Surveying and levelling

TOPICS TO BE COVERED :
Principles of Surveying, Technical
terms.
Calculation of reduced level by Height
of instrument.
Rise & Fall method, Simple problems
in levelling.
SURVEYING & LEVELLING :

SURVEYING
• Surveying is the art of and science of determining the relative positions of
various points or stations on the surface of the earth by measuring the
horizontal and vertical distances, angles, and taking the details of these
points and by preparing a map or plan to any suitable scale.”
• Surveying has been an essential element in the
development of the human environment since
the beginning of recorded history (5000 years ago).
• it is a requirement in the planning and execution of nearly every form of
construction. Its most familiar modern uses are in the fields of transport,
building and construction, communications, mapping, and the definition of
legal boundaries for land ownership.

TYPES OF SURVEYING
• Divisions of Surveying - The approximate shape of the Earth can best
be defined as an oblate tri-axial ovaloid. But, most of the civil
engineering works, concern only with a small portion of the earth
which seems to be a plane surface. Thus, based upon the
consideration of the shape of the earth, surveying is broadly divided
into two types :
Surveying
Plane Surveying Geodetic Surveying

• Plane surveying:
• Earth surface is considered a plane of x-y
dimensions.
Z-dimension (height) referenced to the mass
spherical surface of the earth (Mean Sea
Level).
Most engineering and property survey are
plane survey correction to curvature is made
for long strips (e.g. Highway).
For small projects covering Area less than
200 sq.km. Earth curvature is not counted in
calculating the distances. Earth surface is
considered as plane. (Angular error of 1” in
200 sq. km. area by assuming plane).

• Geodetic surveying:
• Earth surface is considered spherical in
revolution (actually ellipsoid)
Z is referenced to MSL (surface of earth)
Very precise surveys (boundaries and
coastal networks).
When survey extends over a large areas
more than 200 sq. km. and degree of
accuracy is also great. The curvature of
earth is also taken into account. Geodetic
survey is used to provide control points to
which small surveys can be connected.

• All distances and directions are horizontal;
• The direction of the plumb line is same at all points within the
limits of the survey;
• All angles (both horizontal and vertical) are plane angles;
• Elevations are with reference to a datum.

PRINCIPLES OF SURVEYING
The fundamental principles upon which the surveying is
being carried out are
• Working from whole to part.
1.Control points: - triangulation of traversing.
2.Triangulation divided into large triangle.
3.Triangles- subdivided in to small triangles
4.To control and localize minor errors.
5.On the other hand –It we work from the part of the
whole; small errors are magnified & uncontrollable at the
end.

• Deciding the position of any point, with
reference to at least two permanent objects
or stations whose position have already been
well defined.
1.Linear measurement
2.Angular measurements
3.Both the linear and angular measurements.
• E.g. Chain surveying- main lines & stations points are
checked by means of check or tie lines.

WORKING FROM WHOLE TO PART
The purpose of working from whole to part is
• to localize the errors and
• to control the accumulation of errors.
• This is being achieved by establishing a hierarchy of
networks of control points.
• The less precise networks are established within the
higher precise network and thus restrict the errors.
• To minimize the error limit, highest precise network
(primary network) of control points are established using
the most accurate / precise instruments for collection of
data and rigorous methods of analysis are employed to
find network parameters

• Introduction
• The horizontal distance between points, projected
onto a horizontal plane, is required to be measured in
order to prepare plan or map of the area surveyed.
This is done through chain surveying
• In surveying there are several methods for
measurement of distance. These are
1.Direct methods
2.Optical methods
3.Electronic method.
• In any work, the choice of a method depends on many
factors like field condition, accuracy required,
availability of resources (instruments, time, skill, fund
etc). Table 8.1 summarizes the principal methods,
instrument required, precision, use, errors of
measurement of distance
METHODS OF MEASUREMENT

• Direct Measurement When the distance between points / stations are
measured directly, usually by using tape or chain is known as direct
method.
 Chain (or) Tape
 Arrows
 Pegs
 Ranging Rods
 Offset Rods
 Plasterer’s laths and whites
 Plumb bob
Instruments Used in Chain Survey:

Chains: They are formed of straight links of Galvanized mild steel wires.
 Metric
Chain
 Gunter’s
Chain (or)
Surveyor’s
Chain
 Engineer’s
Chain
 Revenue
Chain
 Steel band
(or) band
chain
One tally = 2m
Two tally = 2 X 2m = 4m
Three tally = 3 X 2 m = 6 m

TECHNICAL TERMS USED IN SURVEYING
• Technical Terms :
Survey Stations
Chain Line or survey lines
 Main Survey Line
 Base line
 Tie Lines
 Check Lines
Offsets
 Perpendicular offsets
 Oblique offsets
 long offsets
 short offsets

Survey Stations:
1.Main Stations
2.Subsidiary or tie
Main Stations:
Main stations are the end of the lines, which command the boundaries of the survey
Subsidiary or the tie stations:
Subsidiary or the tie stations are the point selected on the main survey lines, where
it is necessary to locate the interior detail such as fences, hedges, building etc.

• Chain / Survey Lines
Main Survey Line:
The lines joining the main stations are called the main survey line or the
chain lines.
Base Lines:
It is main and longest line, which passes approximately through the
Centre of the field. All the other measurements to show the details of
the work are taken with respect of this line.
Tie or subsidiary lines:
A tie line joins two fixed points on the main survey lines. It helps to
check the accuracy of survey and to locate the interior details. The
position of each tie line should be close to some features, such as
paths, building etc.

