The Moon’s motion always concave towards the Sun.
And thus we see, that although the Moon goes round the Earth in a Circle, with respect to the Earth’s center, her real path in the Heavens is not 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.
How her motion is alternately retarded and accelerated.
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 Semidiameter of her Orbit) behind the Earth at her first Quarter in b, when the Earth is in B; that is, she falls back a space equal to her distance from the Earth. From that time her motion is gradually accelerated to her Opposition or Full at c, and then she is come up as far as the Earth, having regained what she lost in her first Quarter from a to b. 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 Semidiameter of her Orbit; and by that means she comes to e, and is then in conjunction with the 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 towards 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, and which shews the concavity of that part of her path, will be about 400 thousand miles.
A difficulty removed.
[PL. VII.]
268. The Moon’s path being concave to the Sun throughout, demonstrates that her gravity towards the Sun, at her conjunction, exceeds her gravity towards 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 where these two powers would be equally strong, is about 70 thousand miles nearer the Earth than the Moon is at her Change. It may now 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 the Moon is so near the Earth in proportion to the Earth’s distance from the Sun, that she is but very little more attracted by the Sun at that time than the Earth is; and whilst the Earth’s attraction is greater upon the Moon than the difference of the Sun’s attraction upon the Earth and her (and that it is always much greater is demonstrable) there is no danger of the Moon’s leaving the Earth; for if she should fall towards the Sun, the Earth would follow her almost with equal speed. The absolute attraction of the Earth upon a drop of falling rain is much greater than the absolute attraction of the particles of that drop upon each other, or of it’s center upon all parts of it’s circumference; but then the side of the drop next the Earth is attracted with so very little more force than it’s center, or even it’s opposite side; that the attraction of the center of the drop upon it’s side next the Earth is much greater than the difference of force by which the Earth attracts it’s nearer surface and center: on which account the drop preserves it’s round figure, and might be projected about the Earth by a strong circulating wind so as to be kept from falling to the Earth. It is much the same with the Earth and Moon in respect to the Sun; for if we should suppose the Moon’s Orbit to be filled with a fluid Globe, of which all the parts would be attracted towards the Earth in it’s center, but the whole of it much more attracted by the Sun; one part of it could not fall to the Sun without the other, and a sufficient projectile force would carry the whole fluid Globe round the Sun. A ship, at the distance of the Moon, sailing round the Earth on the surface of the fluid Globe, could no more be taken away by the Sun when it is on the side next him, than the Earth could be taken away from it when it is on the opposite side; which could never happen unless the Earth’s projectile motion were stopt; and if it were stopt, the Ship with the whole fluid Globe, Earth and all together, would as naturally fall to the Sun as a drop of rain in calm air falls to the Earth. Hence we may see, that the Earth is in no more danger of being left by the Moon at the Change, than the Moon is of being left by the Earth at the Full: the diameter of the Moon’s Orbit being so small in comparison of the Sun’s distance, that the Moon is but little more or less attracted than the Earth at any time. And as the Moon’s projectile force keeps her from falling to the Earth, so the Earth’s projectile force keeps it from falling to the Sun.
Fig. III.
269. All the curves which Jupiter’s Satellites describe, are different from the path described 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 absolute Path of Jupiter and his Satellites delineated.
Fig. III.
Now let us suppose all these Moons to set out from a conjunction with the Sun, as seen from Jupiter. When Jupiter is at A his first or nearest Moon will be at a, 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 a1, his second the curve b1, his third c1, and his fourth d1. The next day when Jupiter is at C, his first Satellite has described the curve a2 from its conjunction, his second the curve b2, his third the curve c2, and his fourth the curve d2, 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 past 3, being twice stationary, and once retrograde between 3 and 4. The path of this Satellite intersects itself every 421⁄2 hours of our time, 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 it’s path every 3 days 13 hours; as at 4. 7. 11. 14. 18, making only five 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 it’s next conjunction. The fourth Satellite d is always progressive, making neither loops nor angles in the Heavens; but comes to it’s next conjunction at e between the numeral figures 16 and 17, or in 16 days 18 hours. In order to have a tolerably good figure of the paths of these Satellites, I took the following method.