Measuring horizontal and vertical angles

Horizontal angles

1. Ordinary method (single angle). Set up and level the theodolite at O. Set zero (or any reading) on the circle, bisect A using the lower clamp, then release the upper clamp, bisect B and read both verniers. Change face, repeat, and take the mean. The angle AOB = reading on B - reading on A.

2. Repetition method - a single angle is added several times on the circle by using the upper and lower clamps alternately; the angle = total / number of repetitions.

3. Reiteration (direction) method - several directions are observed from a reference object, the angles are found from differences of directions, and sets are taken with different zero settings.

        B
        /
   A --- O          horizontal angle AOB at O

Vertical angles

A vertical angle is the angle between the line of sight and the horizontal plane (an angle of elevation or depression).

1. Set up the theodolite and level it with reference to the altitude bubble. Set the vertical-circle verniers to zero with the vertical clamp and tangent screw, and bring the altitude bubble central with the clip screw; the line of sight is then horizontal.
2. On face left, bisect the target using the vertical clamp and tangent screw. Check that the altitude bubble is central and read both verniers of the vertical circle.
3. Change face (transit the telescope) and repeat. The mean of the two results is the vertical angle; the index error is removed.

$$\text{Index error} = \frac{(FL + FR) - 360^\circ}{2}, \quad \text{Vertical angle} = 90^\circ - Z$$

where $Z = (FL - FR + 360^\circ)/2$ is the mean zenith distance, for a zenith-type circle.

                    target P
                  /
      horizontal /  alpha (angle of elevation)
   O ------------+