C 2 75 ] 
{trailed from the approximate diftance, found before,, 
if the latitudes of the moon and jftar are of the fame 
denomination,^ or added thereto, if they are of dif- 
ferent denominations, gives the true diftance of the 
Moon from the ftar. 
. This rule, though only an approximation, 
is fo very exadt, that even, if the latitude of the Moon 
was 5 , and that of the ftar 1 5 0 , the error would be 
only io 7/ ; and if the latitude' of the Moon be 5 0 , 
and that of the ftar io 3 , the error is only 4" 4 ; and 
if the latitudes be lefs, will be lefs in proportion as 
the fquares of the fines of the latitudes decreafe. 
demonstration. 
Let P [Fig. 7.] reprefentoneof the poles of the eclip- 
and Q^R the places of the Moon and ftar. From 
R let the arch of a great circle R D be drawn per- 
pendicular to P By fpherics, the tangent of 
= tangent of P R x cofine of the angle RPD. 
And, by trigonometry, cofineof QJD or (QP-PD) 
= col- QJ' x cof. PD + fin. QP x fin. PD = 
cof. Qj>xcof. P D 4. fin. Qj> x cof. P D x tan. P D 
= cof - P D x col. QP 4. fin. Q 4 x tan. P D = cof. 
P D X col. o P 4 - fin. 04 x tan. P R x col. P 
.-.cof. QJ) : cof. P D : : cof. QP + fin. Q P x tan. P R 
X cof P : 1. But, by fpherics, cof. QD : cof. P D : : 
cof. R Qj_ cof. P R .-. cof. R Qj cof. PR:: cof. 
Q P 4 fin. QP x tan. P R x cof. P ; i. Whence cof. 
R Q = c °f- PR X cof. QP 4. fin. QP X fin. PR 
X cof. P : Now, by trigonometry, cof (QP — PR) 
N n 2 = cof. 
