Modern Metaphysicians.
M.R.I.A., HON. F.R.S.E, &c.
The Philosophy of Mathematics.


[British Controversialist 22 (1869), pp. 161-176.]

``His name ... will undoubtedly be classed with those of the grandest of all ages and countries, such as Lagrange and Newton.''

- Prof. P. G. Tait.

Space and time are the formal conditions of experience; that is to say, we can have no experience that does not involve both. Whence do we derive our conception of space and time? Evidently we come by them in the course of experience. But apart from those conceptions (which are, for the most part, reflex thoughts) we have a perception of the concrete realities, space and time, in the very act or reception of experience. How do we come by that perception? To this question many answers have been given. At present we are mainly concerned with one; viz., that of Kant, who teaches that this sense-perception is not given, as sensation itself is, in an empirical experience, but that space and time are Anschauungen, i. e., intuitions, envisagings, or perhaps still better, perceptions, which come from within, which we impart to sensation, and whereby, under the stimulus of sensation and the formative energy of the intellect, we constitute that which we call experience.

Kant was first led to this conclusion, under the stress of his mathematics. He saw that arithmetic and geometry were à priori sciences, and as such could not be generalizations from experience. Therefore he inferred that time, which is the basis of arithmetic, and space, which is the basis of geometry, must be known to us as totalities - not indeed independently of experience, but on a higher voucher than that of observation. Till the result of recent mathematical researches had been arrived at, it was somewhat carelessly believed that the Mathematics were allied sciences of quantity. But there were even then extensive spheres of speculation which refused to conform to so arbitrary a definition - spheres in which the leading notion was order rather than quantity. The discoveries of Sir W. R. Hamilton, Professor Cayley, and Professor Sylvester have sent that definition to limbo. Every new discovery goes farther and farther to identify algebra with the science of order in Time and Space, i. e., Tactic.

The linear order of time, and the tri-dimensional order of space, present, as might be anticipated, very striking and important analogies. It is by virtue of these that in Sir W. R. Hamilton's hands Algebra was made to administer to Geometry, and in Professor Sylvester's hands Geometry has been forced to administer to the more pressing wants of Algebra and the Calculus. The most remarkable instance of the latter is Sylvester's theory of Reducible Cyclodes, in which the properties of the continued Involutes of the circle are used as instruments for the resolution of algebraical questions of the utmost difficulty.

Metaphysics and logic have usually been cultivated by one class of minds, and mathematics and physics by another. The certainty of the methods employed by the mathematician and the physicist stands in marked contrast to the explorations, generally not conformable to strict logical method, and offering no analogy to the processes of mathematics, which offer so great a charm to the metaphysician. There has been a great division of labour in these vast fields of research; till at length it came to pass that each class looked upon the other with ill-disguised contempt. De Morgan has been undervalued by the metaphysician, because of the great and really beneficial effect of his mathematics on his logical speculations; and Brodie (the chemist) has been the mark of the mathematicians' satire, on the ground that his mathematics have been spoiled by his metaphysics. Sylvester is too big a man to be laughed at; yet he even has, perhaps, laid himself open to ridicule by the intense philosophical cast of his mathematical works, and the intrusion into them of remarks which savour more of the metaphysician and the poet than of the mathematician. Truth to speak, philosopher and poet he is. Some few minds of past ages have helped to bridge over the gulf; such as Descartes, Leibnitz and Kant. The greatest achievement, however, in this work was destined for the subject of this memoir. Sir William Stirling Hamilton, the Edinburgh Professor of Metaphysics and Logic, in a celebrated article, and in subsequent appendices to it, gave a very decided and somewhat dogmatic opinion adverse to ``The Study of Mathematics as an Exercise of Mind.'' Even while he was disputing the merits of mathematics as a factor in a liberal education, George Boole was elaborating his ``Mathematical Analysis of Logic,'' and indirectly proving that two out of three of the principles of algebra were also principles of his logical calculus; and Sir William Rowan Hamilton, the Dublin Professor of Astronomy, was constructing out of pure metaphysics some of the most marvellous mathematical edifices of which this century can boast. By such means a considerable advance has been effected in the reconciliation of the two great scientific factions - the metaphysical and the mathematical.

