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 classifi disposition 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 application to the problem of five squares (for which, in the paper of 1867, he had given the results, but without demonstration), and to send it in as an essay. In taking this course he acted in accordance with the request of one of the academicians, who pointed out to him that in this way the Academy would be relieved of the embarrassment in which it was placed. No episode could bring out in a more striking light the distance that he had advanced beyond his contemporaries than that a question of which he had given the solution in 1867 as a corollary from more general principles that governed the whole class of investigations to which it belonged, should have been regarded by the French Academy in 1882 as of so much importance as to be worthy to form the subject of their great prize. It af fords, too, a singular illustration of the little attention that works destined to be. His two earliest papers were geometrical, and it was not till 1855 that his first contribution to the theory of numbers was published. For the ten years 1854-64 he devoted himself to this vast subject, and made himself completely master of everything that had ever been published upon it in any language. The results of this enormous amount of research are contained in his report on the theory of numbers, which appear in the British Association volumes between 1859 and 1865. This report, which occupies altogether about two hundred and fifty pages of close printing, is quite unique of its kind, and presents a complete and comprehensive view of the actual state of not only the widest but the most complicated and difficult branch of pure mathematics. It is remarkable for the same perfection of form, condensed mode of statement of processes, and what may be termed come classical attracted in the lifetime of their author. strictly logical order, or even in establishing 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 a logical perfection of form which has rendered in the former theory. His third subject obscurity and unnecessary difficulty. The fact the writings of Gauss open to the charge of 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 num- analogical order; there is nothing so far of bers, or "arithmetic," ie., 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 theour work is but begun, that we are still standory of continuous magnitude.. The aims ing 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 itto 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 succesGauss says studied; in fact, ninety-nine out of every sive steps of the demonstration. 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 sub- 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 pleasure which Euler must have taken in each step of his work; but we are conscious never This is one reason why Henry Smith's writings are difficult to read, for as regards the "arithmetical" knowledge required the senior wrangler is no better off than the schoolboy; but another and more powerful reason is afforded by the very perfection of form that he gave to his work. In this he resembled Gauss, and no words could more exactly describe his own work than those which he has applied to the great German mathematician in the following sentences: * theless that we are at an immense distance from the severe grandeur of design which is characteristic of all Gauss's greater efforts. The preceding criticism, if just, ought not to appear wholly trivial, for though it is quite true that in any mathematical work the substance is immeasurably more important than the form, yet it cannot be doubted that many mathematical memoirs of our own time suffer greatly (if we may dare to say so) from a certain 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 forof an abundant fertility of invention with an gotten long after the chief results and methods absolute rigorousness in demonstration, 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 wealthmatical conceptions as our own we are only of their own creations, and so fascinated by the interest attaching to the results at which they arrived, that they did not greatly care to expend their time in arranging their ideas in a They occur in an article on Gauss, by Mr. R. Tucker, in Nature, April 19, 1877. too apt to forget) that it is the business of mathematical science not only to discover new truths and new methods, but also to establish them, at whatever cost of time and labor, upon a basis of irrefragable reasoning. The μαθηματικὸς πιθανολογῶν has no more right to be listened to now than he had in the days of Aristotle; but it must be owned that since the invention of the "royal roads" of analysis, defective modes of reasoning and of proof have had a chance of obtaining currency which they never had before. It is not the greatest, but it is perhaps not the least, of Gauss's claim to the admiration of mathematicians, that while fully penetrated with a sense of the vastness of the science, he exacted the utmost rigorousness in every part of it, never passed over a difficulty as if it did not exist, and never accepted a theorem as true beyond the limits within which it could actually be demonstrated. matical