IX. 



QUATERNIONS AND THE ALGEBRA OF VECTORS. 



[Nature, vol. XLVII. pp. 463, 464, Mar. 16, 1893.] 



IN a recent number of Nature [vol. xlvii, p. 151], Mr. McAulay 

 puts certain questions to Mr. Heaviside and to me, relating to a 

 subject of such importance as to justify an answer somewhat at 

 length. I cannot of course speak for Mr. Heaviside, although I 

 suppose that his views are not very different from mine on the most 

 essential points, but even if he shall have already replied before this 

 letter can appear, I shall be glad to add whatever of force may belong 

 to independent testimony. 



Mr. McAulay asks : " What is the first duty of the physical vector 

 analyst qud physical vector analyst ? " The answer is not doubtful. 

 It is to present the subject in such a form as to be most easily 

 acquired, and most useful when acquired. 



In regard to the slow progress of such methods towards recognition 

 and use by physicists and others, which Mr. McAulay deplores, it does 

 not seem possible to impute it to any want of uniformity of notation. 

 I doubt whether there is any modern branch of mathematics which 

 has been presented for so long a time with a greater uniformity of 

 notation than quaternions. 



What, then, is the cause of the fact which Mr. McAulay and all 

 of us deplore ? It is not far to seek. We need only a glance at the 

 volumes in which Hamilton set forth his method. No wonder that 

 physicists and others failed to perceive the possibilities of simplicity, 

 perspicuity, and brevity which were contained in a system presented 

 to them in ponderous volumes of 800 pages. Perhaps Hamilton may 

 have intended these volumes as a sort of thesaurus, and we should 

 look to his shorter papers for a compact account of his method. But 

 if we turn to his earlier papers on Quaternions in the Philosophical 

 Magazine, in which principally he introduced the subject to the 

 notice of his contemporaries, we find them entitled " On Quaternions; 

 or on a New System of Imaginaries in Algebra," and in them we 

 find a great deal about imaginaries, and very little of a vector 

 analysis. To show how slowly the system of vector analysis 

 developed itself in the quaternionic nidus, we need only say that 

 the symbols S, Y, and V do not appear until two or three years 

 after the discovery of quaternions. In short, it seems to have been 



