Angle measurement by repetition method

Repetition method. The same horizontal angle (say AOB) is added several times mechanically on the circle, and the circle is read only at the start and at the end. The mean angle = total accumulated angle / number of repetitions.

Procedure (angle AOB, three repetitions)

1. Set up, centre and level the theodolite at O, with the telescope on face left. Using the upper clamp and upper tangent screw, set vernier A to 0°00′00″ (or any convenient reading), and read both verniers (A and B).
2. Loosen the lower clamp, sight A, tighten the lower clamp and bisect A exactly with the lower tangent screw.
3. Loosen the upper clamp and swing the telescope clockwise to B. Clamp it and bisect B with the upper tangent screw. Read the verniers (the first value of the angle may be booked as a check).
4. Loosen the lower clamp, swing the telescope back to A, clamp it and bisect A with the lower tangent screw. The circle reading does not change, because the upper plate is clamped to the circle.
5. Loosen the upper clamp and sight B again; bisect it with the upper tangent screw. The reading now shows the angle added twice.
6. Repeat steps 4 and 5 once more for the third repetition. Read both verniers; the reading now shows the angle added three times.
7. Change face (transit the telescope to face right) and repeat the same three repetitions, usually swinging anticlockwise. Take the mean of both faces.

   A                       B
    \ (upper plate)       /
     \                  /
      -------- O --------
   First: set to A, swing to B (upper clamp)
   Next: lower clamp, back to A, then upper again to B

Computation (example)

For face left: accumulated angle = (187°15′50″ - 0°00′20″ + 367°15′58″ - 180°00′20″)/2 = 187°15′34.0″. Angle (face left) = 187°15′34.0″ / 3 = 62°25′11.33″.

For face right: accumulated angle = 187°15′20.0″, angle = 62°25′06.67″.

$$\text{Mean angle AOB} = \frac{62°25′11.33″ + 62°25′06.67″}{2} = 62°25′09.0″$$

Answer: mean angle AOB = 62°25′09.0″.

The reading error of the circle is divided by the number of repetitions (here 3), so the final angle is more accurate than one from a single reading.