Simultaneously with his growth in fame came the growth of his several friendships. There were, in the first place, his scientific friendships with Herschel, Robinson, and many others with whom he had copious correspondence. In the excellent biography to which I have referred, Hamilton's correspondence with Coleridge may be read, as can also the letters to his lady correspondents, among them being Maria Edgeworth, Lady Dunraven, and Lady Campbell. Many of these sheets relate to literary matters, but they are largely intermingled With genial pleasantry, and serve at all events to show the affection and esteem with which he was regarded by all who had the privilege of knowing him. There are also the letters to the sisters whom he adored, letters brimming over with such exalted sentiment, that most ordinary sisters would be tempted to receive them with a smile in the excessively improbable event of their still more ordinary brothers attempting to pen such effusions. There are also indications of letters to and from other young ladies who from time to time were the objects of Hamilton's tender admiration. We use the plural advisedly, for, as Mr. Graves has set forth, Hamilton's love affairs pursued a rather troubled course. The attention which he lavished on one or two fair ones was not reciprocated, and even the intense charms of mathematical discovery could not assuage the pangs which the disappointed lover experienced. At last he reached the haven of matrimony in 1833, when he was married to Miss Bayly. Of his married life Hamilton said, many years later to De Morgan, that it was as happy as he expected, and happier than he deserved. He had two sons, William and Archibald, and one daughter, Helen, who became the wife of Archdeacon O'Regan.

[PLATE: SIR W. ROWAN HAMILTON.]

The most remarkable of Hamilton's friendships in his early years was unquestionably that with Wordsworth. It commenced with Hamilton's visit to Keswick; and on the first evening, when the poet met the young mathematician, an incident occurred which showed the mutual interest that was aroused. Hamilton thus describes it in a letter to his sister Eliza:—

"He (Wordsworth) walked back with our party as far as their lodge, and then, on our bidding Mrs. Harrison good-night, I offered to walk back with him while my party proceeded to the hotel. This offer he accepted, and our conversation had become so interesting that when we had arrived at his home, a distance of about a mile, he proposed to walk back with me on my way to Ambleside, a proposal which you may be sure I did not reject; so far from it that when he came to turn once more towards his home I also turned once more along with him. It was very late when I reached the hotel after all this walking."

Hamilton also submitted to Wordsworth an original poem, entitled "It Haunts me Yet." The reply of Wordsworth is worth repeating:—

"With a safe conscience I can assure you that, in my judgment, your verses are animated with the poetic spirit, as they are evidently the product of strong feeling. The sixth and seventh stanzas affected me much, even to the dimming of my eyes and faltering of my voice while I was reading them aloud. Having said this, I have said enough. Now for the per contra. You will not, I am sure, be hurt when I tell you that the workmanship (what else could be expected from so young a writer?) is not what it ought to be. . .

"My household desire to be remembered to you in no formal way. Seldom have I parted—never, I was going to say—with one whom after so short an acquaintance I lost sight of with more regret. I trust we shall meet again."

The further affectionate intercourse between Hamilton and Wordsworth is fully set forth, and to Hamilton's latest years a recollection of his "Rydal hours" was carefully treasured and frequently referred to. Wordsworth visited Hamilton at the observatory, where a beautiful shady path in the garden is to the present day spoken of as "Wordsworth's Walk."

It was the practice of Hamilton to produce a sonnet on almost every occasion which admitted of poetical treatment, and it was his delight to communicate his verses to his friends all round. When Whewell was producing his "Bridgewater Treatises," he writes to Hamilton in 1833:—

"Your sonnet which you showed me expressed much better than I could express it the feeling with which I tried to write this book, and I once intended to ask your permission to prefix the sonnet to my book, but my friends persuaded me that I ought to tell my story in my own prose, however much better your verse might be."

The first epoch-marking contribution to Theoretical Dynamics after the time of Newton was undoubtedly made by Lagrange, in his discovery of the general equations of Motion. The next great step in the same direction was that taken by Hamilton in his discovery of a still more comprehensive method. Of this contribution Hamilton writes to Whewell, March 31st, 1834:—

"As to my late paper, a day or two ago sent off to London, it is merely mathematical and deductive. I ventured, indeed, to call it the 'Mecanique Analytique' of Lagrange, 'a scientific poem'; and spoke of Dynamics, or the Science of Force, as treating of 'Power acting by Law in Space and Time.' In other respects it is as unpoetical and unmetaphysical as my gravest friends could desire."

It may well be doubted whether there is a more beautiful chapter in the whole of mathematical philosophy than that which contains Hamilton's dynamical theory. It is disfigured by no tedious complexity of symbols; it condescends not to any particular problems; it is an all embracing theory, which gives an intellectual grasp of the most appropriate method for discovering the result of the application of force to matter. It is the very generality of this doctrine which has somewhat impeded the applications of which it is susceptible. The exigencies of examinations are partly responsible for the fact that the method has not become more familiar to students of the higher mathematics. An eminent professor has complained that Hamilton's essay on dynamics was of such an extremely abstract character, that he found himself unable to extract from it problems suitable for his examination papers.

The following extract is from a letter of Professor Sylvester to Hamilton, dated 20th of September, 1841. It will show how his works were appreciated by so consummate a mathematician as the writer:—

"Believe me, sir, it is not the least of my regrets in quitting this empire to feel that I forego the casual occasion of meeting those masters of my art, yourself chief amongst the number, whose acquaintance, whose conversation, or even notice, have in themselves the power to inspire, and almost to impart fresh vigour to the understanding, and the courage and faith without which the efforts of invention are in vain. The golden moments I enjoyed under your hospitable roof at Dunsink, or moments such as they were, may probably never again fall to my lot.

"At a vast distance, and in an humble eminence, I still promise myself the calm satisfaction of observing your blazing course in the elevated regions of discovery. Such national honour as you are able to confer on your country is, perhaps, the only species of that luxury for the rich (I mean what is termed one's glory) which is not bought at the expense of the comforts of the million."

The study of metaphysics was always a favourite recreation when Hamilton sought for a change from the pursuit of mathematics. In the year 1834 we find him a diligent student of Kant; and, to show the views of the author of Quaternions and of Algebra as the Science of Pure Time on the "Critique of the Pure Reason," we quote the following letter, dated 18th of July, 1834, from Hamilton to Viscount Adare:—

"I have read a large part of the 'Critique of the Pure Reason,' and find it wonderfully clear, and generally quite convincing. Notwithstanding some previous preparation from Berkeley, and from my own thoughts, I seem to have learned much from Kant's own statement of his views of 'Space and Time.' Yet, on the whole, a large part of my pleasure consists in recognising through Kant's works, opinions, or rather views, which have been long familiar to myself, although far more clearly and systematically expressed and combined by him. . . . Kant is, I think, much more indebted than he owns, or, perhaps knows, to Berkeley, whom he calls by a sneer, 'GUTEM Berkeley'. . . as it were, 'good soul, well meaning man,' who was able for all that to shake to its centre the world of human thought, and to effect a revolution among the early consequences of which was the growth of Kant himself."

At several meetings of the British Association Hamilton was a very conspicuous figure. Especially was this the case in 1835, when the Association met in Dublin, and when Hamilton, though then but thirty years old, had attained such celebrity that even among a very brilliant gathering his name was perhaps the most renowned. A banquet was given at Trinity College in honour of the meeting. The distinguished visitors assembled in the Library of the University. The Earl of Mulgrave, then Lord Lieutenant of Ireland, made this the opportunity of conferring on Hamilton the honour of knighthood, gracefully adding, as he did so: "I but set the royal, and therefore the national mark, on a distinction already acquired by your genius and labours."

The banquet followed, writes Mr. Graves. "It was no little addition to the honour Hamilton had already received that, when Professor Whewell returned thanks for the toast of the University of Cambridge, he thought it appropriate to add the words, 'There was one point which strongly pressed upon him at that moment: it was now one hundred and thirty years since a great man in another Trinity College knelt down before his sovereign, and rose up Sir Isaac Newton.' The compliment was welcomed by immense applause."

