Attraction of a Sphere—Form of Celestial Bodies—Terrestrial Gravitation retains the Moon in her Orbit—The Heavenly Bodies move in Conic Sections—Gravitation Proportional to Mass—Gravitation of the Particles of Matter—Figure of the Planets—How it affects the Motions of their Satellites—Rotation and Translation impressed by the same Impulse—Motion of the Sun and Solar System.

It has been proved by Newton, that a particle of matter (N.6) placed without the surface of a hollow sphere (N.7) is attracted by it in the same manner as if the mass of the hollow sphere, or the whole matter it contains, were collected into one dense particle in its centre. The same is therefore true of a solid sphere, which may be supposed to consist of an infinite number of concentric hollow spheres (N.8). This, however, is not the case with a spheroid (N.9); but the celestial bodies are so nearly spherical, and at such remote distances from one another, that they attract and are attracted as if each were condensed into a single particle situate in its centre of gravity (N.10)—a circumstance which greatly facilitates the investigation of their motions.

Newton has shown that the force which retains the moon in her orbit is the same with that which causes heavy substances to fall at the surface of the earth. If the earth were a sphere, and at rest, a body would be equally attracted, that is, it would have the same weight at every point of its surface, because the surface of a sphere is everywhere equally distant from its centre. But, as our planet is flattened at the poles (N.11), and bulges at the equator, the weight of the same body gradually decreases from the poles, where it is greatest, to the equator, where it is least. There is, however, a certain mean (N.12) latitude (N.13), or part of the earth intermediate between the pole and the equator, where the attraction of the earth on bodies at its surface is the same as if it were a sphere; and experience shows that bodies there fall through 16·0697 feet in a second. The mean distance (N.14) of the moon from the earth is about sixty times the mean radius (N.15) of the earth. When the number 16·0697 is diminished in the ratio (N.16) of 1 to 3600, which is the square of the moon’s distance (N.17) from the earth’s centre, estimated in terrestrial radii, it is found to be exactly the space the moon would fall through in the first second of her descent to the earth, were she not prevented by the centrifugal force (N.18) arising from the velocity with which she moves in her orbit. The moon is thus retained in her orbit by a force having the same origin, and regulated by the same law, with that which causes a stone to fall at the earth’s surface. The earth may, therefore, be regarded as the centre of a force which extends to the moon; and, as experience shows that the action and reaction of matter are equal and contrary (N.19), the moon must attract the earth with an equal and contrary force.

Newton also ascertained that a body projected (N.20) in space (N.21) will move in a conic section (N.22), if attracted by a force proceeding from a fixed point, with an intensity inversely as the square of the distance (N.23); but that any deviation from that law will cause it to move in a curve of a different nature. Kepler found, by direct observation, that the planets describe ellipses (N.24), or oval paths, round the sun. Later observations show that comets also move in conic sections. It consequently follows that the sun attracts all the planets and comets inversely as the square of their distances from its 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 (N.25) of the planets, or the times of their revolutions round the sun, are proportional to the cubes of their mean distances from its centre (N.26). Hence the intensity of gravitation of all the bodies towards the sun is the same at equal distances. Consequently, gravitation is proportional to the masses (N.27); 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 (N.28). The satellites also gravitate to their primaries (N.29) according to the same law that their primaries do to the sun. Thus, by the law of action and reaction, each body is itself the centre of an attractive force extending indefinitely in space, causing 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. The earth and a feather mutually attract each other in the proportion of the mass of the earth to the mass of the feather. 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 revolve about an axis, it becomes flattened at the poles and bulges at the equator (N.11), in consequence of the centrifugal force arising from the velocity of rotation (N.30); 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 on all particles at equal distances from the centre of a sphere, the equatorial particles would recede from the centre, till their increase in number balance the centrifugal force by their attraction. Consequently, the sphere would become an oblate or flattened 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. The compression (N.31) or flattening at their poles is, however, 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 (N.32) 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 particles, 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 (N.33) in the axis (N.34) of rotation, and the reaction of that matter on the moon is the cause of a corresponding nutation in the lunar orbit (N.35).

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 or revolving motion, at the same time that it is translated (N.36) 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. The sun rotates about an axis, and modern observations show that an impulse in a contrary direction has not been given to his centre of gravity, for he moves in space, accompanied by all those bodies which compose the solar system—a circumstance which in no way interferes with their relative motions; for, in consequence of the principle that force is proportional to velocity (N.37), the reciprocal attractions of a system remain the same whether its centre of gravity be at rest, or moving uniformly in space. It is computed that, had the earth received its motion from a single impulse, that 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.