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comets inversely as the square of their distances from his centre; the sun, therefore, is the centre of a force extending indefinitely in space, and including all the bodies of the system in its action.
Kepler also deduced from observation, that the squares of the periodic times of the planets, or the times of their revolutions round the sun, are proportional to the cubes of their mean distances from his centre: whence it follows that the intensity of gravitation of all the bodies towards the sun is the same at equal distances; consequently gravitation is proportional to the masses, for, if the planets and comets were at equal distances from the sun, and left to the effects of gravity, they would arrive at his surface at the same time. The satellites also gravitate to their primaries according to the same law that their primaries do to the sun. Hence, by the law of action and re-action, each body is itself the centre of an attractive force extending indefinitely in space, whence proceed all the mutual disturbances which render the celestial motions so complicated, and their investigation so difficult.
The gravitation of matter, directed to a centre, and attracting directly as the mass and inversely as the square of the distance, does not belong to it when considered in mass only; particle acts on particle according to the same law when at sensible distances from each other. If the sun acted on the centre of the earth without attracting each of its particles, the tides would be very much greater than they now are; and would also, in other respects, be very different. The gravitation of the earth to the sun results from the gravitation of all its particles, which, in their turn, attract the sun in the ratio of their respective masses. There is a reciprocal action likewise between the
earth and every particle at its surface; were this not the case, and were any portion of the earth, however small, to attract another portion, and not be itself attracted, the centre of gravity of the earth would be moved in space by this action, which is impossible.
The forms of the planets result from the reciprocal attraction of their component particles. A detached fluid mass, if at rest, would assume the form of a sphere, from the reciprocal attraction of its particles; but if the mass revolves about an axis, it becomes flattened at the poles, and bulges at the equator, in consequence of the centrifugal force arising froin the velocity of rotation,-for the centrifugal force diminishes the gravity of the particles at the equator, and equilibrium can only exist where these two forces are balanced by an increase of gravity; therefore, as the attractive force is the same in all particles at equal distances from the centre of a sphere, the equatorial particles would recede from the centre, till their increase in number balanced the centrifugal force by their attraction : consequently, the sphere would become an oblate spheroid; and a fluid partially or entirely covering a solid, as the ocean and atmosphere cover the earth, must assume that form in order to remain in equilibrio. The surface of the sea is therefore spheroidal, and the surface of the earth only deviates from that figure where it rises above, or sinks below, the level of the sea; but the deviation is so small that it is unimportant when compared with the magnitude of the earth-for the mighty chain of the Andes, and the yet more lofty Himalaya, bear about the same proportion to the earth that a grain of sand does to a globe three feet in diameter. Such is the form of the earth and planets; but the compression or flattening at their poles
is so small, that even Jupiter, whose rotation is the most rapid, and therefore the most elliptical of the planets, may from his great distance, be regarded as spherical. Although the planets attract each other as if they were spheres, on account of their distances, yet the satellites are near enough to be sensibly affected in their motions by the forms of their primaries. The moon, for example, is so near the earth, that the reciprocal attraction between each of her paticles, and each of the particles in the prominent mass at the terrestrial equator, occasions considerable disturbances in the motions of both bodies: for the action of the moon, on the matter at the earth's equator, produces a nutation in the axis of rotation, and the reaction of that matter on the moon is the cause of a corresponding nutation in the lunar orbit.
If a sphere, at rest in space, receive an impulse passing through its centre of gravity, all its parts will move with an equal velocity in a straight line; but if the impulse does not pass through the centre of gravity, its particles, having unequal velocities, will have a rotatory motion at the same time that it is translated in space. These motions are independent of one another; so that a contrary impulse, passing through its centre of gravity, will impede its progress, without interfering with its rotation. As the sun rotates about an axis, it seems probable, if an impulse in a contrary direction has not been given to his centre of gravity, that he moves in space, accompanied by all those bodies which compose the solar system—a circumstance which would in no way interfere with their relative motions; for, in consequence of the principle that force is proportional to velocity, the reciprocal attractions of a system remain the same, whether its centre of grav
ity be at rest, or moving uniformly in space. It is computed that had the earth received its motion from a single impulse, such impulse must have passed through a point about twenty-five miles from its centre.
Since the motions of rotation and translation of the planets are independent of each other, though probably communicated by the same impulse, they form separate subjects of investigation.
A planet moves in its elliptical orbit with a velocity varying every instant, in consequence of two forces, one tending to the centre of the sun, and the other in the direction of a tangent to its orbit, arising from the primitive impulse given at the time when it was launched into space : should the force in the tangent cease, the planet would fall to the sun by its gravity; were the sun not to attract it, the planet would fly off in the tangent. Thus, when the planet is in aphelion, or at the point where the orbit is fartherest from the sun, his action overcomes the planet's velocity, and brings it towards him with such an accelerated motion, that, at last it overcomes the sun's attraction, and shooting past him, gradually decreases in velocity, until it arrives at the aphelion where the sun's attraction again prevails. In this motion the radii vectores, or imaginary lines joining the centres of the sun and the planets, pass over equal areas in equal times.
If the planets were attracted by the sun only, this would ever be their course; and because his action is proportional to his mass, which is much larger than that of all
the planets put together, the elliptical is the nearest approximation to their true motions, which are extremely complicated, in consequence of their mutual attraction, so that they do not move in any known or symmetrical curve, but in paths now approaching to, now receding from, the elliptical form; and their radii vectores do not describe areas exactly proportional to the time. Thus the areas become a test of disturbing forces.
To determine the motion of each body, when disturbed by all the rest, is beyond the power of analysis; it is, therefore, necessary to estimate the disturbing action of one planet at a time, whence the celebrated problem of the three bodies, originally applied to the moon, the earth, and the sun-namely, the masses being given of three bodies projected from three given points, with velocities given both in quantity and direction; and, supposing the bodies to gravitate to one another with forces that are directly as their masses and inversely as the squares of the distances, to find the lines described by these bodies, and their positions at any given instant.
By this problem the motions of translation of all the celestial bodies are determined. It is an extremely difficult one, and would be infinitely more so, if the disturbing action were not very small when compared with the central force. As the disturbing influence of each body may be found separately, it is assumed that the action of the whole system, in disturbing any one planet, is equal to the sum of all, the particular disturbances it experiences, on the general mechanical principle, that the sum of any number of small oscillations is nearly equal to their simultaneous and joint effect.
On account of the reciprocal action of matter, the sta