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Tycho Brahe, however, would not admit of the motion of the earth, because he could not conceive how a body detached from it could follow its motion: he was convinced that the earth was at rest, because a heavy body, falling from a great height, falls nearly at the foot of the vertical.
Kepler, one of those extraordinary men, who appear from time to time, to bring to light the great laws of nature, adopted sounder views. A lively imagination, which disposed him eagerly to search for first causes, tempered by a severity of judgment that made him dread being deceived, formed a character peculiarly fitted to investigate the unknown regions of science, and conducted him to the discovery of three of the most important laws in astronomy.
He directed his attention to the motions of Mars, whose orbit is one of the most eccentric in the planetary system, and as it approaches very near the earth in its oppositions, the inequalities of its motions are considerable; circumstances peculiarly favourable for the determination of their laws.
He found the orbit of Mars to be an ellipse, having the sun in one of its foci; and that the motion of the planet is such, that the radius vector drawn from its centre to the centre of the sun, describes equal areas in equal times. He extended these results to all the planets, and in the year 1626, published the Rudolphine Tables, memorable in the annals of astronomy, from being the first that were formed on the true laws of nature.
Kepler imagined that something corresponding to certain mysterious analogies, supposed by the Pythagoreans to exist in the laws of nature, might also be discovered between the mean distances of the planets, and their revolutions round the sun: after sixteen years spent in unavailing attempts, he at length found that the squares of the times of their sidereal revolutions are proportional to the cubes of the greater axes of their orbits; a very important law, which was afterwards found equally applicable to all the systems of the satellites. It was obvious to the comprehensive mind of Kepler, that motions so regular could only arise from some universal principle pervading the whole system. In his work De Stella Martis, he observes, that 'two insulated bodies would move towards one another like two magnets, describing spaces reciprocally as their masses. If the earth and moon were not held at the distance that separates them by some
force, they would come in contact, the moon describing distance, and the earth the remainder, supposing them to be equally dense.' If,' he continues, the earth ceased to attract the waters of the ocean, they would go to the moon by the attractive force of that body. The attraction of the moon, which extends to the earth, is the cause of the ebb and flow of the sea.' Thus Kepler's work, De Stella Martis, contains the first idea of a principle which Newton and his successors have fully developed.
The discoveries of Galileo on falling bodies, those of Huygens on Evolutes, and the centrifugal force, led to the theory of motion in curves. Kepler had determined the curves in which the planets move, and Hook was aware that planetary motion is the result of a force of projection combined with the attractive force of the sun.
Such was the state of astronomy when Newton, by his grand and comprehensive views, combined the whole, and connected the most distant parts of the solar system by one universal principle.
Having observed that the force of gravitation on the summits of the highest mountains is nearly the same as on the surface of the earth, Newton inferred, that its influence extended to the moon, and, combining with her force of projection, causes that satellite to describe an elliptical orbit round the earth. In order to verify this conjecture, it was necessary to know the law of the diminution of gravitation. Newton considered, that if terrestrial gravitation retained the moon in her orbit, the planets must be retained in theirs by their gravitation to the sun; and he proved this to be the case, by showing the areas to be proportional to the times: but it resulted from the constant ratio found by Kepler between the squares of the times of revolutions of the planets, and the cubes of the greater axes of their orbits, that their centrifugal force, and consequently their tendency to the sun, diminishes in the ratio of the squares of their distances from his centre. Thus the law of diminution was proved with regard to the planets, which led Newton to conjecture, that the same law of diminution takes place in terrestrial gravitation.
He extended the laws deduced by Galileo from his experiments on bodies falling at the surface of the earth, to the moon; and on these principles determined the space she would move through in a second of time, in her descent towards the earth, if acted upon by the earth's attraction alone. He had the satisfaction to find that the action of the
earth on the moon is inversely as the square of the distance, thus proving the force which causes a stone to fall at the earth's surface, to be identical with that which retains the moon in her orbit.
Kepler having established the point that the planets move in ellipses, having the sun in one of their foci, Newton completed his theory, by showing that a projectile might move in any of the conic sections, if acted on by a force directed to the focus, and inversely as the square of the distance: he determined the conditions requisite to make the trajectory a circle, an ellipse, a parabola, or hyperbola. Hence he also concluded, that comets move round the sun by the same laws as the planets.
