THE 

 LONDON. EDINBURGH and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



SUPPLEMENT to VOL. XXXVI. THIRD SERIES. 



LXII. On the Geometrical Interpretation of Quaternions. By 

 W. F. DoNKiN, M.A., S^c, Savilian Professor of Astro- 

 nomy in the University of Oxford*. 



IN a recent paper in this Journal, I gave two examples of 

 the application of quaternions to geometrical problems, 

 in which I made use of a system of interpretation different 

 from that usually employed by Sir W. Hamilton. This 

 system it is my present object to explain. I am not at pre- 

 sent prepared to offer an opinion as to its practical advantages 

 or disadvantages. It is here proposed as possessing a certain 

 theoretic interest, arising from the fact that it represents the 

 method of quaternions as a natural (or rather the natural) 

 extension to tridimensional space of the usual geometrical in- 

 terpretations of symbolical algebra. The general principles 

 employed in the investigation have no pretensions to novelty. 

 The view here taken of symbolical algebra, and of its inter- 

 pretation in plane geometry, is substantially the same as that 

 advocated by the late Mr. Gregory (in the fourteenth volume 

 of the Edinburgh Philosophical Transactions, and in several 

 papers in the Cambridge Mathematical Journal) ; and the 

 extension to geometry of three dimensions rests upon prin- 

 ciples which will be found, I believe, to agree essentially, as 

 far as they go, with those laid down by Sir W. Hamilton 

 in his papers on symbolical geometry in the last-named pe- 

 riodical. In what follows, then, no further reference will in 

 general be made to previous writers. Moreover, to avoid the 

 use of frequent qualifying phrases, propositions will be stated 

 dogmatically which only claim to be accepted as belonging to 

 a consistent system, not as belonging to the only consistent or 

 useful system. 



General symbolical algebra is a calculus of operations. All 

 the symbols employed, including those arithmetical symbols 

 which occur, represent or indicate operations. The subject 



* Communicated by the Author. 



PhiU Mag. S, 3. No. 246, SuppL Vol, S6. 2 K 



