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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 gravity 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 farthest 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 stability of the system depends upon the intensity of the primitive momentum of the planets, and the ratio of their masses to that of the sun-for the
nature of the conic sections in which the celestial bodies move, depends upon the velocity with which they were first propelled in space: had that velocity been such as to make the planets move in orbits of unstable equilibrium, their mutual attractions might have changed them into parabolas, or even hyperbolas, so that the earth and planets might, ages ago, have been sweeping far from our sun through the abyss of space: but as the orbits differ very little from circles, the momentum of the planets, when projected, must have been exactly sufficient to ensure the permanency and stability of the system. Besides, the mass of the sun is vastly greater than that of any planet; and as their inequalities bear the same ratio to their elliptical motions as their masses do to that of the sun, their mutual disturbances only increase or diminish the eccentricities of their orbits by very minute quantities; consequently, the magnitude of the sun's mass is the principal cause of the stability of the system. There is not in the physical world a more splendid example of the adaptation of means to the accomplishment of the end, than is exhibited in the nice adjustment of these forces, at once the cause of the variety and of the order of Nature.
The mean distance of a planet from the sun is equal to half the major axis of its orbit: if, therefore, the planet described a circle round the sun at
its mean distance, the motion would be uniform, and the periodic time unaltered, because the planet would arrive at the apsides or extremities of the major axis at the same instant, and would have the same velocity, whether it moved in the circular or elliptical orbit, since the curves coincide in these points; but, in every other part, the elliptical motion would either be faster or slower than the circular or mean motion. The difference between the two is called the equation of the centre, which consequently vanishes at the apsides, and is at its maximum ninety degrees distant from these points, or in quadratures, where it measures the eccentricity of the orbit, so that the place of a planet in its elliptical orbit is obtained by adding or subtracting the equation of the centre to or from its mean motion.
The orbits of the planets have a very small inclination to the plane of the ecliptic in which the earth moves; and, on that account, astronomers refer their motions to this plane at a given epoch as a known and fixed position. The paths of the planets, when their mutual disturbances are omitted, are ellipses, nearly approaching to circles, whose planes, slightly inclined to the ecliptic, cut it in straight lines passing through the centre of the sun; the points where an orbit intersects the plane of the ecliptic are its nodes. The ascending node of the