dyi 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1749. 



renders eclipses of the sun as useful at least as those of the moon are in 

 that business. 



Let ABC, in the annexed 

 figure, represent half the 

 earth's enlightened disk, 

 AEC a portion of the eclip- 

 tic projected on fgh, the 

 path of the moon's shadow 

 over the disk, ei, the uni- 

 versal meridian ; a the situa- 

 tion of the place at the be- 

 ginning of the eclipse, p its situation at the end, S the centre of the shade at 

 the beginning, and £ its centre at the end of the eclipse. Draw eg, a^, and j3»i 

 perpendicular to the path of the shadow, |3y parallel to it ; join «J' and (3?, and 

 through a draw Oaa perpendicular to ac. 



Then (computing the true places of the sun and moon at the observed times 

 of the beginning and end of the eclipse) we shall have given h the motion of the 

 moon from the sun in her orbit during the time of the eclipse, and a(f=j3£ the 

 semidiameter of the penumbra ; which are to be reduced into such parts as the 

 semidiameter of the disk contains 10000 : the angles bei and beg, being found 

 by methods commonly known, gei their sum or difference will be given. Alsa 

 Ea and Ej3 will be sines of the sun's altitude at the beginning and end of the 

 eclipse respectively ; lEaand ie,3 are the angles at the sun between the vertex of 

 the place and the pole of those times ; which being found, the angle aE,3 their 

 difference, will be known, whence the line aj3 and the angle Eaj3 may be 

 computed. 



The angle GEa is the sum or difference of the known angles gei and ie^x: in 

 the figure the complement of this to a semicircle is Easy ; which being subtracted 

 from Ea|3, leaves the angle yajS, from which, and the line ap, ay and y|3,=^^, 

 may be found. 



Let a = Si — C,n, b = aS = (it, c := ary, and j: = |3n = y^. Then 

 j/ yi — XX =: ne, and V bb — cc — lex — xx ^= SC,, by Eucl. 1. 47. Conse- 

 quently a — n/Tib — XX = V bb — cc — lex = xx; which being reduced, gives 



the quadratic equation xx-\- cx = - ' ~° 7 ~~~ • '^'^'^ equation solved, gives 

 the value of x, from which iC, and m will be also had. In the triangle o^fi, we 

 have a^ and the angle ^«9 = geb given, whence afi and ^0 may be found : con- 

 sequently (JO will be known ; and from the observed time of the beginning of 

 the eclipse, and hourly motion of the moon from the sun, the time when the 

 centre of the shade is at 6 will be had. Lastly, in the triangle Eia, we have 



