The lines marked n, are the meridians; and the daily rotation of the earth is from or toward the left hand.

The bright half of each circle represents day, and the dark one night, sunrise being at the line of separation on the right, noon at the letter n, sunset at the separation of the light, and darkness on the left, and midnight at the middle of the dark side. The dot at the middle of each right-hand half of the light side is nine o'clock in the morning, and that at the top of the left is three o'clock in the afternoon; the smaller circle represents the moon; the bright sides, the half upon which the sun shines, and which shines itself by reflecting the sun's light; and the shaded side is that upon which the sun does not shine, and which is, in consequence, dark, does not shine, or annually appears itself.

The small and large circles which have the same number correspond to each other, as 0 to 0, 1 to 1, and so on; but as it happens that the moon occupies at the first quarter, almost the same position in space which the earth occupies at the first half-quarter, and as the moon occupies at the third quarter nearly the same position which the earth occupies at the following half-quarter, the intermediate points, between large circles 0 and 1, and 3 and 4, are marked double; namely, and 31 for the earth, and 1 and 3 for the noon.

From 0, in the line A, B, to 4 in the line C, D, through the small circles , 1, 11, 2, 21, 3, and 31, is the true path of the moon, in absolute space, during one lunation; and these small circles are the positions of the moon at the respective half quarters.

The centres of the earth and moon, in all the corresponding situations, are joined by straight lines; namely, from 0 to 0, by part of A, B, from 1 to 1, by part of the earth's orbit, from 2 to 2, by the part of the line or 2, from 3 to 3, by part of the earth's orbit, from 4 to 4, by part of the line C, D, and from to, 1 to 1, 2 to 24, and 31 to 34, by separate lines.

All these lines join the two orbits, and they are all nearly of the same length.

The lines o, o, separate the side of the moon, which is visible to the earth, from the side which is invisible.

By observation, it is found that the moon has always the same side toward the earth, so that the two sides of the moon, separated by the lines o, o, are always the same. But the side of o, o, which is turned toward the earth, at O and at 4, is turned toward the sun at 2; therefore, in the progress from 0 to 4, the moon turns once round its centre.

The figure shews only one half of the earth and moon in all the positions ; but if another be made exactly on the opposite side of the paper, or if the reader will look through from the other side to this one, the two views will represent the whole.

From the mere inspection of this figure the following inferences may be drawn.

1. That at the beginning and end of a lunation, the moon is in the same part of the heavens, with regard to east and west, as the sun is, of course, directly south at 12 o'clock mid-day, has its dark side turned directly to the earth, and is, of course, invisible.

2. That at the end of the first half quarter, the moon is 45°, or three hours in time, eastward of the sun, and one-fourth of the light side, and three-fourths of the dark side, are turned toward the earth; and that in this situation the moon will come to the south, and also set three hours after the sun, will have its western side as illuminated, convex or projecting, and its eastern side concave or hollow. As both the boundaries of these extend to the half of the moon's circumference, the former will be a semicircle, and the latter half an ellipse, cut off by the longer diameter, and thus the moon will appear what we call horned.

3. That at the end of the first quarter the moon will appear 90°, or six hours in time, eastward of the sun; will be six hours later in coming to the south and setting, and will have one half of its illuminated side turned toward the sun, the western half of that boundary being a semicircle, and the eastern a straight line.

4. That at the end of the third half quarter the moon will be 135°, or nine hours in time, eastward of the sun, and will be nine hours later in coming to the south and setting; that three-fourths of its light side will be turned toward the earth, and that the western boundary a semi-ellipse, with its convexity toward the dark side.

5. That at the end of the second quarter, or which is the same thing, at half of the entire lunation, the moon will be 180°, or twelve hours in time, eastward of the sun; that is, will be in exactly the opposite quarter of the heavens; that it will be north when the sun is south, east when the sun is west, and so on: and thus the whole of its illuminated side will be turned toward the earth. It will then appear a complete circle, or what we term full moon.

6. That at the end of the fifth half quarter, the moon will be 250° east, or 135° west of the sun, that it will rise, come to the south, or set, fifteen hours after, or nine hours before the sun, that three-fourths of its illuminated side will be turned toward the sun, and the eastern half of the illuminated part will be a semi-circle, and the western part a semi-ellipse, convex toward the west.

7. That at the end of the third quarter, the moon will be 270° east, or 90° west of the sun, that it will rise, come to the south, or set, eighteen hours later, or six hours earlier than the sun, that one half of its illuminated side will be turned toward the sun, and that the boundary of that half will be a semicircle toward the east, and a straight line toward the west.

8. That at the end of the seventh half quarter, the moon will be 315° east, or 45° west of the sun; that it will rise, or come to the south, twentyone hours later, or three hours earlier than the sun; that one-fourth of

its illuminated side will be turned toward the earth; and that the eastern boundary will be a semicircle, and the western one a semi-ellipse, convex toward the east; and thus the moon will appear of the same magnitude as at the end of the first half quartering; the points, or horns, or cusps, as they are called, will be directed toward the west, whereas, at the end of the first half quarter, they are directed toward the east.

9. That at the end of the fourth quarter, the moon will be in precisely the same situation, with regard to the sun, as it was at the beginning of the first quarter.

