172 QUATERNIONS AND THE ALGEBRA OF VECTORS. 



distributed, so long as I had copies to distribute, among those who 

 I thought might be interested in the subject. I may say, however, 

 since I am called upon to defend my position, that I have found 

 the notations of that pamphlet more flexible than those generally 

 used. Mr. McAulay, at least, will understand what I mean by this, 

 if I say that some of the relations which he has thought of sufficient 

 importance to express by means of special devices (see Proc, R.S.E. 

 for 1890-91), may be expressed at least as briefly in the notations 

 which I have used, and without special devices. But I should not 

 have been satisfied for the purposes of my pamphlet with any 

 notation which should suggest even to the careless reader any con- 

 nection with the notion of the quaternion. For I confess that one 

 of my. objects was to show that a system of vector analysis does not 

 require any support from the notion of the quaternion, or, 1 may 

 add, of the imaginary in algebra. 



I should hardly dare to express myself with so much freedom, if 

 I could not shelter myself behind an authority which will not be 

 questioned. 



I do not see that I have done anything very different from what 

 the eminent mathematician upon whom Hamilton's mantle has fallen 

 has been doing, it would seem, unconsciously. Contrast the system of 

 quaternions, which he has described in his sketch of Hamilton's life 

 and work in the North British Review for September, 1866, with the 

 system which he urges upon the attention of physicists in the Philo- 

 sophical Magazine in 1890. In 1866 we have a great deal about 

 imaginaries, and nearly as much about the quaternion. In 1890 we 

 have nothing about imaginaries, and little about the quaternion. 

 Prof. Tait has spoken of the calculus of quaternions as throwing off 

 in the course of years its early Cartesian trammels. I wonder that 

 he does not see how well the progress in which he has led may be 

 described as throwing off the yoke of the quaternion. A characteristic 

 example is seen in the use of the symbol V. Hamilton applies this to 

 a vector to form a quaternion, Tait to form a linear vector function. 

 But while breathing a new life into the formulae of quaternions, 

 Prof. Tait stands stoutly by the letter. 



Now I appreciate and admire the generous loyalty toward one 

 whom he regards as his master, which has always led Prof. Tait to 

 minimise the originality of his own work in regard to quaternions, 

 and write as if everything was contained in the ideas which flashed 

 into the mind of Hamilton at the classic Brougham Bridge. But not 

 to speak of other claims of historical justice, we owe duties to our 

 scholars as well as to our teachers, and the world is too large, and the 

 current of modern thought is too broad, to be confined by the ipse 

 dixit even of a Hamilton. 