The list of titles which we have appended to Sir W. R. Hamilton's name at the head of this paper, might have been greatly extended; but after three we have ``cut it short,'' in deference to the excellent dictum of De Morgan, - ``These things are the distinctions of the individual while he lives, but after his death the honour attaches to those who gave them.'' It is often a fond and foolish fancy which nicknames a great man after an equally great predecessor. More than one Teutonic philosopher has been called ``the German Plato.'' In this manner has Hegel been compared to Aristotle, and Michel Chasles has been sometimes called ``the French Newton.'' So it happened to Hamilton; he was called ``the Irish Lagrange,'' and not without reason; for his mathematical writings, like those of Lagrange, are distinguished by a rare mastery over symbols, and by the purity and beauty of their style. Besides, in Dynamics, Hamilton extended and completed the general equations of Lagrange's ``Mécanique Analytique.'' Such, however, is the rapid advance of mathematics, that already much of Hamilton's work in this department of science has been superseded. The propriety of this appellation (viz., ``the Irish Lagrange'') is, however, questionable, not by reason of the substantive comparison, but of the adjective prefix. An Irish Christian name, such as Rowan, does not make the bearer of it an Irishman, nor yet the accident that he was born in Ireland. By his father's side, Hamilton was Scotch. Scotland was the native country of his grandparents, but Ireland was the country of their adoption [see note]. In Dublin were born to them two sons, one of whom, Archibald, became a solicitor, and married Miss Sarah Hutton. They had one son, and two daughters, one of whom Miss E. M. Hamilton, became eminent as a writer of poetry. The son, William Rowan, was born in Dominick Street, Dublin, on 4th August, 1805. Precosity of a kind and in a degree equally extraordinary marked his intellectual growth. Like Leibnitz he early excelled in languages, philology, poetry, mathematics, and philosophy. At an age when most boys can rarely prattle their native tongue this admirable genius could read thirteen languages! In this department of knowledge his precocity was as great as that of Sir Henry Wotton, who, on graduating at Cambridge at the age of thirteen, passed a good examination in thirteen languages, comprising Latin, Greek, Hebrew, Arabic, Syriac, &c. The order in which Hamilton learned his languages was this, - Latin, Greek, Hebrew, Syriac, Persian, Arabic, Sanscrit, Hindostani, and Malay; the modern languages, viz., French, Italian, Spanish, and German, being, for the most part, acquired later. By his own account, he used to read the first five of these better than German. That his knowledge of these tongues must have been considerable is proved by two facts, - that at the age of seven he stood an examination in Hebrew by Dr. Meredith, of Trinity College, Dublin, and that at fourteen he wrote a letter in Persian to the Persian ambassador, Mirza Abon Hassan Khan, who was then (1819) on a visit in Dublin. His early education he received from an uncle, the Rev. James Hamilton, of Trim, an eminent classical scholar, imbued with a taste for science; but in mathematics, like his countryman, Robert Murphy (of whom see a biographical sketch in the series of papers under the title of ``Toiling Upward,'' in the British Controversialist of September, 1867, pp. 202-206), he was mostly self-taught.

Having accidentally fallen in with a Latin copy of Euclid when about ten years of age, he speedily became immersed in the study of geometry. He had already acquired a liking for and great skill in arithmetical calculations; so much so, that Zerah Colburn, the American calculating prodigy, who was then exhibiting in Dublin, and Hamilton engaged in duels of expertness. He had also acquired a taste for algebra, and between twelve and fifteen made himself familiar with the various other branches of mathematics, pure and applied, not only as taught in the ordinary treatises on these subjects, but in the best works of the authors of greatest note in each.

His course was rapid and brilliant. He had mastered Euclid's ``Elements'' at the age of twelve, when he took up Newton's ``Universal Arithmetic.'' At seventeen, like George Boole, he had mastered Newton's ``Principia;'' thenceforth he gave his days and nights to Laplace's ``Mécanique Céleste.'' At this time he was brought under the notice of Dr. Brinkley, the Andrews' Professor and Royal Astronomer. It happened in this wise. The adventurous youth of seventeen had detected a serious mistake in Laplace, and a friend, Mr. G. Kiernan, laid the case before Brinkley. The veteran astronomer sent for Hamilton, who, on presenting himself, submitted to Brinkley an original paper on a case of Osculation, entitled ``Contacts between Algebraic Curves and Surfaces.'' Brinkley was astounded, and came to the conclusion that his visitor was a mathematical genius of extraordinary power, to whom, accordingly, he gave every encouragement. In the following year, Dr. Brinkley imparted to a friend his deliberate opinion of his protégé, in these memorable words:- ``This young man, I do not say will be, but is, the first mathematician of his age.'' Hamilton's progress was now meteoric. He entered the University of Dublin in 1823, and the following year his first paper was read by Dr. Brinkley before the Royal Irish Academy. It treated of Caustics, the name given in optics to a peculiar species of curve, formed by the intersection of reflected or refracted rays of light; and it displayed so great a mastery of the mathematical theory of optics, and treated the difficult thesis with such splendid originality, that he was invited by the Council to develop it further. It was this paper which ultimately appeared in the ``Transactions,'' under the title of a ``Theory of Systems of Rays.'' It was presented to the Academy in 1827, while Hamilton was still an undergraduate. He soon after achieved the distinction of an optime in Greek and in Physics, which corresponds, though it is hardly equal, to a double seniority at Cambridge. He also obtained a somewhat similar honour for Hebrew, and carried off two Chancellor's medals for English poems, the subjects being ``The Ionian Islands,'' and ``Eustace St. Pierre.'' He had not yet taken his degree, when, on Dr. Brinkley's resigning the Andrews' Professorship of Astronomy, the university elected Hamilton to the vacant chair, and he became Royal Astronomer of Ireland. All this took place in the year 1827, when the young hero was but twenty-one years of age. It thus came to pass that, by virtue of the conditions of Bishop Law's will, the new Professor, though an undergraduate, had to examine candidates for mathematical honours who had already taken their degrees.

