Suppose CA, CB to be semi-axes of the ellipsis; GP, DK, conjugate diameters; PF, QT perpendiculars to those diameters; Qv an ordinate to the diameter GP; and if the parallelogram QvPR be completed, then (by the properties of the conic sections) the rectangle PvG will be to as to ; and (because of the similar triangles QvT, PCF), to as to ; and, by composition, the ratio of PvG to is compounded of the ratio of to , and of the ratio of to , that is, vG to as to . Put QR for Pv, and (by Lem. XII) for ; also (the points P and Q coinciding) 2PC for vG; and multiplying[Pg 115] the extremes and means together, we shall have equal to . Therefore (by Cor. 5, Prop. VI), the centripetal force is reciprocally as ; that is (because is given), reciprocally as ; that is, directly as the distance PC. Q.E.I. The same otherwise. In the right line PG on the other side of the point T, take the point u so that Tu may be equal to Tv; then take uV, such as shall be to vG as to . And because is to PvG as to (by the conic sections), we shall have . Add the rectangle uPv to both sides, and the square of the chord of the arc PQ will be equal to the rectangle VPv; and therefore a circle which touches the conic section in P, and passes through the point Q, will pass also through the point V. Now let the points P and Q meet, and the ratio of uV to vG, which is the same with the ratio of to , will become the ratio of PV to PG, or PV to 2PC; and therefore PV will be equal to . And therefore the force by which the body P revolves in the ellipsis will be reciprocally as (by Cor. 3, Prop. VI); that is (because is given) directly as PC. Q.E.I. COR. 1. And therefore the force is as the distance of the body from the centre of the ellipsis; and, vice versa, if the force is as the distance, the body will move in an ellipsis whose centre coincides with the centre of force, or perhaps in a circle into which the ellipsis may degenerate. COR. 2. And the periodic times of the revolutions made in all ellipses whatsoever about the same centre will be equal. For those times in similar ellipses will be equal (by Corol. 3 and 8, Prop. IV); but in ellipses that have their greater axis common, they are one to another as the whole areas of the ellipses directly, and the parts of the areas described in the same time inversely; that is, as the lesser axes directly, and the velocities of the bodies in their principal vertices inversely; that is, as those lesser axes directly, and the ordinates to the same point of the common axes inversely; and therefore (because of the equality of the direct and inverse ratios) in the ratio of equality. SCHOLIUM.