XI 



was especially the case when he brought carefully prepared diagrams, 

 such as those used in the modular-function division of the plane : 

 these diagrams were made much cleaver by the use of water-colours 

 to distinguish different sets of regions, and their preparation evidently 

 gave him pleasure. 



But as may be surmised his influence as a teacher was overshadowed 

 by his influence as an investigator. Those whom he affected by his 

 lectures belonged for the most part to the mathematical teachers in 

 Cambridge : the number of undergraduates whom he influenced was 

 small, though, when any one of them did come under his influence, 

 the effect was well marked. His starting point in any subject was 

 usually beyond the range of all other than quite advanced students ; 

 but to any able undergraduate who was willing to devote time, not 

 merely to the comprehension of the matter in the lectures but also to 

 collateral reading, the lectures were stimulating and inspiring. This 

 effect was partly due to the easy strength with which he worked, 

 partly to the spirit in which he approached old and new subjects 

 alike ; an independent suggestiveness and a singular freshness 

 marked his views, and gave an added interest to his exposition even 

 of a well-known theory. One reason of this freshness may be found 

 in the fact that his lectures consisted of the current researches u>on 

 which he was engaged at the time ; sometimes, even, a lecture would 

 be devoted to results which he had obtained since the preceding 

 lecture. Though the titles of the courses occasionally recur from one 

 year to another, the same course was never given twice. The new 

 matter in any course, once given, was usually incorporated in a paper 

 or memoir; and when the same subject was nominally lectured upon 

 again, it was a distinct part of the subject old notes were never 

 used a second time. 



It was not alone by his lectures that he acted as professor. 

 Students, seeking help or desiriug to interest him in their work, 

 found him always willing to give them the benefit of his advice, his 

 criticism, and his knowledge. Nor was it merely mathematicians in 

 Cambridge whom he helped in this way. He was continually con- 

 sulted by foreigners, who appreciated the promptness no less than 

 the fulness of information in his replies. 



It frequently happens that a man of genius, great enough to leave 

 a distinct impress of his originality upon his science, finds it irksome 

 to study what others have written. With the growth of all sciences 

 during the last fifty years, especially it may be said with the 

 growth of pure mathematics in that time, the tendency of workers is 

 to become specialists in their own subject and, perhaps, in subjects 

 immediately cognate with it, and to acquire only a slight acquaint- 

 ance with what is being done outside the circle of their limited 

 interests. Not so was Cayley : he was singularly learned in the work 



