have placed it on record for all future | "mathematical good taste," that distin generations. To say that he “made sac-guished all his work. Not only does the rifices” for it would be untrue; such was report contain a complete account of the his love for it that he regarded nothing as wonderful series of discoveries of Gauss a sacrifice; he never thought that there and his pupils and successors, but there was anything worthy to be compared to is also much original matter, though with it, or that a sacrifice in such a cause was characteristic modesty it is but rarely that possible. Not Gauss, nor Euler, nor Ja- it is distinguished in any way from results cobi, nor any mathematician who gave up that are merely quoted.' But the amount to it all the working hours of his life, of original work that he accomplished was cared for it more than he, and his perfect far greater than he could find room for in devotion was such as only a nature so the report, and the splendid advances that beautiful as his could feel. Those only he made in the science were communiwho know how completely his heart was cated to the Royal Society in a series of engrossed by it, how he longed to attack papers between 1860 and 1867. Attention the obstacles that barred the progress of has just been directed to one of these the science, to solve the mysteries that papers by the award to him of the great he felt were within his grasp, and to com- mathematical prize of the French Acadplete his unfinished successes - problems emy. The subject of the prize was the only half worked out, but through which decomposition of a number as a sum he could see his way can appreciate of five squares - a very special case the unselfishness and the sweetness of of the general question of the classifidisposition which made him yield so will- cation of quadratic forms, of which he ingly and gracefully to the wishes of had published the complete solution in friends, and take a leading part not only 1867: Eisenstein had partially solved

. in the business and management of his the question of five squares, and the own university, but wherever the cause of French Academy, in ignorance of Henry learning or science was involved; in fact, Smith's work, proposed the completion of he never refused to give up his time and Eisenstein's solution as the subject for attention to any purpose for which his the prize for 1883. When this subject friends asked his help, and where he was announced last year Henry Smith's thought his services might be of use. time was engrossed by investigations conBut for this he would have been alive nected with his large memoir on elliptic now; the incessant cares and anxieties of functions, and besides having a great his numerous occupations, combined with dislike to become a competitor, especially the exhaustion produced by the severe under the circumstances, he was very remental efforts to which every moment of luctant to leave, even for a short time, the his spare time was devoted, have prema: work he had in hand. At length, howturely closed one of the most perfect and ever, he decided to write out a portion of valuable lives of our age.

his published work, together with its apHis two earliest papers were geometri- plication to the problem of five squares cal, and it was not till 1855 that his first (for which, in the paper of 1867, he had contribution to the theory of numbers given the results, but without demonstra. was published. For the ten years 1854-64 tion), and to send it in as an essay. In he devoted himself to this vast subject, taking this course he acted in accordance and made himself completely master of with the request of one of the academic everything that had ever been published cians, who pointed out to him that in this upon it in any language. The results of way the Academy would be relieved of this enormous amount of research are the embarrassment in which it was placed. contained in his report on the theory of No episode could bring out in a more numbers, which appear in the British striking light the distance that he had adAssociation volumes between 1859 and vanced beyond his contemporaries than 1865. This report, which occupies alto- that a question of which he had given gether about two hundred and fisty pages the solution in 1867 as a corollary from of close printing, is quite unique of its more general principles that governed the kind, and presents a complete and com- whole class of investigations to which it prehensive view of the actual state of not belonged, should have been regarded by only the widest but the most complicated the French Academy in 1882 as of so and difficult branch of pure mathematics. much importance as to be worthy to form It is remarkable for the same perfection the subject of their great prize. It af. of form, condensed mode of statement fords, too, a singular illustration of the of processes, and what may be termed little attention that works destined to be. come classical attracted in the lifetime of strictly logical order, or even in establishing their author.

by irrefragable proof propositions which they He was led by his researches on the instinctively felt, and could almost see, to be theory of numbers to the theory of ellip- true.

With Gauss the case was otherwise. It tic functions, and on this subject he has may seem paradoxical, but it is probably neverpublished since 1864 results scarcely in theless true, that it is precisely the effort after ferior in importance to his achievements the writings of Gauss open to the charge of

a logical perfection of form which has rendered in the former theory. His third subject obscurity and unnecessary difficulty. The fact was modern geometry, in which he was is that there is neither obscurity nor difficulty quite without a rival in England, and of in his writings as long as we read them in the which he showed the same wonderful submissive spirit in which an intelligent schoolmastery. All that he published had refer- boy is made to read his Euclid. Every asserence to one or other of these three sub- tion that is made is fully proved, and the asserjects. Pure mathematics is divisible into tions succeed one another in a perfectly just two great branches -- the theory of oum- analogical order; there is nothing so far of bers, or “arithmetic,” i.e., the theory of which we can complain. But when we have

finished the perusal, we soon begin to feel that discrete magnitude, and algebra, the the.

