122 THE MATHEMATICAL PRINCIPLES [BOOK I. scribed in a given moment of time, that is (by Lem. "VII), as the tangent PR ; that is (because of the proportionals PR to Q,T, and SP to SP X Q,T SY), as ~y — — ; or as SY reciprocally, and SP X Q,T directly ; but SP X QT is as the area described in the given time, that is (by Prop. XIV), in the subduplicate ratio of the latus rectum. Q.E.D. COR. 1. The principal latera recta are in a ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the ve locities. COR. 2. The velocities of bodies, in their greatest and least distances from the common focus, are in the ratio compounded of the ratio of the distan ces inversely, and the subduplicate ratio of the principal latera recta di rectly. For those perpendiculars are now the distances. COR. 3. Arid therefore the velocity in a conic section, at its greatest or least distance from the focus, is to the velocity in a circle, at the same dis tance from the centre, in the subduplicate ratio of the principal latus rec tum to the double of that distance. COR. 4. The velocities of the bodies revolving in ellipses, at their mean distances from the common focus, are the same as those of bodies revolving in circles, at the same distances ; that is (by Cor. 6. Prop. IV), recipro cally in the subduplicate ratio of the distances. For the perpendiculars are now the lesser semi-axes, and these are as mean proportionals between the distances and the latera recta. Let this ratio inversely be compounded with the subduplicate ratio of the latera recta directly, and we shall have the subduplicate ratio of the distance inversely. COR. 5. In the same figure, or even in different figures, whose principal latera recta are equal, the velocity of a body is reciprocally as the perpen dicular let fall from the focus on the tangent. COR. 6. In a parabola, the velocity is reciprocally in the subduplicate ratio.of the distance of the body from the focus of the figure; it is more variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the focus on the tangent of a parabola is in the subduplicate ratio of the dis tance. In the hyperbola the perpendicular is less variable ; in the ellipsis more. COR. 7. In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance from the centre, in the subduplicate ratio of the number 2 to 1 ; in the ellipsis it is less, and in the hyperbola greater, than according to this ratio, For (by Cor. 2 of this Prop.) the velocitv at the vertex of a parabola is ir