BOOK III.] OF NATURAL PHILOSOPHY. 391 shew), it is here to be neglected. The action of the sun, attracting the moon from the earth, is nearly as the moon's distance from the earth ; and therefore (by what we have shewed in Cor. 2, Prop. XLV. Book I) is to the centripetal force of the moon as 2 to 357,45, or nearly so ; that is, as 1 to 178£ f- . And if we neglect so inconsiderable a force of the sun, the re maining force, by which the moon is retained in its orb, will be recipro cally as D2. This will yet more fully appear from comparing this force with the force of gravity, as is done in the next Proposition. COR. If we augment the mean centripetal force by which the moon is retained in its orb, first in the proportion of 177%$ to 178ff, and then in the duplicate proportion of the semi-diameter of the earth to the mean dis tance of the centres of the moon and earth, we shall have the centripetal force of the moon at the surface of the earth ; supposing this force, in de scending to the earth's surface, continually to increase in the reciprocal duplicate proportion of the height. PROPOSITION IV. THEOREM IV. That the moon gravitates towards the earth, and by thejorce oj gravity is continually drawn off from a rectilinear motion, and retained in its orbit. The mean distance of the moon from the earth in the syzygies in semidiameters of the earth, is, according to Ptolemy and most astronomers, 59 : according to Vendelin and Huygens, 60 ; to Copernicus, 60 1 ; to Street, 60