Fundamental of Surveying Measuring Vertical Distance

Chapter Four
Measuring Vertical Distance
4.1 Introduction to Leveling
Leveling is the process of measuring vertical distances above or below a given reference surface and
a point on, near or below the earth’s surface. It can also be defined as the processes of determining
elevation differences between various points on, near or below the surface of the earth. The
elevation of a point is its vertical distance above or below a reference level, called datum. The most
commonly used datum is the mean sea level (M.S.L)

Leveling is an important method of surveying for many engineering works and construction
projects. Leveling is needed for the design of highways, railways, and canals etc and for locating
the gradient lines. The results of leveling can be used to determine the catchments area, volume of
the reservoir.
The vertical direction is parallel to the direction of gravity at any point; it is the direction of a freely
suspended plumb-bob cord or string. The vertical distance of a point above or below a given
reference surface is called the elevation of the point. The most commonly used reference surface for
vertical distance is mean sea level (MSL). (The words altitude, height, and grade are sometimes
used in place of elevation). Vertical distances are measured by surveyors in order to determine the
elevations of points, in a process called running levels, or simply leveling.
The importance of leveling can not be over estimated with a few exceptions, it must always be
considered in every form of design and construction.
The determination and control of elevations constitute a fundamental operation in surveying and
engineering projects. Leveling provides data for determining the shape of the ground and drawing
topographic maps. The elevation of new facilities such as roads, structural foundations, and
pipelines can then be designed. Finally, the designed facilities are laid out and marked in the field by
the construction surveyor.
The surveyors’ elevation marks (such as grade stakes) severs as a reference points from which
building contractors can determine the proper slope (‘’rate of grade’’) of a road the first floor
elevation of a building, the required cut off elevation four foundation piles, the invert elevation for a
storm sewer, and so on.
This chapter covers the fundamentals of leveling, including the types and proper use of leveling
equipment, leveling field procedures and field notes, differential leveling, profile leveling, cross
section leveling and other related topics.

Basic Definitions
The following are a few introductory definitions that are necessary for the understanding of this
chapter
i. A vertical line- is the line parallel to the direction of gravity. At a particular point it is
the direction assumed by a plumb-bob sting if the plumb-bob is allowed to swing freely.
Because of the earth’s curvature, plumb-bob lines at points some distance apart are not
parallel, but in plane surveying they are assumed to be.
ii. A level surface- is a surface of constant elevation that is perpendicular to a plumb line at
every point. It is best represented by the shape that a large body of still water would take
if it were unaffected by tide.

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 Fundamental of Surveying Measuring Vertical Distance

Figure 1
iii. The elevation of a particular point- is the vertical distance above or below a reference
level surface (normally, sea level)
iv. A level line- is a curved line in a level surface all points of which are of equal elevation.
Every element of the line is perpendicular to gravity.
v. A horizontal line- is a straight line tangent to a level line at one point.
vi. Difference in elevation: -is the vertical distance between level surfaces passing through
a point.
4.2 Principles of ordinary leveling
The process of leveling may be direct or indirect
a) Direct leveling is the method of taking a direct measurement, up or down, from one point to
another. It is the method by which differences of height are measured, vertically, from a truly
horizontal line of sight. The method is used by architects, engineers, surveyors, and builders for
lower-order work.
b) Indirect leveling is the method of taking an indirect measurement by observing the angle of
elevation or depression from one point to another. The tangent of this angle when multiplied by the
horizontal distance apart of the points gives their difference in height (after corrections). This
method is used in advanced leveling and higher-order work and is termed trigonometrically leveling.
c) Barometric leveling is a third method of finding the difference in height between two points, by
means of simultaneous readings of barometric pressure at the two points.

When the leveling instrument has been properly levelled, the bubble tube axis and the line of sight
are horizontal and the vertical axis of the instrument is truly vertical. When the telescope of the
instrument is rotated, the line of sight remains in the horizontal plane. In practice, the leveling
instrument is set up at a convenient position, and the one staff is kept at a point of known elevation.
The reading is taken on the staff and the level of the line of sight is determined. Now the 2 nd staff
is kept on the point of the unknown elevation and a reading is taken. The level of the point is
determined from the level of the line of sight already computed and from the 2nd staff reading.

