432 THE MATHEMATICAL PRINCIPLES [HOOK 111. cY to/ Y X cp, or as//? to/ Y, and cY to cjo ; that is; if ph parallel to TN meet FP in h, as FA to FY and F Y to FP ; that is, as Fh to FP or DJO to DP, and therefore as the area Dpmd to the area DPMc?. And, therefore, seeing (by Corol. 1, Prop. XXX) the latter area and AZ2 conjunctly are proportional to the horary motion of the nodes in the circle, the former area and AZ2 conjunctly will be proportional to the horary motion of the nodes in the ellipsis. Q.E.D. COR. Since, therefore, in any given position of the nodes, the sum of all the areas />Drfm, in the time while the moon is carried from the quadra ture to any place tn, is the area mpQ&d terminated at the tangent of the ellipsis Q,E ; and the sum of all those areas, in oTne entire revolution, is the area of the whole ellipsis ; the mean motion of the nodes in the ellip sis will be to the mean motion of the nodes in the circle as the ellipsis to the circle ; that is, as Ta to TA, or 69 to 70. And, therefore, since (by Corol 2, Prop. XXX) the mean horary motion of the nodes in the circle is to 16" 35"7 16iv. 36V. as AZ2 to AT2, if we take the angle 16" 21'" 3iv. 30V. to the angle 16" 35'" 16iv. 36V. as 69 to 70. the mean horary mo tion of the nodes in the ellipsis will be to 16" 21'" 3iv. 30V. as AZ2 to AT2 ; that is, as the square of the sine of the distance of the node from the sun to the square of the radius. But the moon, by a radius drawn to the earth, describes the area in the syzygie-s with a greater velocity than it does that in the quadratures, and upon that account the time is contracted in the syzygies, and prolonged in the quadratures ; and together with the time the motion of the nodes is likewise augmented or diminished. But the moment of the area in the quadrature of the moon was to the moment thereof in the syzygies as 10973 to 11073 ; and therefore the mean moment in the octants is to the excess in the syzygies. and to the defect in the quadratures, as 1 1023, the half sum of those numbers, to their half difference1 50. Wherefore since the time of the moon in the several little equal parts of its orbit is recip rocally as its velocity, the mean time in the -octants will be to the excess of the time in the quadratures, and to the defect of the lime in the syzy gies arising from this cause, nearly as 11023 to 50. But, reckoning from the quadratures to the syzygies, I find that the excess of the moments of the area, in the several places above the least moment in the quadratures, is nearly as the square of the sine of the moon's distance from the quad ratures : and therefore the difference betwixt the moment in any place, and the mean moment in the octants, is as the difference betwixt the square of the sine of the moon's distance from the quadratures, and the square of the sine of 45 degrees, or half the square of the radius ; and the in crement of the time in the several places between the octants and quad ratures, and the decrement thereof between the octants and syzygies, is in the same proportion. But the motion of the nodes, while the moon de scribes the several little equal parts of its orbit, is accelerated or retarded