Check Line:
A check line also termed as a proof line is a line joining the apex of a triangle to some
fixed points on any two sides of a triangle. A check line is measured to check the
accuracy of the framework. The length of a check line, as measured on the ground
should agree with its length on the plan.
A
B
C
D

Offsets
• These are the lateral measurements from the base line to fix the
positions of the different objects of the work with respect to
base line. These are generally set at right angle offsets. It can
also be drawn with the help of a tape.
1) Perpendicular offsets - The measurements are taken at right angle to the survey line called perpendicular
or right angled offsets
2) Oblique offset - The measurements which are not made at right angles to the survey line are called
oblique offsets or tie line offsets.
3) Long offset - Length of offset ≥ 15 m
4) Short offset - Length of offset < 15 m
A1 A2 A3 A4 A5
ө1 ө2
90o

Metric Chains:
 Available in lengths of 5, 10, 20 and 30 m
 Tallies are fixed at every 2 m intervals
 Circular tally is placed at the center of the chain
 Grooves are provided at the ends to facilitate the placement of arrows
 Length of the Chain is engraved on the brass handle of the chain
Chains Contd…

• Gunter’s Chain / Surveyor’s Chain:
 Before Independence, India used to follow FPS
system …in which length is measured in foot
 Length of the Chain is 66’ , consisting of 100 links,
each link being 0.66’.
 10 Gunter’s square chains = 1 acre.
 10 Gunter’s Chains = 1 furlong = 660’
 80 Gunter’s Chains = 1 mile =80X66 = 5280’
10 X 66’
10X66’
Engineer's Chain:
 Length is 100’ with 100 links, each link being 1’
 Brass tag is provided at every 10 links [number of 10 link
segments are indicated on the tags.]
Here, number on the tag represents the
segments of links
Ex. If n=3, length from beginning to that
point = 3 X 10 X 1’ = 30’
1 2 3

Revenue Chain:
 Length of the Chain is 33’ and consists of 16 links. Each link length is
 Used in Cadastral Survey
Steel band or band Chain:
 It consists of a long narrow strip of steel of uniform width (12 – 16 mm width and 0.3
to 0.6 mm thickness)
 They are available in 20 or 30 m lengths
 Brass studs are provided at every 20 cm and it is numbered ate very meter.
 First and last links are subdivided into cm and mm.
 For convenience, steel bands are wound on special steel crosses / metal reels from
which they are unrealed.
Chains Contd…

Testing and Adjusting Chain:
As we use the chain continuously, the length of it may be shortened [bending of the
links, wearing out of the links ] or elongated [stretching of the links, opening of
the rings etc. ].
So, it becomes essential to check the chain length often before using it.
It should be done by constructing a permanent test gauge, with which the chain is
compared.
+ + ++
10 m 10 m 10 m
10 m 20 m 30 m0 m
Permanent Test Gauge
20 cm X 20 cm
Dressed Stones
Temporary gauge station is established by driving two pegs at requisite distance
apart, and inserting nails into their tops to mark exact points.
Overall length of a chain, when measured at 8 kg pull and checked against a steel
tape @20oC shall be within the following limits
20 m Chain = ± 5 mm
30 m Chain = ± 8 mm.

Adjusting the Chain:
If the Chain is found to be long:
 Closing the joints of the rings
 Reshaping the elongated rings
 Removing one or more small circular rings
 Replacing worn out rings
 Adjusting the links at the ends.
If the Chain is found to be short:
 Straightening the links
 Flattening the circular rings
 Replacing one or more small circular rings by bigger ones
 Inserting additional circular rings
 Adjusting the links at the end.

Adjustment of error due to length of chain
Where
l= Design length of the chain
l’= Actual length of the chain
D’ = actual length of the chain line
D = measured length of the chain line

Similarly
For area =
For volume =

TAPES
 Cloth or linen
tape
 Metallic tape
 Steel tape
 Invar tape
Cloth or Linen Tape:
 Closely woven linen, 12 to 15 mm wide varnished to resist moisture.
 Commonly available in 10, 20, 25 and 30 m; 33’, 50’, 66’ and 100’.
 End of tape is provided with small brass ring whose length is included
in the total length of the tape.
They are not used for accurate measurements. WHY?
 It is easily affected by moisture
 Its length gets altered by streaching
 It is likely to sag
 Further, its life is short.

TAPES
 Cloth or linen
tape
 Metallic tape
 Steel tape
 Invar tape
Metallic Tape:
 Metallic tape is nothing but a cloth tape that is reinforced with brass or
copper wires.
 Commonly available in 10, 15, 20, 30 and 50m.
in the total length of the tape and they are supplied in a leather case.

TAPES
 Cloth or linen
tape
 Metallic tape
 Steel tape
 Invar tape
Metallic Tape:
 Metallic tape is nothing but a cloth tape that is reinforced with brass or
copper wires.
 Commonly available in 10, 15, 20, 30 and 50m.
in the total length of the tape and they are supplied in a leather case.
Steel Tape:
 They are made of steel strips having width of 6 – 10 mm.
 Available in lengths of 1, 2, 5, 10, 20, 30 and 50 m.
 These tapes are more durable and accurate than the metallic tape.
in the total length of the tape.
 They are supplied in a leather case or a corrosion resistant metal
case.
 They are used for accurate measurement of distance.

TAPES
 Cloth or linen
tape
 Metallic tape
 Steel tape
 Invar tape
Invar Tape:
 Invar tapes are made of alloy
 Steel - 64%
 Nickel - 36%
 Its coefficient of thermal expansion is very less ≈ 0.000000122/1oC
 It is 6 mm in width and available in lengths of 10, 20, 30, 50 and 100m.
 Invar is soft in nature and so, should be carefully handled to avoid
damage.
 They are used for accurate survey. They can also be used in places
where the temperature varies drastically.

Arrows:
 They are used to mark the position of the ends of the chain on the
ground.
 They are made of steel wires of diameter 4 mm.
 The length of the arrow ranges from 25 to 50 cm.
 One end of the arrow is bent in the form of a loop / circle and the
other end pointed.
Ranging Rods:
 Ranging rods are used to fix up intermediate points on an or establish the
position of a station.
 They are made with well seasoned timber with an iron shoe at the bottom
or of light steel tubes.
 They are circular in C/S with 3 cm Dia. Their lengths vary from 2 to 3 m.
 They are painted alternately with white-red (or) white-black bands. Each
band being 20 cm in length.

