186 Dr. Tiarks on the Reduction of Lunar Distances [Marcu, — 
results, they are uncertain in which calculation the mistake lies, 
and, as I have observed myself, have often neither patience nor 
time enough to find out the errors which are easily committed in 
logarithmic calculations. 
it appeared, therefore, to me, that a method susceptible of an 
easy check, like those which Prof. Gauss has introduced into 
astronomical calculations, would be desirable both for astrono- 
mers and navigators. ‘The method which I propose for this 
purpose seems to deserve notice ; and I can recommend it the 
more as the practical seamen to whom I have had an opportunity 
of communicating it, have found it easy and useful. 
Reduction of the apparent Distances of the Moon from a celestial 
Body to the true Distance. 
Let be | 
D’s altitude yes tis i ©’s orstars altitude a ge at sy 
true =h 
Distance § @Pparent = D H+a+D _ S 
true el 2 a 
and the angle at the zenith in the triangle formed by the moon, 
the celestial body, and the zenith = Z. 
We have immediately the two following equations : 
Sin. A .sin.H + cos. A . cos. H .cos. Z= cos. D 
Sin. 2’ . sin. H’ + cos. ’ . cos. H’ . cos. Z’= cos. D; therefore, 
om it cos, D — sin. hk. sin. H __ cos. D! — sin, 2’. sin. H’ (cx) 
cos. $e 
cos. h. cos. A cos, A’, cos, H/ 
From this is easily derived, 
: s cos, kh’. cos. H’ : 
cos. D’ = sin. h’ . sin. H’ + 
cos. H. cos. H * 
and from this by adding and subtracting on the left cos. H’ .cos.h’ 
ait, ; K cos. h'. cos, H’ ¢ 
cos. D’ = cos. (H’ — h’) + eee Oe: D — cos. (H —h)t 
D+H—A 
——}. sin 
a0 
$cos. D — sin. A.sin, Ht 
but cos. D—cos.(H — h) being equal — 2 sin. ( 
eae 
we have likewise, 
3. a! ' . (D+H—hk 
a it Ye _ 9 603. 4’ .cos. H ( 1G 
cos. D’ = cos (H h’y) —2 aot Tidos EA ae We 
D—Hik 
( 26 
= cos. (H’ i) — 20 E ee . sin. (S — A) . sin. 
Cs EL) s's)s ela deilels » wy andi SOP she Salad tea thee sr ern oni AD 
Suppose 
cos. hk’. ces. H’ . 2 m 
————_——-— sin. (s — a iS _— = COS. @*, OF 
cos. h.cos. H sin. (S h) . sin. (S H) ? 
