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ELEMENTARY
SURVEYING
by
ARTHUR LOVAT HIGGINS, D.Sa,
A.R.C.S., A.M.Inst.C.E.
FORMERLY UNIVERSITY READER IN CIVIL ENGINEERING
UNIVERSITY OF LONDON
Author of
The Field Manual, Higher Surveying, The Transition Spiral,
Phototopography, etc.
WITH DIAGRAMS
LONGMANS, GREEN AND CO.
LONDON i: NEW YORK :: TORONTO
LONGMANS, GREEN AND CO. LTD.
6 & 7 CLIFFORD STREET, LONDON, W.I
NICOL ROAD, BOMBAY, I
17 CHITTARANJAN AVENUE, CALCUTTA, 13
36A MOUNT ROAD, MADRAS, 2
LONGMANS, GREEN AND CO INC,
55 FIFTH AVENUE, NEW YORK, 3
LONGMANS, GREEN AND CO.
215 VICTORIA STREET, TORONTO, I
First published . . . 1943
Second Impression . . 1945
Third Impression . . 1946
Fourth Impression . . 1947
CODE NUMBER 86340
BOOK
PRODUCTION
|WAREO)NOMY|
STANDARD
THIS BOOK IS PRODUCED IN
COMPLETE CONFORMITY WITH THE
AUTHORIZED ECONOMY STANDARDS
MADE AND PRINTED IN GREAT BRITAIN
BY JARROLD AND SONS, LTD. NORWICH
PREFACE
Now that Elementary Surveying is regarded as something more than a
mere adjunct to mathematics and geography, it appeared to the writer
that there might be a place for a little book which aims at opening a
vista of the educational and professional possibilities of the subject,
presenting it as the application of a few general geometrical principles
rather than something akin to a handicraft with each operation an entity.
It is hoped this book will stimulate enthusiasm among those who con-
template entering one of the professions implied in the Introduction
or, otherwise, create an interest in the other man's job.
The text is based largely upon the syllabus in Elementary Surveying
in the General School Examination of the University of London, and
matter outside this curriculum is indicated with an asterisk, suggesting
the introduction to an intermediate course in the subject. Also many
of the questions are taken from papers set by the writer in this particular
examination; and he takes this opportunity of expressing h ; s indebted-
ness to the Senate of that University for their courtesy in permitting
him to reproduce this material.
In addition to the theoretical exercises, a number of field exercises
are added, and these no doubt will suggest lines upon which others
can be devised in keeping with what may be (conveniently) styled
"local" conditions. These examples are short, and anticipate the
adoption of parues of three (four at most) pupils, this organisation,
in the writers opinion, being the only rational way of handling the
subject. Parts of larger surveys or schemes can be allocated to these
parties, who retain their identity as far as is practicable. Prior to
going into the field the routine should be outlined so as to reduce
supervision to a minimum, and, better still, to leave the parties to their
own devices.
The writer takes this opportunity of expressing his indebtedness to
Mr. A. N. Utting, of the Cambridge University Engineering Labora-
tory, for preparing the drawings from which the figures are reproduced,
also his thanks to Mr. S. G. Soal, M.A., of Queen Mary College, for
his kindness in reading the proofs.
In conclusion the writer acknowledges the agency of his wife, whose
influence really led him to undertake this short but pleasant enterprise.
Queen Mary College, ARTHUR LOVAT HIGGINS
cjo King's College,
Cambridge
CONTENTS
CHAPTER PAGE
INTRODUCTION 1
I. FUNDAMENTAL PRINCIPLES 4
Co-ordinates Five fundamental methods Triangulation and traversing
Offsets, locating objects Chains and chaining Sloping distances Other
modes of linear measurement Signals
II. CHAIN SURVEYING 22
Equipment Field book Outline of simple survey Boundary lines
Traversing with the chain
III, PLOTTING PLANS AND MAPS 30
Construction and use of scales Special scales Plotting and finishing
maps Conventional signs Constructing angles; use of protractor and
trigonometrical tables Enlarging maps and plans
IV. FIELD GEOMETRY 48
Reciprocal ranging Perpendiculars and parallels Optical square
Obstructions to measurement and alignment Four classes of obstacles
Checking angles with the tape
V. LEVELLING 60
Classification of methods; historical note Bubble tubes and equivalent
plumb line Telescope Dumpy and other levels Levelling stavesLevelling
practice Two systems of booking levels Curvature and refraction Levelling
difficulties
VI. ANGULAR LEVELLING 79
Methods The clinometer and Abney level Observing heights Barometry
Aneroid barometer
VII. THE COMPASS 86
Historical note The prismatic compass Bearings and azimuths
Magnetic declination and variation Local attraction Fixed and free needle
Traversing with the compass Graphical adjustment of traverse surveys
Compass resection
VIII. PLANE TABLING 102
The plane table and its accessories Primary methods Orientation and
setting Resection; the three-point problem Field work
IX. CONTOURING 113
Nature and uses of contours Horizontal and vertical control Direct and
indirect methods of contour location Combinations of methods Interpola-
tion
X. AREAS AND VOLUMES 123
Areas of simple plane figures Areas of irregular plane figures Methods
Give and take lines Trapezoidal and Simpson's rules Computing scale
Volumes of simple regular solids Cross sections Trapezoidal and pris-
moidal rules Earthwork volumes Use of truncated prisms and contours
__ Longitudinal and cross sections; gradients
XL THEODOLITE SURVEYING ^ 140
' Historical Note The theodolite Circles and verniers Measurement of
angles Theodolite surveys Reducing bearings Latitudes and departures
Adjustment of traverse surveys Miscellaneous problems
TRIGONOMETRICAL TABLE 154
INDEX 155
iv
INTRODUCTION
Surveying may be described as the art of making measurements upon
the earth's surface for the purpose of producing a map, plan, or
estimate of an area. Levelling is combined with surveying when the
project requires that the variations in the surface surveyed shall be
delineated by contour lines, or shown in a vertical section, or used in
the calculation of a volume content.
Surveying may thus be defined as making measurements in the
horizontal plane, and levelling as taking measurements in the vertical
plane.
The converse operation to surveying is setting-out work, or field
engineering, as when a constructional project, such as a railway, high-
way, or reservoir, is pegged out on the ground. Hence it is obvious
that whatever possibilities the future may hold for aerial methods of
surveying, the lowlier methods of ground survey will always be utilised
in the setting-out of works for the use and convenience of man.
Surveying is divided primarily into (1) Geodetical Surveying and
(2) Plane Surveying. In geodesy, the earth is considered a sphere, and
in plane surveying a plane, the approximation being within the per-
missible limits of error for areas up to about 100 square miles. The
former involves a knowledge of spherical trigonometry, and the latter
of plane trigonometry.
Mathematics. The mention of trigonometry introduces aptly the
question as to whe extent of mathematical knowledge necessary in the
various professions in which surveying plays an important part. In
applied science, mathematics is a good servant but a bad master, and
philosophic doubts often overcome enterprise. This suggestion of
more advanced mathematics may cast a shadow over the aspirations
of the reader; but let him be comforted in the thought that few boys
are gifted with real mathematical ability, and not infrequently this is
at the expense of vision and initiative, by one of those balancing feats
of nature, which always settles its account with the least effort. Nor-
mally mathematical knowledge is a slow growth in hard-worked
ground, and many brilliant scientists and engineers would admit that
their knowledge in this connection has grown with mental development
arising from other interests, the complex filling the voids in a wide,
open structure of essential principles.
It is unfortunate that the syllabuses of certain examinations do not
insist upon an elementary knowledge of plane trigonometry. In fact,
a degree may be taken in geography, evading trigonometry by cumber-
some artifices in map projections, while, at the bench the workman
can often use the tablts with facility as merely a part of a day's work.
Therefore get into touch with your trigonometical tables. Four-figure
I
2 ELEMENTARY SURVEYING
tables will suffice when angles are only required to degrees, five figures
when minutes of arc occur, and seven figures whenever seconds are
involved.
In ordinary surveying, such as occurs in connection with estate
management, valuation, building, municipal engineering, town-
planning, and quantity surveying, a knowledge up to and including
the solution of plane triangles is necessary; and the subject is subordi-
nate to mensuration, the application of which demands speed,
accuracy, and orderliness. In civil engineering a knowledge of spherical
trigonometry and the calculus will be desirable, as also is the case in
cartography and hydrography, while geodetic surveying will demand
still more advanced mathematics, particularly knowledge of the theory
of errors.
Errors! What are errors? They are as natural to surveying as colds
and measles are to the young. Scientifically, they are not "mistakes,"
and you make no apology for making them, though you do your
utmost to keep them in their place. The true error in a measured
quantity is never known, simply because the really true measurement
of that quantity is not known. But this is a very advanced argument.
All you know is the "discrepancy" between successive measurements
of the same quantity, all of which may contain error; though, of course,
comparison with a precise standard will convince you whether the
error is great or small. Though you may never aspire to a knowledge
of the theory of errors, you must learn to control and adjust your
errors, always avoiding mistakes with professional contempt by never
dropping a chaining arrow (or an odd ten in calculations) or reading
a foot out on the levelling staff. But this digression is looking years
ahead. You want to know something about the scope of the subject,
which is shown in the following list, where the relative degrees of
accuracy are given in descending order, the demands of accuracy
gradually giving way to the exigencies of speed and time.
TRIGONOMETRICAL SURVEYING, for the preparation of maps of large
extents of territory.
LAND SURVEYING, ranging from the Land Division System of the
United States and extensive topographical surveys and work for
boundary commissions to small areas, such as farms and estates.
HYDROGRAPHICAL SURVEYING, ranging from coast surveys to plans
for harbour works.
ENGINEERING LOCATION SURVEYING, for the construction of highways,
railways, and various public works. Mine surveys are to be included
in this category.
PRELIMINARY AND PARLIAMENTARY SURVEYS, in connection with a
projected scheme, such as the construction of a railway or a waterworks.
PIONEER AND EXPLORATORY SURVEYING, for geological, engineering,
and mining enterprises, also work in connection with archeological
expeditions.
INTRODUCTION 3
SURVEYING, ranging from reconnaissance to maps by
aerial photographic methods. In war, these are carried out in dangerous
situations, and accuracy must be subordinated to speed.
Some writers subdivide the subject in accordance with the i'lstrumeni
used; e.g., The Chain, The Theodolite, The Compass, etc., and others
by the methods, as Photographic Surveying, Tacheometrical Surveying,
Plane Tabling, etc.
ORDNANCE SURVEY MAPS. Most countries issue a series of maps
for the various subdivision of their states and departments, further
sheets showing municipalities, etc., based upon these. In the United
Kingdom, this is done by the Ordnance Survey Department. The
best known of the Ordnance sheets are the Six-Inch, or "County*'
maps, on a scale of 6 inches to the mile or a representative fraction of
1 : 10,560, which is used largely in connection with parliamentaiy plans;
the Twenty-five Inch, approximately 25 inches to the mile, or exactly
1 : 2,500, as used for certain constructional surveys, and (double scale)
in land valuation; and the One-Inch, or I : 63,360, either plain or
coloured, contoured and hill-shaded. Various other maps are obtain-
able, formerly the 1 : 500 "Town" map for certain districts, down to
the latest series for the Land Utilisation Survey.
The commoner Qrciugince sheets should be carefully examined, and
notes made as to the conventional signs used to represent such features
as county, borough, and parish boundaries, roads, marshes, canal
locks, tunnels, etc., etc. Levels are marked on these maps, and, in
addition, the 25-inch gives the areas of enclosures, the well-known bond
indicating that a detached area is included in a given acreage*
CHAPTER I
FUNDAMENTAL PRINCIPLES
In introducing the First Five Principles of Surveying, it may be
advisable for us to recall our acquaintanceship with Co-ordinates, or
"graphs," as you doubtless call them. In Fig. 1 you will recognise
the axes of rectangular (or Cartesian) co-ordinates, with the X and Y
axci corresponding to abscissae x and ordinates y y the origin being at O.
"Positive north and positive east,
Negative south and negative west."
Rectangular co-ordinates are also used in plotting surveys by the
Method of Latitudes and Departures, the four quadrants representing
the four quarters of the compass, as
indicated by the letters, N.E., N.W.,
S.W., S.E.
Possibly you have also met the cubic
parabola, y = O.lx 3 , as plotted with
respect to the axes in Fig. 1. It is not
altogether an intruder here, being a
member of the same family as y=cx*,
which is the transition curve the rail-
way surveyor sets out, in order to ease
the passage of a train from the tangent-
straight to the circular curve against the
effects of centrifugal force on the train's
motion.
Other forms of co-ordinates are used
in surveying; in particular, Polar Co-
ordinates, in which the point P is fixed
with respect to the axes by the distance
OP and the bearing or angle (3. But
there are endless applications of our mathematical principles in applied
science, and each is not a stranger living in the same house.
I. FIRST FIVE PRINCIPLES
In the introduction it was stated that Surveying consisted in making
measurements in the horizontal plane, and Levelling taking measure-
ments in the vertical plane. Actually, in surveying the measurements
consist in fixing the positions of points in the horizontal plane; two
points fix a straight line, and three or more straight lines determine
the plan of a plane figure. If the actual position of a point P is also
found in a vertical plane, vertically above its plan p, the point P is fixed
+x
S.L
- Y
FIG. 1
FUNDAMENTAL PRINCIPLES 5
in space; and this is the basis of topographical surveying, which leads
to a map in which the surface features are delineated, and usually
represented by contour lines.
All surveying operations are based upon these principles, as will
appear in the summaries appended to the following methods.
FIRST METHOD Rectangular Co-ordinates
Here the point p is fixed with respect to the survey line AB by the
distance^/? measured at right angles to AB from the point q (Fig. 2).
Uses. (1) Auxiliary, as in taking right
angle offsets to the boundaries from the
skeleton outline of a survey.
(2) Setting out buildings and certain o
engineering works. 90
(3) Fundamental in important opera-
tions, such as the U.S.A. Lands Survey. A ^ B
Here the X co-ordinates are really FIO. 2
parallels of latitude, and the Y co-ordinates
meridians, guide and principal; and as the area surveyed becomes
extensive account has to be taken of the fact that on a spherical earth
the meridians must converge in order to pass through the poles.
Thus in a few lines our little mathematics has carried us from
mechanics to geography.
SECOND METHOD Focal Co-ordinates
Here the point/? is "tied" by the distances ap, bp, which are measured
respectively from a and 6, known points in the survey line AB (Fig. 3).
Uses. (1) Auxiliary, as in surveying
boundaries with long offsets, particularly in
surveying frontage lines in town surveying.
(2) Basis of all chain surveying, whether
4 'chain triangulation" or traversing. _ r * .
(3) Method of referencing survey stations ^ a b "
on the completion of the field work. 1 10. 3
THIRD METHOD Angular Co-ordinates
Here the point p is fixed with respect to the line AB by the inter-
section of two visual lines, ap, bp, which at known points a and b make
observed angles 6 and 9 respectively with
AB (Fig. 4).
This method is peculiarly applicable to the
locating of inaccessible points and objects,
such as mountain peaks, sounding boats, and /e y / \ p
through the medium of electrical communi- A a L
cation, the position of aeroplanes in flight. FIO. 4
6 ELEMENTARY SURVEYING
Uses. (1) Basis of the method of "intersections" with the plane
table and compass, also the kindred process in ordinary and stereo-
scopic photographic surveying.
(2) Method embodied in range-finders and telemeters, the base ab
being near the observer; and conversely, the principle employed in
tacheometry, the optical measurement of distances, the base being at
the distant point observed.
(3) Basis of all pure triangulation, which may range from a simple
net of triangles to a major and minor system, or even a primary, a
secondary, and a tertiary net, as in the Ordnance Survey of the United
Kingdom.
FOURTH METHOD Polar Co-ordinates
Here the point p is fixed with reference to the survey line AB by the
distance ap measured from a known point a in A B at a known
angle p from that line.
Uses. (1) Method of locating details by "angles and distances."
(2) Method of "radiation" and "pro-
gression" in plane tabling, where the
angles are measured goniographically;
i.e. constructed without account of their
magnitudes.
a b (3) Basis of traversing with the com-
FIG- 5 pass or theodolite, AB being a reference
meridian or N. and S. line.
Inverse polar co-ordinates occur in certain operations, the (dotted)
distance bp being measured instead of ap.
FIFTH METHOD Trilinear Co-ordinates
Here the point p is fixed by 6 and <p, the angles subtended at p by
three visible and mapped points, A, B,
and C (Fig. 6).
Uses. (1) The "three-point problem"
in resection with the plane table, also
with the compass and the theodolite.
(2) Important method in marine
surveying, P being the sounding boat
and A,B,C, three points plotted on the
chart.
(3) Method embodied in resection
FIG. 6 in space in stereoscopic methods of
surveying.
(4) Locating positions by wireless signals from three known trans-
mitting stations.
Now the mere knowledge of these principles is not the sole qualifi-
cation of a surveyor. There is the art or technique of the subject,
FUNDAMENTAL PRINCIPLES 7
which alone is acquired by practice and experience. Primarily, this
consists in judiciously selecting methods and instruments to suit the
objects and nature of the survey. It is not acquired by making a
crazy-patchwork map merely to show that you have used evt*y instru-
ment at your disposal, though of course contingencies may arise in
which it is expedient to depart from the one prevailing method of the
survey. Secondly, the art requires that you shall make all your
measurements with uniform accuracy, never mixing the crude and
precise promiscuously. Unfortunately there are many obsessed with
the idea that rough measurements will accommodate themselves, not
only obligingly, but correctly between points surveyed with great
precision as a basic framework. Thirdly, simplicity and economy are
to be considered with due regard to the strength or rigidity of trm
basic figure or scheme.
TRIANGULATION AND TRAVERSING. There are two primary methods
of making a survey: (1) Triangulation and (2) Traversing.
In triangulation the area is covered as nearly as may be with a
scheme of triangles, and in traversing, by a polygonal outline, also
approximating to the boundary or fences (Figs. 7, 8), the latter being
more applicable to areas devoid of interior detail.
FIG. 7
FIG. 8
Traverses may be closed, as ABCDEA, or open, as indicated by the
dotted lines EefgC, which actually makes a compound traverse with
the boundary survey, which is the case in certain town and park surveys.
In triangulation surveys, only one side, the base, may be measured
(Pure triangulation)^ or certain sides a^d angles (Mixed triangulation)',
or only the sides are measured, as in chain surveys (Chain triangulation).
The strongest figure is that with the fewest sides, hence the triangle;
and the skeleton of a traverse becomes weaker as the number of sides
increases, so that it may be necessary to brace it up with triangles.
STATIONS. The angular points of a triangulation net or traverse
skeleton are called stations, and are usually indicated thus in chain
surveys , and A whenever angular observations are involved.
Commonly stations are referenced with capital letters, A, B, C, etc.;
and if subsidiary stations occur the small letters, a, b, c, etc., are
requisitioned. In extensile surveys it is advisable to retain the letters
for main stations, and to utilise numerals, 1, 2, 3, etc., for the
8 ELEMENTARY SURVEYING
sub-stations. Otherwise the large and small letters are soon exhausted,
and applying dashes (or primes) leads to confusion in the field notes.
The use of the Greek alphabet is not usually successful, as the surveyor's
classical knowledge seldom gets beyond Epsilon, e.
Stations are established in the ground in a manner consistent with
the duration of the field-work. In small surveys, such as you will
undertake, If-in. square pegs will suffice, the error of planting
the picket or flagpole beside it introducing no serious error in
chain surveys. In practice, however, where the work is likely to
last weeks, a metal socket is let into concrete, the flagpole being
inserted in the socket when the station is not occupied. As the
survey grows and lines are thought of in miles, not chains, the
stations become more permanent, and when the distances reach
up to 60 or even 100 miles scaffolds are erected with signals for
day and night observations.
II. FIELD-WORK
The first item in the surveyors' outfit consists of ranging rods,
or poles, commonly in the 6-ft. length, known as pickets, which,
like the 8-ft. and 10-ft. poles, are painted in one-foot lengths,
alternately red, white, and black, and are shod with a steel point.
A bundle of six pickets forms a convenient set for a party.
The longer sizes are more convenient in larger surveys. Flags of
red and white fabric are desirable when visibility is impaired by
distance or background. Pickets can be supplemented by builders'
laths in "ranging out" or "boning out" long lines so as to guide
the chainmen as they follow the ups and downs of the ground.
CHAINS. The standard chain length of 66-ft. or 4 poles, was
introduced by the celebrated mathematician, Edmund Gunter, in
1620. The length is not only convenient to handle, but is such
that ten square chains are comprised in an English acre. Where-
fore Americans style it the "surveyors' " chain in distinction with
the 100-ft. unit, or "engineers' " chain, which is now used ex-
tensively in this country in engineering surveys.
Both chains are made up of one hundred long pieces of steel or
iron wire, each bent at the ends into a ring, and connected with
the ring of the next piece by two or three oval rings, which
afford flexibility to the whole and render the chain less likely to
become entangled or kinked. Two or more swivels are inserted
in the chain so that it may be turned without twisting.
The entire length of the chain is 66 ft. (or 100 ft.) outside the
handles, and the hundredth part of the whole is a link (or a foot),
this decimal division allowing lengths to be written as 8-21 Iks.
(or ft.). Each link, with the exception of those at the handles, is
7-92 in. =0-66 ft. (or 1 ft.), as measured from the middle ring of
Fio. 9 the three connecting rings to the corresponding ring in the next
FUNDAMENTAL PRINCIPLES 9
length. The handle and short link at each end constitute the first and
hundredth link (or foot). Every tenth link is marked with a brass tag
or teller in a system that allows either end to be used as zero, as indi
cated in Fig. 10, where the one finger tag can be 10 or 90, and the
four-finger tag 40 or 60 Iks., or ft., as the case may be.
The Gunter chain is much more easily manipulated than the 100-ft.
length, and on this account some surveyors insist upon 6-in. links in
orde r to increase the flexibility. Sometimes a 50-ft. chain is used for
offsets, or where traffic exists, but the reversion of few tellers makes it
inconvenient to read.
50
Steel wire chains are lighter and more easily manipulated than those
of iron wire, and the three-ring pattern in No. 12 gauge wire by
Chesterman is recommended. Iron wire chains are still used in rough
farm and estate .vork, and though more cumbersome, are more easily
corrected and repaired than steel chains. A 100-ft. chain in iron wire
would be a tough proposition for a young surveyor; and in the language
of Huckleberry Finn, "A body would need the list o' Goliar to cast it.'*
Chaining arrows are used to mark the ends of the chain lengths.
These are preferably made of steel wire, so as to allow the use of a
lighter gauge. A common length is 1 ft. in the pin, but 18 in. or even
2 ft. may be necessary in long grass or stubble. Arrows should be
made conspicuous by tying strips of wide red tape to the rings. They
should be carried on a steel snap ring or in a quiver slung ac^ss the
shoulders. Ten arrows comprise a set, and the number should be
checked from time to time.
Tapes and Bands. In better class work the surveyor often uses a
blue steel band fitted with handles, and wound on a windlass when
not in use. The links or feet are marked with brass studs, a small
plate denoting the tens. Extreme care should be exercised in the use
of these, particularly in the narrow or lighter patterns. Steel bands
are elastic, and it is quite easy to pass the elastic limit and so produce
a permanent set. For this reason, though more especially to keep the
length constant, some surveyors use a tension handle or a spring
balance, in order to apply a constant pull, which may vary from 5 Ib.
10 ELEMENTARY SURVEYING
to 20 lb., according to the cross-sectional area of the steel. Also, the
tapes are easily snapped, and are liable to corrosion, and need wiping
and oiling to preserve them.
The Linen tape is indispensable to the surveyor, civil engineer, and
valuer. These tapes are usually wound into round leather cases, and
are obtainable in the 50-ft., 66-ft., and 100-ft. lengths, showing feet
and inches, and, desirably, links on the reverse side. Similar patterns
are made in bright steel with etched divisions, but these are expensive,
and their use should be restricted to high-class work and experienced
hands. Linen tapes should be dry when wound into their cases; if
dirty, they should be washed, wiped carefully, and allowed to dry.
CHAINING. Happily the following instructions are being given to
young and enthusiastic surveyors who do not regard it infra dig. to get
down (literally, and on one knee) to a job of the first importance.
Good chaining is a great accomplishment, which can be appreciated
only by those who have had good, bad, and indifferent chainmen.
Some surveyors are fortunate enough to have trained chainmen,
whereas the resident engineer is sometimes at the mercy of a contractor's
foreman, who in his wisdom lends him the two men whom he regards
as surplus to requirements.
Let it be assumed that you have ranged out a line between two
station poles, A and B, by standing behind one pole A, sighting the
other B, and directing by hand signals the interpolating of pickets and
laths at intermediate points in the line. Presumably you have agreed
who shall be Leader (L) and who Follower (F) in chaining the line.
First of all, the chain must be cast in the following manner: Remove
the strap from the chain, and unfold five links from each handle, then,
holding both handles in the left hand, throw the chain well forward,
retaining hold of the handles. If the chain has been done up correctly,
no tangles will occur.
Leader (L), on receiving ten arrows, counts them, and drags the
chain forward along the line AB. As he approaches a chain's length
from the Follower (F), he moves slowly, and on receiving the order,
"Halt," turns and faces F with an arrow gripped against the handle.
He bends down in readiness for further instructions. Follower (F),
bending down, holds his handle against the starting-point A. He then
jerks the chain to expel kinks, directs L to tighten or ease his pull,
lines L in with the forward station B, and finally, with hand signals,
directs L to fix Arrow No. 1. Meanwhile, L, holding the chain clear
of his person (and preferably facing F), responds to the orders from F,
and on receiving the final "stick," fixes an arrow firmly as No. 1.
L now takes up the remaining nine arrows, and drags the chain forward
for the second length, which is measured in the same manner, except
that F holds his handle against Arrow No. 1. On receiving the signal
"stick," L fixes Arrow No. 2 and goes forward, dragging the chain.
Meanwhile F takes up Arrow No. 1 and carries it to No. 2. The
FUNDAMENTAL PRINCIPLES II
process is repeated, L inserting arrows, and F collecting them duly.
If the line is long, the leader L calls out "Ten" on fixing his last arrow,
No. 10. The best practice now is to proceed to measure the eleventh
length, the Leader having no arrows. When the eleventh len^ch has
been laid down, L stands on his end of the chain until F comes up
with ten arrows, which he hands to L, who sticks one (1 1 chains) before
dragging the chain forward for the twelfth length.
Folding the chain properly means the saving of much annoyance
when next it is used. Take it up by the middle (50) teller and shake
it out so that it drags evenly on each side of that teller. Transfer it
to the left hand, and place the first five links on each side of the 50-teller,
side by side, two at a time together, turning the links in the palm of
the hand. Now invert the folded portion in the left hand so that the
50-teller hangs down, and, turning it slowly in the palm of the hand,
fold links equidistant from the middle across it, two at a time, not
straight, as at first, but sloping obliquely to the left at the top. Continue
this oblique folding until the handles are reached, and secure it by means
of the strap in this form, which is that of a hyperboloid of revolution.
Testing Chains. The limits of this book preclude various hints as
to the care, correction, and repair of chains. Nevertheless, these should
be tested from time to time. Students in their enthusiasm may un-
wittingly provoke chaining into a tug-of-war, and even the rings of
steel chains will open under the strain. Although this will not occur
as far as you arc concerned, it is essential that a Standard Length be
laid down carefully with a steel tape on stone flags or a concrete
surface, the ends being marked with cuts into the surface, or, better,
by inserting metal plugs cut with a fine cross and filled with solder.
Sometimes startling disclosures are made in checking a chain against
the standard length. Quite easily a chain may be forgotten during a
break for lunch, and an inoffensive ploughman may tun it down, and
be little alarmed at the repair he has made with not more than three
links missing.
Then there is always the danger that a chain of correct length /
has attained an incorrect length / after protracted use. Hence lines of
correct length L are measured as L, an- 1 consequently tiue areas A
are computed as A', but if the chain has been tested and tne incorrect
length / observed, the correct values can easily be reduced by the
following relations:
LQ= -j *L\ AQ\ J~ } A.
'0
Offsets. Offsets are measurements made from the outer survey
lines of a triangulation or traverse skeleton to the boundary of a
property, the root of a hedge, a fence, or a wall, as the case may be
Usually these are taken at right angles to the survey lines, and their
length is limited roughly to 50 links, though some latitude is allowed
in certain circumstances. Whenever necessarily long, they should also
12
ELEMENTARY SURVEYING
be "tied" from another point in the chain as it lies along the survey
line. Offsets are usually measured with the linen tape, though formerly
the offset staff was used. The right angle is estimated, but when the
offset is long, this is best done by "swinging" the tape in the following
manner: A directs B to hold the ring (O) end of the tape at a point
in the boundary or detail, and, pulling the tape out gently, A swings
it over the chain and notes the lowest readings both on the chain and
tape as the respective chainage and length of the offset.
