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same as that of her passage over the meridian. But the actual retardation, being affected by the moon's change in declination, as well as by the inequalities of her motion in right ascension, is subject to greater variation. In the latitude of Philadelphia, the least daily retardation is only about 20 minutes, and the greatest is about 1h. 20m.

The less or greater retardation of the moon's rising, attracts most attention when it occurs at the time of full moon, as it affects the succeeding period of moonlight evenings. When the retardation has its least or nearly its least value, the moon is up or rises early in the evening for five or six days following the full moon: whereas, when the retardation is greatest the moon ceases, in the course of two or three days, to be seen in the early part of the evening.

Supposing the moon's orbit to coincide with the ecliptic, from which it does not greatly deviate, the daily retardation in the moon's rising at full moon, has nearly its least value at the full moon, which occurs near the time of the autumnal equinox. As this is about the period of the English harvest, this rising of the moon is called the Harvest Moon.

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222. Eclipses occur only at New and Full Moon. As an eclipse of the sun is caused by the moon passing between the sun and earth (214), it can only occur when the moon is in conjunction with the sun, that is, at the time of new moon. An eclipse of the moon is caused by the interposition of the earth, between the sun and moon, which prevents, in whole or in part, the illumination of the latter by the former. It must therefore occur when the moon is in opposition, that is, at the time of full moon.

If the moon's orbit coincided with the plane of the ecliptic, there would necessarily be an eclipse of the sun at every new moon, since the moon would in that case pass directly between the sun

and earth; and an eclipse of the moon at every full moon, as the earth would then be directly between the sun and moon. But as the orbit is inclined to the ecliptic, an eclipse can only occur when the moon, at the time of new and full moon, is at, or near one of its nodes. In other cases the moon is too far north or south of the ecliptic, to cause an eclipse of the sun or to be itself eclipsed.


223. Earth's shadow and penumbra. The magnitude of the sun. being far greater than that of the earth, and both being globular bodies, the shadow of the earth must evidently be of a conical form. Let AB and hg, Fig. 37, be sections of the sun and earth by a plane passing through their centres S and E; and let AC and BC, and also, AH and BK, be tangents common to the two sections. Then will gCh be a section of the earth's conical shadow or umbra, as it is frequently called, and EC will be the axis of the shadow. If the plane CEhK, be supposed to revolve round the axis EC, the tangent hK will describe the convex surface of the frustum of a cone, within the whole of which, the light of the sun must be more or less obstructed by the earth. That part of the frustum, which is included between the umbra and convex surface, that is, the part of which HgChK is a section, is called the earth's penumbra.

224. Beginning or end of an eclipse of the moon. An eclipse of the moon is regarded as beginning or ending at the instant her edge touches the earth's shadow. Thus, if mn be a part of the moon's orbit, the eclipse begins when the moon is at a, and ends when she is at e. Prior, however, to the beginning of an eclipse, while the moon is passing from the edge of the penumbra to the edge of the shadow, she must evidently suffer a gradual but increasing diminution of her light. This circumstance renders it difficult, if not impracticable, to observe with accuracy the instant at which the eclipse begins. On account of the gradual increase of the moon's light in passing from the shadow, the same difficulty occurs at the end.

Sometimes the moon, at full moon, though too far north or south of the ecliptic to come in contact with the shadow, may still be

sufficiently near to pass through the penumbra. In this case the moon suffers a diminution of light without being eclipsed.

225. Length of the earth's shadow. The length of the earth's shadow exceeds three times the distance of the moon from the earth. Let us assume the moon to be at one of her nodes at the instant she is in opposition at c. Then will the centres of the sun, earth, and moon, be in the same straight line SC. Put x = moon's horizontal parallax,





♪ = moon's apparent semi-diameter,
8' =



Then the angle SEB
EBg s' -π'.
Hence, (93 G),

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8' and EBg='. Hence, ECg = SEB But, ECg is the parallax of the point C.

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ECg::: Ec: EC,

x' :x :: Ec: EC = Ec


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Hence, since (95, 96 and 100), я always exceeds 53′ and ♪' — π' is always less than 17', it follows that EC must be more than three times Ec, the moon's distance from the earth.

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226. Semi-diameter of the earth's shadow. The apparent semidiameter of the earth's shadow, at the distance of the moon, is called the semi-diameter of the shadow. Thus, the angle bEc is the semi-diameter of the shadow. The point c is called the centre of the shadow.

