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formerly to have been assumed, an approximate one. But, unless the earth's orbit were exactly circular the true place of the star will not be the centre of the hodograph. To enter into further details on this subject we should require geometrical diagrams or analytical symbols. The discoveries we have already described, and the papers and treatises we have mentioned, might well have formed the whole work of a long and laborious life. But, not to speak of the enormous collection of Ms. books, full to overflowing with new and original matter, left by Hamilton, which have been handed over to Trinity College, Dublin, and of whose contents we hope a large portion at least may soon be published, the works we have already called attention to barely form the greater portion of what he has published. His extraordinary investigations connected with the solution of algebraic equations of the Fifth Degree, and his exami. nation of the results arrived at by Abel, Jerrard, and Badano, in their researches on this subject, form another grand contribution to science. There is .. great paper on Fluctuating Functions, a subject which, since the time of Fourier, has been of immense and ever increasing value in physical applications of mathematics. Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of differential equations, which constantly occur in the treatment of physical questions, only a few items have been published, at intervals, in the Philosophical Magazine. Besides all this, Hamilton was a voluminous correspondent. Often a single letter of his occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of his mind, never to be satisfied with a general understanding of a question, he pursued it until he knew it in all its details. He was ever courteous and kind in answering any applications for assistance in the study of his works, even when his compliance must have cost him much valuable time. He was excessively precise and hard to please, with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little, compared with the extent of his investigations. His peculiar use of capitals, italics, and other typographical artifices for the purpose of imitating in writing and type, as closely as possible, the effects of emphasis and pause in a vivá voce lecture, will be evident from almost any of the extracts we have made from his works. To such an extent did he carry this, that some pages
of his Lectures are almost painful to the eve. Hamilton had, at one time, serious intentions of entering the Church, and was, more than once, offered ordination. The followin letter, written to the Editor of the }.} Ecclesiastical Journal, and published in that work, contains a very singular attempt to elucidate one of the grandest questions connected with the Christian religion.
“ON THE ASCENsion of our BLESSED Lord.
“Whitsun Eve, 1842.
“SIR-The meditations of a Christian, at this sacred season, turn naturally on that seeming pause in the operations of divine Providence, when, as at this time, the disciples who had seen their Lord parted from them, and taken up into heaven, were waiting at Jerusalem for the promised coming of the Comforter. , You will judge whether the following remarks, in part confessedly conjectural, but offered (it is hoped) in no presumptuous spirit, may properly occupy any portion of your columns, in connexion with the events which the Church at this SeaSOn COmmemorateS. *- “It may be assumed that your readers are disposed to adopt, in its simplicity, the teaching of the 4th article, that “Christ did truly rise again from death, and took again his body, with flesh, bones, and all things appertaining to the perfection of Man's nature; where with he ascended into heaven, and there sitteth, until he return to judge all Men at the last day. They will not be inclined to explain away the doctrine of the Ascension of the Lord's Humanity, into what some have sought to substitute for it, a ceasing of the Godhead to be manifested in the person of Christ. Far rather will they be ready to believe that the “glorious’ Ascension was the epoch of a more bright manifestation of God in Christ, than any which had been vouchsafed before though perhaps rather to angelic than to human beings; and that no merely figurative, though in part a spiritual sense, is to be assigned to those passages of Holy Writ, which speak of Jesus as having been highly exalted, and seated at the right hand of God. As God, indeed, we know that Heaven, and the Heaven of Heavens, cannot contain him; yet it is also declared that Heaven is His Throne, and Earth is His Footstool: and Scripture and the Church seem to attest alike, that the risen and glorified Humanity of Christ is now in Heaven, as in some holiest place, where God is eminently manifested, and eminently worshipped; his power, his name, and his presence dwelling there.
