12 GEOGRAPHIC TABLES AND FORMULAS. [bull. 214. 
GRAPHIC REDUCTION TO CENTER. 
Approximate closure errors of triangles ma} r be tested in the rield 
before distances have been computed by scaling from the plot the 
distances between stations in miles and the perpendicular distance in 
feet from signal to line joining instrument and distant station. 
Then, since 1 foot at a distance of 40 miles subtends an angle of 1" 
(nearly), 
length of perpendicular in feet X 40 ,. _ 
— c — L , — = — n — — — correction in seconds, 
number ot miles 
Example: Station P. Correction for swing on line B P, 30 miles 
in length from instrument to signal 
_ 3.8 feet X 40 _ 
30 
correction for swing on line A P, 25 miles in length, 
2.6 feet X 40_ ,„ 
25 " '" 
and correction to angle B P A = Q to reduce from instrument to sig- 
nal = 5.1" + 4.2" = 9.3", agreeing closely with the exact compu- 
tation. 
APPROXIMATE SPHERICAL EXCESS IN SECONDS. 
This may be obtained by dividing tin 4 area of the triangle in square 
miles by 75.5. 
SOLUTION OF TRIANGLES. 
Given two sides and included angle, to solve the triangle: 
Fig. 3. — Solution of triangles. 
Let x be an auxiliary angle; then 
tan »=— , or log tan x=\og a— log b; 
tan J (A-B) = tan (^-45°) tan £ (A+B); 
i (A-f-B)-H (A-B) = A; 
t (A+B)-£ (A-B)=B; 
from which remaining parts can be computed. 
