Chapter X: Part 10

264. The Moon’s periodical and synodical revolution may be familiarly represented by the motions of the hour and minute hands of a watch round it’s dial-plate, which is divided into 12 equal parts or hours, as the Ecliptic is divided into 12 Signs, and the year into 12 months. Let us suppose these 12 hours to be 12 months, the hour hand the Sun, and the minute hand the Moon; then will the former go round once in a year, and the latter once in a month; but the Moon, or minute hand must go more than round from any point of the Circle where it was last conjoined with the Sun, or hour hand, to overtake it again: for the hour hand being in motion, can never be overtaken by the minute hand at that point from which they started at their last conjunction. The first column of the annexed Table shews the number of conjunctions which the hour and minute hand make whilst the hour hand goes once round the dial-plate; and the other columns shew the times when the two hands meet at every conjunction. Thus, suppose the two hands to be in conjunction at XII, as they always are; then, at the first following conjunction it is 5 minutes 27 seconds 16 thirds 21 fourths 49-1/11 fifths past I where they meet; at the second conjunction it is 10 minutes 54 seconds 32 thirds 43 fourths 38-1/2 fifths past II; and so on. This, though an easy illustration of the motions of the Sun and Moon, is not precise as to the times of their conjunctions; because, while the Sun goes round the Ecliptic, the Moon makes 12-1/3 conjunctions with him; but the minute hand of a watch or clock makes only 11 conjunctions with the hour hand in one period round the dial-plate. But if, instead of the common wheel-work at the back of the dial-plate, the Axis of the minute hand had a pinion of 6 leaves turning a wheel of 40, and this last turning the hour hand, in every revolution it makes round the dial-plate the minute hand would make 12-1/3 conjunctions with it; and so would be a pretty device for shewing the motions of the Sun and Moon; especially, as the slowest moving hand might have a little Sun fixed on it’s point, and the quickest a little Moon. Besides, the plate, instead of hours and quarters, might have a Circle of months, with the 12 Signs and their Degrees; and if a plate of 29-1/2 equal parts for the days of the Moon’s age were fixed to the Axis of the Sun-hand, and below it, so as the Sun always kept at the 1/2 day of that plate, the Moon-hand would shew the Moon’s age upon that plate for every day pointed out by the Sun-hand in the Circle of months; and both Sun and Moon would shew their places in the Ecliptic: for the Sun would go round the Ecliptic in 365 Days and the Moon in 27-1/3 days, which is her periodical revolution; but from the Sun to the Sun again, or from Change to Change, in 29-1/2 days, which is her synodical revolution.

+-----+-------------------------------+ |Conj.| H. M. S. ʺʹ ʺʺ v p^{ts}. | +-----+-------------------------------+ | 1 | I 5 27 16 21 49-1/11 | | 2 | II 10 54 32 43 38-2/11 | | 3 | III 16 21 49 5 27-3/11 | | 4 | IIII 21 49 5 27 16-4/11 | | 5 | V 27 16 21 49 5-5/11 | | 6 | VI 32 43 38 10 54-6/11 | | 7 | VII 38 10 54 32 43-7/11 | | 8 | VIII 43 38 10 54 32-8/11 | | 9 | IX 49 5 27 16 21-9/11 | | 10 | X 54 32 43 38 10-10/11| | 11 | XII 0 0 0 0 0 | +-----+-------------------------------+

[Sidenote: The Moon’s motion thro’ open space described.]

265. If the Earth had no annual motion, the Moon’s motion round the Earth, and her track in absolute space, would be always the same[58]. But as the Earth and Moon move round the Sun, the Moon’s real path in the Heavens is very different from her path round the Earth: the latter being in a progressive Circle, and the former in a curve of different degrees of concavity, which would always be the same in the same parts of the Heavens, if the Moon performed a compleat number of Lunations in a year.

[Sidenote: An idea of the Earth’s path and the Moon’s.]