Objects buildings in particular are located by finding points on
the chain which are in line with the end walls of houses, as judged by
sighting along these while standing on the chain as it lies on the ground
in the survey line. Fig. 1 1
(left) shows how a build-
ing is fixed by rectangular
offsets, the diagonals be
and ad being sometimes
measured as checks. The
line cd being thus fixed,
the position of the build-
ing may be plotted, and
since it is rectangular, it
may be constructed on the
side cd when the remain-
ing sides have been mea-
sured up with the linen
tape. Fig. 1 1 (right) shows
a common method of
"tying in" a building
which lies obliquely to
the survey line. Here the
points a' and c' are selected so that they are in line with the respective
sides /'#' e'h' 9 of the building and the corners/' and e' are tied with
the lines of #'/', />'/', and c'e', J'e' 9 respectively, the readings a', c', b' ,
and d' being suitably recorded in the field
notes.
At this stage we may consider two simple
instruments which are used to set out the
right angles of long perpendiculars, the
geometrical construction of which will be
dealt with in Chapter IV.
CROSS STAFF. The cross-head is best known
in the open form, shown in Fig. 12, the
more complicated patterns possessing little
to qualify their use. This pattern consists of four metal arms, turned up
at the ends, and cut with vertical sighting slits at right angles. The head
11
FIG. 12
FUNDAMENTAL PRINCIPLES U
is attached to an iron-shod staff, which is planted at the point at which
it is desired to set out the right angle. Two slits are sighted along the
survey line, and the right angle is set out by sighting in a picket through
the other pair of slits. The chief difficulty is that of planting the staff
(or Jacob) truly vertical, but this can be facilitated by the simple artifice
hereafter described.
Cross heads can be constructed in the manual training classes, and even
if metal is not available, quite useful instruments can be made from hard
wood. The best way of ensuring that the staff is vertical is to use a ring-
plummet, which may be improvised as follows. Drill ^-in. holes near the
alternate corners of the hexagonal face of a backnut for IJ-in. gas-pipe, and
drill three corresponding holes in the rim of socket in which the staff is
inserted. Suspend the nut by three threads from the socket; then, when the
staff is vertical, it will appear centrally in the hole of the backnut.
* Optical Square. The optical square belongs to a class of reflective
instruments of which the Sextant is the representative instrument in
modern surveying. The best-known form consists of a circular box
about 2 in. diameter and f in. deep. The lid, though attached, can be
slid round so as to cover the sight-holes and thus protect the mirrors
when out of use. Fig. 13 shows a plan of the square when the lid, or
cover, is removed; h is the
half-silvered horizon glass,
rigidly attached in a frame
to the sole plate, and i the
wholly silvered index glass,
which in some patterns is
adjustable by means of a
key. The three openings
required for sighting are cut
alike in the rims of the case
and cover: a square hole Q
for the Horizon sight, a
similar one O for the Index
sight, and a pin-hole e for Fio. 13
the Eye sight.
The index glass / is set at an angle of 105 to the index sight line Oi,
and, since the angle between the planes of the mirrors is 45, the rays
coming from a pole P fixed at right angles to the survey line AB will
be finally reflected to the eye along the eye-horizon line he, which is
perpendicular to Oi by the optical fact that the angles of incidence and
reflection are equal. Prisms are sometimes used in optical squares,
and a pair of 45 prisms are embodied in the Line Ranger, a device
for interpolating points in survey lines.
A perpendicular is erected in the following manner, the optical
square being inverted if the right angle is to be erected on the left of
a line, AB, as indicated by pickets.
14 ELEMENTARY SURVEYING
Place the square on the top of a short pole interpolated in AB at
the point at which the perpendicular is required. Send out an assistant
to the required side of the line AB, estimating the right angle, as well
as you can. Then, sighting B through the eye-horizon, direct the
assistant to move until you see his picket by reflection vertically above
B, as viewed directly, raising the eye momentarily in obtaining the
coincidence.
III. SLOPING DISTANCES
Already, doubtless, you have been wondering how hills, valleys, and
undulations will affect your measurements. Over two thousand years
ago Government officials were worried about the matter, and quite
possibly at this moment some contractor has a headache about it.
Polybius told those in authority that no more houses could be built
upon a hillside than within the same limits on level ground. Other
economic arguments are that the majority of plants shoot up vertically,
and no more trees or crops can be grown on a hill than on its productive
base, as the horizontal equivalent is called. Exception, however, occurs
in the case of certain creeping plants. There is also the geometrical
argument which contends that a map must represent areas of any
surface on a plane sheet. For instance, a triangle can be plotted with
any three distances, and so the four-sided skeleton of an irregular field
which slopes steeply from one corner will plot as two triangles on a
diagonal as a common base, even though all the measurements are
made on the actual ground surface; but if the other diagonal be
measured likewise, its length will not check with the resulting figure,
being too long or too short, to an extent dependent upon its own slope
and the distortion induced by the other irregularly measured lengths .
Hence, all measurements must be reduced to a common basis, which
for general convenience is the horizontal plane.
Wherefore, an "area" is the superficial content of a horizontal plane
surface of definite extent, and this definition is understood in the
valuation of land. No account is taken of the nature or relief of the
surface, which theoretically is thus "reduced to horizon," or in other
words, projected on to a horizontal plane.
On the other hand, certain exceptions must be admitted, and these
refer to the work of the labourer, which consists of lineal or superficial
measurements, such as mowing, hedging, and ditching.
Let us hope by this time that our contractor has fathomed the reason
why more concrete will be required in constructing the road up Hilly
Rise than the amount he estimated by scaling from the map.
Slopes are expressed either (a) by the vertical angle a the surface
makes with the horizontal, or (b) by the ratio of the vertical rise in
the corresponding horizontal distance, 1 in x 9 say. If the actual sloping
distance is /, the vertical rise d is / tan a, and not / sin , as used
in certain connections; that is, the gradient on a road or railway is
FUNDAMENTAL PRINCIPLES 15
the tangent of the angle of slope, expressed as a fraction; 1 in 12, or
1 in 75, as the case may be.
It is very difficult to assess slopes by eye, and the limit at which
they should be taken into account depends upon the accuracy required
in the work, angles up to 3 or 5 being neglected in ordinary w^rk.
In Fig. 14, it is evident that the horizontal distance b corresponding
to the sloping length / is
6=/.cos a (1)
Corrections are sometimes given in reduction tables, or are engraved
as such on clinometers, being differences
c=/(l cosa) (2)
which are subtracted from the measurements made on the incline.
FIG. 14
Now cos a-V 1 sin 2(X > and if a is verv small COS a= l ~ I sin 2
where sin a d\L Hence
d* (3)
c= 27
the rule used when pegs are driven on steep slopes and their differences d
found by levelling.
Rule (3) shows that if we ignore a difference in height (or in alignment) of
142 ft.= 17 in., in a length of 100 ft., the error in length will not exceed
J in., or 0-01 ft.
Also, in surveying, a correction is of the same magnitude but opposite
sign to the corresponding error. Hence, if we prescribe a ratio of
precision to our chaining, such as 1 : r, it is possible to determine the
slope at which it is necessary to apply a correction.
Thus in rule (2), if the ratio c/I must not exceed 1/r, cos. <x= 11 /r. Hence
if we are to chain to 1 in 1,000 or 1 in 5,000, the angles of slope must not
exceed 2 34' and 1 08' respectively, even assuming that error does not arise
from other sources.
16 ELEMENTARY SURVEYING
There are two general methods of determining horizontal distances
when measuring slopes:
(1) Stepping, by taking such precautions as will ensure that the chain
or tape is stretched out horizontally.
(2) Observing Slopes when taking hypotenusal measurements or
chaining along the actual slope, the angle a or the gradient 1 in A:
being observed, frequently with the clinometer.
(1) STEPPING. In this method it is usual to employ short portions
of the chain, lengths varying from 20 to 50 links, in accordance with
the steepness of the slope and the weight of the chain. In the latter
respect the sagging effect of the chain may be so serious that the tape
must be used in accurate work. Some surveyors insert arrows slantwise
when they require the slope to be taken into account, and sticking
arrows in this way facilitates the use of a plumb-bob, which is far
better than "drop arrows," loaded with a lead plummet, to ensure a
vertical fall.
Let us proceed to measure down the slope from B to A with P and Q
as chainmen, R going outwards to the side of the line with a straight
rod (or picket) in his hand (Fig. 14).
P, at the starting-point B, puts Q into line, holding his handle of
the chain on the ground. Q, gripping a plumb-line at (say) the 40-teller,
exerts a pull, almost invariably holding his end too Ipw. (In fact, the
sense of looking horizontally is badly impaired when working on
slopes.) Hence the advisability of the services of R, whose duty it is
to see that the chain is horizontal. R, standing some distance to the
side of the line, looks for telegraph wires or ridges of roofs, in order
to direct Q in raising or lowering his end of the length PQ. When
no horizontal object can be viewed, R extends his right hand and
balances the rod on his forefinger, and uses this artifice in judging the
horizontal. When "All right" is signalled, Q fixes an arrow and
proceeds for the next length.
Stepping uphill is more difficult, as it requires that both Q and P
must move their ends of the length used, or that P also must be provided
with a plumb-bob.
Stepping has the advantage that it is quick and does not necessitate
any alteration in the field notes, but its use is limited to lines that
involve few or no offsets. When there is much detail, as in surveying
streets or crooked fences, the following method must be used, since
the chain will of necessity lie on the ground for some time.
(2) OBSERVING SLOPES. The instrument most commonly used in this
operation is known as a clinometer, an instrument made in more
forms and types than any other surveying instrument, the compass
included.
At present we need only examine it in its simplest and improvised
form. Take a 5-in. or 6-in. celluloid protractor, insert a stout pin at
FUNDAMENTAL PRINCIPLES 17
its centre o, tie a thread to the pin, and attach a light weight, say a
bunch of keys, at the other end of the thread. Appoint somebody of
your own height to proceed up the slope, directing him into the line A B.
Now sight along the straight edge
of the protractor which should be
held with its plane vertical, and
move it until you see the eyes of
your helpmate; then grip the thread
and the protractor at the edge near
the point g, and, bringing it clown
thus, read the angle, which will be
the complement 90 a in observ-
ing angles of elevation (up the slope)
and /or angles of depression (down
the slope).
Obviously the foregoing process FIG. 15
requires some practice, but it suggests,
failing a proprietary instrument, the lines of constructing a good substitute.
Attach a piece of three-ply, 6 in. x 6 in. to a piece of hard wood, J in. square,
and attach the protractor to the plywood, keeping its zero line parallel to
the upper surface of the wood. Take two brass strips, | in. wide, drill a pin-
hole sight in one, and cut a $ in. square hole in the other. Bend the strips
at right angles, f in. from the pin-hole and the bottom of the square hole
respectively, and attach these sights to the upper face of the wood with
brass screws. Insert a tiny picture-ring in the centre of the protractor, so
that a plummet with a hook attachment can be readily suspended. Finally
make a wooden handle and fix it to the back of the baseboard. Figure
around the outside of the protractor the even slope ratios, tan a, as 1 : 1, 1 : 5;
1:12, etc. As a further refinement, the corrections to surface measurements
can be inscribed in accordance with Rule 2, preferably from the tables in a
surveying manual. Such a device can be used in many connections.
The chief difficulty is knowing when and where to take the slopes,
since these often vary on a hillside or consist of featureless undulations.
What is big in the field is small on a map; and the sense of appreciating
the general trend must be cultivated.
Apart from injudicious selection of slope limits, the chief drawback
to tliis method is the fact that the field notes must show the angles of
slope or their ratios together with the limits of each different slope.
In practice the distances along the survey lines must be duly amended
before plotting, preferably as red ink corrections. Only measurements
along survey lines will be affected; not offsets normally.
LINEAR MEASUREMENT. Since one aim of this little book is to give
a broad view of the subject, a summary of the different methods of
measuring lines will not be out of place, particularly if some idea of
the relative degrees of accuracy are shown. In surface measurements
the precision is influenced mainly by the nature of the ground and
the precautions that are tak"-n at the expense of speed. The ratios for
ordinary chaining are 1 : 750 to 1 ; 1,500, with a fair average of 1 : 1,000
18 ELEMENTARY SURVEYING
for careful work on good ground. A limit of 1 : 50,000 seems reason-
able for surface measurement with steel tapes and every precaution.
In optical and other measurements, instruments and atmospheric
conditions are the controlling factors; and the ratios given are repre-
sentative of average practice.
(a) Pacing, after training (1 : 75 to 1 : 150); lower value in route
surveys.
(b) By Perambulator, in road measurement and exploratory work
(1 : 150 to 1 : 300).
(c) By river launch, towing the patent log (1 : 500 to 1 : 900).
(d) By optical measurement, by tacheometer or range-finder (1 : 300
to 1 : 650).
(e) By sound signals, guns being fired alternately between ships or
the shore and a ship (1 : 500 to 1 : 2,000).
(/) By aeroplane, in controlled flight over ground stations (1 : 1,000
to 1 : 3,000).
(g) By base tapes and compensated bars (1 : 300,000 to 1 : 1,000,000).
Practised pacing is a great asset to the surveyor, and is particularly
useful in reconnaissance, route, and military surveying; but the diffi-
culty of counting is a great handicap, even if stones are transferred
from pocket to pocket at every hundred paces. The passometer, or
pace-counter, is a useful investment, and is to be preferred to the
pedometer, which gives distances, and suffers from the refinements
necessary to setting it to the individual step. Both instruments are
similar and like watches in appearance, the mechanism being operated
by a delicate pendulum device. They should be carried vertically above
the waist; in the vest pocket or clipped inside the collar opening of
the waistcoat. If carried in the trousers pockets, they usually count
only half-paces. They respond to well-defined paces, and not to the
shuffling gait of a celebrated comedian of the silent films: a fact that
may be useful when the user does not want counts to be recorded.
IV. FIELD-CODE
In the writer's youth the text-books gave much sound personal
advice to the surveyor, even on matters of dress and deportment.
Doubtless this would appear superfluous in a modem text-book, even
though sound sense and good taste are not experience, the "obvious"
being evident only after the event. Possibly the line of approach
should be through the medium of a code, which at least has an official
air.
(1) Surveying equipment is expensive, and if damaged or neglected
is likely to impair some other fellow's work. Sheep and cattle are
naturally inquisitive, and range-poles are easily snapped. Horses
masticate flags (and lunch haversacks), cows chew tripods, and two
lambs can overturn an expensive level in two to four minutes.
FUNDAMENTAL PRINCIPLES 19
Wherefore, instruments should never be left unguarded, and, during
recesses, should be left in enclosures, tripods firmly planted, and staves
and poles left on the ground, and never leant against trees or walls.
(2) Instruments should be securely attached to tripods, security in
this respect being checked from time to time. When necessarily exposed
to rain, levels, compasses, and theodolites should be protected with a
waterproof cover, the tennis racket case serving this purpose well.
Wet instruments should be carefully dried. Tripods and poles should
not be shouldered in streets or through doorways, and levels, etc.,
should be carried under the arm, the instrument forward, except in
the open.
(3) Permission should always be asked before entering any field,
yard, or forecourt. Every respect should be given to property. Chaining
or walking through crops of all kinds may lead to a claim for damage.
Hedges must never be opened or cut in order to make stations inter-
visible. Fences should never be climbed in order to shorten journeys;
barbed wire is no respecter of clothing, and the proper way is the
shortest.
(4) Gates should be properly closed and fastened, even for temporary
egress. An open gate may lead to straying cattle, with consequent
damage and expense; and neglect in this respect may lead to the
withdrawal of your permit.
(5) Chaining on paths and highways should be carried out with
extreme caution, and always under supervision. Pedestrians and cyclists
are easily tripped, and a stretched chain may lead to a motor accident.
When only municipal parks are available, special attention should be
given to the conditions of the permit. It should always be remembered
that these are places of recreation; and that undue interest by the public
will soon subside if you work silently and show no resentment.
(6) Field notes should be legible, explicit, and easily interpreted by
a surveyor who has never seen the area. They should be complete
before leaving the field. It may be impossible to supply an omitted
measurement, and the entire work may be rendered invalid.
(7) Stock should be taken of the equipment before leaving the field.
Chains, range-poles, and arrows are easily forgotten when clearing
the ground. Station pegs should be removed. If driven where they
are likely to cause accidents, they should be removed nightly, and the
position of the station carefully referenced.
(8) Shouting instructions is bad taste. In public spaces it provokes
ridicule, and in private lands annoyance or curiosity.
Surveying affords excellent opportunities of trying out the semaphore
code. But a simpler system is desirable: something like the following,
which is suggestive rather thi*n standard.
20 ELEMENTARY SURVEYING
SIGNALS
(a) "Halt." Raise the right arm full length vertically above the head,
the right hand extended.
Directing staff men and chainmen, but obviated by "fix picket" in boning-in.
(b) "Fix. 9 ' Extend the forearms forward horizontally, and depress
the hands briskly.
Ranging out lines and establishing stations. "Fix arrow" is indicated by
depressing the right hand sharply, the sign implying "All right" in short
distances.
(c) "Stay There." Raise both arms full length vertically above the
head, the hands extended.
Directing staffmen to remain while a reading is taken, and generally to
await further instructions.
(d) "Go Ahead." Extend the right arm full length above the head,
and wave it between this and a position horizontally in front of the
body, graduating the motion to the desired forward movement, and
bringing the arm full length to the halt position.
Directing staffmen in levelling and chainmen in fixing stations.
(e) "Right" or "Left" Extend the right or left elbow in the required
direction, and graduate the motion of the forearm to suit the lateral
movement required.
When it is desired to bring staffmen or chainmen round through a con-
siderable distance from their present positions, emphasise the signal by
swinging the arm and body in the required direction, periodically indicating
the required spot with the arm extended.
(/) "Come Nearer" Circle the right arm over the head, slackening
the motion as the required position is approached, and finally bringing
the arm to the halt position.
"Come here" or "Come in" is indicated by bringing the hand to the crown
of the head after every few turns.
(g) "Plumb Staff" "To your Right." Extend the right arm upwards
slightly to the right, and swing the entire body to the right, checking
further movement by thrusting out the left arm. Vice versa in plumbing
to the staffman's left.
Plumbing the staff in levelling and adjusting station poles.
(h) "Higher" Hold the left hand, palm downwards, in front of the
body, and raise the right hand briskly above it; repeat after momentary
pauses, emphasising the motion by raising the body until the signal is
interpreted and obeyed.
The signal implies "Too Low," and instructs the staffman to extend a
telescopic staff or to move to higher ground. The signal may be reversed to
suggest movement to lower ground.
(i) "All Right" Swing both arms from the sides simultaneously,
bringing the hands together above the head several times.
For great distances where the less-emphatic "Fix" would not be recognised.
American surveyors signalled "O.K." for "All Right" fully forty years
before we took it into our vernacular.
FUNDAMENTAL PRINCIPLES 21
CLASS EXERCISES
1 (a). Describe with reference to neat sketches, the following methods of
measuring sloping distances with the chain:
(a) Stepping; (b) Observing slopes.
1 (b). In chaining you are instructed to take into account the slope ot the
ground when it gives rise to an error of measurement of 1 in 1,000 in Land
Surveys and 1 in 3,000 in Town Surveys.
Express as angles or otherwise the slopes corresponding to these errors.
(2 34' or 1 : 22 and 1 29' or 1 : 39.) (G.S.)
1 (c). Describe how you would "reference" a survey station so that you
could re-establish its position if required.
1 (J). A purchaser disputed the area of a field which was stated to be
54 a. 3 r. 24 p., the sale price being 300 per acre. It was proved, however,
that the Gunter chain used was 0-4 link too short; and the court decided
that the excess payment should be refunded to the purchaser.
Calculate the amount of the refund. (G.S.)
(Excess, 0-438 acres; Refund, 131 Ss. Qd.)
1 (e). Describe the optical square, indicating its principles on a neat sketch.
FIELD EXERCISES
Problem 1 (a). Examine and test the assigned chains against a standard
length.
Equipment: Chains, scriber or chalk, rule, and in the absence of a permanent
standard, an accurate steel tape or band.
Problem 1 (b). Investigate the accuracy of chaining by measuring the
line AB . . . times.
Equipment: Chain, arrows, rule and a set of pickets.
Problem 1 (c). Ascertain the average length of the natural pace and assess
the accuracy of careful pacing.
Equipment: Chain, arrows and set of pickets, and desirably a passometer.
Problem 1 (d). Measure up the specified portion of the ... Building.
Equipment: Set of pickets, chain, arrows, and a linen tape.
Problem 1 (e). Set out one of the following in the playing-field:
(a) Tennis court; (b) Hockey ground.
Equipment: Set of pickets, chain (50ft.), arrows, tape, and cross staff.
(On hard surfaces improvised tripods may be used, the feet tied to prevent
opening out. A picket (or plumb-bob) may be inserted in the junction-piece
to which the legs are hinged.)
ORIGINAL PROBLEMS
(e.g. Use of a cycle wheel as a road-measuring perambulator, the strikes
on a gong serving as an improvised trocheameter.)
CHAPTER H
CHAIN SURVEYING
Not the least of the educational values of surveying is the fact that the
introduction to the art is through the medium of the simple chain
survey; something utilitarian as well as instructive, and something that
merges into the complex naturally and unobtrusively.
The execution of an extensive chain survey is the finest training for
the surveyor; though the field of imagination, effort, and resource has
been impaired by the premature inception of the theodolite, which is
often introduced inexpediently. There is a place for everything in the
field; but a proper place. In chain surveys the selection of stations
can be truly pioneer work, since all lengths must be measured, and
reconnaissance in order to obtain inter-visibility becomes a matter of
greatest importance. But the labour is usually rewarded by satisfaction
in the results, which in no small way arises from the fact that all
measurements are of the same order, often the same precision, and not
mixed, as in accurate angles and rough chaining. Sanction to purchase
a theodolite may sound important in the council chamber, but dis-
illusionment is often the lot of the surveyor.
The writer recalls some notes he encountered thirty-five years ago;
the records of a very extensive chain survey carried out in the 'sixties.
A classical piece of work, but a monumental piece of plotting, particu-
larly in view of the fact that page 22 of the duplicated sheets was
missing. But let us proceed, in order that you may foster your own
reminiscences.
Equipment. The usual outfit in work of the present nature will
consist of one or two sets of range-poles or pickets (flags), chain, arrows,
linen tape (cross staff or optical square), pocket compass, and, above
all, the field-book.
(By the way, see that the linen tape is also figured in links when
measurements are to be made in Gunter chains.)
FIELD NOTES. The field notes are entered in a book with stiff covers,
about 7| in. by 4i in., containing plain leaves, opening lengthwise,
and secured with an elastic band. Usually two red lines, about f in.
apart, are ruled centrally down the middle of the page to represent the
survey line, and the notes are recorded up the page, as in looking
forward along the chain to the next forward station. This method of
upward booking should be characteristic of all forms of line notes.
In keeping field notes, scale is relatively unimportant compared with
neatness and clarity of interpretation, particularly in regard to offset
detail. Some of the notes recorded seventy years ago emphasise a
marked decline in handwriting and lettering and general presentation,
22
CHAIN SURVEYING
23
which to-day is not infrequently loose and half legible. Few surveyors
record their notes in precisely the same way, but vary their conventional
signs, though, of course, these follow a more or less general scheme.
(It is now suggested that the reader study the survey of "Conventional
Farm," page 37, and obtain soire idea of representing detail and objects,
improvising clear abbreviations.)
Fig. 16 shows a specimen page of the notes of a chain survey, various
symbols being introduced. Space will not allow discussion of the
24 ELEMENTARY SURVEYING
various points of contention, such as the merits of using a single red
line instead of a pair, whether lines should be numbered or not, etc.
Usually a page is allocated to a line regardless of its length, though
obviously very long lines or lines with much detail will require two or
more pages. Also, two strokes are drawn to denote the end of a line,
even if this is not stated in words. Some insert direction marks at the
stations, without further remark, or with arrow-heads and letters
indicating the directions of the adjacent stations concerned. Frequently
the station chainages are inserted in rings. The crossing of a road or
fence requires that the double lines be imagined as a single line, by
breaking the road or fence; while contact with a fence corner necessi-
tates contact at a red line with the zero offset distance "0" suggested
by the word "At." Right-angle offsets are generally understood, but
when these are long, necessitating "tying," dotted lines are inserted
with the tape measurements figured along them.
Keeping the Field Notes. The keeping, or better, the custody of the field
notes affords no difficulty in actual practice, but is a matter of serious im-
portance in instructional classes, booking being a substantial part of the
training. A class under instruction may appear like a rush of reporters in an
American gangster film, overwhelming the story or the instrument in their
enterprise. On the other hand, the lone keeper of records may be a well-
meaning but irresponsible student, who fails to produce the evidence when
required, and often loses it and the labours of his fellows. A middle
course must be found by deputing a trustworthy student to be responsible
for the "party copy," and at least one other student of that party should
transcribe the notes before leaving the field. Often the "class copy" must
inevitably be the work of several hands, often inadept, and the leader must
keep an eye to the book from time to time. Some object to the indoor
transcription of notes, and even like to see the marks of the field (which
need little cultivation); but a copy is a copy wherever made, and whether in
pencil or waterproof ink. Indelible pencils may serve in official capacities,
but there is no place for them in plotting or surveying, except for marking
stakes.
I. CHAIN TRIANGULATION
Let us consider Fig. 17, bearing in mind the following rules:
(a) As long lines as possible, consistent with short offsets, which
latter should be restricted to 50 links, though even a chain may be per-
missible if it obviates a subsidiary triangle in an unimportant gap.
(b) As few main triangles as possible, consistent with covering the
area without a number of subsidiary triangles for outlying boundaries,
inlying details, etc.
(c) As well-formed triangles as possible, with no angle under 30 or
greater than 120, in the main, but with reasonable latitude in subsidiary
triangles.
(d) As strong check lines as possible, in order to verify all main
triangles with an additional measurement, unless these are otherwise
CHAIN SURVEYING 25
mtvxivwia oy interior fence or road lines. Small or isolated subsidiary
triangles need not be checked.
The diagonal AC in Fig. 17 is selected as the basis of the
work, and is frequently styled
the base line, quite without
qualification. A line alongside
the approach road usually
assumes this capacity in plot-
ting. On AC are built the two
triangles ABC and ADC, which
together comprehend the area
without requiring long offsets
to the boundaries, this difficulty
being obviated by inserting the
triangle efg in the gap.
Now any three lengths will Fio. 17
form a triangle, and if a chain
length is "overlooked" in measuring a line, a plan will certainly follow,
but one of sorts. Hence it is essential to measure a check line, such as
#>, and, in doing this, a pole O should be interpolated so that it is
both in AC and BD, its position being recorded in both these lines: thus:
AB, 6-64 chs. with O at 3-28 chs., and ED, 6-72 chs. with O at 2-04 chs.,
no offsets being taken from these lines. In the case of farms, etc.,
where several fields are included, it is seldom necessary to think about
checks, as these will arise from lines along farm roads and fences
often a check too many in slipshod work.
The field-work may be detailed concisely as follows:
(1) Reconnoitre the ground and select suitable points for the
stations A, B 9 C, D y etc., consulting existing maps, if available, in
the case of large surveys. Select the stations in accordance with the
foregoing rules, aiming at simplicity and strength, and never sacri-
ficing a strong triangle in order to avoid a difficulty. Establish the
stations suitably with pegs, and if necessary fix flags to the station
poles.
(2) Sketch an "index map" on the first page of the field-book, and
insert the survey lines. This item often permits simplification in the
field-book. In large surveys an index to the lines is desirable, so that
in plotting, the lines can be readily found from the numbered pages of
the book.
(3) Proceed to measure the lines and the offsets to the adjacent
boundaries, selecting the order most suitable to prevailing conditions.
Thus in the afternoons, in winter, visibility along the long lines may
become very poor, and these should be measured first. Normally, read
the chain to the nearest link, since in plotting this will introduce an
error of less than 1/200 inch on a scale as large as 1 chain to 1 inch. In
certain connections, particularly with the 100-ft. chain, it is necessary
26 ELEMENTARY SURVEYING
to work more accurately, as calculations may be involved, or certain
portions may be required on a very large scale.
