scure or dark part, is seen to be very irregular and serrated; and its form is often found to vary even during the time of observation. Bright spots are frequently seen on the dark part, near the line of separation. The light of these is observed to spread till they become united to the enlightened part. The whole enlightened surface, also, appears diversified with numerous spots of various shapes and different degrees of brightness.

These appearances clearly prove that the moon's surface must be very irregular, abounding in mountains and valleys; the bright spots seen on the dark part being evidently the elevated tops of mountains, which, being illuminated by the sun, are visible, while the lower part and intervening valleys remain in the dark, till by the moon's farther advance in her orbit they are reached by the

sun's rays.

Luminous spots, entirely unconnected with the phases, or, in other words, spots that are not the reflection of the sun's light, have sometimes been observed on the moon's disc. These are supposed to be volcanoes.

As the line separating the enlightened from the dark part of the disc, is always extremely irregular, it follows that the moon cannot have any large tracts of water; for over a surface of water, the line would evidently be uniform.

212. Height of the lunar mountains. If the distance of the illuminated summit of a mountain from the enlightened part of the disc be observed with a micrometer, and the positions of the sun and moon at the time be obtained by observation or computation, its height may be computed.* Its height may also be found from the measured length of the shadow it casts when in a different position with respect to the sun. Dr. Herschel computed the heights of a number of the lunar mountains. The height of the highest of those he found to be about 12 miles. But, according to Schroeter, Mädler, and other distinguished selenographers, some of the lunar mountains are as high as any of those on our globe. The Appenine range rises to a height of 3 miles. The abysses, however, are more remarkable, some of them being 20 or 30 miles in diameter, and 3 or 4 miles deep.

* See Appendix to Part I, art. 58.

213. Same surface of the moon always towards the earth. The various spots on the moon always occupy nearly the same positions on the disc. Hence it follows that nearly the same surface is always turned towards the earth. Hence, also, if we suppose the moon to be inhabited, the inhabitants on about one half the surface can never see the earth while they remain on that half.

214. Of the moon's atmosphere. The moon sometimes passes between the earth and sun, and, sometimes, between the earth and a star or planet, causing what in the former case is called an eclipse of the sun, and in the latter, an occultation of the star or planet. Assuming the moon to have no atmosphere, the durations of these phenomena may be very accurately computed by means of the known motions and apparent magnitudes of the bodies.

Now, if the moon was surrounded by an atmosphere, such as appertains to the earth, it would, by its action on the rays of light passing through it, produce a sensible effect on the duration of an eclipse of the sun or of an occultation. But no such effect has been observed. It is, therefore, inferred that the moon has no atmosphere; or that, if she has, it must be of very little density.

From a full investigation of the subject, Professor Bessel draws the conclusion, that, if we assume the moon to have an atmosphere constituted like that of the earth, its density at the moon's surface cannot be more than about the 1000th part of that of the earth's, at the earth's surface.*

215. Moon's rotation on her axis. The moon revolves with a uniform motion, from west to east, about an axis nearly perpendicular to the plane of the ecliptic, in the same time that she makes a revolution in her orbit.

Let E, Fig. 36, be the centre of the earth, aa' a part of the moon's orbit, a and a' two successive positions of the moon's centre, and a'D a line parallel to aE. Then, since nearly the same surface of the moon is always turned towards the earth (213), that point in the surface which is at e when the moon's centre is at a, will be at e' or nearly so, when the centre is at a'. Assuming the point

* Astronomische Nachrichten, No. 263. This is an excellent astronomical periodical, originally edited by the late Professor Schumacher, at Altona, and at present conducted by Professor Hansen.

K

to be exactly at e', it must, during the interval, have moved about an axis perpendicular to the plane of the orbit, through the angle Ea'D, which is equal to aEa' the angular motion in the orbit. Hence, the angular motions about the axis and in the orbit being equal, the moon must revolve on her axis in the same time that she makes a revolution in her orbit.

The small changes observed in the position of the spots on the disc, are caused by the inequalities of her motion in her orbit, and an inclination of her axis; and not by any inequality in her rotation. For, assuming the rotation to be uniform, if the moon's motion from a to a' is greater than the mean motion, the angle aEa' must be greater than that through which a spot at e is carried in the same time by the rotation on the axis. Consequently, when the moon's centre is at a', the spot must be to the east of Ea'. If, on the contrary, the motion from a to a' is less than the mean motion, the spot must be to the west of Ea'. An inclination of the axis will, evidently, cause the spots to have an alternate north and south motion.