266. Let a nail in the end of the axle of a chariot-wheel represent the Earth, and a pin in the nave the Moon; if the body of the chariot be propped up so as to keep that wheel from touching the ground, and the wheel be then turned round by hand, the pin will describe a Circle both round the nail and in the space it moves through. But if the props be taken away, the horses put to, and the chariot driven over a piece of ground which is circularly convex; the nail in the axle will describe a circular curve, and the pin in the nave will still describe a circle round the progressive nail in the axle, but not in the space through which it moves. In this case, the curve described by the nail will resemble in miniature as much of the Earth’s annual path round the Sun, as it describes whilst the Moon goes as often round the Earth as the pin does round the nail: and the curve described by the nail will have some resemblance of the Moon’s path during so many Lunations.

[Sidenote: Fig. II.

PL. VII.]

Let us now suppose that the Radius of the circular curve described by the nail in the axle is to the Radius of the Circle which the pin in the nave describes round the axle as 337-1/2 to 1; which is the proportion of the Radius or Semidiameter of the Earth’s Orbit to that of the Moon’s; or of the circular curve _A_ 1 2 3 4 5 6 7 _B_ &c. to the little Circle _a_; and then, whilst the progressive nail describes the said curve from _A_ to _E_, the pin will go once round the nail with regard to the center of it’s path, and in doing so, will describe the curve _abcde_. The former will be a true representation of the Earth’s path for one Lunation, and the latter of the Moon’s for that time. Here we may set aside the inequalities of the Moon’s Moon, and also the Earth’s moving round it’s common center of gravity and the Moon’s: 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 plain upright surface all the way, and leave their tracks visible. And if the chariot should be 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 center of the circular curve described by the pin in the Axle; as the Moon’s path is always concave to the Sun in the center of the Earth’s annual Orbit.

[Sidenote: Proportion of the Moon’s Orbit to the Earth’s.]

In this Diagram, the thickest curve line _ABCD_, 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 east; 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 _abcdef_ represents 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 will be described, with the rest of my Astronomical machinery, in the last Chapter. The Sun is supposed to be in the center of the curve _A 1 2 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 Earth’s path in this scheme, that the Radius of the Moon’s Orbit in the Heavens bears to the Radius of the Earth’s annual path round the Sun; that is, as 240,000 to 81,000,000, or as 1 to 337-1/2.

[Sidenote: Fig. II.]

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 Earth describes the curve _ab_; and is in her first Quarter at _b_ when the Earth is at _B_. As the Earth describes the curve _B 8 9 10 11 12 13 14_ the Moon describes the curve _bc_; and is opposite to the Sun at _c_, when the Earth is at _C_. Whilst 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_. Once more, whilst 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 center of that Circle.

[Sidenote: 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_.

[Sidenote: 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.

[Sidenote: 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.

[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_.

279. Having rectified the Globe, turn it until the patch at _0_, about the beginning of ♓ Pisces on the half _LUI_ of the Ecliptic, comes to the Eastern side of the Horizon; and then keeping the ball steady, set the hour Index to XII, because _that_ hour may perhaps be more easily remembred than any other. Then, turn the Globe round westward, and in that time, suppose the patch _0_ to have moved thence to _1_, 13-1/6 degrees, whilst the Earth turns once round its Axis, and you will see that _1_ rises only about 20 minutes later than _0_ did on the day before. Turn the Globe round again, and in that time suppose the same patch to have moved from _1_ to _2_; and it will rise only 20 minutes later by the hour-index than it did at _1_ on the day or turn before. At the end of the next turn, suppose the patch to have gone from _2_ to _3_ at _U_, and it will rise 20 minutes later than it did at _2_. And so on for six turns, in which time there will scarce be two hours difference: Nor would there have been so much if the 6 degrees of the Sun’s motion in that time had been allowed for. At the first Turn the patch rises south of the East, at the middle Turn due East, and at the last Turn north of the East. But these patches will be 9 hours of setting on the western side of the Horizon, which shews that the Moon will be so much later of setting in that week in which she moves through these two Signs. The cause of this difference is evident; for Pisces and Aries make only an Angle of 15 degrees with the Horizon when they rise; but they make an Angle of 62 degrees with it when they set § 275. As the Signs Taurus, Gemini, Cancer, Leo, Virgo, and Libra rise successively, the Angle increases gradually which they make with the Horizon; and decreases in the same proportion as they set. And for that reason, the Moon differs gradually more in the time of her rising every day whilst she is in these Signs, and less in her setting: After which, through the other six Signs, _viz._ Scorpio, Sagittary, Capricorn, Aquarius, Pisces, and Aries, the rising difference becomes less every day, until it be at the least of all, namely, in Pisces and Aries.

