Bodies in Orbit

Proposition 14 Theorem 6

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.

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

…

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 .

Therefore … is as QT² × SP².

That is, the latus rectum L is in the duplicate ratio of the area QT × SP. Q.E.D.

Corollary

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.

Proposition 15 Theorem 7

The periodic times in ellipses are in the sesquiplicate ratio of their greater axes.

The lesser axis is a mean proportional between the greater axis and the latus rectum. Therefore, the rectangle under the axes is in the ratio compounded of the subduplicate ratio of the latus rectum and the sesquiplicate ratio of the greater axis.

But this rectangle (by Cor. 3. Prop. XIV) is in a ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the periodic time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the sesquiplicate ratio of the greater axis, equal to the ratio of the periodic time. Q.E.D.

Corollary

Therefore the periodic times in whose diameters are equal ellipses are the same as in circles to the greater axes of the ellipses.