proportion with which the ſquare of the diſtance is augmented. Let us now ſee whether, by making a compariſon between the centripetal forces of the Planets, and the force of gravity, we may not by chance find them to be of the ſame kind. Now they will be of the ſame kind if we find on both ſides the ſame laws, and the ſame affections. Let us then firſt conſider the centripetal force of the Moon which is neareſt to us.
The rectilinear ſpaces, which bodies let fall from reſt deſcribe in a given time at the very beginning of the motion, when the bodies are urged by any forces whatſoever, are proportional to the forces. This appears from mathematical reaſoning. Therefore the centripetal force of the Moon revolving in its orbit is to the force of gravity at the ſurface of the Earth, as the ſpace, which in a very ſmall particle of time the Moon, deprived of all its circular force and deſcending by its centripetal force towards the Earth, would deſcribe, is to the ſpace which a heavy body would deſcribe, when falling by the force of its gravity near to the Earth, in the ſame given particle of time. The firſt of theſe ſpaces is equal to the verſed ſine of the arc deſcribed by the Moon in the ſame time, becauſe that verſed ſine meaſures the tranſlation 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 diſtance from the centre of the Earth are given. The laſt ſpace is found by experiments of pendulums, as Mr. Huygens has ſhewn. Therefore by making a calculation we ſhall find that the firſt ſpace is to the latter, or the centripetal force of the Moon revolving in its orbit
will be to the force of gravity at the ſuperficies