280. The Moon goes round the Ecliptic in 27 days 8 hours; but not from Change to Change in less than 29 days 12 hours: so that she is in Pisces and Aries at least once in every Lunation, and in some Lunations twice.

[Sidenote: Why the Moon is always Full in different Signs.

Her periodical and synodical Revolution exemplified.]

281. If the Earth had no annual motion, the Sun would never appear to shift his place in the Ecliptic. And then every New Moon would fall in the same Sign and degree of the Ecliptic, and every Full Moon in the opposite: for the Moon would go precisely round the Ecliptic from Change to Change. So that if the Moon was once Full in Pisces, or Aries, she would always be Full when she came round to the same Sign and Degree again. And as the Full Moon rises at Sun-set (because when any point of the Ecliptic sets the opposite point rises) she would constantly rise within two hours of Sun-set during the week in which she were Full. But in the time that the Moon goes round the Ecliptic from any conjunction or opposition, the Earth goes almost a Sign forward; and therefore the Sun will seem to go as far forward in that time, namely 27-1/2 degrees: so that the Moon must go 27-1/2 degrees more than round; and as much farther as the Sun advances in that interval, which is 2-1/15 degrees, before she can be in conjunction with, or opposite to the Sun again. Hence it is evident, that there can be but one conjunction or opposition of the Sun and Moon in a year in any particular part of the Ecliptic. This may be familiarly exemplified by the hour and minute hands of a watch, which are never in conjunction or opposition in that part of the dial-plate where they were so last before. And indeed if we compare the twelve hours on the dial-plate to the twelve Signs of the Ecliptic, the hour-hand to the Sun and the minute-hand to the Moon, we shall have a tolerably near resemblance in miniature to the motions of our great celestial Luminaries. The only difference is, that whilst the Sun goes once round the Ecliptic the Moon makes 12-1/3 conjunctions with him: but whilst the hour-hand goes round the dial-plate the minute-hand makes only 11 conjunctions with it; because the minute hand moves slower in respect of the hour-hand than the Moon does with regard to the Sun.

[Sidenote: The Harvest and Hunter’s Moon.]

282. As the Moon can never be full but when she is opposite to the Sun, and the Sun is never in Virgo and Libra but in our autumnal months, ’tis plain that the Moon is never full in the opposite Signs, Pisces and Aries, but in these two months. And therefore we can have only two Full Moons in the year, which rise so near the time of Sun-set for a week together as above-mentioned. The former of these is called the _Harvest Moon_, and the latter the _Hunter’s Moon_.

[Sidenote: Why the Moon’s regular rising is never perceived but in
Harvest.]

283. Here it will probably be asked, why we never observe this remarkable rising of the Moon but in harvest, since she is in Pisces and Aries at least twelve times in the year besides; and must then rise with as little difference of time as in harvest? The answer is plain: for in winter these Signs rise at noon; and being then only a Quarter of a Circle distant from the Sun, the Moon in them is in her first Quarter: but when the Sun is above the Horizon the Moon’s rising is neither regarded nor perceived. In spring these Signs rise with the Sun because he is then in them; and as the Moon changeth in them at that time of the year, she is quite invisible. In summer they rise about mid-night, and the Sun being then three Signs, or a Quarter of a Circle before them, the Moon is in them about her third Quarter; when rising so late, and giving but very little light, her rising passes unobserved. And in autumn, these Signs being opposite to the Sun, rise when he sets, with the Moon in opposition, or at the Full, which makes her rising very conspicuous.

