105 
of a Base Line on the Coast of Coromandel. 
The angles SNA, NAF, GAK, are all equal ; for SNA and 
NAF are alternate angles, formed by aright line falling upon 
two parallels (Euclid, 29th, 1st) : and NAF, GAK, are equal 
angles, formed by the mutual intersection of two right lines 
(Euclid, 15th, 1st). For the same reason, the angle NAH is 
equal to KAB, and HAF to GAB. Now, the obtuse angle 
NAB, being known (from observation), its supplement is equal 
to BAK, which, being subtracted from GxAK, leaves the angle 
GAB required ; by means of which, and the hypothenuse AB 
(given by mensuration), we obtain the side AG of the right- 
angled triangle GAB, by plane trigonometry; for as rad.: 
cosine ^GAB : : side AB : side AG. This computed side is 
equal to the opposite side OP of the rectangular parallelogram 
OPAG, which being added to the side NO of the right-angled 
triangle ONA (found by the like analogy from the observed 
angle N, and the measured hypothenuse NA) gives the dis- 
tance NP on the base line. 
Proceeding in like manner, with the last found angle, to find 
the angles of the remaining triangles, and thence their respec- 
tive bases, we shall have the sides of the corresponding rect- 
angles ; the sum of which sides, taken together, will be the true 
measure of the base line NS. 
Supposing no imperfection in the instrument with which 
the angles have been taken, and no error (however small) 
committed in taking the six preceding angles, then the angle 
XES, obtained by this method, will prove exactly equal to the 
angle NSE, which may be had by observation. In the present 
instance their difference amounts to no more than 2' 5". 
In the example before us, the observed angles are as follow, 
from which we may easily deduce the remaining ones required : 
MDCCXCII. P 
