186 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



her apparent diameter in c, bears a less proportion to her horizontal diameter, 

 than the rectangle under the sine of her horizontal parallax and twice the sine of 

 half the apparent latitude cg, to the square of the radius. 



The sine of ce is to the sine of zc as the sine of the horizontal parallax to the 

 radius; and ce, the difference of zc and ze, being very small, the difference of 

 the sines of those arches may be esteemed to bear to the sine of ce, the ratio of 

 the cosine of zc to the radius; and thus the difference of the sines of zc and ze, 

 will be to the sine of zc, as the rectangle under the sine of the horizontal paral- 

 lax and the cosine of zc, to the square of the radius. And in like manner the 

 difference of the sines of zg and zh, will be to the sine of zg, as the rectangle 

 under the sine of the horizontal parallax and the cosine of zg, to the square of 

 the radius. But s. ze is to s. zc, as the moon's horizontal diameter to her ap- 

 parent diameter in c, and s. zh to s. zg, as the moon's horizontal diameter to 

 her apparent diameter in g. Therefore the difference of the apparent diameter 

 in G from the apparent diameter in c, is to the horizontal diameter, as the 

 rectangle under the sine of the horizontal parallax and the difference of the co- 

 sines of zc and zg, to the square of the radius. But in the triangle czg, the 

 difference of zc and zg is less than the third side cg : therefore the chord of the 

 difference of those arches, and much more the difference of their cosines, will be 

 less than the chord of cg, or twice the sine of half cg. Hence the ratio of the 

 augmentation of the apparent diameter in g, to the apparent diameter in c, will 

 be less than the rectangle under the sine of the horizontal parallax and twice the 

 sine of half cg, the apparent latitude, to the square of the radius. 



More accurately, the chord of the difference of zc and zg being to the dif- 

 ference of their cosines, as the radius to the cosine of half their sum, the differ- 

 ence of the moon's apparent diameters in c and g may be considered as nearly 

 bearing to the horizontal diameter, the ratio of the parallelopipedon, whose 

 altitude is the sine of the horizontal parallax, and base the rectangle under the 

 chord of CG and the cosine of zc, to the cube of the radius; the cosine of zc 

 being to the cosine of zb, the distance of the nonagesime degree from the 

 zenith, as the cosine of bc, the apparent distance of the moon from the nona- 

 gesime degree, to the radius. But this difference can never be any sensible 

 quantity. 



Carol. 5. — When the moon is in the longitude of the nonagesime degree, the 

 parallax in longitude ceases, and the apparent latitude is the difference of the 

 moon's apparent distance from the zenith, and the distance of the nonagesime 

 degree from the same. But now since dc is to the horizontal parallax, as the 

 rectangle under the sine of bc and the cosine of zb, to the square of the radius; 

 if an arch be taken to the horizontal parallax, as s. bd X cs. zb to the square of 

 the radius, this arch will differ but little from the parallax in longitude, and is 



