[North British Review, 45 (1866), 37-74]
With the din of controversy ringing in our ears, as the battle of intellectual giants sways now onward and anon back, it is soothing to turn to something of a loftier character. When Homer has had enough of ghastly gashes, described with sickening fidelity and most impartial relish, whether they be inflicted by Achæan or Trojan, his fancy soars to Olympus, where a more imposing but less numerous group, withal strangely resembling in their peculiarities the inferior race, are employed in those calmer discussions which suit their exalted nature. Let us for a while forsake the task of following the keen intellect of a Lowe or a Newman (too often employed in merely bewildering minds of a stamp inferior to their own), and seek repose in the contemplation of something far more elevated and much more subtle,- the character and works of a man of genius.
This term we use, of course, in the most strict and exclusive sense. Unfortunately, like such terms as `gentleman,' `esquire,' etc. etc., it has in modern days been far too laxly employed. There would be no inconvenience in this had we any higher term to apply to those extraordinary instances which are above everyday comparisons, and in fact furnish themselves the only standard by which they can be measured. We think we may assume that real genius always makes itself known; for it is scarcely conceivable that when, as happens some score or two of times in a century, a human being is endowed with it, he should fail to make his way to the very foremost rank, not merely in his own country but, in the world. For genius is something of a loftier order than the lucid, logical, and quick-witted intelligence of the barrister or the mathematician; it involves essentially an unusual amount of the creative or originative power, and it was in this sense that the ancients regarded the higher flights of the imagination; as the term `poet' remains to testify in most civilized languages.
But if to genius be added enormous erudition and untiring energy, we can hardly set limits to our expectations of what its possessor may achieve, if but life and health be granted to him. When such a phenomenon (as he may well be called) occurs, it behoves us common mortals to study and examine him. Everything about him, even, or perhaps especially, his peculiarities, is deserving of the most careful attention.
Scotland has had such men. In the words of one of the most remarkable of them -
`Yet Caledonia claims some native worth,few, of course, in actual number, because they are everywhere rare; but many, when her small population is considered. Such a list as Napier, Watt, Scott, Hunter, Black, Maclaurin, and Cochrane, through perhaps not including even all of the very first rank, forms a galaxy nowhere to be surpassed. But, besides these undoubted Scotsmen, there are others, such as Byron for instance, who are only not Scottish by a sort of legal fiction. This was the case with Hamilton whose name will not only rank with any in the foregoing list, but will undoubtedly be classed with those of the grandest of all ages and countries, such as Lagrange and Newton.
As dull Boeotia gave a Pindar birth;'
His grandfather came over from Scotland to Dublin with two young sons, of whom Archibald became a solicitor in Dublin, James the curate of Trim, county Meath [see note]. A branch of the Scottish family to which they belonged had settled in the north of Ireland in the time of James I., and this seems to have given rise to the common impression that Hamilton was an Irishman. Archibald married a relative of the celebrated Dr. Hutton [see note], and their son, WILLIAM ROWAN HAMILTON, was born in Dublin on the 4th of August 1805. He displayed great talent at a very early age, and when barely three was given in charge to his uncle, who seems to have at once commenced to teach him Hebrew. He made such progress that, at the age of seven, he was pronounced by one of the Fellows of Trinity College, Dublin, to have shown a greater knowledge of the language than many candidates for a Fellowship. At the age of thirteen he had acquired considerable knowledge of at least thirteen languages. Among these, besides the classical and the modern European languages, were included Persian, Arabic, Sanscrit, Hindustani, and even Malay. This singular direction seems to have been given to his studies, partly by the natural bent of his teacher, and partly because his father intended him for the service of the East India Company. He wrote, at the age of fourteen, a complimentary letter to the Persian Ambassador, who happened to visit Dublin; and the latter said he had not thought there was a man in Britain who could have written such a document in the Persian language. Some idea of the nature of his knowledge of these languages may be gathered from the following extract from a letter of his, dated 1859: `I never learned the [German] language as accurately as I did Greek, or Latin, or Hebrew, or Syriac, or Persian (when I was a boy), and am always fancying that I have quite forgotten it (the German aforesaid), until I take up some book or article, and become interested. I have to think of the difference between the significations of the words Kegel and Kugel!' From this time his mathematical tastes seem to have considerably interfered with his study of languages; and though to the end of his life he retained much of the extraordinary learning of his childhood and youth, often reading Persian and Arabic in the intervals of sterner pursuits, he had long abandoned them as a study, and employed them merely as a relaxation.
His mathematical studies seem to have been undertaken and carried to their full development without any assistance whatever, and the result is that his writings belong to no particular `school,' unless, indeed, we consider them to form, as they are well entitled to do, a school by themselves. As an arithmetical calculator he was not only wonderfully expert, but he seems to have occasionally found a positive delight in working out to an enormous number of places of decimals the result of some irksome calculation. It is probably to his powers of mental arithmetic that a relative of his refers when she says: `I remember him a little boy of six, when he would answer a difficult mathematical question, and run off gaily to his little cart.' At twelve he engaged Colburn, the American `calculating boy,' who was then being exhibited as a curiosity in Dublin, and he had not always the worst of the encounter. But, two years before, he had accidentally fallen in with a Latin copy of Euclid, which he eagerly devoured; at twelve he attacked Newton's Arithmetica Universalis. This was his introduction to modern analysis. He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on Analytical Geometry and the Differential Calculus. We give an extract from a letter written by him about this period to his cousin, a young lady, as it shows not only what he was then engaged upon, but how his work impressed him, and prepares us for some of the more striking qualities which he manifested at a later period:-
`TRIM, October 9, '21.
`... Since I came down ... I have been principally employed in reading Science. In studying Conic Sections and other parts of Geometry, I have often been struck with the occurrence of what may be called demonstrated Mysteries,- since, though they are proved by rigidly mathematical proof, it is difficult, if not impossible, to conceive how they can be true. For instance, it is proved that the most minute line can be divided into an infinite number of parts; and that there can be assigned two lines (the Hyperbola and its asymptote), which shall continually approach without ever meeting, although the distance between them shall diminish within any assignable limits.
`If, therefore, within the very domain of that Science which is most within the grasp of human Reason - which rests on the firm pillars of Demonstration, and is totally removed from doubt or dispute, there be truths which we cannot comprehend, why should we suppose that we can understand everything connected with Nature and Attributes of an Infinite Being? For, if ye understand not Earthly things, how shall ye those that are Heavenly?'
About this period he was also engaged in preparation for entrance at Trinity College, Dublin, and had therefore to devote a considerable portion of his time to classics. In the summer of 1822, in his seventeenth year, he began a systematic study of Laplace's Mécanique Céleste. Nothing could be better fitted to call forth such mathematical powers as those of Hamilton, for Laplace's great work, rich to profusion in analytical processes alike novel and powerful, demands from the most gifted student careful and often laborious study. It was in the successful effort to open this treasure-house that Hamilton's mind received its final temper. `Dès-lors il commença à marcher seul,' to use the words of the biographer of another great mathematician. From that time he appears to have devoted himself almost wholly to original investigation (so far at least as regards mathematics), though he ever kept himself well acquainted with the progress of science both in this country and abroad.
Having detected an important defect in one of Laplace's demonstrations, he was induced by a friend to write down his remarks, that they might be shown to Dr. Brinkley (afterwards Bishop of Cloyne, but) who was then Royal Astronomer of Ireland, and an accomplished mathematician. Brinkley seems at once to have perceived the vast talents of young Hamilton, and to have encouraged him in the kindest manner. He is said to have remarked, in 1823, of this lad of eighteen,- `This young man, I do not say will be, but is, the first mathematician of his age.' And their regard was mutual, for Hamilton always mentions his predecessor with esteem and gratitude. Thus, at the conclusion of the earliest paper he presented to the Royal Irish Academy, he says: `Whatever may be the opinion of others as to its value, I have the pleasure to think that my paper is inscribed to the one who will best be able to perceive and appreciate what is original; whose kindness has encouraged, whose advice has strengthened me; to whose approbation I have ever looked as to a reward sufficient to repay me for industry however laborious, for exertion however arduous.' We shall presently see how well these terms are applicable to the grand investigation to which they are appended.
Hamilton laid before Dr. Brinkley, at their first interview, besides the observations on Laplace already mentioned, some original investigations in analytical geometry, connected with the contact of curves and surfaces, and with pencils of rays. He writes to a friend, in 1858, as follows:- `In one of those unpublished papers, which (when I was quite a boy) attracted the notice of Dr. Brinkley, and won for me a general invitation to breakfast here (the Observatory), which I often walked out from Dublin to avail myself of...;' and from this we see how genially this intimacy was commenced. It was of very great consequence to Hamilton, for Brinkley read his papers carefully, approved especially of the optical one, and requested him to develop it further. This was done about a month after, but neither of these papers has yet been published.