“A local translation of Christ's Body being thus believed, it is natural to believe also that this change of place was accomplished in time, and not with that strict instantaneity which may be attributed to a purely spiritual operation. Accordingly we read that at least the first part of the act of Ascension,-the part of which the Apostles were witnesses.--was gradual; their gaze could follow for a while their ascending Lord: nor was it instantly, though it may have been soon, that a cloud received him out of their sight. And to suppose that the remainder of that wonderful translation was effected without occupying some additional time, seems almost as much ‘against the truth of Christ's natural Body,” as that it should be “at one time in more places than one,' which latter notion a rubric of our Book of Common Prayer rejects as error and absurdity. The Cloud which hovered over Bethany was surely not that Heaven where Jesus sitteth at the right hand of God ; and to believe that his arrival, as Man, at the latter, was subsequent to his arrival at the former, seems to be a just as well as an obvious inference, from the Doctrine of the Ascension of His Body. “But how long was it subsequent 2 We dare not, by mere reasoning, attempt to decide this question. That place to which the Saviour has been exalted, and which, although in one sense “Heaven,” is in another sense declared to be “far above all heavens,” may well be thought to be inconceivably remote from the whole astronomical universe; no eye, no telescope, we may suppose, has pierced the mighty interspace: light may not yet have been able to spread from thence to us, if such an effluence as light be suffered thence to radiate. And, on the other hand, it must be owned, that, vast beyond all thought of ours as the interval in space may be, Christ's glorious Body may have been transported over it, in any interval of time, however short. “Reason is silent then: nor can we expect to find, on this point, a clear revelation in Scripture; but do we meet with no indications f Does Holy Writ leave us here entirely without light? I think that it does not: and shall submit to you a view, which it seems to me to suggest. “First, it is clear from Scripture, that the Ascension of Christ had been entirely performed before the Descent of the Spirit on the Day of Pentecost. Thus, in a well-known verse of that sixty-eighth Psalm, which the Church has connected with the Service for Whitsunday, and which St. Paul has quoted in reference to the Ascension; in the first sermon of Peter to the Jews; and in other passages of the Bible: the obtaining of “gifts for men,” the receiving from the Father the promise of the Holy Ghost, is spoken of as a result or consequence of Christ's having ascended up on high, -having been exalted by the right hand of God, having ascended, as did not David, into the Heavens. The act of ascending occupied therefore no longer time than that from Holy Thursday to Whitsunday. “But may it not have been allowed to occupy so long a time as this? No reason d priori can be given against the supposition ; no passage of Scripture, no decision of the Church, so far as I know, is against it. The very close connexion announced, in the texts above alluded to, between the Ascension of Christ into Heaven, and the Descent of the Holy Ghost upon Earth, appears to me an indication in its favour. For the purely spiritual nature of the later descent prevents the necessity, almost the possibility, of our supposing it to have occupied time at all. No sooner, it may reasonably be thought, did
Jesus take his seat at the right hand of God, than the Spirit fell upon the Apostles. The finished work, of ascending up on high, may have been followed instantly by the receiving of gifts for men. “Should this conjecture be admitted, of the Ascension not having been completed till the Day of Pentecost, although commenced ten days before, it might suggest much interesting meditation respecting the “glory,’ the “great triumph,” with which our Saviour Christ was then exalted into God's Kingdom of Heaven. May not the transit from the Cloud to the Throne have been but one continued passage, in long triumphal pomp, through powers and principalities made 3subject? May not the ‘Only Begotten Son' have then again been brought forth into the world,— not by a new Nativity, but (as it were) by Proclamation and Investiture, while the Universe beheld its God, and all the Angels worshipped him? And would not such triumphal progress harmonize well with that Psalm, which has always been referred in a special manner to the Ascension, and which speaks of the everlasting Gates as lifting up their heads, that the King of Glory might come in “Many other reflections occur to me, but I forbear. If anything unscriptural or uncatholic shall be detected by you in the foregoing remarks, or (in the event of you publishing them) by your readers, the pointing it out will be received as an obligation by, Sir, your obedient
Like most men of great originalty, Hamilton generally matured his ideas before putting pen to paper. “He used to carry on,” says his elder son, “long trains of algebraical and arithmetical calculation in his mind, during which he was unconscious of the earthly necessity of eating: we used to bring in a “snack’ and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards. I have been much with him in his periods of mathematical incubation, and would divide them into three, thus:–First, that of contemplation, above indicated. Second, that of construction. In this he committed to paper (or, if nothing else were at hand, as when in the garden, a few formulae written on his finger-nails) the skeleton, afterwards to be clothed with flesh and blood, of the results arrived at. Third, the didactic stage. Having now completely satisfied himself of the correctness of the results (and sometimes having retraced and simplified the method of discovery) he proceeded to consider how to teach it, and this by experiment. I was so long with him in his periods of mathematical incubation that I knew, almost by the tones of his voice and the expression of his eyes, when the didactic period had arrived, and
generally anticipated it by fetching the blackboard to whatever room he might be in. The audience generally consisted of the Observatory assistant and myself. . . . He was not so much teaching, as throwing his mind into a didactic attitude. I amused him once by saying that his lecturing us on equations of the fifth degree reminded me of the lion preparing for action by whetting his claws on the bark of a tree. . He appeared to enjoy intensely arithmetical calculations. I never saw him look so perfectly happy as when running like a sleuth-hound on the track of some unhappy decimal which had marred the work, and unearthing it in its den. . . I cannot otherwise express his attachment to his own Ms. volumes than by saying that he loved them. He once, at a luncheon party of students at the Observatory, ranged some thirty of them on the chimney-piece, and, turning to the students, said, ‘These books represent much of the happiness of my life.’”