(4) Concurrently with the measurement of the lines, take offsets
from the chain as it lies on the ground, sending out the ring end of
the tape to the roots of hedges, fences, walls, etc., and swinging the
tape as already described in estimating the right angles. Widths of
gateways should be figured in addition to the offsets to the posts, and
particular note should be made whenever the chain line crosses fences,
roads, footpaths, and ditches. When a subsidiary triangle is set out,
such as efg, offsets should be taken from eg andfg, though occasionally
this may be done after closing the line on the end station. At least
two corners, fixing the faces of buildings should be located, and these,
like all important measurements, should be "tied" even though they
may be squared off from the survey line. Trees need little attention
when they grow along boundaries, but otherwise their positions should
be found, particularly if isolated or if they are planted along avenues.
Clusters of trees may often be surveyed from a line between them, and
often the general limits and a mere count as to their number is sufficient.
On large scale plans, it is often desirable to represent trees by a con-
ventional circle rather than to obscure the ground with artistic matter.
Offsets should be taken where there is a distinct change in the direction
of a boundary, remembering what is large to the eye is often undis-
tinguishable on the map. Two (three at most) are necessary in the case
of straight fences.
Never take offsets at regular intervals merely in order to use Simp-
son's Rule, which should be restricted to distances scaled from the
plan.
(5) Continue the work on these lines until all the lines are measured
or surveyed, taking care that no important triangle is unchecked and
that no important detail is omitted. Reference two stations of an
important line in order to facilitate re-survey, by tying the stations
with two tape measurements from trees, gate-posts, or prominent
points on buildings. Finally remove all station pegs, poles, and laths.
The plotting of the survey will be discussed in a later chapter.
Hedge and Ditch. There are very few cases in which the surveyor can
tell by mere inspection the precise position of a legal boundary line
between properties. In the case of brick and stone walls, the centre some-
times forms the division line, in which case it is known as a party wall,
while in other cases the wall is built entirely on one property and the
boundary line is then the outer face. Frequently, local inquiries have to
be made as to the positions of stones and marks on parish boundaries.
Also the boundary between properties and parishes may be the centre of
a brook or a stream. When a hedge has a ditch on either side of it, or
none at all, the root is the boundary line if it divides the property of
two different owners A and B. But when a hedge has a ditch (or the
remains of one) the hedge and the ditch usually belong to the same
CHAIN SURVEYING
27
MrA
Fio.
property, the clear side or brow forming the boundary line. Thus, in
Fig. 18, the boundary of Mr. A.'s property is the line XY, while Mr. B.'s
property includes both the hedge and
the ditch. There are exceptions to this
rule. Usually the owner's side is denoted
by a "T" when a hedge is represented
by a mere line on a map.
Commonly all measurements are
taken to the root of the hedge, the
following allowances being made: 5 or
6 to 7 links according as adjacent fields
belong to the same or different owners,
and 7 to 10 links when abutting on public lands. Further discussion
may get us entangled in the Law of Property, and that is best left to
lawyers or chartered surveyors.
II. CHAIN TRAVERSES
Traversing denotes the running of consecutive survey lines more or
less in conformity with the configuration of a wood, pond, or planta-
tion, or a route, road, river, or stream, the two categories representing
the primary classes of traverse surveys:
(a) Open Traverses, and (b) Closed Traverses.
What in themselves are open traverses may occur between triangula-
tion stations, or between the stations of closed traverses, placing the
latter in the category of Compound Traverses.
Strictly, the chain alone is not the ideal method of dealing with a
traverse, which is best surveyed with the compass and chain, or, better,
the theodolite and chain. Needless to say, it would be incongruous
to run a closed traverse around a wood, and then introduce one of
these instruments in order to survey an interior road. Strange things
like this happen in surveying when a proper examination of the ground
is not made.
(a) Open Traverses. In chain traverses it is necessary to fix the
relative positions of the lines AB, EC, CD, etc., by means of ties ab y
cd, etc., whereas otherwise the directions would be determined by the
angles or bearings at A, B, and C. Usually the bearing of the first
line, AB 9 is taken with a pocket compass, as in the case of chain
triangulation, so that a magnetic meridian, or N. and S. line, can be
drawn on the plan.
The tendency is to use ties far too short, or otherwise giving too
acute or oblique intersections, so that the directions of the main
traverse lines may be in error. In the case of roads through woods it
is often extremely difficult to get in ties at all.
Fig. 19 shows the main traverse lines and ties with reference to a
portion of a stream, which by a stretch of imagination may be a road,
28
ELEMENTARY SURVEYING
D
FIG. 19
or even a contour line. The main stations A, B, C, and D are selected
so as to render the offsets short, and the tie stations, 0, b, c, and d,
to fix the angles rigidly, incidentally serving for offset measurements
when close to the stream. The routine
differs little from that detailed for triangu-
lation surveys. The notes, however, should
not terminate with the end station, B, say,
but should include the end b of the tie on
the next line, so as to retain continuity and
avoid omissions. There is certainly much
to be said for the use of the single red line
instead of the pair in work of this nature.
When the traverse is run between stations
more rigidly located by chain or other
triangulation, the traverse lines can be
adjusted to fit between the main stations
by the methods described in Chapter VII.
(b) Closed Traverses. Fig. 20 shows the
foregoing method applied to the case of a
pond. Sometimes certain of the ties afford
a convenient basis for offset measurements,
as, for instance, the line de. Each main line
requires an angle tie, and not infrequently
several main lines are laid down whereas few would suffice. This is
evident in the triangle bed, which not only replaces two main lines,
but doubtless affords a better basis for offset measurements.
The area in Fig. 20 is shown
traversed in the counter-clockwise
direction simply because when a
theodolite is used, "back angles" will
also be the interior angles of the
polygonal skeleton. The principles
and methods of chain triangulation
are also employed in mixed triangu-
lation surveys with the theodolite,
which in the case of Fig. 17 might
obviate the chaining of the diagonals
by the observation of four angles;
and this would be an extravagant
innovation unless great accuracy is
required or obstructions impede the
measurement of AB and CD. Also,
the principles could be extended to
compound chain surveys, such as
those of farms and estates. Some idea of surveying Conventional
Farm might be obtained by inspection of Plate I. Generally, however,
j
D
6
FIG. 20
CHAIN SURVEYING 29
the sketching of lines on diagrammatic surveys is of little value
unless contours and other information are supplied. Examine an
area and you will discover that this is something more than a
diagram. This comment does not apply to plotting from unseen field
notes, such as are given in text-books. Often a field class has to be
abandoned on account of the weather, limitations of time, or an
omission on the part of a member of the party. Hence, extract notes
from text-books must be resorted to. Nevertheless, nothing is so good
as notes brought in from the field.
CLASS EXERCISES
2 (a). Sketch the plan of a farm which consists of six adjacent fields and
a building, the whole area approximating to a rectangle with a road running
along the south boundary. Assuming that the interior fences are low and
the ground fairly level, indicate clearly how you would survey the farm with
the chain, measuring tape, and range-poles only.
2 (b). Draw up a page of a field-book, and insert the imaginary notes of
an important line in the survey in Qu. 2 (a).
2 (c). In measuring a survey line BC, chaining was done on the surface of
the ground, and the slopes taken with the clinometer at the sections indicated.
I 1 : 12 | 1 : 10 | 1:8 |
() 50 245 360 510 720 824 960 1128(C)
Enter these on a page of field notes and make the necessary corrections
for sloping ground.
2 (d). Sketch an isolated wood of irregular shape, containing a road leading
to a quarry; and indicate how you would survey this with the chain, tape, and
poles only.
2 (). Sketch the plan of a street you know, and indicate how you would
survey the frontage lines of buildings and fix other details from a survey line
which runs down the centre of the carriage-way.
FIELD EXERCISES
Problem 2 (a). Survey the (specified) field by chain methods only.
Equipment (which is also (he same in the following problems): Chain, arrows,
set of pickets, pocket compass, and linen tape,
Problem 2 (b). Survey the (specified) pond (wood or plantation) with the
chain, poles and tape only.
Problem 2 (c). Survey the (specified) *oad between the range-poles marked
,4 and B.
Problem 2 (d). Survey the (specified) cottage (gate lodge) and garden.
Problem 2 (e). Survey the (specified) farmyard, and measure up the
buildings.
ORIGINAL PROBLEMS
CHAPTER m
PLOTTING PLANS AND MAPS
It is but natural that the young surveyor is eager to see how his own
efforts show up on paper; and, in deference to his wishes, the present
chapter is inserted somewhat prematurely, possibly overlooking various
difficulties he has encountered. On the other hand, it is desirable to
proceed slowly, in order to take a wider view of the subject of plotting
rather than to distribute it throughout the book, though matters not
of immediate interest may be revised at a second reading. Anyway,
the uses and construction of scales is a matter of primary importance.
I. SCALES
A scale is used to measure straight lines on plans or maps in certain
conventional ratios to the actual lengths of the corresponding lines in
space. Scales may be expressed in the following three ways:
(a) By a Statement, such as 1 Chain to 1 Inch, 6 Inches to 1 Mile, etc.
(b) By a Representative Fraction (R.F.)> such as 1 : R, the denomina-
tor being the number of units in space represented by one scale unit;
the in. or cm., as the case may be. Thus the R.F. of the scales stated
above are respectively 1 : 792 and 1 : 10,560. The method is universal,
applying to all systems of measurement; and most Continental maps are
characterised by even ratios, such as our 25-in. Ordnance sheet, which
is 1 : 2,500, and not, therefore, precisely 25 inches to the mile. The
R.F. is absolutely necessary when two unrelated systems of units are
involved.
(c) By a Divided Line, or map scale, which is usually "open" divided.
Usually (a) and (c) are combined to^ express the scale, and all three
modes are used on the Ordnance maps.
Scales occur in two forms, which are Open Divided or Close Divided,
according as only the first main division or all the main divisions
are subdivided.
(1) As refined or improvised drawing instruments for plotting maps
and plans, and (2) as an important feature of the plan or map for
convenience in scaling measurements and distances.
(1) Office Scales. Office scales are constructc4 of boxwood, celluloid,
or ivory, the flat section bearing two scales, being better than those
of triangular section which carry six scales. Oval section scales carry
four different fets of divisions, and are usually (open) divided for
engineering and architectural plans. Surveying plotting scales are close
divided, and are sometimes provided with a short length of the same
dividing known as an offset scale.
30
PLOTTING PLANS AND MAPS 31
^High-class scales are expensive, and, failing access to these, the students
must content himself with a good 12 in. boxwood rule, improvising wherever
necessary special divisions on strips of drawing-paper. A useful and inex-
pensive item is the protractor scale, 6 in. long, and similar to the so-called
military protractor. The boxwood pattern is the best. One form show
inches with eights and tenths, centimetres, a diagonal scale, giving hundredths
of an inch, and J, i, i (inch) to 1 ft. (1 ch., or 100 ft.), also a scale of chords,
three edges being divided for the construction and measurement of angles.
It is exceedingly useful also in the fe1d, though its principal use is giving fine
measurements through the medium of the dividers when constructing scales
by the methods hereafter described.
(2) Map Scales. Scales of this category are open divided, and are
drawn on the map to facilitate measurement with a paper strip or a
pair of dividers, and to provide against the shrinkage of the paper
over the lapse of year. A "shrunk" scale is made when the surveyor
has omitted to insert a scale on his map, and the paper shows evidence
of shrinkage. It is then necessary to find two prominent points on
the map which &till exist in tru* area; to measure carefully the distanc-;
between them, and then to construct a true scale so that it can be used
in the future, although it carries the statement of the exact scale on
which the survey was plotted.
Scales should be drawn with extreme care, never unduly shor f or
long, and preferably with a single line. A double line with alternate
primary divisions blacked in is olten used. Here, unfortunately, the
artist covers up his inaccuracies, so that often the scale is of little use,
except to the eye of the beholder. Students have a habit at first of
setting off primary divisions, and figuring these with fractional values
and their multiples. This must never be done. The primary divisions
must show integral va ues of the units, however fractional the actual
lengths may be, in inches, etc.
Among the various kinds of scales that may have to be constructed
are Comparative Scales and Time Scales. Comparative scales show
two different systems, such as feet and metres on the same representa-
tive fraction; and time scales show time intervals instead of yards or
metres for a given statement or representative fraction, being used for
pacing, trotting, etc., in military surveying and exploratory mapping.
Constructing Scales. When the division introduces fractions, it is
usual to resort to construction by diagonal division, as shown in Fig. 21 .
TENTH?; I J 3 UNITS
Fio, 21
32 ELEMENTARY SURVEYING
A horizontal line pa is drawn, and at any convenient acute
to it a line pb. When a convenient length has been marked off, as pa,
say 4 in., a drawing scale is placed along pb, and the division is chosen
so that it represents conveniently the number of units (usually fractional
and integral) represented by the 4-in. length of pa. Next, b and a are
joined, and parallels to ba are drawn through the even points of
division, 2' and 1', the subdivisions of pi' giving likewise the sub-
divisions of /?0.
The following examples introduce the types of problems that
commonly arise:
(i) Construct a scale showing chains and tenths, given the statement, say,
10 ft. to 1 mile.
Find the number of chains that are represented by a convenient length,
4 inches, say. Here 4 in. =176 ft. =2-667 chs. Take the decimally-divided
scale and measure off pb 2-67 in. Join ab and draw parallels through 2'
and 1', the primary divisions on pb which give on pa divisions each corre-
sponding to one chain. Write "0" at the end of the first division and sub-
divide the division on the left into 10 parts, as indicated. Extend the scale a
convenient length to the right.
(ii) Construct a scale showing chains and tenths, given the R. F, 1 : 528.
Here 1 in. = 528 in. =44 ft.; and 4 in. =2-667 chs., which is the scale of the
preceding example.
(iii) Construct comparative scales of 1 : 500 showing yards and metres.
Here 1 in. = 500 in. = 13-9 yds., while alternatively 1 cm. = 5 metres. If
pa is still 4 in., pb would have to represent 55-6 yds., and b could con-
veniently be 5-56 in., so that tens of yards would appear as primary divisions,
with single yards on the left. In the metric system, the scale would be con-
structed by merely setting off 2 cm. primary divisions, each to represent
10 metres.
(iv) Construct on the scale of 6 in. to the mile a scale for marching at
100 paces per minute with an average length of 27 in.
27 x 100
Here 100 paces will cover ^ =75 yds. per min., or 375 yds. in 5 min.
while on the given scale 1 in. =29 -3 yds. or 1-28 in. = 375 yds. Hence a
suitable scale would be 6 in. to 7| in. in length, the primary divisions being
5 minute intervals, and the close divisions 1 minute intervals.
(v) Construct the scale stated as 2 chains to 1 inch, omitted from an old
shrunk map, given that a line scaling 5-80 in. was found to be 11-85 chs. on
recent re-measurement.
5*80 x 2
Here 2 chs. are actually represented by . .. =0-98 in. Otherwise the
1 1 *o3
4 in. length of pa in Fig. 21 will represent ^ =8-17 chs., and the true
J'O
scale can be constructed by joining b at 8-17 chs. to a, and drawing parallels
8'7, 7 '6, TO, being the zero of the open divisions.
Mapping Requisites. Apart from the drawing-board, T-square,
set-square, compasses, and dividers, all of which are too well known
to require description, there are certain items which must be discussed
at length.
(1) First a good drawing-pen is necessary for drawing lines in ink,
the common mapping-pen serving for lettering and inserting details.
PLOTTING PLANS AND MAPS
33
Waterproof Indian ink should be used, particularly whenever a colour
wash is to be applied. Ordinary writing-ink should never be used on
plans, nor crayons, which emphasise only bad taste.
(2) A clinograph is preferable to the lever types of parallel rules
for transferring parallels to oblique lines, as in plotting bearings. An
adjustable T-square will also serve the purpose; and often one can
be improvised from a broken T-square, the stock being secured to the
head with an adjustable thumb nut.
(3) Good quality drawing-paper should be used; never the soft
surface material which becomes ragged along inked lines. A sample
of the paper should be tested as to how it will take ink, stand erasures,
even with sand-paper in the event of accidents, and, possibly, how it
will react to water-colours. The sizes that will be used in the present
connection are the Half Imperial (23 in. x 16 in.) and Imperial Sheets
(30 in. x 22 in.).
Always use a hard pencil, HH or HHH, chisel-pointed, and a round-
pointed H or HH for lettering, etc. A pricker is recommended for
marking off scale distances on survey lines.
(4) Finally, the chief item is the beam compass, since the lengthening
bar will extend the use of ordinary compasses only to relatively short
lines. A good quality beam compass should be available, though a
few additional ones could be improvised in the workshop with f-in.
or f-in. square mahogany rods, 18 in. to 24 in. long, by making adjust-
able clips and attaching these to the points and pencil-holders of old
compasses. In an emergency strips of paper, 15 in. X 1J in., might
be used, a stout pin serving as the centre.
Fio. 22
BEAM COMPASS
The beam compass is not only used in plotting chain surveys, but
often in laying down accurate triangulation nets, the sides of the
triangles having been calculated by the Sine Rule from observed angles
and the one measured side, the base.
34 ELEMENTARY SURVEYING
II. PLOTTING THE SURVEY
There are four principal steps in the routine of plotting maps and
plans: (1) Selecting the Scale; (2) Placing the Survey; (3) Constructing
the Triangulation or Skeleton; and (4) Inserting the Detail. Finishing the
map will be discussed later.
(1) In selecting the scale the objects of the survey and the extent
of surface to be represented must be borne in mind. Incidentally,
centimetres and chains or feet must never be mixed, as in 1 cm. to
1 ch., even if this would be geometrically convenient. All "irregulari-
ties" must be avoided; such as 64 ft. to 1 in., simply because a scale
reading to ^ in. is available; or 132 ft. to 1 in., when the scale is
definitely 2 chs. to 1 in. If distances are to be scaled to the nearest
foot, the scale should not be less than 50 ft. to the inch. The tendency
is to use too large a scale, leaving very little margin, while, within
reason, a fairly wide margin is effective.
(2) Only in maps of extensive areas is it desirable that the true
north should be at the top of the sheet, the side border lines being
true north and south lines.
The area should be viewed from the local aspect, and thus the
approaches to the property should appear at the bottom, with roads
approximately parallel to the bottom edge of the paper, regardless of
the position of the meridian needle, true or magnetic. Considerable
thought may be involved in placing a survey on the paper, so as to be
pleasing to the eye, and easily evident to the least intelligent.
(3) In plotting the triangles, a survey line (AB in Fig. 17) is selected
as the base, and this is drawn to scale in the best position on the paper
as can be judged, often from a trial plotting. The beam compass is
then set to the respective scale lengths of the sides, AC, BC, adjacent
to the base, and, with A and B as centres, arcs are swung accordingly,
intersecting at the apex C of the triangle. On this triangle another,
ACD, is constructed likewise; and the process is continued until the
entire framework is completed. Subsidiary triangles, such as efg, in
Fig. 17, can be inserted with the ordinary compasses. Stations should
be indicated by small circles, appropriately lettered A, B, C, Z), etc.
(4) Details of surveys of the present class will be inserted by offsets,
mostly rectangular, long offsets being tied by means of ordinary
compasses. The most rapid method of inserting right-angle offsets is
by the conjoint use of a close divided surveying scale and an offset
scale, the main scale being held in position by a pair of shoe-shaped
weights. Few students will have these latter at hand, and the T-square
and set-square must be brought into service. (A straight edge is better
than a T-square, since it is more easily manipulated.) The plotting
scale, zero at the beginning station of the line, is placed carefully along
the pencilled line, and the points at which offsets were taken are pricked
off; then with the aid of the set-square, short perpendiculars arc erected
PLOTTING PLANS AND MAPS 35
for right-angle offsets, the positions of tie line offsets being indicated
by a short stroke across the survey line. The scale is then applied to
the offset measurements, the ends carefully pricked or pencilled, and
the fences, etc., etc., are inserted with the aid of the set-square, or,
occasionally, a French curve. Likewise, the nearest walls of buildings
are inserted, the corners usually being fixed by ties described with
ordinary compasses.
Finishing Survey Maps. Inserting details is a step closely related to
the finishing of survey maps, and at this stage the imaginary survey
of Conventional Farm should be consulted (page 37). All details
should be carefully outlined in pencil, reducing the use of erasers to
a minimum. In instructional surveys it is usual to insert the survey
lines in very fine red ink lines, red circles being drawn for the stations
of chain surveys and triangles wherever angles are observed. The
station letters should also be inserted in red, but not too conspicuously.
This retention of the skeleton occurs in practice only where construc-
tional work is likely to follow, as in mine surveys particularly. When
a number of subsidiary traverses occur in instructional mapping, the
lines are sometimes shown in another colour: green or blue ink.
Handwriting should be restricted to one detail: the signature.
Free-hand lettering cannot be taught by a text-book, being normally
the outcome of practice and training.
Many have improved themselves in free-hand lettering by practising
to some scheme, such as the following. Write a sentence three times
on the top three lines of a sheet of ruled foolscap; and hence find the
natural slope of the downstrokes. Next, with the aid of the squares,
cover the remainder ot the sheet with a series of strokes, ruled parallel,
about J in. apart, and then rule guide-lines parallel to the paper ruling,
giving the height of the body of the letters. Finally, reproduce the
written sentences by changing script into hand-lettering.
Half an hour's practice a day for a week often has a revolutionary
result. Erect letters emphasise defects far more than slanting, though
an experienced draughtsman can work to any slope with facility.
Titles are best inserted by outlining them first between guide-lines
with a fairly soft pencil, so as to obtain uniform heights and spacing.
Then the squares are used, finally giving the outline in block or other
letters, which are inked with the drawing-pen and the mapping-pen,
and filled in or finished in black ink. The celluloid open stencils are
exceedingly useful in obtaining well-proportioned outlines in pencil.
Unfortunately craftsmanship is at a discount to-day, and the will
seems to be taken for the deed; but the fact remains that draughts-
manship is a great accomplishment, and the master-touch can give an
atmosphere even to a prosaic survey map.
Finally, always test your pen from time to time on a piece of the
same paper as that on the board; and keep a cloth at hand to wipe
the pen when out of temporary use. Never let the ink coagulate in
36 ELEMENTARY SURVEYING
the pen. Also, seize the opportunity of instruction in sharpening a
drawing-pen. Use soft rubbers generally, and clear the rubbings from
the board before inking or colouring.
The General Requirements of a Map or Plan are: (a) The Title,
(b) the Scale, (c) the Meridian Needle, and (d) the Border Lines; also,
possibly, (e) an Explanation or Legend, as to the symbols employed,
and (/) a Terrier, showing the acreage held by various owners. The
surveyor's signature and the date should always be given. Special
requirements include (g) Contours, hachures, or spot-levels, and
(h) Constructional lines and symbols for building and engineering works.
Simplicity is the keynote of modern mapping, and this involves great
skill, as the artist cannot hide defects with trimmings. Early carto-
graphic art was characterised by wonderful embellishments, often in
rich mezzo-tint, and far more entertaining than the prosaic land and
sea: birds, beasts, and equally gigantic men everywhere, even leviathans
basking where America is now known to be. Gradually these have
vanished like prehistoric monsters, giving place to accuracy in the
terrain; in fact, the artistic touch has almost disappeared during the
past fifty years. One of the remains is the touch applied at the bottom
right corners of tree-trunks, indicating the shadows cast by conven-
tional light coming from the top left-hand corner of the plan. This
convention is sometimes used in connection with objects, the lower
and right-hand edges being outlined boldly.
Conventional signs are used to indicate features, such as boundaries,
roads, buildings, lakes and constructional objects, and, obviously, there
is a host of these, certain symbols being varied in accordance with the
scale of the map. Many of those in everyday use are shown in the
Survey of Conventional Farm, where the ownership suggests that the
names are the property of Symbol & Sign (Plate I). The artist never
writes "Tree," "Horse," or "Inn" on a painting, though there would
certainly be some justification for doing this in certain specimens of
modern art. The noun must be used only when duly qualified; as, for
example, "R. Medway," "G. Junction Canal," "G.W.R.," "Beverley
Brook," etc., etc.
Signs are sometimes modified to resemble the object more closely
in large-scale maps, where they become relatively important in the
small area portrayed. Thus, walls become double lines, and hedges
shoot from the root line. Trees should never be drawn taller than
1 in. or f in. When, as in the case of a house and garden, these are
actually large to scale, it is better to represent them with small circles,
with added verdure, if desirable, but never so as to obscure the ground
below.
The title should be printed neatly and compactly in what appears
to be the best position. In certain plans there is a more or less fixed
position for the title. The divided scale line should always be inserted
as well as the statement of the scale for the reasons already mentioned,
38 ELEMENTARY SURVEYING
The meridian needle is best drawn with a star for the true north, and
an arrow-head for the magnetic north. If, as it should be, the map is
dated, the magnetic declination can be found from a book, and the
true north indicated, thus admitting of additions after the lapse of years.
Contour lines are either indicated by alternate dots and dashes, or are
traced in sepia, the contour heights being figured on the high sides, or
in gaps in the contours.
A border line with an appropriate margin gives a finish to the map;
but it is often a finish indeed when a passing student collides with the
head or stock of the T-square. A neat margin usually requires double
lines about ^ in. apart, and the effort demands great courage when
a border-pen is not at hand. A break in the border for outlying details
looks far better than one very close to the edges of the paper. Also,
rounded corners are often preferred to plain right angles.
Stencil plates are convenient for pencilling the outlines of various
features, but, again, these should be small, and letter stencils never of
the size used in directing boxes for passage by rail.
COLOURING. This is a subject that must be introduced diffidently,
since it may lead to the ruination of a nicely plotted plan. Happily,
however, some maps drawn on good paper in waterproof ink have
been resuscitated after a necessary immersion in a tank or bath of
clean water. Students are exceedingly liberal with colour washes:
pastures, green indeed, and soil exceedingly rich, even if the roads are
veritable quagmires. The secret is to apply only the faintest suggestion
of colour. "Use sparingly," in the words on the labels of certain
proprietary articles. Above all, practise on a piece of similar paper
first, and always colour before inking when waterproof ink is not
available. Incline the drawing-board towards you, inserting wood
blocks beneath it. Transfer a pool of colour to the top of the portion
to be tinted, and wash the area over rapidly, lightly sweeping the
surface, and, above all, avoid brushwork, as in painting domestic
objects; garage doors, for instance. Perhaps it is providential that the
demand for colouring is declining in modern practice.
The best water-colours are sold in cakes, which are rubbed down
in saucers and mixed to give any desired shades, always excluding dirt,
dust, and treacherous particles of colour. Numerous conventions arc
in use, the following being fairly common:
Water. Prussian blue, toned from deep at the banks to faint at the
centres of rivers, lakes, etc. A touch is also applied to the conven
tional sign for marshes.
Land. Arable, burnt umber or sepia; Pasture, Hooker's green,
preferably varied in adjoining fields; Trees, Hedges, etc., in green, but
darker.
Buildings. Brick, crimson lake; Timber, India yellow.
Roads. Roman ochre.
Property surrounding the portion for which the survey was made
PLOTTING PLANS AND MAPS 39
is not coloured, but all the conventional signs in black ink are usually
inserted.
Since the present chapter deals mainly with Office Work, opportunity
will be taken to include certain relevant operations.
III. CONSTRUCTING ANGLES
The most obvious method of constructing angles is by means of the
protractor; but it must be borne in mind that the ordinary pattern,
say in the 6-in. size, is not sufficiently accurate except for inserting
details, and never for plotting the skeletons of theodolite traverses.
It would be impossible to plot to nearer than 10 minutes of arc on the
3-in. radius, and when this is extended as a survey line to 12 in., say,
the error would be considerable, though admissible in rough compass
surveys. Accurate celluloid protractors are the best of this category,
as far as constructing angles is concerned. Silver-plated types reflect
light, and are particularly disconcerting in examinations, where the
protractor is usually allowed for plotting compass traverses. There
are, of course, elaborate forms with vernier arms; and there is the
cheaper form of cardboard protractor with an 18-in. open circle, as
used for plotting bearings. Also there is the scale of chords, which is
no more accurate than a small protractor, even though it introduces
a highly important method of constructing and measuring angles
through the medium of the tables headed "Chord."
There are also the trigonometrical tables, preferably those giving
minutes in four- or five-figure trigonometrical ratios. Two methods
of constructing or measuring angles must be considered, for they are
not only useful in plotting angles, but in constructing and measuring
angles in the field when the theodolite is not at hand.