Hamilton had now entered college, and his career there was perhaps unexampled. Amongst a number of competitors of more than ordinary merit, he was first in every subject, and at every examination. His is said to be the only recent case in which a student obtained the honour of an optimé in more than one subject. This distinction had then become vary rare, not being given unless the candidate displayed a thorough mastery over his subject. Hamilton received it for Greek and Physics. How many more such honours he might have attained it is impossible to say; but he was expected to win both the gold medals at the Degree Examination, had his career as a student not been cut short by an unprecedented event. This was his appointment to the Andrews Professorship of Astronomy in the University of Dublin, vacated by Dr. Brinkley in 1827. The chair was not exactly offered to him, as has been sometimes asserted; but the electors, having met and talked over the subject, authorized one of their number, who was Hamilton's personal friend, to urge him to become a candidate, a step which his modesty had prevented him from taking. Thus, when barely twenty-two, he was established at the Dublin Observatory. He was not specially fitted for the post, for, although he had a profound acquaintance with theoretical astronomy, he had paid but little attention to the regular work of the practical astronomer. And it must be said that his time was better employed for himself, his university, and his race, in grand original investigations, than it would have been had he spent it in meridian observations made even with the best of instruments; infinitely better than if he had spent it on those of the Observatory, which, however good in their day, are totally unfit for the delicate requirements of modern astronomy. Indeed there can be little doubt that Hamilton was intended, by the University authorities who elected him to the professorship of Astronomy, to spend his time as he best could for the advancement of science, without being tied down to any particular branch. Had he devoted himself to practical astronomy, they would assuredly have furnished him with modern instruments and an adequate staff of assistants.
But the official duties of the Andrews professor are not confined to the Observatory. He lectures and examines in Trinity College. And in this part of his work Hamilton was unsurpassed, and perhaps unsurpassable. His lectures, sometimes on astronomy, even in its most popular form, anon on his own grand inventions of the hodograph and the quaternion, were admirably lucid, and are said to have almost fascinated his audience. And his examination papers were the despair of the `crammers.' In them there was such an intense novelty and originality, that the experience of forty years could give no inkling of what was coming; the venerable crammers gave up the attempt; and the victory was won by the real intellect of the deserving candidate, not, as it too often is, by the adventitious supply of old material forced into the mere memory of the crammed.
In 1835, being Secretary to the meeting of the British Association, which was held that year in Dublin, he was knighted by the Lord-Lieutenant. But far higher honours rapidly succeeded, among which we may merely mention his election in 1837 to the President's chair in the Royal Irish Academy, and the rare and coveted distinction of being made Corresponding Member of the Academy of St. Petersburg. These are the few salient points (other, of course, than the epochs of his more important discoveries and inventions presently to be considered) in the uneventful life of this great man. Retaining his wonderful faculties unimpaired to the very last, and steadily continuing till within a day or two of his death the task which had occupied the last six years of his life, he died as he had lived, a sincere and humble Christian. He was but sixty years old. What might not that mighty genius have done in a few more years, is a question which all will ask who come to know that he had done for quaternions, and at what a stage in their progress he was removed. He lived long enough so to develop them that their future career is certain; but it is sad to think that he is not to pioneer their next grand and inevitable advance.
In such an article as this we must restrict ourselves to the more important only, or the more distinctive, of Hamilton's discoveries; and in noticing them, and explaining in a brief and popular manner their nature and their applications, we shall adhere, not strictly, but as nearly as possible, to the chronological order.
The germ at least of his first great discovery was contained in one of those early papers which in 1823 he had communicated to Dr. Brinkley. We have already mentioned that Hamilton had considerably developed it; and, under the title of `Caustics,' this paper was presented in 1824, by Brinkley, to the Royal Irish Academy. It was referred, as usual, to a committee. Their report, while acknowledging the novelty and value of its contents, and the great mathematical skill of its author, recommended that before being published, it should be still further developed and simplified by its author. During the next three years the paper grew to an immense bulk, principally by the additional details which had been inserted at the desire of the committee. But it also assumed a much more intelligible form, and the grand features of the new method were now easily seen. Hamilton himself seems not till this period to have fully understood either the nature or the importance of his discovery, for it is only now that we find him announcing his intention of applying his method to dynamics. The paper was now entitled Theory of Systems of Rays, and the first part was printed in 1828 in the Transactions of the Royal Irish Academy. The second and third parts have not yet been printed; but it is understood that their more important contents have appeared in the three voluminous Supplements to the first part which have been published in the Transactions of the Royal Irish Academy, and in the two papers On a General Method in Dynamics, which appeared in the Philosophical Transactions in 1834-5.
To give the popular reader an idea of the nature of the great step taken by Hamilton in these papers is by no means easy, but we may make an attempt. We commence with an extract from an elementary article of his (Dublin University Review, October 1833), not merely because it forms a good introduction to the subject, and gives us some of his own views of his discovery, but also because it is a favourable specimen of his peculiar style:-
`For the explanation of the laws of the linear propagation of light, two principal theories have been proposed, which still divide the suffrages of scientific men.
`The theory of Newton is well known. He compared the propagation of light to the motion of projectiles; and as, according to that First Law of Motion, of which he had himself established the truth by so extensive and beautiful an induction, an ordinary projectile continues in rectilinear and uniform progress, except so far as its course is retarded or disturbed by the influence of some foreign body; so, he thought, do luminous and visible objects shoot off little luminous or light-making projectiles, which then, until they are accelerated or retarded, or deflected one way or another, by the attractions or repulsions of some refracting or reflecting medium, continue to move uniformly in straight lines, either because they are not acted on at all by foreign bodies, or because the foreign actions are nearly equal on all sides, and thus destroy or neutralize each other. This theory was very generally received by mathematicians during the last century, and still has numerous supporters.
`Another theory, however, proposed about the same time by another great philosopher, has appeared to derive some strong confirmations from modern inductive discoveries. This other is the theory of Huygens, who compared the gradual propagation of light, not to the motion of a projectile, but to the spreading of sound through air, or of waves through water. It was, according to him, no thing, in the ordinary sense, no body which moved from the sun to the earth, or from a visible object to the eye, but a state, a motion, a disturbance, was first in one place, and afterwards in another. As, when we hear a cannon which has been fired at a distance, no bullet, no particle even of air, makes its way from the cannon to our ears; but only the aërial motion spreads, the air near the cannon is disturbed first, then that which is a little farther, and last of all the air that touches us. Or like the waves that spread and grow upon some peaceful lake, when a pebble has stirred its surface; the floating water-lilies rise and fall, but scarcely quit their place, while the enlarging wave passes on and moves them in succession. So that great ocean of ether which bathes the farthest stars, is ever newly stirred, by waves that spread and grow, from every source of light, till they move and agitate the whole with their minute vibrations: yet like sounds through air or waves on water, these multitudinous disturbances make no confusion, but freely mix and cross, while each retains its identity, and keeps the impress of its proper origin. Such is the view of Light which Huygens adopted, and which justly bears his name: because, whatever kindred thoughts occurred to others before, he first showed clearly how this view conducted to the laws of optics, by combining it with that essential principle of the undulatory theory which was first discovered by himself, the principle of accumulated disturbance.
`According to this principle, the minute vibrations of the elastic luminous ether cannot perceptibly affect our eyes, cannot produce any sensible light, unless they combine and concur in a great and as it were infinite multitude; and on the other hand, such combination is possible, because particular or secondary waves are supposed in this theory to spread from every vibrating particle, as from a separate centre, with a rapidity of propagation determined by the nature of the medium. And hence it comes, thought Huygens, that light in any one uniform medium diffuses itself only in straight lines, so as only to reach those parts of space to which a straight path lies open from its origin; because an opaque obstacle, obstructing such straight progress, though it does not hinder the spreading of weak particular waves into the space behind it, yet prevents their accumulation within that space into one grand general wave, of strength enough to generate light. This want of accumulation of separate vibrations behind an obstacle, was elegantly proved by Huygens: the mutual destruction of such vibrations by interference, is an important addition to the theory, which has been made by Young and Fresnel. Analogous explanations have been offered for the laws of reflexion and refraction.'
In the time of Euclid it was known that light moves in general in straight lines, and the law of its reflexion was known. So far, therefore, the necessary data for the solution of any optical problem involving any number of successive reflexions was known. But though it was easy enough to apply them to the solution of a particular problem, to found a science on such data was not an easy matter. Huyghens, indeed, was led by the principles of the undulatory theory to make one very general statement. Suppose light to diverge in air from a luminous point, each wave is a sphere surrounding the point as centre, and each ray being a radius of the sphere cuts it at right angles. Thus a series of rays proceeding from a single point have the property of being all cut perpendicularly by a set of surfaces (in this simple case, concentric spheres). After reflexion at a plane mirror, we know that the rays all diverge as if they came from another point, which is called the image of the luminous point. These reflected rays have therefore the property of being cut at right angles by a set of surfaces (in this case spheres with their common centre at the image). If, however, the reflecting surface be not plane, but curved in any manner, do the reflected rays still possess the property of being all intersected at right angles by a series of surfaces? Will they still possess this property after two or more reflections? Huyghens saw that they must, since these surfaces are, on the undulatory theory, the successive waves which have left the source of light. But if this be true, it ought to be capable of proof from the mere data known to Euclid. Malus, celebrated in the modern history of light, and a powerful mathematician, attempted without success to prove the more general of these propositions, and was led by the extreme complexity of his formulæ into errors which induced him to doubt its truth. Another great mathematician, the late Baron Plana, was equally unsuccessful. Before this, however, Hamilton had taken up the question, and had gradually attained the very simple proof of this and other far more general propositions which he gave in his Theory of Systems of Rays. Hamilton's process, when applied to this problem, may be made to depend on two simple propositions, whose truth is evident from ordinary geometry. But, for simplicity, we confine ourselves to the case of one reflexion.
