Fig. IV.
[PL. VII.]
How to delineate the paths of Jupiter’s Moons.
And Saturn’s.
Having drawn their Orbits on a Card, in proportion to their relative distances from Jupiter, I measured the radius of the Orbit of the fourth Satellite, which was an inch and a tenth part; then multiplied this by 424 for the radius of Jupiter’s Orbit, because Jupiter is 424 times as far from the Sun’s center as his fourth Satellite is from his center; and the product thence arising was 4664⁄10 inches. Then taking a small cord of this length, and fixing one end of it to the floor of a long room by a nail, with a black lead pencil at the other end I drew the curve ABCD &c. and set off a degree and an half thereon, from A to T; because Jupiter moves only so much, whilst his outermost Satellite goes once round him, and somewhat more; so that this small portion of so large a circle differs but very little from a straight line. This done, I divided the space AT into 18 equal parts, as AB, BC, &c. for the daily progress of Jupiter; and each part into 24 for his hourly progress. The Orbit of each Satellite was also divided into as many equal parts as the Satellite is hours in finishing it’s synodical period round Jupiter. Then drawing a right line through the center of the Card, as a diameter to all the 4 Orbits upon it, I put the card upon the line of Jupiter’s motion, and transferred it to every horary division thereon, keeping always the said diameter-line on the line of Jupiter’s path; and running a pin through each horary division in the Orbit of each Satellite as the card was gradually transferred along the Line ABCD etc. of Jupiter’s motion, I marked points for every hour through the Card for the Curves described by the Satellites as the primary planet in the center of the Card was carried forward on the line: and so finished the Figure, by drawing the lines of each Satellite’s motion, through those (almost innumerable) points: by which means, this is perhaps as true a Figure of the paths of these Satellites as can be desired. And in the same manner might those for Saturn’s Satellites be delineated.
The grand Period of Jupiter’s Moons.
270. It appears by the scheme, that the three first Satellites come almost into the same line or position every seventh day; the first being only a little behind with the second, and the second behind with the third. But the period of the fourth Satellite is so incommensurate to the periods of the other three, that it cannot be guessed at by the diagram when it would fall again into a line of conjunction with them, between Jupiter and the Sun. And no wonder; for supposing them all to have been once in conjunction, it will require 3,087,043,493,260 years to bring them in a conjunction again: See § [73].
Fig. IV. The proportions of the Orbits of the Planets and Satellites.
271. In Fig. 4th we have the proportions of the Orbits of Saturn’s five Satellites, and of Jupiter’s four, to one another, to our Moon’s Orbit, and to the Disc of the Sun. S is the Sun; M m the Moon’s Orbit (the Earth supposed to be at E;) J Jupiter; 1. 2. 3. 4 the Orbits of his four Moons or Satellites; Sat Saturn; and 1. 2. 3. 4. 5 the Orbits of his five Moons. Hence it appears, that the Sun would much more than fill the whole Orbit of the Moon; for the Sun’s diameter is 763,000 miles, and the diameter of the Moon’s Orbit only 480,000. In proportion to all these Orbits of the Satellites, the Radius of Saturn’s annual Orbit would be 211⁄4 yards, of Jupiter’s orbit 112⁄3, and of the Earth’s 21⁄4, taking them in round numbers.
272. The annexed table shews at once what proportion the Orbits, Revolutions, and Velocities, of all the Satellites bear to those of their primary Planets, and what sort of curves the several Satellites describe. For, those Satellites whose velocities round their primaries are greater than the velocities of their primaries in open space, make loops at their conjunctions § [269]; appearing retrograde as seen from the Sun whilst they describe the inferior parts of their Orbits, and direct whilst they describe the superior. This is the case with Jupiter’s first and second Satellites, and with Saturn’s first. But those Satellites whose velocities are less than the velocities of their primary planets move direct in their whole circumvolutions; which is the case of the third and fourth Satellites of Jupiter, and of the second, third, fourth, and fifth Satellites of Saturn, as well as of our Satellite the Moon: But the Moon is the only Satellite whose motion is always concave to the Sun. There is a table of this sort in De la Caile’s Astronomy, but it is very different from the above, which I have computed from our English accounts of the periods and distances of these Planets and Satellites.
| The Satellites | Proportion of the Radius of the Planet’s Orbit to the Radius of the Orbit of each Satellite. | Proportion of the Time of the Planet’s Revolution to the Revolution of each Satellite. | Proportion of the Velocity of each Satellite to the Velocity of its primary Planet. | ||||
|---|---|---|---|---|---|---|---|
| of Saturn | 1 | As 5322 | to 1 | As 5738 | to 1 | As 5738 | to 5322 |
| 2 | 4155 | 1 | 3912 | 1 | 3912 | 4155 | |
| 3 | 2954 | 1 | 2347 | 1 | 2347 | 2954 | |
| 4 | 1295 | 1 | 674 | 1 | 674 | 1295 | |
| 5 | 432 | 1 | 134 | 1 | 134 | 432 | |
| of Jupiter | 1 | As 1851 | to 1 | As 2445 | to 1 | As 2445 | to 1851 |
| 2 | 1165 | 1 | 1219 | 1 | 1219 | 1165 | |
| 3 | 731 | 1 | 604 | 1 | 604 | 731 | |
| 4 | 424 | 1 | 258 | 1 | 258 | 424 | |
| The Moon | As 3371⁄2 | to 1 | As 121⁄3 | to 1 | As 121⁄3 | to 3371⁄2 | |
CHAP. XVI.
The Phenomena of the Harvest-Moon explained by a common Globe: The years in which the Harvest-Moons are least and most beneficial from 1751, to 1861. The long duration of Moon-light at the Poles in winter.
No Harvest-Moon at the Equator.