CORRESPONDENCE WITH CAYLEY 159 



Since I returned to Edinburgh I have been considering more closely the question 

 of the new edition of my Quaternions and looking up specially Sylvester's papers 

 in the Comptes Rendus and the Phil. Mag. It seems to me from my point of view 

 (which I think is that of Hamilton) that all these things, excellent and valuable 

 as they are, are not Quaternions but developments of Matrices. As I understand 

 Hamilton's quest, it was for a method which should supersede Cartesian methods, 

 wherever it is possible to do so. Hence i, j, k, and their properties, though they 

 were the stepping stones by which Hamilton got his method, are to be discarded 

 in favour of a, q, <f>, etc.: and no problem or subject is a fit one for the introduction 

 of Quaternions if it necessitates the introduction of Cartesian Machinery.... 



The conclusion from this seems to be that I ought, instead of inserting your 

 contributions in the text of my book as it stands, to make a new chapter " On the 

 Analytical view of Quaternions" (or some such title) in which they will form the 

 spinal column. Therein will naturally assemble all the disaffected or lob-sided members, 

 which are not capable of pure quaternionic treatment but which are nevertheless 

 valuable, like the occipital ribs and the anencephalous heads in an anatomical 

 museum. 



Ten days later Cayley replied : 



"I... have not yet written out two further notes which I should like to send 

 you for the new Chapter which (I take it kindly) you do not compare with the 

 Chamber of Horrors at Madame Tussaud's....! need not say anything as to the 

 difference between our points of view; we are irreconcileable and shall remain so: 

 but is it necessary to express (in the book) all your feelings in regard to coordinates ? 

 One remark : I think you do not give your symbol < a sufficiently formal introduction : 

 it comes in incidentally through a particular case, without the full meaning of it 

 being shown. The two notes will be on the equation aq + qb = o and on Sylvester's 

 solution of af + bq + c = o." 



On Oct. 22, 1888, Tait wrote: 



I am very glad to know that you will give me two more of them [i.e. the 

 notes]; especially as I found Sylvester's papers hard to assimilate. A considerable 

 part of each paper seems to be devoted to correction of hasty generalizations in the 

 preceding one ! 



I don't know that my point of view of coordinates is very different from yours, 

 though my sight is vastly inferior. But I can see pretty clearly in the real world, 

 with its simple Euclidean space, by means of the quaternion telescope. Witness 

 a paper of Thomson's which I have just seen in type for the next Phil. Mag. ; 

 where three pages of formulae can easily, and with immense increase of comprehensi- 

 bility, be put into as many lines of quaternions. 



In his reply to this letter Cayley, after indicating his desire to see proofs 

 of Tait's Preface to his coming new edition of his Quaternions, asked : 



