If several bodies revolve about one common centre, and the centripetal force is reciprocally in the duplicate ratio of the distance of places from the centre; I say, that the principal latera recta of their orbits are in the duplicate ratio of the areas, which the bodies by radii drawn to the centre describe in the same time.
For (by Cor. 2, Prop. XIII) the latus rectum L is equal to the quantity QT²/QR in its ultimate state when the points P and Q coincide. But the lineola QR in a given time is as the generating centripetal force; that is (by supposition), reciprocally as SP2. And therefore
QT²/QR is as QT2 × SP2; that is, the latus rectum L is in the duplicate ratio of the area QT × SP. Q.E.D.
COR. Hence the whole area of the ellipsis, and the rectangle under the axes, which is proportional to it, is in the ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. For the whole area is as the area QT × SP, described in a given time, multiplied by the periodic time.