Part 13 (1/2)
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 a.n.a.lytique' of Lagrange, 'a scientific poem'; and spoke of Dynamics, or the Science of Force, as treating of 'Power acting by Law in s.p.a.ce 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 's.p.a.ce 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 a.s.sociation Hamilton was a very conspicuous figure. Especially was this the case in 1835, when the a.s.sociation 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 a.s.sembled 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 circ.u.mstances 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 a.s.sociated 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 ill.u.s.tration 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 cla.s.sics 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 a.n.a.lysis 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 death-bed, twenty-two years later, to his son Archibald, in which he has recorded the circ.u.mstances 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 a.s.sociation, 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 Ca.n.a.l, 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 pa.s.sed 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 establis.h.i.+ng such a system, in which geometrically opposite facts--namely, two lines (or areas) which are opposite IN s.p.a.cE 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 t.i.tle 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 ma.s.s 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:--