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 466-4/10 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.
[Sidenote: 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.
[Sidenote: 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 21-1/4 yards, of Jupiter’s orbit 11-2/3, and of
the Earth’s 2-1/4, 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.
+------------+-----------------+----------------+----------------------+
| | Proportion of | Proportion of | Proportion of |
| | the Radius of | the Time of | the Velocity of |
| The | the Planet’s | the Planet’s | each Satellite |
| Satellites | Orbit to the | Revolution to | to the Velocity |
| | Radius of the | the Revolution | of its primary |
| | Orbit of each | of each | Planet. |
| | Satellite. | Satellite. | |
+------------+-----------------+----------------+----------------------+
| 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 337-1/2 to 1 | As 12-1/3 to 1 | As 12-1/3 to 337-1/2 |
+------------+-----------------+----------------+----------------------+