Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/27

Rh proportion with which the quare of the ditance is augmented. Let us now ee whether, by making a comparion between the centripetal forces of the Planets, and the force of gravity, we may not by chance find them to be of the ame kind. Now they will be of the ame kind if we find on both ides the ame laws, and the ame affections. Let us then firt conider the centripetal force of the Moon which is nearet to us.

The rectilinear paces, which bodies let fall from ret decribe in a given time at the very beginning of the motion, when the bodies are urged by any forces whatoever, are proportional to the forces. This appears from mathematical reaoning. Therefore the centripetal force of the Moon revolving in its orbit is to the force of gravity at the urface of the Earth, as the pace, which in a very mall particle of time the Moon, deprived of all its circular force and decending by its centripetal force towards the Earth, would decribe, is to the pace which a heavy body would decribe, when falling by the force of its gravity near to the Earth, in the ame given particle of time. The firt of thee paces is equal to the vered ine of the arc decribed by the Moon in the ame time, becaue that vered ine meaures the tranlation of the Moon from the tangent, produced by the centripetal force; and therefore may be computed, if the periodic time of the Moon and its ditance from the centre of the Earth are given. The lat pace is found by experiments of pendulums, as Mr. Huygens has hewn. Therefore by making a calculation we hall find that the firt pace is to the latter, or the centripetal force of the Moon revolving in its orbit will be to the force of gravity at the uperficies Rh