The laws of reflexion (that the angles of incidence and reflexion are equal, and that the plane of the incident and reflected rays contains the perpendicular to the reflecting surface) involve the first of these propositions, viz.: that in general a reflected ray takes the shortest path from a given point S to some point, I, of the reflecting surface, and thence to a second given point, P. [This is an inadequate representation of the truth, for the path may be a maximum, or a maximum-minimum; but it would require considerable detail, or the introduction of analytical expressions, to give an exact statement; and we are attempting, not to explain the subject completely but, to give the general reader an idea of what Hamilton did.] Also, when from a given point the shortest straight line is to be drawn to a given surface, it is evident that it must meet the surface at right angles. This is the second proposition above referred to. Now if we measure off along each reflected ray a length, IP, which, together with the length of the corresponding incident ray, SI, from the luminous point, gives a constant sum, V, the extremities, P, of all such lines will form a certain surface, which may also be called V. Thus, the length of the whole course of each ray, from the luminous point to the surface V, is the same. Hence, if any surface be drawn so as to touch V externally at the point P, the length of the ray SIP is less than if for P we put any other point of the new surface, even if, for I, we substitute any other point of the reflecting surface. Hence, keeping I fixed, IP is the shortest line to the new surface, and is therefore, by the second proposition, perpendicular to it, and of course also perpendicular to the surface V which touches it at P. This is Huyghens' proposition. The quantity or expression V is thus seen to contain the complete solution of any such question: for, if its form can be assigned, we have only to draw perpendiculars to the corresponding surface at every point, and these lines represent the reflected rays. And it is obvious that the same method, with similar results, may be applied to any number of successive reflections.
The quantity V, in these simple questions, is the length of the path which has been described by the ray in its passage from the luminous source. If we multiply it by the velocity of light, it becomes, on the corpuscular theory, what is called the Action of the luminous corpuscle; and the first of the above propositions becomes a case of the principle of Least Action in Dynamics. If we divide V by the velocity of light, we get the Time of passage from the luminous point to the surface V, and this, in the undulatory theory, is a minimum. It appears, then, that the law of reflexion is derivable from either theory.
To form the quantity V for a ray refracted from one homogeneous singly refracting medium into another, we must, on the corpuscular theory, multiply the length of each part of the ray by the velocity with which the corpuscle moves along it, and add the two parts; on the undulatory theory, we must divide the length of each part of the path by the corresponding velocity of the wave, and add. These velocities are determinable by direct experiment, and hence the surfaces corresponding to the two values of V can be constructed. These are, in general, perfectly distinct from each other; so that the refraction of light furnishes a decisive test, and has enabled experimenters to pronounce in favour of the undulatory theory. But, as regards Hamilton's method, it matters not which theory we adopt, if in taking the corpuscular theory we use the reciprocal of the velocity as a multiplier instead of the velocity itself.
The exact step, in the above simple example, at which Hamilton's process comes in is the use of the second of the auxiliary propositions. The first of these propositions is, as we have seen, a case of Maupertuis's Least Action, the second gives a faint indication of Hamilton's Varying Action. In the former we suppose the initial and final points fixed, and determine the requisite form of the intervening path. In the latter we suppose in general the extreme points also to be variable, and determine them by the conditions of the problem.
Supposing the reader to have now an idea of the manner in which the solution of an optical question may be arrived at if we know the function V, which Hamilton calls the Characteristic Function, it remains that we should show how V itself may, in any case, be found. But, unfortunately, this does not admit of any such simple explanation, even in a particular case, as that which we have given of the former part of the question. We can only say that Hamilton showed that it was in every case to be determined by means of two partial differential equations, of the first order and second degree: and that these could be at once formed from the data of each particular problem. To the solution, then, of these two equations, the whole difficulty of any optical question is reduced: and, in the paper and its three supplements, many extremely general properties, most of them perfectly novel, are developed at great length. Chasles speaks of the method employed as `dominant toute cette vaste théorie.' But it is quite impossible to give the non-mathematical reader any idea of the full merit of this remarkable series of memoirs, remarkable not merely for the great and original discoveries in which they abound, but also for `a mastery over the management of algebraical symbols which has perhaps never been surpassed.'
It is strange, indeed, that the one particular result of this theory, which, perhaps more than anything else that Hamilton has done, has rendered his name known beyond the little world of true philosophers, should have been easily within the reach of Fresnel and others for many years before; and in no way required Hamilton's new conceptions or methods, although it was by them that he was led to its discovery. This singular result is still known by the name Conical Refraction, which he proposed for it when he first predicted its existence in the third Supplement to his Systems of Rays, read in 1832. To give the reader an idea of its nature, let us suppose light from a brilliant point to fall on a plate of glass, or other singly refracting body, the side next the light being covered by a plate of metal with a very small hole in it. A single ray will thus be admitted into the glass, will be refracted in the ordinary way, and will escape from the plate as a single ray parallel to the direction of incidence. Try the same experiment with a slice of Iceland-spar, or other doubly refracting crystal. In general, the single incident ray will be split into two, which will pursue separate paths in the crystal, but will emerge parallel to each other and to the incident ray. But if a plate of a biaxal crystal be used, Hamilton showed that there are two directions in which if the incident ray fall it will be divided in the crystal, not into two but, into an infinite number of rays, forming a hollow cone. Each of these rays emerges parallel to the incident ray, so that they form on emergence a hollow cylinder of light.
But further suppose the same three substances to be experimented on as follows: place on each side of the plate a leaf of tinfoil, in which a very small hole is pierced, and expose the whole to light, proceeding, not from a point but, from a large surface. The particular ray which passes in glass, and other single refracting bodies, from hole to hole through the plate, comes from one definite point of the luminous body, and emerges from the second hole as a single ray. In uniaxal crystals, and generally in biaxal crystals, two definite and distinct rays from the luminary are so refracted as to pass from hole to hole; and therefore; at emergence, as each passes out parallel to its direction at incidence, we have two emergent rays. But Hamilton showed that there are two directions in every biaxal crystal, such that if the line between the holes be made to coincide with either, the light which passes from hole to hole will belong to an infinite number of different incident rays, forming a cone. On emergence, they will of course again form a cone. Thus the prediction was, that in a plate formed of a bixal crystal, a single ray, incident in a certain direction, would emerge as a hollow cylinder of light; and that light, forced to pass through such a plate in a certain direction, would enter and emerge as a hollow cone.
These two phenomena are deducible at once form the form of the Wave Surface (as it is called) in biaxal crystals, long before assigned by Fresnel: but no one seems to have anticipated Hamilton in closely studying the form of that surface from its equation, certainly not in recognising the fact that it possesses four conical cusps, and also that it has four tangent planes, each of which touches it, not in one point but, in an infinite number of points forming a circle. The reader may get a rough idea of such properties by thinking of the portion of an apple which is nearest to the stalk.
But, besides these very remarkable results, which Hamilton showed must be obtained by proper experimental methods, he predicted others, of perhaps still more decisive character, with reference to the polarization of the light of the cone and cylinder above described. All these results of theory were experimentally verified, at Hamilton's request, in 1833, by Dr. Lloyd, the substance employed being a plate of arragonite.
The step from Optics to Dynamics, in the application of the method of Varying Action, was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on the subject. These display, like the `Systems of Rays,' a mastery over symbols, and a flow of mathematical language (if the expression can be used) almost unequalled. But they contain, what is far more valuable still, the greatest addition which Dynamical Science has received since the grand strides made by Newton and Lagrange. Jacobi and other mathematicians have developed, to a great extent, and as a question of pure mathematics only, Hamilton's processes, and have thus made extensive additions to our knowledge of Differential Equations. But there can be little doubt that we have as yet obtained only a mere glimpse of the vast physical results of which they contain the germ. And though this, of course, is by far the more valuable aspect in which any such contribution to science can be looked at, the other must not be despised. It is characteristic of most of Hamilton's, as of nearly all great discoveries, that even their indirect consequences are of high value.
After the remarks we have made on the Optical Paper, we may dismiss the Dynamical ones very briefly; for the reader who has followed the illustration we gave of an elementary case of the former, will easily understand its bearing on the latter: and, if the Optical example be not understood, we cannot find a Dynamical one which can be presented with any more chance of being intelligible to him. We will merely quote some of Hamilton's own remarks, inserting (in square brackets), a few hints to help the reader:-
`In the solar system, when we consider only the mutual attractions of the sun and of the ten known planets, the determination of the motions of the latter about the former is reduced, by the usual methods, to the integration of a system of thirty ordinary differential equations of the second order, between the co-ordinates and the time; or, by a transformation of LAGRANGE, to the integration of a system of sixty ordinary differential equations of the first order, between the time and the elliptic elements: by which integrations, the thirty varying co-ordinates, or the sixty varying elements, are to be found as functions of the time. In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree: and every other dynamical problem, respecting the motions of any system, however numerous, of attracting or repelling points, (even if we suppose those points restricted by any conditions of connexion consistent with the law of living force,) is reduced, in like manner, to the study of one central function, of which the form marks out and characterizes the properties of the moving system, and is to be determined by a pair of partial differential equations of the first order, combined with some simple considerations. The difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another: and even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from the reduction of the most complex and, probably, of all researches respecting the forces and motions of body, to the study of one characteristic function, the unfolding of one central relation....