A good idea of the process of “incubation " above mentioned is given by the following extract from a letter to a mathematical friend. Hamilton is speaking of one of the most beautiful discoveries contained in his last work on quaternions, the general symbolical solution of a vector equation of the first degree; and he writes, in 1859, the day after the discovery was made :
“While I was walking, on business of another sort, through Dublin yesterday, the question again occurred to me.
“Purae sunt platea, nihil ut meditantibus obstet"— “I nunc, et versus tecum meditare canoros.”
I was not so rash as to attempt the composition of a Sonnet in the streets; though, in acceptance of a challenge from a Lady, long ago, beside whom I was sitting in a Music Room, I did dash off a Sonnet before the performance had ceased. But those days are over:-happily 2 Yes, so far as the getting a little more sense, and less sensibility, is concerned.
“The problem, however, (though not the Sonnet,) haunted me, as it happened, yesterday, while I was walking from the Provost's House to that of the Academy; and though I wrote nothing down, that day, (for I had an immensity of other things to attend to,) I resumed it this morning; and arrived at what gyou might call, in the language of your last, a * * learingly easy” solution (in the sense of
perp gly easy being very UNLABorious, for I do not pretend that the reasoning does not require close attention). . . . So simple does this solution appear, that I hesitate as yet to place entire confidence in it; and therefore, till I have fully written it out, -for at present it is partly mental,—and have given it a complete and thorough reexamination, I hesitate to communicate it to you.”
When Vulcan cleft the labouring brain of Jove, With his keen axe, and set Minerva free, The unimprisoned Maid, exultingly, Bounded aloft, and to the Heaven above Turned her clear eyes, while the grim Workman strove To claim the Virgin Wisdom for his fee, His private wealth, his property to be, And hide in Lemnian eave her light of love. If some new truth, O Friend! thy toil discover, If thine eyes first by some fair form be blest, Love it for what it is, and as a lover Gaze, or with joy receive thine honoured guest: The new found Thought, set free, awhile may hover Gratefully, near thee, but it cannot reSt.