The young professor was at this time one of the ablest and most enthusiastic members of ``The Porch,'' an association of choice spirits connected with the University of Dublin, which met during the winter months for literary improvement, discussion, and social intercourse - a society to which the Dublin University Review, and subsequently The Dublin University Magazine, owed their origin. To the latter of these Hamilton was an extensive contributor, especially of poetical compositions, some of which have great elegance of diction, depth of thought, and reach of imagination.

He now took up his residence at the Observatory, which is at Dunsink, six miles from Trinity College. It is admitted on all hands that Hamilton's professional lectures on astronomy, delivered to the college classes, were the best ever heard within the walls of Trinity College, uniting in themselves consecutive thinking, logical statement, sound philosophy, exact science, moral truth, and splendid poetical imagination - all blending together so thoroughly as to form the highest and most attractive intellectual treat. It was on the peroration of one of these lectures that Mrs. Hemans founded her beautiful poem, ``The Prayer of the Lonely Student.''

This work, and his own original researches, fully occupied the long days of this gifted and industrious man. The routine work of the Observatory was mainly performed by an assistant. Hamilton was better employed thus than in making observations with the obsolete instruments of the Observatory.

In 1833 he married Miss Helen Maria Bayly. By that lady, who survives him, he had two sons and a daughter, who are all alive. Sir W. R. Hamilton was an active member of the British Association, which he joined at its second meeting in 1832, at Oxford. At various early meetings he gave expositions of his dynamical and optical methods; and when, in 1835, it met at Dublin, under the presidency of Dr. Lloyd, Sir W. R. Hamilton held the office of local secretary, and delivered an address, published in the ``Transactions'' of the body, on ``The Power of Social Sympathy as an Impulse for the Promotion of Science.'' On this occasion Lord Normanby, then Viceroy of Ireland, in the library of the University of Dublin, in presence of the Association, conferred on the representative man of Irish science the honour of knighthood - an honour similar to that which, as Dr. Whewell then remarked, had been conferred at another Trinity College, a hundred and thirty years before on another mathematician of the highest excellence in optics and astronomy, Sir Isaac Newton.

In 1837 he was elected President of the Royal Irish Academy, though Archbishop Whately was put in nomination against him. In the meanwhile gold medals had been showered upon him by the learned societies; but although holding the gold medal of the Royal Society, he never became a F. R. S.!

He retained his professorship for eight years only, but he did not resign his appointment (of Royal Astronomer) at the Observatory, where he continued to reside till his death, which happened on September 2, 1865, in his sixtieth year. He was buried on the 7th, in Mount Jerome Cemetery, and the fellows, scholars, and students of Trinity College, together with the council and members of the Royal Irish Academy, and a great number of private friends, followed his remains to their resting-place.

We will now proceed to give a few anecdotes illustrating the character of the man; some extracts from his poetical works shall follow; and lastly, we will endeavour to give our readers a general notion of Hamilton's greatest achievements in science.

``Extremes meet,'' says the proverb; and it so happens that the subtle and the simple are sometimes combined. It was so with our hero. As a rule, all great mathematicians are simple-minded men. We can hardly call to mind an exception. Hamilton used to speak of himself with childlike candour; some might say with excusable vanity; but the phrase would convey a very false impression. On being called ``the greatest British mathematician,'' he earnestly disclaimed the imputation. ``I think you flatter me there,'' said he. ``I should say either Cayley or Sylvester is the greatest mathematician; but if I am not the greater mathematician, perhaps I am the greater man. It is the combination which, in my case, is extraordinary. I am a poet.'' Sir John Herschel is said to have been an astronomer by bent, and a chemist by birth. So Hamilton said, on another occasion, ``I live by mathematics, but I am a poet.'' He had all Wordsworth's amour-propre, and love of conversation. Wordsworth was fond of citing and reciting his poems, which he did with childlike complacency. So Hamilton used to refer with great gusto to his achievements as an orator, and was wont to recite from memory part of an after-dinner speech he had delivered at Oxford in 1832, which was at the time allowed to be a model of oratory.

We quote from a letter written by the Rev. R. P. Graves, curate of Windermere, an account of the meeting of a brilliant company in the Lake District, in July 1844, which can scarcely fail to interest our readers, bringing into view, as it does, a number of men famous for intellect, imagination, eloquence, wit, knowledge, and culture.

