However Molenbroek and Kimura published To Friends and Fellow Workers in Quaternions in Nature dated 7 August 1895 where they proposed an Association they suggested be called "The International Association for Promoting the Calculus of Quaternions." They added a P.S., dated 17 September 1895, where they wrote [5]:-
It has been suggested by friends interested in this matter to enlarge the scope of the proposed Association so as to include all systems allied to quaternions and to Grassmann's "Ausdehnulgslehre." This suggestion we are in full sympathy with. The name of the Association might then be "The International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics."
By the time this paper was written, Kimura was at Yale University in the United States. For a version of their paper To Friends and Fellow Workers in Quaternions see THIS LINK.

The same two authors also published To those Interested in Quaternions and Allied Systems if Mathematics in Science, again proposing the founding of an Association in this paper dated October 1895. For a version of this paper ([6] in the references below), see THIS LINK.

Charles Jasper Joly, President of the Association in 1901-02, wrote about its founding in his President's Address [2]:-
A fair number of mathematicians signified their appreciation of the movement by promising to join the society, and for the time Dr Molenbroek acted as Secretary and Treasurer for Europe, and Mr Kimura for America. Mr Kimura issued papers for the first election of Officers, but unfortunately for the immediate success of the Association the election proved a failure. Professor Tait received a majority of votes for the office of President. He declined to act on the ground of failing health, and suggested that a younger man should be appointed. Mr Kimura was elected secretary, but in the meantime he was obliged to return to his native country, where he felt it was impracticable to carry on the work of organization owing to the distance of his abode from the majority of the members, and the long delays involved in postal communication. Dr Molenbroek, the newly elected Treasurer, lost his health, and he became quite unable to transact the laborious duties of organization. Under these circumstances, Molenbroek and Kimura requested Professor Hathaway "to endeavour to bring the society into more active existence."
For a version of Joly's complete address, see THIS LINK.

Arthur Stafford Hathaway was Professor of Mathematics at Rose Polytechnic Institute, Terre Haute, Indiana. He suggested that a new committee be elected at the meeting of the British Association held in Toronto, Canada, in 1897. This meeting elected the committee for the two years 1897-1898: President, Robert Stawell Ball, Lowndean Professor of Astronomy in the University of Cambridge; General Secretary, Alexander Macfarlane, Professor of Mathematical Physics at Lehigh University, Bethlehem, Pennsylvania; and Treasurer, Pieter Molenbroek. The meeting hoped that Molenbroek's health would improve and allow him to take on this role but, unfortunately, this hope was not fulfilled. Alexander Macfarlane (1851-1913) studied at the University of Edinburgh where he wrote a Ph.D. thesis describing experimental results concerning electricity that had been carried out in Peter Guthrie Tait's laboratory. After teaching at the University of Edinburgh and the University of St Andrews, Macfarlane went to the United States where he was professor of physics at the University of Texas from 1885 to 1894, and then professor of mathematical physics at Lehigh University. He applied quaternions to physics, calling it the 'algebra of physics'.

With Molenbroek unable to undertake his duties, the committee for 1899-1900 was: President, Robert Stawell Ball; General Secretary and Treasurer, Alexander Macfarlane. These officers are listed in the Bulletin of the International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics, published in March 1900. Also listed are eleven National Secretaries, one for each of: Australasia; Canada; France; Germany; Great Britain and Ireland; Holland; Italy; Japan; Russia; Switzerland; and United States.

However, the problems the Association continued to experience are described by Joly in [2]:-
The early misfortunes of the society had left their mark. The scheme had hung fire so long that much of the original enthusiasm was lost, and even its most ardent supporters began to question the possibility of carrying out the object of the Association in any useful way. There seemed to be no likelihood of a substantial increase in the number of members; the funds of the Association would not allow of the extensive publications originally contemplated, and the society appeared to languish in fruitless inactivity.
For more details, see THIS LINK.

The Association was founded in the middle of a controversy between those promoting the 'vector calculus' as proposed by J Willard Gibbs, and those promoting the quaternions. Peter Guthrie Tait, the great champion for the quaternions, described Gibbs as "one of the retarders of quaternion progress" because of his introduction of the vector calculus. However, the Quaternion Association had elected Robert Stawell Ball and Alexander Macfarlane as its two main officers yet they were believers in the vector calculus. One would have to say that the only way the Quaternion Association might have survived was if it transformed itself into a broader Society since, as we all know, the vector calculus completely took over from quaternions in applications.

Hubert Kennedy describes the end of the Quaternion Association in [3]:-
The Association continued in existence until 1913. In the "Bulletin" for that year, Secretary James B Shaw reported the death of Macfarlane just before the Bulletin was completed and noted: "As all the terms of office expire with the end of the current year, this leaves the Association almost in a state demanding a reorganization." The reorganization seems not to have occurred and the Association apparently dissolved.
However, we should not dismiss the quaternions, or the Association, too lightly. One of the achievements of the Association was the publishing Bibliography of Quaternions and Allied Systems of Mathematics in 1904. This 86-page document contained over 1000 references. Members of the Association continued to add new books and papers up to 1913, these additional references being published as supplements.

References (show)

  1. M J Crowe, A history of vector analysis: The evolution of the idea of a vectorial system (Courier Corporation, 1967).
  2. C J Joly, President's address, Bulletin of the International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics (March 1900).
  3. H Kennedy, James Mills Peirce and the cult of Quaternion, Historia Mathematica 6 (1979), 423-429.
  4. S Kimura, Note on Quaternions, Nature 52 (1895), 366.
  5. S Kimura and P Molenbroek, Friends and Fellow Workers in Quaternions, Nature 52 (1895), 545-546.
  6. S Kimura and P Molenbroek, To those Interested in Quaternions and Allied Systems of Mathematics, Science (2) 2 (1895), 524-525.
  7. A Macfarlane, Quaternions, Science (2) 3 (1896), 99-100.
  8. J N Shutt, Quaternions: A case study in the selection of tools for mathematical physics (Worcester Polytechnic Institute, Worcester, Massachusetts, 1986).

Last Updated February 2018