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 diſtances 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 perpendicular 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. 14.) the perpendicular let fall from the focus on the tangent of a parabola is in the ſubduplicate ratio of the diſtance. 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 velocity at the vertex of a parabola is in this ratio, and (by cor. 6. of this prop. and prop. 4.) the ſame proportion holdſ in all diſtances. And hence alſo in a parabola, the velocity is every where equal to the velocity of a body revolving in a circle at half the diſtance; in the ellipſis it is leſs, and in the hyperbola greater.
