190 
HINTS TO TRAVELLERS. 
To clear the Lunar Distance by Natural Cosines. 
Take the sum and difference of the apparent altitudes; also the sum 
and difference of the true altitudes. When the sum of the altitudes is less 
than 90°, add together the natural cosines (Table XXVIII.) of the sum 
and difference of the apparent altitudes; also the natural cosines of the 
sum and difference of the true altitudes. 
When the distance is less than 90°, add together the natural cosine of 
the sum of the apparent altitudes and the natural cosine of the apparent 
distance. When the distance is greater than 90° take their difference, 
multiply this result by the sum of the natural cosines of the true 
altitudes, and divide the product by the sum of the natural cosines of 
the apparent altitudes; the result will be a quantity which call x ; the 
difference between x and the natural cosine of the sum of the true alti¬ 
tudes will be the natimil cosine of the true distance when it is less than 
90°, but when greater than 90°, deduct it from 180°, and the result will 
be the true distance. 
When the sum of the altitudes is greater than 90°, instead of the sums of 
the natural cosines, of the sums and differences of the true and apparent 
altitudes, take their differences; x is found as before, and is to be added 
to the natural cosine of the sum of the true altitudes, and the result will 
be the natural cosine of the true distance. : 
0 
r 
n 
0 / // 
Mars’ App. Alt.. 
.. .. 40 
II 
00 
Mars’ True Alt. 
ys App. Alt. 
.. .. 39 
JI 
OO 
))’s True Alt. 
Sum. 
42 
00 
Sum. 
Difference .. .. 
40 
00 
Difference .. 
Sum of App. Alts. 
0 
.. .. 79 
/ 
42 
00 
Nat. Cosine .. .. 
Diff. of App. Alts. 
.. .. oo 
4 ° 
00 
Nat. Cosine .. .. 
.. .. = *999932 
(ist Term). 
o / rr 
Sum of True Alts.8o 22 29 Nat. Cosine 
Diff. of True Alts. 00 2 21 Nat. Cosine 
(2nd Term) .= i* 167203 
= *167204 
= *999999 
