Part 17

CHAP. XV.

_The Moon’s surface mountainous: Her Phases described: Her path, and the paths of Jupiter’s Moons delineated: The proportions of the Diameters of their Orbits, and those of Saturn’s Moons, to each other; and to the Diameter of the Sun._

[Sidenote: PL. VII.

The Moon’s surface mountainous.]

252. By looking at the Moon with an ordinary telescope we perceive that her surface is diversified with long tracts of prodigious high mountains and deep cavities. Some of her mountains, by comparing their height with her diameter (which is 2180 miles) are found to be three times higher than the highest hills on our Earth. This ruggedness of the Moon’s surface is of great use to us, by reflecting the Sun’s light to all sides: for if the Moon were smooth and polished like a looking-glass, or covered with water, she could never distribute the Sun’s light all round; only in some positions she would shew us his image, no bigger than a point, but with such a lustre as would be hurtful to our eyes.

[Sidenote: Why no hills appear on her edge.]

253. The Moon’s surface being so uneven, many have wondered why her edge appears not jagged, as well as the curve bounding the light and dark places. But if we consider, that what we call the edge of the Moon’s Disc is not a single line set round with mountains, in which case it would appear irregularly indented, but a large Zone having many mountains lying behind one another from the observer’s eye, we shall find that the mountains in some rows will be opposite to the vales in others; and so fill up the inequalities as to make her appear quite round: just as when one looks at an orange, although it’s roughness be very discernible on the side next the eye, especially if the Sun or a Candle shines obliquely on that side, yet the line terminating the visible part still appears smooth and even.

[Sidenote: The Moon has no twilight.

Fig. I.]

254. As the Sun can only enlighten that half of the Earth which is at any moment turned towards him, and being withdrawn from the opposite half leaves it in darkness; so he likewise doth to the Moon: only with this difference, that the Earth being surrounded by an Atmosphere, and the Moon having none, we have twilight after the Sun sets; but the Lunar Inhabitants have an immediate transition from the brightest Sun-shine to the blackest darkness § 177. For, let _tkrsw_ be the Earth, and _A_, _B_, _C_, _D_, _E_, _F_, _G_, _H_ the Moon in eight different parts of her Orbit. As the Earth turns round its Axis, from west to east, when any place comes to _t_ the twilight begins there, and when it revolves from thence to _r_ the Sun _S_ rises; when the place comes to _s_ the Sun sets, and when it comes to _w_ the twilight ends. But as the Moon turns round her Axis, which is only once a month, the moment that any point of her surface comes to _r_ (see the Moon at _G_) the Sun rises there without any previous warning by twilight; and when the same point comes to _s_ the Sun sets, and that point goes into darkness as black as at midnight.

[Sidenote: The Moon’s Phases.]

255. The Moon being an opaque spherical body, (for her hills take off no more from her roundness than the inequalities on the surface of an orange takes off from its roundness) we can only see that part of the enlightened half of her which is towards the Earth. And therefore, when the Moon is at _A_, in conjunction with the Sun _S_, her dark half is towards the Earth, and she disappears as at _a_, there being no light on that half to render it visible. When she comes to her first Octant at _B_, or has gone an eighth part of her orbit from her Conjunction, a quarter of her enlightened side is towards the Earth, and she appears horned as at _b_. When she has gone a quarter of her orbit from between the Earth and Sun to _C_, she shews us one half of her enlightened side as at _c_, and we say, she is a quarter old. At _D_ she is in her second Octant, and by shewing us more of her enlightened side she appears gibbous as at _d_. At _E_ her whole enlightened side is towards the Earth, and therefore she appears round as at _e_, when we say, it is Full Moon. In her third Octant at _F_, part of her dark side being towards the Earth, she again appears gibbous, and is on the decrease, as at _f_. At _G_ we see just one half of her enlightened side, and she appears half decreased, or in her third Quarter, as at _g_. At _H_ we only see a quarter of her enlightened side, being in her fourth Octant, where she appears horned as at _h_. And at _A_, having compleated her course from the Sun to the Sun again, she disappears; and we say, it is New Moon. Thus in going from _A_ to _E_ the Moon seems continually to increase; and in going from _E_ to _A_, to decrease in the same proportion; having like Phases at equal distances from _A_ or _E_, but as seen from the Sun _S_, she is always Full.

[Sidenote: The Moon’s Disc not always quite round when full.]

256. The Moon appears not perfectly round when she is Full in the highest or lowest part of her Orbit, because we have not a direct view of her enlightened side at that time. When Full in the highest part of her orbit, a small deficiency appears on her lower edge; and the contrary when Full in the lowest part of her Orbit.

[Sidenote: The Phases of the Earth and Moon contrary.]

