Horizontal angle by repetition and direction methods

(i) 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
    \                    /
     \                 /
      ------ O ------

Field notes

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.

(ii) Direction (reiteration) method

Direction (reiteration) method. It is used when several angles are to be measured at one station. All the directions are observed from one reference object (RO), and the angles are obtained as differences of directions. The circle is not moved between objects, so many angles are obtained in one round.

Procedure (stations A, B, C observed from O)

1. Set up, centre and level the instrument at O, on face left.
2. Set vernier A at 0°00′00″ (or any value) with the upper clamp, then bisect the reference object A (a well-defined point) using the lower clamp and lower tangent screw. Read both verniers.
3. Loosen the upper clamp and swing clockwise to B; bisect it and read both verniers. Do the same for C, and finally close the horizon by sighting A again. The second reading on A should be nearly the same as the first; any difference is the closing error.
4. Change face (face right). Sight A again and swing anticlockwise through C and B back to A, reading both verniers each time.
5. Take the mean of the two faces for each station. The angle between any two stations = difference of their mean directions.
6. For more accuracy, repeat the whole round (a set) with different initial readings (for 2 sets, start at 0° and 90°), and take the mean of the angles from the different sets.

         B       C
          \     /
           \   /
   A ------ O          RO = A (reference object)

Field notes (example)

The angles are AOB = 100°40′20.0″, BOC = 60°40′10.0″ and COA = 198°39′35.0″. The closing reading on A differs from the first by only 0°00′05.0″, which is small enough to accept, so no adjustment is needed.