``One of those walks deserves a special record, both on account of the distinguished persons whom it united in enjoyment, and the full realization it afforded of all that might be expected of the quality of the enjoyment to which such men were the contributors. The party consisted of Mr. Wordsworth, Archdeacon Hare, Sir William R. Hamilton, Professor Butler, and two ladies, both by name and mental qualities worthy of the association, besides myself. The day was brilliant, and continued so throughout, as we ascended one of the ravines of Loughrigg Fell, opposite to Rydal, crossed over the fell, descended to the margin of Loughrigg Tarn, and returned to the social circle of Rydal Mount by the western side of Grasmere and Rydal lakes, enjoying the perfect view of the former lake to be seen from the green terrace of Loughrigg, and the equally advantageous aspect of Rydal Mere and Nab Scar, which this route presents. I remember that not only poetry and philosophy, with other lighter matters, formed topics of conversation, but that religious subjects also, and especially the doctrine of the resurrection, were spoken of with a reverent and cordial interest. Our eminent countrymen excited admiration from all by the ample share they contributed, in the way both of original remark and brilliantly apposite quotation, to the fund of intellectual treasure then poured forth. The day was additionally memorable as giving birth to an interesting minor poem of Mr. Wordsworth's. When we reached the side of Loughrigg Tarn (which, you may remember, he notes for its similarity, in the peculiar character of its beauty, to the Lago di Nemi - Dianæ speculum) the loveliness of the scene arrested our steps and fixed our gaze. The spendour of a July noon surrounded us and lit up the landscape, with the Langdale Pikes soaring above, and the bright tarn shining beneath; and when the poet's eyes were satisfied with their feast on the beauty familiar to them, they sought relief in the search, to them a happy vital habit, for new beauty in the flower-enamelled turf at his feet. There his attention was attracted by a fair smooth stone, of the size of an ostrich's egg, seeming to embed at its centre, and, at the same time, to display a dark star-shaped fossil of most distinct outline. Upon closer inspection this proved to be the shadow of a daisy projected upon it with extraordinary precision by the intense light of an almost vertical sun. The poet drew the attention of the rest of the party to the minute but beautiful phenomenon, and gave expression at the time to thoughts suggested by it, and which so interested our friend Professor Butler, that he plucked the tiny flower, and, sying `that it should be not only the theme, but the memorial of the thoughts they had heard,' bestowed it somewhere carefully for preservation. This little poem, in which some of these thoughts were afterwards crystallized, commences with the stanza -

`So fair, so sweet, withal so sensitive,
Would that the little flowers were born to live,
Conscious of half the pleasure that they give!' ''

In his early days Hamilton was a staunch Berkeleyan. Full of The Principles of Human Knowledge and The Minute Philosopher, he went to Highgate, and called on Coleridge. The two poets met that once - the veteran who had versified the asses' bridge, and the youth who, with the grace and dignity of poetry, was soon to construct the high priori bridge which should span the gulf of time and space. Of course the talk turned on philosophy. Coleridge soliloquized, as was his wont; at length, having got in a few words edgeways, Hamilton declared his adherence to Berkeley's Principles. Coleridge's answer was, ``Oh, sir, you will grow out of that;'' as accordingly it came to pass. Of late years it was Hamilton's practice to read Plato and Kant as a relaxation from severer labours! In reference to this he alleged that ``change of labour is, to a studious man, a relaxation.''

``Extremes meet'' sometimes with a vengeance, as where the sublime and the ridiculous clash and commingle. Hamilton was a zealous Christian, and a sincere member of the evangelical section of the Established Church. We have hear of a mathematician calculation the number of possible factions in a house where three were divided against two, ``our Lord having given but three cases out of forty-two.'' This was nothing to Hamilton's speculation concerning the Ascension of Christ. Christians do not all hold the Dædalian, or material explanation of the event; e. g., Dr. Horace Bushnell takes the Ascension to be an externalized symbol of a spiritual act. Hamilton held the grossest form of what we may call Dædalianism. He believed that Christ not only visibly departed from earth, but that he travelled through the planetary spaces into the stellar spaces, and beyond the sidereal universe into a celestial realm; and that He performed this journey in ten days - i. e., between the Resurrection and the day of Pentecost. Hence it is easy to determine the lower limit of our Lord's average velocity, since we know the lower limit of distance, say the distance of the star 61 Cygni! We do not affirm that Hamilton pushed his calculation to this point; but obviously this is the mathematian's necessary inference from the assigned premises. Alas! like Newton, he was as weak in theology as he was strong in mathematics. Hamilton's investigation, which was published in 1842, in the Irish Ecclesiastical Journal, provoked a highly gifted sceptic (himself a poet) to reply, ``What an exquisite reductio ad absurdum!''

In our opinion no great mathematician is a mere mathematician. Hamilton was deeply and widely read in metaphysics. One day, as he and Southey were walking in the country, Hamilton fell into one of his Coleridge-like monologues on a point in metaphysics. The resemblance struck Southey, and he said, ``If you had been Coleridge, you would have talked to that ploughman just as you have been talking to me!''