Plumb bob:
 It consists of a string attached at the top of the metal bob.
 As the bob always points towards the gravity, it represents the vertical line.
 They are used to transfer the points on the ground while chaining along a
sloping ground.
 It is further used in the primary adjustments of all the surveying instruments.
Pegs:
 They are used to mark the positions of the survey stations or the end points of a survey
line
 They are made of stout timber. They are generally square in section and tapered at the
end
 They are 22 mm X 25 mm in C/S and 150 mm long.
 These pegs are driven by hammer.

Laths:
 Useful for ranging long lines, also used over uneven ground where the ranging
rod is not visible due to obstructions,
 They are light, cheap, being white; they are easily visible at a great distance.
Usually 1.0m long
Whites:
 When the ranging rod is not available or insufficient, whites are used. These
are thin strip of bamboo and 40 cm to 1 m in length.
 One end is sharp and the other end is split for inserting pieces of white papers.
They are also useful for temporary marking of counter points.
Cross staff:
 The cross staff is used for
a) Finding out foot of the perpendicular from a given point to a line
b) Setting right angle at a given point on a line
.

Technical Terms used in chain Surveying :
Survey Stations
Chain Line or survey lines
 Main Survey Line
 Base line
 Tie Lines
 Check Lines
Offsets
 Perpendicular offsets
 Oblique offsets
 long offsets
 short offsets

Survey Stations:
1.Main Stations
2.Subsidiary or tie
Main Stations:
Main stations are the end of the lines, which command the boundaries of the survey
Subsidiary or the tie stations:
Subsidiary or the tie stations are the point selected on the main survey lines, where it is
necessary to locate the interior detail such as fences, hedges, building etc.
Main Survey Line:
The lines joining the main stations are called the main survey line or the chain lines.
Base Lines:
It is main and longest line, which passes approximately through the centre of the field. All the
other measurements to show the details of the work are taken with respect of this line.
Tie or subsidiary lines:
A tie line joins two fixed points on the main survey lines. It helps to check the accuracy of
survey and to locate the interior details. The position of each tie line should be close to some
features, such as paths, building etc.
Chain / Survey Lines

Offsets
• These are the lateral measurements from the base line to fix the positions of the different
objects of the work with respect to base line. These are generally set at right angle offsets.
It can also be drawn with the help of a tape.
1) Perpendicular
offsets
2) Oblique offset
3) Long offset
4) Short offset
The measurements are taken at right angle to the survey
line called perpendicular or right angled offsets
The measurements which are not made at right angles to
the survey line are called oblique offsets or tie line offsets.
A1 A2 A3 A4 A5
ө1 ө2
90o
Length of offset ≥ 15 m
Length of offset < 15 m

Procedure for carrying Chain Survey:
There are four steps in chain survey:
1) Reconnaissance Survey
2) Marking stations
3) Running Survey Lines
4) Taking offsets
Reconnaissance Survey:
The preliminary inspection of the area to be surveyed is called reconnaissance. The
surveyor inspects the area to be surveyed, survey or prepares index sketch or key plan.
Marking Stations:
Surveyor fixes up the required no stations at places from where maximum possible
stations are possible.
Running Survey Lines:
Then he selects the way for passing the main line, which should be horizontal and clean
as possible and should pass approximately through the centre of work.
 Then ranging roads are fixed on the stations.
 After fixing the stations, chaining could be started.
 Make ranging wherever necessary.
 Measure the change and offset.
 Enter in the field the book.

Selection of Survey Stations:
 Survey stations must be mutually visible
 Survey lines must be few as practically possible so that the frame work
can be plotted conveniently
 The frame work must have one or two base lines. If one base line is
used, it must run along the length and through the middle of the field. If
two base lines are there, it should cross in the form of letter ‘X’
 The lines should run on a level ground as far as possible
 The main lines should form well conditioned triangles
 Each triangle or portion of frame work must be provided with sufficient
check lines

 All the lines from which the offsets are taken should be
placed close to the corresponding surface features so as
to get short offset
 As far as possible, the main survey lines should not pass
through any obstacle.
 To avoid any trespassing, the main survey lines should fall
with in the boubdaries of the property to be surveyed.

Ranging
• When the distance to be measured is more than a tape length, a straight line is
required to be laid between the points/ stations along which measurements are
to be carried out. The process of laying out a straight line between points is
known as ranging.
Direct Ranging When the end stations are inter visible, ranging is being
carried out directly. The intermediate points are placed at distances having
interval less than one tape/chain length. The intermediate points are found by
moving a ranging pole in transverse direction and thus, points are selected in
such a way that the end points and the intermediate points lie in a straight line.
In this method, two flags, one ranging pole and a bunch of pegs are required in
a team of at least one surveyor and one assistant.
Indirect Ranging When the end stations between which a straight line is to
be laid, are not inter visible, indirect method of ranging is being adopted. It is
being carried out either by reciprocal method or by random line method.
• Reciprocal Ranging
• Random Line Method

Ranging :: Direct Ranging
e
B
A
Distance ≤ Chain Length

27 March 2020 GDRCET Basic Civil Engineering
In-Direct Ranging:

Field Book:
Book in which chainage, offsets and sketches of features are entered is called a field book
It is a rectangular book of about 20 cm X 12 cm in size.
It is of two types:
 Single line
 Double line
15
mm
Rules:
Field notes are entered from bottom to
top
No. of chain lines and No. of stations
should be marked

Instructions for booking field notes:
 All the measurements should be recorded as soon as they are taken
 Each chain line, tie line, name of the survey line should be clearly written
 The chainage of the starting station is zero and increases as we proceed forward.
 The notes should be complete, accurate and neat
 Suitable scale is chosen
 Writing should always be from the bottom
 The figure should not be crowded together
 In case of a long survey lines, there should be an entry at the end of every 10 chains.
General requirements/entries
 A layout of the lines
 The details of the lines
 The date of the survey
 A page index of the lines
 Name of the surveyor and its assistants.

