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The revolutions of the satellites of Jupiter about that planet are more regular (p. 386); for they describe circles concentric with Jupiter by equable motions, as exactly as our senses can distinguish. |
And so the satellites of Saturn are revolved about this planet with motions nearly (p. 387) circular and equable, scarcely disturbed by any eccentricity hitherto observed. |
That Venus and Mercury are revolved about the sun, is demonstrable from their moon-like appearances (p. 388); 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; and sometimes they pass over the sun's disk, when directly interposed between the earth and the sun. |
And 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. 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 16d 18h.05′ 13″ is to the time 1d.18h.28′ 36″ as 493½″ to 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 |
And 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. |
And 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. |
And 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. |
Kepler and Bullialdus have, with great care (p. 388), 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 (by Cor. VI, Prop. IV) the centripetal circum-solar force throughout all the planetary regions decreases in the duplicate proportion of the distances from the sun. In examining this proportion, we are to use the mean distances, or the transverse semi-axes of the orbits (by Prop. XV), and to neglect those little fractions, which, in defining the orbits, may have arisen from the in sensible errors of observation, or may be ascribed to other causes which we shall afterwards explain. And thus 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, according to the computation of Kepler, as the numbers 951000, 519650, 152350, 100000, 72400, 38806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 72398, 38585; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the distances drawn from the periodic times, fall in between them. |
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, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393). |
But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon, did thereby augment the horizontal parallax of the moon by about the like number of minutes; that is, by about the 12th or 15th part of the whole parallax. Correct this error, and the distance will be come 60 or 61 semi-diameters of the earth, nearly agreeing with what others have determined. |
Let us, then, assume the mean distance of the moon 60 semi-diameters of the earth, and its periodic time in respect of the fixed stars 27d.7h.43′, as astronomers have determined it. And (by Cor. VI, Prop. IV) 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 3600 to 1, would (secluding the resistance of the air) complete a revolution in 1h.24′ 27″. |
Suppose the circumference of the earth to be 123249600 Paris feet, as has been determined by the late mensuration of the French (vide p. 406); 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. |