Chapter 2 of 9

Levelling & theodolite

In the TNPSC AE Civil syllabus under Engineering Survey · 2 parts

Levelling · Theodolite Surveying & Traversing

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📑 Contents (19 sections)

Part 1 of 2

Levelling

Last reviewed 16 Sept 2026 · 11 min read

Terms

Term	Meaning
Level surface	A surface parallel to the mean spheroidal surface of the earth (e.g. still water surface) — every point is equidistant from the earth's centre
Horizontal plane	Tangent to the level surface at a point
Datum	Reference surface to which elevations are referred — commonly mean sea level (MSL)
Reduced level (RL)	Elevation of a point above (or below) the datum
Benchmark (BM)	A fixed point of known RL — GTS benchmarks (Great Trigonometrical Survey, established by the Survey of India), permanent BMs (by government departments), arbitrary BMs (assumed RL for small works), temporary BMs (at the end of a day's work)
Line of collimation (line of sight)	Line through the intersection of cross hairs and the optical centre of the objective
Height of instrument (HI)	RL of the line of collimation when the instrument is levelled
Back sight (BS)	First staff reading after setting up the instrument — taken on a point of known RL (BM or change point)
Fore sight (FS)	Last staff reading before shifting the instrument
Intermediate sight (IS)	Any reading between BS and FS at the same setup
Change point (turning point, CP)	A point on which both an FS (from one setup) and a BS (from the next setup) are taken

Instruments

Levels

Level	Features
Dumpy level	Telescope rigidly fixed to the vertical spindle; simple, stable, retains adjustments; widely used
Wye (Y) level	Telescope rests in Y-shaped supports and can be removed/rotated — easy to test and adjust, but wears
Tilting level	Telescope can be tilted slightly about a horizontal axis with a fine screw to centre the bubble for each sight — quicker accurate levelling
Automatic (auto) level	A compensator (suspended prisms) automatically makes the line of sight horizontal after rough levelling with a circular bubble — fast and accurate; most common now
Digital level	Reads a bar-coded staff electronically and stores data
Laser level	Rotating laser beam defines a horizontal (or inclined) plane — construction work

Levelling staves

Self-reading staves — read directly by the observer at the instrument: solid, folding (e.g. 4 m in two parts) and telescopic (Sopwith) staves, graduated commonly to 5 mm divisions.
Target staff — a sliding target is moved by the staffman to the line of sight and read by him (long sights, precise work).
Bar-coded (invar) staves for digital levels.

Adjustments of a level

Temporary adjustments (at every setup)

Setting up — tripod firmly set, instrument fixed, approximate levelling by tripod legs.
Levelling up — using foot screws (three-screw head: turn two screws in opposite directions for one axis, then the third screw for the perpendicular axis) until the bubble remains central in all positions.
Elimination of parallax — focus the eyepiece on the cross hairs first (against a light background), then focus the objective on the staff until there is no apparent movement of the image relative to the cross hairs.

Permanent adjustments (dumpy level)

Axis of the bubble tube perpendicular to the vertical axis (so the bubble stays central through a full rotation).
Line of collimation parallel to the axis of the bubble tube — checked by the two-peg test.
Horizontal cross hair perpendicular to the vertical axis.

Two-peg test

Set two pegs A and B about 50–100 m apart. Set the level midway; the difference of staff readings gives the true difference in level (collimation error cancels as sight lengths are equal).
Set the level near one peg (or beyond one peg) and read both staves; the apparent difference differs from the true difference if the line of collimation is inclined.
Compute the error and the correct staff reading on the far peg; adjust the cross hairs (dumpy level) or the bubble (tilting level) accordingly.

Types of levelling

Type	Purpose / method
Simple levelling	Difference in level of two nearby points from one setup
Differential (compound) levelling	Difference in level of points far apart or not visible from one setup — series of setups with change points
Fly levelling	Approximate levelling (long sights, few readings) to carry levels roughly or check benchmark values
Check levelling	Levelling back to the starting benchmark (or to another BM) to check the work
Profile (longitudinal) levelling	RLs of points at regular intervals along a line (road, canal, pipeline) to draw a longitudinal section
Cross-sectioning	RLs along lines perpendicular to the centre line — for earthwork quantities
Reciprocal levelling	Accurate difference in level of two points far apart with an obstacle between (river, valley) — eliminates collimation, curvature and refraction errors
Precise levelling	High-accuracy levelling with special instruments and procedures — establishing benchmarks
Trigonometric levelling	From measured vertical angles and distances (theodolite/total station)
Barometric levelling	From differences in atmospheric pressure — rough, exploratory
Hypsometric levelling	From the boiling point of water (which falls with altitude) — rough

Reduction of levels

FormulaHeight of instrument (collimation) method

H I = R L of BM or CP + B S; R L of a point = H I - (I S or F S)

Arithmetic check: $\sum B S - \sum F S = last RL - first RL$

Quick and less laborious; suitable for profile levelling with many intermediate sights; no check on intermediate RLs.

FormulaRise and fall method

Compare each reading with the previous reading at the same setup:

Previous reading − present reading > 0 → rise; < 0 → fall.
$R L = previous RL + rise$ (or − fall).

