appearance of the dark portion of the young or waning moon completing its crescent (art. 413). 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.

(418.) 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. 401); 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 the velocity of 0°.98565 per diem, in the same time that the moon describes that arc plus a complete revolution, with her velocity of 13°-17640 per diem; and, the times of description being identical, the spaces are to each other in the proportion of the velocities. Let V and v be the mean angular velocities, x the superfluous arc; then V:v::1+x:x; and V―v:v::1:x, whence x is found, and ==the time of describing x, or the difference of the sidereal and synodical periods. 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 29a 12ь 44m 25.87.

x

(419.) 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.

(420.) 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 C D a spherical body, whether earth or moon, illuminated by it. If we join and prolong A C, BD; since AB is greater than CD, these lines will meet in a point 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 Fig. 60.

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 O N P) of the sun's surface, the rest (B O N 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.

(421.) In a lunar eclipse (represented in the upper figure), the moon is seen to enter1 the penumbra first, and, by degrees, get involved in the umbra, the former bordering the latter like a smoky haze. At this period of the eclipse, and while yet a considerable part of the moon remains 1 The actual contact with the penumbra is never seen; the defalcation of light comes on so very gradually that it is not till when already deeply immersed, that it is perceived to be sensibly darkened.

unobscured, the portion involved in the umbra is invisible to the naked eye, though still perceptible in a telescope, and of a dark grey hue. But as the eclipse advances, and the enlightened part diminishes in extent, and grows gradually more and more obscured by the advance of the penumbra, the eye, relieved from its glare, becomes more sensible to feeble impressions of light and colour; and phenomena of a remarkable and instructive character begin to be developed. The umbra is seen to be very far from totally dark and in its faint illumination it exhibits a gradation of colour, being bluish, or even (by contrast) somewhat greenish, towards the borders for a space of about 4' or 5' of apparent angular breadth inwards, thence passing, by delicate but rapid gradation, through rose red to a fiery or copper-coloured glow, like that of dull red-hot iron. As the eclipse proceeds this glow spreads over the whole surface of the moon, which then becomes on some occasions so strongly illuminated, as to cast a very sensible shadow, and allow the spots on its surface to be perfectly well distinguished through a telescope.

(422.) The cause of these singular, and sometimes very beautiful appearances, is the refraction of the sun's light in passing through our atmosphere, which at the same time becomes coloured with the hues of sunset by the absorption of more or less of the violet and blue rays, as it passes through strata nearer or more remote from the earth's surface, and therefore, more or less loaded with vapour. To show this, let A D a be a section of the cone of the umbra, and F Bhf of the penumbra, through their common axis DES, passing through the centres ES of the earth and sun, and let K Mk be a section of these cones at a distance EM from E, equal to the radius of the moon's orbit, or 60 radii of the earth.' Taking this radius for unity, since ES, the distance of the sun, is 23984, and the semidiameter of the sun 111 such units, we readily calculate DE=218, D M=158, for the distances at which the apex of the geometrical umbra lies behind the earth and the moon respectively. We also find for the measure of the angle ED B, 15′ 46′′, and therefore DBE= 89° 44′ 14′′, whence also we get M C (the linear semidiameter of the umbra)=0·725 (or in miles 2864), and the angle CEM, its apparent angular semidiameter as seen from E=41′ 30′′. And instituting similar calculations for the geometrical penumbra we get M K=1·005 (3970 miles), and K EM 57′ 36′′; and it may be well to remember that the doubles of these angles, or the mean angular diameters of the umbra and penumbra, are described by the moon with its mean velocity in 2h 43m, and 3h 47 respectively, which are therefore the respective durations of

The figure is unavoidably drawn out of all proportion, and the angles violently exaggerated. The reader should endeavour to draw the figure in its true proportions.

the total and partial obscuration of any one point of the moon's disc in traversing centrally the geometrical shadow.

(423.) Were the earth devoid of atmosphere, the whole of the phenomena of a lunar eclipse would consist in these partial or total obscurations. Within the space Cc the whole of the light, and within K C and ck a greater or less portion of it, would be intercepted by the solid body Bb of the earth. The refracting atmosphere, however, extends from B, b, to a certain unknown, but very small distance B H, 6 h, which, acting as a convex lens, of gradually (and very rapidly) decreasing density, disperses all that comparatively small portion of light which falls upon it over a space bounded externally by Hg, parallel and very nearly coincident with B F, and internally by a line B z, the former representing the extreme exterior ray from the limb a of the sun, the latter, the extreme interior ray from the limb A. To avoid complication, however, we will trace only the courses of rays which just graze the surface at B, viz: Bz from the upper border, A, and Bv from the lower, a, of the sun. Each of these rays is bent inwards from its original course by double the amount of the horizontal refraction (33') i. e. by 1° 6', because, in passing from B out of the atmosphere, it undergoes a deviation equal to that at entering, and in the same direction. Instead, therefore, of pursuing the courses BD, BF, these rays respectively will occupy the posi tions Bzy, Bv, making, with the aforesaid lines, the angles D Bb, FB v, each 1o 6'. Now we have found DBE = 89° 44' 14" and therefore

D

FBE(=DBE+ angular diam. of )=90° 17′ 17′′, consequently the angles E By and E Bv will be respectively 88° 38′ 14′′ and 89° 11' 17" from which we conclude Ez = 42.03 and Ev v = 88.89, the former falling short of the moon's orbit by 17-07, and the latter surpassing it by 28.89 radii of the earth.

as

(424.) The penumbra, therefore, of rays refracted at B, will be spread over the space v By, that at H over g H d, and at the intermediate points, over similar intermediate spaces, and through this compound of superposed penumbræ the moon passes during the whole of its path through the geometrical shadow, never attaining the absolute umbra Bb at all. Without going into detail as to the intensity of the refracted rays, it is evident that the totality of light so thrown into the shadow is to that which the earth intercepts, as the area of a circular section of the atmosphere to that of a diametrical section of the earth itself, and, therefore, at all events but feeble. And it is still further enfeebled by actual clouds suspended in that portion of the air which forms the visible border of the earth's disc as seen from the moon, well as by the general want of transparency caused by invisible vapour, which is especially effective in the lowermost strata, within three or four miles of the surface, and which will impart to all the rays they transmit, the ruddy hue of sunset, only of double the depth of tint which we admire in our glowing sunsets, by reason of the rays having to traverse twice as great a thickness of atmosphere. This redness will be most intense at the points x, y, of the moon's path through the umbra, and will thence degrade very rapidly outwardly, over the spaces x c, y C, less so inwardly, over xy. And at C, c, its hue will be mingled with the bluish or greenish light which the atmosphere scatters by irregular dispersion, or in other words by our twilight (art. 44). Nor will the phenomenon be uniformly conspicuous at all times. Supposing a generally and deeply clouded state of the atmosphere around the edge of the earth's disc visible from the moon (i. e. around that great circle of the earth, in which, at the moment the sun is in the horizon,) little or no refracted light may reach the moon.' Supposing that circle partly clouded and partly clear, patches of red light corresponding to the clear portions will be thrown into the umbra, and may give rise to various and changeable distributions of light on the eclipsed disc; while, if entirely clear, the eclipse will be remarkable for the conspicuousness of the moon during the whole or a part of its immersion in the umbra.3

2

'As in the eclipses of June 5, 1620, April 25, 1642. Lalande, Ast. 1769.

2 As in the eclipse of Oct. 13. 1837, observed by the author.

As in that of March 19, 1848, when the moon is described as giving "good light" during more than an hour after its total immersion, and some persons even doubted its being eclipsed. (Notices of R. Ast. Soc. viii. p. 132.)