ECLIPSES. ] 



ASTRONOMY. 



973 



the interposition of this body ; and we perceive that if 

 the rays of light, A B, A' B', preserve their straight 

 path they cannot penetrate within the portion B O B', or 

 between the summit of the cone A () A' and the earth, 

 and that this is the form and dimensions of the shadow 

 cast by the globe T. In order that the moon may be 

 eclipsed, it must be placed between the earth T and the 

 summit of the cone O ; and the distance of those two 

 points is easily determined in the following manner : 

 Draw the line T C parallel to O A, and in the two similar 

 triangles, O T B and T S C, we have the proportion S : 

 T S : : T B : O T. The radius of the earth, T B, being 

 taken for unity, the line S C will be equal to the radius 

 of the solar sphere, diminished by the former quantity, 

 or to 111 radii of the earth. As the mean distance, TS, 

 of the earth and sun are equal to 24,000 radii of the 

 earth, we conclude that the distance O T is equal to about 

 216 terrestrial radii ; or, more exactly, that the length 

 of the axis of the shadow is 210 '531 radii of the earth. 

 At the time when the sun is in perigee this will decrease 

 to 212 '896 radii of the earth, and at the time of apogee will 

 be 220-238 radii. As the distance of the moon from the 

 earth never exceeds 64 terrestrial radii, and the least 

 length of the shadow is 212 terrestrial radii, it is obvious 

 that the moon must be obscured when the earth is 

 situated between it and the sun. The breadth of the 

 shadow of the earth at the distance of 108 terrestrial 

 radii will be equal to half the diameter of the earth, and 

 .lerably greater at the distance of only CO radii ; but 

 the exact dimensions of the latter point may be obtained 

 as follows : 



Dimension* of the Shadow. Let M N (Fig. 120) be the 

 Fig. 120. 



Respective Positions of Moon and Shadow. Let A C 

 (Fig. 121) be the great circle of the ecliptic, and B D the 



Fig. 121. 



surface of the celestial sphere, which is here supposed to 

 pasn through the centre of the moon ; this surface will 

 cut the cone of the earth's shadow at M M', and the angle 

 M T M' is the apparent angle of the shadow which it is 

 required to determine. The half of this angle, or M T O, 

 is equal to the angle BMT, diminished by the angle 

 MOT; but the former is the parallax of the moon 

 (since M T is the distance of the moon from the earth), 

 and the latter is equal to the angle ATS (the semi-dia- 

 meter of the sun) diminished by the angle BAT (the 

 parallax of the sun). In order, therefore, to obtain the 

 apparent semi-diameter of the shadow at the distance of 

 the centre of the moon, we must add the parallax of the 

 sun to that of the moon, and subtract the apparent semi- 

 diameter of the son. 



As the moon moves over a space about equal to its 

 diameter in an hour, it follows that it may be entirely 

 within the shadow for two hours. The greatest value of 

 the dimensions of the earth's shadow at the moon, may 

 be readily obtained by taking the greatest parallax of the 

 moon, and the least of the sun ; and as the former is 

 61' 29" at its maximum, and the sun's least serai-dia- 

 meter and corresponding parallax, respectively, are 

 15 46" and 8"-6, it would follow that the greatest semi- 

 diameter of the earth's shadow at the distance of the 

 moon is 45' 52". 



The apparent semi-diameters of the shadow for other 

 distances of the sun and moon may be readily calculated, 

 and are here annexed : 



Sun in perigee 



Sun at mean distance 



Sun in apogee 



Moon in apogee . . 

 ,, at mean distance 

 ,, in perigee . . 



Moon in apogee . . 

 ,, at mean distance 

 in perigee . . 



fMoon in apogee . . 

 at mean distance 

 in perigee . . 



37 42 

 41 31 

 45 20 



37 58 



41 48 

 45 37 



38 14 



42 3 

 45 52 



orbit of the moon, N will be the nodes of this orbit. The 

 shadow O moves along the first circle with a velocity equal 

 to that of the sun, and the moon passes along the second 

 circle with a velocity about thirteen times greater. In 

 order that the moon may meet the shadow, it must 

 happen that the centre of the shadow is sufficiently near 

 the nodes N at the moment of opposition. By taking 

 into consideration the above facts that the apparent 

 diameters of the moon and the earth's shadow vary from 

 one epoch to another and remarking that the distance 

 of the centre of the shadow from the node N is precisely 

 equal to the distance of the centre of the sun from the 

 opposite node of the moon, we find that if the distance 

 of the centre of the sun from the node at the time of 

 full moon be greater than 12 3', there cannot be an 

 eclipse ; and if the distance be less than 9 31', there 

 must certainly be an eclipse. Between these two ex- 

 tremes the case is doubtful ; but a more exact calcula- 

 tion will show whether an eclipse really does occur. 

 PARTIAL ECLIPSES OP THE Moox. When the moon is 

 wholly obscured, as we see it can be, the eclipse is 

 termed total ; but when only a portion of the lunar 

 disc enters into the shadow, the eclipse is partial. 

 In the latter case, the outline of the earth's shadow 

 projected upon the disc of the full moon, clearly 

 shows its round and globular figure, although the 

 diameter of the conical shadow is large in pro- 

 portion to that of the moon. The accompanying 

 diagram (Fig. 122) gives an idea of the relative pro- 

 portions of the shadow of 

 the earth and the disc of 

 the moon at those times, 

 and of the curvature of the 

 circumference of the shadow 

 a b c. But the definition 

 of the earth's shadow is 

 far from being so sharp as 

 is here represented, and, like 

 that of the shadow of any 

 other opaque body, is edged 

 with a cloudy and imperfectly- 

 defined penumbra, the dimen- 

 sions of which may be estimated by the following con- 

 siderations : Let the lines A O' B' C' and A' < )' B C be 

 drawn (Fig. 123), another cone, A O' A', having its 



Fig. 122. 



summit O' between the sun and the earth, is formed, 

 and enveloping the sun and the earth in its opposite 

 parts A O' A', B O' B'. It is clearly seen that all the 

 part situated within the space C 1} B' C', and without 

 the real shadow B () B', would only receive a portion of 

 the rays of the sun coming from the part of the hemi- 

 sphere turned towards it, the other part being hid by the 

 earth ; and that the portion of the sun which is visible 

 is greater according as this is nearer to the exterior sur- 

 face of the space C B B' C', and, on the contrary, smaller 

 as it is nearer to the real shadow of the earth, or B O B'. 



