the sea, hurried in and out with great violence, sometimes lays the shores under water, sometimes leaves them dry, for many miles. Nor is the force of the influx and efflux to be broke till it has raised or depressed the water to forty or fifty feet and more. Thus also long and shallow straits that open to the sea with mouths wider and deeper than the rest of their channel (such as those about Britain and the Magellanic Straits at the eastern entry) will have a greater flood and ebb, or will more intend and remit their course, and therefore will rise higher and be depressed lower. Or the coast of South America it is said that the Pacific sea in its reflux sometimes retreats two miles, and gets out of sight of those that stand on shore. Whence in these places the floods will be also higher but in deeper waters the velocity of influx and efflux is always less, and therefore the ascent and descent is so too. Nor in such places is the ocean known to ascend to more than six, eight, or ten feet. The quantity of the ascent I compute in the following manner.
Let S represent the sun, T the earth (419, 420), P the moon, PAGB the moon's orbit. In SP take SK equal to ST and SL to SK in the duplicate ratio of SK to SP. Parallel to PT draw LM; and, supposing the mean quantity of the circum-solar force directed towards the earth to be represented by the distance ST or SK, SL will represent the quantity thereof directed towards the moon. But that force is compounded of the parts SM, LM; of which the force LM and that part of SM which is represented by TM, do disturb the motion of the moon (as appears from Prop. LXVI, and its Corollaries). In so far as the earth and moon are revolved about their common centre of gravity, the earth will be liable to the action of the like forces. But we may refer the sums as well of the forces as of the motions to the moon, and represent the sums of the forces by the lines TM and ML, which are proportional to them. The force LM, in its mean quantity, is to the force by which the moon may be revolved in an orbit, about the earth quiescent, at the distance PT in the duplicate ratio of the moon's periodic time about the earth to the earth's periodic time about the sun (by Cor. XVII, Prop. LXVI); that is, in the duplicate ratio of 27d.7h.43′ to 365d.6d.9′; or as 1000 to 178725, or 1 to 17829⁄40. The force by which the moon may be revolved in its orb about the earth in rest, at the distance PT of 60½ semi-diameters of the earth, is to the force by which it may revolve in the same time at the distance of 60 semi-diameters as 60½ to 60; and this force is to the force of gravity with us as 1 to 60 60 nearly; and therefore the mean force ML is to the force of gravity at the surface of the earth as 1 60½ to 60 60 17829⁄40, or 1 to