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BACK SIGHT FORE SIGHT

T.P.(TURNING POINT)
B.M.

Setup

Figure 2
4.3 Terms used in direct leveling

The following terms are commonly used in leveling.

1. Station. The station is the point where the staff is held for taking reading (observation) from
a leveling instrument.

The point where the instrument is set up is not important so far the booking of the readings
in a level field book is concerned.

2. Height of the instrument. (H.I): It is the elevation of the line of sight with respect to the
datum. It should be noted that the height of instrument is not the height of the line of sight
above the ground where the leveling instrument is set up.
3. Bench mark (BM): - is a permanent or semi permanent point of known elevation that
provides beginning point for determining the elevation of other points.
4. Reduced level (RL): - It is a level of a point from a reference datum.
5. Back sight (B.S.): It is the reading taken on a staff held at point of known elevation or at the
point whose elevation has already been determined. The back sight is usually the first
reading taken after setting up the instrument. The back sight is taken on a bench Mark
(B.M) for the first setting of the instrument and on a turning point (T.P.) for the subsequent
settings because the level of turning point can be determined before the shifting of the
instrument.

6. Fore sight (F.S.): It is the reading taken on the staff either held at the last point whose
elevation is required or held at the turning point just before shifting the instrument.

7. Intermediate sight (IS): It is the reading taken on a staff held at point whose elevations is
required, but which is not a turning point or the last point.

8. Turning point. (T.P.): For leveling over a long distance, the instrument has to be set up a
number of times. A turning point is the point selected on the route before shifting the
instrument. The turning point should be selected on a firm ground or rock.

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Figure 3

9. Balancing of sight: To reduce the effect of instrumental and other errors, the distance of the
point where a back sight is taken and the distance of the point where a fore sight is taken, as
measured from the instrument station, should be approximately equal. This is known as
balancing of sights.

4.4 Leveling Instruments

4.4.1 Levels
Levels are categorized in to three groups.
1) Dumpy levels 2) Tilting levels 3) and Automatic levels

1. Dumpy levels
In dumpy level, the line of sight is perpendicular to the vertical axis. Once the instrument is leveled
the line of sight becomes horizontal and the vertical axis becomes truly vertical provided the
instrument in adjacent.

2. Tilting levels
It has the telescope that can be tilted about a horizontal axis. This design enables the operator to
quickly and accurately centre the bubble and brings the line of sight in to a horizontal plane. In
tilting levels the line of sight is or should be parallel to the axis of the telescope. It is only horizontal
when the bubble of the spirit level is central.

3. Automatic levels
One of the most significant improvements in leveling instrumentation has been automatic level or
self-leveling levels. It has an internal compensatory that automatically makes horizontal the line of
sight and maintains the position through the application of the force of gravity.

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As soon as the instrument is levelled by a means of a circular bubble, the movable component of
the compensatory swings free to a position that makes the line of sight horizontal. The
compensatory can operate within the range of  10 minutes of arc from the horizontal.

a)Dumpy level b) Tilting level c) Automatic level

Figure 4

4.4.2 Tripods
A tripod is a three- legged stand used to support a level or other surveying instrument during field
measurements. There are two models of tripods.
(1) The extension leg tripod and
(2) The fixed leg tripod.

Figure 5a the extension leg tripod Figure 5b Fixed leg tripod

4.4.3 Level rods (leveling staff)

They are used to measure the vertical distance between a line of sight and a survey point and a
height different between two points.
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Figure 6 leveling staff

4.5 Preparing levels for work

1. Setting up: Set up the tripod at a convenient height and press the tripod feet firmly into the
ground. The tripod head should be approximately horizontal. Fix the instrument on it.