The following three sonnets, by Sir W. R. Hamilton, will serve as samples of his poetic genius:-


O brooding spirit of wisdom and of love,
   Whose mighty wings e'en now o'ershadow me,
   Absorb me in thine own immensity,
And raise me far my finite self above!
Purge vanity away, and the weak care
   That name or fame of me may widely spread:
   And the deep wish keep burning, in their stead,
Thy blissful influence afar to bear,
Or see it borne! Let no desire of ease,
   No lack of courage, faith, or love, delay
   Mine own steps on that high thought-paven way
In which my soul her clear commission sees:
Yet with an equal joy let me behold
Thy chariot o'er that way by others rolled!


Of high Mathesis, with her charm severe,
   Of line and number, was our theme; and we
   Sought to behold her unborn progeny,
And thrones reserved in Truth's celestial sphere;
While views, before attained, became more clear;
   And how the One in Time, of Space the Three,
   Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echos of an ancient strain,
   Some shadowy outlines of old thoughts sublime,
   Gently he smiled to see, revived again,
In later age, and Occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.


When Vulcan cleft the labouring brain of Jove
   With his keen axe, and set Minerva free,
   The unimprisoned maid, exultingly,
Bounded aloft, and to the heaven above
Turned her clear eyes, while the grim workman strove
   To claim the Virgin Wisdom for his fee,
   His private wealth, his property to be,
And hide in Lemnian cave her light of love.
   If some new truth, O Friend! thy toil discover,
   If thine eyes first by some fair form be blest,
Love it for what it is, and as a lover
   Gaze, or with joy receive thine honoured guest:
The new-found thought set free, awhile may hover
   Gratefully near thee, but it cannot rest.

This we must allow is the pearl of Hamilton's sonnets. The last six lines are indeed transcendently lovely.

It may not be uninteresting to the reader to see a sonnet by his sister, E. M. Hamilton, on the nw planet (Neptune), 1846:-

Immortal Newton! did thy glory seem
   A dewdrop quivering in the light of noon,
   Whose prism of spendour was to perish soon
'Neath the strong sunbeams? Did they fear or dream
Thy genius not a spark from the Supreme -
   King of those myriads? Lo! unto the skies
   Men lift their watching and unsleeping eyes -
Waiting for what? - an unborn planet's beam!
And look! in truth the prophesied one breaks
   Forth 'mid its 'lustrious brethren on their sight.
     Welcome! oh, unimaginably far!
     Eloquent planet! truth-attesting star!
In whose deep silence the Eternal speaks -
   ``I am the Prophet - fount of genius and of light!''

Hamilton composed but two separate works for the press: the first of these was the portly octavo volume called ``Lectures on Quaternions,'' 1853 [Preface (historical), pp. 64; contents, pp. lxxii.; text, pp. 736]; and the second, ``Elements of Quaternions,'' 1866, published after his death. His other writings were contributed to magazines and the Transactions of three learned societies, viz., the Royal Irish Academy, the British Association for the Advancement of Science, and the Royal Society. In the Transactions of the Royal Irish Academy are twelve dissertations by Hamilton, some of which would respectively fill a large octavo volume. Those in the Transactions of the British Association for the Advancement of Science are numerous; while there is but one contribution from his pen in the Philosophical Transactions, viz., ``On a General Method in Dynamics,'' 1834-5.

This paper produced quite a sensation among mathematicians; Jacobi, of Königsberg, expanded and extended the purely mathematical portion, several of the most profound mathematicians of France have commented on its principles and elaborated its applications - all of them uniting to praise the affluent genius of the original discoverer. On account of these researches, the rare and much coveted distinction of Honorary Member of the Imperial Academy of St. Petersburg was conferred on the (so-called) ``Irish Lagrange.''

The rest of his writings will be found in the second, third, and fourth series of the London, Dublin, and Edinburgh Philosophical Magazine, the Dublin University Magazine, the Cambridge and Dublin Mathematical Journal, and the Irish Ecclesiastical Journal. The great majority of these miscellaneous writings are on subjects strictly mathematical, of which it would be hopeless to attempt to convey any popular notion, with, perhaps, the exception of his optical predictions. We will give a brief account of these, and then attempt a necessarily meagre and imperfect sketch of the origin of that Algebraical Method by which Hamilton will be remembered so long as mathematics are cultivated among men. Hamilton enjoys the high desert of having proved the truth of the Undulatory Theory of light. According to Newton's hypothesis, light is an aggregate of minute corpuscules, radiating with enormous velocity from their sources. The opposition hypothesis, which was first developed by Huyghens, and continued by Fresnel, Airey [sic], Hamilton, &c., is that light is a minute vibration of an exceedingly rare medium. One of the tests to which it was put was the theoretical determination of the centre spot in Newton's rings. By the undulatory theory it should be black; by Newton's theory it should be white; and black it really is. In the course of his researches Sir W. R. Hamilton lighted on a much more delicate and decisive test than this. In calculating according to the Undulatory Theory, the course of a single ray of light passing, under certain conditions, through a biaxial crystal, he determined two cases of conical refraction. Only two refracted rays had ever been observed; but according to Hamilton's calculations the incident ray ought under one condition to be broken into an infinite number of rays, and to emerge as a cone; and under another condition to be broken into an infinite number of rays, and after forming an internal cone to emerge as a cylinder. The predictions will be found in Vol. xvii. of the Transactions of the Royal Irish Academy, in a paper read January 23 and October 22, 1832. As we understand the matter, no such results would be countenanced by the corpuscular hypothesis. Dr. Humphrey Lloyd undertook to verify these predictions by very delicate experiments on a crystal of arragonite, and for the first time the two cases of conical refraction were rendered visible! Dr. Lloyd's paper, which will be found in the same volume at p. 145, was read January 28, 1833.