`Although LAGRANGE and others, in treating of the motion of a system, have shown that the variation of this definite integral [the Action of the system] vanishes when the extreme co-ordinates and the constant H [the initial energy] are given, they appear to have deduced from this result only the well-known law of least action; namely, that if the points or bodies of a system be imagined to move from a given set of initial to a given set of final positions, not as they do nor even as they could move consistently with the general dynamical laws or differential equations of motion, but so as not to violate any supposed geometrical connexions, nor that one dynamical relation between velocities and configurations which constitutes the law of living force; and if, besides, this geometrically imaginable, but dynamically impossible motion, be made to differ infinitely little from the actual manner of motion of the system, between the given extreme positions; then the varied value of the definite integral called action, or the accumulated living force of the system in the motion thus imagined, will differ infinitely less from the actual value of that integral. But when this well-known law of least, or as it might be called, of stationary action, is applied to the determination of the actual motion of a system, it serves only to form, by the rules of the calculus of variations, the differential equations of motion of the second order, which can always be otherwise found. It seems, therefore, to be with reason that LAGRANGE, LAPLACE, and POISSON have spoken lightly of the utility of this principle in the present state of dynamics. A different estimate, perhaps, will be formed of that other principle which has been introduced in the present paper, under the name of the law of varying action, in which we pass from an actual motion to another motion dynamically possible, by varying the extreme positions of the system, and (in general) the quantity H, and which serves to express, by means of a single function, not the mere differential equations of motion, but their intermediate and their final integrals.'
These extracts give a very good idea, not only of the method itself, but of Hamilton's own opinion of it, though certain phrases employed may reasonably be objected to.
To give the popular reader an idea of the nature of Quaternions, and the steps by which Hamilton was, during some fifteen years, gradually conducted to their invention, it is necessary to refer to the history of a singular question in algebra and analytical geometry, the representation or interpretation of negative and imaginary (or impossible) quantities.
Descartes' analytical geometry and allied methods easily gave the representation of a negative quantity. For it was seen at once to be a useful convention, and consistent with all the fundamental laws of the subject, to interpret a negative quantity as a quantity measured in the opposite direction to that in which positives of the same kind are measured. Thus a negative amount of elevation is equivalent to depth, negative gain is loss, a negative push is a pull, and so on. And no error, but rather great gain in completeness and generality, results from the employment of this convention in algebra, trigonometry, geometry, and dynamics.
But it is not precisely from this point of view that we can readily see our way to the interpretation of impossible quantities. Such quantities arise thus: If a positive quantity be squared, the result is positive; and the same is true of a negative quantity. Hence, when we come to perform the inverse operation, i.e., extract the square root, we do not at once see what is to be done when the quantity to be operated on is negative. When it is positive, its square root may be either a negative or a positive number, as we have just seen. If positive, it is to be measured off in some definite direction, if negative, in the opposite. But how shall we proceed to lay off the square root of a negative quantity? Wallis, in the end of the sixteenth century, suggested that this might be done by going out of the line on which the result, when real, would have been laid down; and his method is equivalent to this:- Positive unity being represented by an eastward line, negative unity will of course be represented by an equal westward line, and these are the two square roots of positive unity. According to Wallis' suggestion a northward and a southward line may now be taken to represent the two square roots of negative unity, or the so-called impossibles or imaginaries of algebra. But the defect of this is that we might have assumed with equal reason any other line (perpendicular to the eastward one) as that on which the imaginary quantities are to be represented. In fact, Wallis' process is essentially limited to plane problems, and has no application to tridimensional space. But, imperfect as this step is, it led at once to another of great importance, the consideration of the length, and direction, of a line independently of one another. And we now see that as the factor negative unity simply reverses a line, while the square root of negative unity (employed as a factor) turns it through a right angle, the one operation may be looked upon as in a certain sense a duplication of the other. In other words, twice turning through a right angle, about the same axis, is equivalent to a reversal; or, negative unity, being taken to imply reversal of direction, may be considered as rotation through two right angles, and its square root (the ordinary imaginary or impossible quantity) may thus be represented as the agent which effects a certain quadrantal rotation. But, as before remarked, the axis of this rotation is indeterminate; it may have any direction whatever perpendicular to the positive unit line. If we fix on a particular direction, everything becomes definite, and we can on the same plan interpret the (imaginary) cube roots of negative unity as factors or operators which turn a line through an angle of sixty degrees positively or negatively. Similarly, any power of negative unity, positive or negative, whole or fractional, obtains an immediate representation. And the general statement of this proposition leads at once (but not by the route pursued by its discoverer) to what is called De Moivre's Theorem, one of the most valuable propositions in plane trigonometry. Warren, Argand, Grassmann, and various others, especially in the present century, vainly attempted to extend this process to space of three dimensions. The discovery was reserved for Hamilton, but was not attained even by him till after fifteen or twenty years of arduous work. And it is a curious fact that it was by speculations totally unconnected with geometry that he was so prepared as to see, almost at the instant of seizing it, the full value of his invention. The frightful complexity of the results to which Warren was led in endeavouring to express as lines the products and quotients of directed lines in one plane, seems to have induced Hamilton to seek for a representation of imaginary quantities altogether independent of geometry. The results of some at least of his investigations are given in a very curious essay, Algebra as the Science of pure Time, communicated to the Royal Irish Academy in 1833, and published, along with later developments, in the seventeenth volume of their Transactions. We quote considerable portions of the introductory remarks prefaced to this Essay, as they show, in a very distinct manner, the logical character and the comprehensive grasp of Hamilton's mind.
`The Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contemplation; according as ease of operation, or symmetry of expression, or clearness of thought, (the agere, the fari, or the sapere,) is eminently prized and sought for. The Practical person seeks a Rule which he may apply, the Philological person seeks a Formula which he may write, the Theoretical person seeks a Thoerem on which he may meditate. The felt imperfections of Algebra are of three answering kinds. The Practical Algebraist complains of imperfection when he finds his Instrument limited in power; when a rule, which he could happily apply to many cases, can be hardly or not at all applied by him to some new case; when it fails to enable him to do or to discover something else, in some other Art, or in some other Science, to which Algebra with him was but subordinate, and for the sake of which and not for its own sake, he studied Algebra. The Philological Algebraist complains of imperfection, when his Language presents him with an Anomaly; when he finds an Exception disturb the simplicity of his Notation, or the symmetrical structure of his Syntax; when a Formula must be written with precaution, and a Symbolism is not universal. The Theoretical Algebraist complains of imperfection, when the clearness of his Contemplation is obscured; when the Reasonings of his Science seem anywhere to oppose each other, or become in any part too complex or too little valid for his belief to rest firmly upon them; or when, though trial may have taught him that a rule is useful, or that a formula gives true results, he cannot prove that rule, nor understand that formula: when he cannot rise to intuition from induction, or cannot look beyond the signs to the things signified.
`It is not here asserted that every or any Algebraist belongs exclusively to any one of these three schools, so as to be only Practical, or only Philological, or only Theoretical. Language and Thought react, and Theory and Practice help each other. No man can be so merely practical as to use frequently the rules of Algebra, and never to admire the beauty of the language which expresses those rules, nor care to know the reasoning which deduces them. No man can be so merely philological an Algebraist but that things or thoughts will at some times intrude upon signs; and occupied as he may habitually be with the logical building up of his expressions, he will feel sometimes a desire to know what they mean, or to apply them. And no man can be so merely Theoretical, or so exclusively devoted to thoughts, and to the contemplation of theorems in Algebra, as not to feel an interest in its notation and language, its symmetrical system of signs, and the logical forms of their combinations; or not to prize those practical aids, and especially those methods of research, which the discoveries and contemplations of Algebra have given to other sciences. But, distinguishing without dividing, it is perhaps correct to say that every Algebraical Student and every Algebraical Composition may be referred upon the whole to one or other of these three schools, according as one or other of these three views habitually actuates the man, and eminently marks the work.
`These remarks have been premised, that the reader may more easily and distinctly perceive what the design of the following communication is, and what the Author hopes or at least desires to accomplish. That design is Theoretical, in the sense already explained, as distinguished from what is Practical on the one hand, and from what is Philological on the other. The thing aimed at, is to improve the Science, not the Art, nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor failures of symmetry in expression. And that confusions of thought and errors of reasoning, still darken the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit of love and honour have studied Algebraic Science, admiring, extending, and applying what has been already brought to light, and feeling all the beauty and consistence of many a remote deduction, from principles which yet remain obscure, and doubtful.
`For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by Euclid in his Elements, two thousand years ago; though he may well desire to see them treated in a clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of the argument. But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes, each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing. It must be hard to found a SCIENCE on such grounds as these, though the forms of logic may build up from them a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them.
`So useful are those rules, so symmetrical are those expressions, and yet so unsatisfactory those principles from which they are supposed to be derived, that a growing tendency may be perceived to the rejection of that view which regarded Algebra as a SCIENCE, in some sense analogous to Geometry, and to the adoption of one or other of those two different views, which regard Algebra as an Art, or as a Language: as a System of Rules, or else as a System of Expressions, but not as a System of Truths, or Results having any other validity than what they may derive from their practical usefulness, or their logical or philological coherence. Opinions thus are tending to substitute for the Theoretical question,- ``Is a Theorem of Algebra true?'' the Practical question,- ``Can it be applied as an Instrument, to do or to discover something else, in some research which is not Algebraical?'' or else the Philological question,- ``Does its expression harmonize, according to the Laws of Language, with other Algebraical expressions?''
`Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is continually engaging mathematicians more and more, and has almost superceded the Study of Geometrical Science, were found at last to be not, in any strict or proper sense, the Study of a Science at all: and if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Elegance, at the expense of Contemplation and Intuition. Indulgence, therefore, may be hoped for, by any one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a SCIENCE of Algebra: a Science properly so called; strict, pure, and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of a priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning.
`The Author of this paper has been led to the belief, that the Intuition of Time is such a rudiment. This belief involves the three following as components: First, that the notion of Time is connected with existing Algebra; Second, that this notion or intuition of Time may be unfolded into an independent Pure Science; and Third, that the Science of Pure Time, thus unfolded, is co-extensive and identical with Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction; the second of a deduction; the third is a joint result of the deductive and inductive processes.'
It would not be easy, in our limited space, and without using algebraic symbols freely, to give the reader more than a very vague idea of the nature of this Essay. What we are most concerned with at present is the bearing of its processes upon the interpretation of imaginary quantities, and even on that we can only say a few words. The step in time from one definite moment to another depends, as is easily seen, solely on the relative position in time of the two moments, not on the absolute date of either. And, in comparing one such step with another, there can be a difference only in duration and direction,- i.e., one step may be longer or shorter than the other, and the two may be in the same or in opposite directions, progressive or retrograde. Here numerical factors, positive and negative, come in. But to introduce something analogous to the imaginary of algebra, Hamilton had to compare with each other, not two, but two pairs or Couples of, steps. Thus, a and b representing steps in time, (a, b) is called a Couple; and its value depends on the order as well as the magnitude of its component steps. It is shown that (-a, -b) is the same couple taken negatively. And the imaginary of common algebra is now represented by that operation on a step-couple which changes the sign (or order of progression) of the second step of the couple, and makes the steps change places. That is, it is the factor or operator which changes (a, b) into (-b, a): for a second application will obviously produce the result (-a, -b). There is, no doubt, here a perfectly real interpretation for the so-called imaginary quantity, but it cannot be called simple, nor is it at all adapted for elementary instruction. The reader will observe that Hamilton, with his characteristic sagacity, has chosen a form of interpretation which admits of no indeterminateness. Unlike Wallis and others, who strove to express ordinary algebraic imaginaries by directions in space, Hamilton gave his illustration by time or progression, which admits, so to speak, of but one dimension. We may attempt to give a rough explanation of this process, for the reader who is not familiar with algebraic signs, in some such way as this:- If an officer and a private be set upon by thieves, and both be plundered of all they have, this operation may be represented by negative unity. And the imaginary quantity of algebra, or the square root of negative unity, will then be represented by a process which would rob the private only, but at the same time exchange the ranks of the two soldiers. It is obvious that on a repetition of this process both would be robbed, while they would each be left with the same rank as the first. But what is most essential for remark here is that the operation corresponding to the so-called imaginary of algebra is throughout regarded as perfectly real.
In 1835 Hamilton seems to have extended the above theory from Couples to Triplets, and even to a general theory of Sets, each containing an assigned number of time-steps. Many of his results are extremely remarkable, as may be gathered from the only published account of them, a brief notice in the Preface to his Lectures on Quaternions. After having alluded to them, he proceeds: `There was, however, a special importance to the consideration of triplets.... This was the desire to connect, in some new and useful (or at least interesting) way, calculation with geometry, through some undiscovered extension, to space of three dimensions, of a method of construction or representation which had been employed with success by Mr. Warren (and indeed also by other authors, of whose writings I had not then heard), for operations on right lines in one plane: which method had given a species of geometrical interpretation to the usual and well-known imaginary symbol of algebra.' After many attempts, most of which launched him, like his predecessors and contemporaries, into a maze of expressions of fearful complexity, he suddenly lit upon a system of extreme simplicity and elegance. The following remarkable interesting extract from a letter gives his own account of the discovery:-
` Oct. 15, '58.
`P.S. - To-morrow will be the fifteenth birthday of the Quaternions. They started into life, or light, full grown, on the 16th of October 1843, 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. I pulled out, on the spot, a pocket-book which still exists, and made an entry, on which, at that very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be 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.
`Less than an hour elapsed, before I had asked and obtained leave, of the Council of the Royal Irish Academy, of which Society I was, at that time, the President,- to read at the next general Meeting, a Paper on Quaternions; which I accordingly did, on November 13, 1843.
`Some of those early communications of mine to the Academy may still have some interest for a person like you, who has since so well studied my Volume, which was not published for ten years afterwards.
`In the meantime, will you not do honour to the birthday, to-morrow, in an extra cup of - ink? for it may be obsolete now to propose XXX,- or even XYZ.'
We must now endeavour to explain, in a popular manner as possible, the nature of the new Calculus. In order to do so, let us recur to the suggestion of Wallis, before described, and endeavour to ascertain the exact nature of its defects. We easily see that one great defect is want of symmetry. As before stated, if we take an eastward line, of proper length, to represent positive unity, an equal westward line represents negative unity; but all lines perpendicular or inclined to these are represented by so-called imaginary quantities. Hamilton's great step was the attainment of the desired symmetry by making all lines alike expressible by so-called imaginary quantities. Thus, instead of 1 for the eastward line, and the square root of negative unity for a northward line, he represents every line in space whose length is unity by a distinct square root of negative unity. All are thus equally imaginary, or rather equally real. The i, j, k mentioned in the extract just given, are three such quantities; which (merely for illustration, because any other set of three mutually rectangular directions will do as well) we may take as representing unit lines drawn respectively eastwards, northwards, and upwards. The square of each being negative unity, we may interpret the effect of such a line (when used as a factor or operator) as a left (or right) handed rotation through one right angle about its direction. The effect of a repetition of the operation is a rotation through two right angles, or a simple inversion. Thus if we operate with i on j we turn the northward line left-handedly through a right angle about an eastward axis, i. e., we raise it to a direction vertically upward. Thus we see that ij = k. But if we perform the operation again, we see that ik or i²j is now the southward line or -j. Thus i² = -1. And similarly with the squares of j and k. It is to be noticed that in all these cases the operating line is supposed to be perpendicular to the operand. Also that we have taken for granted (what is easily proved), that i, j, k may stand indifferently for the unit lines themselves or for the operation of turning through a right angle. Thus, the equation ij = k may either mean (as above) that i acting on the line j turns it into the line k; or that the rectangular rotation j, succeeded by i, is equivalent to the single rotation k. We may easily verify the last assertion by taking i as the operand. j changes it to -k, and i changes this to j. But k turns i into j at once.
Even these simple ideas lead us at once to one of the most remarkable properties of quaternions. When turning the northward line (j) about the eastward line (i), we wrote the operator first,- thus ij = k. Now, the order of multiplication is not indifferent, for ji is not equal to k. ji, in fact, expresses that i (the eastward line) has been made to rotate left-handedly through a right angle about j (the northward line). This obviously brings it to the downward direction, or we have ji = -k. Similar expressions hold for the other products two and two of the three symbols. Thus we have the laws of their multiplication complete. And on this basis the whole theory may be erected. Now any line whatever may be resolved (as velocities and forces are) into so much eastward (or westward), so much northward (or southward) and so much upward (or downward). Hence every line may be expressed as the sum of three parts, numerical multiples of i, j and k respectively. Call these numbers x, y and z, then the line may be expressed by xi + yj + zk. If we square this we find - (x² + y² + z²); for the other terms occur in pairs like xi × yj and yj × xi, and destroy each other; since we have, as above, ij + ji = 0, with similar results for the other pairs of the three rectangular unit-lines. Now x² + y² + z² is the square of the length of the line (by a double application of Euclid I. xlvii.) Hence the square of every line of unit length is negative unity. And herein consists the grand symmetry and consequent simplicity of the method, for we may now write a single symbol such as (Greek letters are usually employed by Hamilton in this sense), instead of the more cumbrous and not more expressive form xi + yj + zk.
We have seen that the product, and consequently the quotient, of two lines at right angles to each other, is a third line perpendicular to both; and that the product or quotient of two parallel lines is a number: what is the product, or the quotient, of two lines not at right angles, and not parallel, to each other? It is a QUATERNION. This is very easily seen thus. Take the quotient of one line by another, and suppose them drawn from the same point: the first line may, by letting fall a perpendicular from its extremity upon the second, be resolved into two parts, one parallel, the other perpendicular, to the second line. The quotient of the two parallel lines is a mere number, that of the two perpendicular lines is a line, and can therefore be expressed as the sum of multiples of i, j, and k. Hence, w representing the numerical quotient of the parallel portions, the quotient of any two lines may be written as w + xi + yj + zk; and, in this form, is seen to depend essentially upon FOUR perfectly distinct numbers; whence its name. In actually working with quaternions, however, this cumbrous form is not neccessary; we may express it as , and being the two lines of which it is the quotient; and in various other equivalent algebraic forms; or we may at once substitute for it a single letter (Hamilton uses the early letters a, b, c, etc.) The amount of condensation, and consequent shortening, of the work of any particular problem which is thus attained, though of immense importance, is not by any means the only or even the greatest advantage possessed by quaternions over other methods of treating analytical geometry. They render us entirely independent of special lines, axes of co-ordinates, etc., devised for the application of other methods, and take their reference lines in every case from the particular problem to which they are applied. They have thus what Hamilton calls an `internal' character of their own; and give us, without trouble, an insight into each special question, which other methods only yield to a combination of great acuteness with patient labour. In fact, before their invention, no process was known for treating problems in tridimensional space in a thoroughly natural and inartificial manner.