2. Leveling up: Level the instrument with the foot screws until the circular bubble is in the
centre.
Figure 7c shows the procedure how to level an automatic level
The movement of the circular bubble can be divided into two directions. One direction is parallel to
line joining any two of the foot screws. The second direction is then parallel to a perpendicular line
from the third foot screw to the line joining the two other foot screws.
Turn the instrument until the telescope axis is parallel to any two foot screws. The screws are held
by the thumb and forefinger of each hand and turned equally and simultaneously in the opposite
direction until the bubble has moved to the line AB (fig. 7a). Line AB is perpendicular to the line
through the two foot screws. By using the third screw the bubble is moved towards the centre of the
circle (fig. 7b). The leveling procedure is completed when the bubble is in the centre of the circle
(Figure 7c).

Left thumb Leveling

screws
R o ta tio n

Right thumb

Bubble

A B A B A B

Circular bubble

Figure7a figure 7b figure 7c

Left thumb rule: the bubble is always moving towards the direction of movement of the left
thumb!

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3. Focusing: First focus the eyepiece until the cross hairs appear sharp and clear then point the
telescope towards the object (staff) and focus until you see clearly the graduation of the staff.

4.6. Methods of Leveling

4.6.1 Differential leveling and its procedures

Differential leveling is required for the determination of the difference of elevation of two points,
which are quite apart. A number of setting ups of the instrument are required in differential leveling.
In fig. 1 BM1 represents a point of known elevation (benchmark) and BM 2 represents a benchmark to
be established some distance away. It is desired to determine the elevation of BM 2. The level is
placed in such a location that a clear rod reading is obtainable, but no attempt is made to keep on the
direct line joining BM1 and BM2. A back sight is taken on BM1.
The rod-man chooses a turning point TP1 at some convenient spot with in the range of the telescope
along the general route BM1 to BM2. It is desirable, but not necessary, that each foresight distance as
I1–TP1 is approximately equal to its corresponding back sight distance as BM 1-I1. The chief
requirement is that the turning point shall be a stable object at an elevation and in a location
favourable to a rod reading of the required precision. The rod is held on a turning point and a fore
sight is taken. The observer then set up the instrument at some favourable point as I 2 and takes a
back sight to the rod held on the turning point TP 1. Then the rod-man establishes the second turning
point TP2 and the observer takes a fore sight. The process is repeated until finally a foresight is
taken on the terminal point BM2.
To check weather there is an error or not the leveling work should always be started from a known
point and should be finished at a known point.
The leveling field book should be checked immediately in the field.

Note: While taking the readings the staff rod has to be held vertically!

H.I.

H.I.=1681.444
H.I.

Line of sight
2.101 1.021
1.000 3.221

1.244 0.662 T.P.2

I 3
I 2
T.P.1 B.M.2
I 1
B.M.1
Elevation of B.M.1 = 1680.20

Figure 8 Compound differential leveling

4.6.2 Profile leveling its procedures

The process of determining the elevation of points at short measured intervals along a fixed line is
called profile leveling. The need of profile leveling arises during the location and construction of
highways, railroads, canal, and sewers.

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 Fundamental of Surveying Measuring Vertical Distance

1 .0 2 1
0 .2 4 1

0 .3 6 1

2 .8 1 1
3 .4 4 0

2 .9 8 8

3 .6 6 1

2 .1 4 4

2 .0 1 2
2 .0 0 0
2 .0 1 1
B.M.3

I 1 T.P.2
I 2
T.P.1

Figure 9 illustrates the steps in leveling for profile

The instrument is set up in some convenient location not necessarily on the line (as at I 1) and the rod
is held on benchmark (BM3). A back sight is taken and the height of instrument is obtained as in
differential leveling.
Readings are then taken with the rod held on the ground at successive stations along the line. These
rod readings, being for points of unknown elevation, are foresights regardless of whether they are in
front or ahead of the level. They are frequently designated as intermediate fore sights to
distinguish them from foresights taken on turning points or benchmarks. The intermediate foresights
subtracted from the height of the instrument give ground elevations of stations. When the rod has
been advanced to a point beyond which further readings to ground points can not be observed, a
turning point (TP1) is selected and a foresight (3.440m) is taken to establish its elevation. The level
is set up in an advanced position (I2) and a back sight (2.988m) is taken on the turning point (TP 1)
just established. Rod readings on ground points are then continued as before.