A more substantial recognition of the labours of Hamilton took place subsequently. He thus describes it in a letter to Mr. Graves of 14th of November, 1843:—

"The Queen has been pleased—and you will not doubt that it was entirely unsolicited, and even unexpected, on my part—'to express her entire approbation of the grant of a pension of two hundred pounds per annum from the Civil List' to me for scientific services. The letters from Sir Robert Peel and from the Lord Lieutenant of Ireland in which this grant has been communicated or referred to have been really more gratifying to my feelings than the addition to my income, however useful, and almost necessary, that may have been."

The circumstances we have mentioned might lead to the supposition that Hamilton was then at the zenith of his fame but this was not so. It might more truly be said, that his achievements up to this point were rather the preliminary exercises which fitted him for the gigantic task of his life. The name of Hamilton is now chiefly associated with his memorable invention of the calculus of Quaternions. It was to the creation of this branch of mathematics that the maturer powers of his life were devoted; in fact he gives us himself an illustration of how completely habituated he became to the new modes of thought which Quaternions originated. In one of his later years he happened to take up a copy of his famous paper on Dynamics, a paper which at the time created such a sensation among mathematicians, and which is at this moment regarded as one of the classics of dynamical literature. He read, he tells us, his paper with considerable interest, and expressed his feelings of gratification that he found himself still able to follow its reasoning without undue effort. But it seemed to him all the time as a work belonging to an age of analysis now entirely superseded.

In order to realise the magnitude of the revolution which Hamilton has wrought in the application of symbols to mathematical investigation, it is necessary to think of what Hamilton did beside the mighty advance made by Descartes. To describe the character of the quaternion calculus would be unsuited to the pages of this work, but we may quote an interesting letter, written by Hamilton from his deathbed, twenty-two years later, to his son Archibald, in which he has recorded the circumstances of the discovery:—

"Indeed, I happen to be able to put the finger of memory upon the year and month—October, 1843—when having recently returned from visits to Cork and Parsonstown, connected with a meeting of the British Association, the desire to discover the laws of multiplication referred to, regained with me a certain strength and earnestness which had for years been dormant, but was then on the point of being gratified, and was occasionally talked of with you. Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, 'Well papa, can you multiply triplets?' Whereto I was always obliged to reply, with a sad shake of the head: 'No, I can only ADD and subtract them,'

"But on the 16th day of the same month—which happened to be Monday, and a Council day of the Royal Irish Academy—I was walking in to attend and preside, and your mother was walking with me along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an UNDERCURRENT of thought was going on in my mind which gave at last a RESULT, whereof it is not too much to say that I felt AT ONCE the importance. An ELECTRIC circuit seemed to CLOSE; and a spark flashed forth the herald (as I FORESAW IMMEDIATELY) of many long years to come of definitely directed thought and work by MYSELF, if spared, and, at all events, on the part of OTHERS if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse—unphilosophical as it may have been—to cut with a knife on a stone of Brougham Bridge as we passed it, the fundamental formula which contains the SOLUTION of the PROBLEM, but, of course, the inscription has long since mouldered away. A more durable notice remains, however, on the Council Books of the Academy for that day (October 16, 1843), which records the fact that I then asked for and obtained leave to read a Paper on 'Quaternions,' at the First General Meeting of the Session; which reading took place accordingly, on Monday, the 13th of November following."

Writing to Professor Tait, Hamilton gives further particulars of the same event. And again in a letter to the Rev. J. W. Stubbs:—

"To-morrow will be the fifteenth birthday of the Quaternions. They started into life full-grown on the 16th October, 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge—which my boys have since called Quaternion Bridge. I pulled out a pocketbook which still exists, and made entry, on which at the very moment I felt that it might be worth my while to expend the labour of at least ten or fifteen years to come. But then it is fair to say that this was because I felt a problem to have been at that moment solved, an intellectual want relieved which had haunted me for at least fifteen years before.

"But did the thought of establishing such a system, in which geometrically opposite facts—namely, two lines (or areas) which are opposite IN SPACE give ALWAYS a positive product—ever come into anybody's head till I was led to it in October, 1843, by trying to extend my old theory of algebraic couples, and of algebra as the science of pure time? As to my regarding geometrical addition of lines as equivalent to composition of motions (and as performed by the same rules), that is indeed essential in my theory but not peculiar to it; on the contrary, I am only one of many who have been led to this view of addition."

Pilgrims in future ages will doubtless visit the spot commemorated by the invention of Quaternions. Perhaps as they look at that by no means graceful structure Quaternion Bridge, they will regret that the hand of some Old Mortality had not been occasionally employed in cutting the memorable inscription afresh. It is now irrecoverably lost.

It was ten years after the discovery that the great volume appeared under the title of "Lectures on Quaternions," Dublin, 1853. The reception of this work by the scientific world was such as might have been expected from the extraordinary reputation of its author, and the novelty and importance of the new calculus. His valued friend, Sir John Herschel, writes to him in that style of which he was a master:—

"Now, most heartily let me congratulate you on getting out your book—on having found utterance, ore rotundo, for all that labouring and seething mass of thought which has been from time to time sending out sparks, and gleams, and smokes, and shaking the soil about you; but now breaks into a good honest eruption, with a lava stream and a shower of fertilizing ashes.

"Metaphor and simile apart, there is work for a twelve-month to any man to read such a book, and for half a lifetime to digest it, and I am glad to see it brought to a conclusion."

We may also record Hamilton's own opinion expressed to Humphrey Lloyd:—

"In general, although in one sense I hope that I am actually growing modest about the quaternions, from my seeing so many peeps and vistas into future expansions of their principles, I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions was for the close of the seventeenth."

Bartholomew Lloyd died in 1837. He had been the Provost of Trinity College, and the President of the Royal Irish Academy. Three candidates were put forward by their respective friends for the vacant Presidency. One was Humphrey Lloyd, the son of the late Provost, and the two others were Hamilton and Archbishop Whately. Lloyd from the first urged strongly the claims of Hamilton, and deprecated the putting forward of his own name. Hamilton in like manner desired to withdraw in favour of Lloyd. The wish was strongly felt by many of the Fellows of the College that Lloyd should be elected, in consequence of his having a more intimate association with collegiate life than Hamilton; while his scientific eminence was world-wide. The election ultimately gave Hamilton a considerable majority over Lloyd, behind whom the Archbishop followed at a considerable distance. All concluded happily, for both Lloyd and the Archbishop expressed, and no doubt felt, the pre-eminent claims of Hamilton, and both of them cordially accepted the office of a Vice-President, to which, according to the constitution of the Academy, it is the privilege of the incoming President to nominate.

In another chapter I have mentioned as a memorable episode in astronomical history, that Sir J. Herschel went for a prolonged sojourn to the Cape of Good Hope, for the purpose of submitting the southern skies to the same scrutiny with the great telescope that his father had given to the northern skies. The occasion of Herschel's return after the brilliant success of his enterprise, was celebrated by a banquet. On June 15th, 1838, Hamilton was assigned the high honour of proposing the health of Herschel. This banquet is otherwise memorable in Hamilton's career as being one of the two occasions in which he was in the company of his intimate friend De Morgan.

In the year 1838 a scheme was adopted by the Royal Irish Academy for the award of medals to the authors of papers which appeared to possess exceptionally high merit. At the institution of the medal two papers were named in competition for the prize. One was Hamilton's "Memoir on Algebra, as the Science of Pure Time." The other was Macullagh's paper on the "Laws of Crystalline Reflection and Refraction." Hamilton expresses his gratification that, mainly in consequence of his own exertions, he succeeded in having the medal awarded to Macullagh rather than to himself. Indeed, it would almost appear as if Hamilton had procured a letter from Sir J. Herschel, which indicated the importance of Macullagh's memoir in such a way as to decide the issue. It then became Hamilton's duty to award the medal from the chair, and to deliver an address in which he expressed his own sense of the excellence of Macullagh's scientific work. It is the more necessary to allude to these points, because in the whole of his scientific career it would seem that Macullagh was the only man with whom Hamilton had ever even an approach to a dispute about priority. The incident referred to took place in connection with the discovery of conical refraction, the fame of which Macullagh made a preposterous attempt to wrest from Hamilton. This is evidently alluded to in Hamilton's letter to the Marquis of Northampton, dated June 28th, 1838, in which we read:—

"And though some former circumstances prevented me from applying to the person thus distinguished the sacred name of FRIEND, I had the pleasure of doing justice...to his high intellectual merits...I believe he was not only gratified but touched, and may, perhaps, regard me in future with feelings more like those which I long to entertain towards him."