We have already said that Hamilton was a metaphysician. He had read Kant in the German, and was early imbued with the doctrine of Kant's ``Transcendental Æsthetic,'' to the effect that space and time, as perceived in the use of the senses, are anschauungen, or perceptions not derived from the sensible world, but from the nature of the sentient person. Both becoming thus known à priori, each should be the sphere of an à priori science. From space, accordingly, we have derived pure geometry. What pure science has time given us? Kant indeed answers this question; but his commentators are yet divided as to the meaning of his answer. We hold that answer to mean that arithmetic rests on pure time, and therefore algebra, so far as it is pure. It seems highly probably that when Hamilton asked himself the question, What pure science has time given us? he did not trouble himself with Kant's answer; but answered it for himself in the most satisfactory way; viz., by eliciting from pure time all the science it was able to give. The result was his magnificent dissertation ``On Algebra as the Science of Pure Time,'' which was the porch to the main edifice, ``The Theory of Conjugate Functions.'' Then papers were read, or at least presented, to the Royal Irish Academy on November 4, 1833, and June 1, 1835, and are printed in vol. xvii. of their ``Transactions;'' following, in fact, the essay on ``Conical Refraction,'' This was a fruit; that a germ.

Time having but one dimension, it is plain that there are but two ways of regarding a given moment of time; we may regard it as past or future to the present moment. We have to consider also the time which has elapsed, or will elapse, between the two moments. Let that moment be past, we look back through such and such an elapsed time; let it be future, we look forward to it through such and such a time which is to elapse. By thus feigning any given moment to be past or future to another, we may consider all moments in couples, and determine formulæ expressing their relations to each other: as for instance, we may say, in words, from A to B is the converse of from B to A; which Hamilton writes, -

(B - A) + (A - B) = 0.

We thus come to consider the passage in thought from one moment to another; B - A is the step from A to B; and A - B is the step from B to A; and these two steps neutralize each other, or generate a null step. In the course of constructing this theory, which at length becomes very complex, Hamilton found he was constructing an algebra of pure time, which ran side by side with common algebra. But there were great differences between them: in particular the expression [sqrt] -1, which in common algebra expresses an impossible operation or result, is possible and real in Hamilton's new algebra. This expression signifies that a square number of quantity may have a negative sign; whereas in arithmetic and algebra all squares are positive. In the algebra of pure time, the couple (a, b) if operated on by the symbol i becomes (-b, a); and (-b, a) if operated on by the same symbol becomes (-a, -b), which is written i² (a, b) = (-a, -b) = (-1) (a, b). If we now separate the symbols of operation from the couple operated upon, we see plainly that i² = -1. Hence we may safely conclude that i is equivalent to the expression [sqrt] -1, and is a perfectly real operation in the algebra of pure time.

From considering couples Hamilton studied triplets, and finally sets of four moments, or of four steps, which he called quaternions. The magic symbol i operating upon a set (a, b, c, d) regarded as two couples would convert it into (- (c, d), (a, b) ); but regarded as a quaternion it converts it into (b, -a, d, -c); and this in like manner becomes (-a, -b, -c, -d) which is - (a, b, c, d); so that here, as in the case of couples, i² = -1. But that is not all. In considering four moments or steps Hamilton found that he needed two other symbols of operation, which he called j and k. The symbol j turns the quaternion (a, b, c, d) into (c, d, -a, -b), and the symbol k turned it into (d, c, -b, -a): whence he concluded that j² = -1 and k² = -1. He thus obtained the following relations between i, j, and k: i² = j² = k² = -1; and i, j and k are three roots of negative unity.

Up to this point there is absolutely no difficulty. We have found that if we consider (say) a set of four steps in time (which set is called a quaternion), we have three kinds of transformation of the quaternion, the symbols of operation being these wonderful letters, i, j and k. Hamilton's intellect now went through the most wonderful change that every came over mind of man. It is only with the utmost difficulty that we can follow it in its lawful mazes.

In this complex system of transformation of moments and steps in time, Hamilton discerned the dim outline of tridimensional space! In these three symbols, i, j, k, he saw the three rectangular axes of solid geometry; and it at length occurred to him that in this algebra of pure time lay couched a new and most powerful, because natural, geometry. So he thenceforth bent his genius to the enormous task of constructing a new algebra of space, or triple algebra (as De Morgan calls it). He constructs three rectangular axes; suppose one directed northwards, another eastwards, and another upwards, representing three unit lines, i, j, and k. If we turn the northward line about the eastward line as axis, we operate with i on j, and thus get ij = k. Performing the operation a second time we get i²j = ik = -j; or i² = -1. Similarly we get j² = -1, and k² = -1, AS BEFORE, and we have attained to the perfect symmetry of representation - all three axes being represented by a negative root of unity. With this for foundation Hamilton constructs his great and imperishable theory of quaternions, or algebra of pure space.