Arithmetic checks: $\sum B S - \sum F S = \sum rise - \sum fall = last RL - first RL$

More laborious but provides a complete check on all readings (including intermediate sights) — preferred for accurate work such as fly and check levelling.

Curvature and refraction

Over long sights, the horizontal line of sight departs from the level surface.

Curvature makes staff readings too large (the level line curves downward away from the line of sight) — correction is negative.
Refraction bends the line of sight downward towards the earth, making readings smaller — correction is positive; about one-seventh of the curvature correction.

FormulaCurvature and refraction corrections (

d

in km, corrections in m)

Curvature: $C_{c} = - \frac{d ^{2}}{2 R} = - 0.07849 d^{2}$
Refraction: $C_{r} = + \frac{1}{7} \times 0.07849 d^{2} = + 0.01121 d^{2}$
Combined: $C = - 0.06728 d^{2}$

Distance to the visible horizon from a height $h$ (m):

d = \frac{h}{0.06728} = 3.855 h km

Reciprocal levelling

Used when the instrument cannot be placed midway between two points (e.g. across a wide river).

Set the level near A; read the staff at A ( $a_{1}$ ) and at B ( $b_{1}$ ).
Set the level near B; read the staff at A ( $a_{2}$ ) and at B ( $b_{2}$ ).

True difference in level = \frac{( b _{1} - a _{1} ) + ( b _{2} - a _{2} )}{2}

(positive → B is lower than A.) The method eliminates errors due to collimation, curvature and refraction (if refraction is the same at both times). The combined error $e = \frac{( b _{1} - a _{1} ) - ( b _{2} - a _{2} )}{2}$ .

Sensitivity of the bubble tube

The sensitivity is the angle through which the line of sight tilts when the bubble moves by one division — a more sensitive bubble moves more for a small tilt.

FormulaSensitivity of bubble tube

α = \frac{s}{n D} radians = \frac{s}{n D} \times 206265 seconds

R = \frac{n l D}{s}

$s$ = difference in staff readings for bubble movement of $n$ divisions; $D$ = distance from instrument to staff; $l$ = length of one bubble division; $R$ = radius of curvature of the bubble tube.

Sensitivity increases with larger radius of curvature, larger diameter of the tube, longer bubble, lower viscosity and smooth interior surface.

Errors in levelling

Type	Examples
Instrumental	Line of collimation not parallel to bubble axis (collimation error — eliminated by equal back and fore sight distances), sluggish bubble, defective staff graduations, loose tripod
Natural	Curvature and refraction, wind (vibration), sun (unequal expansion of instrument and bubble), settlement of tripod or staff on soft ground
Personal	Imperfect levelling, parallax, staff not held vertical (reading too large), wrong reading or booking, bubble not central at the time of reading, change point not firm

Permissible closing errors (commonly quoted, $K$ = distance in km): rough levelling about $\pm 100 K$ mm; ordinary levelling about $\pm 24 K$ mm; accurate levelling about $\pm 12 K$ mm; precise levelling about $\pm 4 K$ mm. The closing error is distributed to intermediate points in proportion to distance (or number of setups).

Worked examples

Worked ExampleExample 1 — height of instrument and rise and fall methods

The following readings were taken with a level (BM = 100.000 m): 1.585 (BS on BM), 1.965 (IS at P1), 2.325 (FS at CP1), 1.215 (BS at CP1), 0.855 (IS at P2), 1.035 (FS at P3). Find the RLs by both methods and apply checks.

Solution — HI method.

Station	BS	IS	FS	HI	RL
BM	1.585			101.585	100.000
P1		1.965			99.620
CP1	1.215		2.325	100.475	99.260
P2		0.855			99.620
P3			1.035		99.440

Check: $\sum B S - \sum F S = 2.800 - 3.360 = - 0.560 = 99.440 - 100.000$ ✓

Rise and fall method: BM→P1 fall 0.380 (99.620); P1→CP1 fall 0.360 (99.260); CP1→P2 rise 0.360 (99.620); P2→P3 fall 0.180 (99.440). Check: $\sum rise - \sum fall = 0.360 - 0.920 = - 0.560$ ✓

Worked ExampleExample 2 — curvature and refraction

Find the combined correction for a sight of 2 km, and the distance to the visible horizon from a lighthouse 25 m high.

Solution. $C = - 0.06728 \times 2^{2} = - 0.269 m$ $d = 3.855 25 = 19.3 km$

Worked ExampleExample 3 — reciprocal levelling

With the level near A: staff at A = 1.625, at B = 2.545. With the level near B: staff at A = 0.920, at B = 1.810. Find the true difference in level and the combined error.

Solution. True difference $= \frac{( 2.545 - 1.625 ) + ( 1.810 - 0.920 )}{2} = \frac{0.920 + 0.890}{2} = 0.905 m$ (B lower than A) Combined error $= \frac{0.920 - 0.890}{2} = 0.015 m$

Worked ExampleExample 4 — sensitivity of bubble tube

With the staff 100 m away, the bubble was moved through 2 divisions (each 2 mm) and the staff readings changed by 0.010 m. Find the sensitivity and radius of the bubble tube.

