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Earth's moving round their common centre of gra- PLATE vity: all which, if they were truly copied in this experiment, would not sensibly alter the figure of the paths described by the nail and pin, even though they should rub against a plane upright surface ali the way, and leave their tracks visibly upon it. And if the chariot were driven forward on such a convex piece of ground, so as to turn the wheel several times round, the track of the pin in the nave would still be concave toward the centre of the circular curve described by the pin in the axle: as the Moon's path is always concave to the Sun in the centre of the Earth's annual orbit.
In this diagram, the thickest curve-line ABCDE, with the numeral figures set to it, represents as much of the Earth's annual orbit as it describes in 32 days from west to cast; the little circles at a, b, c, d, e, shew the Moon's orbit in due proportion to the Earth's; and the smallest curve ab c d e f rcpresents the line of the Moon's path in the heavens for 32 days, accounted from any particular new Moon at a. The machine Fig. 5th, is for delineating the Moon's path, and shall be described, with the rest of my astronomical machinery in the last chapter. The Sun is supposed to be in the centre of the curve A 12 3 4 5 6 7 B, &c. and the small dotted circles upon it, represent the Moon's orbit, of which the radius is in the same proportion to the Propofi Earth's path in this scheme, that the radius of the Moon's Moon's orbit in the heavens bears to the radius of orbit to the Earth's annual path round the Sun: that is, as Earth's. 240,000, to 81,000,000*, or as 1 to 337.
When the Earth is at A, the new Moon is at a; and in the seven days that the Earth describes the curve 1 2 3 4 5 6 7, the Moon in accompanying the Fig. 1I. Earth describes the curve ab; and is in her first quarter at 6 when the Earth is at B. As the Earth
* For the true distances, see p. 138.
Moon's motion al.
describes the curve B 8 9 10 11 12 13 14, the Moon describes the curve b c; and is at c, opposite to the Sun, when the Earth is at C. While the Earth describes the curve C 15 16 17 18 19 20 21 22, the Moon describes the curve cd; and is in her third quarter at d when the Earth is at D. And lastly, while the Earth describes the curve D 23 24 25 26 27 28 29, the Moon describes the curve de; and is again in conjunction at e with the Sun when the Earth is at E, between the 29th and 30th day of the Moon's age, accounted by the numeral figures from the new Moon at A. In describing the curve abcde, the Moon goes round the progressive Earth as really as if she had kept in the dotted circle A, and the Earth continued immoveable in the centre
of that circie. The And thus
we see that, although the Moon goes round the Earth in a circle, with respect to the ways con- Earth's centre, her real path in the heavens is not ward the very different in appearance from the Earth’s path.
To shew that the Moon's path is concave to the Sun, even at the time of change, it is carried on a little farther into a second lunation, as to f.
267. The Moon's absolute motion from her change to her first quarter, or from a to b, is so much slower than the Earth's, that she falls 240 thousand miles (equal to the semi-diameter of her orbit) behind the Earth at her first quarter in b, when the
Earth is at B; that is, she falls back a space equal How her to her distance from the Earth. From that time her motion
motion is gradually accelerated to her opposition or is alternately re. full at c, and then she is come up as far as the Earth, tarded and having regained what she lost in her first quarter accelerated.
from a to 6. From the full to the last quarter at d, her motion continues accelerated, so as to be just as far before the Earth at d, as she was behind it at her first quarter in b. But from d to e her motion is retarded, so that she loses as much with respect to the Earth as is equal to her distance from it, or to the semi-diameter of her orbit; and by that means
she comes to e, and is then in conjunction with the PLATE Sun as seen from the Earth at E. Hence we find, that the Moon's absolute motion is slower than the Earth's from her third quarter to her first; and swifter than the Earth's from her first quarter to her third; her path being less curved than the Earth's in the former case, and more in the latter. Yet it is still bent the same way toward the Sun; for if we imagine the concavity of the Earth's orbit to be measured by the length of a perpendicular line Cg, let down from the Earth's place upon the straight line bgd at the full of the Moon, and connecting the places of the Earth at the end of the Moon's first and third quarters, that length will be about 640 thousand miles; and the Moon when new only approaching nearer to the Sun by 240 thousand miles than the Earth, is the length of the perpendicular let down from her place at that time upon the same straight line, all which shews that the concavity of that part of her path, will be about 400 thousand miles.