257. ’Tis plain by the Figure, that when the Moon changes to the Earth, the Earth appears Full to the Moon; and _vice versâ_. For when the Moon is at _A_, _New_ to the Earth, the whole enlightened side of the Earth is towards the Moon: and when the Moon is at _E_, _Full_ to the Earth, it’s dark side is towards her. Hence a _New Moon_ answers to a _Full Earth_, and a _Full Moon_ to a _New Earth_. The _Quarters_ are also reversed to each other.

[Sidenote: An agreeable Phenomenon.]

258. Between the third Quarter and Change, the Moon is frequently visible in the forenoon, even when the Sun shines; and then she affords us an opportunity of seeing a very agreeable appearance, wherever we find a globular stone above the level of the eye, as suppose on the top of a gate. For, if the Sun shines on the stone, and we place ourselves so as the upper part of the stone may just seem to touch the point of the Moon’s lowermost horn, we shall then see the enlightened part of the stone exactly of the same shape with the Moon; horned as she is, and inclining the same way to the Horizon. The reason is plain; for the Sun enlightens the stone the same way as he does the Moon: and both being Globes, when we put ourselves into the above situation, the Moon and stone have the same position to our eyes; and therefore we must see as much of the illuminated part of the one as of the other.

[Sidenote: The nonagesimal Degree, what.]

259. The position of the Moon’s Cusps, or a right line touching the points of her horns, is very differently inclined to the Horizon at different hours of the same days of her age. Sometimes she stands, as it were, upright on her lower horn, and then such a line is perpendicular to the Horizon: when this, happens, she is in what the Astronomers call _the Nonagesimal Degree_; which is the highest point of the Ecliptic above the Horizon at that time, and is 90 degrees from both sides of the Horizon where it is then cut by the Ecliptic. But this never happens when the Moon is on the Meridian, except when she is at the very beginning of Cancer or Capricorn.

[Sidenote: How the inclination of the Ecliptic may be found by the position of the Moon horns.

PL. VII.]

260. The inclination of that part of the Ecliptic to the Horizon in which the Moon is at any time when horned, may be known by the position of her horns; for a right line touching their points is perpendicular to the Ecliptic. And as the Angle that the Moon’s orbit makes with the Ecliptic can never raise her above, nor depress her below the Ecliptic, more than two minutes of a degree, as seen from the Sun; it can have no sensible effect upon the position of her horns. Therefore, if a Quadrant be held up, so as one of it’s edges may seem to touch the Moon’s horns, the graduated side being kept towards the eye, and as far from the eye as it can be conveniently held, the arc between the Plumb-line and that edge of the Quadrant which seems to touch the Moon’s horns will shew the inclination of that part of the Ecliptic to the Horizon. And the arc between the other edge of the Quadrant and Plumb-line will shew the inclination of the Moon’s horns to the Horizon at that time also.

[Sidenote: Fig. I.

Why the Moon appears as big as the Sun.]

261. The Moon generally appears as large as the Sun; for the Angle _vkA_, under which the Moon is seen from the Earth, is the same with the Angle _LkM_, under which the Sun is seen from it. And therefore the Moon may hide the Sun’s whole Disc from us, as she sometimes does in solar Eclipses. The reason why she does not eclipse the Sun at every Change shall be explained afterwards. If the Moon were farther from the Earth as at _a_, she could never hide the whole of the Sun from us; for then she would appear under the Angle _NkO_, eclipsing only that part of the Sun which lies between _N_ and _O_: were she still further from the Earth, as at _X_, she would appear under the small Angle _TkW_, like a spot on the Sun, hiding only the part _TW_ from our sight.

[Sidenote: A proof of the Moon’s turning round her Axis.]

262. The Moon turns round her Axis in the time that she goes round her orbit; which is evident from hence, that a spectator at rest, without the periphery of the Moon’s orbit, would see all her sides turned regularly towards him in that time. She turns round her Axis from any Star to the same Star again in 27 days 8 hours; from the Sun to the Sun again in 29-1/2 days: the former is the length of her sidereal day, and the latter the length of her solar day. A body moving round the Sun would have a solar day in every revolution, without turning on it’s Axis; the same as if it had kept all the while at rest, and the Sun moved round it: but without turning round it’s Axis it could never have one sidereal day, because it would always keep the same side towards any given Star.

[Sidenote: Her periodical and synodical Revolution.]

263. If the Earth had no annual motion, the Moon would go round it so as to compleat a Lunation, a sidereal, and a solar day, all in the same time. But, because the Earth goes forward in it’s orbit while the Moon goes round the Earth in her orbit, the Moon must go as much more than round her orbit from Change to Change in compleating a solar day as the Earth has gone forward in it’s orbit during that time, _i. e._ almost a twelfth part of a Circle.

[Sidenote: Familiarly represented.

A Table shewing the times that the hour and minute hands of a watch are in conjunction.

A machine for shewing the motions of the Sun and Moon.

PL. VII.]

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.]