* (1) Chord Method. The following method is actually that which
would result if a table of chords were included in the tables, as they
are in the more precise, such as Chambers' Seven Figure Mathematical
Tables, where the values refer to a unit chord. But the unit may be
conveniently 10 in. in plotting and 100 ft. in field construction, which
merely means that the decimal point is moved respectively one or two
places to the right. Of course centimetres
could be used consistently, but centimetres
and inches must never be mixed.
Anyway, the sine of an angle is always
at hand, and the sine is a kind of half-
brother to the chord, as will be seen in
Fig. 23, where an angle 6 is to be set
out at a station A, being measured from
Ab.
(1) Set the beam compass at 10 in.
exactly, and with A as centre, swing FIG. 23
40 ELEMENTARY SURVEYING
an arc of convenient length be. (2) Find in the table of chords the
unit value for the angle 0, and multiply this by 10 for the length
of the chord be in inches. Otherwise, look up the sine of the half-angle,
J0, and find the value for the hypotenuse Ab, which, being 10 in., gives
the chord be as 20 times the tabular value of the sine in inches. (3) Swing
an arc with the chord be as radius about 6, cutting the arc be at c.
Join Ac for the required angle bAc=0.
The alternative method in (2) follows from the fact that
Thus, for 0=44 20', |0=22 10', and sin.J0=0-3773; whence for
a 10-in. (cm.) radius, the chord 6c=2x3'773=7'55 in. (cm.).
In the field, the angle could be measured by inserting arrows at
b and c by swinging a 100-ft. radius, measuring the chord bc > halving
be at </, and finding J0 from W=^4fr.sin|0.
*(2) Tangent Method. Although applicable to angles, the principle
is par excellence in plotting the bearings in traverse surveys, a subject
that will be treated with reference to the compass in due course. The
method involves the table of tangents, and is applied with a base of
10, the base now taking the place of the hypotenuse. Consider the
method with reference to the closed traverse ABCD of Fig. 24.
(1) Through a point O in the
centre of the paper, rule a vertical
line and a horizontal line, N.S.
and W.E. respectively; and make
ON= OS- O W= OE= 10 in., and
draw parallels so as to form an
outer square of 20 in side exactly.
Obviously, on a half-imperial
sheet, centimetres must supersede
inches unless some larger con-
venient decimal scale is at hand.
Assume the central vertical to be
a north and south line, the four
interior squares representing the
four quadrants: N.E., S.E., S.W., N.W.
Draw up a table showing the Traverse Lines, their Lengths and
Bearings, also the Tangents of bearings under 45 and the Co-tangents
of bearings over 45.
Incidentally, a bearing is under 90, and is measured from the N.
or the S. point, being defined by N. or S. in front of the magnitude
and E. or W. following, as described on page 90. Also the tangent
of 45 is unity, so that when a bearing exceeds this value, the com-
plement, the co-tangent, must be introduced: Cot. p=tan (90 p).
(3) Plot the direction line for each traverse line in its proper quadrant,
measuring ten times the tabular value of the tangent or co-tangent
along the outer side of a small square, and joining the point thus found
PLOTTING PLANS AND MAPS 41
tc rthe centre of the large square. Tangent distances are scaled outwards
from the north and south points on the upper and lower sides respec-
tively, while co-tangent distances are scaled from the east and west
points, upwards or downwards, accordingly.
(4) Draw the ^parallel to each direction line in its correct position
on the paper, using the clinograph or adjustable T-square.
Whole circle bearings (or azimuths), styled bearings in military
surveying and applied geography, are plotted in a similar manner, the
lines falling within quadrants which exhibit the angles as bearings
proper, values under 90 being expressed by p, over 90 as 180 p,
over 180 as p 180, and over 270 as 360 p, corresponding
respectively with N.E., S.E., S.W., and N.W. bearings.
IV. ENLARGING AND REDUCING MAPS
Frequently enlarged or reduced copies of maps and plans are
required. In practice, this is usually done mechanically, with the aid
of the pantagraph, or photographically ; as in photo-engraving. When
the necessary instruments are not available, recourse must be made to
Graphical Methods, the best known of which is that of (1) Proportional
Squares, though often the method of (2) Angles and Distances serves
as an excellent substitute.
(1) Proportional Squares. This method consists in covering the
original map with a network of squares (otherwise called a grid or
graticules), either actually on the map or on a superimposed sheet of
tracing-paper. These squares are then reproduced proportionally
larger or smaller on a clean sheet of paper, and the lines of the survey
are inserted with reference to the sides of the squares by plotting
distances in the proportions they bear to the sides of the squares of
the original. In a great many cases of enlargement and reduction the
scale of the copy is either a simple multiple or sub-multiple of the scale
of the original, and the squares of the original can be made a convenient
mapping unit, 1 in., say, while those of the copy will be simply so
many inches, or so much of an inch; say, 2 in. and f in. respectively.
But cases constantly arise in which the given statements or representa-
tive fractions, or both, and the squares of the copy involve complex
fractions of the mapping unit. The same inconvenience arises when
squares with sides representing chains, hundreds of feet, or other even
units of measurement, are used instead of convenient mapping units,
inches, centimetres, etc. It is always advisable to ascertain if one or
other of these units, a mapping or a field unit, will lead to simple
square dimensions in both the original and the copy; for various simple
relationships, not evident at si^ht, are often discovered by such pro-
cedure. As a rule, field units are to be preferred when mapping units
introduce squares of equally inconvenient dimensions; but even they
cannot be considered unless suitable direct relationship to inches, or
42
ELEMENTARY SURVEYING
simple parts thereof, exists in one or other of the given statements'or
representative fractions, as the case may be.
Let us consider the matter with reference to the three cases that
may arise:
(a) When Statements are Given. Suppose it be required to enlarge
a map from 1 chain to 1 in. to 40 ft. to 1 in. Here 1-in. squares on
the original will require 66/40, or 1'65-in. squares on the enlargement.
Thus, both convenient mapping and field units are inherent in the
smaller scale. But if reduction from 5 ft. to the mile to 4 ch. to 1 in.
is required, 2-in. squares, representing 2-66 chs. on the original will
necessitate 0-66-in. squares on the copy, while 2 chs. represented by
1 J in. on the original, will merely require -in. squares on the copy.
(b) When Representative Fractions are Given. Suppose it be required
to reduce a portion of the Ordnance 1 : 1,056 sheet to the engineering
scale of 1 : 1,200. Here feet are the units in view, although the original
scale is directly related to chain units, 1 in. representing 88 ft., or f in.
one chain. But all relations between the given scales will introduce
fractional dimensions in the squares of either the original or the copy,
and, in general, squares representing 100 ft. on the copy would be
preferred to 1-in. squares on the original.
(c) When Representative Fractions and Statements are Given. Let it
be required to enlarge the 25-in. Ordnance sheet to a scale of 100 ft.
to 1 in. Here 1 : 2,500 corresponds to a scale of 25-344 in. to the
mile, or 1 in. to 208-296 ft., and since one of the given scales is simply
connected with hundreds of feet, the enlargement can be made with
equal facility with either field or mapping units. Thus with 1 -in. squares
400 800 1200 >600 2000 2400
400 800 1200 1600 2000 2400
Fio. 25
PLOTTING PLANS AND MAPS
43
or^tne original, squares of 2-08 in. side will be required on the copy,
while 2-in. squares representing 200 ft. on the copy will require squares
of 0-96 in. side on the original. The latter case is illustrated in Fig. 25.
Once the original map and the copy sheet have been covered with
suitable squares, the plotting is quite simple, the intersections of fences,
etc., with the sides of the squares, and the positions of points, etc., in
the squares, being judged by eye with reference to the corners. As in
all graphical methods, the use of proportional compasses facilitates
plotting and raises the accuracy of the work.
One of the slotted limbs of these double-pointed compasses is
graduated for a series of proportions between opposite pairs of points,
the compasses being set for the desired proportion by changing the
sliding block so that the index line coincides with the mark figured
with that proportion on the graduated limb.
(2) Angles and Distances. The following method
is particularly suitable for areas in which important
detail is sparse or localised, and the accuracy of
reproduction is highly important.
Describe a circle of any convenient size in the
centre of the area to be enlarged or reduced, and
through its centre o draw a reference meridian ns,
and rays to important points, such as p, q, and
r. If necessary, produce these rays to cut the circle
in a, b, and c respectively.
Describe a circle of the same radius on a clean
sheet of paper, and insert the meridian.
The copy is assumed to be superposed over the
original in Fig. 27, capitals superseding the small
letters.
Measure the chord distances na, nb, nc, etc., on
the original, and
set them off as
NA, NB, NC,
etc., on the circle of the copy.
Draw rays through A, B, C, etc.,
in the latter, and along these rays
set off the computed distances of
the selected points P, Q> and R in
the proportion that the scale of the
copy is greater or less than the
scale of the original. Having thus R
fixed the ruling points, fill in the
intervening detail by eye.
The use of this method is not advised, unless a simple proportion,
or one readily obtained with proportional compasses, exists between
the scale of the original and the copy.
Fio. 26
44
ELEMENTARY SURVEYING
CLASS EXERCISES
3 (a). The 1 : 10,000 Service Map of France is to be used in the following
connections:
(a) Laying Decauville track with measurements both in metres and feet.
(b) Reconnaissance, with pacing at the rate of 100 paces of 30 inches
per minute.
Construct the respective "comparative" and "time" scales.
3 (b). Construct the following scales to the representative fraction of 1 : 1250 :
(a) Reading to 10 ft., with main divisions of 100 ft.
(b) Reading to single metres, with main divisions of 10 metres. (G.S.)
(0-96 in. to 100 ft.; 0'8 cm. to 10 m.)
3 (c). A survey map dated 1860 is stated to be on a scale of 4 chains to
1 inch, although no scale is drawn. Believing that the paper had shrunk
considerably, a surveyor found two prominent points on the map that are
still existing: he measured the distance between these and found it to be
15*39 chains whereas it scaled only 15'20 chains on the map.
Construct a scale for the old map, suitable for measuring lengths up to
20 chains. (G.S.)
(20 ch. represented by 4*94 in.)
3 ((/). The scale of an old French map is 1,000 toises to 1 French inch.
You wish to copy the map on the scale of 1 mile to 1 inch by the method of
squares. If you draw }-in. squares on the old map, what size must they be
on *he new one, given that 1 toise was 72 French inches. Draw a scale of
yards for the new map (0-284 inches.)
3 (e). Enlarge the plan shown in Fig. 27 (e) to a scale twice the size of that
of the figure.
OFFICE EXERCISES
Problem 3 (a). Plot the survey from the notes given on Plate II. (G.S.)
Problem 3 (6). Plot the survey from the notes given on Plate III. (G.S.)
Problem 3 (c). ditto Plate IV. (G.S.)
Problem 3 (d). Plot your survey of (specified area) and finish it in the
prescribed manner.
Problem 3 (e). Enlarge the specified portion of the assigned map to a
scale of ...
ORIGINAL PROBLEMS
PLATE II
The above pages of Field Notes refer to a survey in which only the chain,
tape, and range-poles were used, all measurements being in feet.
Plot the survey on a scale of 50 ft. to 1 inch.
PLATE III
Ltne.
The above pages of Field Notes refer to a Chain Survey of a meadow,
all measurements being in links.
Plot the survey on a scale of 1 chain to 1 inch, placing the Magnetic
N. and S. line parallel to the short edges of the paper with A 2J inches from
the lower and right-hand edges,
s
3
1 =3 T3
1 ~2 <* G
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_o t- T3
D * * .
1JB w S
W *^ fl> O
' s o a 5
S M
.3
*
, ^J * O
i S -S -a
P-1S
*- -o .
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-I
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-gsllfi'
CHAPTER IV
FIELD GEOMETRY
It may be well at this stage to consider a number of problems, some
of which you may have already encountered in the field; and the title
of the present chapter must be understood to include also the applica-
tions of geometrical principles in dealing with obstructed survey lines,
more commonly, however, when only the chain, tapes, and poles are
at your disposal.
The subject of ranging out survey lines strictly should have preceded
their measurement, though, on the other hand, many cases of ranging
out lines become matters of obstructed distances. There is no geometry
in the following artifice.
Reciprocal Ranging. It often happens that a hill or high ground
intervenes so that the end stations, A and B y are not visible from each
other; and it is necessary to interpolate pickets in the line AB in order
to guide the chainmen. Reciprocal ranging is also useful in inter-
polating additional stations in a survey line without going to the end
stations, A or B, in order to direct the boning-out. It is also convenient
on level ground in foggy weather when the station poles can be seen
for about only five-eighths of the distance AB.
Observers a and
b, each holding a
picket, place them-
selves on the ridge
of the hill, in the
line between A and
B as nearly as they
can guess, and so
that one can see the
other and the station
beyond him. Ob-
server a looks to b,
and by signals, puts
Z>'s picket in line with
B. Observer b then
Observer a repeats
moved by b to #,
PLAN
FIG. 28
looks to A, and put a's picket in line at a'.
his operation from a', and is then himself
(not shown). In this manner the two alternately line in each other,
gradually approaching the straight line between the stations A, B, till
at last they find themselves exactly in it at a" and 6", as shown
in Fig. 28 (b).
FIELD GEOMETRY. The primary operations in field geometry consist
48
FIELD GEOMETRY 49
in *(i) Laying down Perpendiculars, and (ii) Running in Parallels. The
work presents little difficulty when a theodolite is available, though
there are instances when even this would be of little use, and the
surveyor must resort to purely linear methods. Underlying the use
of linear methods is the fact that right angles must be reduced to a
minimum, since these can never be set out precisely with auxiliary
instruments, such as the optical square or the cross staff, while the
construction of right angles is a tedious matter with the chain alone,
particularly when these are but part of a method. These facts should
be borne in mind also when measuring obstructed distances.
I. PERPENDICULARS
Perpendiculars may have to be (a) Erected at given points in Survey
Lines, or (b) Let fall from given external points to Survey Lines.
These cases will be considered separately, p and p' denoting respec-
tively internal and external points with respect to a survey line AB.
(1) The 3:4:5 Method. It is almost a matter of propriety that the
subject should be introduced by the application of Pythagoras' Theorem,
introducing the fact that the square on the hypotenuse of a right-angled
triangle is equal to the sum of the squares on the other two sides.
The longest perpendicular is usually required, and this can be laid down
with the following combinations of the available unit lengths:
(152025) ft. with the 50-ft. chain; and (334455) ft. /Ik. with
the 100 ft. /Ik. chain.
Commonly, however, the method is applied in the following manner
with the Gunter chair (Fig. 29):
(a) Internal Points. (1) Measure
from the given point p a length of
30 Iks. (ft.) along the survey line AB
to a point q. (2) Hold one handle
of the chain at p and at q its 90-
Ik. (ft.) ring. (3) Pull out the chain A .
evenly by the teller 40 Iks. (ft.)
from p until it takes up the form
shown in Fig. 29. (4) Fix an arrow
at /?', a point in the required perpendicular.
(b) External Points. The method is not directly applicable to this
case, and would involve calculation by similar triangles, as in the
following cumbersome construction:
(i) Erect a perpendicular pp', as above, at any convenient point p
in AB. (2) Line in a point Q in AB with p' and the given external
point P'. (3) Measure the sides pQ, p'Q, and P'Q, and calculate the
position of P, the foot of the required perpendicular from QP= 7^
50 ELEMENTARY SURVEYING
Tfie following methods, though detailed with reference to the cfiatn
and its limitations, have a wide range of application when long lengths
of cord or wire are used.
As in the foregoing method, the steel tape cannot be subjected to sharp
bends, and on this account two steel tapes would need to be tied together.
Also wires or cords could not be used in the 3:4:5: method, since figured
lengths are required. But in the following, light steel wires or strong cord
(such as sea-fishing line) or (in an emergency) the linen tape, can all be used
to advantage, provided a safe and uniform pull is applied. In fact, a small
spring balance is desirable in precise work, care always being that the elastic
limit is not passed. The chief advantage is that lengths may be bisected by
merely doubling the cord or wire back upon itself. Various methods must
be improvised for marking lengths. For instance, an electric wiring con-
nector may be taken from its porcelain container, sawn in two, and each half
secured in its temporary position on the wire by means of a pocket screw-
driver. One index secured in the middle of the whole length, and an adjust-
able one on each side are desirable. Small pipe clips, as used in chemistry,
are convenient in the case of cord lines. Loops should be made at the ends,
and the total length made a convenient multiple of a standard length, care-
fully tested. The erecting of a long perpendicular at a point in a straight wall
becomes a simple matter, and lengths up to 300 ft. can be laid down
expeditiously and accurately; but in all accurate work arrows should be
lined in, either by eye or with the aid of the stretched cord or wire.
(2) Chord-Bisection Method. In general, this is the best linear method
of (a) erecting perpendiculars at given points in survey lines.
(a) Internal Points. (1) Hold or secure the ends of the chain in the
survey line AB at a and b, points 15 to 40 ks. (ft.) equidistant from the
given point p. (2) Pull out the chain evenly, and fix an arrow at the
50-teller to indicate p\ a point on the required perpendicular.
When long perpendiculars are required, it is advisable to lay down
a second triangle, such as acb or ac'b, so as to obtain a second point
on the perpendicular (Fig. 30). This, of course, applies more particu-
larly when working with 200 ft., etc.,
lengths of wire or cord, and in this
case an outer index should be set at
a convenient distance for the loop from
the lengths ap, bp.
(b) External Points. (1) With one
end of the chain at the external point
/?', swing an arc about p 1 ', cutting the
survey line AB in the points a and b.
(2) Measure the length ab, and bisect
it at p. Otherwise obviate measure-
ment by laying down a second triangle
(by the preceding method), preferably
Fl - 30 on the opposite side of AB, as acb,
and lining an arrow at p with those at p' and c (Fig. 30)
This method is in effect, the process of swinging the tape "to and
fro" in measuring short offsets, the lowest readings on tape and survey
FIELD GEOMETRY
51
Ho. 31
lint being at the point of tangency to the arc. Incidentally, the length
of the chain or tape should at least be 5 per cent greater than the
length of the required perpendicular.
(3) Semicircle Method, In general this is the best linear method
of (b) laying down perpendiculars from given external points, and
though it might be extended to Case (a), there is little to commend
such practice under ordinary circumstances.
The construction is based upon the theorem that the angle in a
semicircle is a right angle.
(b) External Points. (1) Select a point a in the survey line AB so
that the line is at an angle between
30 and 60 with the direction of
the given external point /?'. (2) Measure
the distance ap', and bisect it at the centre
o. (3) With one end of the chain held or
secured at o, swing an arc of radius op',
cutting the survey line at p, the foot of the
required perpendicular (Fig. 31).
II. PARALLELS
A parallel may be required (a) through a given external point, or
(b) at a given distance from the survey line AB.
The second case reduces to the first by setting off the given perpen-
dicular distance pp' from AB, preferably as a part of the construction.
(1) By Right-angle OJfsets. (a) Through Given Points. The following
method would be inapplicable to the present case with a theodolite,
since it involves the dropping of a perpendicular from a given point;
and the method of alternate angles would be used, as with all angle-
measuring instruments. In lower grade work it is easily effected by
trial with the optical square or the cross staff, or the chain might be
used alone, though the setting-out of two right angles should be the
limit in this, as in other constructions.
(1) Let fall a perpendicular from the
given point p' on to the survey line AB.
(2) At any convenient point q in AB,
erect a perpendicular qq', making qq'~pp'.
The line through the points p f and q' is
the required parallel. (3) Test the accuracy A-
of the construction by the equality of
the diagonals qp', pq r (Fig. 32).
(b) At Given Distances. The following construction is expressly
suited to the use of the theodolite, sextant, etc., and the cross staff
and optical square may be used likewise, except in accurate work.
Occasionally it might be the best method when only the chain and
arrows are at hand, but it has the defect of requiring two right angles.
P'
FIG. 32
52
ELEMENTARY SURVEYING
FIG. 33
(1) Erect perpendiculars at convenient points, p, q, in the sunfcy
line, and along each of these set out the given distance pp' and qq' of
the required parallel. (2) Test the accuracy of the construction by
erecting a third perpendicular of the same length, and noting the
alignment of its extremity with the points p' and q' .
With the theodolite, only the perpendicular pp' might be erected
and a perpendicular to pp' set out at p', thus making the alternate
angles each 90 in the following method.
D r (2) By Alternate Angles. This purely
angular method should supersede all other
methods in the case of (a) parallels through
given points when an angle measuring
instrument is at hand. A convenient point
8 q is selected in AB, and the angle pqp'
=OL is measured. Then at /?' the angle
q'p'q is set out equal to a. (b) At a given
distance pp', a right angle to AB is set at/?, and at p' another right angle,
pp'q', would be measured off, giving the parallel p'q' (Fig. 33).
The method might be effected by purely linear means by swinging
arcs about q and p' with 50-ft. or 100-ft. radii, and then measuring the
chord r's' equal to the chord rs, thus tying the alternate angles a.
It would seldom be, if ever, used in laying down a parallel with the
chain or tape at a given distance from AB.
(3) By Similar Triangles. The use of similar triangles is recommended
when only the chain and poles are available.
(a) Through Given Points. (1) From a convenient point q in the
survey line AB, measure the distance to the given point p', and bisect
qp' at o. (2) From another convenient point r in AB, run in a line
through o to s', making os' equal to ro.
The line through p' and s* is the re- rf D * r ,
quired parallel (Fig. 34). If it is incon-
venient to bisect qp' at o, select o
appropriately and measure qo and op' \
also ro, and then make os ' equal to
or xp'o.
qo
This also applies to the case
r P
FIG. 34
when the point p' is not very distant from
the survey line AB. In this case the point o
o p* is selected outside the parallels, and is the
common apex of similar triangles, whose
bases are on the survey line and the
parallel. Again, it is convenient to swing
an arc about o so that the triangles are
___ -. B ^ sosce ^ es (Fig- 35).
P 9 (b) At Given Distances. (1) At any given
FIG. 35 point p in AB, erect a perpendicular pp 1
FIELD GEOMETRY 53
eqilal in length to the given distance, thus fixing ths point p'. (2) Then
proceed as before, running the line from q through o' to r', making
o'r' equal to qo'. The line through p' and r' is the required parallel.
The use of an external apex o' is dealt with on similar lines.
III. MEASURING ANGLES BY LINEAR METHODS
Contingencies often arise when it is necessary to construct or measure
angles in the field; and in the absence of a theodolite, this may often
be done with sufficient accuracy by means of the tape, with the aid
of a table of sines or chords, as described on page 39. Imagine
yourself in charge of setting out certain engineering works, and the
theodolite has not arrived, no tables are at hand, and the foreman is
none too amiable because his men are held up for the want of an
angle to fix a direction. The science of surveying is to know funda-
mental principles, and the art, to apply them with dexterity and accuracy
in any emergency. You know that a radian is the angle subtended at
the centre of a circle by an arc equal in length to the radius of the
circle, the angle thus being 57-3. Hence, if
a radius of 57-3 ft. (Iks.) is described about
the angular point A, the length around ihe
arc will be the magnitude of the angle in
degrees and decimals. Suppose that the fore-
man requires a direction fixed by an angle
0=36 15'. Well, merely centre the ring end 36
of the tape on A, and swing an arc from r with
a radius of 57-3 ft. (Iks.), at the same time inserting arrows (or clean
twigs) around the arc /-,?; then measure carefully round the arrows
from r with the tape, and fix ^ at a distance of 36 ft. 3 in. (36J Iks.) so
as to give the required direction AC.
IV. OBSTRUCTED DISTANCES
Although the stations of a survey should be selected so as to avoid
obstacles as far as possible, a survey line of great importance, even if
it be obstructed in some way, must not be discarded for others less
suitable to the general scheme. In chain surveys, particularly, it often
happens on hilly ground that the end stations of a line are readily
intervisible, but when the chainmen work into a hollow they find
themselves confronted by a pond or even a building. Also an essential
extension of a survey line may introduce similar difficulties.
Now obstacles may (i) impede the chaining of a survey line only
or, in addition, (ii) prevent the alignment or prolongation of the line.
Impeded chaining means that a geometrical construction must be
resorted to in order that the distance may be determined, while, in
addition, broken alignment will require additional construction in order
that the direction of the line may be re-established after the obstacle
54
ELEMENTARY SURVEYING
is passed. Broken alignment requires that two points shall be established
beyond the obstacle, or one point and an angle shall be likewise fixed,
the angle often being a right angle.
Detached, or Isolated Obstacles, are of two classes: (a) those which
impede chaining only, such as ponds, lakes, and low plantations, and
(b) those which impede both measurement and alignment, such as
buildings and woods.
Continuous Obstacles likewise fall into two classes: (c) those which
impeded chaining only, such as rivers and canals, and (d) those which
impede both measurement and alignment, such as high boundary walls
and blocks of buildings.
The first two classes (a) and (b) introduce the same basic con-
structions, and may therefore be treated together, keeping in view
the fact that (b) will require one extra point or an angle, in order that
the line may be continued on the far side of the obstacle. Classes
(c) and (d) differ essentially, and must therefore be considered separately.
The best-known methods will be considered with reference to an
obstructed survey line AB y A being on the "working" side of the
obstacle and B on the "distant" side, as suggested by progress from
A to B. Also right angles will be blacked in, or otherwise indicated,
in the diagrams.
Obstacles of Classes (d) and (b). (1) By Right-angle Offsets. This
method is best adapted to close sites, one side of the obstacle being
impassable. Its use at once suggests the theodolite, particularly in
precise work, since a number of right angles are involved, and these
cannot be set out very accurately with the cross staff or optical square.
(d) Erect ab, dc, equal perpendiculars to the survey line A3, at
points a and d in that line, on opposite sides of the obstacle. Measure
be, which should be equal in length to the obstructed distance ad.
If necessarily carried out with the
chain and range-poles only, some
check will be desirable; say, for
instance, prolonging be to e, and
comparing the lengths of the
B diagonals ed and cf after having
measured df equal to ce (Fig. 37).
(b) At a erect ab, a perpendic-
ular to the survey line AB; at b, set
out be, a perpendicular to ab\ at c erect cd, a perpendicular to be,
measuring cd equal to ab; and finally re-establish the alignment at d
by setting out dB perpendicular to cd. Otherwise, or as a check,
obtain a second point /on the required prolongation by producing be
to e, erecting ef perpendicular to ce, and measuring ef equal to cd.
Since it is never advisable to set out two consecutive right angles
by linear methods, this construction should not be attempted with the
chain and poles alone.
FIG. 37
FIELD GEOMETRY
55
(*2) By One Random Line. Though best modified to the interpolation
of points in woods, etc., this method can be used in fairly close sites
when the necessary deviation from the survey line AB is not great.
The necessary right angles should never be set out with the chmn
alone, though in average work the optical square and cross staff may
be used. It is modified to a purely linear method by running a second
random line, which requires that the obstacle shall be passable on
both sides.
(a) Select a point e as the apex of a right angle included between
two lines that meet the survey line A B, one on each side of the obstacle.
Sighting from e at a point a in
AB, set out a right angle, thus
fixing the point d on the distant
side of the obstacle. Measure
the lengths ae, ed, and calculate
the obstructed distance from ad
- V(oe)M-(*rf) 2 . (Fig. 38.)
(b) From a, a point on the
working side, run a random line
ag; and at c, a point in it, erect a perpendicular to meet the survey line at
/?, a point also on the working side of the obstacle. Measure ab, be, and
ac, and calculate the values of ab\ac and be lac. At e and g, points in
the random line, ae and ag units respectively from a, erect perpendiculars
erf, gf 9 having calculated their respective lengths from ae.cbjac and
ag.cblac. The points rf and / thus determined are on the required
prolongation of the survey line, and the impeded distance, ad or af, is
calculated from ad~ .ae or af=- .ag accordingly.
ac ac
(3) By Equilateral Triangles. This purely linear method commends
itself by its simplicity, though it is somewhat extravagant of space and
time, as regards obstacles of Class (a). Also, it demands extreme care
in prolonging the angular ties when applied to obstacles of Class (b),
where otherwise it is a most useful method (Fig. 39).
(d) On the working side of the survey line AB, lay down the side ac
of an equilateral triangle abc of
side L, L being 50 ft., 66 ft., or any
convenient unit. Produce ab, one
of its sides, to rf, a point con-
veniently clear of the obstruction,
and on de lay down a tie triangle,
also of side L, and produce the
side dfto meet the survey line AB
on the distant side of the obstacle
at a point h. Then ad ah.
(b) Proceed in the above manner, and produce df to h, making dh
equal to ad. Then on gh as base construct an equilateral triangle ghj
FIG. 39
56
ELEMENTARY SURVEYING
of side L (if possible) in order to obtain a second pointy on the disfant
side of the obstacle. Re-establish the direction of the line by sighting
through j and h in the direction thus given to B.