268. The Moon's path being concave to the Sun A difficulthroughout, demonstrates that her gravity toward ty removthe Sun at her conjunction, exceeds her gravity to. ward the Earth. And if we consider that the
quantity of matter in the Sun is almost 230 thousand times as great as the quantity of matter in the Earth, and that the attraction of each body diminishes as the square of the distance from it increases, we shall soon find, that the point of equal attraction between the Earth and the Sun, is about 70 thousand miles nearer the Earth than the Moon is at her change. It may then appear surprising that the Moon does not abandon the Earth, when she is between it and the Sun, because she is considerably more attracted by the Sun than by the Earth at that time. But this difficulty vanishes when we consider, that a common impulse on any system of bodies affects
not their relative motions; but that they will continue to attract, impel, or circulate round one another, in the same manner as if there were no such impulse. The Moon is so near the Earth, and both of them so far from the Sun, that the attractive power of the Sun may be considered as equal on both: and therefore the Moon will continue to circulate round the Earth nearly in the same manner as if the Sun did not attract them at all. For bodies in the cabin of a ship, may move round, or impel one another in the same manner when the ship is under sail, as when it is at rest; because they are all equally affected by the common motion of the ship. If by any other cause, such as the near approach of a comet, the Moon's distance from the Earth should happen to be so much increased, that the difference of their gravitating forces toward the Sun should exceed that of the Moon toward the Earth; in that case the Moon when in conjunction, would abandon the Earth, and be either drawn into the Sun or comet,
or circulate round about it. Fig. III. 269. The curves which Jupiter's satellites de
scribe, are all of different sorts from the path describ. ed by our Moon, although these satellites go round Jupiter as the Moon goes round the Earth. Let ABCDE, &c. be as much of Jupiter's orbit as he describes in 18 days from A to T; and the curves a, b, c, d, will be the paths of his four moons going
round him in his progressive motion. The abso.
Now let us suppose all these moons to set out lute path of Jupiter from a conjunctis
a conjunction with the Sun, as seen from Jupiand his ter at A; then his first or nearest moon will be at a, satellites delineat
his second at b, his third at c, and his fourth at d. At the end of 24 terrestrial hours after this conjunction, Jupiter has moved to B, his first moon or satellite has described the curve a l, his second the curve b 1, his third c 1, and his fourth d l. The next day, when Jupiter is at C, his first satellite has
described the curve a 2, from its conjunction, his PLATZ second the curve b 2, his third the curve c
2, and his fourth the curve d 2, and so on. The numeral figures under the capital letters shew Jupiter's place in his path every day for 18 days, accounted from A to T'; and the like figures set to the paths of his satellites, shew where they are at the like times. The first satellite, almost under C, is stationary at +, as seen from the Sun; and retrograde from + to 2: at 2 it appears stationary again, and thence it moves forward until it has passed 3, and is twice stationary and once retrograde between 3 and 4.The path of this satellite intersects itself every 42 hours, making such loops as in the diagram at 2. 3. 5. 7. 9. 10. 12. 14. 16. 18, a little after every conjunction. The second satellite b, moving slower, barely crosses its path every 3 days 13 hours; as at 4. 7. 11. 14. 18. making only 5 loops and as many conjunctions in the time that the first makes ten. The third satellite c, moving still slower, and having described the curve c 1. 2. 3. 4. 5. 6. 7, comes to an angle at 7, in conjunction with the Sun, at the end of 7 days 4 hours; and so goes on to describe such another curve 7. 8. 9. 10. 11. 12. 13. 14, and is at 14 in its next conjunction. The fourth satellite d is always progressive, making neither loops nor angles in the heavens; but comes to its next conjunction at e between Fig. III. the numeral figures 16 and 17, or in 16 days 18 hours. In order to have a tolerable good figure of the paths of these satellites, I took the following method.
Having drawn their orbits on a card, in proportion to their relative distances from Jupiter, I mea- Fig. IV. sured the radius of the orbit of the fourth satellite, which was an inch and ó parts of an inch; then multiplied this by 424 for the radius of Jupiter's orbit, because Jupiter is 424 times as far from the Sun's centre as his fourth satellite is from his cen. tre, and the product thence arising was 48338