The Theodolite and its
Application
This booklet was written originally in
German by Herr" 0. Trutmann and is
published by Wild Heerbrugg Ltd.,
Heerbrugg, Switzerland, Optical Preci-
sion Instrument Makers. It is not in-
tended as an advanced manual on sur-
veying, but has been prepared, as a ser-
vice to our customers, in the form of an
elementary description of simple tech-
niques and instrumental construction.
Note:
The text and examples refer to surveys
measured in degrees and metres. The
reader who is accustomed to working
with grades or feet should have no dif-
ficulty in following the calculations and
booking sheets and, if he wishes, can
gain valuable experience by converting
the units to those with which he is more
familiar and then recomputing some or
all of the examples.
In most cases logarithms have been
used for the calculations, as it is con-
sidered that the new surveyor is able to
get a better feeling for the handling of
numbers when using this method. It
would be excellent practiceforthe reader
to recompute some of the examples by
means of natural functions and with a
desk calculator .
The English version of this booklet has
been made by A. H. Ward, F.R.I.C.S.
Contents
Page
1 Introduction 7
1.1 Triangulation. 8
1.2 Trigonometric Levelling 8
1.3 Traversing 8
2 Instruments 11
2.1 Basic Principles 11
2.2 Wild T2 Universal Theodolite 12
2.3 Wild T3 Precision Theodolite 14
2.4 Repetition Theodolites and Tacheometers 14
2.4.1 Scale Microscope 14
2.4.2 Optical Micrometer, 16
2.5 Wild TO Compass Theodolite 16
2.6 General Specifications 16
2.6.1 Coincidence Levels 16
2.6.2 Telescope Image 16
2.6.3 Field of View 18
2.6.4 Magnification 18
2.7 The Major Features of Theodolites 18
2.7.1 The Arrangement of the Axes 19
2.7.2 Testing and Adjusting 21
2.7.2.1 Standing Axis Error 21
2.7.2.2 Horizontal Collimation Error 21
2.7.2.3 Vertical Index Error, 22
2.7.2.4 Sighting Error 23
2.7.2.5 Graduation Errors 23
2.8 Choice of the Most Suitable Instrument 24
3 The Adjustment of Errors 26
3.1 Introduction 26
3.2 Types of Error 26
3.2.1 Gross Errors 26
3.2.2 Accidental (Random) Errors 26
3.2.3 Systematic Errors 27
4
3.2.4 Example 27
3.3 Average Error 28
3.4 Mean Square Error (Standard Deviation) 28
3.5 Probable Error 29
3.6 The Propagation of Errors 29
3.6.1 Summation 29
3.6.2 Multiplication 30
4 Co-ordinates 31
4.1 Definitions 31
4.2 The Four Quadrants 32
4.3 Calculation of Co-ordinates using Bearing and Distance 34
4.4 Calculation of Bearings and Distances from Co-ordinates 35
4.5 Practical Examples 37
5 Angle Measurement 39
5.1 The Repetition Method 39
5.2 The Reiteration Method 40
5.3 The Direction Method 40
5.4 The Sector Method 43
6 Optical Distance Measurement 44
6.1 Vertical Staves 44
6.1.1 The Reichenbach Principle 44
6.1.2 Self-Reducing Instruments 46
6.2 Horizontal Staves or Bars 48
6.2.1 The Richards Optical Wedge 48
6.2.2 Self-Reducing Tacheometers 51
6.3 Horizontal Subtense Bar 52
6.3.1 Subtense Distance Measurement 52
6.3.2 Measurement of Sub-divided Sections 54
6.3.3 Auxiliary Base Method 54
7 Simple Surveying Techniques 56
7.1 Triangulation 56
7.1.1 Base Extension 56
7.1.2 The Triangulation Network 58
7.1.3 Choice of Instrument 58
7.1.4 Orientation of the Network 60
7.1.5 Calculation of the Network 60
7.1.5.1 Adjusting the Angles 62
7.1.5.2 Solution of Base Triangles 62
7.1.5.3 Solution of Main Triangles 64
7.1.5.4 Calculation of Bearings and Co-ordinates 65
7.1.5.5 Intersection 67
7.1.5.6 Resection 69
7.1.6 Co-ordinate List 72
7.2 Trigonometric Levelling 73
7.2.1 The Curvature of the Earth 73
7.2.2 Refraction -74
7.2.3 Measuring Procedure 75
7.3 Traversing 77
5
7.3.1 General 77
7.3.2 Angle Measurement ]9
7.3.3 The Traverse Sides 82
7.3.4 The Calculation of Co-ordinates 83
7.3.4.1 Angle Adjustment 83
7.3.4.2 Bearings 83
7.3.4.3 Co-ordinate Differences 83
7.3.4.4 Calculation of Elevations 86
7.3.4.5 Gross Errors 86
7.3.4.6 Systematic Errors 86
7.3.5 Connection to Inaccessible points 87
7.3.6 The Three Tripod Method 88
7.4 Compass Traverses 89
7.4.1 The Propagation of Errors 92
7.4.2 Leap-frog Stations 94
7.5 Detail Surveys 94
7.5.1 Offset Surveys and the Optical Square 94
7.5.2 Polar Co-ordinates 98
7.5.3 Cadastral Surveys. 98
7.5.4 Topographic Surveys 100
7.5.5 Cartographic Maps 102
8 Care of Instruments 107
6
1. Introduction
Directions are measured with the theodolite to two or more points, together with the
inclinations of these directions, all referred to the horizontal plane passing through
the observation point. From these measurements horizontal and vertical angles are
obtained.