Mistakes in Chaining
• Adding or dropping a full length of chain
• Adding or dropping a part of the length of chain
• Other points incorrectly taken as 0 or 30 meter marks on chain
•
• Reading numbers incorrectly
• Calling numbers incorrectly or not clearly

Assignment-1
Q.1) Explain the principles of surveying? Differentiate between
plane and geodetic surveying?
Q.2) Discuss in brief the principle of Chain Surveying?
Q.3) Write short notes on the instruments used in chain surveying?
Q. 4) Write the various factors to be considered while deciding
Survey stations
Q. 5) Explain Base line, Check line, tie line and oblique offsets with
a neat sketch?
Q.6) Define the terms
i) Survey stations ii) Chain or survey line
Q.7) what is ranging? Explain direct and indirect ranging?
Due date for submission: 2 March 2011.
NOTE: Your Assignments Contain marks which shall be consider for your Internal Assessment.

LEVELLING
• It a branch of survey in which
The elevations of given points on/above
or below the ground with respect to an
assumed datum are determined
Points at a given elevation or at different
elevations with respect to a given datum
are established.

Basic Definitions in Levelling :
Level Surface
Level Line
Horizontal Plane
Horizontal line
Vertical Line
Datum
Elevation
Vertical angle
Mean Sea Level
Bench Mark
or

Level Surface
Level Line
Horizontal Plane
Horizontal line
Vertical Line
Datum
Elevation
Vertical angle
Mean Sea Level
Bench Mark
Level line is defined as a line that lie in a level
surface. It is perpendicular to the direction of gravity
at that point.

Level Surface
Level Line
Horizontal Plane
Horizontal line
Vertical Line
Datum
Elevation
Vertical angle
Mean Sea Level
Bench Mark
Horizontal plane through a point is a plane tangential to
the level surface at that point. It is perpendicular to the
direction of gravity at that point.

Level Surface
Level Line
Horizontal Plane
Horizontal line
Vertical Line
Datum
Elevation
Vertical angle
Mean Sea Level
Bench Mark
Vertical line is a line normal to the level line. It is
normally considered to be plumb line.
Datum: Datum is a surface with reference to which
elevations are referred.
Usually MSL is taken as Datum

Level Surface
Level Line
Horizontal Plane
Horizontal line
Vertical Line
Datum
Elevation
Vertical angle
Mean Sea Level
Bench Mark
Bench Mark: It is a fixed point of reference whose
elevation with respect to a datum is known.
It is by using this BM, we determine the elevations of
all other points.
Bench Mark
GTS Bench
Mark
Permanent
Bench Mark
Arbitrary
Bench Mark
Temporary
Bench Mark

Bearings and Angles:
Direction of a survey line can be represented by:
θ
A
B
C
θ
A
B
C
N
Φ
(a) Between the two lines (b) With reference to a given direction
Bearing: It is defined as the angle of a line with reference to a particular direction.
This particular direction with reference to which angel is measured is known as meridian
All bearings are angles where as all angles are not Bearings

Meridian:
Fixed direction on the surface of the earth with reference to which the directions of the
survey lines are expressed is known as Meridian
Meridian
True MeridianMagnetic Meridian Arbitrary Meridian
Direction indicated
by a freely
suspended and
properly balanced
magnetic needle
unaffected by local
attractive force is
called magnetic
meridian.
True meridian at a
place is a direction
indicated by an
imaginary circle
passing around the
earth through that
place and the two
geographical poles.
For small surveys, any
temporary direction
shall be taken as fixed
direction and the
angles of the lines are
measured with respect
to this. This temporary
direction i.e. assumed
is termed as arbitrary
meredian.

27 March 2020 Basic Civil Engineering
Bearing:
The Horizontal angle between the reference meridian and the survey line in clock-wise
direction is known as bearing.
Bearing
True BearingMagnetic Bearing Arbitrary Bearing
The Horizontal
angle between the
Magnetic meridian
and the survey line
in clock-wise
direction is known
as Magnetic
bearing.
The Horizontal
angle between the
true meridian and
the survey line in
clock-wise direction
is known as true
bearing.
The Horizontal angle
between the arbitrary
meridian and the
survey line in clock-wise
direction is known as
arbitrary bearing.
The above classification is based on the reference direction

Observing Bearing:
• The compass centered over station A of the line AB and is leveled.
• Having turned vertically the prism and sighting vane, raise or lower the prism
until the graduations on the rings are clear and look through the prism.
• Turn the compass box until the ranging rod at the station B is bisected by hair
when looked through the prism.
• Turn the compass box above the prism and note the reading at which the hair
line produced appears to cut the images of the graduated ring which gives the
bearing of line AB.

Relationships between bearings
True bearing = Magnetic bearing ± Declination
Dip and Declination:
θ Φ
True meridian (TM)
Magnetic meridian (MM)
ΦE
TM
MM
ΦW
TM
MM
If MM is towards east it is +ve and
If MM is towards west, it is -ve

Magnetic bearing of a line AB is S 28030’ E. Calculate the true bearing if the
declination is 7030’ west
We know that TB = MB± Declination
As the declination is towards west, it is –ve.
TB = [180- 28030’ ]- 7030’

Fore Bearing & Back Bearing:
 Every line has two bearings one observed at each end of the
line.
 The bearing of the line in the direction of progress of the
survey is called Fore Bearing (FB), while the bearing in the
opposite direction is called Back Bearing (BB).
 Therefore BB of a line differs from FB by exactly 180o.

