Here, again, we find it impossible, from this experiment, to determine the exact law which regulates the attractive power of the individual particles composing the earth, but we do demonstrate the fact that the earth's gravity is not concentrated at its centre, but dwells, according to some law, in all the atoms which compose its mass; and this law, we shall prove hereafter, is none other than the great law of universal gravitation.

It is impossible to form a just idea of the vast importance which attaches to the grand discovery of Newton. It worked out, instantly and absolutely, a complete revolution in the whole science of astronomy. Previous to the discovery of the law of universal gravitation, all the observations upon the stars and planets, which had been accumulating for so many centuries, could only be regarded as so many isolated facts, having no specific relation the one to the other. The planets were independent orbs, moving through space in orbits peculiar to themselves, and only united by the single fact that the sun constituted the common centre of revolution. The discovery made by Newton converted this scheme of isolated worlds into a grand mechanical system, wherein each orb was dependent upon every other, each satellite affecting every other, and the whole complex scheme gravitating to the common centre, which exerted a predominant power over each and every one of these revolving worlds.

Those eccentric bodies which we denominate comets, whose abrupt appearance in the heavens with their glowing trains of light, whose rapid movements and sudden disappearance have excited such a deep interest in all ages of the world, were found not to be exempt, as we shall hereafter show, from the empire of gravitation.

CHAPTER X.

THE LAWS OF MOTION AND GRAVITATION APPLIED TO A SYSTEM OF THREE REVOLVING BODIES.

A System of two Bodies.-Quantities required in its Investigation.-Five in number.-Sun and Earth.-Sun, Earth, and Moon, as Systems of Three Bodies.-The Sun supposed Stationary.-Changed Figure of the Moon's Orbit. Sun Revolving changes the Position of the Moon's Orbit. -Solar Orbit Elliptical.-Effects resulting from the Inclination of the Moon's Orbit. Moon's Motion above and below the Plane of the Ecliptic.-Revolution of the Line of Nodes.-Sun, Earth, and Planet, as the Three Bodies.-Perturbations destroy the Rigour of Kepler's Laws.-Complexity thus introduced.-Infinitesimal Analysis.-Difference between Geometrical and Analytical Reasoning.

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We shall now present, as clearly as we can, without the aid of mathematical reasoning, the application of the laws of motion and gravitation to the circumstances arising in a system of three bodies mutually affecting each other. We will commence even with a simpler case, and suppose a solitary planet to exist, subjected to the attractive power of one sun, and that these are the only bodies in the universe. Let us consider what quantities are demanded to render it possible for the mathematician to take account of the circumstances of motion which will belong to this solitary world.

First of all, it is evident that the quantity of matter contained in the sun, or its exact weight, must be known; for the energy or power of the sun varies directly as its mass; and two suns, so related that the weight of one is tenfold greater than that of the other, the heavier one will exert a power of attraction tenfold greater than the lighter

one.

In the second place, we must know the distance of the planet from the sun, for the power of the sun's attraction decreases as the square of the distance at which it operates,

increases; so that, if at a distance of unity it exerts an attractive force which we may call one, at a distance two this force will be diminished to one-fourth; at a distance three to one-ninth; at a distance four to one-sixteenth; at a distance ten to the one-hundredth part of its first value.

In the third place, the mass or weight of the planet must be known; for not only does the sun attract its planet, but in turn the planet attracts the sun, and the intensity of this attraction, which affects the motion of the planet as well as that of the sun, depends exclusively upon the mass or weight of the planet.

In the fourth place, we must know the intensity of the impulsive force which is employed to start the planet in its orbit; for, upon the intensity of this force will the initial velocity of the planet depend, and we see readily that the form of the orbit as to curvature will depend upon the initial velocity. The greater this velocity, the more nearly will the curvature of the orbit coincide with the straight line in which the planet would have moved, in case it had been operated upon by the impulsive force alone.