our work is but begun, that we are still standory of continuous magnitude. The aimsing on the threshold of the temple, and that and methods and processes of the two there is a secret which lies behind the veil, and branches are quite distinct. The algebra. is as yet concealed from us. . . . No vestige ical branch, which admits of application appears of the process by which the result it. to physics and to all the exact sciences, is self was obtained, perhaps not even a trace of the one that has been most generally the considerations which suggested the successtudied; in fact, ninety-nine out of every sive steps of the demonstration. Gauss says hundred mathematical papers relate to it. more than once that, for brevity, he only gives A characteristic of Henry Smith's work, his propositions. Pauca sed matura were the

the synthesis, and suppresses the analysis of no less than of Gauss's, is the “arithmet words with which he delighted to describe the ical” mode of treatment that runs through character which he endeavored to impress upon the whole of it, no matter what the subo his mathematical writings. If, on the other ject; and his great command over the hand, we turn to a memoir of Euler's, there is processes of the science of number is a sort of free and luxuriant gracefulness about everywhere conspicuous.

the whole performance which tells of the quiet This is one reason why Henry Smith's pleasure which Euler must have taken in each writings are difficult to read, for as regards step of his work ; but we are conscious neverthe "arithmetical" knowledge required theless, that we are at an immense distance the senior wrangler is no better off than from the severe grandeur of design which is

characteristic of all Gauss's greater efforts. the schoolboy ; but another and more The preceding criticism, if just, ought not to powerful reason is afforded by the very appear wholly trivial, for though it is quite perfection of form that he gave to his true that in any mathematical work the subwork. In this he resembled Gauss, and stance is immeasurably more important than no words could more exactly describe his the form, yet it cannot be doubted that many own work than those which he has ap. mathematical memoirs of our own time suffer plied to the great German mathematician greatly (if we may dare to say so) from a cerin the following sentences : *

tain slovenliness in the mode of presentation;

and that (whatever may be the value of their If we except the great name of Newton (and contents) they are stamped with a character of the exception is one which Gauss himself would slightness and perishableness which contrasts have been delighted to make) it is probable strongly with the adamantine solidity and clear that no mathematician of any age or country hard modelling, which (we may be sure) will has ever surpassed Gauss in the combination keep the writings of Gauss from being for. of an abundant fertility of invention with an gotten long after the chief results and methods absolute rigorousness in denonstration, which contained in them have been incorporated in the ancient Greeks themselves might have en treatises more easily read, and have come to vied. It may be admitted, without any dis- form a part of the common patrimony of all paragement to the eminence of such great working mathematicians. And we must never mathematicians as Euler and Cauchy, that they forget (what in an age so futile of new mathewere so overwhelmed with the exuberant wealth matical conceptions as our own we are only of their own creations, and so fascinated by the too apt to forget) that it is the business of interest attaching to the results at which they mathematical science not only to discover new arrived, that they did not greatly care to ex. truths and new methods, but also to establish pend their time in arranging their ideas in a them, at whatever cost of time and labor, upon

a basis of irrefragable reasoning. • They occur in an article on Gauss, by Mr. R. The μαθηματικός πιθανολογων has no more Tucker, in Nature, April 19, 1877.

right to be listened to now than he had in the

days of Aristotle; but it must be owned that matical paper that showed talent and since the invention of the “royal roads” of originality, but was ill-arranged and inanalysis, defective modes of reasoning and of complete, that it was “worthy to have proof have had a chance of obtaining currency found a place in Gauss's waste paper bas. which they never had before. It is not the ket;” and it might, indeed, be truly said greatest, but it is perhaps not the least, of Gauss's claim to the admiration of mathe- that much of the best work of Henry

Smith's contemporaries was only worthy maticians, that while fully penetrated with a sense of the vastness of the science, he exacted of a place in his waste-paper basket. the utmost rigorousness in every part of it, The cold severity of his writings forms never passed over a difficulty as if it did not a curious contrast to the brilliant gaiety exist, and never accepted a theorem as true of his manner, and future generations beyond the limits within which it could actually who will know him only from his works be demonstrated.