A minute investigation of the subject, founded on successive accurate observations of the positions of the spots, proves that the rotation on her axis is uniform.

216. Inclination of moon's axis. From the investigation mentioned in the preceding article, it is found that the axis is nearly perpendicular to the plane of the ecliptic. The plane of the moon's equator, that is, the plane through her centre, perpendicular to the axis, makes with the plane of the ecliptic, an angle of 11°; and it intersects the latter in a line parallel to the line of the nodes.

217. Moon's librations. The alternate east and west motion of the moon's spots, produced by the inequalities of her motion in her orbit (215), is called the moon's libration in longitude; and the alternate north and south motion, depending on the inclination of the axis, is called the libration in latitude.

The diurnal motion of the observer, by which his position with reference to the radius vector Ea is changed, as from c to d, produces a slight apparent motion in the spots. This is called the diurnal or parallactic libration.

In consequence of these librations of the moon, small portions

of the surface to the east and west, and also to the north and south, alternately come into view and disappear.

218. Moon's passage over the meridian. The moon's mean daily motion in right ascension, which is the same as in longitude, is greater than that of the sun by more than 12°. Hence, if on any given day, we suppose the moon to be on the meridian at the same instant with the sun, she will, at the end of 24 hours, when the sun has again returned to the meridian, be more than 12° to the east, and will not, therefore, arrive at the meridian till nearly an hour later. On the next day she would arrive at the meridian nearly two hours later than the sun. Thus, her passage of the meridian is retarded from day to day. about 52 minutes.

The mean retardation is

in consequence of the inequalities in the moon's motion in right ascension, depending in part on the inequalities of her motion in her orbit, but more on the inclination of the orbit to the equator, the daily retardation in her passage over the meridian is subject to considerable variation. It varies from about 38 to 66 minutes.

219. To find the time of the moon's passage over the meridian on a given day.

Let A and A' be the right ascension of the moon and sun respectively, at noon of the given day, expressed in time, and reduced to seconds, m and m' their hourly variations in right ascension, also, in seconds of time, and let t be the required time of her passage over the meridian, in hours. Then, at the time t, we have the moon's right ascension = A + tm, and the sun's A' + tm'. Hence, as the moon is on the meridian at the time t, if the latter right ascension be subtracted from the former, the remainder will be the time t in seconds; or, being divided by 3600, it will be the time t in hours. Consequently,

=

The time of the moon's passage over the meridian of Greenwich,

is given in the Nautical Almanac for every day in the year. It may easily be found for any other meridian by proportion.

Cor. If A and m be taken to represent the right ascension and hourly variation in the right ascension of a planet, the above formula will give the time of its passage over the meridian. For a fixed star, mo, and the formula becomes,

220. To find the time of the moon's rising or setting.

To find the true semi-diurnal arc of a body (183), the declination or polar distance when the body is in the horizon is required. For the sun, this is usually assumed to be the same as on the meridian, as it changes but little during the interval. But for the moon, the change is too great to be neglected. We may, We may, however, by using the moon's meridian declination, or the declination* at an estimated time of rising or setting, obtain an approximate value of the semi-diurnal arc. This, subtracted from the time of passage over the meridian, or added to it, gives an approximate time of the rising or setting.

Then, to find the true time, let the declination be found for the approximate time, and again compute the semi-diurnal arc. This requires to be still further corrected, on account of the moon's increase in right ascension during the interval between her being on the meridian and in the horizon. To obtain this second correction, let D the difference between the times of two consecutive passages over the meridian, one of which precedes, and the other follows, the required time of rising or setting. Then, as 24h. : semi-diurnal arc last computed :: D: the correction; which being added to this semi-diurnal arc, gives its true value very nearly. This applied to the time of the passage over the meridian, gives the time of rising or setting.

221. Daily retardation of the moon's rising or setting. The average daily retardation of the moon's rising or setting is the

* The moon's right ascension and declination are given in the Nautical Almanac for every hour.