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.

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.

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 291⁄2 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.

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.

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 491⁄11 fifths past I where they meet; at the second conjunction it is 10 minutes 54 seconds 32 thirds 43 fourths 381⁄2 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 121⁄3 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 121⁄3 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 291⁄2 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 271⁄3 days, which is her periodical revolution; but from the Sun to the Sun again, or from Change to Change, in 291⁄2 days, which is her synodical revolution.

Conj.H.M.S.ʺʹʺʺv pts.
1I5271621491⁄11
2II10543243382⁄11
3III1621495273⁄11
4IIII2149527164⁄11
5V2716214955⁄11
6VI32433810546⁄11
7VII38105432437⁄11
8VIII43381054328⁄11
9IX4952716219⁄11
10X543243381010⁄11
11XII00000