The small triangles may be laid down on the other side of AB, if
more convenient, as indicated by the dotted lines in Fig. 39.
(4) By Two Random Lines. Although more complicated than the
preceding method, the following construction, also wholly linear, gives
a stronger figure, but involves calculations and also requires that the
obstacle be passable on both sides (Fig. 40).
(a) From a point a in the
working side of the survey
line AB, measure the lines
ae, ag, one on either flank
of the obstacle, and con-
veniently beyond it. Line
in with the points e and g
the point / in AE on the
distant side of the obstacle.
From a point b in ae on the
working side, lay down a
parallel to eg, calculating
the position of din ag from
ablae=ad/ag. Measure along AB the length ac, and calculate the
impeded distance of from either (ae\ab)ac or (ag/acl)ac.
(b) Run in the lines ae, ag, as in the preceding case, and produce
them conveniently. From a point b on the working side of the obstacle,
measure any convenient line across to d in ag, noting the reading where
the survey line is crossed at c. Measure also ab, ac, and ad; and calculate
the value of ab\ad and either bcjab or cd\ad. Measure from a along
ag to any two convenient points g and k\ calculate ag(ab\ad) and
ak(ab/ad), and measure these distances from a along ae to the points
e and h respectively. Determine / and j, two points on the required
prolongation, by measuring either from e and h along eg and hk
FIG. 40
Fio. 41
FIELD GEOMETRY 57
dis\^nces respectively equal to ae (be jab) and ah (be jab), or in the opposite
direction, distances equal to ag (cdjad) and ak (cd/ad) respectively. The
aC ac
obstructed distance a/is equal to either -r.ae or --j.ag.
Obstacles of Class (c). The four constructions shown in Fig. 41 are
based upon the relations of similar triangles, and are expressly applic-
able to continuous obstacles of the present class. Each requires two
right angles, which in ordinary work might be set out with the optical
square or the cross staff or by means of the chain.
Method (A). At a point b, erect a perpendicular be to the survey
line AB. At c, lay down a perpendicular to the visual line cd to meet
the survey line at a point a. Measure ab and ac, and calculate the
(ac) 2
impeded distance from ad=--~ (Fig. 41 A).
ab
Method (B). At a point a erect ab, a perpendicular to AB, and in it
determine a point c, visually in line with the point e, the mid point
of ab, and a point d on the distant side of the obstruction. Measure be,
a length equal to the impeded distance ad (Fig. 4 IB).
Method (C). If the survey line AB is at an angle to the river, lay
down ab at a convenient angle and produce it backwards, making ae
equal to ab. Erect a perpendicular to this line at each of its extremities
b and e, and determine where each of these lines intersects the survey
line; namely, the points c and d. Measure ac, a length which is equal
to the impeded distance ad. (Fig. 41 C).
Method (D). Erect a perpendicular of length ac at a convenient
point a in the survey line AB. Erect a perpendicular at b, another
point in AB, and in thk perpendicular find a point e in line with c and
a point d in A B on the distant side of the obstacle. Measure ab, ac
4 . _. , acXab /r ,. Air^\
and be. Then ad=-^ (Fig. 4 ID).
be~ac
Instead of perpendiculars at a and b, these may be parallels at any
convenient angle, the same expression holding for the length ad.
Obstacles of Class (d). This class includes the most difficult cases
that arise in land surveying: obstacles that in many cases may be
essayed with the theodolite or compass, though not always expediently,
and, failing these, must be negotiated by some artifice especially
adapted to the circumstances.
(i) High Boundary Walls. Obtain a piece of J-in. board, about 9 in.
wide and at least 4 in. longer than the thickness of the wall. Fix two
2-in. xl-in. battens on the underside, close along the short edges, and
along the centre line, parallel to the long edges, drive 4-in. wire nails
straight up through each batten to serve as a pair of sights.
Using a ladder, place the board on the coping course with the wall
between the battens and the nail points uppermost. From the distant
side of the wall, sight the pickets in the direction of A, and shift the
58 ELEMENTARY SURVEYING
board until the sights are both exactly in line with the pickets. Secure
the board in this position by means of wood wedges. Now with the
ladder on the working side of the wall, instruct the chainmen to fix
pickets on the distant side, lining them in towards B by means of the
nail sights.
The objection to the method is that the line is prolonged through
the medium of plain sights seldom more than 15 in. apart.
A more accurate, though laborious, method would be to procure
a straight scaffold pole, 20 ft. long, and, with the aid of a cord stretched
centrally down the length of the pole, to insert four picture rings; one
near each end and one about 7 ft. from each end. Balance the pole
across the wall with the rings downwards, and suspend a plumb-bob
from each of the rings, a, b, c, d, say. In windy weather it will be
necessary to damp the vibrations of the bobs by immersing them in
buckets of water. Next, standing back to the wall on the working
side, look towards A, and instruct the chainmen to move the pole until
two adjacent plumb-lines, b, a, come exactly into line with the pickets
already inserted. Finally, go to the distant side of the wall, and with
the back to it sight through the other plumb-lines, c, d, and direct the
chainmen to fix pickets in the direction of B.
(ii) Two High Walls enclosing Roads, etc. When two walls of about
the same height, and no great distance apart, cross the survey line, it is
often possible to bone out by "mutual ranging." Observer on wall M
puts his pole and that of an observer on the wall TV in line with the
pickets interpolated on the working side from A. Observer on N then
sights in poles towards B on the side beyond M in line with his pole
and that held by the observer on M.
Much art has been lost, not in surveying alone, by evading obstacles.
CLASS EXERCISES
4 (a). Show with reference to neat sketches what you consider to be the
best method of dealing with each of the following obstructions in a survey
line, when only the chain and poles are at your disposal:
(a) Pond, passable on one side only.
(b) Isolated building. (G.S.)
4 (b) (a). You are surveying in foggy weather, and it is possible only to
see the pole at B for five-eights of the distance AB. Describe, with reference
to a sketch, how you would proceed to measure the line AB.
(b) In the same survey the line FG must cross a river 35 yds. in width.
Describe, with reference to a sketch, how you would measure the length FG
with only the chain and range-poles at your disposal. (G.S.)
*4 (c). Describe, with neat sketches, how you would overcome the follow-
ing difficulties when only the chain tape, and range-poles are at your disposal:
(a) Chaining between stations when the line is obstructed by a building
passable on one side only.
(b) Measuring a line between stations when the line is crossed by a road
which is fronted by boundary walls 12 ft. in height.
(c) Interpolating a subsidiary station in a survey line without ranging from
the end stations, which are 24 chains apart. (U.L.)
FIELD GEOMFTRY
59
f4 (d). Describe with sketches how you would overcome the following
difficulties in chain surveying:
(a) Ranging a line over a hill between stations when the latter are not
mutually intervisible, but are both visible for a considerable distance on the
hill itself.
(b) Chaining between stations when the line is obstructed by a detacned
building, passable on both sides.
(c) Measuring a line between stations when a boundary wall 12 ft. in
height crosses the line. (U.L.)
4 (e). Describe two methods of measuring angles by means of a tape.
FIELD EXERCISES
Problem 4 (a). The range-poles A and B are the end stations of a survey
line which is obstructed by the (specified) building. Determine the length
of AB.
Equipment: Chain, arrows, and set of pickets.
Problem 4 (b). The pickets A,B, and C,D, represent stations on the
opposite banks of an imaginary river too wide to be chained across. Using
two different methods, find the lengths of CD and AB.
Equipment ', as in 4 (a).
^Problem 4 (c). Find the error that would result in measuring with the
tape the three angles of the triangle, as indicated by the pickets A, B, and C.
Check the work by the "radian" method.
Equipment: Tape (steel or linen), arrows, and table of Sines or Chords.
^Problem 4 (d). Determine the perpendicular distance of the (specified)
inaccessible point from the survey line indicated by the range poles A and B.
Equipment: Chain, arrows (cross staff), and set of pickets.
The selection of points is indicated by the numbers on Fig. 42 (d), the lines
actually measured being crossed (//) and chained in the order suggested by
the numbers at the ends of the lines.
*Problem 4 (e). Run a line through the given point parallel to the in-
accessible survey line ind : cated by the range-poles A and B.
Equipment, as in 4 (d).
The selection of stations is indicated by the numbers on Fig. 42 (e), which
also suggest the order in which the auxiliary parallels (shown) thick are run
in obtaining the required parallel 04.
Fio. 42
FIG. 42 (e)
CHAPTER V
LEVELLING
Levelling is the art of determining the differences in elevation of points
on the earth's surface for the purposes of (a) tracing contour lines,
(b) plotting vertical sections to represent the nature of that surface,
and (c) establishing points at given elevations in constructional projects.
The methods of levelling may be divided into the following cate-
gories:
(1) Gravitational Levelling, (2) Angular Levelling, and (3) Hypso-
metrical Levelling.
Gravitational methods include Spirit Levelling, as usually under-
stood in practice; Angular methods, the application of trigonometry
or tacheometry, and hypsometry, those methods which depend upon
variations of the pressure of the atmosphere, as utilised in the baro-
meter, the boiling-point thermometer, and the altimeter as used in
finding heights in aircraft navigation. The three systems in the general
sense represent three degrees of accuracy in descending order: precise
to accurate, accurate to moderate, and moderate to approximate. At
the same time they represent in ascending order their applications to
small, medium, and great differences of elevation, which the writer
prefers to designate "Reduced Levels," "Elevations," and "Altitudes"
accordingly.
I. PRINCIPLES OF LEVELLING
Levelling is surveying in the vertical plane, and the systems of
vertical co-ordinates involved are respectively: (1) Rectangular Co-
ordinates, (2) Inverse Polar Co-ordinates, while (3) has obviously no
geometrical basis (see page 82).
Fundamentally, all levelling is based upon gravitation since the
ruling levels of all methods are based upon spirit levelling.
In practice all elevations are referred to some
"datum." This may be some assumed level
plane, known as a local datum, or it may be
some level spherical surface, such as that of
the Ordnance Survey, which is the "approxi-
mate mean water at Liverpool." Points of
reference to the datum are known as bench
marks, which are figured on maps conveniently
with the elevation above datum.
Now a Level Line is strictly a line concentric
with the earth's mean figure as given by mean sea level, acd in
Fig. 43. Since a plumb-line is a vertical line, always tending to point
60
LEVELLING
61
to tiie centre of the earth, a Horizontal Line is a tangent to the earth's
curve, as ab. We must see horizontally, and a levelling instrument con-
strains us to look horizontally. Hence a surveyor at a sights along ab,
and the distance be is the earth's curvature, which, being 8 inches per
(mile) 2 , would not be detected by ordinary instruments.
Only gravitational levelling will be treated in the present chapter,
and some applications of angular levelling will be given in the following
chapter, where reference to hypsometry will also be made.
HISTORICAL DEVELOPMENT. The basis of gravitational levelling is the
plumb-line, or plummet, and, by a stretch of the imagination, the bubble
of a spirit level is a plumb-bob with an exceeding long line, making it so
accurate and sensitive that its vibrations could never be nulled. In
fact, a way of specifying the accuracy or sensitiveness of a bubble
tube is by its "equivalent" plumb-line, which may be 300 ft. in length
or more.
Plumb-line Level. Let us consider the primitive instrument shown
in Fig. 44. Here a builders' square is attached to the top of a vertical
stake B, which is driven into the ground, the stock being adjustable
and secured in a horizontal position by means of a thumb nut c. In
the head of A, at d is a hook,
at e a ring, and from d a
plumb-bob P is suspended.
On top of the stock arc fitted
two sights A, A, of equal
height. Now if the stock is
so adjusted and clamped at c
so that the plumb-line passes
centrally through the ring,
the sights A, A, will be truly
horizontal, and the eye at E
will be constrained to look
horizontally, which is the basis of levelling. If, then, a vertical staff,
divided into feet and tenths, were moved in the direction in which the
stock points the readings taken from E on the staff would show the
relative heights of the ground at the different staff stations, and, by
subtraction, the differences of elevation.
Another way of introducing the principle would be to take a mirror,
about IJ-in. square, remove the silver above a diagonal, and mount
the mirror in a metal frame so that the line of demarcation (say A, A)
is horizontal. Then if the frame were suspended from the uppermost of
its corners (v, say) and a heavy weight were suspended from the lowest
corner, the plane of the mirror would be truly vertical. Hence if the
instrument were held near the eye, the pupil would be seen by reflec-
tion, and above the silver edge the vision would be horizontal, so that
a sight on a staff could be taken. Such an instrument could be con-
structed in the metal-work classes.
> I? . "? >...
._, 2
#
1
A
e -
"
B
. P
S
KV
I I
\
FIG. 44
62
ELEMENTARY SURVEYING
FIG. 45
The principle is employed in two well-known instruments:/ the
Reflecting Level and De Lisle' s Clinometer.
Water Level. Suppose a |-in. glass tube were turned up at the ends
and fitted by means of a knuckle joint to the top of a stand or tripod,
the tube being almost full of water, as shown in Fig. 45. Then if the
eye E is placed near one end,
looking over the menisci, a hori-
zontal sight is obtained, and a
vertical levelling staff could be
read as before. Again this is the
principle of gravitation, in that
water at rest has found its own
level. The principle survives in the
water level, a simple instrument
used for transferring levels in
spaces so confined that the use
of any other instrument would be
impossible.
This instrument consists of a
pair of glass tubes, like test tubes,
but with a short open glass pipe
sealed into the bottom, fine lines being etched on the tubes. Attached
to the pipes are the ends of a length of rubber tubing, the whole being
nearly filled with water. The open ends are plugged until the tubes
approximate to the same height, so that the levels can be transferred
when the water reaches the etched marks.
Telescope. Doubtless a water level was attached to a metal sighting
tube provided with a pinhole eye-sight and a horizontal hair line at
the open end, the whole being mounted on a tripod. Still there was
the limitation that the naked eye could not estimate to a tenth of a foot
on a staff at distances exceeding about 150 ft. But as soon as a water-
level tube was mounted upon a telescope, the range of sighting would
be increased by 20 times, or 1/100 ft. could be read at distances up to
300 ft. %
Fig. 46 is a longitudinal section of an ordinary, or draw tube, tele-
scope, the type still found in the majority of surveying instruments.
A is the outer tube and B the inner draw tube, which is moved by
means of the rack and pinion R by an external focusing screw at the
side of the body, but here hidden from view. O is the double-convex
object glass, which throws an inverted image of the levelling staff on
the plane of the diaphragm D. The diaphragm, which is supported by
the screws d, d, consists of one horizontal line and two vertical lines,
either etched finely on glass or actually spider webs. E is the eyepiece,
which magnifies the image through the medium of two plano-convex
lenses, giving a magnifying power in the ratio of the focal length / of
the object glass to the focal length/, of the eyepiece. Thus the staff is
LEVELLING
63
seer* between the two vertical lines of the diaphragm, and the reading
is taken at the horizontal line (or crosswire), the numbers on the staff
FIG. 46
being also seen inverted. Incidentally the landscape is also seen in-
verted, as is also the case with most surveying instruments. Rarely an
inverting eyepiece is used: a tube fitted with four lenses instead of two.
At first sight an inverting eyepiece appears an investment; though, on
the other hand, the surveyor would think something was radically
wrong if he lost his habit of seeing things upside down. Also there is
an adage, "More glass, less light."
Since the end of the Great War of 1914-18, the internal focusing
telescope has superseded the foregoing pattern in instruments of recent
manufacture. In this type, the distance between the objective O and
the diaphragm is fixed, and instead of the focusing screw moving the
draw tube it moves a double concave (or negative) focusing lens. This
leads to a more compact telescope, and one less susceptible to con-
structional defects, being on the whole an improvement, though, on
the other hand, many surveyors of great field experience are inveter-
ately conservative, and repeat among other things the slogan, "More
glass, less light."
Bubble Tubes. A water-level tube on the top of the telescope leaves
much to be desired, apart from sensitiveness. Hence it is superseded
by the bubble tube, or phial, which is usually filled with pure alcohol
so that a bubble of vapour is contained when the ends of the tube are
sealed. Bubble tubes must be curved, either bent bodily or ground
internally to a curve, or they would represent plumb-lines of infinite
length, far too sensitive for mundane matters. Cheap bubble tubes,
such as those fitted in carpenters' levels, are usually bent; but in all
proper surveying instruments they are ground, often with such pre-
cision that each of the small division marks could be used for measuring
small vertical angles, even as small as 5-seconds of arc. All good bubble
tubes are costly and demand respect, if not for the skill in producing
them, for the cost in replacing them when broken.
The idea is that the vapour bubble, being lighter than its spirit, rises
to the uppermost point of the curve of the tube. Hence, if marks fixing
its length are etched on the tube, the tube can be mounted or inset in
a block of metal or wood, which, tried on a truly horizontal surface,
will register the deviation fronc! the horizontal when laid on any other
plane surface.
64 ELEMENTARY SURVEYING
Now the radius of the curve to which the bubble tube is beyt or
ground is the length of the equivalent plumb-line. Suppose that a
vertical staff is sighted at a horizontal distance D from a levelling
instrument with the bubble out of the centre and nearer the eyepiece.
Then if the bubble be moved an equal distance from its central position
towards the objective end, the staff reading will alter accordingly, and
the difference of the staff readings will be the intercept s. Hence, if R
is the radius of the bubble tube and n the number of divisions of
length v through which the bubble has travelled, it follows that
Measure six divisions of a bubble tube with a diagonal scale and
find the length in feet of a single division, v. The rest is simple. Feet
must go with feet, even though it is a privilege of youth to mix units
indiscriminately.
In recent years great improvements have been made in the produc-
tion of bubble tubes; in particular, the type in which the bubble has a
constant length, even in tropical climates.
Thus the modern spirit levelling instrument is evolved. To-day it is
made in three predominant forms, all of which embody the same
essential principles, differing only in manipulation and adjustment.
For nearly 100 years the Dumpy Level has characterised British practice,
and must therefore be our representative instrument.
THE DUMPY LEVEL. The Gravatt level was called "dumpy," because
it was more compact than its immediate predecessor, the "Y" level,
so called because the telescope rested in crutches of this form.
The dumpy level, like most levels,
consists of four primary parts: (1)
the Telescope\ (2) the Level Tube\
(3) the Limb, and (4) the Levelling
Head.
In the instrument shown in Fig.
47, the limb is really a casing
around the telescope, and termina-
ting in a vertical conical spindle,
which rotates in the levelling head.
The objective end of the telescope
is covered with a ray shade, or sun
cap, which is used for cutting out
the glare of the sun. Underneath
the telescope will be seen a clamp and its slow motion screw, not always
fitted (or wanted) in levels, but provided here in order that the telescope
may be moved gradually round about the vertical axis of rotation,
particularly when the diaphragm is "webbed" with fine metal points,
instead of spider lines or lines on glass (page 62). The diaphragm
webbing is variously styled the crosswires, the cross hairs, or lines.
FIG. 47
LEVELLING 65
Tie levelling head of the model shown is of the Tribrach, or Three
Screw pattern. Earlier instruments were provided with Four Screw
Parallel Plates. These plates had an awkward habit of "locking," but
virtue was found in this vice when the instrument was in skilful hands.
The lower tribrach sprang or parallel plate is bored internally and
threaded so that the instrument can be mounted upon its stand, or
tripod, which may be of the solid round form, or framed, more like
that of a photographic camera.
The sight line is known as (i) the "line of collimation " and is the line
between the centres of the object-glass and the horizontal cross wire;
also (ii) the bubble line is an imaginary line tangential (or axial) to an
undistorted bubble, being horizontal when the bubble is in the middle
of its run.
(i) The one condition essential to accurate levelling is that the line of
collimation shall be parallel to the bubble line, so that when the bubble
is centralised by means of the foot or plate screws, the line of collimation
will be horizontal.
(ii) Another condition that has somewhat fallen into abeyance by
the restoration of the old principle of the tilting screw in the modern
level is that the bubble should "traverse," that is, remain in the middle
of its run for all directions in which the telescope may point. Traversing
is not a necessity, but a great convenience, particularly in trial and
error work, such as contouring. In fact few levels traversed per-
fectly, but a slight touch to the foot screw soon put matters right.
Certainly the tilting screw does the same thing, but there is a difference,
apart from the prejudice of experience.
The levelling up of the instrument and the focusing of the telescope
preparatory to taking observations with any mounted instrument are
known as Temporary Adjustments, as distinct from Permanent
Adjustments, which are those by which the correct relations are made
between the Fundamental Lines, as (i) and (ii) arc called in the present
connection.
Caution. Nothing is so disconcerting to a student as a proper level out of
adjustment; a fact that is evinced in closing a circuit of levels upon the
starting-point. The causes and effects of maladjustments should be kept in
mind. Apart from accidents, instruments may be thrown out of adjustment
by numerous causes, seen and unseen, such as forcing them into their cases,
staffmen sitting on the cases, careless transport, storage under extreme con-
ditions of temperature, etc., even undue admiration in a pawnbroker s
shop in the interim between pledging and redemption.
The essential relation between the bubble line and the line of collimation
is usually restored by means of the antagonistic pair of capstan-headed
diaphragm screws, which must be treated with care; and one with less than
the engineering degree standard of training should not undertake the
making of permanent adjustments apart, of course, from men ot practical
experience. Less than forty years ago it was regarded as the next ottence to
capital crime for a capable assistant engineer or pupil to attempt to adjust
even a dumpy level. Even this was considered "a matter for the maker by
66 ELEMENTARY SURVEYING
men who have left very substantial monuments behind them.
to-day, the outlook has changed, and there may be a surveyor in the locality
who will undertake the adjustment with a little persuasion, for as you will
doubtless find, "traditions die hard." The maker will certainly adjust the
instrument most satisfactorily, but this involves the trouble of careful pack-
ing and certain risks of transport, though the motor car has removed the
possibility of the case being thrown out of the luggage van.
In this connection, it must be known if it is the level that is really at fault.
There is little difficulty in testing the adjustment by the "two peg'* method,
which is the only absolute field test as to the accuracy of any levelling instru-
ment. The various ways of applying this test are described in most text-
books on surveying, and are detailed at length in the writer's Field Manual.
In the present connection the best plan would be to introduce the method of
"Reciprocal Levelling" the process used to eliminate instrumental defects
and the errors of curvature and refraction in the very long but necessary
sights, such as occur in levelling across a wide valley or river.
(1) Select two points, A and B, on a fairly level piece of ground at an esti-
mated distance of about 4 chains or 300 feet apart; and at these points drive
pegs firmly in grassland or chalk crosses on concrete or asphalt surfaces.
(2) Set up the level near one point, A, say, so that when the staff is held
vertically on that peg or point it will be possible to measure directly up to
the eyepiece a staff reading a^ (3) Sight through the telescope, and read at
the horizontal wire the (same) staff held vertically on the peg or mark B,
noting the reading b^ when the bubble is central. (4) Now set the level up
likewise near B, and measure the staff reading b 2 up to the centre of the eye-
piece. (5) Sight through the telescope and observe at the horizontal wire the
reading a 2 on the staff now held on the peg or point A when the bubble is
central. (6) Find the differences (a^-b^) and (a 2 -6 2 ), and if these are equal
the level is in adjustment; but if this is not the case, the error E^\ ((a l ~b l )~-
(a 2 -b 2 )), which is corrected by means of the diaphragm screws, gently
slackening one screw and taking up the slackness more gently with the other
in moving the diaphragm over an image, which is also real.
LEVELLING STAVES. Levelling staves are made in two forms: tele-
scopic and folding. Telescopic staves have the advantage that they are
heaviest at the bottom and are not top-heavy like the folding patterns,
but this embodies the disadvantage that the uppermost, the third length,
is very narrow and therefore more difficult to read. The former are
made commonly in 14-ft. lengths, though 16-ft. and even 18-ft. are
obtainable, while the latter are usually 10-ft. or 12-ft. when extended
fully in all cases.
The type of staff used mainly in this country is the Self-reading
staff, so called because it is read from the telescope; but in America
another form is also used, the Target Rod (the Boston, the Philadelphia,
and the New York patterns), a target being set by the staffman under
the directions of the surveyor at the instrument. Target staves are
sometimes used in precise work in this country, and apparently they
were seen at the time Alice in Wonderland was written.
Although numerous "readings," or modes of division, have been
designed, the prevalent one is the Sopwith "ladder," shown in Fig. 48.
This shows (a) Primary divisions in feet, the numerals of which are
shown in red on the left of the staff; (b) Secondary divisions of tenths
LEVELLING
67
.17
of ft foot, which are indicated alternately in black figures of that
height on the right of the staff; and (c) Steps, or blocks; subdivisions
alternately black and white, each one-hundredth of a
foot in height. Feet are read at the tops of the red
figures in line with the wider black spaces, which here,
as at all tenths, denote the pointings of the secondary
portions. The tops and bottoms of the alternate black
figures are also in line with the tops of these wider black
spaces. Sometimes a black diamond and a dot are
placed at the bottom of the middle shorter black space
to denote each half of a tenth of a foot reading. Also
small red numerals are painted at intervals along the staff
to provide against the event that a large red numeral
does not appear in the field of view of the telescope.
A real levelling instrument must be available if only for
demonstration purposes. An older model can be purchased
at a reasonable figure. Military instruments will not serve
the purpose of an engineer's level satisfactorily, even though
they may prove excellent substitutes for theodolites.
Impiovised Levels. A number of sighted levels and staves
reading to tenths might be constructed if necessary. For
instance, light brass tubing, about U-in. diameter, could be
cut into 10-in. lengths; a circular disc with a pin-hole centre
could be soldered in as an eyesight at one end, and at the
other end a horse hair could be stretched across from small
holes in the horizontal diameter, with two similar hairs
vertically, so that the three represent diaphragm webbing.
A rectangular plate soldered at this end with its upper edge
across the horizontal diameter, would serve the purpose of
the horizontal web, as in the case of Abney levels. A spirit
level, about 4 in. or 5 in. long, in a metal container, could be
attached to the top of the sighting tube by means of adjust-
able clips, so that the bubble could be set central when the
line of sight is established truly horizontally in the manner
described in the Two Peg test.
The chief difficulty, however, is the means of attaching
the level to the tripod in such a way that the bubble can
be adjusted. This may mean a piece of work for a fitter,
though much can be done with thumbscrews and J-in. plate,
if a drill and screw taps are available. Otherwise some stiff
form of ball joint could be improvised. Light frame tripods
are easily constructed from 1-in. square ash or pine, six 5-ft.
lengths being required. Pairs are screwed together and
pointed at the toes (desirably shod), while the tops are
opened out to fit on J-in. bolts through lugs projecting from
a triangular plate, similar to that into which the levelling
thumbscrews are threaded.
Improvised Staves. Folding staves can be constructed from
two 5-ft. lengths of well-seasoned pine, both 3 in. wide, the
lower being 1 i-in. thick and the upper 1-in., so as to avoid top heaviness.
These should be given two coats of white paint, and then painted to show
alternate tenth-of-a-foot black blocks across, black numerals, also one-tenth,
FIG. 48
68 ELEMENTARY SURVEYING
being inserted at the edges of the white spaces. The lengths should be secured
together with a strong brass butt hinge, and a bolt should be fitted at the
back to retain the upper length in position when extended. The bolt should
shoot from the upper length into the loop on the lower, otherwise the staff-
man's ringers are exposed to great risks.
Otherwise, and particularly when proper levels are used, staff papers for
renovating old staves could be purchased from any of the surveying instru-
ment makers, who supply these, plain or varnished, with full instructions,
Since these arc usually divided for telescopic staves, the lower 5-ft. length
should be planed out with a rectangular channel, so that a narrower upper
length will lie in it when folded, thus protecting the divisions in transport.
It is desirable that the backs and sides of the wood should be lightly stained
and varnished, and especially that a brass sole plate should be fitted; 10-ft.
readings, painted as above on fabric, could be used, but these are best
attached to a staff with the zero exactly at the foot.
TEMPORARY ADJUSTMENTS. Although these are a part of the field
routine with both the level and the theodolite, they will be detailed
here to avoid interruptions in the procedure of the Practice of Levelling,
the important subject of the following subsection. A note will be
added in order to avoid repetition when the subject arises in connection
with the theodolite.
Consider the three (and four) dots, lettered A, (A) and B, B, in Fig. 49
to represent respectively the plan(s) of the tribrach and plate screws
of a surveying instrument.
B ^B For brevity, these will be
called "foot-screws,'* and the
A A remarks relative to the four
O P* ate screws will be enclosed
in brackets. The small o in the
f centre is the plan of the verti-
D cal axis about which the tele-
B scope or instrument rotates.