Hamilton was in the habit, from time to time, of commencing the keeping of a journal, but it does not appear to have been systematically conducted. Whatever difficulties the biographer may have experienced from its imperfections and irregularities, seem to be amply compensated for by the practice which Hamilton had of preserving copies of his letters, and even of comparatively insignificant memoranda. In fact, the minuteness with which apparently trivial matters were often noted down appears almost whimsical. He frequently made a memorandum of the name of the person who carried a letter to the post, and of the hour in which it was despatched. On the other hand, the letters which he received were also carefully preserved in a mighty mass of manuscripts, with which his study was encumbered, and with which many other parts of the house were not unfrequently invaded. If a letter was laid aside for a few hours, it would become lost to view amid the seething mass of papers, though occasionally, to use his own expression, it might be seen "eddying" to the surface in some later disturbance.

The great volume of "Lectures on Quaternions" had been issued, and the author had received the honours which the completion of such a task would rightfully bring him. The publication of an immortal work does not, however, necessarily provide the means for paying the printer's bill. The printing of so robust a volume was necessarily costly; and even if all the copies could be sold, which at the time did not seem very likely, they would hardly have met the inevitable expenses. The provision of the necessary funds was, therefore, a matter for consideration. The Board of Trinity College had already contributed 200 pounds to the printing, but yet another hundred was required. Even the discoverer of Quaternions found this a source of much anxiety. However, the board, urged by the representation of Humphrey Lloyd, now one of its members, and, as we have already seen, one of Hamilton's staunchest friends, relieved him of all liability. We may here note that, notwithstanding the pension which Hamilton enjoyed in addition to the salary of his chair, he seems always to have been in some what straitened circumstances, or, to use his own words in one of his letters to De Morgan, "Though not an embarrassed man, I am anything rather than a rich one." It appears that, notwithstanding the world-wide fame of Hamilton's discoveries, the only profit in a pecuniary sense that he ever obtained from any of his works was by the sale of what he called his Icosian Game. Some enterprising publisher, on the urgent representations of one of Hamilton's friends in London, bought the copyright of the Icosian Game for 25 pounds. Even this little speculation proved unfortunate for the purchaser, as the public could not be induced to take the necessary interest in the matter.

After the completion of his great book, Hamilton appeared for awhile to permit himself a greater indulgence than usual in literary relaxations. He had copious correspondence with his intimate friend, Aubrey de Vere, and there were multitudes of letters from those troops of friends whom it was Hamilton's privilege to possess. He had been greatly affected by the death of his beloved sister Eliza, a poetess of much taste and feeling. She left to him her many papers to preserve or to destroy, but he said it was only after the expiration of four years of mourning that he took courage to open her pet box of letters.

The religious side of Hamilton's character is frequently illustrated in these letters; especially is this brought out in the correspondence with De Vere, who had seceded to the Church of Rome. Hamilton writes, August 4, 1855:—

"If, then, it be painfully evident to both, that under such circumstances there CANNOT (whatever we may both DESIRE) be NOW in the nature of things, or of minds, the same degree of INTIMACY between us as of old; since we could no longer TALK with the same degree of unreserve on every subject which happened to present itself, but MUST, from the simplest instincts of courtesy, be each on his guard not to say what might be offensive, or, at least, painful to the other; yet WE were ONCE so intimate, an retain still, and, as I trust, shall always retain, so much of regard and esteem and appreciation for each other, made tender by so many associations of my early youth and your boyhood, which can never be forgotten by either of us, that (as times go) TWO OR THREE VERY RESPECTABLE FRIENDSHIPS might easily be carved out from the fragments of our former and ever-to-be-remembered INTIMACY. It would be no exaggeration to quote the words: 'Heu! quanto minus est cum reliquis versari, quam tui meminisse!'"

In 1858 a correspondence on the subject of Quaternions commenced between Professor Tait and Sir William Hamilton. It was particularly gratifying to the discoverer that so competent a mathematician as Professor Tait should have made himself acquainted with the new calculus. It is, of course, well known that Professor Tait subsequently brought out a most valuable elementary treatise on Quaternions, to which those who are anxious to become acquainted with the subject will often turn in preference to the tremendous work of Hamilton.

In the year 1861 gratifying information came to hand of the progress which the study of Quaternions was making abroad. Especially did the subject attract the attention of that accomplished mathematician, Moebius, who had already in his "Barycentrische Calculus" been led to conceptions which bore more affinity to Quaternions than could be found in the writings of any other mathematician. Such notices of his work were always pleasing to Hamilton, and they served, perhaps, as incentives to that still closer and more engrossing labour by which he became more and more absorbed. During the last few years of his life he was observed to be even more of a recluse than he had hitherto been. His powers of long and continuous study seemed to grow with advancing years, and his intervals of relaxation, such as they were, became more brief and more infrequent.

It was not unusual for him to work for twelve hours at a stretch. The dawn would frequently surprise him as he looked up to snuff his candles after a night of fascinating labour at original research. Regularity in habits was impossible to a student who had prolonged fits of what he called his mathematical trances. Hours for rest and hours for meals could only be snatched in the occasional the lucid intervals between one attack of Quaternions and the next. When hungry, he would go to see whether any thing could be found on the sideboard; when thirsty, he would visit the locker, and the one blemish in the man's personal character is that these latter visits were sometimes paid too often.

As an example of one of Hamilton's rare diversions from the all- absorbing pursuit of Quaternions, we find that he was seized with curiosity to calculate back to the date of the Hegira, which he found on the 15th July, 622. He speaks of the satisfaction with which he ascertained subsequently that Herschel had assigned precisely the same date. Metaphysics remained also, as it had ever been, a favourite subject of Hamilton's readings and meditations and of correspondence with his friends. He wrote a very long letter to Dr. Ingleby on the subject of his "Introduction to Metaphysics." In it Hamilton alludes, as he has done also in other places, to a peculiarity of his own vision. It was habitual to him, by some defect in the correlation of his eyes, to see always a distinct image with each; in fact, he speaks of the remarkable effect which the use of a good stereoscope had on his sensations of vision. It was then, for the first time, that he realised how the two images which he had always seen hitherto would, under normal circumstances, be blended into one. He cites this fact as bearing on the phenomena of binocular vision, and he draws from it the inference that the necessity of binocular vision for the correct appreciation of distance is unfounded. "I am quite sure," he says, "that I SEE DISTANCE with EACH EYE SEPARATELY."

The commencement of 1865, the last year of his life saw Hamilton as diligent as ever, and corresponding with Salmon and Cayley. On April 26th he writes to a friend to say, that his health has not been good for years past, and that so much work has injured his constitution; and he adds, that it is not conducive to good spirits to find that he is accumulating another heavy bill with the printer for the publication of the "Elements." This was, indeed, up to the day of his death, a cause for serious anxiety. It may, however, be mentioned that the whole cost, which amounted to nearly 500 pounds, was, like that of the previous volume, ultimately borne by the College. Contrary to anticipation, the enterprise, even in a pecuniary sense, cannot have been a very unprofitable one. The whole edition has long been out of print, and as much as 5 pounds has since been paid for a single copy.

It was on the 9th of May, 1865, that Hamilton was in Dublin for the last time. A few days later he had a violent attack of gout, and on the 4th of June he became alarmingly ill, and on the next day had an attack of epileptic convulsions. However, he slightly rallied, so that before the end of the month he was again at work at the "Elements." A gratifying incident brightened some of the last days of his life. The National Academy of Science in America had then been just formed. A list of foreign Associates had to be chosen from the whole world, and a discussion took place as to what name should be placed first on the list. Hamilton was informed by private communication that this great distinction was awarded to him by a majority of two-thirds.

