Sept. 20, 1883] 



NATURE 



48: 



giving a new proof of some of the results, or point- 

 ing out that some of them were capable of greater gene- 

 ralisation. By his services in this way he has made 

 himself so widely popular that if European mathema- 

 ticians had to elect themselves a head I could not name 

 any one likely to have a larger number of votes. 



With respect to Cayley as an original inquirer, his 

 special merit has in my opinion been truly seized by Mr. 

 Glaisher, who has described him as the greatest living 

 master of algebra. While, as I have said, no part of 

 mathematics comes amiss to him, he is always happiest 

 when he can translate his theorems into pure algebra and 

 show that a proposed result is but the expression of an 

 algebraical fact. In this respect he differed from H. J. 

 Smith, by whose recent loss English mathematics has 

 so terribly suffered, who was entirely arithmetical in his 

 thoughts and work. 



Mathematicians, like chess-players, may be divided into 

 the book-learned and the original, the highest amount of 

 excellence being attained by those who combine great 

 knowledge of books with the power to strike into new 

 paths of their own. Of this I have spoken already. But 

 there is another division of chess-players, the solid 

 and the brilliant, some being full of ingenious devices 

 which, however, will not bear a careful examination ; 

 others being quite free from mistake but wooden in their 

 style. Cayley combines the excellences of the two kinds 

 in a very high degree, though his merits in the one 

 respect appear to me to be more marked than in the 

 other. Men weak in power of calculation have often 

 exhibited beautiful exercises of ingenuity in their attempts 

 to arrive at results by some shorter process. Such a 

 master of algebra in all its forms as Cayley was not to be 

 dismayed by any amount of calculation, and he therefore 

 has been able to trample down many a difficulty which 

 an inferior in this respect might have evaded by some 

 ingenious oblique method. 



As Cayley is not afraid of hard work himself, so it is 

 necessary for the readers of his papers not to be easily dis - 

 couraged by formidable calculations. But in my opinion it 

 is not this so much that makes Cayley's papers difficult to 

 read as the fact that he usually proceeds by the synthetic, 

 not the analytic, method. It usually happens that a 

 mathematical inquirer begins by proposing to himself 

 some comparatively simple question. By the time he 

 has found the answer to it, the subject opens on him ; 

 the first question suggests others, the theorem first dis- 

 covered is found to admit of wide generalisations, and 

 perhaps it may be found that these could have been 

 arrived at in quite another way. When the time comes 

 for the inquirer to publish his results to the world, the 

 most attractive course is to take his readers by exactly 

 the same road he has travelled himself, beginning with 

 the simple problem which first attracted attention, and 

 leading on step by step to the highest results arrived at . 

 Cayley on the contrary usually begins by trying to estab- 

 lish at once the highest generalisation he has reached, 

 writing down equations and proceeding to make calcu- 

 lations as to the good of which he has not taken his 

 readers into his confidence. The consequence is that 

 few master his papers but those who have found a clue 

 to them by some previous work in the same direction. 



I fancy that the difficulty of Cayley's papers is to be 



accounted for by his having had comparatively little experi- 

 ence in teaching mathematics until rather late in life, and 

 then only to students of the highest order. He lectured for a 

 few years at Trinity after taking his degree, but I dare say 

 that he did wisely in going to the bar instead of making a 

 livelihood by mathematical teaching at Cambridge, for 

 one who loved mathematics so much for its own sake, would 

 hardly sympathise with the many whose only object in 

 coming to him would be to learn how they could success- 

 fully get through an examination. On his return to Cam- 

 bridge he possibly would have extended his influence more 

 widely if he had taken what may seem the lazier course of 

 giving the same series of lectures year after year. But 

 Cayley preferred to give his classes his latest and highest 

 work, and each year has taken for his subject that of the 

 memoir on which he was for the time engaged. The 

 result has been that he has been brought little in contact 

 with any but the most advanced students, who alone could 

 profit by such instruction, nor even they, indeed, unless 

 they were as high-minded as himself, and were content to 

 spend a great amount of time and labour on work that 

 could not "pay" at the great University examination. 



As I have spoken of Cayley's lectures I ought not to 

 omit to mention the honour done him by the heads of the 

 Johns Hopkins University of Baltimore, Maryland, an 

 institution which numbers among its professors, as head 

 of its mathematical department, Cayley's distinguished 

 friend and fellow worker, Sylvester. They invited Cayley 

 to go over to lecture at Baltimore in the winter session 

 of 18S2. He accepted a proposal in every way so 

 flattering, and lectured at Baltimore in the months of 

 January to May, 1882, returning to England in June. 

 His subject was Elliptic and Abelian functions, and^ his 

 lectures, in which he considered from an algebraic point 

 of view the geometrical theories of Clebsch and Gordan, 

 were given for publication to the American Journal of 

 Mathematics, and are likely to form a classic memoir on 

 the subject. 



As I have said so much of Cayley's mathematical 

 labours, it will probably be expected that I should speak a 

 little less vaguely, and endeavour to explain more par- 

 ticularly the nature and progress of his discoveries ; yet it 

 is not easy to make the history of discovery in the higher 

 branches of pure mathematics readable even for so select 

 a class as the subscribers to Nature. It requires but a 

 small stock of technical knowledge to enable a reader to 

 follow with interest a history of mechanical inventions, or 

 of discoveries admitting of useful practical applications, 

 or of the skilled organisation of labour ; but what is to 

 be said of the work done by a solitary student in his 

 closet, the result of which will not so much as cheapen 

 one yard of calico ? 



It would be out of place if I were to take trouble here 

 to show that pure mathematics have after all added much 

 to the material wealth of the world. My subject is the 

 life of a great artist who has had courage to despise the 

 allurements of avarice or ambition, and has found more 

 happiness from a life devoted to the contemplation of 

 beauty and truth than if he had striven to make himself 

 richer, or otherwise push himself on in the world. We 

 do not classify painters according to the numbers capable 

 of appreciating their respective productions. On the 

 contrary, we can unde rstand that it is often the lowest 



