Last reviewed 16 Sept 2026 · 10 min read
The theodolite
A theodolite measures horizontal and vertical angles accurately. It is also used for prolonging lines, setting out angles and curves, levelling (trigonometric) and, with stadia hairs, measuring distances (tacheometry).
Types
- Transit theodolite — the telescope can be revolved through 180° in a vertical plane about its horizontal axis (standard today).
- Non-transit theodolite — telescope cannot be transited (obsolete).
- By reading system: vernier theodolites (least count commonly 20″), micrometer/optical theodolites (1″ or better), electronic digital theodolites (with digital angle display); total stations combine electronic theodolites with EDM.
Main parts
Trivet and tribrach with levelling screws, lower plate (carrying the horizontal circle) with lower clamp and tangent screw, upper plate (carrying verniers/reading system) with upper clamp and tangent screw, plate levels, standards (A-frame) supporting the horizontal (trunnion) axis, telescope, vertical circle with vertical circle vernier/index and altitude bubble, plumb bob or optical plummet, tripod.
Technical terms
| Term | Meaning |
|---|---|
| Vertical axis | Axis about which the instrument rotates in a horizontal plane |
| Horizontal (trunnion) axis | Axis about which the telescope rotates in a vertical plane |
| Line of collimation | Line through the intersection of cross hairs and optical centre of objective |
| Centring | Setting the vertical axis exactly over the station mark |
| Transiting (plunging, reversing) | Rotating the telescope 180° about the horizontal axis |
| Swinging | Rotating the telescope about the vertical axis — right swing (clockwise) or left swing (anticlockwise) |
| Face left (telescope normal) | Vertical circle on the left of the observer when sighting |
| Face right (telescope inverted) | Vertical circle on the right of the observer |
| Changing face | Transiting and swinging so that the face changes |
Temporary adjustments
- Setting up over the station with tripod.
- Centring — plumb bob or optical plummet over the station mark; shifting head for fine centring.
- Levelling up — using plate levels and foot screws: bubble central parallel to two foot screws, then perpendicular using the third screw; repeat until central in all positions.
- Focusing — eyepiece for cross hairs, objective for the object — elimination of parallax.
Measurement of angles
Horizontal angle (simple method)
Set the vernier to 0°, sight the first station using the lower clamp; release the upper clamp, sight the second station, read the angle. Repeat on the other face and take the mean.
Repetition method
Used to measure a single horizontal angle to a finer degree of accuracy than the least count.
- Measure the angle once and do not reset the vernier; with the lower clamp, sight the first station again.
- Using the upper clamp, sight the second station — the reading accumulates the angle twice.
- Repeat for a set number of repetitions (e.g. 3 on face left and 3 on face right, with left and right swings).
- Angle = (final reading) ÷ (number of repetitions) — adding full circles if the reading passes 360°.
Errors eliminated/reduced: errors of eccentricity of verniers, errors due to inadequate least count (reading error distributed), errors of graduations (different parts of the circle used), errors of collimation and trunnion axis (face left + face right), errors due to slip partly — accuracy improves.
Reiteration (direction) method
Used when several angles are to be measured at a station: directions of all stations are read successively from a reference station, closing back on it (the horizon is closed; sum of angles = 360°). Repeated with the circle set at different initial readings on both faces.
Vertical angles
Angle of elevation (above the horizontal) or depression (below). Measured using the vertical circle with the altitude bubble central; face left and face right readings are averaged to eliminate index error.
Other operations
- Magnetic bearing of a line — using a trough or tubular compass attached to the theodolite.
- Deflection angle — angle a line makes with the prolongation of the preceding line (right or left) — used in route surveys.
- Prolonging a straight line — by double sighting (face left and face right, taking the mean point) to eliminate collimation error.
- Setting out angles, ranging a line, locating the intersection of two lines.
Fundamental lines and permanent adjustments
| Required relationship | Adjustment/test |
|---|---|
| Axis of plate levels perpendicular to the vertical axis | Plate level test — bubble central in all positions |
| Line of collimation perpendicular to the horizontal axis | Collimation test — prolonging a line by face left and right (spire test variants) |
| Horizontal axis perpendicular to the vertical axis | Spire test — sighting a high point and a low point on both faces |
| Axis of altitude level parallel to line of collimation (vertical circle index correct) | Vertical index test — two-peg-type test for zero reading when line of sight is horizontal |
| Vertical cross hair in a plane perpendicular to the horizontal axis | Cross hair test |
Errors eliminated by face-left and face-right observations
| Eliminated by averaging both faces | Not eliminated by changing face |
|---|---|
| Collimation error (line of sight not perpendicular to horizontal axis) | Error due to vertical axis not being truly vertical (imperfect levelling / plate level error) |
| Horizontal (trunnion) axis error (not perpendicular to vertical axis) | Graduation errors (reduced by using different parts of the circle) |
| Index error of vertical circle | Personal and natural errors |
| Eccentricity of verniers (by reading both verniers) |
Theodolite traversing
Methods
| Method | Description |
|---|---|
| Included angles method | Interior (or exterior) angles measured at each station — closed traverses; common for boundaries |
| Deflection angles method | Deflection angles measured — open traverses such as roads, railways, canals |
| Direct angle (angle to the right) method | Clockwise angles from the back station to the forward station |
| Fast needle method | Magnetic bearings measured with the theodolite compass, carried forward with the circle clamped |
| Loose needle method | Magnetic bearing observed independently at each station |
Checks for a closed traverse: sum of interior angles ; sum of exterior angles ; sum of deflection angles (right − left) . Angular misclosure is distributed equally among the angles (if all measured with equal care).