In August he was still at work on the table of contents of the "Elements," and one of his very latest efforts was his letter to Mr. Gould, in America, communicating his acknowledgements of the honour which had been just conferred upon him by the National Academy. On the 2nd of September Mr. Graves went to the observatory, in response to a summons, and the great mathematician at once admitted to his friend that he felt the end was approaching. He mentioned that he had found in the 145th Psalm a wonderfully suitable expression of his thoughts and feelings, and he wished to testify his faith and thankfulness as a Christian by partaking of the Lord's Supper. He died at half-past two on the afternoon of the 2nd of September, 1865, aged sixty years and one month. He was buried in Mount Jerome Cemetery on the 7th of September.

Many were the letters and other more public manifestations of the feelings awakened by Hamilton's death. Sir John Herschel wrote to the widow:—

"Permit me only to add that among the many scientific friends whom time has deprived me of, there has been none whom I more deeply lament, not only for his splendid talents, but for the excellence of his disposition and the perfect simplicity of his manners—so great, and yet devoid of pretensions."

De Morgan, his old mathematical crony, as Hamilton affectionately styled him, also wrote to Lady Hamilton:—

"I have called him one of my dearest friends, and most truly; for I know not how much longer than twenty-five years we have been in intimate correspondence, of most friendly agreement or disagreement, of most cordial interest in each other. And yet we did not know each other's faces. I met him about 1830 at Babbage's breakfast table, and there for the only time in our lives we conversed. I saw him, a long way off, at the dinner given to Herschel (about 1838) on his return from the Cape and there we were not near enough, nor on that crowded day could we get near enough, to exchange a word. And this is all I ever saw, and, so it has pleased God, all I shall see in this world of a man whose friendly communications were among my greatest social enjoyments, and greatest intellectual treats."

There is a very interesting memoir of Hamilton written by De Morgan, in the "Gentleman's Magazine" for 1866, in which he produces an excellent sketch of his friend, illustrated by personal reminiscences and anecdotes. He alludes, among other things, to the picturesque confusion of the papers in his study. There was some sort of order in the mass, discernible however, by Hamilton alone, and any invasion of the domestics, with a view to tidying up, would throw the mathematician as we are informed, into "a good honest thundering passion."

Hardly any two men, who were both powerful mathematicians, could have been more dissimilar in every other respect than were Hamilton and De Morgan. The highly poetical temperament of Hamilton was remarkably contrasted with the practical realism of De Morgan. Hamilton sends sonnets to his friend, who replies by giving the poet advice about making his will. The metaphysical subtleties, with which Hamilton often filled his sheets, did not seem to have the same attraction for De Morgan that he found in battles about the quantification of the Predicate. De Morgan was exquisitely witty, and though his jokes were always appreciated by his correspondent, yet Hamilton seldom ventured on anything of the same kind in reply; indeed his rare attempts at humour only produced results of the most ponderous description. But never were two scientific correspondents more perfectly in sympathy with each other. Hamilton's work on Quaternions, his labours in Dynamics, his literary tastes, his metaphysics, and his poetry, were all heartily welcomed by his friend, whose letters in reply invariably evince the kindliest interest in all Hamilton's concerns. In a similar way De Morgan's letters to Hamilton always met with a heartfelt response.

Alike for the memory of Hamilton, for the credit of his University, and for the benefit of science, let us hope that a collected edition of his works will ere long appear—a collection which shall show those early achievements in splendid optical theory, those achievements of his more mature powers which made him the Lagrange of his country, and finally those creations of the Quaternion Calculus by which new capabilities have been bestowed on the human intellect.



LE VERRIER.



The name of Le Verrier is one that goes down to fame on account of very different discoveries from those which have given renown to several of the other astronomers whom we have mentioned. We are sometimes apt to identify the idea of an astronomer with that of a man who looks through a telescope at the stars; but the word astronomer has really much wider significance. No man who ever lived has been more entitled to be designated an astronomer than Le Verrier, and yet it is certain that he never made a telescopic discovery of any kind. Indeed, so far as his scientific achievements have been concerned, he might never have looked through a telescope at all.

For the full interpretation of the movements of the heavenly bodies, mathematical knowledge of the most advanced character is demanded. The mathematician at the outset calls upon the astronomer who uses the instruments in the observatory, to ascertain for him at various times the exact positions occupied by the sun, the moon, and the planets. These observations, obtained with the greatest care, and purified as far as possible from the errors by which they may be affected form, as it were, the raw material on which the mathematician exercises his skill. It is for him to elicit from the observed places the true laws which govern the movements of the heavenly bodies. Here is indeed a task in which the highest powers of the human intellect may be worthily employed.

Among those who have laboured with the greatest success in the interpretation of the observations made with instruments of precision, Le Verrier holds a highly honoured place. To him it has been given to provide a superb illustration of the success with which the mind of man can penetrate the deep things of Nature.

The illustrious Frenchman, Urban Jean Joseph Le Verrier, was born on the 11th March, 1811, at St. Lo, in the department of Manche. He received his education in that famous school for education in the higher branches of science, the Ecole Polytechnique, and acquired there considerable fame as a mathematician. On leaving the school Le Verrier at first purposed to devote himself to the public service, in the department of civil engineering; and it is worthy of note that his earliest scientific work was not in those mathematical researches in which he was ultimately to become so famous. His duties in the engineering department involved practical chemical research in the laboratory. In this he seems to have become very expert, and probably fame as a chemist would have been thus attained, had not destiny led him into another direction. As it was, he did engage in some original chemical research. His first contributions to science were the fruits of his laboratory work; one of his papers was on the combination of phosphorus and hydrogen, and another on the combination of phosphorus and oxygen.

His mathematical labours at the Ecole Polytechnique had, however, revealed to Le Verrier that he was endowed with the powers requisite for dealing with the subtlest instruments of mathematical analysis. When he was twenty-eight years old, his first great astronomical investigation was brought forth. It will be necessary to enter into some explanation as to the nature of this, inasmuch as it was the commencement of the life-work which he was to pursue.

If but a single planet revolved around the sun, then the orbit of that planet would be an ellipse, and the shape and size, as well as the position of the ellipse, would never alter. One revolution after another would be traced out, exactly in the same manner, in compliance with the force continuously exerted by the sun. Suppose, however, that a second planet be introduced into the system. The sun will exert its attraction on this second planet also, and it will likewise describe an orbit round the central globe. We can, however, no longer assert that the orbit in which either of the planets moves remains exactly an ellipse. We may, indeed, assume that the mass of the sun is enormously greater than that of either of the planets. In this case the attraction of the sun is a force of such preponderating magnitude, that the actual path of each planet remains nearly the same as if the other planet were absent. But it is impossible for the orbit of each planet not to be affected in some degree by the attraction of the other planet. The general law of nature asserts that every body in space attracts every other body. So long as there is only a single planet, it is the single attraction between the sun and that planet which is the sole controlling principle of the movement, and in consequence of it the ellipse is described. But when a second planet is introduced, each of the two bodies is not only subject to the attraction of the sun, but each one of the planets attracts the other. It is true that this mutual attraction is but small, but, nevertheless, it produces some effect. It "disturbs," as the astronomer says, the elliptic orbit which would otherwise have been pursued. Hence it follows that in the actual planetary system where there are several planets disturbing each other, it is not true to say that the orbits are absolutely elliptic.

At the same time in any single revolution a planet may for most practical purposes be said to be actually moving in an ellipse. As, however, time goes on, the ellipse gradually varies. It alters its shape, it alters its plane, and it alters its position in that plane. If, therefore, we want to study the movements of the planets, when great intervals of time are concerned, it is necessary to have the means of learning the nature of the movement of the orbit in consequence of the disturbances it has experienced.

We may illustrate the matter by supposing the planet to be running like a railway engine on a track which has been laid in a long elliptic path. We may suppose that while the planet is coursing along, the shape of the track is gradually altering. But this alteration may be so slow, that it does not appreciably affect the movement of the engine in a single revolution. We can also suppose that the plane in which the rails have been laid has a slow oscillation in level, and that the whole orbit is with more or less uniformity moved slowly about in the plane.