A comparison of the magnitude of the orbits of the satellites and the periods of their revolutions, with the same quantities relatively to the planets, made known to him the respective masses and densities of the sun and of planets accompanied by satellites, and the intensity of gravitation at their surfaces. He observed, that the satellites move round their planets nearly as they would have done, had the planets been at rest, whence he concluded that all these bodies obey the same law of gravitation towards the sun: he also concluded, from the equality of action and re-action, that the sun gravitates towards the planets, and the planets towards their satellites; and that the earth is attracted by all bodies which gravitate towards it. He afterwards extended this law to all the particles of matter, thus establishing the general principle, that each particle of matter attracts all other particles directly as its mass, and inversely as the square of its distance.
These splendid discoveries were published by Newton in his Principia, a work which has been the admiration of mankind, and which will continue to be so while science is cultivated.
Referring to that stupendous effort of human genius, La Place, who perhaps only yields to Newton in priority of time, thus expresses himself in a letter to the writer of these pages:
'Je publie successivement les divers livres du cinquième volume qui doit terminer mon traité de Mécanique Céleste, et dans lequel je donne l'analyse historique des recherches des géomètres sur cette matière. Cela m'a fait relire avec une attention particulière l'ouvrage incomparable des Principes Mathématiques de la philosophie naturelle de Newton, qui contient le germe de toutes ces recherches. Plus
j'ai étudié cet ouvrage, plus il m'a paru admirable, en me transportant surtout à l'époque où il a été publié. Mais en même tems que j'ai senti l'élégance de la méthode synthétique suivant laquelle Newton a présenté ses découvertes, j'ai reconnu l'indispensable nécessité de l'analyse pour approfondir les questions très difficiles qu'il n'a pu qu'effleurer par la synthèse. Je vois avec un grand plaisir vos mathématiciens se livrer maintenant à l'analyse; et je ne doute point qu'en suivant cette méthode avec la sagacité propre à votre nation, ils ne soient conduits à d'importantes dé
The reciprocal gravitation of the bodies of the solar system is a cause of great irregularities in their motions; many of which had been explained before the time of La Place, but some of the most im portant had not been accounted for, and many were not even known to exist. The author of the Mécanique Céleste therefore undertook the arduous task of forming a complete system of physical astronomy, in which the various motions in nature should be deduced from the first principles of mechanics. It would have been impossible to accomplish this, had not the improvements in analysis kept pace with the rapid advance in astronomy, a pursuit in which many have acquired immortal fame ; that La Place is pre-eminent amongst these, will be most readily acknowledged by those who are best acquainted with his works.
Having endeavoured in the first book to explain the laws by which force acts upon matter, we shall now compare those laws with the actual motions of the heavenly bodies, in order to arrive by analytical reasoning, entirely independent of hypothesis, at the principle of that force which animates the solar system. The laws of mechanics may be traced with greater precision in celestial space than on earth, where the results are so complicated, that it is difficult to unravel, and still more so to subject them to calculation: whereas the bodies of the solar system, separated by vast distances, and acted upon by a force, the effects of which may be readily estimated, are only disturbed in their respective movements by such small forces, that the general equations comprehend all the changes which ages have produced, or may hereafter produce in the system; and in explaining the phenomena it is not necessary to have recourse to vague or imaginary causes, for the law of universal gravitation may be reduced to calcu
lation, the results of which, confirmed by actual observation, afford the most substantial proof of its existence.
It will be seen that this great law of nature represents all the phenomena of the heavens, even to the most minute details; that there is not one of the inequalities which it does not account for; and that it has even anticipated observation, by unfolding the causes of several singular motions, suspected by astronomers, but so complicated in their nature, and so long in their periods, that observation alone could not have determined them but in many ages,
By the law of gravitation, therefore, astronomy is now become a great problem of mechanics, for the solution of which, the figure and masses of the planets, their places, and velocities at any given time, are the only data which observation is required to furnish. We proceed to give such an account of the solution of this problem, as the nature of the subject and the limits of this work admit of.