Fore and Back bearing:
N
N
N
N
N
N
ө1
ө3
ө2
ө4
ө5
ө6
ө7
ө9
ө8
ө10
A
B
C
D
E
F
Line Fore Bearing Back Bearing
AB Ө1 Ө2
BC Ө3 Ө4
CD Ө5 Ө6
DE Ө7 Ө8
EF Ө9 Ө10

Relationship between Fore bearing and back bearing:
N
N
ө1
ө1
ө2
A
B
0o to 180o [OR] 1st and 2nd Quadrant

Relationship between Fore bearing and back bearing:
N
N
ө1
ө1
ө2
So,
Back Bearing = Fore Bearing ± 180o
N
N
ө1
ө2
ө1
A
B
B
A
0o to 180o [OR] 1st and 2nd Quadrant 180o to 360o [OR] 3rd and 4th Quadrant

There are two systems commonly used to express the bearing.
 WHOLE CIRCLE BEARING:
In this system the bearing of a line measured with the magnetic
north in clockwise direction. The value of bearing thus varies from
0o to 360o.
 QUADRANTAL SYSTEM:
In this system the bearing of a line is measured eastward or
westward from north or south whichever is near. The directions can
be either clock wise or anti clockwise depending upon the position
of the line.

Whole Circle Bearing and Quadrantal / Reduced Bearing
E
N
W
S
ө1
ө2
ө3
ө4
C
B
A
D
E
Line WCB
AB θ1
AC θ1
AD θ1
AE θ1
The horizontal angle which a line makes with the magnetic meridian in the clock wise
direction is known as Whole Circle Bearing [WCB]

Whole Circle Bearing and Quadrantal / Reduced Bearing
E
N
W
S
Φ1
Φ2
Φ3
Φ4
C
B
A
D
E
Line QB
AB N Φ1 E
AC S Φ2 E
AD S Φ3 W
AE N Φ4 W
The horizontal angle which a line makes with the magnetic meridian in the clock wise
or anti0clock wise direction from Magnetic North or Magnetic south is known as
Quadrantal Bearing [QB]

W E
N
S
ө1
ө2
ө3
ө4
C
B
A
D
E Line WCB
AB θ1
AC Θ2
AD Θ3
AE θ4
QB
N θ1E
S [180-θ2] E
S [θ3 – 180] W
N [ 360-θ4] W
Conversion of WCB to QB/ Reduced Bearing

Convert the following
WCB to QB
65o35’
140o20’
255o10’
336o40’
Convert the following
QB to WCB
N 56o30’ E
S 32o15’ E
S 85o45’ W
N 15o10’ W
Find the back bearings of the following
observed fore bearing of lines
AB 63030’
BC 112045’
CD 203045’
DE 320030’
A
B
C
D
E

The bearings of the sides of a closed traverse A, B, C, D, E, A are as follows
Side F.B. B.B.
AB 107015’ 287015’
BC 220 2020
CD 281030’ 101030’
DE 189015’ 9015’
EA 124045’ 304045’ N
N
N
N
N
A
B
E
C
D
220
281030’
189015’
124045’
287015’
2020
101030’
9015’
θA
θB
θC
θDθE
A
θA
N
= θA =360- [ BB of EA – FB of AB ]A
= 360-[304045’ - 107015’ ]
=360-197030’
=162O 30’

Side F.B. B.B.
AB 107015’ 287015’
BC 220 2020
CD 281030’ 101030’
DE 189015’ 9015’
EA 124045’ 304045’ N
N
N
N
N
A
B
E
C
D
220
281030’
189015’
124045’
287015’
2020
101030’
9015’
θA
θB
θC
θDθE
θB = 360- [BB of AB – FB of BC ]
= 360- [287015’ - 220 ]
= 94045’
N
220
287015’
θB
B

Side F.B. B.B.
AB 107015’ 287015’
BC 220 2020
CD 281030’ 101030’
DE 189015’ 9015’
EA 124045’ 304045’ N
N
N
N
N
A
B
E
C
D
220
281030’
189015’
124045’
287015’
2020
101030’
9015’
θA
θB
θC
θDθE
θC = [FB of CD – BB of BC ]
= [281030’ – 2020 ]
= 79030’
N
C
281030’
2020
θC

Side F.B. B.B.
AB 107015’ 287015’
BC 220 2020
CD 281030’ 101030’
DE 189015’ 9015’
EA 124045’ 304045’ N
N
N
N
N
A
B
E
C
D
220
281030’
189015’
124045’
287015’
2020
101030’
9015’
θA
θB
θC
θDθE
θD = [FB of DE – BB of CD ]
= [189015’ – 1010 30’ ]
= 87045’
N
D 189015’
101030’
θD

Side F.B. B.B.
AB 107015’ 287015’
BC 220 2020
CD 281030’ 101030’
DE 189015’ 9015’
EA 124045’ 304045’ N
N
N
N
N
A
B
E
C
D
220
281030’
189015’
124045’
287015’
2020
101030’
9015’
θA
θB
θC
θDθE
θE = [FB of EA – BB of DE ]
= [124045’ – 90 15’ ]
= 115030’
N
E
124045’
9015’
θE

Side F.B. B.B.
AB 107015’ 287015’
BC 220 2020
CD 281030’ 101030’
DE 189015’ 9015’
EA 124045’ 304045’ N
N
N
N
N
A
B
E
C
D
220
281030’
189015’
124045’
287015’
2020
101030’
9015’
θA
θB
θC
θDθE
θA = 162030’
θB= 94045’
θC = 79030’
θD = 87045’
θE = 115030’
Sum of Internal angles = 5400
Sum of the internal angles of a closed traverse with n sides = [ (2 X n) -4] *900
In the present case, no. of sides = 5.
i.e. n = 5.
So, [ (2 X 5) -4] *900 = 5400
Check

Side F.B.
AB 70030’
BC 1320
CD 56000’
DE 215030’
EA 310000’
N
N
N
D
N
C
N
A
B
E
70030’

Bench Marks
GTS Bench
Mark
Grand Trigonometrical Survey [GTS] Bench Mark
100
100 km2
They are established
by “Survey of India”
with very high degree
of precision with
respect to MSL.
MSL

Bench Marks
Permanent
Bench Mark
GTS Bench Mark
These bench
marks are
established
between GTS
bench marks.
They are
marked/ located
on tops of
culverts, piers of
bridges, kilometer
stones, Railway
Platforms.
They are
established by
Survey of India
(or) PWD of that
area.
Permanent Bench Mark

Bench Marks
Arbitrary
Bench Mark
These bench marks are selected on some
permanent objects and their elevations are
arbitrarily assumed.
These bench marks are used in small scale
leveling operations.
Arbitrary bench marks are not related to GTS
or Permanent Bench marks.
Temporary Bench Mark
These bench marks are left at the end of a day’s leveling
operation. The leveling operation for the next day may be
continued with respect to the bench mark left previous day. Such
bench marks do not have any use later.

