PRESIDENTIAL ADDRESS, B.A., 1871. 139 



1837) and Boole and Cayley. This method was 

 greatly advanced by Gregory, who first gave to its 

 working-power a secure and philosophical founda- 

 tion and so prepared the way for the marvellous 

 extension it has received from Boole, Sylvester 

 and Cayley, according to which symbols of opera- 

 tion become the subjects not merely of algebraic 

 combination, but of differentiations and integrations 

 as if they were symbols expressing values of vary- 

 ing quantities. An even more marvellous develop- 

 ment of this same idea of the separation of symbols 

 (according to which Gregory separated the 

 algebraic signs + and from other symbols or 

 quantities to be characterised by them, and dealt 

 with them according to the laws of algebraic com- 

 bination) received from Hamilton a most astonish- 

 ing generalisation, by the invention actually of 

 new laws of combination, and led him to his 

 famous " Quaternions," of which he gave his 

 earliest exposition to the Mathematical and 

 Physical Section of this Association, at its meeting 

 in Cambridge in the year 1845. Tait has taken up 

 the subject of quaternions ably and zealously, and 

 has carried it into physical science with a faith 



