1S4 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



cz, Gz, CT, ET, intersecting each other, the ratio of s. zc to s. ce is compounded 

 of the ratio of s. zg to s. gh and of the ratio of s. sht to s. et.* But, ce and 

 GH being the parallaxes in altitude at the respective distances from the zenith 

 zc, ZG, s. zc is to s. CE as s. zg to s. gh : therefore the sine of ht will be equal 

 to the sine of et, and the arches ht, et together make a semicircle : whence 

 ET is equal to hl. 



Carol. — The arch kh being drawn, the parallax in longitude, when the moon 

 is in h, will be to hl, as rad. to s. kh, or the cosine of the latitude ; and ef, 

 or its equal hl, to cd, as s. ke to the radius. Therefore the moon's parallax in 

 longitude, when in h, is to the parallax in longitude, when she appears in the 

 ecliptic, as the sine of ke to the sine of kh, that is, as the cosine of the latitude, 

 when the moon appears in the ecliptic, to the cosine of her l.ititude in h. 



Prop. 3. — When the moon appears out of the ecliptic, if her latitude is small, 

 the difference of the moon's latitude, when the moon appears in the ecliptic under 

 the same apparent longitude, if both latitudes are on the same side of the ecliptic, 

 otherwise their sum, will be to the moon's apparent latitude, nearly as the sine of 

 the moon's distance from the zenith, when appearing in the ecliptic under the 

 same apparent longitude, to the sine of the corresponding apparent distance. 



Fig. 6. When the moon appears out of the ecliptic in g, the four great 

 circles cz, gz, ct, et, intersecting each other as before, the ratio of s. 

 cz to s. ZE will be compounded of the ratio of s. cg to s. eh, or of cg to 

 EH in these small arches, and of the ratio of s. ht to s. gt, which last 

 ratio, when the latitude is small, and ht near a quadrant, is nearly the 

 ratio of equality. Now, in the triangle ekh, the arch eh exceeds the dif- 

 ference of ke and kh, that is, the difference of the latitudes, when both the 

 latitudes are on the same side of the ecliptic, and their sum, when the lati- 

 tudes are on the opposite sides. But here the excess will be inconsiderable. 

 Therefore if an arch x be taken, whose sine shall be to the sine of the difference, 

 or sum of the latitudes, as s. zc to s. ze, x shall be nearly equal to cg, the ap- 

 parent latitude in g. 



Carol. 1 . If the arches de, bz be continued to k, the pole of the ecliptic, the 

 4 great circles cb, cz, dk, bk, will intersect each other, and s. bd will be to the 

 sine of Bc, in the ratio compounded of the ratio of s. ze to s. zc, and of s. dk 

 to s. EK, the least of which ratios, the arch de being small, and dk a quadrant, 

 is nearly the ratio of equality : therefore s. bd is to s. bc nearly as s. ze to s. zc; 

 so that s. BD will be to s. bc nearly as the difference of the moon's true latitude, 

 when she appears in g, from her latitude de, with which she would appear in 

 the ecliptic, if the points h and e are both on the same side of the ecliptic, or as 



* Ptolem. Almag. L. i. c. 12. Menel. Spheric. L. iii. pr. 1. — Orig. 