But let him speak for himself. The following passage is extracted from a letter to a mathematical friend, who was at the time engaged in studying the new calculus:-
... `Whatever may be the future success ... of Quaternions as an Instrument of Investigation, they furnish already, to those who have learned to read them (,) a powerful ORGAN OF EXPRESSION, especially in geometrical science, and in all that widening field of physical inquiry, to which relations of space (not always easy to express with clearness by the Cartesian Method) are subsidiary, or rather are indispensable.'
To follow up the illustration we began with, it may be well merely to mention here that a quaternion may in all cases be represented as a power of a line. When a line is raised to the first, third, or any odd integral power, it represents a right-angled quaternion, or one which contains no pure numerical part; when to the second, fourth, or other even integral power, it degenerates to a mere number; for all other powers it contains four distinct terms. Compare this with the illustration already given, leading to De Moivre's theorem; and we see what a grand step Hamilton supplied by assigning in every case a definite direction to the axis about which the rotation takes place.
There are many other ways in which we can exhibit the essential dependence of the product or quotient of two directed lines (or Vectors as Hamilton calls them), on four numbers (or Scalars), and the consequent fitness of the name Quaternion. It may interest the reader to see another of them. Let us now regard the quaternion as the factor or operator required to change one side of a triangle into another; and let us suppose the process to be performed by turning one of the sides round till it coincides in direction with the other, and then stretching or shortening it till they coincide in length also. For the first operation we must know the axis about which the rotation is to take place, and the angle or amount of rotation. Now the direction of the axis depends on two numbers (in Astronomy they may be Altitude and Azimuth, Right Ascension and Declination, or Latitude and Longitude); the amount or rotation is a third number; and the amount of stretching or shortening in the final operation is the fourth.
Among the many curious results of the invention of quaternions, must be noticed the revival of fluxions, or, at all events, a mode of treating differentials closely allied to that originally introduced by Newton. The really useful, but over-praised differential coefficients, have, as a rule, no meaning in quaternions; so that, except when dealing with scalar variables (which are simply degraded quaternions), we must employ in differentiation fluxions or differentials. And the reader may easily understand the cause of this. It lies in the fact that quaternion multiplication is not commutative; so that, in differentiating a product, for instance, each factor must be differentiated where it stands; and thus the differential of such a product is not generally a mere algebraic multiple of the differential of the independent quaternion-variable. It is thus that the whirligig of time brings its revenges. The shameless theft which Leibnitz committed, and which he sought to disguise by altering the appearance of the stolen goods, must soon be obvious, even to his warmest partisans. They can no longer pretend to regard Leibnitz as even a second inventor when they find that his only possible claim, that of devising an improvement in notation, merely unfits Newton's method of fluxions for application to the simple and symmetrical, yet massive, space-geometry of Hamilton.
One very remarkable speculation of Hamilton's is that in which he deduces, by a species of metaphysical or à priori reasoning, the results previously mentioned, viz., that the product (or quotient) of two parallel vectors must be a number, and that of two mutually perpendicular vectors a third perpendicular to both. We cannot give his reasoning at full length, but will try to make part of it easily intelligible.
Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them; for by changing the sign of both factors the product is unchanged, whereas, as the whole system has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be wholly or partly a number, because on inverting one of them the sign of that number ought to change; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and from the symmetry of space ought to leave the number unchanged. Hence the product of two perpendicular vectors must be a vector, and an easy extension of the same reasoning shows that it must be perpendicular to each of the factors. It is easy to carry this further, but enough has been said to show the character of the reasoning.
It is characteristic of Hamilton that he fancied he saw in the quaternion, with its scalar and vector elements, the one merely numerical, the other having reference to position in space, a realization of the Pythagorean Tetractys
as it is called in the Carmen Aureum.
Of course, so far as mere derivation goes, it is hard to see any difference between the Tetractys and the Quaternion. But we are almost entirely ignorant of the meaning Pythagoras attached to his mystic idea, and it certainly must have been excessively vague, if not quite so senseless as the Abracadabra of later times. Yet there is no doubt that Hamilton was convinced that Quaternions, in virtue of some process analogous to the quasi-metaphysical speculation we have just sketched, are calculated to lead to important discoveries in physical science; and, in fact he writes -
`Little as I have pursued such [physical] Studies, even in books, you may judge from my Presidential Addresses, pronounced on the occasions of delivering Medals (long ago), from the chair of the R.I.A., to Apjohn and to Kane, that physical (as distinguished from mathematical) investigations have not been wholly alien to my somewhat wide, but doubtless very superficial, course of reading. You might, without offence to me, consider that I abused the license of hope, which may be indulged to an inventor, if I were to confess that I expect the Quaternions to supply hereafter, not merely mathematical methods, but also physical suggestions. And, in particular, you are quite welcome to smile, if I say that it does not seem extravagant to me to suppose, that a full possession of those à priori principles of mine, about the multiplication of vectors - including the Law of the Four Scales, and the Conception of the Extra-spatial Unit,- which have as yet been not much more than hinted to the public,- MIGHT have led, (I do not at all mean that in my hands they ever would have done so,) to an ANTICIPATION of something like the grand discovery of OERSTED; who, by the way, was a very à priori (and poetical) sort of man himself, as I know from having conversed with him, and received from him some printed pamphlets, several years ago. It is impossible to estimate the chances given, or opened up, by any new way of looking at things; especially when that way admits of being intimately combined ... with calculation of a most rigorous kind.'
This idea is still further developed in the following sonnet, which gives besides a good idea of his powers of poetical composition. It is understood to refer to Sir John Herschel, who had, at a meeting of the British Association, compared the Quaternion Calculus to a Cornucopia, from which, turn it as you will, something new and valuable must escape.
Or 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 of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes 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.
Whatever may be the future of Quaternions, and it may possibly far surpass all that its inventor ever dared to hope, there can be but one opinion of the extraordinary genius, and the untiring energy of him who, unaided, composed in so short a time two such enormous treatises as the Lectures (1853), and the Elements of Quaternions (1866). As a repertory of mathematical facts, and a triumph of analytical and geometrical power, they can be compared only with such imperishable works as the Principia and the Mécanique Analytique. They cannot be said to be adapted to the wants of elementary teaching, but we are convinced that every one who has a real liking for mathematics, and who can get over the preliminary difficulties, will persevere till he finishes the work, whichever of the two it may be, he has commenced. They have all that exquisite charm of combined beauty, power, and originality which made Hamilton compare Lagrange's great work to a `scientific poem.' And they conduct the mathematician to a boundless expanse of new territory of the richest promise, the cultivation of which cannot be said to have been more than commenced, even by labour so unremitting, and genius so grand, as Hamilton brought to bear on it.
The unit vectors of the quaternion calculus are not the only roots of unity which Hamilton introduced into practical analysis. In various articles in the Philosophical Magazine he developed the properties of groups of symbols analogous to the i, j, k of quaternions, but more numerous, and gave various applications of them. These groups have, generally, a direct connexion with the `Sets' with which we was occupied just before the invention of the quaternions: and it would be vain to attempt to explain their nature to the general reader. But we must say a few words about another, and most extraordinary, system which Hamilton seems to have invented about 1856, and which has no connexion whatever with any previous group. Unfortunately, Hamilton has published but a page or two with reference to them, yet that little is enough to show that the probability of their becoming, at some future time, of great importance in the study of crystals and polyhedra in general. The subject is capable of indefinite extension; but Hamilton seems to have carefully studied only one particular system, which depends mainly on two distinct and non-commutative fifth roots of positive unity, which, for ease of reference, we will call, with their inventor, and . Although nothing more practical than an ingenious `puzzle' has yet resulted from these investigations, their singular originality and (if we may use the word) oddity, and the wonderful series of new transformations which they suggest to the mathematician, render them well worthy of further study and development. Some idea of a small class of their properties may be derived from the consideration of a pentagonal dodecahedron (a solid enclosed by twelve faces, each of which has five sides). The number of edges of this solid is thirty; as we may see by remarking that, if we could five edges for each of the twelve faces, each edge will have been taken twice. Also, since three edges meet in each corner, and since each edge passes through two corners, we shall get three times too many corners by counting two for each edge. That is, there are twenty corners. Now, in that one of Hamilton's systems which he most fully worked out, the operators and , applied to any edge of the pentagonal dodecahedron, change it into one of the adjoining edges. Thus, going along an edge, to a corner, we meet two new edges, that to the right is derived from the first by the operator , that to the left by . Every possible way of moving along successive edges of such a solid may therefore be symbolized by performing on the first edge the successive operations and in any chosen order. And, as the reader may easily convince himself by trial, such a group of twenty operations as this, consisting of the series , , , , , , , , , , taken twice, brings us back to the edge we started from, after passing through each corner once, and only once. Such results as these, however, are far more easily obtained by analysis. Upon this mathematical basis Hamilton founded what he called the Icosian Game, an elegant, and in some cases difficult puzzle. As the dodecahedron would be a clumsy article to handle, besides having the disadvantage of permitting the players to see only half of its edges at once, Hamilton substituted for it the annexed plane diagram, [Note: For the use of this diagram we are indebted to Mr. Jacques of Hatton Garden, London, who has purchased the copyright of the game.] which is somewhat distorted by projection (the eye being supposed to be placed very near to the middle of one face), in order to prevent any two of the lines which represent the edges from crossing each other. The game is played by inserting pegs, numbered 1, 2, 3,... 20, in successive holes, which are cut at the points of the figure representing the corners of the dodecahedron; taking care to pass only along the lines which represent the edges. It is characteristic of Hamilton that he has selected the 20 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. Thus, let BCDFG be the given five, we may complete the series by following the order of the consonants; or we may take the following order (after G) HXWRSTVJKLMNPQZ. If LTSRQ, or ZBCDM, be given there are four solutions. If fewer than five be fixed at starting there are, of course, more solutions. This is only the simplest case of the game. Puzzles without number, and of a far higher order of difficulty, can be easily suggested after a little practice, but even more readily by the proper mathematical processes. Thus, BCD may be given, the problem being to insert all the pegs in order, and end at a given hole. If that hole be M, it is impossible; if T, there is one solution; if J, two; and, if R, four. Again, certain initial points being given, finish with a given number of pegs. Thus, given KJV, finish with the eighth. The other five are TSNML, for when we have got to L no other peg can be inserted. If LKJ be given the others are VWRST. Similarly to finish with any additional number short of 18.