The following final extract, from a letter written in 1858, gives a very clear insight into the view Hamilton took of his own discoveries, and of the comparative value which he attached to methods and results. There is no doubt that, in the case of quaternions at least, he sought mainly to improve his methods, and almost studiously avoided the treatment of new subjects; and the result is, that in his hands alone the development attained is extraordinary:
“I reminded the R. I. A. that, so long ago as 1831, I had communicated to that body an Extension of (what is usually called) Herschel's Theorem: namely, the following extended Formula. . . By making the two particular assumptions . . . my formula becomes . . . which is, if I remember rightly, one form of Herschel's Theorem. . . . In speaking of “Herschel's Theorem,' I believe that I follow an usage, which of course he did not originate, but against which he has never complained. In my own case, however, I did complain, although (as I hope) gently, that a much less general formula of mine, which had indeed occurred in the same short paper of 1831 . . . had been cited, in a then recent number of the Cambridge and Dublin Mathematical Journal, under the title of “ Hamilton's Theorem.” What I meant was merely this;–that although I had no desire to have any theorem of mine so named, yet it was scarcely just, in my opinion, to select, out of a single and short paper, a formula which involved only one functional characteristic, one symbol of operation, and one ultimately evanescent variable; and by the manner in which the formula so selected was mentioned, or by the title under which it was cited, to ignore, or even virtually to reject, the much more general equation, which (as you see) involved two functional signs, two operators, and two ultimately evanescent variables. So, don't cite anything as “Hamilton's Theorem,” if you wish not to tread upon my corns ! I hope, indeed, that it may not be considered as unpardonable vanity or presumption on my part, if, as my own taste has always led me to feel a greater interest in methods than in results, so it is by METHODS, rather than by any THEoREMs, which can be separately quoted, that I desire and hope to be remembered. Nevertheless it is only human nature, to derive some pleasure from being cited, now and then, even about a “Theorem,” especially when . . . the quoter can enrich the subject, by combining it with researches of his own.”
In concluding, we have only to express a hope that we have rendered intelligible to the general reader, though perhaps in but of degree, at all events the nature of some of the grand investigations of this illustrious man. Of course there will ever be many who, though (or perhaps because) totally incapable of understanding anything lofty or difficult, will sneer out over such pages the cui bono of ignorance. They cannot see one of the sources of the vastness of modern commerce in Newton meditating about gravity, another in Watt patching a trumpery model. To their narrow vision the designer of a new easy-chair or the inventor of a new sauce, a lucky speculator or a sensation-novelist, even, it may be, a mountebank assuming the guise of a philosopher, is the grandest of the human race; but, while science lasts, the name of Hamilton will hold an honoured position among those of her few greatest sons.
We have endeavoured to give, in brief compass, a trustworthy account of Hamilton and his works. Of himself the account is easy, being mainly quoted from his correspondence. In our account of his works we have endeavoured, so far as we could, to avail ourselves of extracts from his writings. In several cases this was impossible; and we must warn the reader not to judge of the importance of the subject by the extremely small fragments which we have been forced to give as popularly intelligible specimens. Many of the preceding extracts are taken
from letters which we have received from Hamilton himself. We have derived some assistance from articles, or sketches, in the Dublin University Magazine (Jan. 1842), the Gentleman's Magazine (Jan. 1866,) and the Monthly Notices of the Royal Astronomical Society (Feb. 1866). The last of these, especially, is an admirable tribute to Hamilton's memory, but is somewhat marred by inaccuracies in the note on the nature of quaternions. And we must express our obligations to W. E. Hamilton, Esq., C. E., the elder son of Sir William, for many facts and documents; and for his kindness in verifying the statements we have made as to his father's ancestry and early history.
We are glad to hear that the author of the first of these sketches, the Rev. R. P. Graves, one of Hamilton's oldest friends, and brother of his former colleague in the University the Bishop of Limerick, is about to write his biography. The prospect of such a volume leaves us but one wish to express, that the authorities of Trinity College may publish, as speedily as possible, if not all, at least all that is most valuable in, the Mss. of the most distinguished among the many great men who, as students and professors, have shed lustre on the University of Dublin.
We conclude with an extract from the Opening Address (Session 1865–6) of the President of the Royal Society of Edinburgh, of which Hamilton was an Honorary Fellow.
“Sir John Herschel once wrote thus:– ‘Here whole branches of continental discovery are unstudied, and, indeed, almost unknown even by name. It is vain to conceal the melancholy truth. We are fast dropping behind. In mathematics we have long since drawn the rein and given over a hopeless race, etc.” Hamilton, while second to none, was one of the earliest of that brilliant array of mathematicians, who, since Herschel wrote, have removed this stigma, and well-nigh reversed the terms of his statement. Another was the late Professor Boole. . . . Their death has made a gap in the ranks of British science which will not soon be filled; and our sorrow is but increased by the recollection that they have been removed in the full vigour of their intellect, and when their passion for work was, if possible, stronger than ever.”
