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system has been hitherto insensible, which seems to indicate that their masses are inconsiderable. It is possible, however, that the minute errors of our best tables depend upon it. An exact theory of the perturbation of the planets, compared with very precise observations, is the only means of ascertaining this point, so important to the system of the world.

CHAP. V.

Of the Perturbations of the Motion of the Moon.

THE MOON is attracted at the same time Moon

by the Sun and by the Earth, but its motion round the Earth is only disturbed by the difference of the action of the Sun upon these two bodies: if the Sun was at an infinite distance, it would act equally upon them, and in the direction of parallel lines; their relative motion, therefore, would not be affected by an action which was common to both; but its distance, though very great compared with that of the Moon, cannot be considered as infinite: the Moon is alternately nearer and farther from the Sun than the Earth, and the straight line joining the centres of the Sun and Moon, forms angles more or less acute with the radius vector of the Earth. Thus the Sun

acts unequally and in different directions on the Earth and Moon; and from this diversity of action, inequalities must necessarily arise in the lunar motion, depending on the respective positions of the Moon and Sun. To determine these, we must at the same time consider the mutual actions and motions of these three bodies, the Sun, the Earth, and the Moon. This constitutes the famous problem of the three bodies, the exact solution of which surpasses the powers of analysis; but from the proximity of the Moon, compared with its distance from the Sun, and from the comparative smallness of its mass, an approximation may be obtained extremely near the truth. Nevertheless, the most delicate analysis is necessary to investigate all the terms, whose influence becomes sensible; of this the first steps that were made in this analysis afford sufficient proof.

Euler, Clairaut, and Dalembert, who resolved this problem nearly about the same time, agreed in finding by the theory of gravitation, the motion of the lunar peri

gee only half as great as it appears to be from observation. From which Clairaut concluded that the law of attraction was not quite so simple as had been imagined; and he supposed it to consist of two parts, one varying inversely as the squares of the distances, and sensible only at the great distance of the planets from the Sun, and that the other, increasing in a greater ratio as the distance diminished, became sensible at the distance of the Moon from the Earth. This conclusion was vehemently opposed by Buffon: he maintained that since the primordial laws of nature should be the most simple possible, they could only depend on one modulus, and their expression, therefore, must consist of one single term. This consideration should no doubt lead us not to complicate the law of attraction, except in case of extreme necessity; at the same time our ignorance respecting the nature of this force, does not permit us to pronounce with certainty as to the simplicity of its expression. However this may be, the metaphysician was in the right this

time in his contest with the mathematician, who retracted his error on making this important discovery, that by carrying on the approximation farther than had been done at first, the law of attraction, reciprocally as the squares of the distances, gave the motion of the lunar perigee, exactly conformable to observation, which has since been confirmed by all those who have occupied themselves on this subject. It is impossible without the assistance of analysis, to explain the connection of all the inequalities of the Moon's motion with the combined action of the Sun and Earth upon this satellite. We shall observe, that the theory of universal gravitation has not only explained the motion of the node and of the perigee of the lunar orbit, together with the three great inequalities known by the names of variation, evection, and annual equation, all which astronomers had already recognized; but it has likewise developed a great number of others less considerable, which it would have been almost impossible to have found and ascer

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