R
3 Fig.1
I n Fig. 1, 0 is the point of observation, from which points P" P2 and P3 are sighted.
The tilting axis of the theodolite telescope lies in the horizontal plane passing
through 0. It follows that the difference between the directions OP, and OP2 gives
the horizontal angle CX1-2 and that the difference between the directions OP2 and
OP3 gives the horizontal angle CX2.-3. The vertical angle is defined as the slope or
inclination of the line of sight, in relation to the horizontal plane through 0. As the
lines of sight to points P, and P2 both lie above this plane, the vertical angles [3,
and (:\2, respectively, are both positive and, as the line of sight to point [33 lies below
the horizontal plane, the vertical angle [33 is negative. P1', P2' and P3' represent the
vertical projections, on to the horizontal plane, of P" P2 and P3, respectively, and
OP1', OP2', OP3' are the horizontal projections of the slope distances OP" OP2 and
OP3, respectively. If the slope distance OP, is known, and also the vertical angle [31,
then the horizontal distance OP,' and the difference in elevation P1-P1' can be cal-
culated. This elevation difference then also represents the difference between 0
and P1. For the computing of co-ordinates and for mapping, only these horizontal
projections of the lines of sight can be used.
The accuracy of angle measurement depends, to a great extent, on the diameter of
the theodolite's horizontal and vertical circles. The circle diameter, in turn, influences
the size and weight of the instrument as other parts (such as the base plate, the
standards and the telescope) must be designed with corresponding dimensions.
The accuracy requirements for angle measurement vary considerably, when all
types of surveying are considered, and, as a result, so do theodolites. In order to
appreciate these requirements, and at the same time, have a glance at the great
importance of angle measurement let us first consider, very briefly, how a survey is
carried out.
The normal practice in all surveys is to work from the whole to the part. For large
areas (national or ordnance surveys) a geodetic base is measured first and serves
as the basis for the establishment of a uniformly accurate network of control points
covering the whole survey area. This consists initially of large triangles which are
broken down progressively into smaller and smaller figures until, finally, the survey
7
of topographical detail may be undertaken, without difficulty, by reference to the
dense coverage of control points. Triangulation and traversing are used to obtain
this coverage.
1.1 Triangulation
The principles of triangulation are simple. If the length of Qne side and the size of
two angles of any triangle are known, then the third angle and the other two sides
of the triangle can be calculated. If all three corners of the triangle are accessible,
the third angle is also measured, thus providing a check on the angles, as their sum
m ust be 1800.
If, therefore, at the ends of a comparatively short, but most precisely measured, base
AB the angles to C and D are measured, the distance CD and the positions of these
two points in a co-ordinate system can be calculated. Similarly, the positions of
points E and F can be found by angular observation to them from C and D, respec-
tively (Fig. 2).
This base extension is continued until the sides of the triangles correspond to the
size required in first order triangulation. The whole area to be surveyed is then
covered with large triangles, in which only the angles are measured. Depending on
the size of the area, the topographical features and the development, the side lengths
of the triangles normally vary between about 20 miles and 60 miles (30 km and
100 km) (Fig.3). By means of suitably distributed astro fixes and azimuths, this wide-
spread network is oriented geographically and is computed with absolute values
with the aid of special error adjustment techniques.
The basic survey system is developed with a closer concentration of points by
breaking doy/n the large triangles into ones with shorter sides. The triangulation
goes to second and then third order and, finally, to the fourth order detail triangula-
tion, in which the density and positioning of the control points are adapted to suit
the amount of detail in the area to be surveyed. Eventually there may be approximately
3 to 8 points per square mile (1 to 3 points per square kilometre), where the develop-
ment of the country needs the establishment of fourth order control.
Since the introduction of electronic distance measuring instruments, such as the
Wild Dl SO Distomat, it is now possible, instead of measuring angles, to measure
directly the sides of triangles up to 95 miles (150 km) long with geodetic accuracy.
This system is called trilateration. In a modern survey network the two methods
can be combined, according to the type of terrain.
1.2 Trigonometric Levelling
Triangulation gives only the horizontal relationship between points. Differences in
elevation between them can be calculated by vertical angle measurement. Broadly
speaking these are the principles of triangulation. From this it will be seen that the
basis of the whole control system of a territorial survey, apart from one or more base
line measurements, depends almost entirely upon the measurement of angles,
coupled recently with electronic distance measurements. Trigonometric leveling is
discussed in greater detail in para. 7.2.
1.3 Traversing
Even if triangulation has been continued to the 4th order, it is possible that the re-
quired density of control has not yet been reached to enable the detail survey to be
undertaken with ease and the necessary accuracy. The network must be broken down
still further but, at this stage, the principles of triangulation are put aside in favour of
traversing. This consists of connecting the triangulation stations by means of a
series of points, whose positions are determined by measuring the angles at the
8