CHAPTER IV 

 QUATERNIONS 



TAIT'S quaternion work was unique ; and his influence in the development 

 of the calculus was second only to that of the great originator himself. He 

 alone of all Hamilton's contemporaries seems to have been able to grasp 

 the real significance of the method by direct perusal of Hamilton's Lectures. 

 The extraordinary seventh " Lecture " bristled with novelties and difficulties. 

 In grappling with these in his later Cambridge days Tait saw the value of 

 quaternions as an instrument of research. But it was not till he was settled 

 in Belfast that he began to make headway. 



On August ii, 1858, Dr Andrews wrote Hamilton a note introducing 

 his young mathematical colleague as one who " had been directing his 

 attention of late to Quaternions, and is anxious to be allowed to correspond 

 with you on that subject" 



In a cordial response to this letter Hamilton speaks of having recently 

 turned his attention to " differential equations and definite integrals in 

 connection with old but revived researches of my own (I do not mean, just 

 now, those which Jacobi has enriched by his comments)." He enclosed, no 

 doubt to test the powers of his would-be correspondent, a number of questions, 

 some of which Tait answered in his second letter of August 20. 



The first letter, of date August 19, must ever be regarded as of great 

 historic importance. It began a remarkable correspondence, which brought 

 Hamilton himself back to the study and further development of the subject, 

 culminating finally in the production of both Hamilton's Elements and Tail's 

 Elementary Treatise, 



After thanking Hamilton for the very kind manner in which he had 

 responded to Andrews' request, Tait continued : 



I attacked your volume on Quaternions immediately on its appearance, and 

 easily mastered the first 6 lectures but the portions I was most desirous of under- 

 standing, viz. the physical applications of the method, have given me very considerable 

 trouble; and, but for your offered assistance, I am afraid I should have had to 

 relinquish all hopes of using Quaternions as an instrument in investigation, on 



