THE

SYSTEM OF THE WORLD.

BOOK IV.

OF THE THEORY OF UNIVERSAL GRAVITA

TION.

Opinionum commenta delet dies, naturæ judicia confirmat.

CIC. DE NAT. DEOR.

HAVING, in the preceding Books, explained the laws of the celestial motions, and those of the action of forces producing motion, we have now to compare them together, to learn what forces animate the solar system, to arrive without the assistance of any hypothesis, but by strict geometrical reasoning, at the principle of universal gravitation from which they are derived. It is in the celestial regions that the laws of mechanics are observed

with the greatest precision; on the earth so many causes tend to complicate their result, that it is very difficult to unravel them, and still more difficult to submit them to calculation. But the bodies of the solar system, separated by immense distances and subject to the action of a principal force, whose effect is easily calculated, are not disturbed in their respective motions by forces sufficiently considerable to prevent us from including under general formulæ, all the changes which a succession of ages has produced or may hereafter produce in the system. There is no question here of vague causes, which cannot be submitted to analysis, and which the imagination modifies at pleasure to accommodate them to the phenomena. The law of universal gravitation has this inestimable advantage, that it may be reduced to calculation, and by a comparison of its results with observation, it presents the most certain method of verifying its existence. We shall see that this great law of nature represents all the celestial phenome

na even in their minutest details, that there is not one single inequality of their motions which is not derived from it, with the most admirable precision, and that it explains the cause of several singular motions, just perceived by astronomers, and which were either too complicated or too slow for them to recognize their law. Thus, so far from having to fear that new observations will disprove this theory, we may be assured before-hand, that they will only confirm it more and more; and we may be assured that its consequences are equally certain as if they actually had been observed. The most profound geometry was indispensable to establish these theories: I have collected them in my Treatise of Celestial Mechanics. I shall confine myself here to present the principal results of this work, indicating the steps that led to them, and explaining the reasons as far as can be done without the assistance of analysis.

F

CHAP. I.

Of the Principle of Universal Gravitation.

Of all the phenomena of the solar system, the elliptic motion of the planets and of the comets seems the most proper to conduct us to the general law of the forces by which they are animated. Observation has shewn that the areas described by the, radii vectores of the planets and comets about the Sun are proportional to the times. Now we have seen in Chap. II. of the preceding Book, that for this to take place, the force which deflects the path of these bodies from a right line must constantly be directed towards the origin of their radii vectores. The tendency of the planets and comets to the Sun is therefore a necessary consequence of the proportionality of these areas to the times in which they are described.

To determine the law of this tendency, let us suppose the planets moved in circular orbits, which supposition does not greatly differ from the truth. The squares of their real velocities will then be proportional to the squares of the radii of these orbits divided by the squares of the times of their revolutions. But by the law of Kepler the squares of these times are to each other as the cubes of their radii. The squares of the velocities are therefore reciprocally as these radii. It has been shewn above that the central forces of several bodies moving in circular orbits, are as the squares of the velocities, divided by the radii of the circumferences described; the tendency therefore of the planets to the Sun is, reciprocally, as the squares of the radii of their orbits supposed circular. This hypothesis, it is true, is not rigorously exact, but the constant relation of the squares of the times to the cubes of the greater axes of their orbits, being independant of their excentricities, it is natural to think it would subsist also in the

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