and again illuminate the earth. It is no more than a white cloud does standing off upon the clear blue sky. By day, the moon can hardly be distinguished in brightness from such a cloud; and, in the dusk of evening, clouds catching the last rays of the sun appear with a dazzling splendour, not inferior to the seeming brightness of the moon at night. That the earth sends also such a light to the moon, only probably more powerful by reason of its greater apparent size*, is agreeable to optical principles, and explains the appearance of the dark portion of the young moon completing its crescent (art. 350.). For, when the moon is nearly new to the earth, the latter (so to speak) is nearly full to the former; it then illuminates its dark half by strong earth-light; and it is a portion of this, reflected back again, which makes it visible to us in the twilight sky. As the moon gains age, the earth offers it a less portion of its bright side, and the phenomenon in question dies away.

(353.) The lunar month is determined by the recurrence of its phases: it reckons from new moon to new moon; that is, from leaving its conjunction with the sun to its return to conjunction. If the sun stood still, like a fixed star, the interval between two conjunctions would be the same as the period of the moon's sidereal revolution (art. 338.); but, as the sun apparently advances in the heavens in the same direction with the moon, only slower, the latter has more than a complete sidereal period to perform to come up with the sun again, and will require for it a longer time, which is the lunar month, or, as it is generally termed in astronomy, a synodical period. The difference is easily calculated by considering that the superfluous arc (whatever it be) is described by the sun with his velocity of 0°.98565 per diem, in the same time that the moon describes that arc plus a complete revolution, with her velocity of

The apparent diameter of the moon is 32′ from the earth; that of the earth seen from the moon is twice her horizontal parallax, or 1° 54'. The apparent surfaces, therefore, are as (114)2: (32)2, or as 13 : 1 nearly.

13° 17640 per diem; and, the times of description being identical, the spaces are to each other in the proportion of the velocities.* From these data a slight knowledge of arithmetic will suffice to derive the arc in question, and the time of its description by the moon; which, being the excess of the synodic over the sidereal period, the former will be had, and will appear to be 29d 12h 44m 25.87.

(354.) Supposing the position of the nodes of the moon's orbit to permit it, when the moon stands at A (or at the new moon), it will intercept a part or the whole of the sun's rays, and cause a solar eclipse. On the other hand, when at E (or at the full moon), the earth O will intercept the rays of the sun, and cast a shadow on the moon, thereby causing a lunar eclipse. And this is perfectly consonant to fact, such eclipses never happening but at the exact time of the full moon. But, what is still more remarkable, as confirmatory of the position of the earth's sphericity, this shadow, which we plainly see to enter upon and, as it were, eat away the disc of the moon, is always terminated by a circular outline, though, from the greater size of the circle, it is only partially seen at any one time. Now, a body which always casts a circular shadow must itself be spherical.

(355.) Eclipses of the sun are best understood by regarding the sun and moon as two independent luminaries, each moving according to known laws, and viewed from the earth; but it is also instructive to consider eclipses generally as arising from the shadow of one body thrown on another by a luminary much larger than either. Suppose, then, A B to represent the sun, and CD a spherical body, whether earth or moon, illuminated by it. If we join and prolong AC, BD; since A B is greater than CD, these lines will meet in a point

*Let V and v be the mean angular velocities, ≈ the superfluous arc; then V:v:: 1+x: x; and V-v: v:: 1: x, whence is found, and the time of describing, or the difference of the sidereal and synodical periods. We shall have occasion for this again.

ข

E, more or less distant from the body CD, according to its size, and within the space CED (which represents

a cone, since CD and A B are spheres), there will be a total shadow. This shadow is called the umbra, and a spectator situated within it can see no part of the sun's disc. Beyond the umbra are two diverging spaces (or rather, a portion of a single conical space, having K for its vertex), where if a spectator be situated, as at M, he will see a portion only (A ON P) of the sun's surface, the rest (BON P) being obscured by the earth. He will, therefore, receive only partial sunshine; and the more, the nearer he is to the exterior borders of that cone which is called the penumbra.