TBM
Day -2
TBM
Day 3
TBM
Day -1
TBM
Day 4

Instruments used for Levelling
 Level
 Level Staff
 Plumb Bob

Focusing Screws
Foot Screws
Upper parallel plate (Tri-branch)
Telescope
Object piece /
Object end
Eye Piece
Longitudinal Bubble
Traverse Bubble Tube
Lower parallel plate (Trivet)
Bubble tube
adjusting
screw
Level
Head

Digital level
There are fundamentally two
types of automatic levels.
First, the optical one whose
distinguishing feature is self-
leveling i.e., the instruments gets
approximately leveled by means
of a circular spirit level and then it
maintains a horizontal line of
sight of its own.
Second, the digital levels whose
distinguishing features are
automatic leveling, reading and
recording
Digital Level

Automatic Level
1. Base Plate
2. Horizontal Circle
3. Eyepiece
4. Circular Bubble
5. Sighting Pointer
6. Objective Lens
7. Focusing Knob
8. Fine Motion Drive
9. Footscrew
10. Bubble Mirror

Leveling Staffs
The staff is simply a large ruler,
available in lengths of 3, 4 or 5
metres and usually made of wood
or aluminium.

Level Staff
Self reading Target Staff
Solid Staff
Folding Staff
Telescopic Staff
Solid staff is available as a single unit
with no joints or hinges.
Smallest division on the staff is 5 mm.
These are made of well seasoned
wood.
They are available in both Foot and
Meter.
Their length are upto 3 m.

Level Staff
Self reading
Solid Staff
Folding Staff
Telescopic Staff
 Folding staff is available in two
pieces each of 2 m in length hinged
together so that it can be folded to a
single piece.
 Width of such staff is 75 mm and 18
mm thick.
 Staff has two handles, one on each
section, for folding the staff.
 They are more convenient to handle
and also to transport.

Level Staff
Self reading
Solid Staff
Folding Staff
Telescopic Staff
 Telescopic staff consists of three
pieces which can be extended
to the full length of 4 m.
 The upper piece is a solid piece
while the lower two pieces are
hollow from inside.
 The over all length of the staff
thus becomes 1.5 m when the
staff is not in use.

Level Staff
Self reading Target Staff
Solid Staff
Folding Staff
Telescopic Staff
 Target staff is a solid staff having a
sliding target equipped with vernier.
The rod is graduated in feet, tenths and
hundredths, and the vernier of the
target enables the reading to be taken
up to a thousandth part of the feet.
 It is used for long distance sighting.

 Setting up the level
 Leveling the instrument
 Removal of parallax
Temporary Adjustment of Dumpy Level
At each setting of a level instrument, temporary adjustment is required to be
carried out prior to any staff observation. It involves some well defined
operations which are required to be carried out in proper sequence.
It consists of
Setting
Leveling
Focusing

During Setting, the tripod stand is set up at a convenient height having its head
horizontal (through eye estimation). The instrument is then fixed on the head by rotating
the lower part of the instrument with right hand and holding firmly the upper part with left
hand. Before fixing, the leveling screws are required to be brought in between the
tribrach and trivet. The bull's eye bubble (circular bubble), if present, is then brought to
the centre by adjusting the tripod legs.
Leveling of the instrument is done to make the vertical axis of the instrument truly
vertical. It is achieved by carrying out the following steps:
Step 1: The level tube is brought parallel to any two of the foot screws, by rotating the
upper part of the instrument.
Step 2: The bubble is brought to the centre of the level tube by rotating both the foot
screws either inward or outward. (The bubble moves in the same direction as the left
thumb.)
Step 3: The level tube is then brought over the third foot screw again by rotating the
upper part of the instrument.
Step 4: The bubble is then again brought to the centre of the level tube by rotating the
third foot screw either inward or outward.
Step 5: Repeat Step 1 by rotating the upper part of the instrument in the same quadrant
of the circle and then Step 2.
Step 6: Repeat Step 3 by rotating the upper part of the instrument in the same quadrant
of the circle and then Step 4.
Step 7: Repeat Steps 5 and 6, till the bubble remains central in both the positions.
Step 8: By rotating the upper part of the instrument through 180 ° , the level tube is
brought parallel to first two foot screws in reverse order. The bubble will remain in the
centre if the instrument is in permanent adjustment.

Focusing is required to be done in order to form image through objective lens
at the plane of the diaphragm and to view the clear image of the object
through eye-piece. This is being carried out by removing parallax by proper
focusing of objective and eye-piece. For focusing the eye-piece, the telescope
is first pointed towards the sky. Then the ring of eye-piece is turned either in
or out until the cross-hairs are seen sharp and distinct. Focusing of eye-piece
depends on the vision of observer and thus required whenever there is a
change in observer. For focusing the objective, the telescope is first pointed
towards the object. Then, the focusing screw is turned until the image of the
object appears clear and sharp and there is no relative movement between
the image and the cross-hairs. This is required to be done before taking any
observation.

Removal of Parallax
 Eye piece is focused on to the cross hairs.
 The image of the level staff should fall in the plane of the cross hair.
 If the above condition is not satisfied there shall be errors in readings
 So, it is essential to establish the afarsaid condition before taking readings.
1. Focusing
the eye
piece.
2. Focusing
the
objective
 To eliminate the parallax error, white paper is place in front
of the eye piece and then it is focused by the screw for
distinct vision of the cross hair.
 To bring the image of the staff in the plane of the cross hair.
 Telescope is directed towards the staff and the focusing
screw is turned till the image appears clear and sharp.

