QUATEKNIONS AND THE ALGEBRA OF VECTORS. 171 



versant. I should tell him that a vector algebra is so far from being 

 any one man's production that half a century ago several were already 

 working toward an algebra which should be primarily geometrical and 

 not arithmetical, and that there is a remarkable similarity in the results 

 to which these efforts led (see Proc. A.A.A.8. for 1886, pp. 37, ff.) [this 

 vol. p. 91, ff.]. I should call his attention to the fact that Lagrange 

 and Gauss used the notation (a/3y) to denote precisely the same as 

 Hamilton by his S(a/3y), except that Lagrange limited the expression 

 to unit vectors, and Gauss to vectors of which the length is the secant 

 of the latitude, and I should show him that we have only to give up 

 these limitations, and the expression (in connection with the notion 

 of geometrical addition) is endowed with an immense wealth of 

 transformations. I should call his attention to the fact that the 

 notation [?V 2 ], universal in the theory of orbits, is identical with 

 Hamilton's V(/o 1 /o 2 ), except that Hamilton takes the area as a vector, 

 i.e., includes the notion of the direction of the normal to the plane of 

 the triangle, and that with this simple modification (and with the 

 notion of geometrical addition of surfaces as well as of lines) this 

 expression becomes closely connected with the first-mentioned, and is 

 not only endowed with a similar capability for transformation, but 

 enriches the first with new capabilities. In fact, I should tell him 

 that the notions which we use in vector analysis are those which he 

 who reads between the lines will meet on every page of the great 

 masters of analysis, or of those who have probed deepest the secrets of 

 nature, the only difference being that the vector analyst, having regard 

 to the weakness of the human intellect, does as the early painters 

 who wrote beneath their pictures " This is a tree," " This is a horse." 



I cannot attach quite so much importance as Mr. McAulay to 

 uniformity of notation. That very uniformity, if it existed among 

 those who use a vector analysis, would rather obscure than reveal 

 their connection with the general course of modern thought in 

 mathematics and physics. There are two ways in which we may 

 measure the progress of any reform. The one consists in counting 

 those who have adopted the shibboleth of the reformers; the other 

 measure is the degree in which the community is imbued with the 

 essential principles of the reform. I should apply the broader 

 measure to the present case, and do not find it quite so bad as 

 Mr. McAulay does. 



Yet the question of notations, although not the vital question, is 

 certainly important, and I assure Mr. McAulay that reluctance to 

 make unnecessary innovations in notation has been a very powerful 

 motive in restraining me from publication. Indeed my pamphlet on 

 Vector Analysis, which has excited the animadversion of quater- 

 nionists, was never formally published, although rather widely 