will find it hard to believe what will be These words describe not only Henry recorded of their author. In conversaSmith's views, but the quality of his own tion and correspondence he was so lightwork. He did not care that his papers some and gay, and whimsical in the exshould be "easy to read,” but he did care pression of his affection for his formulæ; that they should be imperishable; and the but the printed pages show nothing but words “adamantine solidity” express bet. stern dignity and power, without a trace ter than any others could do the character of his own bright fancy or a word to show of the work he has left. To the "slove how he loved his work or the pleasure it enly" way in which much of the mathe- had been to him. matics of our time is presented to the His victories were won by the hardest world he had the strongest dislike; and of intellectual conflicts, in which for the he spared no time or pains that all his time his whole heart and soul and powers own work should be as complete in its were entirely and absolutely absorbed. It details as in its main results, and that it was in his wide interests and sympathies, should be as perfect in form as in sub- the pleasure of intercourse with others, stance. He wished that what he did and the love of all that was good and culshould be done for all time, and that it tivated, that he found relief from these . should also receive from his own hand severe mental efforts. Had he not been the form which it was to retain. The gifted with a disposition that gave him order in which results are best and most the keenest sympathy with every human logically displayed is not as a rule that in interest, that attracted him to society and which they are most easily followed; and, endeared him to his friends, that gave besides this, his writings are rendered him, in fact, his other noble life – the life more difficult by the fact that he did not the world knew — his fierce devotion to allow bimself to publish “steps ” in his the subject he loved would have ended his work, in order to assist the reader, when days long since. they were not required by the logic. An. His extreme modesty forbade him ever other point that should also be noticed is to speak of his work except to those who this: mathematicians usually work at knew of it and appreciated it, and even whatever interests them, and publish pa. then he generally referred to it only in his pers of various degrees of importance, own light way; but there were times when some relating to the boundaries of the he made no attempt to conceal the intense subject and others only to quite elemento delight he had felt in the discovery of ary matters; but it was not so in his case. principles that he knew must remain land. He directed his efforts only to'acknowl. marks in science. As time went on, and edged difficulties in science, victory over engagements and duties thickened upon which would produce a real advance. He him, he became more and more haunted severely restricted himself to such ques- and oppressed by the mass of work that tions, and was never tempted to deviate lay unfinished in his study. “I have from his course by anything that inter- twenty papers embedded in my note-book. ested him on the way.

I extricated and published seven last For all these reasons the standard of year,” he gave as a reason for being excellence of his writings is far above obliged to decline to undertake a fresh that of other great mathematicians. His piece of work. But in spite of this conpublished mathematical papers occupy stant anxiety, he continued to read new perhaps twelve hundred pages; but this mathematical literature on its appearance amount would have been tripled had he — all that related to his own subjects and been less exacting in the quality of his a vast amount besides with the same work. Clerk Maxwell said of a mathe. | avidity and ease as of old; and the still unsolved mysteries of the subject and the any trace of egotism or dogmatism, his endeavor to discern indications of the kindliness and generosity, the delicate lines that future discovery would take ex. gaiety of his wit, the brilliance of his con. ercised even a greater fascination over versation and his powers of conciliation. bim than ever. Only three months before It is strange to notice how entirely what his death, referring to the opinion (ex. has been written of him and his character pressed by a speaker at the Balfour Me by those who were unaware of his mathmorial Meeting at Cambridge) that a ematical eminence applies also to him as man's most original ideas came to him a mathematician. That the “note" of before he was thirty, he said that in his personal ambition was absent from bis own case he was certain that not only had composition is equally true, whether we bis power of seeing and understanding regard his public or his mathematical things uninterruptedly increased all career. He was well content to leave his through his life, but that his thoughts works to tell their own tale when the and ideas and “invention” had under: proper time should come, and he cared gone a corresponding progression and not that they should bring him fame or development. A glance through his note-honor in his lifetime. In this there was books affords striking evidence of this, no trace of cynicism; no such feeling for the later entries are especially rich could exist in his nature. He worked at in suggestions for future researches and bis subjects simply for the love of them, in "guesses " at what the results may be and he had no desire to make them the found to be.

means of drawing attention to himself. His power of reading rapidly- as one Science can indeed boast few characters might "skim a novel – new mathemati. so perfect as his. It has been sometimes cal publications of the most difficult kind, said of him that he was too fond of comseizing the ideas and grasping the proc- promises. If this be so it may be partly esses as if by intuition, was a truly won explained by his moderation and dislike derful gift. if the bare truth were told of extremes ; but a truer reason may be with regard to the accuracy and extent of found in the quickness and breadth of his : his acquaintance with the actual state of intellect and sympathy, which enabled him mathematics, taken in its very widest to understand and appreciate both sides sense, it would seem simply incredible to of every question, and prevented him from any one who knew how much of his life ever pressing home a victory. was devoted to other occupations, how An article on Henry Smith could not be great and varied was his knowledge in closed more fitly than by the concluding other directions, and how vast is the words of the notice in the Athenæum : range and how rapid the progress of the “ No one, probably, has ever had a larger sciences with which he showed this per. circle of private friends to lament his fect familiarity.

loss. He had all the gifts which win and But little space remains in which to preserve attachment; not only sincerity, speak of his attainments and influence in constancy, depth of feeling, and liveliness other fields, or of his personal and social of sympathy, but a sweetness and nobility gifts; but these are far more widely known of nature to which no words can render than are the works that will give him his adequate testimony;". permanent place in the world's history.

