BOOK I1I.J OF NATURAL PHILOSOPHY. 431 pm an arc which it describes in the least moment of time; N and n tlw nodes joined by the line N//, ; pK and ink perpendiculars upon the axis Q,, produced both ways till they meet the circle in P and M, and the line of the nodes in D and cl. And if the moon, by a radius drawn to the earth, describes an area proportional to the time of description, the horary motion of the node in the ellipsis will be as the area pDdm and AZ2 conjunctly. For let PF touch the circle in P, and produced meet TN in F; arid pj touch the ellipsis in p, and produced meet the same TN in /, and both tangents concur in the axis TQ, at Y. And let ML represent the space which the moon, by the impulse of the above-mentioned force 3IT or 3PK, would describe with a transverse motion, in the meantime while revolving in the circle it describes the arc PM ; and ml denote the space which the moon revolving in the ellipsis would describe in the same time by the im pulse of the same force SIT or 3PK ; and -let LP and Ip be produced till they meet the plane of the ecliptic in G and g, and FG and /"^ be joined, of which FG produced may cut pf, pa; and TQ, in c, e, and R respect ively ; and/0" produced may cut TQ in r. Because the force SIT or 3PK in the circle is to the force SIT or 3/?K in the ellipsis as PK to /?K, or as AT to «T, the space ML generated by the former force will be to the space ml generated by the latter as PK to p"K ; that is, because of the similar figures PYK/? and FYRc, as FR to cR. But (because of the similar triangles PLM, PGF) ML is to FG as PL to PG. that is (on ac count of the parallels L/r, PK, GR), as pi to pe, that is (because of the similar triangles plm, cpe), as lm to ce ; and inversely as LM is to lm, or as FR is to cR, so is FG to ce. And therefore if fg was to ce as/// to cY, that is, as fr to cR (that is, as fr to FR and FR to cR conjunctly, that is, as/T to FT, and FG to ce conjunctly), because the ratio of FG to ce, expunged on both sides, leaves the ratios fg to FG and/T to FT, fg would be to FG as/T to FT; and, therefore, the angles which FG and/£- would subtend at the earth T would be equal to each other. But these angles (by what we have shewn in the preceding Proposition) are the motions of the nodes, while the moon describes in the circle the arc PM, in the ellipsis the arc jt?w; and therefore the motions of the nodes in the circle and in the ellipsis would be equal to each other. Thus, I say, it cex /Y would be, if fg was to cc as/Y to cY, that is, if/,r was oqual to— ^ . But because of the similar triangles/?/?, cep, fg is to cc as//? to cp ; anJ therefore/?- is equal to— - — ; and therefore the angle which fg sub tends in fact is to the former angle which FG subterds. that is to say, the motion of the nodes in tl;^ ellipsis is to the motion of the same in the circle aa this/^ or- —to the forrer/o- or — — , that is, as//? X