if we trace by observation, from month to month, the point where it traverses the ecliptic, we shall find that the nodes of its orbit are in a continual state of retreat upon the ecliptic. Suppose O to be the earth, and A bad that portion of the plane of the ecliptic which is intersected by the moon, in its alternate passages through it, from south to north, and vice versa; and let ABCDEF be a portion of the moon's orbit, embracing a complete sidereal revolution. Suppose it to set out from the ascending node, A; then, if the orbit lay all in one plane, passing through O, it would have a, the opposite point in the ecliptic, for its descending node; after passing which, it would again ascend at A. But, in fact,

D

b

B

E

A

its real path carries it not to a, but along a certain curve, A B C, to C, a point in the ecliptic less than 180° distant from A; so that the angle A O C, or the arc of longitude described between the ascending and the descending node, is somewhat less than 180°. It then pursues its course below the ecliptic, along the curve C D E, and rises again above it, not at the point c, diametrically opposite to C, but at a point E, less advanced in longitude. On the whole, then, the arc described in longitude between two consecutive passages from south to north, through the plane of the ecliptic, falls short of 360° by the angle AO E; or, in other words, the ascending node appears to have retreated in one lunation on the plane of the ecliptic by that amount. To complete a sidereal revolution, then, it must still go on to describe an arc, E F, on its orbit, which will no longer, however, bring it exactly back to A, but to a point somewhat above it, or having north latitude.

(408.) The actual amount of this retreat of the moon's node is about 3' 10" 64 per diem, on an average, and in a period of 6793.39 mean solar days, or about 18.6 years, the ascending node is carried round in a direction contrary to the moon's motion in its orbit (or from east to west) over a whole circumference of the ecliptic. Of course, in the middle of this period the position of the orbit must have been precisely

reversed from what it was at the beginning. Its apparent path, then, will lie among totally different stars and constellations at different parts of this period; and this kind of spiral revolution being continually kept up, it will, at one time or other, cover with its disc every point of the heavens within that limit of latitude or distance from the ecliptic which its inclination permits; that is to say, a belt or zone of the heavens, of 10° 18′ in breadth, having the ecliptic for its middle line. Nevertheless, it still remains true that the actual place of the moon, in consequence of this motion, deviates in a single revolution very little from what it would be were the nodes at rest. Supposing the moon to set out from its node A, its latitude, when it comes to F, having completed a revolution in longitude, will not exceed 8'; which, though small in a single revolution, accumulates in its effect in a succession of many: it is to account for, and represent geometrically, this deviation, that the motion of the nodes is devised.

(409.) The moon's orbit, then, 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 the apogee of the moon will occur where the perigee occurred before.

(410.) The best way to form a distinct conception of the moon's motion is to regard it as describing an ellipse about the earth in the focus, and, at the same time, to regard this ellipse itself to be in a twofold state of revolution; 1st, in its own plane, by a continual advance of its axis in that plane; and 2dly, by a continual tilting motion of the plane itself, exactly similar to, but much more rapid than, that of the earth's equator produced by the conical motion of its axis described in art 317.

(411.) As the moon is at a very moderate distance from us (astronomically speaking), and is in fact our nearest neighbour, while the sun and stars are in comparison immensely beyond it, it must of necessity happen, that at one time or other it must pass over and occult or eclipse every star and planet within the zone above described (and, as seen from the surface of earth, even somewhat beyond it, by reason of parallax, which may throw it apparently nearly a degree either way from its place as seen from the centre, according to the observer's station). Nor is the sun itself exempt from being thus hidden, whenever any part of the moon's disc, in this her tortuous course, comes to overlap any part of the space occupied in the heavens by that luminary. On these occasions is exhibited the most striking and impressive of all the occasional phenomena of astronomy, an eclipse of the sun, in which a greater or less portion, or even in some rare conjunctures the whole, of its disc is obscured, and, as it were, obliterated, by the superposition of that of the moon, which appears upon it as a circularly-terminated black spot, producing a temporary diminution of daylight, or even nocturnal darkness, so that the stars appear as if at midnight. In other cases, when, at the moment that the moon is centrally superposed on the sun, it so happens that her distance from the earth is such as to render her angular diameter less than the sun's, the very singular phenomenon of an annular solar eclipse takes place, when the edge of the sun appears for a few minutes as a narrow ring of light, projecting on all sides beyond the dark circle occupied by the moon in its centre.

