effective, and has formed the basis of all subsequent researches relative to the same object. It is founded on the theory of the variation of arbitrary constants; but in integrating the differential expressions of the comet's elements the usual mode of procedure is not adhered to. In the lunar and planetary theories the integrations are so effected, that they conduct to formulæ by means of which the place of the disturbed body may be readily calculated for any assignable instant whether past or future. A similar course not being practicable in the theory of comets, on account of the peculiarities to which we have already referred, Lagrange proposed to substitute instead of it what has been termed the method of quadratures. This consists in dividing the orbit of the comet into a number of distinct arcs, and then summing up the effect of perturbation for each arc. By this process a fresh set of elements is obtained at the end of each summation, and these form the basis of computation for the following arc. The application of this process to the whole of the orbit would demand an enormous amount of calculation, but Lagrange shewed that such a course was not necessary. When the comet is near the perihelion, the method of quadratures is indispensable; but, when it is traversing the superior part of its orbit, the ordinary method of integration may be practised on account of the great distance of the comet compared with the distance of the disturbing body.

In 1770 a comet appeared, the circumstances connected with which led to very interesting results. The observations on it seemed inexplicable by any parabolic orbit that could be devised, until Lexell finally shewed that they might be all reconciled with the supposition of the comet revolving in an ellipse in a period of nearly six years. The opinion of Lexell was confirmed in the most satisfactory manner by Burchardt, who for this purpose undertook a careful discussion of all the observations. Strange to say, the comet never afterwards appeared, notwithstanding the shortness of its period. In order to account for this fact, Lexell remarked that it had been always invisible until the year 1770, but that in 1767 it passed so near Jupiter that it was thrown by the powerful disturbance of that planet into a new orbit and thereby rendered visible; and that in 1779 it again passed so near the planet as to be thrown into another orbit and rendered invisible. Laplace undertook an analytical investigation of this interesting question, and he found that the disturbing action of Jupiter would be capable of producing the singular effects ascribed to it by Lexell *. No doubt, therefore, can exist on the cause either of the appearance or the subsequent disappearance of the comet.

As this comet approached very near the earth, it offered a favourable example for ascertaining to what extent comets in general affect the other bodies of the system. The result of its action on the earth would be to diminish the force of that body towards the sun, and thereby to lengthen the sidereal year. No such change has been detected by astronomers, whence we may conclude that the masses of comets are very small. Laplace found that, if the sidereal year had increased 1 since the year 1770, the mass of the comet would not have exceeded 5th part of the earth's mass. It is evident, however, that the mass is much less than this, for, although the comet passed through the system of Jupiter's satellites both in 1767 and in 1779, it did not produce the slightest perceptible derangements in the motions of any of those bodies.

Laplace, after having investigated with the most brilliant success every subject connected with the system of the world, finally conceived the design

* Méc. Cél. liv. ix. chap. ii.

of uniting in one great work all the discoveries that had been effected in Physical Astronomy. This design appears realised in the Mécanique Céleste, one of the most stupendous monuments of the human intellect which modern civilization can boast of. It consists of five quarto volumes, which were published at different times in the course of the author's life. The first and second volumes appeared in 1799; the third in 1803; the fourth in 1805; and the fifth in 1825. The whole work is divided into sixteen books. Ten of these occupy the first four volumes; the remaining six contained in the last volume may be considered as supplementary to the others. The first book of this immortal production treats of the laws of equilibrium and motion; the second, of the law of universal gravitation and of the centres of gravity of bodies; the third, of the figures of the celestial bodies; the fourth, of the oscillations of the sea and the atmosphere; the fifth, of the motions of the celestial bodies about their centres of gravity; the sixth, of the theory of the planets; the seventh, of the lunar theory; the eighth, of the theory of the satellites of Jupiter, Saturn, and Uranus; the ninth, of the theory of comets; the tenth, of the theory of refraction and other points relative to the system of the world. In the first book of the fifth volume or the eleventh of the whole work, Laplace considers the figure and rotation of the earth. The twelfth book treats of the attraction and repulsion of spheres and of the laws of the equilibrium and motion of elastic fluids; the thirteenth, of the oscillations of the fluids which cover the surfaces of the planets; the fourteenth, of the motions of the celestial bodies about their centres of gravity; the fifteenth, of the motions of the planets and comets; the sixteenth, of the motions of the satellites. Besides the five volumes above mentioned, Laplace composed at different times. supplements to several of the books.

The physical theory of the planetary system is exhibited in the Mécanique Céleste in a state of almost complete developement. No material progress has in consequence been effected in this branch of astronomy since the publication of that immortal work. One discovery of a very remarkable character has indeed been recently added to the long list of triumphs which adorn its history; but during the present century geometers have been mainly occupied in correcting and extending previous results, in improving the methods of investigation, and in illustrating the more obscure points of the theory.

