Use and working principle of subtense bar

Use of the subtense bar

A subtense bar is used for the indirect measurement of horizontal distances where chaining is difficult or slow: across rivers and valleys, and over marshy, steep, densely vegetated or rough ground, or in traffic. It is also used for setting out, for measuring base lines or traverse sides, and for checking tape measurements. With a 1" theodolite it gives results of about 1 in 3000 to 1 in 10,000 on short lines.

Working principle

A subtense bar is a horizontal bar of fixed length $l$ (usually 2 m) with targets at its ends. It is mounted on a tripod and set perpendicular to the line of sight with the help of a small sighting telescope on the bar. The small horizontal angle $\beta$ subtended by the targets at the theodolite is measured (to seconds, by repetition), and the distance is computed from it.

 Theodolite                       Subtense bar
     O  <------------ D ---------->  A |
       \  beta                       | l (2 m)
         \_____________________________ B

Horizontal distance

From the isosceles triangle $OAB$:

$$\frac{l}{2}=D\tan\frac{\beta}{2}\ \Rightarrow\ D=\frac{l}{2}\cot\frac{\beta}{2}\ \approx\ \frac{l}{\beta}\ (\beta\ \text{in radians})$$

Because the bar is horizontal and $\beta$ is measured in the horizontal plane, $D$ is the horizontal distance directly, even when the line of sight is inclined; no slope correction is needed. If the difference in level is wanted, the vertical angle $\alpha$ to the bar is also read, and $V=D\tan\alpha$.