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equal to what it would acquire by falling freely through a right line equal in height to half the radius of the Earth, it would revolve round the globe without ceasing, supposing it met with no resistance; and at every revolution would pass through the point from which it set out. The same reasoning holds good, only making the necessary changes in the proportions, if the ball, instead of setting off from a point on the surface of the Earth, should be projected from a point above the surface, and distant from it a mile, two miles, or any other height. We may carry it therefore in idea as far as the Moon, or suppose it to be the Moon itself, which in fact revolves circularly round the Earth: and then, by the velocity with which the Moon moves, we shall find the ratio of the force, that retains it in it's orbit, or which is continually occasioning it's deflection from the rectilinear path, to the gravity which occasions bodies here below to fall to the ground.
Now according to astronomical and geometrical observations, the radius of the Earth is equal to 4000 english miles: the mean distance of the Moon from the Earth, or the mean radius of the lunar orbit, is equal to sixty times the radius of the Earth: and the Moon performs it's revolution round the Earth in 27 days, 7 hours, 43 minutes. From these data we find, 1st, the whole circumference of the lunar orbit, and the length of the arc, which the
In these calculations I omit small quantities, which would lengthen them uselessly, for all, that is necessary here, is to show the grounds of the method.
axis of the ellipsis, the radius vector, and the arc passed through: whence it may be concluded, that the primary planet is impelled toward the Sun, or the satellite toward it's primary, by a force inversely proportional to the square of the distance of the revolving body from the centre to which it tends. Thus were furnished the means of comparing the gravitations of any planet in any two points of it's orbit. But this was not sufficient: it was necessary, likewise, to know how to compare the gravitations of two different planets; for it might happen, that in different planets gravitation might not follow the inverse ratio of the squares of the distances, which would have deprived the principle of it's generality, and of it's most essential advantages.
The second law of Kepler completes this theory, and reduces all the gravitations to uniformity: it proves, that all the primary planets are impelled toward the Sun by one and the same force, varying in the inverse ratio of the squares of the distances. Thus, for example, the tendency of Mars toward the Sun is to that of Jupiter, as the square of Jupiter's distance from the Sun is to the square of that of Mars. It is the same also with the satellites in respect to their primary planets.
Gravitation is reciprocal between all the bodies in the universe. As the primary planets gravitate toward the Sun, and the satellites toward their primary planets, the Sun in it's turn gravitates toward the planets, and these toward their satellites. A stone, that falls on the surface of the Earth, is attracted by the terrestrial globe, and attracts this globe
in it's turn. The attraction exerted by every body is proportional to it's mass: for there is no reason why the attractive power should exist in one par-, ticle of a body, rather than in another: it is common to all, and the sum of the attraction is proportional to the mass. If two bodies, therefore, be in free space they will move toward each other, passing through spaces inversely proportional to their masses. Hence we see, in our example, that, in consequence of the enormous disproportion of masses, the tendency of the globe of the Earth toward the stone must appear nothing, in comparison of that of the stone toward the globe. As to the diminution, which the power of gravitation undergoes, in proportion as the distance increases, this can become sensible only when the distance is very great. Hence two bodies falling to the surface of the Earth from different heights, but both moderate, are acted upon by gravitations apparently equal; and the two heights passed through are proportional to the squares of the times, as Galileo first found. But this law does not hold, when the two heights differ considerably; as for instance, if one were 100 feet, and the other equal to the radius of the lunar orbit; for from the Earth to the Moon gravitation diminishes in the ratio of 3600 to 1.
From the reciprocal attraction which two planets, as the Earth and Moon, exert on each other, it follows, that the Earth must approach the Moon, at the same time as the Moon approaches the Earth, so that the motion of the Moon is performed round a movable point. It does not, however, on this account
follow any other laws, than if the Earth were fixed: for if we seek generally the curves described by two bodies, which, in conscquence of their mutual attractions, traverse paths toward each other inversely proportional to their masses, and which are projected through space in any given directions, we shall find, that these bodies describe four curves similar to each other; namely, each one round the other body considered as immovable, and each one round their common centre of gravity; which besides may be either at rest, or move uniformly in a right line.
If in the heavens there were only two bodies, turning round each other by virtue of an original impulsion, and of the newtonian attraction, constantly acting, they would move in a manner strictly conformable to the laws of Kepler. But if there be more than two bodies, which is actually the case in nature, the elliptical motion of the former two will be altered every instant by the attractions of the others. Of these inequalities I shall speak hereafter; but I shall first consider some particular applications, that have been made of the principle of attraction, to problems of another kind.
Among these problems, the first that presents itself is the question concerning the figure of the Earth, as far as it depends on the laws, of hydrostatics. Huygens, as has been mentioned, had explained the experiment of Richer at Cayenne, by the combination of the centrifugal force with a constant primitive gra vitation always directed toward the centre of the Earth. Instead of this gravitation Newton substituted the result of all the particular attractions, which the molecules
molecules of the terrestrial globe exert on each other. At present there can be no doubt,' which of these two laws of gravitation is to be preferred: the principle of Newton is avowed by nature; let us examine the use he has made of it, and the great scope he gave to his theory.
Newton tacitly supposes, without offering any demonstration of it, that the Earth, originally fluid and homogeneous, has acquired the figure of an oblate elliptical spheroid, in consequence of the mutual attraction of it's parts, and the centrifugal force. He calculates the weight of the central equatorial column, and of the central polar. From the weight of the former he subtracts the sum of the centrifugal forces of all the molecules that compose it, and makes the remainder equal to the weight of the polar column: whence he deduces the ratio of the equatorial diameter to the polar to be nearly as 230 to 229.
Beside the difference of the hypotheses, which Huygens and Newton had adopted respecting the nature of primitive gravity, they determined the figure of the Earth by different methods. Huygens set out with this principle, that the result of the primitive gravity and the centrifugal force must be every where perpendicular to the surface of the fluid: Newton, on the contrary, conceived, that the columns in the direction of the axes of the spheroid must reciprocally counterpoise each other. These two prineiples appear equally necessary at the same time; one, to establish an equilibrium at the surface of the fluid; the other, in the interiour of the mass. Hence Bouguer