In reviewing the progress of Physical Astronomy since the close of Newton's career, it is impossible not to be struck with the truth of the remark, that great occasions always call forth from the bosom of society suitable minds to cope with the emergencies of the times, and to triumph over opposing difficulties. It has been said that Newton appeared on the theatre of the world when the materials of the magnificent structure erected by him had been already amassed by the persevering industry of preceding ages. The true state of the planetary system had been unfolded by the successive labours of Hipparchus, Copernicus, and Kepler; the laws of motion had been fully established by Galileo, and his successors Huygens, Wallis, and Wren: nay, the all-pervading principle which animates and controls the bodies of the universe had been dimly surmised by more than one philosopher. There was still, however,

wanting some master-spirit to detect the mutual dependence of these disjointed principles, and by a mighty effort of generalization to reduce all the phenomena of the heavens under one dominant law of nature. The prospect of achieving this grand result was the alluring motive by which

genius was invited to the study of celestial physics in the seventeenth century; and to the English philosopher was assigned the immortal honour of its realisation. We may discern a similar adaptation of intellectual power to existing exigencies in the period which elapsed between the publication of the Principia and the appearance of the Mécanique Céleste. Newton had fully established the principle of gravitation by his own unaided efforts; but he bequeathed a vast heritage of profound research to his successors. With a sagacity unexampled in the history of the human mind, he detected the agency of this principle in all the grand phenomena of the planetary system, and by the aid of a sublime geometry of his own invention he succeeded in reducing to calculation a number of its most hidden results. It still remained, however, for geometers to ascertain the effects of the minute perturbations which ensue from the mutual action of the planets, to invent formulæ enabling the astronomer to determine their positions throughout all ages, both past and future; and, finally, to solve the momentous question whether the planets are gradually being absorbed into the sun, or whether the system is so constituted that they will revolve in permanent orbits round the central body. Such were the magnificent problems which Newton's discoveries suggested to his successors. We have seen what mighty energies were awakened by these problems and with what brilliant success their solution was effected. Perhaps no period of history can exhibit an array of mathematical genius, equal to that which adorned the eighteenth century. The labours of Euler, D'Alembert, Clairaut, Lagrange, and Laplace will fill many a bright page in the annals of science, and their names will be for ever associated with that of the illustrious founder of Physical Astronomy, whose reputation they have so much enhanced by their sublime discoveries.

Among the various circumstances which are calculated to excite our admiration, while reviewing this portion of the history of Physical Astronomy, not the least remarkable are the resources of the transcendental analysis; by means of which the geometer has been enabled to unravel the most complicated relations of the system of the world, and to decipher in the anomalous movements of the celestial bodies the constant operation of one all-pervading principle. In vain would the human mind have ever attempted to penetrate into the more recondite parts of the theory of gravitation without the aid of this powerful instrument of research. Its assiduous cultivation was, therefore, essentially necessary for the developement of that theory; and, on this account, the pure analysts, such as Leibnitz and the two eldest Bernoullis, deserve to be associated with those who have more directly contributed to the progress of the science. The discovery of the system of the world by Newton," says Delambre, "was a fortunate event for geometers. Never could the transcendental analysis find a worthier or a more sublime theme. Whatever progress

is made in it, the original discoverer will always maintain his rank. Lagrange, who often asserted Newton to be the greatest genius that ever existed, used to remark also- and the most fortunate; we do not find more than once a system of the world to establish.' It has required a hundred years of labours and discoveries to construct the edifice of which Newton laid the foundation; but all is ascribed to him, and he is supposed to have pursued the whole extent of the career on which he entered with an éclat so well calculated to encourage his successors."*

• Mémoires de l'Institut, 1812, p. xlv: (Notice sur la vie et les ouvrages de Lagrange.)