Solution. $α = \frac{0.010}{2 \times 100} \times 206265 = 10. 3^{''}$ per division $R = \frac{2 \times 0.002 \times 100}{0.010} = 40 m$

Frequently tested points

RL relative to datum (MSL); GTS benchmarks by Survey of India.
BS — first reading after setup (known RL); FS — last reading before shifting; CP has both FS and BS.
Parallax removed by focusing the eyepiece first, then the objective.
Two-peg test checks line of collimation parallel to bubble axis.
Automatic level — compensator; tilting level — tilting screw; digital level — bar-coded staff.
HI method: $R L = H I - r e a d in g$ ; check $\sum B S - \sum F S$ only; rise and fall checks all readings.
Curvature $- 0.07849 d^{2}$ ; refraction $+ 0.01121 d^{2}$ (1/7); combined $- 0.06728 d^{2}$ ; visible horizon $3.855 h$ km.
Reciprocal levelling eliminates collimation, curvature and refraction errors.
Equal BS and FS distances eliminate collimation and curvature errors.
Staff not vertical → reading too large.
Sensitivity $α = s / (n D)$ rad; increases with radius of the tube.

Common MistakeCommon mistakes

Applying the arithmetic check of the HI method to intermediate sights (it does not check them).
Adding the curvature correction to staff readings (it is subtractive).
Taking the difference of readings in reciprocal levelling from one setup only.

Revision SummaryChapter summary

Levelling determines relative elevations with respect to a datum using levels and staves, with benchmarks as references.
Temporary adjustments (levelling, parallax removal) are made at each setup; permanent adjustments are checked by tests such as the two-peg test.
Differential, fly, check, profile, cross-section, reciprocal, precise and trigonometric levelling serve different purposes.
RLs are reduced by the height of instrument or rise and fall methods, with arithmetic checks.
Curvature, refraction, collimation and other errors are minimised by balanced sights, reciprocal observations and careful procedure.

Part 2 of 2

Theodolite Surveying & Traversing

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 $= (2 n - 4) \times 90°$ ; sum of exterior angles $= (2 n + 4) \times 90°$ ; sum of deflection angles (right − left) $= 360°$ . Angular misclosure is distributed equally among the angles (if all measured with equal care).

Traverse computations

FormulaLatitudes and departures

For a line of length $l$ and whole circle bearing (or reduced bearing) $θ$ :

Latitude L = l cos θ, Departure D = l sin θ

Latitude: + northing, − southing
Departure: + easting, − westing

For a closed traverse, $\sum L = 0$ and $\sum D = 0$ ideally.

FormulaClosing error

e = (\sum L)^{2} + (\sum D)^{2}, tan δ = \frac{\sum D}{\sum L}

Relative precision (accuracy) $= \frac{e}{perimeter}$ , expressed as 1 in $\frac{perimeter}{e}$ .

Balancing the traverse

FormulaBalancing rules

Bowditch's rule (compass rule) — when angular and linear measurements are of equal precision:

C_{L} = - \sum L \times \frac{l}{\sum l}, C_{D} = - \sum D \times \frac{l}{\sum l}

Transit rule — when angles are measured more precisely than lengths:

C_{L} = - \sum L \times \frac{∣ L ∣}{\sum ∣ L ∣}, C_{D} = - \sum D \times \frac{∣ D ∣}{\sum ∣ D ∣}

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 $\sum L = 0$ and $\sum D = 0$ provide two equations:

Length and bearing of one line omitted — the missing line closes the traverse: $L = - \sum L_{k n o w n}$ , $D = - \sum D_{k n o w n}$ .
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

Worked ExampleExample 1 — latitudes, departures and closing error

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

$e = 0.18 0^{2} + 0.13 8^{2} = 0.227 m$ ; perimeter = 870.5 m → relative precision ≈ 1 in 3840

Worked ExampleExample 2 — Bowditch correction

Find the Bowditch corrections to the latitude and departure of line AB in Example 1.

Solution. $C_{L} = - 0.180 \times \frac{250}{870.5} = - 0.052 m$ ; $C_{D} = + 0.138 \times \frac{250}{870.5} = + 0.040 m$ Corrected: latitude 124.948 m, departure 216.546 m.

Worked ExampleExample 3 — omitted measurement

In the traverse of Example 1, suppose the length and bearing of DA were not measured. Find them.

Solution. From AB, BC, CD: $\sum L = - 160.885$ , $\sum D = + 81.339$ For DA: $L = + 160.885$ , $D = - 81.339$ Length $= 160.88 5^{2} + 81.33 9^{2} = 180.28 m$ Reduced bearing $= tan^{- 1} (81.339/160.885) = 26°49.6'$ in the NW quadrant → WCB $= 360° - 26°49.6' = 333°10.4'$

Worked ExampleExample 4 — repetition method

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 $= l cos θ$ , departure $= l sin θ$ ; closing error $(\sum L)^{2} + (\sum D)^{2}$ .
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 $\sum L = 0$ and $\sum D = 0$ .

Common MistakeCommon mistakes

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.

Revision SummaryChapter summary

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.

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