Note: While taking the readings the staff rod has to be held vertically!

Sample Field book for profile leveling

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Plotting the profile

Draw a straight line AB to represent a total horizontal distance between the end stations to a
convenient scale. The distances between the consecutive points are marked there on. Verticals are
drawn at each point to and their elevations plotted along these verticals. Each ground point is thus
plotted by a Cartesian coordinate's i.e. horizontal distance as X- ordinate and elevations as Y-
ordinates. The end points of all verticals are joined by straight lines to show the profile of the
ground. The elevations of the datum line may be assumed as the elevation of the first point. But for
easy calculation of difference between the elevation of the datum line and the elevation of different
points, it is always preferred to assume an elevation having full meters. Generally, horizontal scale
is adopted as 1cm=10m and the vertical scale is kept 10 times the horizontal scale i.e. 1cm=1m so
that the inequalities of the ground may be shown clearly.

Profile drawing
Horizontal scale 1:1000
Vertical scale 1:100
4.6.3 Cross sectional leveling
Cross sections at right angles to the center line are run on either side for the purpose of determining
the lateral lay out line of the ground surface. The cross sections are taken at every 20m or 30m
stations depending up on the nature of the ground. They are numbered consecutively from the point
of commencement of the longitudinal profile and are set out at right angles to it with a chain, a tape
and an optical square. The distances are measured right or left from the center line peg. The lengths
the cross section also depends upon the topography of the ground and the type of the project.

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Sample of cross sectional field book

The recordings of the readings of a longitudinal sections and cross sections with its number may be
entered separately. The full descriptions of each cross section whether it is on the left or on the right
of the center line must be written in the field book. Sometimes surveyors prefer to enter the readings
of both the longitudinal and cross sectional sections simultaneously as shown in the table above.

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Plotting of cross sections

A horizontal line is drawn and different points of cross section are plotted on convenient scale,
keeping the central peg of the profile in the center. Perpendiculars are drawn at each point. A
convenient datum level is assumed and difference of elevation of each point and the datum are
plotted along the perpendiculars. The points so obtained are joined by straight lines to get the
desired sectional elevation. In a cross sectioning leveling the horizontal and vertical scales is
generally kept the same i.e. 1cm=1m. It may be noted that the elevations of cross datum lines for
different sections may be kept different to have the ordinates fairly short.

4.6.4 Reciprocal leveling

When a line of levels crosses a broad body of water it is impossible to balance the back sight and
foresight distances, it is necessary to take sights much longer than permissible. Under such a
measurement errors due to curvature and refraction become significant. To obtain the best results we
should have to use the procedure termed as reciprocal leveling.

level surface level surface

c d a b

BM1 BM1 A
water A water

Set up 2 set up 1
Figure 10

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 Fundamental of Surveying Measuring Vertical Distance

The elevation of survey point A is to be determined by leveling from BM1. At a set up near BM1, a
back sight is taken on BM1 a fore sight on A. The difference in elevation is computed as (BS – FS).
Next the level is set up near point A. Assuming that atmospheric refraction remains constant during
the time between the two set ups, the correct differences in elevation is computed as the mean of the
two measured differences.
 a  b   c  d  
El of A  El of BM1
2
Example:
If a = 1.442 m
b = 1.911 m
c = 1.768 m
d = 2.325 m
And El of BM1 = 1980.40 m so:
E1A = 1980.04 – 0.513 = 1979.887m

4.6.5 Trigonometrical levelling

Trigonometrical levelling is used where difficult terrain, such as mountainous areas, precludes the
use of conventional differential levelling. It may also be used where the height difference is large
but the horizontal distance is short such as heighting up a cliff or a tall building. The vertical angle
and the slope distance between the two points concerned are measured. Slope distance is measured
using electromagnetic distance measurers (EDM) and the vertical (or zenith) angle using a
theodolite. When these two instruments are integrated into a single instrument it is called a ‘total
station’. Total stations contain algorithms that calculate and display the horizontal distance and
vertical height, This latter facility has resulted in trigonometrical levelling being used for a wide
variety of heighting procedures, including contouring. However, unless the observation distances are
relatively short, the height values displayed by the total station are quite useless, if not highly
dangerous, unless the total station contains algorithms to apply corrections for curvature and
refraction.

