however, not to have been until 1852 that he addressed a memoir to the Royal Society, of which he was elected a Fellow in the same year.

His mathematical activity during this period was the more surprising, as he was able to devote to these studies only a limited portion of his time. He had been elected a Fellow of Trinity College in 1842; but as he was not willing to take Holy Orders, this was but a temporary provision, for he could only hold his Fellowship for seven years after his Master's degree. It became necessary for him therefore to look out for some profession more remunerative than mathematics, and very soon after taking his Master's degree he became a pupil of the eminent conveyancer, Mr. Christie. It is said that when offering himself as a pupil he modestly suppressed all mention of his antecedents, and that Mr. Christie was much surprised to find out on cross-examining him that he had to do with a Senior Wrangler and Fellow of Trinity. However this may be, he soon became Mr. Christie's favourite pupil, as indeed was not wonderful in the case of one who possessed a very clear head, immense capacity for work, and the power of throwing his whole mind into the work on which he was at the time engaged. After he was called to the bar he never had occasion to look elsewhere for business, for Mr. Christie was always glad to supply him with as much conveyancing work as he was willing to undertake. I have been told that some of his drafts were made to serve as models for students. But nothing that her wealthy rival had to offer could seduce Cayley into unfaithfulness to his first love, Mathematics. For Mathematics he always jealously reserved a due portion of time free from the encroachments of his business relations with Law, and it was during the time of his legal practice that some of his most brilliant mathematical discoveries were made. At last he obtained release from the embarrassment of a divided allegiance. By placing Lady Sadler's trusts on a new footing and founding the Sadlerian Professorship, his University was able to invite him to return, and he gladly accepted what was at the time a very modest provision, but which would enable him to give his whole time to the pursuits most congenial to him. Some time after his return to Cambridge his pecuniary position was improved. His College, which on his return had speedily made him an honorary Fellow, after a time reelected him to a foundation Fellowship, necessarily a very rare distinction, since the reelection of an ex-Fellow involves the exclusion of the claims of a younger candidate. Later still, in the course of University legislation about Professorships, the position of the Sad. lerian Professorship was improved. But these things could not have been foreseen at the time that Cayley accepted the office.

It was in 1863 that, after fourteen years of chamber life in Lincoln's Inn, he married and settled permanently in Cambridge. He never would own to any regret when his friends spoke to him of the prospects of professional advancement which he sacrificed by not remaining at the bar. He knew what mode of life would best promote his own happiness, and he had strength of mind to follow it without troubling his head about the riches or honours a different course might bring. His mathematical work gave him pleasure which he never found in law; and in his hatred of unnecessary words he was once wicked enough

to say that the object of law was to say a thing in the greatest number of words, and of mathematics to say it in the fewest. But, jesting apart, the University had no reason to regret the legal training and knowledge which he had acquired during his absence from it. It has much added to his usefulness as a member of the Council of the Senate, where his opinion has carried the greatest weight, and it has enabled him to be particularly useful both to his College and to the University in the drafting of new statutes and in the necessary preliminary deliberations. At the last contested Parliamentary election Cayley presided at one of the three polling places, and gave universal satisfaction, hearing patiently the arguments on both sides on all disputed points, and then promptly making a decision in a few words in such a way as to inspire general confidence.

But after all it is as a mathematical professor that Cayley is eminently "the right man in the right place." No one could be better fitted to discharge the duties prescribed for the Sadlerian Professor, "to explain and teach the principles of pure mathematics, and to apply himself to the advancement of the science." It is seldom that one man so well combines the two qualifications here indicated, viz. power to teach what is known already, and ability to extend the boundaries of knowledge. It constantly happens that men of great originality of genius find it irksome to study what has been done by others. And now every department of science has so enlarged its borders that it has not only become impossible for one man to master the whole circle of the sciences; but even a single department, such as pure mathematics, includes under it so great a variety of subjects that most men are content to be specialists, and, devoting themselves to their favourite topic, are satisfied with a very superficial knowledge of other branches. Cayley is quite as distinguished for the amount and universality of his reading as for his power of original work, and may fairly count as the most learned mathematician of the present day. I suppose that, if all European mathematicians could be subjected to a tripos competition, no matter who might come out first on the "problem" papers, Cayley would be far ahead in the "book work." And his tastes are so catholic that no form of mathematics comes amiss to him. I remember how we in Dublin were struck by his proficiency in pure geometry, a subject then much cultivated with us, but which we had been accustomed to look on as too little esteemed at Cambridge.

This wideness of knowledge has made Cayley invaluable as a mathematical referee. To several scientific societies (the Royal Society, the Mathematical Society, the Royal Astronomical Society, the Cambridge Philosophical Society) he has long been a principal adviser as to the merits of mathematical papers presented for publication, no one being more willing to take the trouble of examining such papers, or being better able to pronounce how much of their contents is new or important. And no one could te more ready and obliging with his advice to private students who have desired to interest him in their investigations, and to be assured by him that no unscrupulous predecessor has plagiarised their discoveries. Repeatedly have foreign mathematicians expressed their surprise at the rapidity with which he has dealt with such inquiries, an answer commonly coming by return of post, probably

giving a new proof of some of the results, or pointing out that some of them were capable of greater generalisation. By his services in this way he has made himself so widely popular that if European mathematicians 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 discouraged 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 discovered 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 calculations 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 experience 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 successfully get through an examination. On his return to Cambridge 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. 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.

The

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 1882. 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 particularly 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. do not classify painters according to the numbers capable of appreciating their respective productions. On the contrary, we can understand that it is often the lowest

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