There are three methods of measuring horizontal angles: 1. Ordinary Method. 2. Repetition Method. 3. Reiteration Method.

**1. Ordinary Method (Fig. 9.6.):**

**To measure horizontal angle AOB:**

(i) Set up the theodolite at station-point O and level it accurately.

(ii) Set the vernier A to the zero or 360° of the horizontal circle so do this, loosen the upper clamp and tum the upper plate until the zero of vernier A nearly coincides with the zero of the horizontal circle. Tighten the upper clamp and turn its tangent screw to bring the two zeros into exact coincidence.

**(iii) ****Loosen the lower clamp: **

Turn the instrument and direct the telescope approximately to the left hand object (A) by sighting over the top of the telescope. Tighten the lower clamp and bisect A exactly by turning the lower tangent screw. Bring the point A into exact coincidence with the point of intersection of the cross-hairs at diagram by using the vertical circle clamp and tangent screws.

Alternatively bring the vertical cross-hair exactly on the lowest visible portion of the arrow or the ranging rod representing the point A in order to minimise the error due to non- verticality of the object.

(iv) Having sighted the object A, see whether the vernier A still reads zero. This is done to detect the error caused by turning the wrong tangent screw. Read the vernier B and record both vernier readings.

(v) Loosen the upper clamp and turn the telescope clockwise until the line of sight is set nearly on the right hand object (B). Then tighten the upper clamp and by turning its tangent screw, bisect B exactly. In this operation, the lower clamp and its tangent screws should not be touched.

**(vi) ****Read both verniers: **

The reading of the vernier A which was initially set at zero gives the value of the angle AOB directly and that of the other venier B by deducting 180°. The mean of the two vernier readings (after deducting 180° from the reading on vernier B gives the value of the required angle AOB.)

(vii) Change the face of the instrument and repeat the whole process. The mean of the two vernier readings gives the second value of the angle ABC which should be approximately or exactly equal to the previous value.

(viii) The mean of the two values of the angle AOB, one with the face left and the other with the face right, gives the required angle free from all instrumental errors.

**Note:**

The vernier A is initially set to zero for convenience only. It may be set at any other reading, and the difference between the initial and the final readings of the vernier A will give the value of the angle AOB.

**2. Repetition Method:**

This method is used for very accurate work. In this method; the same angle is added several times mechanically and the correct value of the angle is obtained by dividing the accumulated reading by the number of repetitions. The number of repetitions made usually is six, three with the face left and three with the face right. In this way, angles can be measured to a finer degree of accuracy than that obtainable with the least count of the vernier.

However, it cannot be said that any desired degree of accuracy can be obtained by increasing the number of repetitions considerably because the errors due to frequent clamping etc. are introduced. There is therefore, no advantage in increasing the number of observations beyond a certain limit. Three repetitions with face left and three repetitions with face right are quite sufficient except in cases of very precise work.

**To measure the horizontal angle AOB by repetition: **

(i) Set up the theodolite at station -point O and level it accurately. (The face of the instrument should be left.)

(ii) Set the vernier A to zero or 360° by using the upper clamp and its tangent screw. Then loosen the lower clamp, direct the telescope to the left hand object A, and bisect A exactly by using the lower clamp and its tangent screw.

(iii) Check the reading of the vernier A and see whether it still reads zero, and then read the other vernier B.

(iv) Loosen the upper clamp, turn the telescope clock-wise and bisect the right hand object (B) exactly by using the upper clamp and its tangent screw.

(v) Read both verniers- The object of reading the verniers is to obtain the approximate value of the angle. (Suppose the mean reading is 50°4′).

(vi) Loosen the lower clamp and turn the telescope clock-wise until the object (A) is sighted again. Bisect A accurately using the lower tangent screw. Check the vernier readings which must be the same as before.

(vii) Loosen the upper clamp, turn the telescope clock-wise and again sight towards B. Bisect B accurately by using the upper tangent screw.

The verniers will now read twice the value of the angle (It should he approximately 100 °8′).

(viii) Repeat the process until the angle is repeated the required number of times (usually 3). Read both verniers. The final readings after n repetition should be approximately n x (50°4′). Divide the sum by the number of repetitions and the result thus obtained gives the correct value of the angle AOB.

(ix)** **Change the face of the instrument (now the face will be right). Repeat exactly in the same manner and find another value of the angle AOB.

(x) The average of the two values of the angle thus obtained gives the required precise value of the angle (AOB).

The observations are recorded in the tabular form as given in Table 9.1.

**Errors Eliminated by Measuring the Horizontal Angles by Repetition: **

**(i) Errors eliminated by changing face of theodolite: **

(a) Error due to the line of collimation not being perpendicular to the horizontal axis of the telescope.

(b) Error due to the horizontal axis of the telescope not being perpendicular to the vertical axis.

(c) Error due to the line of collimation not coinciding with the axis of the telescope.

**(ii) Errors eliminated by reading both verniers and averaging the readings: **

(a) Error due to the axis of the vernier-plate not coinciding with the axis of the main scale plate.

(b) Error due to the unequal graduations.

**(iii) Error eliminated by measuring the angle on different parts of the circle: **

(a) Error due to unequal graduations.

(iv) The errors in the pointing tend to compensate each other and the remaining error is minimised by the division.

(v) The error due to dishevelment of the bubble can be minimised by taking precautions in levelling.

**3. Reiteration Method (Fig. 9.7):**

Reiteration is another precise and comparatively less tedious method of measuring the horizontal angles. It is generally preferred when several angles are to be measured at a particular station. This method consists in measuring the several angles successively, and finally closing the horizon at the starting point. The final reading of the vernier A should be the same as its initial reading. If not, the discrepancy is equally distributed among all the measured angles.

Suppose it is required to measure the angles AOB, BOC and COD.

**Then to measure these angles by reiteration method: **

(i) Set up the instrument over station point O and level it accurately.

(ii) Set the vernier A to 0 or 360° by using the upper clamp and its tangent screw.

(iii) Direct the telescope to some well-defined object (P) or say, the station point A, which is known as the ‘Reference object’. Bisect it accurately by using the lower clamp and its tangent screw. Check the reading at vernier A which should still be 0 or 360° and note the reading at vernier B.

(iv) Loosen the upper clamp and turn the telescope clockwise until the point B is exactly sighted by using the upper tangent screw. Read both verniers. The mean of the two vernier readings (after deducting 180° from the reading at vernier B) will give the value of the angle AOB.

(v) Similarly bisect C and D successively, read both verniers at each bisection, find the values of the angles BOC and COD.

(vi) Finally, close the horizon by sighting towards the reference object (P) or the station-point A.

(vii) The vernier A should now read 360°. If not, note down the error. This error occurs due to slip etc.

(viii) If the error is small, it is equally distributed among the several observed angles. If large, the readings should be discarded and a new set of readings be taken.

(ix) Change the face of the instrument.

(x) Set the vernier A to a reading other than 0°, say, 60° or 90°). This is done to avoid errors of graduation.

(xi) Again measure the angles in the same manner by turning the telescope this time in the counter-clockwise direction to compensate or slip and errors due to twisting of the instrument.

(xii) Close the horizon and apply the necessary correction to all the angles as before.

(xiii) The mean of the two results for each angle is taken as its true value.