F IO . 49 In the following instructions
it must be remembered that
aptitude in levelling up an instrument cannot be acquired from mere
words: there is that little something else which practice alone gives.
(a) Setting up the Instrument. (1) Plant the tripod firmly with the
telescope at a convenient height for sighting, and press the toes of the
legs into soft ground, or place them in crevices in hard surfaces, always
so that the lower sprang (or parallel plate) is fairly horizontal. (2) Turn
the telescope so that it lies with its eyepiece over the screw A; then, by
means of this screw A (and (A)) bring the bubble to the middle of its
run (in the case of a pair of screws A, (A), working these equally in
opposite directions). The bubble will move towards the screw that is
worked in the clockwise direction as viewed from above. (3) Turn the
telescope through a right angle so that it lies parallel to (or over) the
other screws, and by means of these screws B, B, bring the bubble to
the middle of its run, working the screws equally in opposite directions.
LEVELLING 69
(4}%Return the eyepiece over the screw A, and by means of this screw
(and its opposite fellow (A)) restore the bubble to the middle of its run
if necessary.
If the level tube is in adjustment, the bubble will remain central for
a complete rotation of the telescope, or the deviation will be so small
that a mere touch to the foot screw nearest to the eyepiece will set
matters right, for the bubble must always be central when reading
the staff.
*Whcn a bubble departs considerably from its mid-position on repeating the
foregoing routine, it suggests that adjustment of the level tube is necessary;
but this must not be attempted, since in most patterns of dumpy levels this
would derange the all-important parallelism of the bubble line and line of
collimation.
After all, a "traversing" bubble is a convenience, not a necessity. Some
modern levels are levelled approximately on similar lines, though often with
the aid of an auxiliary circular bubble, which is brought to the centre of a
circle etched on the glass cover. The main bubble is then set to its mid-
position for every sight by means of a tilting screw. *- ,
Usually, in the case of theodolites, two small plate levels are fitted at
right angles to each other, and these can be set parallel to the lines B, B y
and A, (A), thus avoiding the necessity of turning the telescope through a
right angle in the second step.
(b) Focusing the Telescope. The foci of the object-glass and eyepiece
must both be in the plane of the cross wires; otherwise the accuracy
of the reading will be impaired by "visual parallax." Parallax can be
detected by moving the eye up and down when sighting the staff or a
station and noting if the cross hair appears fixed to the (inverted) image
or if it moves relative to that image. The latter condition denotes
parallax, which in many cases is due to incorrect focusing of the object-
glass with the focusing screw, and not to the oft-innocent eyepiece.
Usually it is better first to point the telescope to the clear sky with the
focusing tube well in, and then move the eyepiece with a screwing
motion until the cross wires are seen clearly and sharply. But our
instrument is levelled up. Hence we had better look at a sheet of white
paper held obliquely in front of the telescope and set the eyepiece when
sighting this. Now direct the telescope towards the levelling staff, and
by means of the focusing screw obtain a clear image of the staff. Test
for parallax, but try refocusing with the screw before moving the
eyepiece to eliminate parallax.
A perennial source of annoyance in an instructional class is the focusing
of the telescope to suit the real and unreal idiosyncrasies of many eyes, and
the fellow with spectacles might often oblige by removing them. Inexperi-
enced surveyors are always tampering with the eyepiece, and in a class
seventy per cent of the eyepiece adjustments are unnecessary, leading to wear
if not damage to the instrument.
70 ELEMENTARY SURVEYING
II. PRACTICE OF LEVELLING
Even now \ve cannot proceed until we acquaint ourselves with a few
more terms and definitions. On page 60 we saw that the Datum is the
plane or surface to which elevations are referred, and that the "reduced
level" is the elevation of a point above this datum surface (or below in
the case of soundings at sea). Reduced levels are connected with the
datum through the medium of "benchmarks," which may be official
or local according as the Ordnance datum is adopted, or any convenient
horizontal plane of reference is assumed, the latter serving in the case
of small or instructional surveys. Ordnance benchmarks are indicated
thus, B.M. A 62-3, on the official maps, and are likewise indicated by the
symbol alone cut into the walls of buildings, etc., the centre of the
horizontal bar being the reference line. (By the way, this symbol was
taken from the armorial bearings of an early chief of the Ordnance
Department, and it has no connection whatever with Dartmoor.) An
interesting excursion after studying an Ordnance sheet would be a
search for the benchmarks indicated in a given area. When the motor
hunts of twenty years ago were the thrill of "the bright young people,"
it was observed that a benchmark hunt in Richmond Park would be
equally exciting. Benchmarks on posts and boundary stones have a
ghostlike habit of disappearing and reappearing.
Benchmarks improvised in small jobs or on the cessation of a day's
levelling are known as Temporary Benchmarks (T.B.M.).
All levelling operations must begin at a benchmark, which may be
temporary with an assumed value (50-0 or 100-0) if an Ordnance B.M.
is not in the immediate neighbourhood; and all levelling operations
must close on a benchmark, even for the day. The state recognition of
a benchmark may give a sense of dignity, but this has no effect upon
the work, except that legal requirements may demand due respect for
the Ordnance datum.
Finally, levelling is peculiar in that the point at which the staff is
held is the station, and not the position of the instrument, which may
be anywhere within sight and reason.
Backsight. A backsight (B.S.) is a reading taken on a staff held at
a point of known elevation. It is the first reading taken on setting up
the level anywhere, and is taken on a benchmark at the beginning of
all levelling operations.
^Backsighting is equivalent to measuring up from the datum, for if
the reduced level of the staff station be known, say, 50-0, and the
observed backsight reading is 4-24, the height of the horizontal plane
in which the line of collimation revolves is 54-24, the contraction for
feet being understood (and, therefore, unprofessional). Hence the
rule: Add the backsight to the reduced level for the Height of Plane
of Collimation (//.P.C.), or "height of Instrument" (//./.) or even
Collimation, " as it is variously styled.
LEVELLING 71
'Foresight. A foresight (F.5.) is a reading taken on the staff held on
a point of unknown elevation in order to ascertain what distance that
point is below the plane of collimation, and thus to determine the
reduced level of the ground at the foot of the staff. It is the last reading
taken before removing the level anywhere, and is taken on a benchmark
at the close of a day's operations.
Foresighting is equivalent to measuring down from the horizontal
plane of collimation, for if the reduced level of the plane of collimation
is 54-24, and the foresight reading is 5-26, the reduced level of the foot
of the staff is 54-24 5-26=48-98. Hence the rule: Subtract the fore-
sight from the height of collimation for the reduced level of the staff
station.
Now these two terms in no way denote direction, for often a back-
sight and a foresight are taken in the same direction.
Incidentally, the original method of reducing levels ignores the plane
of collimation, and merely conceives the difference of the back and
foresight readings as a Rise or a Fall, the difference 5-26 4-24 --- 1-02
denoting that the ground has fallen from 50-00 to 48-98.
Backsights and foresights are taken on firm ground, embedded
stones, or even footplates, since both the continuity and the accuracy
of the work depends upon these.
Intermediate Sights. An intermediate sight (Int.S.) is virtually a
foresight taken solely in order to ascertain the reduced level of a point
or to establish a point thereat to a given reduced level. It has the
algebraical sense of a foresight, but not the importance of one, being
often booked to the nearest tenth, especially on rough ground. All
readings between the backsights and the foresights are "intermediates."
Change Points. A change point, or turning point, is a staff station on
which two staff readings are taken; a foresight prior to removing the
level and a backsight in order to fix the new collimation height on
again setting up the level. Occasionally, the term "Shift" is used
colloquially, but this involves risk, since the cold (or weary) staffman
may misinterpret "That's a Shift" as a welcome command. The
importance of hard points for shifts is again emphasised.
A change point is characterised by two features: (i) that two staff
readings are taken at it, and (ii) that these readings must appear in
the same line in the level book, simply because they refer to the same
point.
It is unnecessary to say that it is bad form to note a change point in
the level book when it is thus evident.
Level Books. There are two methods of booking level notes: (1) The
Rise and Fall System, and (2) The Collimation System. These will be
considered together as we run our first line of levels. Common to both
books are columns for B.S., Int.S., F.S., and "Remarks," a column
for distances being provided when measurements are made between
staff stations, as in running vertical sections along a line. All level
72 ELEMENTARY SURVEYING
notes should read down the page, the notable exception occurring
in the American method of contouring and cross sectioning on
railways.
Level books should be simple and adapted to the immediate demands
of the work in hand and not complex or wasteful so as to comprehend
all the various work that may arise.
But let us draw up a page for each system, adding two columns for
Rise and Fall in the former, and one for Collimation in the latter. Now
let us move on to the benchmark on the wall of the "Spotted Dog"
(B.M. 50-0). The staffman already stands there with the foot of the staff
held exactly at the centre of the cross-bar, which is about a couple of
feet above the ground. He will find it easier in the open when he stands
behind the staff with the foot between his feet, holding it vertically
with a hand on each side, never covering the divisions with his fingers.
<
e.s.
i) (b) (<
4-24 (54-24) INTS|4-!4 FS
: L
3-64 ^'tflNTS
1)
4-02
(c
S2J
/u. "
. A
47 10
B.M -
^^ux^.cLJ*^^
*-^ ~ v <-***~v***w - ^ ,^ ' '^'^^'^^^yM^'^^^'^^Kn
BM50-0 A 48-98 B 48-60^*
T
FIG. 50
Fig. 50 shows the instrument levelled up at A, a convenient distance to
the east of the licensed premises. From here a reading 4-24 is taken
on the staff (a) still painfully held on B.M. 50*0. This is entered as a
backsight in both books, while the staffman moves to (b). The reduced
level of the B.M. is also recorded and a note as to its location in the
Remarks column.
(Added to 50-0 the B.S. of 4-24 gives the height of collimation
(54-24) shown above the level, and also booked in the Collimation
column of System (2).) The reading taken on the staff held at (b) is
an intermediate sight of 4-14, which is entered as such in the proper
columns. The readings at (a) and (b) suggest that the ground has risen
4-244-14, and 0-10 rise is entered in the appropriate column of
System (1), where, added to 50-0, the reduced level of the point (a) it
gives 50-10 as the reduced level of (b), which is entered in the column
provided. (In System (2), the intermediate sight is merely subtracted
from the collimation height of 54-24, giving the value 50-10, which is
booked in the column for reduced levels.) The staffman is waiting
at (c). Tell him that is a change point so that he can make a firm
footing for the staff. This is a foresight of 5-26, which is duly recorded.
In System (1), a fall of 5-26 4- 14 1-12 is observed and recorded, and
subtracted from 50-10 to give the recorded reduced level of 48-98.
(In System (2) the reduced level of (c) is found by merely subtracting
LEVELLING
73
5-^6 from the collimation height of 54-24 for the recorded reduced level
of 48-98 of this change point.)
Instruct the staffman to turn the face of the staff towards the position
you indicate as the second position B of the level. Set up the instru-
ment at B and level it carefully.
Now take a backsight on (c\ and check it before booking it as 3-64;
and above all enter it in the same line as 5-26 and 48-98. (In System (2)
a new collimation is established, and the backsight must be added to
the reduced level of (c) for the new collimation height, which is booked
as 52-62 in the column provided.) Direct the staffman to hold the staff
on the point (d). This is certainly an intermediate sight of 4-02, and is
entered as such in both books. In System (1) a fall from (c) to (d) of
4-023-64=0-38 is recorded and subtracted from 49-88 for the reduced
level of 48-60. (In System (2) the reading 4-02 is subtracted from the
new collimation height of 52-62 for this reduced level, which is recorded
as 48-60.) Direct the staffman to go to that mark on the step at the
church gate, as indicated by (e). It was a temporary B.M. of 47-12,
interpolated during the main drainage scheme. Record this foresight
of 5-52 in both books. In System (1) this shows a fall from (d) to (e) of
1-50 which, subtracted from 48-60, gives the reduced level on the
T.B.M. of 47-10 against 47-12. Excellent work for a first effort! In
System (2) the reading 5-52 is subtracted from 52-62 for the reduced
level 47- 10.
The error of 0-02 would represent fair work with an engineers' level, but
an error of 0-10 to 0-20 ft. might be expected with a sighting tube level-
tenths of feet only being read on the staff.
The notes of the line of levels run as in Fig. 50 are recorded on the
appended forms:
(1) Rise and Fall System
B.S.
Int. S.
F.S.
Rise
Fall
Reduced
Level
Remarks
4-24
50-00
B.M. "Spotted Dog"
4-14
0-10
50-10
P.M.
3-64
5-26
1-12
48-98
4-02
0-38'^
48-60
5-52
1-50
47-10
T.B.M. 47-12, Chuich
Gate
10-78
7-88
0-10
3-00
0-10
50-00
47-10
Fall 2-90
2-90 -2-90
74
ELEMENTARY SURVEYING
(2) Collimation System
B.S.
Collimation
M.S.
F.S.
Reduced
Level
Remarks
4-24
54-24
50-00
B.M. "Spotted Dog"
4-14
50-10
P.H.
3-64
52-62
5-26
48-98
4-02
48-60
5-52
47-10
T.B.M. 47-12, Church
Gate
7-88 10-78 50-00
7-88 47-10
Fall 2-90 - 2-90
Checking the Book. The figures at the bottoms of the columns are
the checks; two common to both systems and a third in the Rise and
Fall System:
(1) Diffs. of sums of RS.'s and F.S.'s
=(2) Diff. of first and last reduced levels
(3) Diffs. of sums of Rises and Falls.
These are merely checks on the arithmetic, and never on the levelling
work, though they have often raised the surveyor's spirits until he
discovers that a drastic mistake has occurred outside. On the other
hand, more than one line of levels has been run again unnecessarily
when the arithmetical check would have shown what a simple slip in
addition can do.
Choice of Systems. In the following points of comparison it must be
remembered that these refer to the booking and not to levelling
operations, which are identical:
(1) In the Rise and Fall System the remainder of the reduced levels
may depend upon the reduction of a single intermediate sight. But
there is a check upon the intermediates, whereas in the Collimation
system any intermediate sight may be wrongly reduced without affect-
ing the remainder of the levels. Age and habit are apt to exaggerate
the merits of the Rise and Fall System; and it is a matter of visualisation
as to whether rises and falls are more readily evident in a figure than
in the field.
(2) Also in the Rise and Fall System there is either one more addition
or subtraction in each reduction whenever intermediate sights are
taken, and thus there is a considerable saving in bookwork in the
Collimation system when numerous spot levels occur.
(3) Then the spcprid decimal place from backsights and foresights
LEVELLING 75
mu|t be carried through the intermediates in the Rise and Fall System
unless direct subtraction between backsights and foresights are made.
Whereas in the Collimation system the intermediates can be taken only
to the nearest tenth when desired, without giving thought to the back-
sights and foresights, which are necessarily read to the hundredth of
a foot.
(4) Finally, the Collimation System has the indisputable merit of
emphasising the relatively greater importance of backsights and fore-
sights in the field, but whether this system is more scientific, being
closely related to fundamental principles, is again a matter of opinion.
Levelling Operations. It would be unfair to dismiss the subject
without a word as to what all the fuss is about. Hence the following
summary of the applications of spirit levelling.
(1) Check Levels. If a sewage disposal scheme, or other works, is
under construction, it would be necessary to have numerous temporary
benchmarks, based upon the Ordnance datum, at convenient points
throughout the area. A main circuit is established and levels are run
carefully round, checking on the starting-point; cross lines are run
through the benchmarks in the middle of the area, closing on the outer
benchmarks of the system. Usually this is carried out by accurate or
precise spirit levelling.
(2) Flying Levels. Suppose that there had been no T.B.M. at the
Church Gate in Fig. 50, it would be necessary either to run levels
forward to the next Ordnance B.M., or back to the B.M. on the wall
of the Spotted Dog. Flying levels consist only of backsights and fore-
sights and are run solely to check the accuracy of the work.
(3) Section Levels. When a highway, railway, or other scheme is
projected, it is necessary to run levels along straight lines or around
curves for the purposes of preparing a longitudinal section from which
the gradients and earthwork volumes can be estimated. Cross sections
are also run in connection with roads and railways at right angles to
the longitudinal sections, and, similarly in connection with surveys for
reservoirs, etc. Sections require that the distance between the staff
stations shall be measured. These distances are best recorded in a
"Distance" column rather than in the remarks, though this is often
done in cross-sectioning.
(4) Spot Levels. Spot levels are intermediates taken in areas reserved
for building or the construction of public works. Sometimes contour
lines are interpolated between spot levels. In these, as in much high-
way and railway surveying, the surface levels are taken to the nearest
tenth of a foot.
LEVELLING DIFFICULTIES. The length of sight with a telescopic
levelling instrument should not exceed 5 chs. or 350 ft., and as far as
possible the lengths of the foresights should equal the lengths of the
backsights, either individually or in sum, in order that the small errors of
adjustment may not affect the accuracy of the work. When exceedingly
76 ELEMENTARY SURVEYING
long sights are necessary, as in sighting across a wide river, the
method of Reciprocal Levelling, as suggested on page 66, should be
resorted to, but preferably with the use of a target staff. The averages
of the differences of level as observed in each direction is taken as the
true difference of level, since this average eliminates instrumental errors
and the effects of the earth's curvature and atmospheric refraction.
The effect of curvature c is indicated by be in Fig. 43, where it is
evident that it increases the staff reading and thus makes very distant
points appear too low. The effect c is 8 inches per mile, varying as the
square of the distance. It is therefore about 0-01 ft. in 10 chs., a distance
at which no ordinary staff could be read directly. Refraction reduces
the effect of curvature, bending ab so that b is depressed r = } (be)
towards c. The value of r is really uncertain, as refraction becomes
very capricious near the horizontal. Anyway, the matter is largely
academic in spirit levelling, and the net correction c r is taken at
f(Z)) 2 ft., the distance D being in statute miles. Refraction is a very
important correction in trigonometrical levelling and astronomy.
In conclusion, there is bound to be something omitted, possibly a
difficulty that will be encountered the first time the level is taken into
the field. But the difficulties that arise in levelling are legion, and
could not be summarised in a book of this nature. Nevertheless a few
hints may be given among others.
(1) When reading near the top of the staff, ensure that it is truly
vertical by instructing the staffman to wave it gently to and fro towards
you so that you can record the lowest reading.
(2) When working up and down a steep hill, avoid very short tele-
scopic sights by setting up the level to the sides of the line and zigzag
thus so as to obtain as nearly as possible a balance of the total lengths
of backsights and foresights.
(3) When sighting the staff very near to a telescopic level, instruct
the staffman to hold a piece of paper against the staff as a target from
which the reading can be taken directly. A target improvised in this
way is necessary in taking very long sights, also in testing the adjust-
ments of the level.
(4) When a benchmark is considerably above the level, as under an
arch, invert the staff (foot on the B.M.) and record this (and regard it)
as a negative backsight or foresight, as the case may be.
(5) When a board fence crosses the line, drive a spike through to
support the staff on each side and regard the spike as a change point.
Also a lake of still water too wide to be sighted across can be regarded
as a single change point if pegs are driven flush with the water surface.
(Incidentally, this suggests a method of checking the accuracy of the
collimation adjustment of a level.)
(6) When a wall is encountered, drive pegs in the line on either side
and measure with the staff to the top of the wall, which is regarded
a change point.
LEVELLING
77
CLASS EXERCISES
3 da). Draw up the headings of a specimen page of the following level
books:
(a) Rise and Fall System; (b) Colllmation System.
The following readings were recorded in running a line of levels, the nearest
tenth of a foot being taken in the case of intermediate sights:
(B.M. 62-4) (B.M. 63-3)
3-12, 2-4, 1-8, 0-94; 2-84, 3-1, 3-6, 4-12
Reduce these in the system you prefer, stating the reasons for your choice.
(No error.) (G.S.)
5 (b). In taking the following readings with a dumpy level, the surveyor
started at a benchmark and returned to it, in order to check his work. He
took staff readings on A and B as points for temporary benchmarks in both
the outward and homeward directions.
Record and reduce the levels on a page of a level book, and indicate where
a mistake was made in reading the staff.
B.S.
Int.S.
F.S.
Reduced
Level
Remarks
2-34
76-42
B.M.
5-24
(Outwards)
4-62
9-63
8-52
Points
7-64
5-88
7-24
Point B
4-26
4-32
8-82
0-38
(Homewards)
5-44
10-89
6-88
10-17
Point B
12-66
7-92
Point A
1-47
76-42
B.M.
(Read 9-92 for last F.S. on A.)
(G.S.)
5 (r\ The following levels were taken along the bed of a water course.
Reduce the levels and rind the rates of inclination along the bed of the water
B.S.
Int.S.
F.S.
Reduced
Level
Distance
(ft.)
Remarks
2-43
37-43
B.M. (Mill Ho.)
8-86
Section A
7-64
180
6-46
7-22
230
4-70
420
3-34
540
Section B
0-82
38-28
B.M. (Culvert)
(1 in 148; 1 in 119; 1 in 108; 1 in 88.)
(G.S.)
78 ELEMENTARY SURVEYING
5 (d). The following staff readings were taken in levelling down a hill
between benchmarks 76-4 and 43-8:
(B.M. 76-4) 3-44, 6-78, 12-44; 2-06, 5-66, 11-74;
1-04, 3-68, 7-22, 9-16, 12-88 (B.M. 43-8.)
Interpret these notes in a level book of the "Collimation" System. (U.LJ
5 (e). A surveyor runs flying levels down a hill from a temporary bench-
mark (162*40) to an Ordnance B.M. (123*4), recording his staff readings as
follows:
1*62 11-44 12-68 12*80 8*64
2-86 0-82 1-24
Prepare a page of a level book, and on it record and reduce the above
readings. (G.S.)
(O.B.M. reduces to 123'38.)
FIELD EXERCISES
Problem 5 (a). The points marked A, B, C, D, etc., around the (specified)
building are selected as temporary benchmarks, the assumed reduced level
of A being 100-0. Determine the reduced levels of these and find the error
in closing the circuit on A.
Equipment: Level on tripod, staff and chalk.
Problem 5 (b). Run the levels for a longitudinal section between the
stations indicated by the range-poles A and B.
A convenient B.M. ( ) is ...
Equipment: Level on tripod, staff, chain, and arrows.
Problem 5 (c). The pickets A and B indicate the direction for a proposed
drain, and surface levels at 50 ft. intervals are required. Submit these on an
appropriate form, and check the book.
Equipment: as in 5 (b).
Problem 5 (d). Find the reduced levels of the survey which is being made
by Group ( ).
Equipment: as in 5 (a).
Problem 5 (e). Test the accuracy of adjustment of the assigned level by
the Reciprocal Method.
Equipment: Level on tripod, staff, two pegs, and a mallet.
ORIGINAL PROBLEMS
CHAPTER VI
ANGULAR LEVELLING
In the preamble to Chapter V it was stated that the methods of angular
levelling are based on Inverse Polar Co-ordinates, though funda-
mentally they are dependent upon a horizontal line, such as AB, which
is determined by gravitational methods in which the plumb-line may
take the form of a weighted sector or the guise of a spirit level.
The term inverse polar co-ordinates is coined somewhat loosely, for
both in trigonometrical and tacheometrical levelling, the relation
between the vertical height H and the horizontal distance D is simply
H= >.tan. a (1)
where a is the vertical angle of elevation if above the horizontal sight
line AB, or of depression if below AB. "Acclivity" and "declivity" are
terms used synonymously with these. (Fig. 51.)
In angular levelling the horizontal
distance AB is determined by tri-
angulation, being found graphically
by the intersection of rays on the
plane table or by photo-inter-
sections in photogrammetry. In
tacheometry, the height // is found
from the intercept observed on a
staff at B, but directly or indirectly
the horizontal distance D is involved.
(a) Base known or accessible. Fig. 51 shows the case of an accessible
base, and Fig, 52 the case of triangulation, the right-hand lines corre-
sponding with those in Fig. 51.
It is evident that the method requires some instrument for measuring
vertical angles, and this may be one of the numerous forms of clino-
meters, the sextant, or the theodolite, the accuracy thus increasing
from low to high.
If a is 45 in (1), H= D, since tan 45- 1.
This fixed relation is embodied in the Apecometer, which is a simple
instrument for measuring heights of trees and buildings, the bases of
which are accessible to direct linear measurement. This little instru-
ment is essentially an optical square which reflects at 45 instead of at
90, being held edgewise in sighting. The observer sights a point near
the foot of the object and moves along AB until he finds a point X
from which the top C can also be seen. Then XB=H, the height h of
the eye being afterwards added.
The Brandis "Hypsometer" is really a clinometer for finding heights
generally, various reducing data being inscribed on the instrument
79
FIG. 51
80
ELEMENTARY SURVEYING
The "Dendrometer" is another form of instrument used in connection
with a 10-ft. rod. Some of these devices are exceedingly handy in
forestry and preliminary survey. Road tracers are clinometers on
stands used in connection with sighting targets, and very large clino-
meters are mounted on tripods, various scales being engraved on the
plumbing sector.
Also various improvised forms could be suggested, as, for example,
the principle of Fig. 24. Here the N.E. quadrant could be a frame,
levelled with a bubble, and a sighting-arm could be pivoted at the
centre 0, so that vertical angles could be read as such, or their slope
ratios, on the outer edges of the square. This is the ... After all,
we are not finding the height of the Tower of Babel.
Failing a better instrument, we have our improvised clinometer
(page 17).
Fig. 52 illustrates the case of triangulation.
With the theodolite, the horizontal distance AB (or DB) is calculated
from the observed angles 8 and p, and the base AD by (Angular
Co-ordinates). In the case of the plane table B is fixed by intersecting
rays drawn first from A and then from
B t the end stations of a measured base.
Since the horizontal distance AB (or
DB) is known or plotted, as the case
may be, the height H can be found as
above, graphically or by calculation,
even though B is inaccessible as it
so often is. The height is calculated
in the case of the theodolite, and for
great distances curvature and refraction
are important considerations. H may
also be calculated in plane tabling;
but since this method is graphical, the height may be found by setting
off a right angle at B, and constructing the angle 6 at A, so as to
determine the point C, EC being the height H to scale. In Practical
Geometry, this is known as rabatting the triangle ABC into the horizon-
tal plane or, in other words, Fig. 52 is seen, not as an elevation, but as a
plan, with AB the horizontal projection of AC. Laussedat, the pioneer
of photogrammetry (1854), introduced this graphical process in deter-
mining the heights of points from pairs of photographs taken from
the ends of bases, such as AD. Our few principles go a long way.
Mention must be made of the India Pattern Clinometer, which is
specially adapted to work with the plane table, the board of which
serves as a base for the instrument. A pin-hole sight is used in con-
junction with a sighting index, which can be set to the observed vertical
angles or their tangents, sometimes by means of a rack and pinion
movement.
(b) Base Inaccessible. Frequently it is necessary to determine the
ANGULAR LEVELLING
81
eleyation of a point, the base of which is inaccessible, and it would be
inexpedient to resort to triangulation, as in Fig. 52. In this case it
is necessary to measure a base AD of length L in the vertical plane of
the elevated point C and the instrument stations A and D from which
the two angles and 9 are observed. Consider Fig. 53, the general
Fio. 53
case in which the slope of the ground is appreciable, giving instrumental
heights /2 A and h D on a staff held as near as possible to the base of the
object, with h^h h D algebraically.
Then for the height of the point C above the instrument at A:
AB=H A cot 0.
But
L=ABBD=H A cot 7/ D cot 9.
// A ^=// D - -h, and
L h cot 9
//A = -
cot 6 cot 9
If the ground is level, or nearly so, /*= 0, and H A =-
(2)
(2d)
cot cot 9
and if the instrument is divided with slope ratios, r horizontally to
1 vertically, these are co-tangents, and (2d) becomes // A where
'A ^B
r A and r B are the ratios for the angles observed at A and B in Fig. 53.
Often, however, the tangents of the angles of slope are shown; otherwise
gradients, 1 vertically in r horizontally. Also, if the point S, the staff
station, is of known elevation above datum, the reduced level will
be H A +d.
~Abney Level. Mention must be made of what is possibly, the best
known of all clinometers, the Abney level; an instrument which extends
the principle of the reflecting spirit level to the measurement of vertical
angles, the primary function of all clinometers.
The sighting tube in Fig. 54 is square in section, and is provided
with a pin-hole sight on the right, and, axial with this, the edge of a
82 ELEMENTARY SURVEYING
sighting plate at the object end.