Fundamental Lines
 Bubble tube axis
 Vertical axis
 Optical axis
 Line of collimation
Line tangential to the curved surface of the
bubble tube is called bubble tube axis.

Fundamental Lines
 Vertical axis
 Optical axis
Axis about which the telescope rotates is called
vertical axis.

Fundamental Lines
 Vertical axis
 Optical axis
The straight line passing through the optic center of
eye piece and optic center of the object lens is
called Optical axis.

Fundamental Lines
 Vertical axis
 Optical axis
The straight line passing through the intersection of
the cross wires and the optic center of the object
lens is called Line of collimation.

Relationship between Fundamental Axis
 Line of collimation should be parallel to the bubble tube axis
 Line of collimation should coincide with the optical axis of the
telescope
 Bubble tube axis should be perpendicular to the vertical axis of the
instrument.

Permanent Adjustments of the level
 There exist a relationship between the fundamental lines.
 Instruments when used for a period of time gets disturbed and fail
to satisfy the conditions/relations.
 In such case we need to perform permanent adjustments.
 They are done by the manufacturer and it is a part of advanced
surveying.

Technical Terms used in Leveling
Height of Instrument
Back Sight
Fore Sight
Intermediate Sight
Change Point
Back Sight : It is the sight / reading taken on the level
staff held on the point of known elevation i.e. Bench Mark.
Fore Sight : It is the last reading taken on the level staff
kept at a station from the instrument station before
shifting the instrument.
Intermediate Station : Readings taken
after back sight and before fore sight for
a particular set up of the instrument is
know as IS.
Change Point : This is also known as
Turning point. This is the point on which
both FS and BS are taken. After taking
FS, the instrument is shifted at other
convenient point and BS is taken on the
staff held at the same point.
Height of instrument (HI) – It is the elevation of
the line of sight of the telescope.

F
A [BM]
B C E
D
IP - 1
IP - 2
IS-2
IP - 3
A,B,C,D,E,F – Survey stations
IP-1,2,3 : Instrument positions
FS- Fore sight
BS- Back Sight
IS- Intermediate Sight
Station BS IS FS Remarks
A (BM) BS-1 IP-1
B BS-2 FS-1 IP-1 TO IP-2
C IS-1 IP-2
D IS-2 IP-2
E BS-3 FS-2 IP-2 TO IP-3
F FS-3 IP-3

Methods of Levelling
Simple (or) Direct leveling
Differential leveling
Fly leveling
Profile leveling
Cross sectioning
Reciprocal leveling
A + 200.00
B
2.7m
RL of A : +200.00 m
Height of Instrument (station) : +200.00 + 2.7 m
Fore Sight of B = 0.3 m
Back sight of A = 2.7 m
RL of B : +202.7 – 0.3 = 202.4 m
Simple or Direct Levelling
is used for finding the
level difference between
two stations that are
nearer.

Differential leveling
A + 200.00
B
Differential Levelling:
 If the distance between point whose difference in
elevations is to be determined is large, then it is not
possible to take the readings on A and B from a
single setup.
 In this case, instrument is set at more than one
position, each shifting facilitated by a change point.
CP-3
CP-2
CP-1

Fly leveling
Fly Levelling:
 If the work site is away from the permanent bench mark,
surveyor starts the work with the BS on the bench mark. He
proceeds towards the site by taking fore sights and back
sights on a number of change points till he establishes a
temporary bench mark in the site.
 This type of levelling in which only BS and FS are taken, is
called fly levelling, whose purpose is to connect a permanent
bench mark with temporary bench mark or vice versa.
 Thus Differential levelling and fly levelling differ only in the
purpose.

Methods of Levelling Profile leveling
Profile Levelling:
 This is known as longitudinal sectioning.
 In projects like highways, railways, sewer lines, irrigation canals etc…, profile of
the ground along them are required.
 In such cases, at regular intervals, readings are taken along their length and they
are then plotted to get the profile.
 In this case, instrument is set at more than one position, each shifting facilitated
by a change point.
Longitudinal Profile of the road

Methods of Levelling Cross sectioning
Cross Sectioning:
 In projects like highways, railways, sewer lines, irrigation canals etc…, in addition
to longitudinal profile of the ground, cross section profile is also essential.
 These profiles help in calculating the earth works involved in the projects.
 In such cases, at regular intervals, readings are taken along their chain line for
longitudinal profile and in addition to this, at each station on chain line,
readings are taken at close intervals on either side for cross sectioning.
0
20
40 60
140
80
120100
Station Distance in m Readings RL Remarks
L C R BS IS FS

Reciprocal Levelling
Reciprocal Levelling:
 This is a type of levelling in which the difference
between two stations separated by an obstruction is
determined.

Level Field Book
Station Distance in m Readings RL Remarks
L C R BS IS FS

Methods of Booking and Reducing the Levels
Height of Instrument
Method
Rise and Fall
Method

Height of Instrument Method
 In this method, Height of instrument for first setting of the
instrument is calculated as
HI = RL of Bench Mark + Back sight
 From HI, subtract intermediate sight and Fore sight to compute the
RL of intermediate stations and change points.
 Add back sight to RL of change point to get new height of
instrument.
 Similarly, compute RL of other Intermediate stations and change
points.
 Finally find the sum of BS, FS.
Check:
   pointfirstofRLpointLastofRLFSBS

Rise and Fall Method
 In this method, Height of instrument is not calculated.
 Difference of level between consecutive points is found by
comparing the staff readings on the two points for the same
instrument setting.
 Difference between their staff readings indicate a rise or fall
according as the staff reading at the point is smaller or greater than
at the preceding point.
 RL of the stations are calculated by either adding or subtracting the
rise or fall between the two stations to the RL of previous station.
    FallRisepointfirstofRLpointLastofRLFSBS
Arithmetic Check / Check