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The semi-diameter of the earth's shadow is equal to the sum of the moon's and sun's horizontal parallaxes, less the sun's apparent semi-diameter.

Put S bEc=

semi-diameter of the earth's shadow.

Ebg - ECg and (225), ECg — §

since bЕe (81

s'. = x + x'

л'), or S Taking = 57' 1", ' 9", and 8' 16' 1", which are their я mean values, very nearly, we have the mean value of S 41' 9".



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л', we have S

Cor. Since S = 41' 9", we have 2S 82′ 18′′. The diameter of the shadow is, therefore, more than twice the moon's apparent diameter, and consequently, the moon may be entirely enveloped in the shadow.

Scholium. In obtaining the above expression for the semi-diameter of the shadow, the shadow is assumed to be limited by those rays of the sun which are tangents to the sun and earth. It is, however, found that the observed duration of an eclipse always exceeds the duration computed on this supposition. This is accounted for, by assuming that most of those rays which pass near the surface of the earth, are absorbed by the lower strata of the atmosphere. The extent of the obstruction to the passage of the light being thus increased, the diameter of the shadow, and consequently the duration of the eclipse, must also be increased.

In consequence of the difficulty in ascertaining the exact time of beginning and end of the eclipse (224), astronomers have differed as to the amount of the correction that should be made. According to the observations and computations of Dr. Mædler, a German astronomer, who has recently given particular attention to the subject, the computed semi-diameter should be increased by about a 50th part. Hence, S = = x + x' π

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227. Moon visible when entirely enveloped in earth's shadow. Another effect of the action of the earth's atmosphere is perceptible in eclipses of the moon. Those rays from the sun which enter the atmosphere and are so far from the surface as not to be absorbed, have their directions changed, and leave the atmosphere with a greater inclination to the axis of the shadow. In this way a sufficient quantity of light is generally thrown on the moon to render her visible, even when in the middle of the shadow. She appears, while in the shadow, with a dull red or copper-coloured light.

228. Moon's angular distance, from the centre of the earth's shadow, at the beginning or end of a lunar eclipse. This distance is equal to the sum of the semi-diameters of the earth's shadow and


For, as the eclipse begins when the moon's centre is at a, the angle aEc expresses her distance from the centre of the shadow at that time. But aEc is equal to the sum of bEc and aEb; or, angular distance, aEcS + d.

*Astr. Nach. Nos. 256, 286 and 338.

When the moon is first entirely in the shadow, or when she begins to emerge from it, her angular distance from the centre of the shadow will evidently be, S — §.

Cor. When at, or near, the time of full moon, the moon's angular distance from the centre of the shadow does not become less than S, there evidently cannot be an eclipse; and when it does become less, there must be an eclipse.

229. Lunar ecliptic limits. Referring the points and orbit to the celestial sphere, let c' Fig. 34, be the place of the centre of the earth's shadow in the ecliptic, and M' the place of the moon's centre in her orbit NF, when the angular distance c'M' is perpendicular to the orbit and is equal to S +8. Then it is evident, that, according as the distance of the centre of the shadow from the node N, or of the sun from the opposite node, is greater or less than Ne', the least distance of the centres of the moon and shadow must be greater or less than S. Hence, it follows (228 Cor.), that there can never be an eclipse of the moon when the distance of the sun from the nearest node is greater than the greatest value of Ne', and that there must always be one when this distance is less than the least value of Ne'. The greatest and least values of Ne' are, therefore, called the lunar ecliptic limits. Similar quantities for eclipses of the sun are called solar ecliptic limits.

Now, it is known, both from observations and from investigations in physical astronomy, that at the time of the syzygies the inclination of the moon's orbit has always nearly its greatest value of 5° 17′. Taking, therefore, this value of c'NM' and the greatest and least value of S +8, which, including the correction of S, are about 63′ 17′′ and 53' 8", the right angled spherical triangle c'M'N gives for the greatest value of Ne' or the greater limit 11° 32', and for the less limit 9° 40′.

Taking into view the inequalities in the motions of the sun, moon and nodes, other limits corresponding to the mean motions, have been obtained. These are very convenient in determining when eclipses of the moon may or must occur. According to Delambre, if at the time of mean full moon, the mean longitude of the sun differs more than 12° 36' from that of the nearest node, there cannot be an eclipse; but if it differs less than 9o, there must be

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