paper that showed talent and originality, but was ill-arranged and incomplete, that it was worthy to have found a place in Gauss's waste-paper bas ket;" and it might, indeed, be truly said that much of the best work of Henry Smith's contemporaries was only worthy of a place in his waste-paper basket. The cold severity of his writings forms a curious contrast to the brilliant gaiety of his manner, and future generations who will know him only from his works will find it hard to believe what will be recorded of their author. In conversation and correspondence he was so lightsome and gay, and whimsical in the expression of his affection for his formulæ ; but the printed pages show nothing but stern dignity and power, without a trace of his own bright fancy or a word to show how he loved his work or the pleasure it had been to him. These words describe not only Henry Smith's views, but the quality of his own work. He did not care that his papers should be "easy to read," but he did care that they should be imperishable; and the words "adamantine solidity" express better than any others could do the character of the work he has left. To the "slovenly" way in which much of the mathe matics of our time is presented to the world he had the strongest dislike; and he spared no time or pains that all his own work should be as complete in its details as in its main results, and that it should be as perfect in form as in substance. He wished that what he did should be done for all time, and that it should also receive from his own hand the form which it was to retain. The order in which results are best and most logically displayed is not as a rule that in which they are most easily followed; and, besides this, his writings are rendered more difficult by the fact that he did not allow himself to publish "steps" in his work, in order to assist the reader, when 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 element- 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 which would produce a real advance. He severely restricted himself to such questions, and was never tempted to deviate from his course by anything that interested him on the way. His victories were won by the hardest of intellectual conflicts, in which for the time his whole heart and soul and powers were entirely and absolutely absorbed. It was in his wide interests and sympathies, the pleasure of intercourse with others, and the love of all that was good and cultivated, that he found relief from these. severe mental efforts. Had he not been gifted with a disposition that gave him the keenest sympathy with every human interest, that attracted him to society and endeared him to his friends, that gave him, in fact, his other noble life - the life the world knew his fierce devotion to the subject he loved would have ended his days long since. For all these reasons the standard of excellence of his writings is far above that of other great mathematicians. His published mathematical papers occupy perhaps twelve hundred pages; but this amount would have been tripled had he been less exacting in the quality of his work. Clerk Maxwell said of a mathe engagements and duties thickened upon him, he became more and more haunted and oppressed by the mass of work that lay unfinished in his study. "I have twenty papers embedded in my note-book. I extricated and published seven last year," he gave as a reason for being obliged to decline to undertake a fresh piece of work. But in spite of this constant anxiety, he continued to read new mathematical literature on its appearance all that related to his own subjects and a vast amount besides with the same avidity and ease as of old; and the still T unsolved mysteries of the subject and the any trace of egotism or dogmatism, his kindliness and generosity, the delicate gaiety of his wit, the brilliance of his conversation and his powers of conciliation. It is strange to notice how entirely what has been written of him and his character by those who were unaware of his mathematical eminence applies also to him as a mathematician. That the "note of personal ambition was absent from his composition is equally true, whether we regard his public or his mathematical career. He was well content to leave his works to tell their own tale when the proper time should come, and he cared not that they should bring him fame or honor in his lifetime. In this there was no trace of cynicism; no such feeling could exist in his nature. He worked at his subjects simply for the love of them, and he had no desire to make them the means of drawing attention to himself. Science can indeed boast few characters so perfect as his. His power of reading rapidly-as one might "skim a novel -new mathematiIt 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 great and varied was his knowledge in other directions, and how vast is the range and how rapid the progress of the sciences with which he showed this perfect familiarity. An article on Henry Smith could not be closed more fitly than by the concluding words of the notice in the Athenæum: "No one, probably, has ever had a larger circle of private friends to lament his loss. He had all the gifts which win and preserve attachment; not only sincerity, constancy, depth of feeling, and liveliness of sympathy, but a sweetness and nobility of nature to which no words can render adequate testimony." J. W. L. GLAISHER, From The Cornhill Magazine. NO NEW THING. But little space remains in which to speak of his attainments and influence in other fields, or of his personal and social gifts; but these are far more widely known than are the works that will give him his permanent place in the world's history. 