In short periods of time the changes in the shapes and positions of the planetary orbits, in consequence of their mutual attractions, are of no great consequence. When, however, we bring thousands of years into consideration, then the displacements of the planetary orbits attain considerable dimensions, and have, in fact, produced a profound effect on the system.

It is of the utmost interest to investigate the extent to which one planet can affect another in virtue of their mutual attractions. Such investigations demand the exercise of the highest mathematical gifts. But not alone is intellectual ability necessary for success in such inquiries. It must be united with a patient capacity for calculations of an arduous type, protracted, as they frequently have to be, through many years of labour. Le Verrier soon found in these profound inquiries adequate scope for the exercise of his peculiar gifts. His first important astronomical publication contained an investigation of the changes which the orbits of several of the planets, including the earth, have undergone in times past, and which they will undergo in times to come.

As an illustration of these researches, we may take the case of the planet in which we are, of course, especially interested, namely, the earth, and we can investigate the changes which, in the lapse of time, the earth's orbit has undergone, in consequence of the disturbance to which it has been subjected by the other planets. In a century, or even in a thousand years, there is but little recognisable difference in the shape of the track pursued by the earth. Vast periods of time are required for the development of the large consequences of planetary perturbation. Le Verrier has, however, given us the particulars of what the earth's journey through space has been at intervals of 20,000 years back from the present date. His furthest calculation throws our glance back to the state of the earth's track 100,000 years ago, while, with a bound forward, he shows us what the earth's orbit is to be in the future, at successive intervals of 20,000 years, till a date is reached which is 100,000 years in advance Of A.D. 1800.

The talent which these researches displayed brought Le Verrier into notice. At that time the Paris Observatory was presided over by Arago, a SAVANT who occupies a distinguished position in French scientific annals. Arago at once perceived that Le Verrier was just the man who possessed the qualifications suitable for undertaking a problem of great importance and difficulty that had begun to force itself on the attention of astronomers. What this great problem was, and how astonishing was the solution it received, must now be considered.

Ever since Herschel brought himself into fame by his superb discovery of the great planet Uranus, the movements of this new addition to the solar system were scrutinized with care and attention. The position of Uranus was thus accurately determined from time to time. At length, when sufficient observations of this remote planet had been brought together, the route which the newly-discovered body pursued through the heavens was ascertained by those calculations with which astronomers are familiar. It happens, however, that Uranus possesses a superficial resemblance to a star. Indeed the resemblance is so often deceptive that long ere its detection as a planet by Herschel, it had been observed time after time by skilful astronomers, who little thought that the star-like point at which they looked was anything but a star. From these early observations it was possible to determine the track of Uranus, and it was found that the great planet takes a period of no less than eighty-four years to accomplish a circuit. Calculations were made of the shape of the orbit in which it revolved before its discovery by Herschel, and these were compared with the orbit which observations showed the same body to pursue in those later years when its planetary character was known. It could not, of course, be expected that the orbit should remain unaltered; the fact that the great planets Jupiter and Saturn revolve in the vicinity of Uranus must necessarily imply that the orbit of the latter undergoes considerable changes. When, however, due allowance has been made for whatever influence the attraction of Jupiter and Saturn, and we may add of the earth and all the other Planets, could possibly produce, the movements of Uranus were still inexplicable. It was perfectly obvious that there must be some other influence at work besides that which could be attributed to the planets already known.

Astronomers could only recognise one solution of such a difficulty. It was impossible to doubt that there must be some other planet in addition to the bodies at that time known, and that the perturbations of Uranus hitherto unaccounted for, were due to the disturbances caused by the action of this unknown planet. Arago urged Le Verrier to undertake the great problem of searching for this body, whose theoretical existence seemed demonstrated. But the conditions of the search were such that it must needs be conducted on principles wholly different from any search which had ever before been undertaken for a celestial object. For this was not a case in which mere survey with a telescope might be expected to lead to the discovery.

Certain facts might be immediately presumed with reference to the unknown object. There could be no doubt that the unknown disturber of Uranus must be a large body with a mass far exceeding that of the earth. It was certain, however, that it must be so distant that it could only appear from our point of view as a very small object. Uranus itself lay beyond the range, or almost beyond the range, of unassisted vision. It could be shown that the planet by which the disturbance was produced revolved in an orbit which must lie outside that of Uranus. It seemed thus certain that the planet could not be a body visible to the unaided eye. Indeed, had it been at all conspicuous its planetary character would doubtless have been detected ages ago. The unknown body must therefore be a planet which would have to be sought for by telescopic aid.

There is, of course, a profound physical difference between a planet and a star, for the star is a luminous sun, and the planet is merely a dark body, rendered visible by the sunlight which falls upon it. Notwithstanding that a star is a sun thousands of times larger than the planet and millions of times more remote, yet it is a singular fact that telescopic planets possess an illusory resemblance to the stars among which their course happens to lie. So far as actual appearance goes, there is indeed only one criterion by which a planet of this kind can be discriminated from a star. If the planet be large enough the telescope will show that it possesses a disc, and has a visible and measurable circular outline. This feature a star does not exhibit. The stars are indeed so remote that no matter how large they may be intrinsically, they only exhibit radiant points of light, which the utmost powers of the telescope fail to magnify into objects with an appreciable diameter. The older and well-known planets, such as Jupiter and Mars, possess discs, which, though not visible to the unaided eye, were clearly enough discernible with the slightest telescopic power. But a very remote planet like Uranus, though it possessed a disc large enough to be quickly appreciated by the consummate observing skill of Herschel, was nevertheless so stellar in its appearance, that it had been observed no fewer than seventeen times by experienced astronomers prior to Herschel. In each case the planetary nature of the object had been overlooked, and it had been taken for granted that it was a star. It presented no difference which was sufficient to arrest attention.

As the unknown body by which Uranus was disturbed was certainly much more remote than Uranus, it seemed to be certain that though it might show a disc perceptible to very close inspection, yet that the disc must be so minute as not to be detected except with extreme care. In other words, it seemed probable that the body which was to be sought for could not readily be discriminated from a small star, to which class of object it bore a superficial resemblance, though, as a matter of fact, there was the profoundest difference between the two bodies.

There are on the heavens many hundreds of thousands of stars, and the problem of identifying the planet, if indeed it should lie among these stars, seemed a very complex matter. Of course it is the abundant presence of the stars which causes the difficulty. If the stars could have been got rid of, a sweep over the heavens would at once disclose all the planets which are bright enough to be visible with the telescopic power employed. It is the fortuitous resemblance of the planet to the stars which enables it to escape detection. To discriminate the planet among stars everywhere in the sky would be almost impossible. If, however, some method could be devised for localizing that precise region in which the planet's existence might be presumed, then the search could be undertaken with some prospect of success.

To a certain extent the problem of localizing the region on the sky in which the planet might be expected admitted of an immediate limitation. It is known that all the planets, or perhaps I ought rather to say, all the great planets, confine their movements to a certain zone around the heavens. This zone extends some way on either side of that line called the ecliptic in which the earth pursues its journey around the sun. It was therefore to be inferred that the new planet need not be sought for outside this zone. It is obvious that this consideration at once reduces the area to be scrutinized to a small fraction of the entire heavens. But even within the zone thus defined there are many thousands of stars. It would seem a hopeless task to detect the new planet unless some further limitation to its position could be assigned.

It was accordingly suggested to Le Verrier that he should endeavour to discover in what particular part of the strip of the celestial sphere which we have indicated the search for the unknown planet should be instituted. The materials available to the mathematician for the solution of this problem were to be derived solely from the discrepancies between the calculated places in which Uranus should be found, taking into account the known causes of disturbance, and the actual places in which observation had shown the planet to exist. Here was indeed an unprecedented problem, and one of extraordinary difficulty. Le Verrier, however, faced it, and, to the astonishment of the world, succeeded in carrying it through to a brilliant solution. We cannot here attempt to enter into any account of the mathematical investigations that were necessary. All that we can do is to give a general indication of the method which had to be adopted.

