SOLAR ECLIPSES.] 



ASTRONOMY. 



979 



sun, and k the distance of the moon from the sun. The 

 sun's horizontal parallax at the moon is equal to the 

 sun's horizontal parallax at the earth, increased by the 

 ratio of the distances, and diminished by the ratio of 

 the diameters of the sun and earth ; or the sun's hori- 

 zontal parallax at the moon = sun's horizontal paral- 



lax on the earth (p) X p X 



yp r being the moon's true 



semi-diameter, and R the earth's true semi-diameter. 

 Substituting those values in the preceding expression, 



moon's shadow will be 



the semi-diameter of the 



-- X -j- -\-p ^r> -JT- The parallaxes of the sun and 

 moon may be used instead of the ratio of their distances 



p 



in t!iis expression, and as p = -^ and P 



,. K P 

 sequently j- = p - As 



T> 



_ , 



* 

 : 2T? 1 we 



> con- 



s+ 



K 



fc 



SP 



r; and the apparent semi-diameter 



of the moon's shadow 



a D p 



2 X P i; 



We perceive 



this that when the moon's diameter is equal to that 

 ". sun, the apparent diameter of the moon's shadow 

 is nothing, or the apex of the cone just reaches to the 

 earth. When the moon's diameter ia less than that of 

 the sun, the expression is negative, and the shadow can- 

 not reach the earth's surface. The formulae for the 

 dimensions of the penumbra may be adapted to the 

 moon's shadow in a similar manner. If the apparent 

 semi-diameter of the earth, or the moon's horizontal 

 parallax, be added to the apparent somi-diameter of the 

 shadow, the distance of the centre of the moon's shadow 

 and the centre of the earth is found ; and the limits of the 

 distance of the moon from her nodes, when an eclipse can 

 happen, will readily be obtained, and is found to be 17 27'. 

 It is thus seen that the same formulae apply to the cal- 

 culation of both lunar and solar eclipses, in so far as the 

 times at which, the earth in general would become enve- 

 loped in the shadow of the moon, are concerned ; but this 

 would only have reference to an observer situated on the 

 moon, who would thus be enabled to foretel and note the 

 duration, times, and magnitude of an eclipse, the instant 

 at which the penumbra entered upon the enlightened 

 disc of the earth, and the period of total eclipse. In 

 Figure 135 we perceive FI<*. 135. 



the phenomena which 

 would follow from the 

 interposition of the 

 moon between the sun 

 and earth ; the fictitious 

 observer situated at L, 

 as the moon moved in 

 the direction of the ar- 

 row, carrying with it the real shadow and the penumbra, 

 would first see the penumbra touch the disc of the earth 

 T, then (if the eclipse were total) the contact of 

 the real shadow ; and the above formulae would give 

 the instant at which those appearances would 

 happen. The two cones, that of the real shadow 

 and the penumbra, would continue to traverse 

 the portion of the hemisphere turned towards 

 the moon, covering, successively, the various parts 

 of its disc, advancing on new regions, and leaving 

 those which it had previously shadowed. Finally, 

 the cone of the shadow, then that of the penumbra, 

 would again become tangents to its surface ; and the 

 instant at which this would take place, would mark the 

 end of the eclipse. But in the case of a solar eclipse on 

 the earth's surface, the time at which it became generally 

 visible on the earth, or the period of first contact, would 

 not be of such interest to any observer, as the time at 

 which it would become visible, or total, at his own 



particular locality ; and the position of the observer at dif- 

 ferent parts of the moon was not taken into consideration 

 in the foregoing calculations, the latitude and longitude 

 of the several places on the moon being a matter of in- 

 difference to an observer on the earth. But to an obser- 

 ver on the earth it is a matter of great consequence to 

 know the exact time at which the eclipse would com- 

 mence at his own station (for it will not begin for him 

 until such period), and also the extent and duration of 

 tho shadow of the moon, when it passes over a known 

 station of any assigned latitude and longitude. The cal- 

 culation of an eclipse of the sun is, therefore, more 

 difficult than that of the moon, because more is required 

 to be determined. The foregoing formulse may be ap- 

 plied to the case of an eclipse of the sun, viewed from the 

 centre of the earth, to which they will exactly apply. But 

 to the position of an observer on the surface of the earth, 

 various corrections of parallax have to be applied on 

 account of the angle which he makes with the centre of 

 the earth, whether in latitude or longitude, as the various 

 phases of the eclipse at the different parts of the earth, 

 by which it is visible at some parts and not at others, 

 are due altogether to parallax. In calculating a solar 

 eclipse for any assigned position, it is necessary to correct 

 the angular distances, or the longitudes and latitudes, 

 for the effects of parallax. This requires a long and in- 

 tricate calculation ; but the principal appearances and 

 phenomena, as likewise the times of contact, greatest 

 eclipse, duration, and magnitude, may be determined in 

 the following simple manner : 



GEOMETRICAL OR GRAPHICAL CONSTRUCTION OF A SOLAR 

 ECLIPSE. If an observer be supposed to be situated at 

 the centre of the sun, and the moon be interposed be- 

 tween the sun and earth, he will observe the moon pass- 

 ing across the disc of the earth in the same manner as 

 an observer on the earth sometimes perceives the satel- 

 lites of Jupiter traversing the bright disc of the latter 

 planet. In the same manner, likewise, as an observer 

 on the earth perceives the equator and axis of the sun 

 inclined at different angles to the north and south points 

 and to the ecliptic, at the various seasons of the year (pages 

 929, 930, Figs. 27 and 28), the observer at the sun would 

 perceive the poles and equator of the earth change their 

 situations in respect to the ecliptic, but in a greater de- 

 gree, the solar equator being only inclined at an angle of 

 7 to the ecliptic, whilst that of the earth is inclined at 

 23|. At the time of the vernal equinox, the plane of 

 the equator passes through the sun, and the north and 

 south poles of the earth will consequently appear to an 

 observer at the sun situated at the margin 01 the disc, 

 the parallels of latitude describing straight lines, as in 

 Fig. 136. At the summer solstice, the north pole of the 

 earth will be turned towards the sun, the south pole will 

 be invisible, and the parallels of latitude will, to an ob- 

 server on the sun, be projected into ellipses, as at Fig. 

 137- At the autumnal equinox, the parallels of latitude 

 will again appear as straight lines, but the poles of the 

 earth will be on the opposite side of the poles of the 

 ecliptic, as at Fig. 138. Finally, at the winter solstice, 

 the south pole will be turned towards an observer on the 

 sun, and the parallels of latitude projected into ellipses, 



South. 



South. 



South. 



South. 



as in the former case, but in the contrary direction 

 139). 



By projecting in this manner the surface of the earth 

 at the various seasons of the year, as it appears to a 

 spectator on the sun, we can represent any parallels of 

 latitude. By doing so for the day on which the eclipso 

 is expected to occur, we can mark along the projection, 

 the position of the place at the different hours of the 

 day ; and the moon's apparent path across the earth's 



