1 64 PETER GUTHRIE TAIT 



And we see at once that it consists (or may be regarded as consisting) of two 

 independent and commutative factors, its Tensor and its Versor. 



But then comes the life of the whole; the recognition of the fact that when 

 POQ is a right angle, the versor of a-jp may be treated as in all respects lawfully 

 equivalent to the unit vector drawn perpendicular to p and to a. Thus every unit 

 vector is a quadrantal versor, and conversely. And, further, every versor is a power 

 of a unit vector. Thus, if the angle QOP be, in circular measure, A and if T be 

 the unit vector above defined, we have 



U - = T**l' = cosA + r sin A. 

 P 



Thus the separation of <r/p into the sum of its scalar and vector parts. 



(b) The second is physical rather than mathematical. We think of vectors in 

 a homogeneously strained solid, and if OP be strained into OQ we write a- = tpp. 



We recognise the conjugate strain <f>' and see the criterion of the pure strain 

 in <f> = <f>', as well as the general relation 



where m is the factor by which volume is increased. Thus we have the linear and 

 vector function, or (as you call it) the Matrix, with its fundamental characteristic. 

 Finally we have Nabla, which is defined by the equation 



expressing total differentiation so far as the shift, or displacement, d is concerned. 



In all this there is no reference whatever to anything Cartesian : and no more 

 need there be such in any application or development of these principles. And 

 I have always not merely allowed but proclaimed that, in the eyes of the mathematician, 

 Qns. have the fatal defect of being confined to Euclidian space. But this is one of 

 their great recommendations to the physicist.... 



I should like to know at your convenience when and how the notion of the 

 Matrix came to you : and whether Hamilton's simple case of it was an anticipation 

 or an application of the general theory. 



In response to this, Cayley sent Tait an article he had written out 

 for the Messenger of Mathematics on " Coordinates versus Quaternions," 

 remarking in a covering letter, " I do not know what has made me write 

 it just now, but it puts on record the views which I have held for many 

 years past and which have not been before published." 



He also expressed his dissatisfaction with Tail's sarcastic reference to 

 " Trilinear Coordinates " in the Preface to his Treatise on Quaternions, 

 and added in a postscript to his letter : 



" I certainly did not get the notion of a matrix in any way through quaternions : 

 it was either directly from that of a determinant; or as a convenient mode of 

 expression of the equations 



x' = ax + by 



