Part 18

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.

[Sidenote: 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.

[Sidenote: 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 42-1/2 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.

[Sidenote: 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 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 | +------------+-----------------+----------------+----------------------+

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._

[Sidenote: No Harvest-Moon at the Equator.]

273. It is generally believed that the Moon rises about 48 minutes later every day than on the preceding; but this is true only with regard to places on the Equator. In places of considerable Latitude there is a remarkable difference, especially in the harvest time; with which Farmers were better acquainted than Astronomers till of late; and gratefully ascribed the early rising of the Full Moon at that time of the year to the goodness of God, not doubting that he had ordered it so on purpose to give them an immediate supply of moon-light after sun-set for their greater conveniency in reaping the fruits of the earth.

[Sidenote: But remarkable according to the distance of places from it.]

In this instance of the harvest-moon, as in many others discoverable by Astronomy, the wisdom and beneficence of the Deity is conspicuous, who really ordered the course of the Moon so, as to bestow more or less light on all parts of the earth as their several circumstances and seasons render it more or less serviceable. About the Equator, where there is no variety of seasons, and the weather changes seldom, and at stated times, Moon-light is not necessary for gathering in the produce of the ground; and there the moon rises about 48 minutes later every day or night than on the former. At considerable distances from the Equator, where the weather and seasons are more uncertain, the autumnal Full Moons rise very soon after sun-set for several evenings together. At the polar circles, where the mild season is of very short duration, the autumnal Full Moon rises at Sun-set from the first to the third quarter. And at the Poles, where the Sun is for half a year absent, the winter Full moons shine constantly without setting from the first to the third quarter.

[Sidenote: The reason of this.]

It is soon said that all these Phenomena are owing to the different Angles made by the Horizon and different parts of the Moon’s orbit; and that the Moon can be full but once or twice in a year in those parts of her orbit which rise with the least angles. But to explain this subject intelligibly we must dwell much longer upon it.

[Sidenote: PLATE III.]

274. The [59]plane of the Equinoctial is perpendicular to the Earth’s Axis: and therefore, as the Earth turns round its Axis, all parts of the Equinoctial make equal Angles with the Horizon both at rising and setting; so that equal portions of it always rise or set in equal times. Consequently, if the Moon’s motion were equable, and in the Equinoctial, at the rate of 12 degrees from the Sun every day, as it is in her orbit, she would rise and set 48 minutes later every day than on the preceding: for 12 degrees of the Equinoctial rise or set in 48 minutes of time in all Latitudes.

[Sidenote: Fig. III.]

275. But the Moon’s motion is so nearly in the Ecliptic that we may consider her at present as moving in it. Now the different parts of the Ecliptic, on account of its obliquity to the Earth’s Axis, make very different Angles with the Horizon as they rise or set. Those parts or Signs which rise with the smallest Angles set with the greatest, and _vice versâ_. In equal times, whenever this Angle is least, a greater portion of the Ecliptic rises than when the Angle is larger; as may be seen by elevating the pole of a Globe to any considerable Latitude, and then turning it round its Axis in the Horizon. Consequently, when the Moon is in those Signs which rise or set with the smallest Angles, she rises or sets with the least difference of time; and with the greatest difference in those Signs which rise or set with the greatest Angles.

[Sidenote: Fig. III.

The different Angles made by the Ecliptic and Horizon.]

But, because all who read this Treatise may not be provided with Globes, though in this case it is requisite to know how to use them, we shall substitute the Figure of a Globe; in which _FUP_ is the Axis, ♋_TR_ the Tropic of Cancer, _LT_♑ the Tropic of Capricorn, ♋_EU_♑ the Ecliptic touching both the Tropics which are 47 degrees from each other, and _AB_ the Horizon. The Equator, being in the middle between the Tropics, is cut by the Ecliptic in two opposite points, which are the beginnings of ♈ Aries and ♎ Libra. _K_ is the Hour circle with its Index, _F_ the North pole of the Globe elevated to the Latitude of _London_[60], namely 51-1/2 degrees above the Horizon; and _P_ the South Pole depressed as much below it. Because of the oblique position of the Sphere in this Latitude, the Ecliptic has the high elevation _N_♋ above the Horizon, making the Angle _NU_♋ of 62 degrees with it when ♋ Cancer is on the Meridian, at which time ♎ Libra rises in the East. But let the Globe be turned half round its Axis, till ♑ Capricorn comes to the Meridian and ♈ Aries rises in the East, and then the Ecliptic will have the low elevation _NL_ above the Horizon making only an Angle _NUL_ of 15 degrees, with it; which is 47 degrees less than the former Angle, equal to the distance between the Tropics.