An illustrious living philosopher of France, when alluding to the discovery of universal gravitation, has said, that no Frenchman can reflect without an aching heart on the small participation of his own country in that memorable achievement. If an Englishman could be supposed to be equally sensitive, he has ample reason to regret the inglorious part his country played during the long period which marked the developement of the Newtonian theory. With the exception of Maclaurin and Thomas Simpson (the former of whom certainly contributed towards the solution of one of the great problems of the system of the world, and the latter at least gave ample proofs of his capacity for such researches), hardly any individual of these islands deserves even to be mentioned in connexion with the history of physical astronomy during that period. This deplorable fact has been generally attributed to the pertinacity with which the English mathematicians adhered to the synthetic method of investigation, the resources of which had been already completely exhausted by Newton; and also to their perseverance in employing the fluxional calculus of that great geometer; which, besides being less commodious in point of notation, did not at any time attain the high state of perfection which, at a comparatively early stage of its history, distinguished the rival invention of Leibnitz. The feeling of veneration which they naturally cherished towards their illustrious countryman was doubtless the main cause of their injudicious attachment to his peculiar methods of research; but it was also fostered in a strong degree by the unhappy quarrel which arose between them and the continental mathematicians relative to the original invention of the infinitesimal calculus. Were it not for the mutual estrangement which then ensued between both parties, it would have been an easy task to transfer the improvements of the differential calculus to the fluxional calculus of Newton, which, in point of fact, was identical with it; and by this means the analysts of England might have advanced at an equal pace with those on the continent. When the unpleasant feeling, just referred to, finally died away, the intellectual energies of England were already directed towards objects diametrically opposed to contemplative science; and the few persons who still cultivated mathematics, perceiving how far the analysts on the continent had advanced beyond them in the improvements of the infinitesimal calculus, appear to have abandoned in despair all intentions of original research in physical astronomy, contenting themselves merely with timid dissertations on the Principia. At the beginning of the present century there was hardly an individual in this country who possessed an intimate acquaintance with the methods of investigation which had conducted the foreign mathematicians to so many sublime results. It is gratifying to reflect that a vigorous attempt has been made since that time to recover for England her due position in the physico-mathematical world. Notwithstanding the disadvantages under which they laboured, the geometers of this country have already given ample proof that it was not from any natural defect of intellectual ability that their fathers were compelled so long to remain silent spectators of the triumphs of their neighbours. The most recondite parts of analysis are now studied with ardour and success by a number of talented persons; and England, in the present day, can boast of some of the most distinguished mathematicians of Europe. The late Professor Woodhouse, of Cambridge, deserves to be mentioned as the person who laboured most zealously in

• M. Arago.

removing from the minds of his countrymen the prejudices they had so long cherished against the analytical methods that were in universal use on the continent. In the sequel we shall have the pleasure of noticing occasionally some of the results which may be considered as the first fruits of this salutary innovation.

CHAPTER X.

Variation of the Mean Distances of the Planets.-Researches of Poisson.-The Theory of Planetary Perturbation resumed by Lagrange and Laplace.-Uniformity of the re. sults arrived at by these Geometers.-The General Theory of the Variation of Arbitrary Constants established by Lagrange.-Researches of Poisson on this subject.Death of Lagrange.-Researches of Modern Geometers on the Theory of Perturbation. Method of Hansen.-Developement of the Perturbing Function.- Burchardt.Binet. New Methods devised for obtaining the coefficients of the Perturbing Function.-Secular Inequalities of the Planets.-Researches of Le Verrier.-Theory of the Moon.-Irregularities of the Epoch.-Equation of Long Period. — Researches of Damoiseau, Plana, and Carlini.-Lunar Tables calculated by means of the Theory of Gravitation.—Researches of Lubbock and Poisson.—Reduction of the Greenwich Observations. Discovery of the True Cause of the Irregularities of the Moon's Epoch, by Hansen.-Researches for the purposes of determining the Value of the Moon's Mass. THE first important discovery which distinguished the progress of phy sical astronomy in the nineteenth century related to the variation of the mean distances of the planets from the sun. We have already given an account of the researches of Lagrange on this point of the planetary theory, and have mentioned the remarkable conclusion at which he arrived. It appeared that the mean distance of a planet, when disturbed in its elliptic orbit by the other bodies of the system, was not subject to any variations which increased constantly with the time, but was merely affected by a series of periodic inequalities depending on the relative places of the planets in their orbits. Lagrange shewed that this theorem was true for all powers of the eccentricities and inclinations; but his investigation did not extend beyond a first approximation, and, therefore, was limited to terms of the first order with respect to the disturbing forces*. The interesting question, therefore, still remained to be examined, whether a repetition of the approximation would introduce into the expression of the mean distance any terms increasing with the time, and thereby occasioning a secular inequality in the mean motion. This problem was first attacked by a young geometer, who was destined to pursue a brilliant career in the physico-mathematical sciences. In the year 1808, Poisson, who was then only twenty-five years of age, communicated a memoir to the Institute, in which he investigated the variation of the mean distance, carrying the approximation to the square of the perturbing forces. By means of a most elaborate analysis, he succeeded in shewing that the repetition of the ap proximation would introduce into the formula for the variation of that element, only a class of terms depending on sines and cosines of angles, increasing proportionally with the time. It followed, then, that so far as

Mémoire sur les Inégalités Seculaires des Moyens Mouvemens des Planètes. This memoir was read at the Institute, in June 1808, and was subsequently published in the eighth volume of the Journal de l'Ecole Polytechnique.