Tangential method distance and elevation

In the tangential method the stadia hairs are not used. The telescope is pointed at two targets (vanes) on a staff, a known distance $s$ apart, and the two vertical angles are measured. The distance is calculated from the tangents of these angles. It is used when the telescope has no stadia hairs, or when the distance is too large for the stadia intercept to be read accurately.

                   _ B  (upper target)
               _ -  |
        theta1 -    | s
    O  ----------   A  (lower target)  
        theta2 \_   |
 axis ----------\__ |       horizontal
                  D

Let $D$ be the horizontal distance of the staff from the instrument axis, and $V$ the vertical height of the lower target above (or below) the axis.

Horizontal distance and vertical distance

For two angles of elevation, $\theta_1$ (upper target) and $\theta_2$ (lower target), with the vanes $s$ apart:

$$D=\frac{s}{\tan\theta_1-\tan\theta_2},\qquad V=D\tan\theta_2$$ $$\text{RL of staff station}=\text{RL of axis}+V-h$$

where $h$ is the height of the lower vane above the ground at the staff station. For two angles of depression, $\alpha_1$ (lower target) $>\alpha_2$ (upper target):

$$D=\frac{s}{\tan\alpha_1-\tan\alpha_2},\qquad \text{RL}=\text{RL of axis}-D\tan\alpha_2-s-h$$

For one angle of elevation $\theta$ (upper target) and one of depression $\alpha$ (lower target): $D=s/(\tan\theta+\tan\alpha)$ and RL $=$ RL of axis $-D\tan\alpha-h$.

The method needs only the vertical circle of the theodolite and works even at long distances, but the angles must be measured carefully, because $\tan\theta_1-\tan\theta_2$ is small and a small error in either angle causes a large error in $D$.