ART. III–1. The Book of Ballads. Edited by Box GAULTIER. Seventh Edition. Edinburgh, 1861.
2. Firmilian. Edinburgh, 1854.
3. Tales from Blackwood. Edinburgh.
SINCE the days of the prince of biographers, the wise and warm-hearted Plutarch of Chaeronea, very little has been done in literature for that parallelism which was so essential a part of his biographical theory. To take men of eminence, and place them in juxtaposition; to observe their points of similarity, and of dissimilarity in similarity, so that each should be separately more intelligible from the comparison of him with the other; —this, the Plutarchian idea, has been less fruitful than might have been expected, considering the just popularity of Plutarch from the days of Montaigne downwards. Bishop Hurd deserves the praise of having advocated its study, and of having suggested some material for the purpose; and Coleridge, in what he called the “landing-places” of his Friend, so far followed it up, that he made most ingenious suggestive comparisons between Luther and Rousseau, and between Erasmus and Voltaire. We are not going to deal just now with men of such magnitude; but we must be allowed to congratulate ourselves on having a good opportunity of applying the doctrine in the case of a group of distinguished contemporaries recently taken away. Within about a twelvemonth three humorists have been blotted from the roll of living British men of letters: Professor Aytoun, Mr. Thomas Love Peacock, and the Reverend Frank Mahony— better known as Father Prout. Each of these men represented one of the three kingdoms: Aytoun, our own bonnie Northern land; Peacock, England; and Mahony, Ire. land. They were all humorists. They were all lyrists. They were all more or less Bohemian and eccentric in the exercise of their gifts. They were all men of classical education. They were all men of strongly marked national type. Finally, they had this, too, in common, that they never became exactly popular, that is, universally popular in the sense in which Thackeray or Jerrold were so, but enjoyed their chief reputation among the cultivated classes. Every generation has writers of this peculiar type—writers often of higher powers and attainments than many who are better known, but who, somehow, never pass the line which divides those who are distinguished from those who are famous. It is curious to reflect that De Quincey never had a tithe as many readers as Mr. Harrison Ainsworth, and that Mr. Tupper is some fifty
times as well known as Henry Taylor. But this is one of the eternal phenomena of literature which never discourages real men of letters, while it ought to teach critics that perhaps their most important duty is to help to make known those whom the world has not learned to know for itself. If we propose to glance now at what was done by the three gentlemen just mentioned, for their generation, our object is partly to induce readers to become better acquainted with them at first-hand. Professor Aytoun's works are, indeed, well known in Scotland, but might be better known in the South and in Ireland. Peacock, in spite of the admirable wit and cleverness of his tales, is, we suspect, little appreciated out of London. Father Prout is loved and honoured by own countrymen, and in the literary world of the metropolis his name is a household word; but, elsewhere, few know how much, enjoyment may be got from his pages. We should like to see the reputation of these brilliant men counter-changed, as the heralds say— the Scoth and Irish reputations crossing into each other—and the English intermingling with both. We are no friends to excessive centralization. Indeed, we cherish national individualism as one of the conditions of literary variety, raciness, and colour, . But nationality without intercommunion has a constant tendency to degenerate into provincialism; and provincialism preserves national traits not as living things, but as petrifactions. The intellectual life of every country ought to blow over into other lands like a wind. The north wind is necessary to keep the south cool, and the south wind is necessary to keep the north from freezing. Now, it so happens, as has been already briefly hinted, that each of our three humorists had a strong flavour of his own country about him. In an age when so many Scotchmen emigrate, Aytoun devoted his life to Scotland. He formed himself on native models, and attached himself to a native school of literature. His humour—and it is humour with which we have to do in this paper—was essentially Scotch; that is to say, hearty or even vehement in expression sometimes, but dry to the taste; shrewd and thoughtful at bottom; and based on charac. ter rather than light and brilliant. He did not shine in epigram. His prose style wanted clearness, terseness, grace. His strong point both as writer and talker was humour proper, fun, a perception of the ludicrous; but a perception of the ludicrous from a Scot's point of view, in which the intellectual rather than the moral pleasure to be derived from it is the predominant object sought. Peacock, again, was eminently English in