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4.6.6 Inverted sights

Figure 3.22 shows inverted sights at B, C and D to the underside of a structure. It is obvious from
the drawing that the levels of these points are obtained by simply adding the staff readings to the
HPC to give B = 65.0, C = 63.0 and D = 65.0; E is obtained in the usual way and equals 59.5.
However, the problem of inverted sights is completely eliminated if one simply treats them as
negative quantities.

Fig. 3.22 Inverted sights

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4.7 Level field books and arithmetic check

A level field book or a level book is used for booking and reducing the levels of various points on
the surface of the earth. Booking and reduction of the levels may be done by following two methods.
1. Rise and fall method.
2. Height of collimation method.
1. Rise and Fall Method- In this method, the difference of level between two consecutive points for
each setting of the instrument is obtained by comparing their staff readings. The difference between
their staff readings indicates a rise if the back staff reading is more than the fore sight and a fall if it
is less than the fore sight. The rise and fall worked out for all the points give the vertical distance of
each point relative to the preceding one. If the R.L of the back staff point is known, then RL, of the
following point may be obtained by adding its rise or subtracting its fall from the RL of proceeding
as the case may be. The specimen page of a level book illustrating the method of booking staff
readings and calculating R.Ls. of stations by the rise and fall method is shown under.

Rise and fall method of reduction of levels

Arithmetic checks- The difference between the sum of the back sights and sum of the fore sights
should be equal to the difference of the sum of raises and the sum of falls and should also be equal
to the difference between the reduced levels of the last point and that of the first point i.e.
ΣBF–ΣF.S. = ΣRise–ΣFall =Last R.L. –First R.L
7.475–7.395 = 4.6 15– 4.535 = 100.080— 100.000 = 0.080
In this method of reduction a complete check on intermediate sights is provided because these are
included for calculating the rises and falls.

The 'raise' and 'fall' may be replaced by 'elevation difference' (Δh) in one column so that raise and
fall will be positive and negative respectively on the same column. The following table shows a
portion of the level book in which the rise and fall are replaced by Δh:

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 Fundamental of Surveying Measuring Vertical Distance

2. Height of Collimation Method- In this method, height of the instrument (HI) is calculated for
each setting of the instrument by adding back sight (BS.) to the elevation of the B.M. The reduced
level of the first station is obtained by subtracting its fore sight from the instrument height (HI.). For
the second setting of the instrument, the height of the instrument is calculated by adding the back
sight taken on the first station to its reduced level. The reduced level of the last point is obtained by
subtracting the fore sight of the last point from the height of instrument at the last setting.
If an intermediate sight is observed to an intermediate station, its reduced level is obtained by
subtracting its foresight from the height the instrument for its setting. The specimen page of a level
field book illustrating the method of booking the staff readings and calculating R.Ls. of the stations
by the height of collimation method is shown under.

Height of instrument method of reduction of levels

Arithmetic checks- The difference between the sum of the back sights and the sum of the fore
sights should equate to the difference between the R.L of last station and the R.L. of the first station
i.e.
SB.S–SFS=Last R.L– First R.L
7.475–7.395= 100.080–100.000=0.080.
In this method there is no check on intermediate sights.

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4.8 Testing levels

1. The axis of the circular bubble should be parallel to the instrument’s vertical axis
Test: By rotating the leveling screws bring the bubble into the central position. Rotate the
instrument halfway around. If the bubble remains centred the condition is fulfilled. If it has been
displaced, correct one-half of the error with the use of adjusting screws (screw drivers) of the bubble
and half of the error by means of the leveling screws. Repeat the same method of procedure until the
bubble remains centred.