IP-1
S-1
A
Station BS IS FS
A X

IP-1
A
A-1
Station BS IS FS
A X
A1 X

IP-2
A
A-1
Station BS IS FS
A X
A1 X x

Station BS IS FS
A X
A1 X X
A2 x
IP-2
A
A-2

Station BS IS FS
A X
A1 X X
A2 X X
IP-3
A-2

Station BS IS FS
A X
A1 X X
A2 X X
A3 X
IP-3
A-3

Station BS IS FS
A X
A1 X X
A2 X X
A3 X X
IP-4
A-3

IP-4
A-4
Station BS IS FS
A X
A1 X X
A2 X X
A3 X
A4 X

IP-5
A-4
Station BS IS FS
A X
A1 X X
A2 X X
A3 X
A4 X X

IP-5
B
Station BS IS FS
A X
A1 X X
A2 X X
A3 X
A4 X X
B X

The following staff readings were observed successively with a level, the instrument
having been moved after 3rd, 6th and the 8th readings. 2.228, 1.606, 0.988, 2.090, 2.864,
1.262, 0.602, 1.982, 1.044, 2.684 m. [CSVTU, 2009]

Station BS IS FS HI RL
A 2.228
The following staf readings were observed successively with a level, the instrument having been moved after 3rd, 6th
and the 8th readings. 2.228, 1.606,0.988,2.090,2.864,1.262,0.602,1.982,1.044,2.684 m.
A
B C
E
D
GF

A 2.228
B 1.606
A
B C
E
D
GF

A 2.228
B 1.606
C 0.988
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
D 2.864
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
D 2.864
E 1.262
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
D 2.864
E 0.602 1.262
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.982
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.044 1.982
A
B C
E
D
GF

A 2.228
B 1.606
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 200.0+2.228 = 202.228 200.00
B 1.606
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684
and the 8th readings. 2.228, 1.606,0.988,2.090,2.864,1.262,0.602,1.982,1.044,2.684 m. The RL of BM is 200.00 m
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 202.228 – 0.988 = 201.24
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864 203.33-2.864=200.466
E 0.602 1.262
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864 203.33-2.864=200.466
E 0.602 1.262 203.33-1.262=202.068
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864 203.33-2.864=200.466
E 0.602 1.262 202.068+0.602=202.67 203.33-1.262=202.068
F 1.044 1.982
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864 203.33-2.864=200.466
E 0.602 1.262 202.068+0.602=202.67 203.33-1.262=202.068
F 1.044 1.982 202.67-1.982=200.688
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864 203.33-2.864=200.466
E 0.602 1.262 202.068+0.602=202.67 203.33-1.262=202.068
F 1.044 1.982 200.688+1.044=201.732 202.67-1.982=200.688
G 2.684
A
B C
E
D
GF

A 2.228 2000+2.228 = 202.228 200.00
B 1.606 202.228 – 1.606 = 200.622
C 2.090 0.988 201.24+2.09=203.33 202.228 – 0.988 = 201.24
D 2.864 203.33-2.864=200.466
E 0.602 1.262 202.068+0.602=202.67 203.33-1.262=202.068
F 1.044 1.982 200.688+1.044=201.732 202.67-1.982=200.688
G 2.684 201.732-2.684 =199.048
SUM 5.964 6.916
A
B C
E
D
GF
   pointfirstofRLpointLastofRLFSBS
Abs(5.964-6.916) = abs(199.048-200)

Solve the above problem by Rise and Fall method

Stati
on
BS IS FS Rise Fall RL
A 2.228 200.00
B 1.606
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684
A
B C
E
D
GF

Rise and Fall
A
B C
E
D
GF
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684

Rise and Fall
A
B C
E
D
GF
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988 0.618 201.24
D 2.864
E 0.602 1.262
F 1.044 1.982
G 2.684

Rise and Fall
A
B C
E
D
GF
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988 0.618 201.24
D 2.864 0.774 200.466
E 0.602 1.262
F 1.044 1.982
G 2.684

Rise and Fall
A
B C
E
D
GF
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988 0.618 201.24
D 2.864 0.774 200.466
E 0.602 1.262 1.602 202.068
F 1.044 1.982
G 2.684

Rise and Fall
A
B C
E
D
GF
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988 0.618 201.24
D 2.864 0.774 200.466
E 0.602 1.262 1.602 202.068
F 1.044 1.982 1.38 200.688
G 2.684

Rise and Fall
A
B C
E
D
GF
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988 0.618 201.24
D 2.864 0.774 200.466
E 0.602 1.262 1.602 202.068
F 1.044 1.982 1.38 200.688
G 2.684 1.64 199.048

Rise and Fall
A
B C
E
D
GF
Abs(5.964-6.916) = abs ( 2.842 – 3.794) = abs(199.048-200)
Stati
on
A 2.228 200.00
B 1.606 0.622 200.622
C 2.090 0.988 0.618 201.24
D 2.864 0.774 200.466
E 0.602 1.262 1.602 202.068
F 1.044 1.982 1.38 200.688
G 2.684 1.64 199.048
SUM 5.964 6.916 2.842 3.794
    FallRisepointfirstofRLpointLastofRLFSBS

Station BS IS FS RL
A 0.865 560.5
B 1.025 2.105
C 1.58
D 2.23 1.865
E 2.355 2.835
F 1.76
Station BS IS FS RL
A 1.622
B 1.874 0.354
C 2.032 1.78
D 2.362
E 0.984 1.122
F 1.906 2.824
G 2.036 83.50
The following consecutive readings were taken with a level and 5 m leveling staff on
continuously sloping ground at a common interval of 20 m:
0.385, 1.030, 1.925, 2,825, 3.730, 4.685, 0.625, 2.005, 3.110, 4.485. the reduced level
of the first point was 208.125 m. Rule out a page of level field book and enter the
readings. Calculate the RL of the points and also find the gradient of the line joining the
first and the last point.
Numericals

Surveying and levelling

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