Beyond

this he will see the whole sun, and be in full illumination. All these circumstances may be perfectly well shown by holding a small globe up in the sun, and receiving its shadow at different distances on a sheet of paper.

(356.) In a lunar eclipse (represented in the upper fi. gure), the moon is seen to enter the penumbra first, and, by degrees, get involved in the umbra, the former surrounding the latter like a haze. Owing to the great size of the earth, the cone of its umbra always projects far beyond the moon; so that, if, at the time of the eclipse, the moon's path be properly directed, it is sure to pass through the umbra. This is not, however, the case in solar eclipses. It so happens, from the adjustment of the size and distance of the moon, that the extremity of

her umbra always falls near the earth, but sometimes attains and sometimes falls short of its surface. In the former case (represented in the lower figure), a black spot, surrounded by a fainter shadow, is formed, beyond which there is no eclipse on any part of the earth, but within which there may be either a total or partial one, as the spectator is within the umbra or penumbra. When the apex of the umbra falls on the surface, the moon at that point will appear, for an instant, to just cover the sun; but, when it falls short, there will be no total eclipse on any part of the earth; but a spectator, situated in or near the prolongation of the axis of the cone, will see the whole of the moon on the sun, although not large enough to cover it, i. e. he will witness an annular eclipse.

(357.) Owing to a remarkable enough adjustment of the periods in which the moon's synodical revolution, and that of her nodes, are performed, eclipses return after a certain period, very nearly in the same order and of the same magnitude. For 223 of the moon's mean synodical revolutions, or lunations, as they are called, will be found to occupy 6585.32 days, and nineteen complete synodical revolutions of the node to occupy 6585-78. The difference in the mean position of the node, then, at the beginning and end of 223 lunations, is nearly insensible; so that a recurrence of all eclipses within that interval must take place. Accordingly, this period of 223 lunations, or eighteen years and ten days, is a very important one in the calculation of eclipses. It is supposed to have been known to the Chaldeans, under the name of the saros; the regular return of eclipses having been known as a physical fact for ages before their exact theory was understood.

(358.) The commencement, duration, and magnitude of a lunar eclipse are much more easily calculated than those of a solar, being independent of the position of the spectator on the earth's surface, and the same as if viewed from its center. The common center of the umbra and penumbra lies always in the ecliptic, at a point opposite to the sun, and the path described by the

moon in passing through it is its true orbit, as it stands at the moment of the full moon. In this orbit, its position, at every instant, is known from the lunar tables and ephemeris; and all we have, therefore, to ascertain, is, the moment when the distance between the moon's center and the center of the shadow is exactly equal to the sum of the semidiameters of the moon and penumbra, or of the moon and umbra, to know when it enters upon and leaves them respectively.

(359.) The dimensions of the shadow, at the place where it crosses the moon's path, require us to know the distances of the sun and moon at the time. These are variable; but are calculated and set down, as well as their semidiameters, for every day, in the ephemeris, so that none of the data are wanting. The sun's distance is easily calculated from its elliptic orbit; but the moon's is a matter of more difficulty, for a reason we will now explain.

(360.) The moon's orbit, as we have before hinted, is not, strictly speaking, an ellipse returning into itself, by reason of the variation of the plane in which it lies, and the motion of its nodes. But even laying aside this consideration, the axis of the ellipse is itself constantly changing its direction in space, as has been already stated of the solar ellipse, but much more rapidly; making a complete revolution, in the same direction with the moon's own motion, in 3232.5753 mean solar days, or about nine years, being about 3° of angular motion in a whole revolution of the moon. This is the phenomenon known by the name of the revolution of the moon's apsides. Its cause will be hereafter explained. Its immediate effect is to produce a variation in the moon's distance from the earth, which is not included in the laws of exact elliptic motion. In a single revolution of the moon, this variation of distance is trifling; but in the course of many it becomes considerable, as is easily seen, if we consider that in four years and a half the position of the axis will be completely reversed, and