Some patterns are telescopic, but
the innovation is questionable, as
with any hand instrument, except
the sextant. Inside the tube is a
mirror, the upper edge of which
leans towards the object end, the
FIG. 54 mirror being half-silvered, with the
line of division either vertical, or coincident with the line of vision on the
horizontal edge of the silver. In the reflecting spirit level, the bubble
tube is fixed on the top of the sighting tube, being embedded in an
open recess, so that the bubble can be read by reflection, its image
in contact with the horizontal sight on a levelling staff. This hand
level is used extensively in route contouring in America, being strapped,
or otherwise attached, to the top of a 5-ft. staff, known as a "Jacob."
In the Abney level, however, the bubble tube is carried on an axis
which forms the centre of a graduated arc fixed to the sighting tube.
An index arm is also fixed to this axis, and the bubble tube and arm
are turned by the little wheel in the front of the figure. Thus, for any
inclination of the line of sight, the bubble is moved so as to give the
reflected coincidence that corresponds to its middle position, and the
vertical angle of the observed point is read on the vernier of the index
arm. The arc is also provided with graduations giving the tangents
of these angles, or gradients, or the corrections to be made in chaining
slopes. In the latter connection, it is usually sufficient to sight the
eyes of an assistant of one's own stature. Abney levels can be used
as they lie on the boards of plane tables, also in similar connections
in various mechanical experiments.
An objection to clinometers and other hand instruments is the
difficulty of keeping them steady when taking observations. In this
respect, it is well to note that a "bipod" of surprising steadiness can
be improvised by inserting the knob end of a walking-stick in the
left-hand jacket pocket, and gripping the stick at the height of the eye,
with the right hand, the thumb and one or two fingers supporting
the instrument.
*BAROMETRY. Although barometrical levelling is outside the scope
of Elementary Surveying, this chapter affords the temptation of
introducing the third mode of levelling to those who may proceed
further in the subject, with a view to engineering, geography, or aerial
navigation. In 1647, Pascal demonstrated that the variation in the
density of the atmosphere with changes in altitude might be applied
to the determination of heights; and this was made possible by Torri-
celli's invention of the mercurial barometer, the readings of which
are found to decrease in geometrical progression as the altitudes
increase in arithmetical progression. Thus, the barometer and the
boiling-point thermometer (also alias the Hypsometer) are strictly the
ANGULAR LEVELLING
83
preserves of Physics; and more than one experienced surveyor considers
this to be the proper place. Eminent physicists prepared tables with
different initial assumptions, and the surveyors were not infrequently
bewildered with apparently confusing data and corrections. Often the
wrong tables with the right instrument, on top of no knowledge of
physics. The barometer as a meteorological instrument is not the
barometer as a surveying instrument.
The portable form of barometer is known as the aneroid, which
merely signifies "no liquid."
Possibly you may have heard the story of the fair young examinee,
who was asked how she would find the height of a tower if she had
a pocket aneroid. Her answer was to the effect that she would unpick
her jumper, let down the "thing like a watch," and then measure the
length of wool paid out with her ruler. Whatever the examiner
thought, she was a born surveyor, for she was evidently aware that
the wool would have broken had the instrument been sufficiently large
and sensitive to respond to a difference in elevation of (say) 180 ft.
Also, she may have seen a similar method used in transferring levels
down the shaft of a mine, where tenths of feet matter, even if they do
not in travellers' stories.
Incidentally, the altimeter used in connection with air survey cameras
is a form of aneroid, but, being small, will not give absolute heights
to within 200 ft. Usually a statoscope, or differential aneroid, is used in
addition, so that the variations can be more accurately determined.
The surveying aneroid in itself is an ingenious piece of work; and
its idiosyncrasies are no fault of the maker. Household barometers
are meteorological instruments, and often an excellent solution to the
problems of presents, or prizes at sports meetings. But in the field a
surveying model, never less than 4 in., should be used, and always
with respect for the instructions supplied by the maker. For instance,
the working range should always be taken to about 2,000 ft. less than
the limit engraved on the fixed altitude scale. The aneroid is indispens-
able in exploratory and pioneer work, and good results will follow if
the instrument is used with care and understanding.
The principle of the instrument, as given in respect to the diagram
of Fig. 55, is exceedingly simple.
FIG. 55
The aneroid consists of a circular metal case C with a glass cover c,
the base plate carrying the entire mechanism and the cover the dial.
Fixed to the base plate B is the all-important vacuum chamber V,
84 ELEMENTARY SURVEYING
which is circular and corrugated, and constructed of German silf/er.
The walls of this chamber are under 10 to 15 Ib. per sq. in. of suction,
and would immediately collapse under the outside pressure except for
the material support of the mainspring M, which is fixed to the bridge-
piece m. Now variations in the outside atmospheric pressure are as
tiny weights in the pans of a delicate balance, and these induce pulsa-
tions in the vacuum, which are accompanied by movements of the
mainspring. These movements are transmitted and magnified by means
of the compensated lever L, which transmits them to the crank system /.
A second crank of this system / transmits them to the chain s, which
turns the drum D and the indicator /, the motion being resisted by the
hairspring d, keeping the chain s taut. The pulsations are thus finally
read as altitudes (ft.) and pressures (in.) on the dial A.
The only correction that has to be considered with the aneroid is
for the temperature of the intermediate air. Compensation for tempera-
ture refers to the instrument and not to this correction. Care must be
taken to ascertain the initial temperature for which the instrument is
divided; say, 32 F. or 50 F.
The peculiarities of the instrument should be studied, preferably by
comparison with a standard mercurial pattern; also the results should
be compared with those of a boiling-point thermometer for absolute
altitudes.
Possibly you have observed the tapping of the glass of the barometer
in the vestibule of an hotel. This is not a religious rite, if carried out
with the solemnity of one. It is merely to eliminate "stiction," which
is statical friction with a following here the mechanism ending in the
spindle of the indicating needle.
In addition, there is a "lag" effect, analogous to that which occurs
in other connections, the instrument being sluggish in responding to
a descent after an ascent. Hence, when a scries of journeys is made
uphill and downhill, the greater importance should be attached to the
mean value reduced from the ascents. Surveyors often work to the
height (of barometer) in inches, as though it were the mercurial form,
reducing the altitudes either by formulae or by means of tables. On
returning from the peak station B to the base station A, they can
deduce the probable height in inches at A at the instant the reading
was taken at B. It is possible to make an approximate correction by
comparison of the height and altitude scales.
CLASS EXERCISES
6 (a). You are required to find the height of the bottom of a tank on a
water tower, which is surrounded by a high hedge about 25 ft. from the tower.
The tank is fitted with a gauge and the zero (0) of this is level with the bottom
of the tank.
The following vertical angles were read with a clinometer at A and B
respectively, A t B t and being in the same vertical plane:
13 20
ANGULAR LEVELLING 85
In each case the eye was 5 ft. above A and B, between which the ground
was level. AB measured 185 ft. and the reduced level of A was 64-6.
Determine the height of the bottom of the tank above Ordnance datum.
(186-41 ft.) (G.S.)
*6 (b). During a plane table survey, sights were taken to points A, U, C,
D, and E with a clinometer, which was 4-5 ft. above the table station O, a
peg at a reduced level of 155-5. The horizontal distances scaled from O to
the observed stations were as follows:
A B C D E
760' 420' 315' 880' 1260'
+7 +14i +16 -10 + 12*
Determine graphically or otherwise the reduced levels of A, B, C, /), and E.
(249-0; 264-3; 245-9; 0-6; 336-5 ft.).
*6 (c). Outline three essentially different modes of levelling, one applicable
to each of the following:
(a) Small differences of reduced level;
(b) Medium elevation;
(c) Great altitude.
6 (d). How can an explorer in unknown country obtain rough determina-
tions of absolute heights? Explain fully, showing the limitations of the
methods you suggest.
*6 (e). Draw a sectional view of an aneroid barometer, explaining carefully
how the instrument functions in determining altitudes.
FIELD EXERCISES
Problem 6 (a). Determine the heights of the accessible points indicated on
the (specified) building.
Equipment: Clinometer, chain, and arrows.
Problem 6 (b). Determine the height of the spire on the (specified) building.
Equipment: Clinometer, chain, arrows, and set of pickets.
Problem 6 (c). Determine the difference in elevation of the two (specified)
points, P and Q.
Equipment: as in 6 (b).
Problem 6 (d). Supply Group . . . with the elevations of the stations of the
plane table survey they have in hand.
Equipment: Clinometer, preferably India pattern or Abney level.
"Problem 6 (e). Determine the height of (specified) hill with the aneroid,
(during excursion or field class in the country).
ORIGINAL PROBLEMS
CHAPTER VII
THE COMPASS
The compass may be defined as an instrument in which a magnetic
needle assumes a more or less definite line of reference from which
angular direction lines known as bearings can be measured.
The origin of the compass is lost in antiquity, to adopt the common
phrase. All that is known is that the mariner's compass was used by
the Italians or the Portuguese in the twelfth century A.D., and that there
are indications that it was known in China in the eighth century B.C.
Compasses are made in at least fifty-seven varieties, ranging from the
tiny charms on watch-chains to the most elaborate forms of mining
dials, excluding, of course, the types used in navigation.
As surveying instruments, compasses are made in three forms at the
present time: (1) Occasional Compasses, as found in pocket dials,
trough compasses, etc., incidental to the plane table, theodolite, and
even old pattern levels. (2) Reconnaissance Compasses, including the
service forms of luminous compasses, and the prismatic non-luminous
form, as used extensively on preliminary and route surveys. (3) Sur-
veying Compasses, usually mounted, and fitted with a pair of vertical
sights. In America, the last class stands as designated. These instru-
ments were used extensively on land surveys, and had the merit that
their accuracy was consistent with that of chaining. Harmony between
measurements, both angular and linear, is essential in surveying;
but the compass brought harmony of another kind, giving us "Dixie,"
when those two English surveyors, Mason and Dixon ran the disputed
boundary between Maryland and Pennsylvania in 1767. In England
the surveying compass faded out as the "compass circumferentor,"
on the advent of the small theodolite, but reappeared with some slight
alterations as the "mining dial," the compass needle still holding a
prominent place.
The prismatic compass is the type best known to British surveyors,
and this will be described in detail in order to emphasise the points
essential to a good compass.
^Prismatic Compass. This instrument was invented in 1814 by Captain
Kater, whose famous pendulum is a source of anxiety in most physical
laboratories. It is the most convenient instrument for rapid traverses,
particularly in dense forests and jungles. The characteristic feature
is the prism reading, which enables the surveyor to observe bearings
without resting his compass on the ground or a wall, or deputising
someone to read the divisions for him.
The prismatic compass consists of four main parts: (1) The Compass
Box, (2) the Dial, (3) the Prism, and (4) the Window.
86
THE COMPASS
87
{!) The box, a metal case, 2| in. to 6 in. diameter, carries the needle
pivot or bearing at the centre of its base. On the rim is fitted the
prism P and, diametrically
opposite, the vertical sighting
window V. Under the win-
dow at C, a pin is inserted
to actuate a light check spring
B', which, touching the dial,
damps its oscillations (or fixes
the reading with a very doubt-
ful degree of accuracy). A
glass cover is fitted to the box,
and a metal lid is provided
to protect this glass when
the instrument is out of Fio. 56
use.
(By the way, "Boxing the Compass" does not refer to the last step;
but means, in nautical language, calling the "32 points," or rhumbs,
in order from north by way of east.)
(2) The dial, which is carried by the magnetic needle A, is made of
card in small instruments and of aluminium in larger patterns. It is
figured from to 360 in the clockwise direction, but has its numbers
reversed (as seen in a looking-glass) and advanced 180 so that bearings
can be read directly through the prism P, as though they appeared
at the forward or window end of the dial. (Remember forward end of
the needle for forward bearings.) The needle is usually mounted with
a bearing centre B of agate or chrysolite. (This can wear the pivot and
impair accuracy if the needle is not raised by its Lift during transport.)
(3) The prism is cut to 45 on one face, and to 90 on the other two faces,
which are worked to a convex surface so as also to give magnification
of the numbers on the dial. The prism box is provided with a sight
slit S, and is hinged to a projection, which, for focusing, can be slid
up and down by means of a thumb-nail stud T. The hinge H is fitted
so that the prism box can be folded back for compactness when out
of use. Sometimes a ring is fitted under the prism box for attaching
the instrument to the person as a precaution against accidents.
(4) The window consists of an open frame fitted with a central
vertical hair F, which, in sighting, is used in conjunction with the
sight slit, S. The window-frame is hinged, and, when turned down
for compactness, lifts the needle A from its pivot, the base pressing
down the outer end of the lifting lever L.
Additional parts may include (a) the mirror M for sighting points
considerably above or below the horizon; (b) sunshades, which may
be placed in front of the mirror in solar observations; and (c) a tripod,
which is desirable with the heavier patterns.
USE. (1) Remove the cover and open out the prism and window,
88 ELEMENTARY SURVEYING
and, holding the compass as level as possible, focus the prism by raiding
or lowering its case until the divisions appear sharp and clear. If
necessary, lower the needle on to its pivot. (2) Holding the compass
box with the thumb under the prism at T and the forefinger near the
stud C, sight through the slit S and the hair-line V at the object
or station, lowering the eye to read the required bearing as soon as
the dial comes to rest naturally (or by cautiously damping its swings
by pressing the stud C). v -
The bearing read will be a "forward" bearing and normally a "whole
circle" bearing, a clockwise angle between and 360.
Military Compasses. Although these are made in various forms, the
service patterns are usually of the prismatic class, the box being about
2 in. diameter, provided with a finger-ring under the prism. The more
conspicuous differences from the larger pattern just described are as
follows: (1) An external ring, divided to 360 in the counter-clockwise
direction, is fitted around the circumference of the box. (2) A movable
glass cover, provided with a luminous patch (#), is fitted in a milled
ring over the box and secured by means of a clamping screw. (3) Two
sets of dial divisions, both figured to 360, the inner set being normal
for direct readings; i.e. without the use of the prism. On top of the
dial is a luminous pointer (b), which is used in connection with the
patch (a). (4) The lid carries a circular window, placed eccentrically.
Down this window is scribed a vertical line which serves as a sighting
vane in daylight operations. At the extremities of this line are two
patches (c), which are also used in connection with (a) and (c) in
night operations.
The walking-stick "bipod" described on page 82 (Chapter VI) is
also useful with this instrument.
Now if a compass were supported in gimbals and constantly in view,
as on a ship, it would be possible for the observer to keep on a given
bearing from A to B. But on foot or horseback this is impossible,
and if B is not a visible landmark or station, the observer will soon
find himself moving in a direction parallel to AB. Hence, in marching
on merely a given bearing at night it is often necessary to work to a
selected series of stars, or, failing this, to work in conjunction with
two men by mutual alignment in order to maintain the direction. Some
knowledge as to the identity of conspicuous stars and their apparent
positions and movement is obviously necessary.
Some advice might be found in a story of the last war, when an
old lady was informed that the officers were not holding an egg-and-
spoon race, as she surmised, but were undergoing instruction in
marching with the compass.
The advantages of the compass as a surveying instrument may be
summarised generally, its simplicity and portability being recognised.
(1) Running rapid traverses without regard to preceding lines, a
more or less fixed line of reference for forward bearings existing at
THE COMPASS
'89
all ^tations. (2) Running lines through forests where obstructions
impeding the line of sight are more easily overcome than with other
instruments. (3) Facility of fixing positions by resection on two or
three points, already mapped. (4) Retracing lines which were run with
the compass before the introduction of the theodolite; an application
of the surveying compass in the U.S.A. (5) Facility with which bear-
ings lend themselves to the use of latitudes and departures, particularly
if the circle is divided in the "quadrant system."
Its disadvantages are (1) that lines of great length cannot be run
with great accuracy unaided by a telescope; (2) that at best the
method is not precise, since at best bearings cannot be read within
five to ten minutes (of arc) under the most favourable conditions; and
(3) that the needle is unreliable, and that local attraction may render
rapid work impracticable or impossible.
II. BEARINGS
The axis of the compass needle serves as a reference line known as
a magnetic meridian, or n and s line. This line differs from the
true meridian, as would be given by a line between the observer O and
the north pole, by a horizontal angle known as the magnetic declination,
of which more will be said later.
The true meridian NS is shown faintly and the magnetic meridian
boldly as ns in Fig. 57, where the angle NOnis the declination, being
to the west, as it is in this country to-day.
But the Pole Star actually rotates
about the north pole, making an angle
with the earth's centre of about 1
at the present time, so that its direction
lixes the true north only when it is
vertically above or below the pole,
or at upper or lower transit, in astro-
nomical language.
In general, bearings are horizontal
angles measured from the north and
south points of reference meridians,
and may be true or magnetic bearings
accordingly.
Now we encounter the true British
profusion of terminology; but once
and for all let us classify the two
modes of observing bearings as
"Azimuths" and "Bearings," but
keeping in mind the synonymous uses of the terms.
(a) Whole Circle Bearings (W.C.B.). or simply Azimuths, are angles
measured clockwise from the north point from to 360. Most
geographical and army and air force text-books define these simply as
90
ELEMENTARY SURVEYING
bearings, but not altogether without reason. The graduated circles of
British theodolites and compasses are divided in the Whole Circle System,
to 360* clockwise, which are read directly, but have to be reduced
to the angles we have adopted as bearings.
(The true meridian MS may be forgotten a while, since we are working
with the compass at present, and booking magnetic bearings.)
Thus the azimuth of OA is a; of OB, P; of OC, Y; of OD, 8; as simply
read on the circle, being respectively 30, 140, 230, and 300 in Fig. 57.
(b) Reduced Bearings (R.B.), nautical bearings, or simply Bearings,
are horizontal angles measured from the north and south point, in
either direction from to 90, the angular value being preceded by
the initial letter N or S, and followed by the terminal letter E or W.
They are read directly on circles divided in the Quadrant System,
to 90 to to 90 to 0, which obviates reduction in later methods.
Otherwise they are readily reduced from observed whole circle bear-
ings as follows:
(a) a p Y S
(e.g.) 30 140 230 300
(b) N.(a)E. S.(180-p)E. S.( Y -180)W. N.(360-S)W.
(e.g.) N.30E. S.40E. S.50W. N.60W.
(2) Although azimuths are read directly, it is now desirable to work
with bearings, not only because back bearings are readily evident, but
that azimuths have little future in front of them until they are converted
to bearings.
Every line has two bearings: a forward bearing and a backward
bearing, the forward bearing being understood in plotting. Thus the
forward bearing of a line AB, as suggested by progress from A to B,
is merely P or N. p E., and the back bearing, as suggested by sighting
from B to A, is 180+ p, or S. p W. Hence, with azimuths we merely
add or subtract angles exceeding 90 to or from 180 (or 360), while
with bearings we merely interchange the initial letters (N. and S.) and
the terminal letters (E. and W.).
Thus, in Fig. 58, the forward bearing of AB is N. p E., and its back
bearing S. p W.; the forward bearing of BC is S. Y E., and its back
bearing N. Y W. . . . with the
forward bearing of DA, N. a W,
and its back bearing S. a E.
Possibly you will appreciate
the process better if you write
30 for p, 40 for Y, 50 for 8,
and 60 for a.
Also, the forward azimuth of
AB is simply p and its back
bearing 180+ p; the forward
azimuth of BC (180- Y ) and
Fio. 58 its tack azimuth (360 y) . , .
THE COMPASS 91
witfy the forward azimuth of DA (360 a) and its back azimuth
(180- a), though these would be read merely as 30 and 210, 140
and 320 . . . and 300 and 120 on ordinary whole circle divisions.
The conjoint use of forward and backward bearings is an important
artifice known as working "fixed needle" in the presence of magnetic
disturbances.
(3) MAGNETIC DECLINATION. The magnetic declination, the hori-
zontal angle between the magnetic and the true meridian at any place,
varies at different times and at different points on the earth's surface.
In physics, lines of equal declination on maps are known as Isogonic
Lines, the lines of zero declination being Agonic Lines. The term
expresses the fact that magnetic and true meridians are not coincident,
and imply some magnitude at any given date; but in military surveying
and mathematical geography, the term "variation" is often used
exclusively and synonymously with the term declination. This is
incorrect, since the magnitude at any given place and date is itself
subject to variations, or changes, which arc of the following three
kinds: (a) Secular, (b) Annual, and (c) Diurnal.
(a) Secular variations occur with the lapse of years, centuries
revealing that the motion is periodic in character, something like that of
a pendulum impelled to oscillate in a complex vibration. Thus, at
Greenwich, the declination was 11 36' East in 1580; in 1663 (three
years before the Annus Mirabilis) it was zero; and in 1818 it reached
its most westerly value of 25 41 '. Since then it has decreased steadily
with an increasing annual movement, and is now 10 45' W. (1940).
Its value for any year can be found from some reference work, such
as Whitakefs Almanack.
{b) Annual variations are cyclical changes in which the year is the
period, the variation being greatest at springtime, decreasing to mid-
summer, and then increasing during the following nine months. At
most places it amounts to less than a minute, and is therefore of secondary
importance. Annual variation is totally different from the progressive
angular change due to secular variation.
(c) Diurnal variations are the more or less regular changes in the
needle from hour to hour, leading to a total difference of ten minutes
in any, one day near London. This variation differs for different
localities and for different seasons of the year, being less in winter
than in summer, when it may amount to 15' at places. The cause is
attributed to the influence of sunlight.
There arc also irregular fluctuations which seem to coincide with
the appearance of the aurora borealis, earthquakes, and volcanic
eruptions, the needle becoming extremely capricious. Hence the mine
surveyor relies upon the notices issued as to magnetic storms by the
appropriate department of the Royal Observatory.
(4) LOCAL ATTRACTION. There are also disturbances of the magnetic
needle which can be attributed to Acts of Man. Local Attraction
92 ELEMENTARY SURVEYING
(facetiously known as the feminine of magnetic interference) denotes
the influence which renders compass bearings inaccurate in the neigh-
bourhood of certain bodies, particularly iron, steel, and certain iron
ores, or even nickle, chromium, and manganese.
Thus disguised steel spectacles, keys (cigarette cases), and knives
may cause trouble, and the unassuming chain and arrows are not
always above suspicion. Also, steel helmets and box-respirators have
been known to have been overlooked; and, by the way, a well-meant
cleaning of the glass cover may electrify it so as to attract the needle.
Outside these avoidable sources there are enough fixed sources to
fill a catalogue. Fences, manhole covers, railway metals, trolley wires,
steel structures, etc., in view; and, unseen, underground pipes, ironwork,
etc., etc.
By this time you have doubtless agreed that the compass is best
suited to exploratory work, and the safest place for it is a forest,
desert, or jungle. On the contrary, local attraction can be "bypassed,"
and this a mere detail of the mine surveyor's work.
Suppose we return to Station A of Fig. 58. Here we observe the
forward bearing p of AB as N. 30 E. (30). Next we proceed to B.
Here we find the back bearing, BA, is also p, as S. 30 W. (210).
Hence it is fairly safe to assume that neither A nor B is under magnetic
influence, and with confidence we take the forward bearing r of BC,
reading S. 40 E. (or 140). Then we proceed to C, and, lo! the back
bearing of CB, is Y ', not N. 40 W. (320), but N. 45W. (315).
Knowing that B was immune, C is suspected, local attraction causing
the needle to assume the dotted position n^^
Anyway, let us take the forward bearing 8' of CD (which we believe
should be S) as S. 50 W. (230), even though we record S. 45 W. (225).
Also, let us proceed to D. Here we observe a back bearing, S", as
N. 55 E. (55), and suspect that D is also unduly influenced. But here
we are near to our starting-point, A, which we know to be immune.
Hence we take a forward bearing, DA, of ', and this is N. 55W.
(305 C ). Therefore we hurry to A, and observe a, the back bearing AD,
as S. 60 E. (120).
Now we review our notes and see that our false reference meridian
lies 5 to the west at Z>, and 5 to the east at C.
Also it is evident that the exterior angle at C=
and that the interior angle at C is this value subtracted from 360,
or 360 -(180 +40 +50 )-360 -(180 +45 +45)-90 . But the
interior angle at D is simply (60 +50 )-(55 +55)-110 .
In fact, it would not have mattered if magnetic interference had
existed at all the stations. The record of forward and back bearings
would have enabled us to find the true interior angles of the polygon
with just one geometrical defect. The polygon would be orthomorphic
(correct shape), but it would be displaced on the drawing-paper by
THE COMPASS 93
the error in the magnetic meridian we assume for our first station.
Hence, it is desirable that one station should be unaffected, and this
can be ascertained by taking a bearing at an intermediate point in
a line and noting if this agrees with one of the end bearings.
This process is known as working "fixed needle." Free need^
traversing is the normal method by forward bearings. No regard is
thus taken of back bearings, so that each line is independent; but if
local attraction exists, the configuration of the traverse will be incorrect.
Finally, the sum of the interior angles should be equal to twice the
number N of right angles as the figure has sides, less four right angles,
or, algebraically, (2N 4)90. Naturally you will seldom obtain this
sum, for, apart from local attraction, there are those natural ailments,
errors of observation; and the error of closure, as it is called, may be
from (J to 1)\/N". degrees difference from (2N 4)90, according to
the size and quality of the compass.
A simple polygon should be traversed with a compass influenced
by suspended keys, or, better, the shadow of a steel helmet, if privileged
to wear one.
III. TRAVERSING
A good compass would have been exceedingly welcome when we
were running the open traverse of the stream in Fig. 19, or fixing the
traverse angles around the pond in Fig. 20. This would have obviated
the use of those terrible ties, essential to fixing directions when only
the chain is at hand. Also the field notes would have been simplified,
since the mere entry "N. 12 E.," etc., would have superseded the
entries relative to the lengths and positions of the angle ties. At the
same time, it is always desirable in all traverses to keep a tabular
record, showing Line, Length, Bearing, with further columns for future
calculations if likely to be required.
However, an illustrative example of the application of Polar Co-
ordinates, ihe fourth principle of surveying, is desirable.
Now traverse surveys of quiet country lanes are very interesting,
particularly if they are run between definite landmarks shown on, say,
the 25-in. Ordnance sheets.
Starting at a point, or station A on the roadside, a sight is taken
on a distant point B 9 a heap of stones, or a mark, and the forward
bearing of AB is observed. Then the distance AB is carefully paced,
aided desirably by that useful present, the passometer. (Notes as to
buildings, width of road, etc., are jotted down.) Also, if there is a
prominent landmark, say, a church spire, a bearing should be taken to
this, in order to serve as a check on the work.
From B, another station C is seen, possibly at a definite point near
the bend of the road. The bearing of BC is observed, and the distance
BC is paced, notes with estimated distances to objects being recorded
incidentally. Possibly from B, the church spire may again be visible,
94
ELEMENTARY SURVEYING
FIG. 59
but, if not, it may be sighted from C. So the open traverse is made,
until the end F is reached, which might be a guide-post on the grass
verge near the cross-roads.
En route, various inaccessible objects, such as hill-tops, might be
located by "intersections," or Angular Co-ordinates, the third principle
of surveying. If a clinometer were at hand, the heights of conspicuous
points might be found by the method outlined in Chapter V.
A pocket prismatic compass has been in view in this traverse. If
a larger pattern were available the distances might be measured with
a convenient length of R.E. tracing tape. This is fairly safe in traffic.
But, of course, chaining thus makes it a two-man job.
Now the explorer actually does the same thing, measuring distances
by time on foot, or horseback, etc. His distances will be very much
greater, and the scale of plotting very much smaller; but the principles
are the same. In fact, Tom Sawyer would see our prosaic country lane
as a rough mountain valley, the little river, the Mississippi, the church,
the Grand Canyon, and the twin hills as Council Bluffs.
But we are anxious to see how our first effort works out on paper,
plotting it, say, on the scale of 1 : 2,500. The work will be simplified
if we summarise our notes in tabular form, as follows:
Line
Length
fr
Compass
Bearing
Station
Notes
AB
930
N. 85 E.
A
St. Mary Ch., N. 52 E.
BC
755
S. 66 E.
B
CD
750
N. 64 E.
C
St. Mary Ch., N. 27 W.
DE
623
N. 62 E.
D
High Jinks, S. 54 E.
EF
1052
N. 83 E.
E
H.J., S. 19 E.; L.J., S. 43
E.
F
Low Jinks, S. 14 W.
Adjustment of Traverse Surveys. Although the following process
more particularly applies to closed traverse polygons, it is also used
THE COMPASS
95
in correcting open traverses, when these begin and end at points
accurately surveyed, as in the case of the triangulation stations of a
main survey, or definite points on the 6-in. or other Ordnance sheets.
