The Planets Move in Ellipses

Proposition 13 Theorem 13

The planets move in ellipses which have their common focus in the sun’s centre. By radii drawn to that centre, they describe areas proportional to the times of description.

The weights of the planets towards the sun are reciprocally as the squares of their distances from the sun’s centre.

• If the sun were at rest and the other planets did not mutually act one upon another, their orbits would be ellipses, having the sun in their common focus.

They would describe areas proportional to the times of description, by Prop, I and XI, and Cor. 1, Prop. XIII, Book I.

But the mutual actions of the planets one upon another are so very small, that they may be neglected; and by Prop. LXVI, Book I, they less disturb the motions of the planets around the sun in motion than if those motions were performed about the sun at rest.

The action of Jupiter on Saturn is not to be neglected.

The force of gravity towards Jupiter is to the force of gravity towards the sun (at equal distances, Cor. 2, Prop. VIII) as 1 to 1067.

Therefore in the conjunction of Jupiter and Saturn, because the distance of Saturn from Jupiter is to the distance of Saturn from the sun almost as 4 to 9, the gravity of Saturn towards Jupiter will be to the gravity of Saturn towards the sun as 81 to 16 × 1067 or, as 1 to about 211.

Hence arises a perturbation of the orb of Saturn in every conjunction of this planet with Jupiter, so sensible, that astronomers are puzzled with it.

As the planet is differently situated in these conjunctions, its eccentricity is sometimes augmented, sometimes diminished; its aphelion is sometimes carried forward, sometimes backward, and its mean motion is by turns accelerated and retarded.

Yet the whole error in its motion about the sun, though arising from so great a force, may be almost avoided (except in the mean motion) by placing the lower focus of its orbit in the common centre of gravity of Jupiter and the sun (according to Prop. LXVII, Book I), and therefore that error, when it is greatest, scarcely exceeds two minutes; and the greatest error in the mean motion scarcely exceeds two minutes yearly.

But in the conjunction of Jupiter and Saturn, the accelerative forces of gravity of the sun towards Saturn, of Jupiter towards Saturn, and of Jupiter towards the sun, are almost as 16,81, and … or 156609.

Therefore, the difference of the forces of gravity of the sun towards Saturn, and of Jupiter towards Saturn, is to the force of gravity of Jupiter towards the sun as 65 to 156609, or as 1 to 2409.

But the greatest power of Saturn to disturb the motion of Jupiter is proportional to this difference; and therefore the perturbation of the orbit of Jupiter is much less than that of Saturn’s.

The perturbations of the other orbits are yet far less, except that the orbit of the earth is sensibly disturbed by the moon.

The common centre of gravity of the earth and moon moves in an ellipsis about the sun in the focus thereof, and, by a radius drawn to the sun, describes areas proportional to the times of description.

But the earth in the mean time by a menstrual motion is revolved about this common centre.