Mr Marrat on a New Method of Working Lunars. 99 
Agaiii, tan MZ : tan MN : : rad.: cos ZMN, and consider- 
ing M as a>plane triangle, by the principles of plane trigo- 
nometry we have M w : M n : : rad. : cos ZMN ; therefore, by 
equality, tan MZ : tan MN : : M w : M w* In the same man- 
ner, we have tan SZ : tan SN : : Ss : so. Now, M n, and s o, 
are the corrections ; and it is evident from the figure, that when 
the angle M is acute, M n must be subtracted from the appa- 
rent distance, but added when that angle is obtuse ; also when 
the angle S, at the sun or star is acute, s o must be added, but 
subtracted when it is obtuse. After the observation is made, 
a figure may be drawn by the hand, sufficiently exact for 
ascertaining the nature of the angles at M and S ; and this is 
the only ambiguity which it is necessary to attend to in solv- 
ing the problem by this method. 
Example. — On the 7th of June, sea account, at G*' 37' P. M. 
in Longitude 120° W. by account, the observed distance of the 
moon’s farthest limb from Aldebaran, was 39° 7' 4", the observed 
altitude of the star 43° 18', and the observed altitude of the 
moon’s lower limb 52° 52' Required the true distance ? 
The apparent altitude of the star is found to be 43° 14', that 
of the moon 53° 04', and the apparent distance of the moon and 
star 38° 52', also the moon’s horizontal parallax 54' 35". Now, 
in the triangle MZS, we have MZ, the moon’s co-altitude 
= 36° 56, ZS = 46° 46', and MS = 38° 52', to find the neces- 
sary corrections. It is well known that M w is equal to the dif- 
ference between the moon’s parallax in altitude and the refrac- 
tion, and that S ^ is the difference between the sun’s parallax 
and refraction, where the sun and moon are observed ; but S s 
is equal only to the refraction, when a star and the moon are 
observed, or 1925". 
In this example, M is found to be 32' 05", or 1925", and 
S .s = 1', or 60", 
ZS —46° 46' 2138 ^. 62 '- MS 
ZM - 36 56 19 26 - ^ MS 
2 1 83 42 
I Sum, 41 51 
2) 9 50 
tan i MS, 19° 26' ... 9.54754 
tan I Sum, 41 51 ... 9.95215 
tan I DifF. 4 55 ... 8.93462 
' 18. 88677 
tan 4 DifF. seg. 12M9' ... 9.33923 