The analogy of signs, in passing from pure time to pure space, is very easily apprehended. The analogy of time to linear space is self-evident; so is that of + and - (plus and minus) in time to + and - as indicating the directions of a straight line. The transition from + to -, or from - to + in space implies the rotation of a line through 180°; but this is impossible in time, since a point in time can look but two ways: to the past or to the future. Now the symbols i², j², k², do respectively turn + into - in operating on moments or steps in time; i, j, and k respectively indicating an operation of half the extent, cannot make any subject-moment or subject-step rotate at all, so it cannot turn it, though a quadrant. Accordingly, in Hamilton's ``Algebra of Pure Time,'' where the couple (a, b) is the subject, the effect of i upon it is to reverse one only of the moments or steps, and transpose them. So, where the quaternion (a, b, c, d) is the subject, the effect of i, j, or k upon it is to reverse two only of the moments or steps, and effect a transposition among them. In each case the double operation changes the sign of the set from + to -. In pure space, on the contrary, the act of the rotation becomes possible, and i, j, and k respectively effect a quarter revolution on the subject-line, and in that quadrantal rotation is subtlely involved a complex act strictly analogous to the effect of i, j, or k on a set of four moments or steps. In the theory of quaternions, then, as an algebra of space, the term quaternion is still significant. Its use, however, has there reference rather to the fact that four data are required to turn one directed straight line into another, viz., the ratio of their lengths (which is a number, or algebraic quantity), the angle between them, and the two elements which determine the plane in which they lie. When these two straight lines are at right angles to each other, or parallel to each other, the quaternion degenerates; and in the one case is a triplet, in the other a number, or algebraic quantity. The general problem of turning line into line is that which is met and thoroughly solved by this masterly calculus.

The moment when Sir W. R. H. seized the fundamental equations of the theory of quaternions, as an instrument for geometrical investigation, was in the course of the 16th October, 1843 (to quote from a private letter published by Professor Tait), ``as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge, which my boys have since called the Quaternion Bridge; that is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k, exactly such as I have used them ever since.''

We do not propose to attempt further explanation of it, the details being unsuited to the popular pages of the British Controversialist. We have given enough to show the nature of the speculation and its stupendous importance. In the course of developing this theory, Hamilton struck upon a very curious fact, viz., that the differential calculus of Leibnitz is inapplicable to it; so that he was constrained to have recourse to the abandoned theory of fluxions, as left by Newton, in which fact that Anti-Leibnitz party see on of the revenges brought about by time, for the long-successful fraud of the great German! ``Justice is sure though slow.''

``Rarò antecedentem scelestum
Deseruit pede Poena claudo.''

The whole history of mathematics, so far is it has been studied by us, does not afford another instance of such a growth as the theory of quaternions, where the calculus grew in the most orderly fashion out of exact metaphysics. The success, too, promises to be commensurate with the merit of the performance; for already is the theory of quaternions an accepted part of the university curriculum of Cambridge, Dublin, and Edinburgh.

Professor Tait, in an admirable memoir of Hamilton, which we shall specify at the end of this paper, describes two of Hamilton's mechanical inventions, viz., The Icosian Game and The Hodograph. As we are acquainted with these only through Professor Tait's paper, we will give his own words. ``The Icosian game is played on a plane diagram, which represents a distorted projection of a pentagonal dodecahedron (a solid enclosed by twelve faces, each of which has five sides). This diagram consists of 30 straight lines (representing the edges of the dodecahedron), and 20 points where 3 straight lines meet. `The game is played by inserting pegs numbered 1, 2, 3,... 20, in sucessive holes, which are cut at the points of the figure representing the corners of the dodecahedron, taking case to pass only along the lines which represent the edges. It is characteristic of Hamilton that he has selected the twenty consonants of our alphabet to denote these holes. When five pegs are placed in any five successive holes, it is always possible in two ways, sometimes in four, to insert the whole twenty, so as to form a continuous circuit.... This is only the simplest case of the game,' Mr. Jaques, of Hatton Garden, has the copyright.

