A vernier theodolite reads to 20 seconds. So how do you measure an angle more precisely than 20 seconds with an instrument that cannot read finer than that?
The repetition method answers exactly that question, and it does so with a trick that is genuinely clever. This post covers both methods of measuring horizontal angles, and the measurement of vertical angles.
The Two Methods
Horizontal angles with the help of a theodolite are now generally measured by two methods:
- Repetition method
- Reiteration method
1. Repetition Method
Used to measure a horizontal angle to a finer degree of accuracy than that of the least count of the vernier.
The Procedure
- The angle is measured two or more times without setting the vernier to zero.
- The angle is found by dividing the final vernier reading by the number of observations.
Angle = Final vernier reading ÷ Number of observations
Why It Beats the Least Count
This is the idea worth understanding properly, because it seems to get something for nothing.
The key phrase is “without setting the vernier to zero”. Ordinarily you would zero the instrument, measure the angle, read it, then zero again. In the repetition method you do not reset — each new measurement of the same angle is added on top of the accumulated reading.
Measure a 30° angle six times and the vernier ends up reading about 180°. Divide by six and you recover the angle.
Now consider the error. Your reading error is limited by the least count — say 20 seconds — but you only make that error once, on the final reading. Dividing by six divides that error by six as well.
Reading once but dividing by n reduces the reading error to one nth of the least count. Six repetitions of a 20″ instrument give roughly 3″ precision.
This is why the accumulation matters. If you read and reset each time, you would incur the full least-count error at every reading and averaging would help far less. By letting the angle build up on the circle, the instrument’s coarseness is spread across many angles but read only once.
Recall How the Two Plates Make This Possible
Repetition depends entirely on the two concentric spindles described in the parts of a theodolite.
- Clamp the upper plate to the lower and swing both together — the reading travels with you unchanged.
- Clamp the lower plate and release the upper — the reading changes as you turn.
Alternating between these two states is what lets the same angle be laid down repeatedly around the circle without ever returning to zero. Without the twin-spindle design, the repetition method would be impossible.
2. Reiteration Method
Also known as the direction method or the method of series.
- Suitable for the measurement of the angles of a group having a common vertex point.
- All the angles are measured successively, and the horizon is closed.
Closing the Horizon
Closing the horizon provides a check, as a closed horizon should give a reading of 360° on the vernier.
How the Check Works
Imagine standing at a station with five targets around you. You sight the first, then swing to the second, third, fourth, fifth — and finally back to the first.
You have now turned through a complete circle. The angles you measured must therefore add up to exactly 360°, because that is what a full turn is.
If your readings sum to 359° 58′ instead, you know immediately that 2 minutes of error has crept in somewhere — and you know it before leaving the station, while there is still time to remeasure.
This is the same principle as the closed traverse check in compass surveying, and the same principle as fore-and-back bearings: arrange the work so that a known total must come out, and any shortfall reveals error.
Comparing the Two Methods
| Repetition Method | Reiteration Method | |
|---|---|---|
| Also called | — | Direction method, method of series |
| Purpose | Measure one angle more precisely than the least count | Measure several angles around a common vertex |
| Vernier zeroed between readings? | No — readings accumulate | Angles measured successively |
| Result obtained by | Dividing the final reading by the number of observations | Reading directions and differencing them |
| Built-in check | — | Closed horizon must read 360° |
| Best when | High precision needed on a single angle | Many angles radiate from one station |
Choosing Between Them
The two methods answer different questions, so they are not really rivals.
Ask “how precisely can I measure this one angle?” and the answer is the repetition method, which trades time for accuracy beyond the instrument’s least count.
Ask “how do I efficiently measure six angles from this station and know they are right?” and the answer is reiteration, which measures them all in one sweep and checks itself by closing at 360°.
Measurement of Vertical Angles
A vertical angle is the angle which the inclined line of sight to an object makes with the horizontal.