284. At the Equator, the North and South Poles lie in the Horizon; and therefore the Ecliptic makes the same Angle southward with the Horizon when Aries rises as it does northward when Libra rises. Consequently, as the Moon at all the fore-mentioned patches rises and sets nearly at equal Angles with the Horizon all the year round; and about 48 minutes later every day or night than on the preceding, there can be no particular Harvest Moon at the Equator.

285. The farther that any place is from the Equator, if it be not beyond the Polar Circle, the Angle gradually diminishes which the Ecliptic and Horizon make when Pisces and Aries rise; and therefore when the Moon is in these Signs she rises with a nearly proportionable difference later every day than on the former; and is for that reason the more remarkable about the Full, until we come to the Polar Circles, or 66 degrees from the Equator; in which Latitude the Ecliptic and Horizon become coincident, every day for a moment, at the same sidereal hour (or 3 minutes 56 seconds sooner every day than the former) and the very next moment one half of the Ecliptic containing Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini rises, and the opposite half sets. Therefore, whilst the Moon is going from the beginning of Capricorn to the beginning of Cancer, which is almost 14 days, she rises at the same sidereal hour; and in autumn just at Sun-set, because all that half of the Ecliptic in which the Sun is at that time sets at the same sidereal hour, and the opposite half rises: that is, 3 minutes 56 seconds, of mean solar time, sooner every day than on the day before. So whilst the Moon is going from Capricorn to Cancer she rises earlier every day than on the preceding; contrary to what she does at all places between the polar Circles. But during the above fourteen days, the Moon is 24 sidereal hours later in setting; for the six Signs which rise all at once on the eastern side of the Horizon are 24 hours in setting on the western side of it: as any one may see by making chalk-marks at the beginning of Capricorn and of Cancer, and then, having elevated the Pole 66-1/2 degrees, turn the Globe slowly round it’s Axis, and observe the rising and setting of the Ecliptic. As the beginning of Aries is equally distant from the beginning of Cancer and of Capricorn, it is in the middle of that half of the Ecliptic which rises all at once. And when the Sun is at the beginning of Libra, he is in the middle of the other half. Therefore, when the Sun is in Libra and the Moon in Capricorn, the Moon is a Quarter of a Circle before the Sun; opposite to him, and consequently full in Aries, and a Quarter of a Circle behind him when in Cancer. But when Libra rises Aries sets, and all that half of the Ecliptic of which Aries is the middle. And therefore, at that time of the year the Moon rises at Sun-set from her first to her third Quarter.

[Sidenote: The Harvest Moons regular on both sides of the Equator.]

286. In northern Latitudes, the autumnal Full Moons are in Pisces and Aries; and the vernal Full Moons in Virgo and Libra: in southern Latitudes just the reverse because the seasons are contrary. But Virgo and Libra rise at as small Angles with the Horizon in southern Latitudes as Pisces and Aries do in the northern; and therefore the Harvest Moons are just as regular on one side of the Equator as on the other.

287. As these Signs which rise with the least Angles set with the greatest, the vernal Full Moons differ as much in their times of rising every night as the autumnal Full Moons differ in their times of setting; and set with as little difference as the autumnal Full Moons rise: the one being in all cases the reverse of the other.

[Sidenote: The Moon’s Nodes.]

288. Hitherto, for the sake of plainness, we have supposed the Moon to move in the Ecliptic, from which the Sun never deviates. But the orbit in which the Moon really moves is different from the Ecliptic: one half being elevated 5-1/3 degrees above it, and the other half as much depressed below it. The Moon’s orbit therefore intersects the Ecliptic in two points diametrically opposite to each other: and these intersections are called the _Moon’s Nodes_. So the Moon can never be in the Ecliptic but when she is in either of her Nodes, which is at least twice in every course from Change to Change, and sometimes thrice. For, as the Moon goes almost a whole Sign more than round her Orbit from Change to Change; if she passes by either Node about the time of Change, she will pass by the other in about fourteen days after, and come round to the former Node two days again before the next Change. That Node from which the Moon begins to ascend northward, or above the Ecliptic, in northern Latitudes, is called the _Ascending Node_; and the other the _Descending Node_, because the Moon, when she passes by it, descends below the Ecliptic southward.