Let us suppose that a planet is revolving outside Uranus, at a distance which is suggested by the several distances at which the other planets are dispersed around the sun. Let us assume that this outer planet has started on its course, in a prescribed path, and that it has a certain mass. It will, of course, disturb the motion of Uranus, and in consequence of that disturbance Uranus will follow a path the nature of which can be determined by calculation. It will, however, generally be found that the path so ascertained does not tally with the actual path which observations have indicated for Uranus. This demonstrates that the assumed circumstances of the unknown planet must be in some respects erroneous, and the astronomer commences afresh with an amended orbit. At last after many trials, Le Verrier ascertained that, by assuming a certain size, shape, and position for the unknown Planet's orbit, and a certain value for the mass of the hypothetical body, it would be possible to account for the observed disturbances of Uranus. Gradually it became clear to the perception of this consummate mathematician, not only that the difficulties in the movements of Uranus could be thus explained, but that no other explanation need be sought for. It accordingly appeared that a planet possessing the mass which he had assigned, and moving in the orbit which his calculations had indicated, must indeed exist, though no eye had ever beheld any such body. Here was, indeed, an astonishing result. The mathematician sitting at his desk, by studying the observations which had been supplied to him of one planet, is able to discover the existence of another planet, and even to assign the very position which it must occupy, ere ever the telescope is invoked for its discovery.

Thus it was that the calculations of Le Verrier narrowed greatly the area to be scrutinised in the telescopic search which was presently to be instituted. It was already known, as we have just pointed out, that the planet must lie somewhere on the ecliptic. The French mathematician had now further indicated the spot on the ecliptic at which, according to his calculations, the planet must actually be found. And now for an episode in this history which will be celebrated so long as science shall endure. It is nothing less than the telescopic confirmation of the existence of this new planet, which had previously been indicated only by mathematical calculation. Le Verrier had not himself the instruments necessary for studying the heavens, nor did he possess the skill of the practical astronomer. He, therefore, wrote to Dr. Galle, of the Observatory at Berlin, requesting him to undertake a telescopic search for the new planet in the vicinity which the mathematical calculation had indicated for the whereabouts of the planet at that particular time. Le Verrier added that he thought the planet ought to admit of being recognised by the possession of a disc sufficiently definite to mark the distinction between it and the surrounding stars.

It was the 23rd September, 1846, when the request from Le Verrier reached the Berlin Observatory, and the night was clear, so that the memorable search was made on the same evening. The investigation was facilitated by the circumstance that a diligent observer had recently compiled elaborate star maps for certain tracts of the heavens lying in a sufficiently wide zone on both sides of the equator. These maps were as yet only partially complete, but it happened that Hora. XXI., which included the very spot which Le Verrier's results referred to, had been just issued. Dr. Galle had thus before his, eyes a chart of all the stars which were visible in that part of the heavens at the time when the map was made. The advantage of such an assistance to the search could hardly be over-estimated. It at once gave the astronomer another method of recognising the planet besides that afforded by its possible possession of a disc. For as the planet was a moving body, it would not have been in the same place relatively to the stars at the time when the map was constructed, as it occupied some years later when the search was being made. If the body should be situated in the spot which Le Verrier's calculations indicated in the autumn of 1846, then it might be regarded as certain that it would not be found in that same place on a map drawn some years previously.

The search to be undertaken consisted in a comparison made point by point between the bodies shown on the map, and those stars in the sky which Dr. Galle's telescope revealed. In the course of this comparison it presently appeared that a star-like object of the eighth magnitude, which was quite a conspicuous body in the telescope, was not represented in the map. This at once attracted the earnest attention of the astronomer, and raised his hopes that here was indeed the planet. Nor were these hopes destined to be disappointed. It could not be supposed that a star of the eighth magnitude would have been overlooked in the preparation of a chart whereon stars of many lower degrees of brightness were set down. One other supposition was of course conceivable. It might have been that this suspicious object belonged to the class of variables, for there are many such stars whose brightness fluctuates, and if it had happened that the map was constructed at a time when the star in question had but feeble brilliance, it might have escaped notice. It is also well known that sometimes new stars suddenly develop, so that the possibility that what Dr. Galle saw should have been a variable star or should have been a totally new star had to be provided against.

Fortunately a test was immediately available to decide whether the new object was indeed the long sought for planet, or whether it was a star of one of the two classes to which I have just referred. A star remains fixed, but a planet is in motion. No doubt when a planet lies at the distance at which this new planet was believed to be situated, its apparent motion would be so slow that it would not be easy to detect any change in the course of a single night's observation. Dr. Galle, however, addressed himself with much skill to the examination of the place of the new body. Even in the course of the night he thought he detected slight movements, and he awaited with much anxiety the renewal of his observations on the subsequent evenings. His suspicions as to the movement of the body were then amply confirmed, and the planetary nature of the new object was thus unmistakably detected.

Great indeed was the admiration of the scientific world at this superb triumph. Here was a mighty planet whose very existence was revealed by the indications afforded by refined mathematical calculation. At once the name of Le Verrier, already known to those conversant with the more profound branches of astronomy, became everywhere celebrated. It soon, however, appeared, that the fame belonging to this great achievement had to be shared between Le Verrier and another astronomer, J. C. Adams, of Cambridge. In our chapter on this great English mathematician we shall describe the manner in which he was independently led to the same discovery.

Directly the planetary nature of the newly-discovered body had been established, the great observatories naturally included this additional member of the solar system in their working lists, so that day after day its place was carefully determined. When sufficient time had elapsed the shape and position of the orbit of the body became known. Of course, it need hardly be said that observations applied to the planet itself must necessarily provide a far more accurate method of determining the path which it follows, than would be possible to Le Verrier, when all he had to base his calculations upon was the influence of the planet reflected, so to speak, from Uranus. It may be noted that the true elements of the planet, when revealed by direct observation, showed that there was a considerable discrepancy between the track of the planet which Le Verrier had announced, and that which the planet was actually found to pursue.

The name of the newly-discovered body had next to be considered. As the older members of the system were already known by the same names as great heathen divinities, it was obvious that some similar source should be invoked for a suggestion as to a name for the most recent planet. The fact that this body was so remote in the depths of space, not unnaturally suggested the name "Neptune." Such is accordingly the accepted designation of that mighty globe which revolves in the track that at present seems to trace out the frontiers of our system.

Le Verrier attained so much fame by this discovery, that when, in 1854, Arago's place had to be filled at the head of the great Paris Observatory, it was universally felt that the discoverer of Neptune was the suitable man to assume the office which corresponds in France to that of the Astronomer Royal in England. It was true that the work of the astronomical mathematician had hitherto been of an abstract character. His discoveries had been made at his desk and not in the observatory, and he had no practical acquaintance with the use of astronomical instruments. However, he threw himself into the technical duties of the observatory with vigour and determination. He endeavoured to inspire the officers of the establishment with enthusiasm for that systematic work which is so necessary for the accomplishment of useful astronomical research. It must, however, be admitted that Le Verrier was not gifted with those natural qualities which would make him adapted for the successful administration of such an establishment. Unfortunately disputes arose between the Director and his staff. At last the difficulties of the situation became so great that the only possible solution was to supersede Le Verrier, and he was accordingly obliged to retire. He was succeeded in his high office by another eminent mathematician, M. Delaunay, only less distinguished than Le Verrier himself.

Relieved of his official duties, Le Verrier returned to the mathematics he loved. In his non-official capacity he continued to work with the greatest ardour at his researches on the movements of the planets. After the death of M. Delaunay, who was accidentally drowned in 1873, Le Verrier was restored to the directorship of the observatory, and he continued to hold the office until his death.

The nature of the researches to which the life of Le Verrier was subsequently devoted are not such as admit of description in a general sketch like this, where the language, and still less the symbols, of mathematics could not be suitably introduced. It may, however, be said in general that he was particularly engaged with the study of the effects produced on the movements of the planets by their mutual attractions. The importance of this work to astronomy consists, to a considerable extent, in the fact that by such calculations we are enabled to prepare tables by which the places of the different heavenly bodies can be predicted for our almanacs. To this task Le Verrier devoted himself, and the amount of work he has accomplished would perhaps have been deemed impossible had it not been actually done.

