(by cor. 3. prop. 6.) that is, (becauſe is given) directly as PC. Q. E. I.
Cor. 1. And therefore the force is as the diſŧance 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. 4.) but in ellipſes that have their greater axe 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 inverſly; 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 axis inverſely; and therefore (becauſe of the equality of the direct and inverſe ratio's) in the ratio of equality.
Scholium.
If the ellipſis by having its centre removed to an infinite diſtance degenerates into a parabola, the body will move in this parabola; and the force, now tending to a centre infinitely remote, will become equable. which is Ga1ileo's theorem. And if the parabolic ſection of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in like manner as in the