289. The Moon’s oblique motion with regard to the Ecliptic causes some difference in the times of her rising and setting from what is already mentioned. For whilst she is northward of the Ecliptic, she rises sooner and sets later than if she moved in the Ecliptic: and when she is southward of the Ecliptic she rises later and sets sooner. This difference is variable even in the same Signs, because the Nodes shift backward about 19-2/3 degrees in the Ecliptic every year; and so go round it contrary to the order of Signs in 18 years 225 days.

290. When the Ascending Node is in Aries, the southern half of the Moon’s Orbit makes an Angle of 5-1/3 degrees less with the Horizon than the Ecliptic does, when Aries rises in northern Latitudes: for which reason the Moon rises with less difference of time whilst she is in Pisces and Aries than there would be if she kept in the Ecliptic. But in 9 years and 112 days afterward, the Descending Node comes to Aries; and then the Moon’s Orbit makes an Angle 5-1/3 degrees greater with the Horizon when Aries rises, than the Ecliptic does at that time; which causes the Moon to rise with greater difference of time in Pisces and Aries than if she moved in the Ecliptic.

291. To be a little more particular, when the Ascending Node is in Aries, the Angle is only 9-2/3 degrees on the parallel of _London_ when Aries rises. But when the Descending Node comes to Aries, the Angle is 20-1/3 degrees; this occasions as great a difference of the Moon’s rising in the same Signs every 9 years, on the parallel of _London_, as there would be on two parallels 10-2/3 degrees from one another, if the Moon’s course were in the Ecliptic. The following Table shews how much the obliquity of the Moon’s Orbit affects her rising and setting on the parallel of _London_ from the 12th to the 18th day of her age; supposing her to be Full at the autumnal Equinox; and then, either in the Ascending Node, highest part of her Orbit, Descending Node, or lowest part of her Orbit. _M_ signifies morning, _A_ afternoon; and the line at the foot of the Table shews a week’s difference in rising and setting.

+--------+---------------+---------------+---------------+---------------+ | | Full in her | In the | Full in her | In the lowest | | | Ascending | highest part | Descending | part of her | | | node. | of her Orbit. | node. | Orbit. | | Moon’s +---------------+-------+-------+-------+-------+-------+-------+ | Age | Rises | Sets | Rises | Sets | Rises | Sets | Rises | Sets | | | at | at | at | at | at | at | at | at | | | H. M. | H. M. | H. M. | H. M. | H. M. | H. M. | H. M. | H. M. | +--------+-------+-------+-------+-------+-------+-------+-------+-------+ | | _A_ | _M_ | _A_ | _M_ | _A_ | _M_ | _A_ | _M_ | | 12 | 5 15 | 3 20 | 4 30 | 3 15 | 4 32 | 3 40 | 5 16 | 3 0 | | 13 | 5 32 | 4 25 | 4 50 | 4 45 | 5 15 | 4 20 | 6 0 | 4 15 | | 14 | 5 48 | 5 30 | 5 15 | 6 0 | 5 45 | 5 40 | 6 20 | 5 28 | | 15 | 6 5 | 7 0 | 5 42 | 7 20 | 6 15 | 6 56 | 6 45 | 6 32 | | 16 | 6 20 | 8 15 | 6 2 | 8 35 | 6 46 | 8 0 | 7 8 | 7 45 | | 17 | 6 36 | 9 12 | 6 26 | 9 45 | 7 18 | 9 15 | 7 30 | 9 15 | | 18 | 6 54 | 10 30 | 7 0 | 10 40 | 8 0 | 10 20 | 7 52 | 10 0 | +--------+-------+-------+-------+-------+-------+-------+-------+-------+ | Dif. | 1 39 | 7 10 | 2 30 | 7 25 | 3 28 | 6 40 | 2 36 | 7 0 | +--------+-------+-------+-------+-------+-------+-------+-------+-------+