162 PETER GUTHRIE TAIT 



a blank form to be filled up. But Sp<pp = i is strictly kinematical, and defines an 

 ellipsoid (or other central conicoid) with reference to a strain in this case a pure 

 strain the conception of which is vividly realistic. 



gives no physical suggestion at all. 



I should have said, in my paper, that we have to thank Cartesian processes 

 for the idea of an Invariant. In pure quaternions you have them always, so that 

 they present no feature for remark, p itself is an Invariant just as much as V is. 

 But what do you say to my little three term formula (on p. 95) which is equivalent 

 to 189 Cartesian terms? 



Cayley seems to have made no immediate reply to this letter ; but 

 on June 6, 1894, in a short note on other matters he threw in the 

 remark : 



" I wish you would tell me in what sense you consider Quaternions to be a method: 

 I do not see that they are so, in the sense in which coordinates are a method ; and 

 I consider them rather as a theory." 



Tait replied on June 10, 1894: 



As to your question about Quaternions, I fear that I do not quite catch your 

 meaning, so far at least as regards the technical distinction between a " Theory " and 

 a " Method." From my point of view, Quaternions are a mode of representing 

 geometrical or physical facts in such a clear way that one can see their mutual 

 relations and their consequences. They assist me in these in the same sort of way 

 as a figure or a model does, in the case of a knot or a complex surface : or as an 

 experiment of a crucial kind does. In fact they help one to think. I look upon 

 them as contrasted with, rather than as related to, numerical work whether by 

 logarithms or by definite integrals. These in themselves do not help you to think, 

 though they are vitally important when you wish to measure ; and though the working 

 of them out may require very much thought. I fear this, in its turn, will not be very 

 comprehensible to you : but I have not been in the habit of dealing with such 

 classes of questions ; or at all events of trying to express, in language, my notions 

 about them. 



The discussion now entered upon a more definite phase, and on 

 June 1 8, Cayley wrote : 



" Considering coordinates and quaternions each as a method I should formulate 

 the relation between them as follows : We seek to determine the position of a 

 variable point P in space in regard to a fixed point O. Thro" draw the rectangular 

 axes Ox, Oy, Oz. 



" Then 



(coordinates) the position is determined by the coordinates x, y, z. 

 (quaternions) the position is determined by means of the vector a ( = OP). 