If the moon's orbit were in the same plane with the ecliptic, the sun and moon would always be in precisely the same part of the heavens at every new moon, in precisely opposite parts of the heavens, at every full moon, and in precisely the same circle of the heavens perpetually, so that although the times of their rising and setting would be continually varying throughout the month, the time of their presence, above or below the horizon, would be the same in every twenty-four hours; that is, supposing, also, that the axis of the earth's diurnal motion were at right angles to the ecliptic. But, in consequence of the obliquity of that axis, and in consequence, also, of the obliquity of the moon's orbit to the ecliptic, great apparent inequalities are introduced.

But the variation produced by the obliquity of the ecliptic is not all, for the orbit of the moon, as has been said, forms an angle of 5° 9 3′′ with that; and the two intersect each other at two points, called the nodes of the moon's orbit. These nodes have a motion towards all points in the heavens, retrograde on in the direction opposite to that in which the moon moves. This motion is completed in between 18 and 19 years, or in a little more than 6793 days.

In consequence of the obliquity of the ecliptic to the axis of the earth's revolution, the angle which the ecliptic makes with the horizon, at the earth's surface, must be continually changing; and, therefore, the points of the ecliptic and equator, which are intersected by the same meridian, will not come to the horizon at the same time, except at the equinoctial points. The half of the ecliptic which leans towards the horizon will, of course, make a smaller angle with that than the equator does; and the portion which leans from the equator will make a greater angle. In consequence of this, a much greater length of the ecliptic will come to the horizon in the same time, in the first of these positions than there does in the last. In the northern hemisphere, when the beginning of Aries is at the east side of the horizon, the horizon and ecliptic must nearly coincide: and when the first of Libra is at the east side of the horizon, the horizon and ecliptic make the greatest angle possible. Now, the motion of the moon in the ecliptic is nearly uniform, at the rate of about 12° eastward, in twenty-four hours. Therefore, when the moon appears at or near the beginning of Aries, the difference of time between its rising one day, and the following will

be the smallest possible; and when the moon is in the first of Libra, the time between its rising the one day and the following, will be the greatest possible. The full moon is always opposite the sun; and the sun appears in Aries, in March, and in Libra, in September; consequently the full moon must appear in Libra, in March, and in Aries, in September. Therefore, if there happen to be a full moon about the 22d of September, that is, about the autumnal equinox, that full moon must rise for several successive nights at very nearly the same hour. This phenomenon is called the HARVEST MOON, and its uniformity is a maximum when the moon is then so situated that its orbit makes the least possible angle with the horizon.

Although the moon performs a revolution round an ellipse, having the same mean diameter as twice her distance from the earth, in 27 days, 7 hours, and 4 minutes, as has been stated; yet, in consequence of the motion of the earth extending this ellipse into the epicycloid represented in the figure, the whole period from one change to another is, on an average, 29 days, 12 hours, 44 minutes, 12 seconds, very nearly.

The surface of the moon is exceedingly irregular; the substance of which it is composed is denser than the earth, it possesses an atmosphere, and there are mountains upon it of great elevation, and some of them volcanic; but whether it contains, or indeed can contain, any substance like that water which covers so much of the earth's surface, is a point upon which astronomers are not agreed.

The motions and appearance of the moon during one lunation, may be, perhaps, as well understood by referring to the lower figure on the second copper-plate, where S, is the sun; O, P, Q, R, the earth's orbit made circular for the sake of simplicity; E is the earth; and each of the smaller circles round E represents the moon's orbit (also drawn a circle for the sake of simplicity.) On the inner circle, k, l, m, &c., represent the moon as it would appear to a spectator placed over the centre of the moon's orbit, at the beginning of each half quarter; and K, L, M, &c., on the outer circle, represent the moon as it appears at the same time to the inhabitants of the earth.

The earth and the moon are bodies tarough which light cannot penetrate; and they are both illuminated by the sun, therefore they must cast shadows in the direction opposite to the sun; and, as each of them is smaller than the sun, their shadows must terminate in points. In the preceding figure BD is the earth's orbit during one month, as before; the large circles represent the earth at the several quarters of a lunation: and the small circles represent the moon at the corresponding parts of the same, as denoted by the corresponding figures. The light of the sun comes upon them all, from the centre where the lines B A and C D would meet, and, therefore, the earth must there cast shadows in the direction of the S S, and the moon in the direction s s. Now in the positions oo and 4 4, both of which represent the new moon, the shadow of the moon would necessarily, if the sun, moon, and earth were exactly in the same straight line fall upon the earth. Or, to the earth's inhabitants, the moon would seem to come in between them and the sun. This would form what is called an eclipse of the sun. But in the situations 1, 1, 2, 2, and 3, 3, or in any intermediate situation, the shadow of the moon would not fall in the direction of the earth; or, in other words, the moon would not come between the sun and earth so as to occasion an eclipse of the sun.

Again, if the earth and moon were in the situation 2, 2; that is, if it were full moon, and if the sun, earth, and moon were exactly in the same straight line, the moon would pass through the earth's shadow; or, in other words, the earth would come between the sun and moon, and thus deprive it of the