In the fifth place, before we can completely master the circumstances of motion to the planet, we must know the direction in which the impulse is applied; for upon this direction it is manifest that the figure of the orbit will depend. If the impulsive force be applied in a direction passing through the sun's centre, and toward the sun, it is clear that the planet will simply fall to the sun in a straight line. If it met with no resistance it would pass through and beyond the sun's centre, until its velocity would be entirely overcome by the attraction of gravitation, when it would stop, fall again to the sun, and thus vibrate for ever in a right line. In case the direction of the impulse is oblique to the line joining the planet and the sun (the angle falling within certain limits of value), then the planet will describe an elliptical figure in its revolution around the sun, and will return precisely to the point of departure to repeat the same identical curve, with the same velocities precisely at each of

the points of its orbit, in the same exact order for ever. In examining the peculiarities which distinguish the movements of this revolving body, we shall find as a necessary consequence of the laws under which it moves, that its motion must be slowest at that point of its orbit where it is farthest from the sun. Leaving this point as it approaches the sun, its velocity must rapidly increase, and will reach its maximum at the perihelion of its orbit, where, being nearest to the sun, it will move with its swiftest velocity. Receding now from the centre of attraction, it will lose its velocity by the same degrees with which it was augmented, and will again pass its aphelion with its slowest velocity. Thus we perceive that the movements of a single planet revolving about the only sun in existence are marked with great simplicity; and in case the mathematician knows precisely the five quantities already named, viz.: the sun's mass, the planet's distance, the planet's mass, the intensity of the impulsive force, and the direction of this force, it is not at all difficult to determine all the circumstances of motion of the planet, and to predict its place in its orbit with absolute precision at the end of ten thousand revolutions.

We will not at present attempt to show how these five quantities may be obtained. These determinations belong to the department of instrumental astronomy, a subject which will be treated after closing what we have to say on the application of the law of gravitation to the movements of a system of three bodies.

In case the planets had been formed of a material such as to be attracted by the sun, but not to attract each other, and if the satellites had been composed of a material such as to be attracted by their primaries only; then the elements of the orbits of all these revolving bodies would have remained for ever absolutely invariable. So soon, then, as accurate observation should have furnished the five quantities required in determining the circumstances of motion in any revolving body, mathematical computation would have fitted an invariable orbit to each one of these bodies,

and would have furnished by calculation the exact place of each one of these bodies in all coming time. The whole system would have been one of perfect equilibrium; aud, although complexity would have presented itself apparently in the interlacing revolutions of these revolving worlds, yet absolute simplicity, combined with short periodical changes, would have restored each one of these bodies to the exact position occupied when first launched in its orbit.

This, however, is not the case of nature. The sun not only attracts the planets, but also attracts their satellites. The primary planets not only attract their satellites, but attract each other; and thus not a single body exists in the whole universe, which is not dependent upon every other.

We have already seen that, in case the sun with one planet were the only objects in existence, having traced the planet in one single revolution round the sun, the variations of motion thus developed would be repeated without the slightest change in any succeeding revolution for all coming time.

Suppose this solitary planet to be the earth; and that from a knowledge of the weight of the sun, the distance of the earth from the sun, the weight of the earth, the intensity of the impulsive force, and the direction in which that force is applied to start the earth in its orbit, we determine the elements of its orbit. The form of this orbit, its magnitude, and position in space, will remain absolutely invariable; and the changes of motion in the first revolution will be repeated exactly in all succeeding revolutions. Let us now add to our system of two bodies a third body, as the moon. In case the sun had no existence, or was removed to an infinite distance, then the circumstances of motion in the moon, once determined, would remain absolutely invariable; but the moment we unite the three bodies, the sun, earth, and moon, into a system of three orbs, mutually dependent upon each other, the perfection and simplicity which marks a system of two bodies is for ever destroyed; and modifications are at once introduced into the motion of the earth revolving