[Sidenote: Least and greatest, when.]

276. The smallest Angle made by the Ecliptic and Horizon is when Aries rises, at which time Libra sets: the greatest when Libra rises, at which time Aries sets. From the rising of Aries to the rising of Libra (which is twelve [61]Sidereal hours) the angle increases; and from the rising of Libra to the rising of Aries it decreases in the same proportion. By this article and the preceding, it appears that the Ecliptic rises fastest about Aries and slowest about Libra.

+------+-----------+--------+---------+ | | Signs | Rising | Setting | | | | Diff. | Diff. | | Days | +--------+---------+ | | Degrees | H. M. | H. M. | +------+-----------+--------+---------+ | 1 | ♋ 13 | 1 5 | 0 50 | | 2 | 26 | 1 10 | 0 43 | | 3 | ♌ 10 | 1 14 | 0 37 | | 4 | 23 | 1 17 | 0 32 | | 5 | ♍ 6 | 1 16 | 0 28 | | 6 | 19 | 1 15 | 0 24 | | 7 | ♎ 2 | 1 15 | 0 20 | | 8 | 15 | 1 15 | 0 18 | | 9 | 28 | 1 15 | 0 17 | | 10 | ♏ 12 | 1 15 | 0 22 | | 11 | 25 | 1 14 | 0 30 | | 12 | ♐ 8 | 1 13 | 0 39 | | 13 | 21 | 1 10 | 0 47 | | 14 | ♑ 4 | 1 4 | 0 56 | | 15 | 17 | 0 46 | 1 5 | | 16 | ♒ 1 | 0 40 | 1 8 | | 17 | 14 | 0 35 | 1 12 | | 18 | 27 | 0 30 | 1 15 | | 19 | ♓ 10 | 0 25 | 1 16 | | 20 | 23 | 0 20 | 1 17 | | 21 | ♈ 7 | 0 17 | 1 16 | | 22 | 20 | 0 17 | 1 15 | | 23 | ♉ 3 | 0 20 | 1 15 | | 24 | 16 | 0 24 | 1 15 | | 25 | 29 | 0 30 | 1 14 | | 26 | ♊ 13 | 0 40 | 1 13 | | 27 | 26 | 0 50 | 1 7 | | 28 | ♋ 9 | 1 0 | 1 58 | +------+-----------+--------+---------+

[Sidenote: Quantity of this Angle at London.]

277. On the Parallel of _London_, as much of the Ecliptic rises about Pisces and Aries in two hours as the Moon goes through in six days: and therefore whilst the Moon is in these Signs, she differs but two hours in rising for six days together; that is, 20 minutes later every day or night than on the preceding. But in fourteen days afterwards, the Moon comes to Virgo and Libra; which are the opposite Signs to Pisces and Aries; and then she differs almost four times as much in rising; namely, one hour and about fifteen minutes later every day or night than the former, whilst she is in these Signs; for by § 275 their rising Angle is at least four times as great as that of Pisces and Aries. The annexed Table shews the daily mean difference of the Moon’s rising and setting on the Parallel of _London_, for 28 days; in which time the Moon finishes her period round the Ecliptic, and gets 9 degrees into the same Sign from the beginning of which she set out. So it appears by the Table, that while the Moon is in ♍ and ♎ she rises an hour and a quarter later every day than the former; and differs only 24, 20, 18 or 17 minutes in setting. But, when she comes to ♓ and ♈, she is only 20 or 17 minutes later of rising; and an hour and a quarter later in setting.

278. All these things will be made plain by putting small patches on the Ecliptic of a Globe, as far from one another as the Moon moves from any Point of the celestial Ecliptic in 24 hours, which at a mean rate is [62]13-1/6 degrees; and then in turning the globe round, observe the rising and setting of the patches in the Horizon, as the Index points out the different times in the hour circle. A few of these patches are represented by dots at _0_ _1_ _2_ _3_ &c. on the Ecliptic, which has the position _LUI_ when Aries rises in the East; and by the dots _0_ _1_ _2_ _3_, &c. when Libra rises in the East, at which time the Ecliptic has the position _EU_♑: making an angle of 62 degrees with the Horizon in the latter case, and an angle of no more than 15 degrees with it in the former; supposing the Globe rectified to the Latitude of _London_.