2. The horizontal cross hair should be perpendicular to the instruments vertical axis
Test: To test for this condition set a leveling rod (staff) at 30-40m away from the instrument and
take readings by the two ends of the horizontal cross-hair at the same position of the leveling rod. If
the readings agree then the condition is fulfilled. If they differs by more than 2mm rotate the
diaphragm by using the adjusting screws until the readings are in agreement.

3. The telescope’s line of sight should be true level for instruments with compensators
This test involves leveling by two-rod technique (peg test). Two wooden pegs are driven in to the
ground at a distance d of approximately 50m-70m apart. With the level set up at the mid-point
between the pegs, a back sight reading (BS) “a” is taken at the rod at peg A, and a fore sight reading
(FS) “b” is taken at peg B (see fig.3). Assuming that the error due to the inclined line of sight is x1
the correct height difference h between the two pegs is computed as follows:

h = (a + x1) – (b + x1) = a – b

d d

x1 x1

b
a
B
A

Figure 11

Next the level is moved near to peg A and the back sight (BS) reading “c” is taken at the rod at peg
A (fig 4). Then a fore sight (FS) reading “d” is taken at the rod held at B. Let x2 represent the error
in fore sight (FS) reading due to inclination of the line of sight. Then x2 can be computed from the
following:

2d

x2
c d

A B

Figure 12

h=c-(d+x2) = (a-b)

X2 = (c-d)-(a-b)

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The line of sight is inclined downward if x2 is positive and upward if x2 is negative. The error
introduced by this source can be completely eliminated by balancing the back sight (BS) and fore
sight (FS) distances.

4.9. Curvature and Refraction

The nature of the earth's curvature and atmospheric refraction affect leveling operations. The
magnitude of the curvature C in relation to the radius R of the earth and the tangent distance d can
be derived as follows:

a) Curvature d

B'
(R+C) 2 = R2+d2 
C
B
 R2+2RC+C2 = R2 +d2
level line

earths surface
C is very small when compared with R so R

d2
C 
2R , R=6370km

 d2   2 
C   km  C   d  1000  m  0.0785  d 2
 12740   12740 
   

Figure 13

b) Refraction

Refraction will cause the line of sight to be deflected downward by a small angle. Pressure,
temperature, latitude, humidity affect refraction and its value is not constant. Its value is taken as
1/7 curvature, or 14% of the curvature of the earth, and is opposite in effect to that of curvature.

c) Combined correction
Combined correction = 0.0785  d 2 
1
7

 0.0785  d 2 
=
6
7

 0.0785  d 2 
C & R = 0.0673 d 2 meters d is distance in km, and the result must be in m.

If the distance d is smaller than 120m curvature and refraction errors are negligible. The error
introduced by this source can be completely eliminated by balancing the back sight (BS) and fore
sight (FS) distances.

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4.10 Difficulties in leveling

1. Staff station above the line of sight

2. Staff station much below the line of sight

3. Staff to near the instrument

If the staff is too near the instrument, the graduations are not visible. If not visible, move a sheet of
white paper up or down on the staff, until its one edge is bisected by the line of sight. Read the staff
corresponding to the edge of the paper with naked eyes and note the staff reading.
4.11 Common Leveling Mistakes
i. Misreading the rod- unless the instrument man is very careful, he or she
may occasionally read the rod incorrectly; as, for instance, 3.72m instead of 4.72m. This
mistake most frequently occurs when the line of sight to the rod is partially obstructed by
leaves, grass and so on.
ii. Moving Turning points- A careless road man causes a serious leveling
mistake if he or she moves the turning points. The rodman holds the rod at one point while
the instrument man takes the foresight reading, and then while the level is being moved to a
new position, the Rodman may puts the level rod down while he or she does something else.

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iii. Field Note mistakes- To prevent the recording of incorrect values, the
instrument man should call out the reading as he or she reads& records them.
iv. Mistakes with extended rod- When readings are taken on the extended
portion of the level rod, it is absolutely necessary to have the two parts adjusted properly.

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