J. W. L. GLAISHER. Of all that has been written of him since his death there is scarcely a word with which his friends will not all agree. Universal tribute has been paid to his brilliant genius, to the ungrudging manner in which

From The Cornhill Magazine, he freely devoted to the common good

NO NEW THING. gifts that, had he employed them in any

CHAPTER XXXV. way for his personal ambition, would have early won him a European reputation, to the serenity of heart which "enabled him The ways of deceit are seldom ways to wear the gifts of genius with sobriety of pleasantness; and Edith Winnington and to use them nobly and well, without soon found that the part which she had seeking to expend them in the purchase set herself to play was so full of difficulty of fame, or of wealth, or of advancement,” and discomfort as to be very nearly into his moderation, his insight into human supportable. In the first place, Mr. Stannature, his gentleness and modesty, his niforth, who abhorred crooked dealing

invincible wisdom,” his freedom from above all things, was as clumsy a fellow.

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conspirator as ever a poor girl was afflicted Mrs. Winnington immediately went with. If he would have simply turned through a sort of pantomime of dropping and fled whenever Mrs. Winnington en- down dead. Verbal comment would, she tered the room, the maintenance of the felt, be absurdly inadequate to the occaplot would have been less hopeless; but sion, and for some minutes she would do this he would not do. He seemed to nothing but gasp and groan. When, howthink that, having taken upon himself to ever, she did recover the use of her delude a fellow-creature, it behoved him tongue she employed it with all that vigor to make believe a great deal; and, instead of which she was a mistress. She scoldof chatting naturally about vivisection, as ed, she entreated, she wept copiously ; of yore, he took to paying wild and im- finally she declared that Edith was a silly probable compliments, to jerking out girl, who did not know her own mind, and pretty phrases, too evidently learned by that she herself would make it ber busiheart, and to suggesting agreeable projects ness to console poor Mr. Stanniforth, and with an indescribably sheepish air, while to let him know that his rejection was not Mrs. Winnington sat staring at him, as if meant to be taken seriously. she had some faint idea that he was going Thus there was nothing for it but to out of his mind.

explain that Mr. Stanniforth stood in no Nor was this exasperating conduct the need of consolation; and so, by degrees, worst of what Edith had to endure at the the whole truth came out, and Mrs. Winhands of her well-meaning friend. From nington received the crushing intelligence the moment of that meeting with Walter that not only was another to bear away Brune at the Botanical Gardens, Tom had the prize, but that that other's chance of made up his mind to bring about the happy doing so was the result of Edith's offiunion of Miss Winnington with the young ciousness in enlightening the wretched man whom he hoped some day to call his man as to the state of his own affections. brother-in-law. This end, no doubt, might There is no saying what might not have be achieved in many ways, it being evi- happened to the offender afier this, if a dently only a question of money; but it ring at the door-bell had not caused an was important to ascertain, before pro abrupt suspension of hostilities. ceeding to action, whiat Walter's tastes “ Not at home!”gasped Mrs. Winning. were, what

areer he considered himself ton from the sofa. “Go and tell them, best fitted for, and how a comfortable in not at home!” come could be provided for him without But either Edith was too late, or she wearing too much the appearance of a thought only of effecting her own escape; gift. Mr. Stanniforth would have been for the next moment Colonel Kenyon was very glad, therefore, if Miss Winnington announced, and, striding into the room, would kindly have taken him into her full beheld the foe with whom he had come confidence, and the nods and winks and to wage war prostrate upon her sofa, oracular speeches in which he indulged, dishevelled as to her hair, and very red by way of encouraging her thereto, were and swollen as to her eyes and nose. indeed enough to have tried the patience “ How do you do?" said Mrs. Winning. of Job. Edith could not tell him that she ton. “ I don't know why they let you had broken off all relations with Walter; come up. I am not in a state to receive neither could she by any means make him visitors. I am very ill indeed." understand that the subject was painful “Oh!” said Hugh, a good deal discon. to her; and, what with Tom's provoking certed; for he felt that the force of his stupidity upon the one hand and her attack must now be greatly weakened. mother's suspicious acuteness upon the “What is the matter with you? Gout other, she began at length to ask herself again ?" whether it would not be a great deal better "I believe," answered Mrs. Winning, to basten the inevitable hour, to sever ton impressively, “that I am about to the hair which sustained the impending die.” sword, and have done with it.

“Oh, I don't think so; you don't look The courage of despair came to Edith's like it at all," said Hugb, with conspicu. aid one morning, when her mother had ous lack of sympathy. been subjecting her to a more than usually “I do not know," rejoined Mrs. Win severe course of interrogation, and with nington, “what I may look like, but I a calmness which astonished herself, she know what I feel. However, I have no said,

wish to weary you with my complaints “ I think I had better tell you that I I have never talked about my health, noi have refused Mr. Stanniforth."

taken care of it, as you are aware. Per

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