VOL. LXI.] PHILOSOPHICAL TKANSACTIOKS. 1'85 



the sum of those latitudes, when h and e are on different sides of the ecliptic, to 

 the moon's visible latitude. 



Corol. 1. — The moon's apparent diameter, is to her horizontal diameter, as 

 the sine of her apparent distance from the zenith to the sine of her true distance. 

 Therefore, when the moon is in c, her apparent diameter is to her horizontal 

 diameter, as s. zc to s. ze, and s. zc being to s. ze nearly as s. bc to s. bd; the 

 moon's apparent diameter in c will be to her horizontal diameter, nearly as s. bc 

 to s. bd. Again, the ratio of s. cg to s. eh, is compounded of the ratio of s. zg 

 to s. ZH, and of the ratio of s. ct to s. et; and is also compounded of the ratio 

 of s. 7C to s. ZE, and of the ratio of s. gt to s. th ; but the sine of et is equal 

 to the sine of th, the arches et and th composing a semicircle ; also the sine of 

 CT there differs little from the sine of gt; therefore s. zg is to s. zh, that is, 

 the moon's apparent diameter, when in g, to her horizontal diameter, nearly as 

 s. zc to s. ZE, or nearly as s. bc to s. bd. 



Corol. 3. — In all latitudes of the moon, eh will not greatly exceed the differ- 

 ence, or sum of the moon's latitude in h, and the latitude with which she would 

 appear in the ecliptic. Therefore the ratio of s. zc to s. ze being compounded 

 of the ratio of s. cg to s. eh, and of the ratio of s. ht to s. gt, if x be taken, 

 that its sine be to the sine of the difference or sum of the latitudes, as s. zc to 

 ZE, s. X will be nearly to s. cg as s. ht to s. gt. Hence the difference of s. x 

 and s. Gc will be to s. cg nearly as the difference of s. ht and s. gt to s. gt, ht 

 not sensibly differing from tl. Now ft and tl together make a semicircle, 

 and the sum of fg and gl is twice the difference of tl from a quadrant, and the 

 difference between fg and gl equal to twice the difference of tg from a quadrant, 

 also the difference between the sines of tl and tg is equal to the difference of 

 the versed sines of the difference of those arches from quadrants ; and further 

 the rectangle under the sines of two arches is equal to the rectangle under half 

 the radius, and the difference of the versed sines of the sum and difference of 

 those arches: therefore the difference of the sines of x and of cg will be to the 

 sine of cg, as the rectangle under the sine of half fg and the sine of half gl, to 

 the rectangle under half the radius and the sine of gt ; and in these small arches 

 the difference of x and cg, will be to cg, nearly as the rectangle under the sines 

 of FG and gl to the rectangle under twice the radius and the sine of gt, or even 

 twice the square of the radius; this difference being to be added to x, when the 

 moon's apparent latitude, and that by which she would appear in the ecliptic, 

 are on the same side of the ecliptic, otherwise deducted from x for the final 

 correction of the apparent latitude. And in the last place, this correction will 

 be always so small in quantity, that in computing it of may be safely substituted 

 for GL. 



Corol. 4. — The excess of the moon's apparent diameter, when seen in g, above 



VOL. XIII. B B 



