Avams.—1o calculate Distances by Reciprocal Vertical Angles, 189 
If no logarithmic or trigonometrical tables are at hand, the difference of 
altitude may be found as follows :— 
As :00000485 represents the value of sin 1’ are 1" or tang 1" (true to 
the last figure), and as the tangents of small angles vary very nearly as the 
number of seconds contained in the angle, we may substitute for the tangent 
of the angle the number of seconds multipled by ‘00000485. 
In practice, the operation may be shortened by combining the two 
multipliers together; thus, 00000485 x 117 = :0005675. 
(Norz.—In order to show how very nearly the sines, arcs, and tangents 
agree for the first two degrees, their values at two degrees are given, for the 
sake of comparison, 
Thus sin 22 = 
are 2° =  :0349066 
tang? — 0349208 | 142 = 8" 
Diff. 
0348995 | 71 = 14 
Therefore, the arc of 2° = sin 2° 00’ 01%’, and the tangent of 
2° = arc of 2° 00’ 08", 
Also, in obtaining the tangent of 2° by multiplying -00000485 x 60 
X 60 x 2, the result is ‘0349200, or just 1 of a second below the true value. 
Similarly the tangent of 1°, found in the same manner, is ‘0174600, or 
just 1” above its true value ; but the value used for tang 1’, viz., 00000486, 
is slightly in excess of its true value, which is -0000048481368, etc.) 
Then the difference of altitude may be found by the following rules :— 
Case 1.—When one angle is in an elevation. 
RuLe.—Take the difference of the observed vertical angles, and also half 
the sum, both reduced to seconds ; multiply them together, and their product 
by ‘0005675; the result will be the difference of altitude between the two 
stations in feet. 
Case 2.— When both angles are depressions. 
Rvrz.—Take the sum of the observed verticle angles, and also half the 
differenee, both reduced to seconds, multiply them together, and their pro- 
duct by -0005675 ; the result will be the difference of altitude between the 
two stations in feet. 
Or, if zenith distances are used, the following general rule will apply in 
all cases :— : 
Rurr.—Subtract 180? from the sum of the observed zenith distances and 
reduce the remainder to seconds; then take half the difference of the 
Observed zenith distances and reduce it to seconds ; multiply the two 
quantities together, and the product by *0005675, and the result will be the 
difference of altitude between the two stations in feet, 
