That these forces decrease in the duplicate proportion of the distances from the centre of every planet appears by Cor. vi., Prop. iv., Book I.[3] for the periodic times of the satellites of Jupiter are one to another in the sesquiplicate proportion of their distances from the centre of this planet. Cassini assures us that the same proportion is observed in the circum-Saturnal planets. In the circum-solar planets Mercury and Venus, the same proportional holds with great accuracy.
That Mars is revolved about the sun is demonstrated from the phases which it shows and the proportion of its apparent diameters; for from its appearing full near conjunction with the sun and gibbous in its quadratures,[4] it is certain that it travels round 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 orbit than Mars is likewise revolved about the sun with a motion nearly equable as well in distance as in the areas described, I infer from Mr. Flamsted's observations of the eclipses of the innermost satellite; and the same thing may be concluded of Saturn from his satellite by the observations of Mr. Huyghens and Mr. Halley.
If Jupiter was viewed from the sun it would never appear retrograde or stationary, as it is seen sometimes from the earth, but always to go forward with a motion nearly uniform. And from the very great inequality of its apparent geocentric motion we infer—as it has been previously shown that we may infer—that the force by which Jupiter is turned out of a rectilinear course and made to revolve in an orbit 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, 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.
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. Kepler and Bullialdus have with great care determined the distances of the planets from the sun, and hence it is that their tables agree best with the heavens. And in all the planets, in Jupiter and Mars, in Saturn and the earth, as well as in Venus and Mercury, the cubes of their distances are as the squares of their periodic times; and, therefore, the centripetal circum-solar force throughout all the planetary regions decreases in the duplicate proportion of the distances from the sun. Neglecting those little fractions which may have arisen from insensible errors of observation, we shall always find the said proportion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury from the sun, drawn from the observations of astronomers, are (Kepler) as the numbers 951,000, 519,650, 152,350, 100,000, 70,000, 38,806; or (Bullialdus) as the numbers 954,198, 522,520, 152,350, 100,000, 72,398, 38,585; and from the periodic times they come out 953,806, 520,116, 152,399, 100,000, 72,333, 38,710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the differences drawn from the periodic times fall in between them.
Earth as a Centre
That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus:
The mean distance of the moon from the centre of the earth is, we may assume, sixty semi-diameters of the earth; and its periodic time in respect of the fixed stars 27 days 7 hr. 43 min. Now, it has been shown in a previous book that a body revolved in our air, near the surface of the earth supposed at rest, by means of a centripetal force which should be to the same force at the distance of the moon in the reciprocal duplicate proportion of the distances from the centre of the earth, that is, as 3,600 to 1, would (secluding the resistance of the air) complete a revolution in 1 hr. 24 min. 27 sec.
Suppose the circumference of the earth to be 123,249,600 Paris feet, then the same body deprived of its circular motion and falling by the impulse of the same centripetal force as before would in one second of time describe 151⁄12 Paris feet. This we infer by a calculus formed upon Prop. xxxvi. ("To determine the times of the descent of a body falling from a given place"), and it agrees with the results of Mr. Huyghens's experiments of pendulums, by which he demonstrated that bodies falling by all the centripetal force with which (of whatever nature it is) they are impelled near the surface of the earth do in one second of time describe 151⁄12 Paris feet.