Traverse computations
For a line of length and whole circle bearing (or reduced bearing) :
- Latitude: + northing, − southing
- Departure: + easting, − westing
For a closed traverse, and ideally.
Relative precision (accuracy) , expressed as 1 in .
Balancing the traverse
Bowditch's rule (compass rule) — when angular and linear measurements are of equal precision:
Transit rule — when angles are measured more precisely than lengths:
Other methods: graphical (Bowditch) adjustment, third rule, Crandall's method and least squares adjustment.
Coordinates
- Consecutive coordinates — latitude and departure of each line relative to its starting point.
- Independent (total) coordinates — coordinates of each station relative to a common origin; obtained by cumulative addition of corrected consecutive coordinates.
- Gale's traverse table — a standard tabular format for computing bearings, latitudes, departures, corrections and independent coordinates.
Omitted measurements
When some measurements of a closed traverse are missing (not measured or lost), they can be computed since and provide two equations:
- Length and bearing of one line omitted — the missing line closes the traverse: , .
- Length of one line and bearing of another omitted, or lengths of two lines omitted, or bearings of two lines omitted — solved by trigonometric relations (sometimes after joining known points with a closing line).
Worked examples
A closed traverse ABCDA has the following data:
| Line | Length (m) | WCB |
|---|---|---|
| AB | 250.0 | 60°00′ |
| BC | 180.0 | 150°00′ |
| CD | 260.0 | 240°00′ |
| DA | 180.5 | 333°10′ |
Find the closing error and relative precision.
Solution.
| Line | Latitude | Departure |
|---|---|---|
| AB | +125.000 | +216.506 |
| BC | −155.885 | +90.000 |
| CD | −130.000 | −225.167 |
| DA | +161.065 | −81.477 |
| Sum | +0.180 | −0.138 |
; perimeter = 870.5 m → relative precision ≈ 1 in 3840
Find the Bowditch corrections to the latitude and departure of line AB in Example 1.
Solution. ; Corrected: latitude 124.948 m, departure 216.546 m.
In the traverse of Example 1, suppose the length and bearing of DA were not measured. Find them.
Solution. From AB, BC, CD: , For DA: , Length Reduced bearing in the NW quadrant → WCB
An angle was measured by repetition 6 times; the initial reading was 0°00′00″ and the final reading after six repetitions was 243°28′30″. Find the angle.
Solution. Angle = 243°28′30″ ÷ 6 = 40°34′45″
Frequently tested points
- Transit theodolite — telescope revolves 180° vertically; vernier least count commonly 20″.
- Face left = vertical circle on the left of the observer.
- Temporary adjustments: setting up, centring, levelling up, focusing (parallax).
- Repetition method — single angle, higher accuracy; reiteration — several angles at a station.
- Face left and face right eliminate collimation, trunnion axis and index errors; not the vertical axis error.
- Spire test — horizontal axis perpendicular to vertical axis.
- Deflection angles for open (route) traverses; included angles for closed traverses.
- Latitude , departure ; closing error .
- Bowditch's rule ∝ length of line (equal angular and linear accuracy); transit rule ∝ latitude/departure (angles more precise).
- Consecutive vs independent coordinates; Gale's traverse table.
- Omitted measurements from and .
- Believing that changing face eliminates errors due to imperfect levelling.
- Using the transit rule when lengths and angles are equally precise.
- Forgetting signs of latitude and departure when bearings are in SE, SW or NW quadrants.
- The transit theodolite measures horizontal and vertical angles and performs many field operations.
- Temporary adjustments and methods such as repetition and reiteration give accurate angles.
- Permanent adjustments maintain the fundamental axis relationships; face-left/face-right observations eliminate several instrumental errors.
- Theodolite traverses are computed through latitudes and departures, closing error and relative precision.
- Traverses are balanced by Bowditch's or transit rules, converted to independent coordinates, and omitted measurements are computed from closure conditions.