Page:Newton's Principia (1846).djvu/522

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. And 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:—

The greatest elongation of the satelites from the centre of Jupiter as seen from the sun.					The periodic times of their revolutions.
1st 2d 3d 4th	′ 1 3 4 8	″ 48 01 46 13½	or or or or	″ 108 181 186 493½	d 1 3 7 16	h 18 13 03 18	′ 28 17 59 5	″ 36 54 36 13

Whence the sesquiplicate proportion may be easily seen. For example; the 16^d 18^h.05′ 13″ is to the time 1^d.18^h.28′ 36″ as 493½″ $\scriptstyle \times {\sqrt {493{\frac {1}{2}}''}}$ to 108 $\scriptstyle \times {\sqrt {108''}}$, 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:—

Distance of the	1st	2d	3d	4th
By Galileo " Simon Marius " Cassini Borelli, more exactly	6 6 5 5⅔	10 10 8 8⅔	16 16 13 14	28 26 23 24⅔

After the invention of the micrometer:—

By Townley " Flamsted More accurately by the eclipses

5,51 5,31 5,578

8,78 8,85 8,876

13,47 13,98 14,159

24,72 24,23 24,903