We have been thus explicit on this apparently trivial matter, because we do not know of any other game of skill which is so closely allied to mathematics, and because the analysis employed, though very simple, is more startlingly novel than even that of the quaternions. The i, j, k of quaternions can, as we have seen, be represented by three definite unit-lines at right angles to each other. How can we represent geometrically the or the of this new calculus, either of which produces precisely the same effect whatever edge of whatever face of the dodecahedron it be applied to?
Another very elegant invention of Hamilton's, and one which appears to have been suggested to him by his quaternion investigations, is the Hodograph, which supplies 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 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 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 the orbit of every planet and comet, however excentric its path may be, is a circle. A star, therefore, in consequence of aberration, appears to describe 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 assumed, an approximate one. But, unless the earth's orbit were exactly circular, the true place of the star will not be the centre of the hodograph. To enter into further details on this subject we should require geometrical diagrams or analytical symbols.
The discoveries we have already described, and the papers and treatises we have mentioned, might well have formed the whole work of a long and laborious life. But, not to speak of the enormous collection of MS. books, full to overflowing with new and original matter, left by Hamilton, which have been handed over to Trinity College, Dublin, and of whose contents we hope a large portion at least may soon be published, the works we have already called attention to barely form the greater portion of what he has published. His extraordinary investigations connected with the solution of algebraic equations of the Fifth Degree, and his examination of the results arrived at by Abel, Jerrard, and Badano, in their researches on this subject, form another grand contribution to science. There is also his great paper on Fluctuating Functions, a subject which, since the time of Fourier, has been of immense and ever increasing value in physical applications of mathematics. Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of differential equations which constantly occur in the treatment of physical questions, only a few items have been published, at intervals, in the Philosophical Magazine. Besides all this, Hamilton was a voluminous correspondent. Often a single letter of his occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of his mind, never to be satisfied with a general understanding of a question, he pursued it until he knew it in all its details. He was ever courteous and kind in answering any applications for assistance in the study of his works, even when his compliance must have cost him much valuable time. He was excessively precise and hard to please, with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little, compared with the extent of his investigations. His peculiar use of capitals, italics, and other typographical artifices for the purpose of imitating in writing and type, as closely as possible, the effects of emphasis and pause in a vivâ voce lecture, will be evident from almost any of the extracts we have made from his works. To such an extent did he carry this, that some pages of his Lectures are almost painful to the eye.
Hamilton had, at one time, serious intentions of entering the Church, and was, more than once, offered ordination. The following letter, written to the Editor of the Irish Ecclesiastical Journal, and published in that work, contains a very singular attempt to elucidate one of the grandest questions connected with the Christian religion.
`ON THE ASCENSION OF OUR BLESSED LORD.
`Whitsun Eve, 1842.
`SIR,- The meditations of a Christian, at this sacred season, turn naturally on that seeming pause in the operations of divine Providence, when, as at this time, the disciples who had seen their Lord parted from them, and taken up into heaven, were waiting at Jerusalem for the promised coming of the Comforter. You will judge whether the following remarks, in part confessedly conjectural, but offered (it is hoped) in no presumptuous spirit, may properly occupy any portion of your columns, in connexion with the events which the Church at this season commemorates.
`It may be assumed that your readers are disposed to adopt, in its simplicity, the teaching of the 4th article, that ``Christ did truly rise again from death, and took again his body, with flesh, bones, and all things appertaining to the perfection of Man's nature; wherewith he ascended into Heaven, and there sitteth, until he return to judge all Men at the last day.'' They will not be inclined to explain away the doctrine of the Ascension of the Lord's Humanity, into what some have sought to substitute for it,- a ceasing of the Godhead to be manifested in the person of Christ. Far rather will they be ready to believe that the ``glorious'' Ascension was the epoch of a more bright manifestation of God in Christ, than any which had been vouchsafed before though perhaps rather to angelic than to human beings; and that no merely figurative, through in part a spiritual sense, is to be assigned to those passages of Holy Writ, which speak of Jesus as having been highly exalted, and seated at the right hand of God. As God, indeed, we know that Heaven, and the Heaven of Heavens, cannot contain him; yet it is also declared that Heaven is His Throne, and Earth is His Footstool: and Scripture and the Church seem to attest alike, that the risen and glorified Humanity of Christ is now in Heaven, as in some holiest place, where God is eminently manifested, and eminently worshipped; his power, his name, and presence dwelling there.
`A local translation of Christ's Body being thus believed, it is natural to believe also that this change of place was accomplished in time, and not with that strict instantaneity which may be attributed to a purely spiritual operation. Accordingly we read that at least the first part of the act of Ascension,- the part of which the Apostles were witnesses,- was gradual; their gaze could follow for a while their ascending Lord: nor was it instantly, though it may have been soon, that a cloud received Him out of their sight. And to suppose that the remainder of that wonderful translation was effected without occupying some additional time, seems almost as much ``against the truth of Christ's natural Body,'' as that it should be ``at one time in more places than one,'' which latter notion a rubric of our Book of Common Prayer rejects as error and absurdity. The Cloud which hovered over Bethany was surely not that Heaven where Jesus sitteth at the right hand of God; and to believe that his arrival, as Man, at the latter was subsequent to his arrival at the former, seems to be a just as well as an obvious inference, from the Doctrine of the Ascension of His Body.
`But how long was it subsequent? We dare not, by mere reasoning, attempt to decide this question. That place to which the Saviour has been exalted, and which, although in one sense ``Heaven,'' is in another sense declared to be ``far above all heavens,'' may well be thought to be inconceivably remote from the whole astronomical universe; no eye, no telescope, we may suppose, has pierced the mighty interspace: light may not yet have been able to spread from thence to us, if such an effluence as light be suffered thence to radiate. And, on the other hand, it must be owned, that, vast beyond all thought of ours as the interval in space may be, Christ's glorious Body may have been transported over it, in any interval of time, however short.
`Reasoning is silent then; nor can we expect to find, on this point, a clear revelation in Scripture: but do we meet with no indications? Does Holy Writ leave us here entirely without light? I think that it does not: and shall submit to you a view, which it seems to me to suggest.
`First, it is clear from Scripture, that the Ascension of Christ had been entirely performed before the Descent of the Spirit on the Day of Pentecost. Thus, in a well-known verse of that sixty-eighth Psalm, which the Church has connected with the Service for Whitsunday, and which St. Paul has quoted in reference to the Ascension; in the first sermon of Peter to the Jews; and in other passages of the Bible: the obtaining of ``gifts for men,'' the receiving from the Father the promise of the Holy Ghost, is spoken of as the result or consequence of Christ's having ascended up on high,- having been exalted by the right hand of God,- having ascended, as did not David, into the Heavens. The act of ascending occupied therefore no longer time than that from Holy Thursday to Whitsunday.
`But may it not have been allowed to occupy so long a time as this? No reason à priori can be given against the supposition; no passage of Scripture, no decision of the Church, so far as I know, is against it. The very close connexion announced, in the texts above alluded to, between the Ascension of Christ into Heaven, and the Descent of the Holy Ghost upon Earth, appears to me an indication in its favour. For the purely spiritual nature of the latter descent prevents the necessity, almost the possibility, of our supposing it to have occupied time at all. No sooner, it may reasonably be thought, did Jesus take his seat at the right hand of God, than the Spirit fell upon the Apostles. The finished work, of ascending up on high, may have been followed instantly by the receiving of gifts for men.