The superb success which had attended Le Verrier's efforts to explain the cause of the perturbations of Uranus, naturally led this wonderful computer to look for a similar explanation of certain other irregularities in planetary movements. To a large extent he succeeded in showing how the movements of each of the great planets could be satisfactorily accounted for by the influence of the attractions of the other bodies of the same class. One circumstance in connection with these investigations is sufficiently noteworthy to require a few words here. Just as at the opening of his career, Le Verrier had discovered that Uranus, the outermost planet of the then known system, exhibited the influence of an unknown external body, so now it appeared to him that Mercury, the innermost body of our system, was also subjected to some disturbances, which could not be satisfactorily accounted for as consequences of any known agents of attraction. The ellipse in which Mercury revolved was animated by a slow movement, which caused it to revolve in its plane. It appeared to Le Verrier that this displacement was incapable of explanation by the action of any of the known bodies of our system. He was, therefore, induced to try whether he could not determine from the disturbances of Mercury the existence of some other planet, at present unknown, which revolved inside the orbit of the known planet. Theory seemed to indicate that the observed alteration in the track of the planet could be thus accounted for. He naturally desired to obtain telescopic confirmation which might verify the existence of such a body in the same way as Dr. Galle verified the existence of Neptune. If there were, indeed, an intramercurial planet, then it must occasionally cross between the earth and the sun, and might now and then be expected to be witnessed in the actual act of transit. So confident did Le Verrier feel in the existence of such a body that an observation of a dark object in transit, by Lescarbault on 26th March, 1859, was believed by the mathematician to be the object which his theory indicated. Le Verrier also thought it likely that another transit of the same object would be seen in March, 1877. Nothing of the kind was, however, witnessed, notwithstanding that an assiduous watch was kept, and the explanation of the change in Mercury's orbit must, therefore, be regarded as still to be sought for.

Le Verrier naturally received every honour that could be bestowed upon a man of science. The latter part of his life was passed during the most troubled period of modern French history. He was a supporter of the Imperial Dynasty, and during the Commune he experienced much anxiety; indeed, at one time grave fears were entertained for his personal safety.

Early in 1877 his health, which had been gradually failing for some years, began to give way. He appeared to rally somewhat in the summer, but in September he sank rapidly, and died on Sunday, the 23rd of that month.

His remains were borne to the cemetery on Mont Parnasse in a public funeral. Among his pallbearers were leading men of science, from other countries as well as France, and the memorial discourses pronounced at the grave expressed their admiration of his talents and of the greatness of the services he had rendered to science.



ADAMS.



The illustrious mathematician who, among Englishmen, at all events, was second only to Newton by his discoveries in theoretical astronomy, was born on June the 5th, 1819, at the farmhouse of Lidcot, seven miles from Launceston, in Cornwall. His early education was imparted under the guidance of the Rev. John Couch Grylls, a first cousin of his mother. He appears to have received an education of the ordinary school type in classics and mathematics, but his leisure hours were largely devoted to studying what astronomical books he could find in the library of the Mechanics' Institute at Devonport. He was twenty years old when he entered St. John's College, Cambridge. His career in the University was one of almost unparalleled distinction, and it is recorded that his answering at the Wranglership examination, where he came out at the head of the list in 1843, was so high that he received more than double the marks awarded to the Second Wrangler.

Among the papers found after his death was the following memorandum, dated July the 3rd, 1841: "Formed a design at the beginning of this week of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus, Which are as yet unaccounted for, in order to find whether they may be attributed to the action of an undiscovered planet beyond it; and, if possible, thence to determine the elements of its orbit approximately, which would lead probably to its discovery."

After he had taken his degree, and had thus obtained a little relaxation from the lines within which his studies had previously been necessarily confined, Adams devoted himself to the study of the perturbations of Uranus, in accordance with the resolve which we have just seen that he formed while he was still an undergraduate. As a first attempt he made the supposition that there might be a planet exterior to Uranus, at a distance which was double that of Uranus from the sun. Having completed his calculation as to the effect which such a hypothetical planet might exercise upon the movement of Uranus, he came to the conclusion that it would be quite possible to account completely for the unexplained difficulties by the action of an exterior planet, if only that planet were of adequate size and had its orbit properly placed. It was necessary, however, to follow up the problem more precisely, and accordingly an application was made through Professor Challis, the Director of the Cambridge Observatory, to the Astronomer Royal, with the object of obtaining from the observations made at Greenwich Observatory more accurate values for the disturbances suffered by Uranus. Basing his work on the more precise materials thus available, Adams undertook his calculations anew, and at last, with his completed results, he called at Greenwich Observatory on October the 21st, 1845. He there left for the Astronomer Royal a paper which contained the results at which he had arrived for the mass and the mean distance of the hypothetical planet as well as the other elements necessary for calculating its exact position.

[PLATE: JOHN COUCH ADAMS.]

As we have seen in the preceding chapter, Le Verrier had been also investigating the same problem. The place which Le Verrier assigned to the hypothetical disturbing planet for the beginning of the year 1847, was within a degree of that to which Adams's computations pointed, and which he had communicated to the Astronomer Royal seven months before Le Verrier's work appeared. On July the 29th, 1846, Professor Challis commenced to search for the unknown object with the Northumberland telescope belonging to the Cambridge Observatory. He confined his attention to a limited region in the heavens, extending around that point to which Mr. Adams' calculations pointed. The relative places of all the stars, or rather star-like objects within this area, were to be carefully measured. When the same observations were repeated a week or two later, then the distances of the several pairs of stars from each other would be found unaltered, but any planet which happened to lie among the objects measured would disclose its existence by the alterations in distance due to its motion in the interval. This method of search, though no doubt it must ultimately have proved successful, was necessarily a very tedious one, but to Professor Challis, unfortunately, no other method was available. Thus it happened that, though Challis commenced his search at Cambridge two months earlier than Galle at Berlin, yet, as we have already explained, the possession of accurate star-maps by Dr. Galle enabled him to discover the planet on the very first night that he looked for it.

The rival claims of Adams and Le Verrier to the discovery of Neptune, or rather, we should say, the claims put forward by their respective champions, for neither of the illustrious investigators themselves condescended to enter into the personal aspect of the question, need not be further discussed here. The main points of the controversy have been long since settled, and we cannot do better than quote the words of Sir John Herschel when he addressed the Royal Astronomical Society in 1848:—

"As genius and destiny have joined the names of Le Verrier and Adams, I shall by no means put them asunder; nor will they ever be pronounced apart so long as language shall celebrate the triumphs Of science in her sublimest walks. On the great discovery of Neptune, which may be said to have surpassed, by intelligible and legitimate means, the wildest pretensions of clairvoyance, it Would now be quite superfluous for me to dilate. That glorious event and the steps which led to it, and the various lights in which it has been placed, are already familiar to every one having the least tincture of science. I will only add that as there is not, nor henceforth ever can be, the slightest rivalry on the subject between these two illustrious men—as they have met as brothers, and as such will, I trust, ever regard each other—we have made, we could make, no distinction between then, on this occasion. May they both long adorn and augment our science, and add to their own fame already so high and pure, by fresh achievements."

Adams was elected a Fellow of St. John's College, Cambridge, in 1843; but as he did not take holy orders, his Fellowship, in accordance with the rules then existing came to an end in 1852. In the following year he was, however, elected to a Fellowship at Pembroke College, which he retained until the end of his life. In 1858 he was appointed Professor of Mathematics in the University of St. Andrews, but his residence in the north was only a brief one, for in the same year he was recalled to Cambridge as Lowndean Professor of Astronomy and Geometry, in succession to Peacock. In 1861 Challis retired from the Directorship of the Cambridge Observatory, and Adams was appointed to succeed him.

The discovery of Neptune was a brilliant inauguration of the astronomical career of Adams. He worked at, and wrote upon, the theory of the motions of Biela's comet; he made important corrections to the theory of Saturn; he investigated the mass of Uranus, a subject in which he was naturally interested from its importance in the theory of Neptune; he also improved the methods of computing the orbits of double stars. But all these must be regarded as his minor labours, for next to the discovery of Neptune the fame of Adams mainly rests on his researches upon certain movements of the moon, and upon the November meteors.

