the "crammers." In them there was such an serted at the desire of the committee. But intense novelty and originality, that the expe- it also assumed a much more intelligible rience of forty years could give no inkling of form, and the grand features of the new what was coming; the venerable crammers method were now easily seen. Hamilton gave up the attempt; and the victory was won himself seems not till this period to have by the real intellect of the deserving candi- fully understood either the nature or the imdate, not, as it too often is, by the adventitious portance of his discovery, for it is only now supply of old material forced into the mere that we find him announcing his intention of memory of the crammed. 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. In 1835, being Secretary to the meeting of the British Association, which was held that year in Dublin, he was knighted by the LordLieutenant. 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 what 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 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 reerred, as usual, to a committee. Their re fort, 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 in 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 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 aerial 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. tion of any optical problem involving any number of successive reflexions were 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 of the sphere cuts it at right angles. Thus point as centre, and each ray being a radius 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 mathemati "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 unician, the late Baron Plana, was equally unform 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 by 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 solu successful. 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 Hamil ton 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 And it is obvious that the same method, with similar results, may be applied to any number of successive reflections. rays. 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 favor 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' 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 opti cal 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 dis coveries 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 particu- | lar 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 singly 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. ne. Thus the prediction was, that in a plate formed of a biaxal 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 from 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 recognizing 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 Dynami "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 re duced, 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 vary ing 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 characterises 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 coordinates and the constant II [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 geome trically 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, cond order, which can always be otherwise the differential equations of motion of the sefound. It seems, therefore, to be with reason that LAGRANGE, LAPLACE, and POISSON have in the present state of dynamics. A different spoken lightly of the utility of this principle estimate, perhaps, will be formed of that other principle which has been introduced in the present paper, under the name of the law of tual motion to another motion dynamically posvarying action, in which we pass from an acsible, 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 the 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 to be a useful convention, and consistent negative quantity. For it was seen at once 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, |