`Should this conjecture be admitted, of the Ascension not having been completed till the Day of Pentecost, although commenced ten days before, it might suggest much interesting meditation respecting the ``glory,'' the ``great triumph,'' with which our Saviour Christ was then exalted into God's Kingdom of Heaven. May not the transit from the Cloud to the Throne have been but one continued passage, in long triumphal pomp, through powers and principalities made subject? May not the ``Only Begotten Son'' have then again been brought forth into the world,- not by a new Nativity, but (as it were) by Proclamation and Investiture,- while the Universe beheld its God, and all the Angels worshipped Him? And would not such a triumphal progress harmonize well with that Psalm, which has always been referred in a special manner to the Ascension, and which speaks of the everlasting Gates as lifting up their heads, that the King of Glory might come in?
`Many other reflections occur to me, but I forbear. If anything unscriptural or uncatholic shall be detected by you in the foregoing remarks, or (in the event of you publishing them) by your readers, the pointing it out will be received as an obligation by, Sir, your obedient servant,
`W[illiam] R[owan] H[amilton].'
Like most men of great originality, Hamilton generally matured his ideas before putting pen to paper. `He used to carry on,' says his elder son, `long trains of algebraical and arithmetical calculation in his mind, during which he was unconscious of the earthly necessity of eating: we used to bring in a ``snack'' and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts wen on soaring upwards. I have been much with him in his periods of mathematical incubation, and would divide them into three, thus:- First, that of contemplation, above indicated. Second, that of construction. In this he committed to paper (or, if nothing else were at hand, as when in the garden, a few formulæ written on his finger-nails) the skeleton, afterwards to be clothed with flesh and blood, of the results arrived at. Third, the didactic stage. Having now completely satisfied himself of the correctness of the results (and sometimes having retraced and simplified the method of discovery) he proceeded to consider how to teach it, and this by experiment. I was so long with him in his periods of mathematical incubation that I knew, almost by the tones of his voice and the expression of his eyes, when the didactic period had arrived, and generally anticipated it by fetching the black-board to whatever room he might be in. The audience generally consisted of the Observatory assistant and myself.... He was not so much teaching, as throwing his mind into a didactic attitude. I amused him once by saying that his lecturing us on equations of the fifth degree reminded me of the lion preparing for action by whetting his claws on the bark of a tree.... He appeared to enjoy intensely arithmetical calculations. I never saw him look so perfectly happy as when running like a sleuth-hound on the track of some unhappy decimal which had marred the work, and unearthing it in its den.... I cannot otherwise express his attachment to his own MS. volumes than by saying that he loved them. He once, at a luncheon party of students at the Observatory, ranged some thirty of them on the chimney-piece, and, turning to the students, said, ``The books represent much of the happiness of my life.'' '
A good idea of the process of `incubation' above mentioned is given by the following extract from a letter to a mathematical friend. Hamilton is speaking of one of the most beautiful discoveries contained in his last work on quaternions, the general symbolical solution of a vector equation of the first degree; and he writes, in 1859, the day after the discovery was made:-
`While I was walking, on business of another sort, through Dublin yesterday, the question again occurred to me.``Puræ sunt plateæ, nihil ut meditantibus obstet''-
``I nunc, et versus tecum meditare canoros.''
I was not so rash as to attempt the composition of a Sonnet in the streets; though, in acceptance of a challenge from a Lady, long ago, beside whom I was sitting in a Music Room, I did dash off a Sonnet before the performance had ceased. But those days are over:- happily? Yes, so far as the getting a little more sense, and less sensibility, is concerned.
`The problem, however, (though not the Sonnet,) haunted me, as it happened, yesterday, while I was walking from the Provost's House to that of the Academy; and though I wrote nothing down, that day, (for I had an immensity of other things to attend to,) I resumed it this morning; and arrived at what you might call, in the language of your last, a ``perplexingly easy'' solution (in the sense of being very UNLABORIOUS, for I do not pretend that the reasoning does not require close attention).... So simple does this solution appear, that I hesitate as yet to place entire confidence in it; and, therefore, till I have fully written it out,- for at present it is partly mental,- and have given it a complete and thorough re-examination, I hesitate to communicate it to you.'
We give here, as curiously applicable to Hamilton himself, another of his sonnets,- `those fourteen-lined productions,' as he says, `to which I attach but little value on the artistic side, though some of them are associated with happy or mournful moments; and which at all events may, to a man's self, serve as instruments of culture, and may have some social or other interest to those who know him chiefly as a writer, or thinker, on subjects of a very different kind.'
TO ADAMS (Discoverer of Neptune.)
Sonnet on Unselfishness in the Pursuit of Truth and Beauty.
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.
The following final extract, from a letter written in 1858, gives a very clear insight into the view Hamilton took of his own discoveries, and of the comparative value which he attached to methods and results. There is no doubt that, in the case of quaternions at least, he sought mainly to improve his methods, and almost studiously avoided the treatment of new subjects; and the result is, that in his hands alone the development attained is extraordinary:-
`I reminded the R. I. A. that, so long ago as 1831, I had communicated to that body an Extension of (what is usually called) Herschel's Theorem: namely, the following extended Formula.... By making the two particular assumptions ... my formula becomes ... which is, if I remember rightly, one form of Herschel's Theorem. ... In speaking of ``Herschel's Theorem,'' I believe that I follow an usage, which of course he did not originate, but against which he has never complained. In my own case, however, I did complain, although (as I hope) gently, that a much less general formula of mine, which had indeed occurred in the same short paper of 1831 ... had been cited, in a then recent number of the Cambridge and Dublin Mathematical Journal, under the title of ``Hamilton's Theorem.'' What I meant was merely this; - that although I had no desire to have any theorem of mine so named, yet it was scarcely just, in my opinion, to select, out of a single and short paper, a formula which involved only one functional characteristic, one symbol of operation, and one ultimately evanescent variable; and by the manner in which the formula so selected was mentioned, or by the title under which it was cited, to ignore, or even virtually to reject, the much more general equation, which (as you see) involved two functional signs, two operators, and two ultimately evanescent variables. So, don't cite anything as ``Hamilton's Theorem,'' if you wish not to tread on my corns! I hope, indeed, that it may not be considered as unpardonable vanity or presumption on my part, if, as my own taste has always led me to feel a greater interest in methods than in results, so it is by METHODS, rather than by any THEOREMS, which can be separately quoted, that I desire and hope to be remembered. Nevertheless it is only human nature, to derive some pleasure from being cited, now and then, even about a ``Theorem,'' especially when ... the quoter can enrich the subject, by combining it with researches of his own.'
In concluding, we have only to express a hope that we have rendered intelligible to the general reader, though perhaps in but small degree, at all events the nature of some of the grand investigations of this illustrious man. Of course there will ever be many who, though (or perhaps because) totally incapable of understanding anything lofty or difficult, will sneer out over such pages the cui bono of ignorance. They cannot see one of the sources of the vastness of modern commerce in Newton meditating about gravity, another in Watt patching a trumpery model. To their narrow vision the designer of a new easy-chair or the inventor of a new sauce, a lucky speculator or a sensation-novelist, even, it may be, a mountebank assuming the guise of a philosopher, is the grandest of the human race; but, while science lasts, the name of Hamilton will hold an honoured position among those of her few greatest sons.
We have endeavoured to give, in brief compass, a trustworthy account of Hamilton and his works. Of himself the account is easy, being mainly quoted from his correspondence. In our account of his works, we have endeavoured, so far as we could, to avail ourselves of extracts from his writings. In several cases this was impossible; and we must warn the reader not to judge of the importance of the subject by the extremely small fragments which we have been forced to give as popularly intelligible specimens. Many of the preceding extracts are taken from letters which we have received from Hamilton himself. We have derived some assistance from articles, or sketches, in the Dublin University Magazine (Jan. 1842), the Gentleman's Magazine (Jan 1866), and the Monthly Notices of the Royal Astronomical Society (Feb. 1866). The last of these, especially, is an admirable tribute to Hamilton's memory, but is somewhat marred by inaccuracies in the note on the nature of quaternions. And we must express our obligations to W. E. Hamilton, Esq., C.E., the elder son of Sir William, for many facts and documents; and for his kindness in verifying the statements we have made as to his father's ancestry and early history.
We are glad to hear that the author of the first of these sketches, the Rev. R.P. Graves, one of Hamilton's oldest friends, and brother of his former colleage in the University the Bishop of Limerick, is about to write his biography. The prospect of such a volume leaves us but one wish to express, that the authorities of Trinity College may publish, as speedily as possible, if not all at least all that is most valuable in, the MSS. of the most distinguished among the many great men who, as students and professors, have shed lustre on the University of Dublin.
We conclude with an extract from the Opening Address (Session 1865-6) of the President of the Royal Society of Edinburgh, of which Hamilton was an Honorary Fellow.
`Sir John Herschel once wrote thus:- ``Here whole branches of continental discovery are unstudied, and, indeed, almost unknown even by name. It is vain to conceal the melancholy truth. We are fast dropping behind. In mathematics we have long since drawn the rein and given over a hopeless race, etc.'' Hamilton, while second to none, was one of the earliest of that brilliant array of mathematicians, who, since Herschel wrote, have removed this stigma, and well-nigh reversed the terms of his statement. Another was the late Professor Boole.... Their death has made a gap in the ranks of British science which will not soon be filled; and our sorrow is but increased by the recollection that they have been removed in the full vigour of their intellect, and when their passion for work was, if possible, stronger than ever.'