The vertical angle is measured by moving the telescope in the vertical circle.
Reading the Definition Carefully
Note that a vertical angle is measured from the horizontal, not from the vertical. This trips students up, since the name suggests otherwise.
The consequences:
- A horizontal line of sight gives a vertical angle of 0°.
- Sighting upwards gives a positive vertical angle, called an angle of elevation.
- Sighting downwards gives a negative vertical angle, called an angle of depression.
This is also why levelling can be done with a theodolite, as noted in the introduction — levelling is simply working with a vertical angle of zero, that is, a horizontal line of collimation.
Which Parts Are Involved
Vertical angle measurement uses a different set of components from horizontal measurement, and it is worth keeping them separate:
| Horizontal Angles | Vertical Angles | |
|---|---|---|
| Circle used | Graduated circle on the lower plate | Vertical circle |
| Read against | Verniers on the upper plate | Verniers on the index frame |
| Axis of rotation | Vertical axis | Horizontal axis |
| Motion called | Swinging | Rotating in the vertical circle |
Quick Revision Notes
- Two methods of measuring horizontal angles: repetition and reiteration.
- Repetition method measures an angle to a finer accuracy than the least count of the vernier.
- The angle is measured two or more times without setting the vernier to zero.
- Angle = final vernier reading ÷ number of observations.
- Precision improves because the reading error is incurred once but divided by n.
- Reiteration method is also called the direction method or method of series.
- It suits a group of angles having a common vertex point.
- All angles are measured successively and the horizon is closed.
- A closed horizon must read 360° on the vernier, which provides the check.
- A vertical angle is the angle the inclined line of sight makes with the horizontal.
- It is measured by moving the telescope in the vertical circle.
Mistakes Students Commonly Make
- Zeroing the vernier between repetitions. The whole method depends on not resetting.
- Forgetting to divide by the number of observations at the end.
- Thinking repetition simply averages readings. It accumulates the angle so that a single reading error is divided by n.
- Confusing the two methods. Repetition is for one angle precisely; reiteration is for several angles at one station.
- Omitting the alternative names for reiteration — direction method and method of series.
- Saying the closed horizon should read 180°. It must read 360°, a complete turn.
- Measuring a vertical angle from the vertical. It is measured from the horizontal.
Conclusion
The repetition method squeezes more accuracy out of an instrument than its least count allows, by letting the same angle accumulate around the circle and reading only once at the end — so the reading error is divided by the number of repetitions. The reiteration method takes a different approach, sweeping through every angle at a station in one series and checking itself by closing the horizon at exactly 360 degrees. And vertical angles, measured on the vertical circle from the horizontal rather than from the vertical, complete the theodolite’s basic repertoire.
Frequently Asked Questions
What are the two methods of measuring horizontal angles?
The repetition method and the reiteration method.
What is the repetition method used for?
To measure a horizontal angle to a finer degree of accuracy than the least count of the vernier allows.
How is the repetition method carried out?
The angle is measured two or more times without setting the vernier to zero, so the readings accumulate. The angle is then found by dividing the final vernier reading by the number of observations.
Why does the repetition method improve accuracy?
Because the angle accumulates on the circle while the reading error, limited by the least count, is incurred only once on the final reading. Dividing by the number of observations divides that error by the same factor.
What is the reiteration method also called?
The direction method or the method of series.
When is the reiteration method suitable?
For the measurement of the angles of a group having a common vertex point, where several angles radiate from one station.
What is closing the horizon?
Measuring all the angles successively around the station and returning to the first target, so that a complete circle has been turned.
What check does closing the horizon provide?
A closed horizon should give a reading of 360 degrees on the vernier. Any shortfall or excess reveals the amount of error before leaving the station.
What is a vertical angle?
The angle which the inclined line of sight to an object makes with the horizontal. It is measured by moving the telescope in the vertical circle.
Is a vertical angle measured from the vertical?
No. Despite the name, it is measured from the ho