(412.) A solar eclipse can only happen when the sun and moon are in conjunction, that is to say, have the same, or nearly the same, position in the heavens, or the same longitude. It appears by art. 409. that this condition can only be fulfilled at the time of a new moon, though it by no means follows, that at every conjunction there must be an eclipse of the sun. If the lunar orbit coincided with the ecliptic, this would be the case, but as it is inclined to it at an angle of upwards of 5°, it is evident that the conjunction, or equality of longitudes, may take place when the moon is in the part of her orbit too remote from the ecliptic to permit the discs to meet and overlap. It is easy, however, to assign the limits within which an eclipse is possible. To this end we must consider, that, by the effect of parallax, the moon's apparent edge may be thrown in any direction, according to a spectator's geographical station, by any amount not exceeding her horizontal parallax. Now, this comes to the same (so far as the possibility of an eclipse is concerned) as if the apparent diameter of the moon, seen from the earth's centre, were dilated by twice its horizontal parallax; for if, when so dilated, it can touch or overlap the sun, there must be an eclipse* at some part or other of the earth's surface. If, then, at the moment of the nearest conjunction, the geocentric distance of the centres of the two luminaries do not exceed the sum of their semidiameters and of the moon's horizontal parallax, there will be an eclipse. This sum is, at its maximum, about 1° 34′ 27′′. In the spherical triangle SNM, then, in which S is the sun's centre, M the moon's, SN the ecliptic, M N the moon's orbit, and N the node, we may

suppose the angle NSM a right angle, SM=1° 34′ 27′′, and the angle M N S=5° 8′ 48′′, the inclination of the orbit. Hence we calculate S N, which comes out N 16° 58'. If, then, at the moment

of the new moon, the moon's node is farther from the sun

*The sun's parallax is here neglected.

in longitude than this limit, there can be no eclipse; if within, there may, and probably will, at some part or other of the earth. To ascertain precisely whether there will or not, and, if there be, how great will be the part eclipsed, the solar and lunar tables must be consulted, the place of the node and the semidiameters exactly ascertained, and the local parallax, and apparent augmentation of the moon's diameter due to the difference of her distance from the observer and from the centre of the earth (which may amount to a sixtieth part of her horizontal diameter), determined; after which it is easy, from the above considerations, to calculate the amount overlapped of the two discs, and their moment of contact.

(413.) The calculation of the occultation of a star depends on similar considerations. An occultation is possible, when the moon's course, as seen from the earth's centre, carries her within a distance from the star equal to the sum of her semidiameter and horizontal parallax; and it will happen at any particular spot, when her apparent path, as seen from that spot, carries her centre within a distance equal to the sum of her augmented semidiameter and actual parallax. The details of these calculations, which are somewhat troublesome, must be sought elsewhere."

(414.) The phenomenon of a solar eclipse and of an occultation are highly interesting and instructive in a physical point of view. They teach us that the moon is an opaque body, terminated by a real and sharply defined surface intercepting light like a solid. They prove to us, also, that at those times when we cannot see the moon, she really exists, and pursues her course, and that when we see her only as a crescent, however narrow, the whole globular body is there, filling up the deficient outline, though unseen. For occultations take place indifferently at the dark and bright, the visible and invisible outline, whichever happens to be towards the direction in which the moon is moving; with this only difference, that a star occulted by the bright limb, if the

* Woodhouse's Astronomy, vol. i. See also Trans. Ast. Soc. vol. i. p. 325