1. Eclipses of the has been said that these are caused by the moon passing through the earth's shadow.,_ Before this explanation, can be accepted, it must be shown that that shadow extebds as far as the moon. This 14 easily &One. Supposing the earth to have no atmosphere, then the shadow is the cone marked in shade in fig. 1, whose apex is at 0; and the question is, whether the distance OT from the apex to the earth's center exceeds the moon's average distance from the earth. Drawing TB, SA, from the centers of the earth and sun respectively, perpendicular to the line OBA, touching both spheres, and the line TC parallel to the line OBA, we have from the similar triangles OTB, TSC, the pro portion OT : TB : : TS : SC. Now, we know that TS, the (mean) distance of the sun, is equal to about 24,000 times TB; also, from the construc tion, AC = TB; and we know that SA = 112 times TB, whence it fol lows that SC = 111 times TB. The above proportion, then, gives OT = 216 times TB, since Iriv. = 216 nearly. But the moon's average distance is only 60 times TB (the earth's radius). Hence it appears that the length of the earth's shadow is almost four times the average distance of the moon, and that the moon can enter it. Further, it is clear that, should it do so, it may be totally obscured; for it must enter at a point much nearer T than half the distance OT, which is 108 times TB; and every where within that distance it might be shown the breadth of the shadow is much greater than the moon's disk. But one consideration now remains to be stated to complete the proof of the theory of lunar eclipses. It was mentioned that they only occur at full moon, and we know that to be the only time when the earth is between the sun and moon, and so has a chance of throwing her shadow upon it. Why they do not occur every full moon, will be explained in treating of the prediction of eclipses.
In the foregoing explanation, we proceeded on the assumption that the earth has no atmosphere. If the assumption were correct, the earth's shadow would be darker and narrower than it is, and the phenomena of E. shorter in duration, but more strik ing. The effect of the atmospheric refraction (see REFRACTION) is to bend the rays which are incident on the atmosphere in towards the axis of the cone of the earth's shadow, those which pass through the lowest strata of the air being most refracted, and converging to a point in the line OT (see fig. 1), at a distance equal 42 radii of the earth from the earth's center. Accordingly, the moon, which, as we have seen, crosses the shadow at a distance of about 60 radii, never enters that part of it which is com pletely dark; thus, she never loses her light entirely, but appears of a distinct red dish color resembling tarnished copper—an appearance caused by the atmospheric refraction, in the same way as the ruddy color of the clouds at sunset. There is another reason why the phenomena of a lunar eclipse are less striking than, from the explanation given relative to fig. 1, might be expected. Every shadow cast by the sun's rays necessarily has a penumbra, or envelope, on both sides of the half-shadow. In the case before us (fig. 2), suppose a cone having its apex 0' between the sun and earth, and enveloping each of them respectively in its opposite halves, CO'C' and AO'A' (fig. 2). It is clear that from every point in the shaded part of the cone CO'C', and without the shadow BOB', a portion of the sun will be visible—and a portion only—the por tion increasing as the point approaches either of the lines CB. C'B; and diminishing as it approaches the lines BO, B'O. In other words, the illumination from the sun's rays is only partial within the space referred to, and diminishes from its extreme boundary lines towards the lines BO, B'O. When then, the moon is about to suffer eclipse, it first loses brightness on entering this penumbra; so that when it enters the real shadow, the contrast is not between one part of it in shade and the other in full brilliancy, but between a part in shade and a part in partial shade. Ou its emersion, the same contrast is presented between the part in the umbra and the part in the penumbra. What we should expect on this geometric view of the earth's shadow, actually happens. From the breadth of the penumbra, it happens that the moon may
fall wholly within it before immersion in the umbra commences; and so softly do the degrees of light shade into one another, that it is impossible to tell when any remarkable point on the moon's surface leaves the penumbra to pass into the umbra, or the reverse.
2. Prediction of Lunar Eclipses.—We said that lunar E. only happen at full moon. They do not happen every full moon, because the moon's orbit is inclined to the eclip tic, on which the center of the earth's shadow moves at at angle of 5° 9' nearly. Of course, if the moon moved on the ecliptic, there would be an eclipse every full moon; but from the magnitude of the angle of inclination of her orbit to the ecliptic, an eclipse can only occur on a full moon happening when the moon is at or near one of her nodes, or the points where her orbit intersects the ecliptic. An eclipse clearly can happen only when the centers of the circle of the earth's shadow and of the moon's disk approach within a distance less than the sum of their apparent semi-diameters; and this 'sum is very small; so that except when near the nodes, the moon, on which ever side of the ecliptic she may be, may pass above or below the shadow without enter ing it in the least. The moon's average diameter is known to be 31' 25" .7, and from the Nautical Almanac we may ascertain its exact amount for any hour—its variations all taking place between the values 29' 22" and 33' 28". As for the diameter of the circle of the shadow, it is easily found by geometric construction and calculation, and is shown to vary between 1° 15 32" and 1° 31' 36"; and its value for any time may be found from the Nautical Almanac, to which value astronomers usually add 1' as a correction for its calculation proceeding on the assumption that the earth has no atmosphere. Starting from these elements, it is a simple problem in spherical trigonometry—which may be solved approximately by plane trigonometry by supposing the moon and the earth's shadow to move for a short time near the node in straight lines—to fix the limits within which the shadow and moon must concur to allow of an eclipse. Recollecting that the earth's shadow on the ecliptic is at the opposite end of the diameter from the sun, and that therefore as it nears one node the sun must approach the other—the sun and shadow being always equidistant from the opposite nodes—we find, from the solution of the above problem: 1. That. if, at the time of full moon, the distance of the sun's center from the nearest node be greater than 12° 3', there cannot be an eclipse. 2. If at that time the distance of the sun's center from the nearest node be less than 9° 31', there will certainly be an eclipse. 3. If the distance of the sun's center from a node be between these values, it is doubtful whether there will be an eclipse, and a detailed calculation must be resorted to, to ascertain whether there will be one or not. Into the nature of that calculation we shall not attempt here to enter; suffice it to say that, knowing from the Nautical Almanac the true time of the sun and moon being in opposition, the true distance of the moon from the node at the time of mean opposition, with the true place of the sun at that time, as well as the moon's latitude, we may, by means of these elements, combined with the obliquity of the moon's path and her motion relative to that of the sun, not only fix whether there shall be an eclipse or not, but predict its exact magnitude, duration, and phases. It may here be men tioned, that before the laws of the solar and lunar motions were discovered with any thing like accuracy, the ancients were able to predict lunar E. with tolerable correctness by means of the lunar cycle (see SOLAR CYCLE) of 18 Julian years and 11 days. Their power of doing so turned on this, that in 223 lunations the moon returns almost exactly to the same position in the heavens. If she did return to exactly the same position, then, by simply observing the E. which occurred during the 223 lunations, we should know the order in which they would recur in all time coming.