The periodic time of the moon is the interval required for one circuit of its orbit. This interval is known with accuracy at the present day, and by means of the ancient eclipses the period of the moon's revolution two thousand years ago can be also ascertained. It had been discovered by Halley that the period which the moon requires to accomplish each of its revolutions around the earth has been steadily, though no doubt slowly, diminishing. The change thus produced is not appreciable when only small intervals of time are considered, but it becomes appreciable when we have to deal with intervals of thousands of years. The actual effect which is produced by the lunar acceleration, for so this phenomenon is called, may be thus estimated. If we suppose that the moon had, throughout the ages, revolved around the earth in precisely the same periodic time which it has at present, and if from this assumption we calculate back to find where the moon must have been about two thousand years ago, we obtain a position which the ancient eclipses show to be different from that in which the moon was actually situated. The interval between the position in which the moon would have been found two thousand years ago if there had been no acceleration, and the position in which the moon was actually placed, amounts to about a degree, that is to say, to an arc on the heavens which is twice the moon's apparent diameter.

If no other bodies save the earth and the moon were present in the universe, it seems certain that the motion of the moon would never have exhibited this acceleration. In such a simple case as that which I have supposed the orbit of the moon would have remained for ever absolutely unchanged. It is, however, well known that the presence of the sun exerts a disturbing influence upon the movements of the moon. In each revolution our satellite is continually drawn aside by the action of the sun from the place which it would otherwise have occupied. These irregularities are known as the perturbations of the lunar orbit, they have long been studied, and the majority of them have been satisfactorily accounted for. It seems, however, to those who first investigated the question that the phenomenon of the lunar acceleration could not be explained as a consequence of solar perturbation, and, as no other agent competent to produce such effects was recognised by astronomers, the lunar acceleration presented an unsolved enigma.

At the end of the last century the illustrious French mathematician Laplace undertook a new investigation of the famous problem, and was rewarded with a success which for a long time appeared to be quite complete. Let us suppose that the moon lies directly between the earth and the sun, then both earth and moon are pulled towards the sun by the solar attraction; as, however, the moon is the nearer of the two bodies to the attracting centre it is pulled the more energetically, and consequently there is an increase in the distance between the earth and the moon. Similarly when the moon happens to lie on the other side of the earth, so that the earth is interposed directly between the moon and the sun, the solar attraction exerted upon the earth is more powerful than the same influence upon the moon. Consequently in this case, also, the distance of the moon from the earth is increased by the solar disturbance. These instances will illustrate the general truth, that, as one of the consequences of the disturbing influence exerted by the sun upon the earth-moon system, there is an increase in the dimensions of the average orbit which the moon describes around the earth. As the time required by the moon to accomplish a journey round the earth depends upon its distance from the earth, it follows that among the influences of the sun upon the moon there must be an enlargement of the periodic time, from what it would have been had there been no solar disturbing action.

This was known long before the time of Laplace, but it did not directly convey any explanation of the lunar acceleration. It no doubt amounted to the assertion that the moon's periodic time was slightly augmented by the disturbance, but it did not give any grounds for suspecting that there was a continuous change in progress. It was, however, apparent that the periodic time was connected with the solar disturbance, so that, if there were any alteration in the amount of the sun's disturbing effect, there must be a corresponding alteration in the moon's periodic time. Laplace, therefore, perceived that, if he could discover any continuous change in the ability of the sun for disturbing the moon, he would then have accounted for a continuous change in the moon's periodic time, and that thus an explanation of the long-vexed question of the lunar acceleration might be forthcoming.

The capability of the sun for disturbing the earth-moon system is obviously connected with the distance of the earth from the sun. If the earth moved in an orbit which underwent no change whatever, then the efficiency of the sun as a disturbing agent would not undergo any change of the kind which was sought for. But if there were any alteration in the shape or size of the earth's orbit, then that might involve such changes in the distance between the earth and the sun as would possibly afford the desired agent for producing the observed lunar effect. It is known that the earth revolves in an orbit which, though nearly circular, is strictly an ellipse. If the earth were the only planet revolving around the sun then that ellipse would remain unaltered from age to age. The earth is, however, only one of a large number of planets which circulate around the great luminary, and are guided and controlled by his supreme attracting power. These planets mutually attract each other, and in consequence of their mutual attractions the orbits of the planets are disturbed from the simple elliptic form which they would otherwise possess. The movement of the earth, for instance, is not, strictly speaking, performed in an elliptical orbit. We may, however, regard it as revolving in an ellipse provided we admit that the ellipse is itself in slow motion.

It is a remarkable characteristic of the disturbing effects of the planets that the ellipse in which the earth is at any moment moving always retains the same length; that is to say, its longest diameter is invariable. In all other respects the ellipse is continually changing. It alters its position, it changes its plane, and, most important of all, it changes its eccentricity. Thus, from age to age the shape of the track which the earth describes may at one time be growing more nearly a circle, or at another time may be departing more widely from a circle. These alterations are very small in amount, and they take place with extreme slowness, but they are in incessant progress, and their amount admits of being accurately calculated. At the present time, and for thousands of years past, as well as for thousands of years to come, the eccentricity of the earth's orbit is diminishing, and consequently the orbit described by the earth each year is becoming more nearly circular. We must, however, remember that under all circumstances the length of the longest axis of the ellipse is unaltered, and consequently the size of the track which the earth describes around the sun is gradually increasing. In other words, it may be said that during the present ages the average distance between the earth and the sun is waxing greater in consequence of the perturbations which the earth experiences from the attraction of the other planets. We have, however, already seen that the efficiency of the solar attraction for disturbing the moon's movement depends on the distance between the earth and the sun. As therefore the average distance between the earth and the sun is increasing, at all events during the thousands of years over which our observations extend, it follows that the ability of the sun for disturbing the moon must be gradually diminishing.

[PLATE: CAMBRIDGE OBSERVATORY.]

It has been pointed out that, in consequence of the solar disturbance, the orbit of the moon must be some what enlarged. As it now appears that the solar disturbance is on the whole declining, it follows that the orbit of the moon, which has to be adjusted relatively to the average value of the solar disturbance, must also be gradually declining. In other words, the moon must be approaching nearer to the earth in consequence of the alterations in the eccentricity of the earth's orbit produced by the attraction of the other planets. It is true that the change in the moon's position thus arising is an extremely small one, and the consequent effect in accelerating the moon's motion is but very slight. It is in fact almost imperceptible, except when great periods of time are involved. Laplace undertook a calculation on this subject. He knew what the efficiency of the planets in altering the dimensions of the earth's orbit amounted to; from this he was able to determine the changes that would be propagated into the motion of the moon. Thus he ascertained, or at all events thought he had ascertained, that the acceleration of the moon's motion, as it had been inferred from the observations of the ancient eclipses which have been handed down to us, could be completely accounted for as a consequence of planetary perturbation. This was regarded as a great scientific triumph. Our belief in the universality of the law of gravitation would, in fact, have been seriously challenged unless some explanation of the lunar acceleration had been forthcoming. For about fifty years no one questioned the truth of Laplace's investigation. When a mathematician of his eminence had rendered an explanation of the remarkable facts of observation which seemed so complete, it is not surprising that there should have been but little temptation to doubt it. On undertaking a new calculation of the same question, Professor Adams found that Laplace had not pursued this approximation sufficiently far, and that consequently there was a considerable error in the result of his analysis. Adams, it must be observed, did not impugn the value of the lunar acceleration which Halley had deduced from the observations, but what he did show was, that the calculation by which Laplace thought he had provided an explanation of this acceleration was erroneous. Adams, in fact, proved that the planetary influence which Laplace had detected only possessed about half the efficiency which the great French mathematician had attributed to it. There were not wanting illustrious mathematicians who came forward to defend the calculations of Laplace. They computed the question anew and arrived at results practically coincident with those he had given. On the other hand certain distinguished mathematicians at home and abroad verified the results of Adams. The issue was merely a mathematical one. It had only one correct solution. Gradually it appeared that those who opposed Adams presented a number of different solutions, all of them discordant with his, and, usually, discordant with each other. Adams showed distinctly where each of these investigators had fallen into error, and at last it became universally admitted that the Cambridge Professor had corrected Laplace in a very fundamental point of astronomical theory.