Plantary Orbits

The revolution of the earth means that centripetal forces are actually directed to the bodies of the sun, of the earth, and other planets.

The moon revolves about our earth. A line drawn from the earth’s center to the moon coincides with the lunar cycles, from the moon’s velocity compared with its diameter.

The moon’s motion is slower when its diameter is less (and therefore is farther from earth). Its motion is faster when its diameter is greater.

The revolutions of Jupiter’s moons are more regular for they describe circles concentric with Jupiter by equable motions.

And so the satellites of Saturn are revolved about this planet with motions nearly circular and equable, scarcely disturbed by any eccentricity hitherto observed.

Venus and Mercury are revolve around the sun. This is demonstrable from their moon-like appearances.

• When they shine with a full face, they are in those parts of their orbs which in respect of the earth lie beyond the sun
• When they appear half full, they are in those parts which lie over against the sun
• When horned, in those parts which lie between the earth and the sun

Sometimes, they pass over the sun’s disk, when directly interposed between the earth and the sun.

Venus, with a motion almost uniform, describes an orb nearly circular and concentric with the sun.

But Mercury, with a more eccentric motion, makes remarkable approaches to the sun, and goes off again by turns.

But it is always swifter as it is near to the sun, and therefore by a radius drawn to the sun still describes areas proportional to the times.

Lastly, that the earth describes about the sun, or the sun about the earth, by a radius from the one to the other, areas exactly proportional to the times, is demonstrable from the apparent diameter of the sun compared with its apparent motion.

These are astronomical experiments; from which it follows, by Prop. I, II, III, in the first Book of our Principles, and their Corollaries (p. 212, 213, 214), that there are centripetal forces actually directed (either accurately or without considerable error) to the centres of the earth, of Jupiter, of Saturn, and of the sun. In Mercury, Venus, Mars, and the lesser planets, where experiments are wanting, the arguments from analogy must be allowed in their place.

That those forces (p. 212, 213, 214) decrease in the duplicate proportion of the distances from the centre of every planet, appears by Cor. VI, Prop. IV, Book 1; for the periodic times of the satellites of Jupiter are one to another (p. 386, 387) in the sesquiplicate proportion of their distances from the centre of this planet.

This proportion has been long ago observed in those satellites; and Mr. Flamsted, who had often measured their distances from Jupiter by the micrometer, and by the eclipses of the satellites, wrote to me, that it holds to all the accuracy that possibly can be discerned by our senses.

He sent me the dimensions of their orbits taken by the micrometer, and reduced to the mean distance of Jupiter from the earth, or from the sun, together with the times of their revolutions, as follows:

Whence the sesquiplicate proportion may be easily seen. For example; the 16d 18h.05′ 13″ is to the time 1d.18h.28′ 36″ as 493½″ … neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

After the invention of the micrometer:—

The periodic times of those satellites, by the observations of Mr. Flamsted, are 1d.18h.28′ 36″ | 3d.13h.17′ 54″ | 7d.3h.59′ 36″ | 16d.18h.5′ 13″, as above.

The distances thence computed are 5,578 | 8,878 | 14,168 | 24,968, accurately agreeing with the distances by observation.

Cassini assures us (p. 388, 389) that the same proportion is observed in the circum-saturnal planets. But a longer course of observations is required before we can have a certain and accurate theory of those planets.

In the circum-solar planets. Mercury and Venus, the same proportion holds with great accuracy, according to the dimensions of their orbs, as determined by the observations of the best astronomers.

That Mars is revolved about the sun is demonstrated from the phases which it shews, and the proportion of its apparent diameters (p. 388, 389, and 390); for from its appearing full near conjunction with the sun, and gibbous in its quadratures, it is certain that it surrounds the sun.

And since its diameter appears about five times greater when in opposition to the sun than when in conjunction therewith, and its distance from the earth is reciprocally as its apparent diameter, that distance will be about five times less when in opposition to than when in conjunction with the sun; but in both cases its distance from the sun will be nearly about the same with the distance which is inferred from its gibbous appearance in the quadratures. And as it encompasses the sun at almost equal distances, but in respect of the earth is very unequally distant, so by radii drawn to the sun it describes areas nearly uniform; but by radii drawn to the earth, it is sometimes swift, sometimes stationary, and sometimes retrograde.

That Jupiter, in a higher orb than Mars, is likewise revolved about the sun, with a motion nearly equable, as well in distance as in the areas described, I infer thus.

Mr. Flamsted assured me, by letters, that all the eclipses of the inner most satellite which hitherto have been well observed do agree with his theory so nearly, as never to differ therefrom by two minutes of time; that in the outmost the error is little greater; in the outmost but one, scarcely three times greater; that in the innermost but one the difference is indeed much greater, yet so as to agree as nearly with his computations as the moon does with the common tables; and that he computes those eclipses only from the mean motions corrected by the equation of light discovered and introduced by Mr. Romer. Supposing, then, that the theory differs by a less error than that of 2′ from the motion of the outmost satellite as hitherto described, and taking as the periodic time 16d. 18h.5′ 13″ to 2′ in time, so is the whole circle or 360° to the arc 1′ 48″, the error of Mr. Flamsted’s computation, reduced to the satellite’s orbit, will be less than 1′ 48″; that is, the longitude of the satellite, as seen from the centre of Jupiter, will be determined with a less error than 1′ 48″. But when the satellite is in the middle of the shadow, that longitude is the same with the heliocentric longitude of Jupiter; and, therefore, the hypothesis which Mr. Flamsted follows, viz., the Copernican, as improved by Kepler, and (as to the motion of Jupiter) lately corrected by himself, rightly represents that longitude within a less error than 1′ 48″; but by this longitude, together with the geocentric longitude, which is always easily found, the distance of Jupiter from the sun is determined; which must, therefore, be the very same with that which the hypothesis exhibits. For that greatest error of 1′ 48″ that can happen in the heliocentric longitude is almost insensible, and quite to be neglected, and perhaps may arise from some yet undiscovered eccentricity of the satellite; but since both longitude and distance are rightly determined, it follows of necessity that Jupiter, by radii drawn to the sun, describes areas so conditioned as the hypothesis requires, that is, proportional to the times.

The same thing may be concluded of Saturn from his satellite, by the observations of Mr. Huygens and Dr. Halley; though a longer series of observations is yet wanting to confirm the thing, and to bring it under a sufficiently exact computation.

For if Jupiter was viewed from the sun, it would never appear retrograde nor stationary, as it is seen sometimes from the earth, but always to go forward with a motion nearly uniform (p. 389). And from the very great inequality of its apparent geocentric motion, we infer (by Prop. III Cor. IV) that the force by which Jupiter is turned out of a rectilinear course, and made to revolve in an orb, is not directed to the centre of the earth. And the same argument holds good in Mars and in Saturn. Another centre of these forces is therefore to be looked for (by Prop. II and III, and the Corollaries of the latter), about which the areas described by radii intervening may be equable; and that this is the sun, we have proved already in Mars and Saturn nearly, but accurately enough in Jupiter. It may be alledged that the sun and planets are impelled by some other force equally and in the direction of parallel lines; but by such a force (by Cor. VI of the Laws of Motion) no change would happen in the situation of the planets one to another, nor any sensible effect follow: but our business is with the causes of sensible effects. Let us, therefore, neglect every such force as imaginary and precarious, and of no use in the phænomena of the heavens; and the whole remaining force by which Jupiter is impelled will be directed (by Prop. III, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tycho, we place the earth in the centre of the system, or the sun with Copernicus: and we have already proved that these distances are true in Jupiter.