``The Hodograph is a contrivance for giving `a graphic representation of the velocity and acceleration in every case of motion of a particle. The easiest illustration we can give of this is a special case, the hodograph of the earth's motion in its orbit. In consequence of the fact that light moves with a finite, though very great, velocity, its apparent direction, when it reaches the eye, varies with the motion of the spectator. The position of a star in the heavens appears to be nearer than it really is to the point towards which the earth is moving; in fact, the star seems to be displaced in a direction parallel to that in which the earth is moving, and through a space such as the earth would travel over in the time occupied by light in coming from the star. This is the phenomenon detected by Bradley, and known as the aberration of light. Thus, the line joining the true place of the star with its apparent place represents at every instant, by its length and direction, the velocity of the earth in its orbit. We are now prepared to give a general definition. The hodograph, corresponding to any case whatever of the motion of a point, is formed by drawing, at every instant, from a fixed point, lines representing the velocity of the moving point in magnitude and direction. One of the most singular properties of the hodograph, discovered by Hamilton, is that the hodograph of every planet or comet, however excentric its path may be, is a circle. A star, therefore, in consequence of aberration, appears to be described as an exact circle surrounding its true place, in a plane parallel to the plane of the ecliptic; not merely, as seems formerly to have been assured, an approximate one. But, unless the earth's orbit were exactly circular, the true place of the star will not be the centre of this hodograph.''

Hamilton's last contribution to the Philosophical Magazine (fourth series, vol, xxvii., p. 124) was ``On Röber's Construction of the Heptagon.'' As no diagram is given, it is almost impossible to understand the construction. The heptagon (or regular figure of seven equal sides) cannot be constructed by Euclid's allowance of means - viz., right lines and circles only. Röber's construction, employing no other means, is a very close approximation to the true heptagon. The seventh part of two right angles is 25° 42' 41'' 4...; Röber's approximation to it is 25° 42' 41'' 39.... For all practical purposes Röber's approximation (which, after all, may not be very complex) is all that can be desired. He seems to have had a touch of the Egyptian mania of John Taylor and Piazzi Smyth; for he believed that some of their temples, as the Temple at Edfu, were designedly constructed as to convey to the initiated, and thus perpetuate, this very method of describing the heptagon. The style of Sir W. R. Hamilton's paper is provoking enough. The greatest living Algebraist of our day was both perplexed and provoked by it, and at last threw down the number in which it appeared, with the remark, ``Why can't Hamilton write like any one else?'' Hamilton's practical illustration of Röber's construction, in order to exhibit to the popular mind the closeness of the approximation, is worthy of him. ``Let us imagine,'' he says, ``a series of seven successive chords inscribed in a circle, according to the construction in question, and inquire how near to the initial point the final point would be. The answer is, that the last point would fall behind the first, but only by about half a second (more exactly by 0.'' 506). If, then, we suppose, for illustration, that these chords are seven successive tunnels, drawn eastward from station to station on the equator of the earth, the last tunnel would emerge to the west of the first station, but only by about fifty feet.

After publishing this, Hamilton's time was almost wholly taken up in completing his last work, ``The Elements of Quaternions.'' He had actually finished the text, and revised all the proof sheets, and was working on the admirable index which is appended to the treatise, when he fell into a dangerous illness, of a gouty nature, we believe, but seriously affecting his brain. He recovered from this attack in a measure, and regained his usual mental vigour. But his vitality was exhausted, and he died at his Observatory, only three days before the meeting of the British Association, (held in Birmingham in September, 1865), in which he had fully intended to have taken part.

Four memoirs of him have been published. The first appeared, when he was by thirty-seven, in the number of the Dublin University Magazine for January, 1842; i. e., a year and three-quarters before the invention of Quaternions. This was from the pen of Hamilton's friend, (brother of the present Bishop of Limerick), the Rev. R. P. Graves, and was accompanied by a portrait. The next two memoirs were published immediately after his death, viz., one on the number of the Gentleman's Magazine for January, 1866, from the pen of Professor De Morgan; and one in the Monthly Notices of the Royal Astronomical Society, February 9th, 1866, by their then president, C. Pritchard, Esq. The last and most able, though not the most detailed, recording Hamilton's life and writings, was from the pen of Prof. P. G. Tait, of Edinburgh, and appeared in the number of the North British Review, for September, 1866.

Besides the copper-plate engraving prefixed to the first of these memoirs, there is a daguerreotype of Sir William, and his lady and family, in the possession of his widow; and there are marble busts of him in the possession of Lord Dunraven, and Lord Talbot de Malahide. All these have been photographed as cartes de visite.

The brain of Hamilton was enormous. The forehead, inadequately shown in these portraits, was broad and massive. The eye was full and imaginative, and the orbits protruded with phrenological power, giving hints of linguistic, artistic, and mathematical talent. ``Casualty'' was large, ``Comparison'' and ``Veneration'' very large. The development is of the highest type of pure mathematical power, of which the heads of Cayley, Sylvester, and W. K. Clifford, are also remarkable examples. The last, though but a young man of six and twenty, is of the foremost rank as a geometer, and gives promise of being the Hamilton of the future. May he, like his great predecessor, labour, and be fruitful. It is understood that a life of Hamilton, by the Rev. R. P. Graves, will shortly be published. It will certainly be of the highest interest, as well for its literary merits as for the unique and splendid genius whose life, character, intellect, and works it will commemorate.


D.R. Wilkins
School of Mathematics
Trinity College, Dublin