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 he freely devoted to the common good gifts that, had he employed them in any 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 CHAPTER XXXV. HONORS DIVIDED. conspirator as ever a poor girl was afflicted with. If he would have simply turned and fled whenever Mrs. Winnington entered the room, the maintenance of the plot would have been less hopeless; but this he would not do. He seemed to think that, having taken upon himself to delude a fellow-creature, it behoved him to make believe a great deal; and, instead of chatting naturally about vivisection, as of yore, he took to paying wild and improbable compliments, to jerking out pretty phrases, too evidently learned by heart, and to suggesting agreeable projects with an indescribably sheepish air, while Mrs. Winnington sat staring at him, as if she had some faint idea that he was going out of his mind. Nor was this exasperating conduct the worst of what Edith had to endure at the hands of her well-meaning friend. From the moment of that meeting with Walter Brune at the Botanical Gardens, Tom had made up his mind to bring about the happy union of Miss Winnington with the young man whom he hoped some day to call his brother-in-law. This end, no doubt, might be achieved in many ways, it being evidently only a question of money; but it was important to ascertain, before proceeding to action, what Walter's tastes were, what career he considered himself best fitted for, and how a comfortable income could be provided for him without wearing too much the appearance of a gift. Mr. Stanniforth would have been very glad, therefore, if Miss Winnington would kindly have taken him into her full confidence, and the nods and winks and oracular speeches in which he indulged, by way of encouraging her thereto, were indeed enough to have tried the patience of Job. Edith could not tell him that she had broken off all relations with Walter; neither could she by any means make him understand that the subject was painful to her; and, what with Tom's provoking stupidity upon the one hand and her mother's suspicious acuteness upon the other, she began at length to ask herself whether it would not be a great deal better to hasten the inevitable hour, to sever the hair which sustained the impending sword, and have done with it. The courage of despair came to Edith's aid one morning, when her mother had been subjecting her to a more than usually severe course of interrogation, and with a calmness which astonished herself, she said, "I think I had better tell you that have refused Mr. Stanniforth." Mrs. Winnington immediately went through a sort of pantomime of dropping down dead. Verbal comment would, she felt, be absurdly inadequate to the occa sion, and for some minutes she would do nothing but gasp and groan. When, however, she did recover the use of her tongue she employed it with all that vigor of which she was a mistress. She scolded, she entreated, she wept copiously; finally she declared that Edith was a silly girl, who did not know her own mind, and that she herself would make it her busi ness to console poor Mr. Stanniforth, and to let him know that his rejection was not meant to be taken seriously. Thus there was nothing for it but to explain that Mr. Stanniforth stood in no need of consolation; and so, by degrees, the whole truth came out, and Mrs. Winnington received the crushing intelligence that not only was another to bear away the prize, but that that other's chance of doing so was the result of Edith's offi ciousness in enlightening the wretched man as to the state of his own affections. There is no saying what might not have happened to the offender after this, if a ring at the door-bell had not caused an abrupt suspension of hostilities. "Not at home!" gasped Mrs. Winnington from the sofa. "Go and tell them, not at home!" But either Edith was too late, or she thought only of effecting her own escape; for the next moment Colonel Kenyon was announced, and, striding into the room, beheld the foe with whom he had come to wage war prostrate upon her sofa, dishevelled as to her hair, and very red and swollen as to her eyes and nose. "How do you do?" said Mrs. Winnington. "I don't know why they let you come up. I am not in a state to receive visitors. I am very ill indeed." "Oh!" said Hugh, a good deal discon. certed; for he felt that the force of his attack must now be greatly weakened. "What is the matter with you? Gout again?" "I believe," answered Mrs. Winning. ton impressively, "that I am about to die." 66 Oh, I don't think so; you don't look like it at all," said Hugh, with conspicu. ous lack of sympathy. "I do not know," rejoined Mrs. Win nington, "what I may look like, but I know what I feel. However, I have no wish to weary you with my complaints. II have never talked about my health, not taken care of it, as you are aware. Per |