132 PETER GUTHRIE TAIT 



taken possession of a very large book, which book I call A. 1859, and which is to 

 relate entirely to quaternions. As yet, in it, I have confined myself to a new 

 discussion of FIRST PRINCIPLES." 



Tail's reply to this constituted the greater part of his letter 23. He 

 said : 



Many thanks for your kind and flattering letter....! applaud your purpose of 

 publishing a practical " Manual of Quaternions." I may mention to you that I had 

 been thinking of attempting something of the kind (but of course a very elementary 

 work) if the idea met with your approval but that was of course before I heard 

 that you intended doing anything of the kind yourself. There was one feature of 

 my dawning idea which might suit you that was to get it printed as one of 

 Macmillan's Cambridge series of which my Treatise on Dynamics forms a portion. 

 It would thus be directly introduced to the largest body of mathematicians in this 

 country.... Another feature would have been (and without this no book takes in 

 Cambridge) numerous examples of the great simplicity of the new method....! merely 

 mention my own half-developed scheme to show you that I think your present 

 proposal an excellent one, and perhaps to give you a useful hint or two with the 

 object of Quaternionizing my own University. 



In letter xvi of date April 10, 1859, Hamilton referred in a 

 remarkably prescient manner to the part which Tail was destined to 

 play in the development of quaternions. He wrote : 



" Let me be permitted to congratulate YOU (as well as myself most sincerely 

 do I add this last objective case) on your having taken up the Quaternions. 

 They will owe MUCH to you ; but I think that you will owe something to them. 

 This may be only the natural vanity of an author ; but I believe that an early 

 appreciation of genius wins a corresponding appreciation, in its turn, from mankind, 

 for itself; even if not accompanied, as in your case it is, and will be, by independent 

 acts of discovery'.' 



These extracts show unmistakably that the mathematical world owes 

 more to Tait than has yet been revealed. It was he who fired Hamilton 

 with the ambition to write his second great Treatise on Quaternions. As 

 we read the correspondence, and especially Hamilton's long chapter-like 

 letters, we see some of the leading features of the Elements taking 

 shape. Had Hamilton lived to write the Preface to the unfinished 

 Elements he probably would have mentioned explicitly the value of 

 the Tait Correspondence. All we have, however, in published form is a 

 footnote towards the close of the unfinished work, where Tait is 

 spoken of as one "eminently fitted to carry on, happily and usefully, this 

 new branch of mathematical science ; and likely to become in it, if the 

 expression may be allowed, one of the chief successors to its inventor." 



