BOOK III.] OF NATURAL PHILOSOPHY. 433 in the duplicate proportion of the time ; for that motion, while the moou describes PM, is (cceteris parilms) as ML. and ML is in the duplicate proportion of the time. Wherefore the motion of the nodes in the syzygj-es, in the time while the moon describes given little parts of its orbit, is diminished in the duplicate proportion of the number H07.'J to the num ber 11023: and the decrement is to the remaining motion as 100 to 10973 ; but to the whole motion as 100 to 11073 nearly. But the decre ment in the places between the octants and syzygies, and the increment in the places between the octants and quadratures, is to this decrement nearly as the whole motion in these places to the whole motion in the syzygies, and the difference betwixt the square of the sine of the moon's distance from the quadrature, and the half square of the radius, to the half square of the radius conjunctly. Wherefore, if the nodes are in the quadratures, and we take two places, one on one side, one on the other, equally distant from the octant and other two distant by the same interval, one from the syzygy, the other from the quadrature, and from the decrements of the motions in the two places between the syzygy and octant we subtract the increments of the motions in the two other places between the octant and the quadrature, the remaining decrement will be equal to the decre ment in the syzygy, as will easily appear by computation ; and therefore the mean decrement, which ought to be subducted from the mean motion of the nodes, is the fourth part of the decrement in the syzygy. The whole horary motion of the nodes in the syzygies (when the moon by a ra dius drawn to the earth was supposed to describe an area proportional to the time) was 32" 42"' ?iv. And we have shewn that the decrement of the motion of the nodes, in the time while the moon, now moving with greater velocity, describes the same space, was to this motion as 100 to 1.1073; and therefore this decrement is 17'" 43iv. 11 v. The fourth part of which 4'" 25iv. 48V. subtracted from the mean horary motion above found, 16" 21//; 3iv. 30V. leaves 16" 16'" 37iv. 42V. their correct mean ho rary motion. If the nodes are without the quadratures, and two places are considered, one on one side, one on the other, equally distant from the syzygies, the sum of the motions of the nodes, when the moon is in those places, will be to the sum of their motions, when the moon is in the same places and the nodes in the quadratures, as AZ2 to AT2. And the decrements of the motions arising from the causes but now explained will be mutually as the motions themselves, and therefore the remaining motions will be mu tually betwixt themselves as AZ2 to AT2 ; and the mean motions will be as the remaining motions. And, therefore, in any given position of the nodes, their correct mean horary motion is to 16" 16'" 37iv. 42V. as AZ2 to AT2 ; that is, as the square of the sine of the distance of the nodes from the syzygies to the square of the radius. ' 28