178 QUATERNIONS AND VECTOR ANALYSIS. 



original paper (p. 226), " No finer argument in favour of the real 

 quaternion vector analysis can be found than in the tangle and the 

 jangle of sections 91 to 104 in the Elements of Vector Analysis." 

 Now I am quite ready to plead guilty to the tangle. The sections 

 mentioned, as is sufficiently evident to the reader, were written at 

 two different times, sections 102-104 being an addition after a couple 

 of years. The matter of these latter sections is not found in its 

 natural place, and the result is well enough characterised as a tangle. 

 It certainly does credit to the conscientious study which Prof. Knott 

 has given to my pamphlet, that he has discovered that there is a 

 violent dislocation of ideas just at this point. For such a fault of 

 composition I have no sufficient excuse to offer, but I must protest 

 against its being made the ground of any broad conclusions in regard 

 to the fundamental importance of the quaternion. 



Prof. Knott next proceeds to criticise or at least to ridicule my 

 treatment of the linear vector function, with respect to which we read 

 in the abstract : " As developed in the pamphlet, the theory of the 

 dyadic goes over much the same ground as is traversed in the last 

 chapter of Kelland and Tait's Introduction to Quaternions. With 

 the exception of a few of those lexicon products, for which Prof. 

 Gibbs has such an affection, there is nothing of real value added to 

 our knowledge of the linear vector function." It would not, I think, 

 be difficult to show some inaccuracy in my critic's characterisation 

 of the real content of this part of my pamphlet. But as algebra is 

 a formal science, and as the whole discussion is concerning the best 

 form of representing certain kinds of relations, the important question 

 would seem to be whether there is anything of formal value in my 

 treatment of the linear vector function. 



Now Prof. Knott distinctly characterises in half a dozen words the 

 difference in the spirit and method of my treatment of this subject 

 from that which is traditional among quaternionists, when he says 

 of what I have called dyadics "these are not quantities, but operators" 

 (Nature, vol. xlvii, p. 592). I do not think that I applied the word 

 quantity to the dyadics, but Prof. Knott recognised that I treated 

 them as quantities not, of course, as the quantities of arithmetic, or 

 of ordinary algebra, but as quantities in the broader sense, in which, 

 for example, quaternions are called quantities. The fact that they 

 may be operators does not prevent this. Just as in grammar verbs 

 may be taken as substantives, viz., in the infinitive mood, so in algebra 

 operators especially such as are capable of quantitative variation- 

 may be regarded as quantities when they are made the subject of 

 algebraic comparison or operation. Now I would not say that it is 

 necessary to treat every kind of operator as quantity, but I certainly 

 think that one so important as the linear vector operator, and one 



