VII. 



ON THE R6LE OF QUATERNIONS IN THE ALGEBRA 



OF VECTORS. 



[Nature, vol. XLIII. pp. 511-513, April 2, 1891.] 



THE following passage, which has recently come to my notice, in 

 the preface to the third edition of Prof. Tait's Quaternions seems 

 to call for some reply : 



" Even Prof. Willard Gibbs must be ranked as one of the retarders 

 of quaternion progress, in virtue of his pamphlet on Vector Analysis, 

 a sort of hermaphrodite monster, compounded of the notations of 

 Hamilton and of Grassmann." 



The merits or demerits of a pamphlet printed for private distri- 

 bution a good many years ago do not constitute a subject of any great 

 importance, but the assumptions implied in the sentence quoted are 

 suggestive of certain reflections and inquiries which are of broader 

 interest, and seem not untimely at a period when the methods and 

 results of the various forms of multiple algebra are attracting so much 

 attention. It seems to be assumed that a departure from quaternionic 

 usage in the treatment of vectors is an enormity. If this assumption 

 is true, it is an important truth ; if not, it would be unfortunate if it 

 should remain unchallenged, especially when supported by so high an 

 authority. The criticism relates particularly to notations, but I 

 believe that there is a deeper question of notions underlying that of 

 notations. Indeed, if my offence had been solely in the matter of 

 notation, it would have been less accurate to describe my production 

 as a monstrosity, than to characterize its dress as uncouth. 



Now what are the fundamental notions which are germane to a 

 vector analysis ? (A vector analysis is of course an algebra for 

 vectors, or something which shall be to vectors what ordinary algebra 

 is to ordinary quantities.) If we pass over those notions which are so 

 simple that they go without saying, geometrical addition (denoted 

 by + ) is, perhaps, first to be mentioned. Then comes the product of 

 the lengths of two vectors and the cosine of the angle which they 

 include. This, taken negatively, is denoted in quaternions by Sa/3, 

 where a and /3 are the vectors. Equally important is a vector 

 at right angles to a and /3 (on a specified side of their plane), and 



