VOL. LXl.] PHILOSOPHICAL TRANSACTIONS. 183 



The arch hd might have been computed by the versed sine of the angle hid. 

 But, if a table of natural sines is not at hand, the arch ad may be found loga- 

 rithmically thus : 



Take half the sum of the 4 first logarithms in the preceding computation of 

 HD, viz 19.53364 



Deduct the sine of half di » ih 7-76675 



the remainder is 11. 76689 



This remainder sought in the table of logarithmic tangents gives 

 the correspondent sine 9.99994 



This sine deducted from the first number leaves the sine of half hd, 

 that is, 19° 59' O* 9.53370 



Prop. 1. — In fig. 5, 6, let bca be the ecliptic, e the moon appearing in the 

 ecliptic in c, from the place of the earth, whose zenith is z ; b the nonagesime 

 degree, the arch zb being perpendicular to the ecliptic, zec the circle of alti- 

 tude ; ED the moon's latitude, the arch de being perpendicular to the ecliptic 

 CB ; and dc the parallax in longitude ; then de is to the horizontal parallax, as 

 the sine of zb, the distance of the nonagesime degree from the zenith, or the 

 altitude of the pole of the ecliptic, to the radius ; also dc is to the moon's hori- 

 zontal parallax, as sin. bc X cos. zb to the square of the radius. The arch ce 

 is to the moon's horizontal parallax, as sin. zc to radius, and de is to ce as sin. 

 zb to sin. zc; whence by equality de is to the horizontal parallax as sin. zb to 

 the radius. Again, sin. zb is to radius, as the tangent of zb to the secant of 

 zbk therefore de is to the horizontal parallax, as tan. of zb to sec. zb : but dc 

 is to DE as sin. bc to tan. zb ; whence by equality dc is to the horizontal paral- 

 lax, as sin. bc to the sec. zb, or as sin. bc X cos. zb to the square of the radius. 



Corol. — If the point s be taken 90 degrees from the apparent place of the 

 moon, and the arch sz be drawn, in the spherical triangle sbz, thecs. zb X cs. 

 BOS, that is, cs. zb X s. bc is equal to rad. X cs. zs: therefore dc is to the hori- 

 zontal parallax as cs. zs, or the sine of the distance of s from the horizon, to the 

 radius. And if the point s is taken in consequence of the moon, it will be above 

 the horizon, when the nonagesime degree is also in consequence of the moon ; 

 otherwise below. 



Prop. 1. — Let G be the apparent place of the moon out of the ecliptic in the 

 circle of latitude ck, k being the pole of the ecliptic, and h her true place. Then 

 EF, the distance of the moon from the circle of her apparent latitude, when she 

 is seen in the ecliptic, is equal to hl, her distance from the circle of her apparent 

 latitude, when her apparent place is g. If a great circle eht be drawn through 

 E and H, till it meet the circle of the apparent latitude in t, the 4 great circles 



* The same may be concluded from the s. hd being to s. td as s. hid to s. ihd. — Orig. 