Rough and medium grade compass traverses are adjusted graphically,
but those made with accurate surveying compass are adjusted arith-
metically, as is usually the case with the theodolite, the correction
being carried out through the rectangular co-ordinates, the latitudes,
and departures of the lines.
There is only one rational method of graphical adjustment, and this
is based upon the method devised by the celebrated mathematician,
Nathaniel Bowditch (1807). This requires that the correction to each
traverse line shall be in the proportion that the line bears to the total
length of the lines, or the perimeter of the traverse. This process affects
both bearings and lengths alike, and was devised for the compass, the
theodolite, as we understand it, being beyond the dreams of the land
surveyor at that time. Yet to-day many surveyors use the method
implicitly in the arithmetical adjustment of theodolite traverses; and
often wisdom would be folly when this blissful ignorance achieves its
end.
The graphical process is as follows when applied to the traverse of
Fig. 59, where A and F are the points or stations on the Ordnance
sheet, from which a tracing has been made for the sake of econony.
b c d e t
FIGS. 60 and 61
Let AbcdefbQ the traverse as plotted to the given scale with a good
protractor, showing fF as the error of closure on the point F on the
Ordnance sheet.
Draw parallels to the direction fF through b, c, d, and e, in Fig. 60.
Set off on the same scale (or some convenient fraction thereof) the
consecutive lengths of the traverse lines along the horizontal base Af
in Fig. 61; and at / erect a perpendicular to Af equal to the error
of closure fF. Join AF, and erect perpendiculars to the base Af y giving
bB y cC, dD, eE, the corrections to be made at the stations 6, c, d, and e.
Obviously these corrections are in the required ratios of the lengths
96 ELEMENTARY SURVEYING
of the sides of the sum of these lengths, and they would be the fame
if Af were one-half, one-quarter, etc., the scale value of the total length
A to /in Fig. 60.
Finally, set off the corrections Bb, Cc> Dd, and Ee along the parallels
at b, c, d, and e, so as to obtain the adjusted traverse ABCDEF, as
indicated in thick lines in Fig. 60.
The method is applied in a similar manner to a closed traverse
polygon. Copy the outline of the traverse ABCD in Fig. 20, but with
A', say J in. above and to the left of A, giving an error of closure AA'.
Then proceed as above, but with the horizontal base divided only into
four lengths, AE, EC, CD, and DA'. No! Run round an irregular
pentagon with the compass and chain, and then see what you have to
say about it.
IV. COMPASS RESECTION
Although the method of Trilinear Co-ordinates, as understood in
the "three-point problem," is usually associated with the plane table,
it has numerous applications with the compass in exploratory and
preliminary surveys.
In theory a point P can be fixed by the bearings observed from P
to any two visible and mapped stations or points, A, B, the magnetic
north serving as the third point.
But, apart from the sluggishness
or other defects of the compass,
the magnetic declination might
vary considerably over the area,
and the direction assumed for
the magnetic meridian might
be true only for a part of the
area if the latter were very ex-
tensive. Hence, it is advisable
to observe three points, A y B,
and C, and by subtracting their bearings from P, to find the angles &
and 9 as subtended by AB and BC at P, being p a and 9, y p, as shown
in Fig. 61. The direction of the magnetic meridian np thus becomes
of little importance, since Q and 9 would be the same whatever the
extent of local attraction. The rays from P to A, B, and C are repro-
duced on a piece of tracing-paper with the aid of a protractor; and
the tracing is shifted over the map of the area until the rays pass
through A, B, and C at the same time. Then if P on the tracing is
pricked through to the map the required position is fixed.
The solution will fail if A, B, C, and P are all on the circumference
of a circle, P thus having an indefinite number of positions.
The method is particularly useful in fixing positions at which
observations for altitudes have been carried out with the aneroid or
the boiling-point thermometer; and in selecting positions for stations
in an extensive scheme of triangulation.
THE COMPASS 97
Here is an idea for an adventure story in your magazine, with
mystery introduced through the medium of a complex code, which
gives clues both to points and bearings in finding the hidden secret.
Incidentally, this fifth principle of surveying is also the basis of
fixing the position P of a wireless receiving station with reference to
three transmitting stations of known wave-length, A, B, and C, the
positions of which are shown on a map. A directional frame aerial
at P is fitted with a horizontal circle, so that the direction of the vertical
plane of the aerial can be determined when it is turned edgewise
towards the transmitting stations so as to receive the signals at maxi-
mum strength. From the divided circle the angles 6 and 9 are found,
and P is determined in a manner similar to that described for compass
re-section.
In practice this is not so simple as it appears to be, for there is
"local attraction" above and below, and all along the paths of the
wireless waves.
OFFICE AND CLASS EXERCISES
7 (A). Plot the survey from the notes given on Plate IV. (G.S.)
7 (B). Plot the survey from the notes given on Plate V. (G.S.)
7 (a). The following bearings were taken with a prismatic compass in an
open traverse ABCDE through an area in which magnetic interference was
suspected:
AB, 39; BA, 219; BC, 84; CB, 267; CD, 122; DC, 294; DE, 129;
ED, 314.
State the values of the corrected magnetic bearings with which you would
plot the survey.
(C and D affected J and + 5; CD, 1 19; DC, 299; DE 9 134) (G.S.)
7 (b). Draw an equilateral triangle of 6 in. side to represent a triangle
ABC, with A, B, and C running in clockwise order, CB being horizontal and
2 in. above the bottom of the page.
A represents a spire; B, a coastguard station; and C, a castle.
A smuggler hurriedly buries some treasure at a point O, and observes the
following compass bearings from O to A, B, and C respectively:
36 135 230
Show how he would find the position of O with a compass bearing and
distance, both from B, the scale of the man being 1 in 1,000.
(Tracing paper will be supplied if required.) (G.S.)
(4-85 in., or 405 ft. and 315 from B.)
1 (c). Describe any form of prismatic compass, giving (if possible) a
sectional view.
State concisely what you know about the following:
(a) Magnetic declination and variation.
(b) Magnetic interference in surveying. (G.S.)
7 (d). The following compass traverse was run between two stations, A
and B, which were fixed by triangulation, B being respectively 475 ft. and
1,200 ft. due N. and due E. of A with reference to the true meridian.
Both forward and backward bearings were taken as local attraction was
suspected.
98
ELEMENTARY SURVEYING
Line
Length
(ft.)
Magnetic bearings
Observed
Corrected
Ab
510
N. 34 E.
bA
be
510
195
S. 34 W.
N. 79 E.
cb
cd
195
540
S. 84 W.
S. 70 E.
dc
dB
540
370
N.75W.
S. 58 E.
Bd
370
N. 58 W.
Plot the traverse with the corrected true bearings, taking the magnetic
declination to be 13 W., and using a scale of 1 in 2,400. If necessary, adjust
the traverse lines to fit between the main stations A and B.
(Place the true north parallel to the short edges of the Answer Book and
assume A about 2J in. from the S. W. corner of the page.)
(Station c affected +5. Closing error from 60 to 75 ft. reasonable.)
*7 (e). Three wireless transmitting stations A, B, and C, are situated in
clockwise order at the vertices of an equilateral triangle of 6 in. side on a map
plotted to a scale of 20 miles to the inch.
An explorer has mapped three stations as P, Q t and R, with the following
magnetic bearings and distances:
PQ, 43-8 mis., 62; QR, 37-8 mis., 30|, P being 34-8 mis. on i\ line 60
N.E. of C.
He then uses his wireless receiver to check his positions at P, Q, and R,
and by means of a directional aerial he ascertains the magnetic bearings of
three transmitting stations he identifies as A, B, and C.
Observer's
Magnetic Bearings to Transmitting
Stations
A
B
C
P
Q
R
20i
354
314i
103
I25J
157
237
243
228
(a) Plot the traverse on the scale stated from the given distances and
bearings, of P, Q, R.
(b) Plot the positions of P, Q, and R, as determined by the wireless signals,
using the tracing paper supplied.
(Stations P, Q, and R, located by wireless signals approximately 2-4 mis. N.,
3*2 mis. S., and 5-9 mis. N. of respective survey traverse stations.)
PLATE V
AC
s AF
2.0 o
3
O
The above pages of Field Notes refer to a Compass-Chain survey of a
Copse, all measurements being in feet.
Plot the survey on a scale of 50 ft. to 1 inch, placing the true N. and
S. parallel to the short edges of the paper, with A about 4 inches from the
lower and left-hand edges.
100
ELEMENTARY SURVEYING
PLATE VI
The following notes refer to the survey of a meadow which was mainly
under water except alongside a stream running through the area. In
consequence a straight backbone ABCD was run near the southern
bank of the stream up to Station C, where the stream bears N.E.,
entering the river near Station h. From the stations A, B, C, D, com-
pass bearings were taken in order to fix the boundaries, which were
straight, except along the bank of the river on the north of the area.
Plot the triangulation network for the survey on a scale of 50 feet
to 1 inch, using a protractor and scale. In doing this, place the Magnetic
North parallel to the short edges of the paper with A l\ in. from the
left-hand short edge and 6\ in. above the bottom edge of the paper.
Add as much of the detail as possible, following the notes given on the
right of the notes.
N.B. The bearings are measured E. and W. from the N. and S. points.
(G.S.)
COMPASS SURVEY OF BRAY'S PIECE
Station
Sighting
to
Magnetic
Bearing
Length
(ft.)
Notes
A
e
f
B
N. 2 30' E.
N. 48 15'E.
N. 90 00' E.
S. 64 30' E.
S. 2 30' W.
AB
410
/, A and e, 12 ft. E. of fence to
Bray's Lane; 10 ft. footpath on
E. side and 30 ft. carriage way.
A, B, C, //, 15 ft. from S. bank of
Mill Brook, 12 ft. wide.
k
I
B
e
f
g
C
N. 55 30' W.
N. 900'W.
N. 5215'E.
N. 90 00' E.
S. 00' E.
S. 00' E.
S. 6530'W.
BC
271
<% / g, h along S. bank of River
Dee; 30 ft. wide.
15 ft. S. of Mill Brook/Bears N.E.
Wall corner, Grove Mill.
Fence corner, Grove Mill.
/, k, Iron fence, Grove Mill.
j
k
I
C
g
h
D
N. 24 00' W.
N. 37 30' E.
N. 90 00' E.
S. 6330'E.
S. 44 00' W.
S. 56 00' W.
CD
237
S. bank of R. Dee.
S. bank of R. Dee/Also at Crown
Inn wall. D also at this wall.
//D/, straight wall, Crown Inn.
ji, straight wall, Grove Mill.
kj, short wall, Grove Mill.
i
j
k
D
h
i
N. 22 00' W.
S. 22 00' E.
Straight wall, Crown Inn.
Straight wall, Crown Inn.
Details: Centre of culvert, 210 ft. from / (21 ft. from A): diameters,
12 ft. inside; 15 ft. outside. Footbridge, 4ft. wide near B.
Bridge over R. Dee: 35 ft. clear span: 40 ft. between walls.
THE COMPASS 101
FIELD EXERCISES
Problem 7 (a). Survey the (specified) pond (or wood) by means of the
chain and compass.
Equipment: Compass, chain, compass, arrows, tape, and a set of picket*.
Problem 7 (b). Run an open traverse of the (specified) road (or stream).
Equipment: As in 1 (a).
Problem 7 (c). Make a compass-pacing traverse between . . . (two specified
places).
Equipment: Small prismatic compass (clinometer and passometer).
Problem 7 (d). Determine the error in the sum of the interior angles of
the polygon indicated by the range poles A, B, C, D, and E.
Equipment: Compass.
Problem 7 (<?). Determine the distance and height of the (specified inac-
cessible points) from and above the station indicated at the range pole A.
Equipment: Compass, two pickets, chain, arrows, and clinometer.
ORIGINAL PROBLEMS
(e.g. Measuring the interior angles of a polygon when the compass needle
is influenced by an attached key).
(e.g. Finding the treasure buned at P from the bearings of A, B, and C,
as obtained from Group . . . (the smuggling party).)
CHAPTER VIII
PLANE TABLING
Although it scarcely needs any introduction to-day, the plane table
may be described as a drawing-board mounted upon a tripod to form
a table upon which surveys can be plotted concurrently with the field
work through the medium of a combined sighting device and plotting
scale.
Hence angles are not observed in magnitude, as in the case of any
goniometer, or angle measurer, such as the compass, sextant, and
theodolite, but instead arc constructed directly, so that the instrument
is a goniography or angle plotter.
Suppose you insert two pins vertically at the ends of a ruler, and
place this on a table at a point O\ then, using these pins as sights,
you glance through them first to one corner A of the room, then to
the other, B, the angle A OB could be constructed if lines were drawn
along the edge of the ruler. That is the sole geometrical principle of
the plane table.
The Table is made in numerous forms and sizes, ranging from small,
light patterns with a simple board and thumb-nut attachment to the
tripod, to elaborate boards with every refinement for levelling the
drawing surface and rotating the board, even to carrying a continuous
roll of drawing-paper.
The sighting device may also range from a simple Sight Rule with
vertical eye-sights to a Telescopic Alidade, which may be simple or the
upper part of a complete transit theodolite mounted upon a rule.
Sight rules are often engraved with a scale on each edge, but the
base rules of alidades
are seldom divided, thus
necessitating the use of
independent plotting
scales.
A simple plane table
is shown in Fig. 63.
Common to all patterns
are the spirit levels and
compass. In simple pat-
terns the spirit level for
levelling the board is sometimes fitted as a cross bubble to the
sight rule, a separate trough compass being supplied In more
expensive patterns a large circular compass sometimes carries two
large bubbles at right angles to each other. Simple tables are levelled
up solely by manipulation of the legs of the tripod, but high-class
~~
FIG. 63
PLANE TABLING 103
models are fitted with a tribrach levelling base. Also the clamping
of the board in an important position is effected solely with
that unsatisfactory device, the thumb-nut, on the one hand, while
on the other, a refined clamp and slow motion is provided. In the
writer's opinion, no board and tripod can be too good in practice,
and a telescopic alidade is essential, though the tendency is to make
this unduly elaborate.
In larger models, a Plumbing Bar is supplied, so that a mark on a
station peg can be transferred up to the board through the medium of
a plumb-bob attached to the undcr-arm of the plumbing bar.
Useful plane tables arc readily constructed from half-imperial drawing
boards, battened rather than framed patterns. A hard wood or metal plate
is secured to the centre of the under face of the board, and a long screw
with a thumb-nut is inserted in this, exactly at the centre of the board. The
thumb-nut serves to secure the board to the top of the tripod and to clamp
it in any desired position in the field. When it is really necessary to construct
tripods, these should be of the framed pattern, as suggested on page 67.
The triangular plate, or headpiece, to which the halves of the tripod legs are
secured, should be drilled to take the screw which protrudes from the under-
face of the board. Wooden camera tripods, though they possess the merit
of telescopic poitability, arc seldom rigid enough; and size should not be
sacrificed in order to lighten the board. A 3-in. or 4-in. bubble in a metal case
will serve for levelling purposes, and the compass should be of a type that
the N. and S. line of the case may be easily transferred to the plotting paper.
Incidentally a waterproof satchel should be made, or improvised from
American cloth. Sight rules should be purchased. Boxwood patterns in
I he 10-in. or 12-in. size should be selected, with one of the scales showing
inches, tenths, etc., however it may be specified: 10, or 100, as the case may
be. Some of the scales engraved on sight rules are of little use in classes, the
work being kept normally to 50-ft. (Iks.), or 100-ft. (Iks.) to the inch.
The 1 : 2,500 means access to the 25-in. Ordnance sheets, and these are very
expensive in classes, while the 6 in. to 1 mile would suggest a man's job on
a half imperial drawing board. Plausible as these scales may sound, their
use impedes the work in instructional classes.
Terms and Definitions. On small scales, or in lower grade work,
the board is regarded a point; and only on very large scales is the
plumbing bar justified. Scale plays an important part in plane tabling.
It is a kind of denominator which fixes the speed of the work; and, in
a way, speed multiplied by accuracy equals a constant; "more accuracy
less speed." Scale also suggests two terms we must know before
going into the field: (a) Orienting the Board, and (b) Setting the Board.
(a) Orienting the board means turning it on the top of the tripod
so that plotted lines are parallel to (or coincident with) the corre-
sponding lines in the field.
(6) Setting the board means orienting it roughly by placing the
north end of the compass box over the north end of a magnetic needle
drawn on the board, and turning the board until the compass needle
comes to rest in the common magnetic meridian.
Military sketch-boards are set in this manner, as also are certain
rough plane table maps. On small scales, the error from orientation
104 ELEMENTARY SURVEYING
will be very small, but on large and even medium scales the defects of
the compass will soon be evident.
Perhaps you will understand the distinction in the following emer-
gency You are stranded at the junction / of five roads, where the
guide-post has been removed as a war-time precaution. But you have
a map of the locality; say 1 in 10,000, or the 6 in. to 1 mile. You find
your position on this without difficulty, and you also see that church C
in the distance. Then, if you spread the map flat, and turn it while
looking along the line between / and C on the map until the church
comes into view, the map will be oriented, strictly, if not accurately.
But if the country around is thickly wooded, you will need a compass
in addition. You lay your map on the ground, an d place the compass
upon it with the north end mark ^on the compass box exactly over the
north end of the magnetic meridian of the map. Then you turn the
map slowly until the needle comes to rest in the common magnetic
meridian. If only the true meridian is shown on the map, a magnetic
meridian must be pencilled across it at the declination for the date
and place. This is "setting'* strictly, though both processes are referred
to as such in military surveying.
Notation. The edge along which rays are drawn on the paper in
plane tabling is known as the "fiducial edge of the alidade," which we
will contract to "ruling edge/* "Centring the alidade" (or sight rule)
at a point or station on the plan means placing the ruling edge over
the plotted position of that point. "Centring the sight rule" is facili-
tated by inserting a bead-headed pin at the point, and keeping the
edge in contact with this pin.
As far as possible capital letters will denote stations in the field,
such as A, B, C, etc., and the corresponding small letters will indicate
the corresponding points on the board, as a y b, c, etc.
METHODS OF PLANE TABLING. It is not without reason that the
plane table is considered the simplest and best instrument for demon-
strating the principles of surveying, tKough, even in this capacity, it
is seldom treated as a versatile demonstrator. A student from the
Orient is said to have observed that the plane table is the best of all
surveying instruments because there are only two things to be remem-
bered about it. If he meant, as he presumably did, the processes of
intersection and three-point resection, he had the academic outlook
fairly well assessed. Apart from economic and climatic considerations,
there is a place for all the five principles in practice, which is not all
solving the three-point problem for resection's sake.
Now there are three primary methods of surveying with the plane
table:
(1) Radiation; (2) Intersection; and (3) Progression, or Traversing.
Although seldom used in entire surveys, they are commonly used
in filling in the details of triangles and polygons surveyed by more
accurate methods.
PLANE TABLING
105
(1) Radiation, (a) Reconnoitre the ground, making an index sketch,
or adding notes to one copied from an existing map. Select as the
station O a point from which all points to be surveyed are visible,
say, A, B, C, and D, as in Fig. 64.
(b) Level the table over the station
O, and, referring to the index sketch,
clamp the board in the best position
for placing the survey on the paper.
Fix a pin in the board at o, to repre-
sent 6, selecting this point so that
the entire survey can be plotted on
the proposed scale. (Using the com-
pass, insert a magnetic N. and S. line
in a convenient place to serve as a
dated meridian, and in large surveys
in roughly orienting the board.)
(c) Centre the sight rule against
D c the pin, and, sighting the stations A,
FlG * ^ J5, C, etc., in order, draw rays to-
wards them, but only round the margins, and not in the body of the
paper, which is the place for the map. Reference these A, B, C, etc.,
in the margins.
(d) Chain the radial distances from O to A, B, C, etc., and set them
off to scale as oa, ob y oc, etc. Connect ab, be, cd, etc., with firm lines,
if these are actually straight boundaries.
In practice, however, this method is used for details, such as inserting
contours, where the radial distances may be found also by means of
the tacheometer, or on
small scales, even by
pacing. In simple sur-
veys it is sometimes
used as an auxiliary
method to progression.
(2) Intersection, (a)
Prepare an index sketch,
as stated above, in-
cidentally obtaining
some idea as to the
distances and lengths in-
volved.
(b) Select a suitable
situation for the base
line PQ, observing that all points to be plotted must be visible from
both ends of the base. Chain the one direct linear measurement, the
base PQ with great care, using a steel tape if one is available. Fix a
pole at Q.
Q
FIG. 65
106 ELEMENTARY SURVEYING
(c) Set up the table, centring and levelling the board over one end
of the base P. Clamp the board in the most convenient position 'for
placing the survey, and, having carefully selected the position for the
base pq, fix a pin at /?, to represent P, the station occupied. (Insert a
magnetic meridian by means of the compass.)
(d) Centre the sight rule against the pin at /?, and sight at A, B, C,
etc., salient points in the survey, drawing rays near the margins and
referencing them accordingly A, B, C, etc. Sight the pole at Q with
the ruling edge still centred against the pin at p. Draw a ray towards
Q, and along it jet off PQ to the scale adopted to represent the measured
Fix a pole at P on vacating the station.
Set up the table, and centre and level the board over the other
end of the base, Q. Fix a pin at q to represent Q. Orient the board
by sighting with the ruling edge along qp to the pole at P, and clamp
the board thus. Centre the sight rule against the pin at q, and, sighting
the points A, B, C, etc., draw rays towards them to intersect the corre-
sponding rays from P in a, b, c, etc. Avoid intersections that are very
oblique, or very acute, bearing in mind the rule for all triangulation
-30 to 120 at the point intersected.
The chief objection to intersections as a sole method of surveying
is the difficulty of selecting a base so proportioned that definite inter-
sections will result, and of plotting that base with respect to both scale
and position so that the resulting map is neither absurdly small nor
so unduly large that certain intersections fall outside the limits of the
paper. The method has been used with some measure of success by
fixing stations by intersections around the boundaries, and then
measuring between them in order to take offsets. Too often, however,
the more accurate chain measurements will not agree with the inter-
sected positions of the stations, which must be adjusted. Here we
have fair linear measurements not mixing with poor angular measure-
ments; and, as hinted before, the surveyor's headaches are not all due
to eye-strain. In general, the method of intersections is best used as
an auxiliary to some other method, particularly for locating inaccessible
objects, such as mountain peaks, points across rivers, etc., etc., also
outlying and broken boundaries.
It is particularly interesting to note that plane tabling by intersections
is analogous to ground photographic surveying, particularly as regards
determining elevations, the clinometer being used in conjunction with
sight rules and the vertical arc with telescopic alidades. The India
pattern clinometer is especially suited to determining elevations in
plane table surveys, the tangents of the vertical angles being read
directly. Here the elevations above the table are found from V D
tan a, as described in Chapter VI, D being the horizontal distance to
the observed point as scaled on the board.
But there is this great difference between plane tabling and photo-
graphic surveying. Plane table surveys are plotted almost entirely in
PLANE TABLING NT/
the field, and the field work is protracted at the saving of office work,
whereas the field work in photographic surveying is brief, but at the
expense of protracted office work. Thus photographic surveying is
especially adapted to observations in exposed or dangerous situations.
In fact, a photographic survey was being made at Sedan at the time
the city capitulated in 1871. While the topic is still before us, it might
be noted that the plane table was the forerunner of the elaborate
plotting machines now used in connection with aerial surveys.
(3) Progression, (a) Prepare an index sketch of the area, as described
with reference to radiation, incidentally considering the first traverse
line on the proposed scale. No difficulty arises in this respect when
filling in the details of a previously surveyed polygon by more accurate
methods.
(Z?) Select and establish the stations, A, B, C, etc., bearing in mind
that if the boundaries are straight, these may be more distant from
the boundaries, but if the fences are undulating, short offsets must be
used, as in ordinary land surveying.
(c) Set up the table over one of the stations A, and ascertain from
the index sketch, the best position for the board and a point indicating
the station A. As before, fix a pin at a, and on vacating A, remove the
pin to the next forward station,
6, etc. (Using the compass, draw a
line in the magnetic meridian. Here
this will merely serve as a magnetic
meridian for the finished survey;
but, in general, it assists in orienting
when it is necessary to resort to
resection.)
(d) Centre and level up the table
over station A, and clamp the
board in the most suitable position.
Sight back on the rear station D
with the ruling edge centred against
the pin at a. Draw a line towards D.
Sight at the forward station B, still keeping the sight rule centred on A.
Draw a line towards B.
(While the rule is still centred on a, draw a fine or dotted line
towards C as a check sight. This will intersect later with the line drawn
from B towards C, fixing c as a check. But checks must take second
place to the main measurements, though they are very helpful in
checking unseen movements of the board.)
(e) Locate detail near A by radial distances, using the sight rule, if
the fences are straight, but if the fences are crooked, measure offsets
in the usual way while chaining from A to B. (Radial distances are
measured just as though station A were station O in the first method.)
Measure AB, and plot it to scale as ab on the board. (Offset detail may
TIG. 66
10* ELEMENTARY
be plotted either in the field or the office, and time and weather are
the deciding factors.) Fix a pole at A and proceed to station B. *
(/) Centre and level up the board over the next forward station B.
Orient the plan by sighting back at A with the ruling edge along ba.
Clamp the board thus. Sight at the next forward station C with the
sight rule centred on b, and draw a line. (Note that this line will
intersect the dotted line from a, giving a check. Incidentally draw a
dotted fine line towards D for a similar check on d\ but regard this as
a check, never letting it supersede a chained measurement except when
a serious movement has occurred.) Locate fence corners, buildings,
etc., near B by radial distances, if otherwise offsets have not been
taken. Fix a pole at B on vacating this station.
(g), (/i) Occupy in order the stations C, D, etc., in order, levelling
and orienting the board and chaining and plotting the traverse lines
EC, CD, etc., as detailed in (f).
Progression is the best method of making purely plane table surveys,
but it is seen at its best in traverses of roads and rivers, particularly in
exploratory work, where intersections are invaluable in fixing lateral
detail, mountain peaks, and the like. The value of radiation can only
be assessed by surveying contour points and other features from a
table set up at stations previously traversed by means of the theodolite
and chain. Combined with tacheometric measurement the method
still has a well-deserved place in topography.
Resection. The characteristic feature of resection is that the point p
plotted is the station P occupied by the table. Strictly there are two
general cases of plotting p from not less than two visible and plotted
points, called known points: (a) when the line through P and A, one of
the known points, is drawn, and (b) when P is no way connected with
any known point, A, B, or C. The former is simple resection, and p is
plotted by orienting the board by sighting along the line drawn through
a towards P and fixing p by a sight through b to B. The second intro-
duces the well-known "three-point" problem, which is often regarded
as resection proper.
Now it often happens that a point of excellent command and general
usefulness to the survey as a whole is not a station, and this can be
occupied and plotted at once if three known points are visible. Simple
resection would possibly involve a return journey to A, a great distance
often, in order to draw a ray through A towards P. There are many
occasions when resort to the method is expedient. But the three-point
problem should never be resorted to for resection's sake; for, after ail,
it is merely incidental in actual work, even though it may be made a
matter of great academical moment.
The three-point problem can be solved (1) By trial, (2) Mechanically,
(3) Graphically, and (4) Analytically, the last applying more particularly
to the theodolite.
Actually there is little to commend trial methods, beyond that an
PLANE TABLING
109
expert can readily eliminate a small "triangle of error" which may
result from the mechanical method. The paper is the property of the
map and not the place for a confusion of efforts at trial solutions. It
is difficult enough to keep the paper clean without cultivating dirt from
erasures of unnecessary lines. Graphical methods may be good in
expert hands, but the following mechanical artifice will meet the
demands of most cases.
Mechanical Solution. (Place the compass with its box N. and S. line
along the meridian drawn on the map, and turn the board until the
needle comes to rest in the common meridian. Otherwise set the board
by eye.)
(1) Fasten a piece of tracing-paper on the board, and, as near its
correct position as can be estimated, assume a point p' to represent
the station occupied by the table.
(2) Centring the sight rule on this point /?', sight successively at the
three points A, B> and C, and draw rays accordingly along the ruling
edge. Unfasten the tracing-paper.
(3) Move the tracing-paper about on the board until the three rays
pass through a, b, and c, the plotted positions of A, B, and C, then
prick through the point /?', obtaining its true position on the map,
say/?.
(4) Orient the board by sighting through p and a to A, and check
by sighting through b and c to B and C respectively.
If the check rays do not pass exactly
through p, they will form a "triangle of
error." The marine surveyor performs this
operation with a three-armed protractor
known as a "station pointer."
The value of the plane table is often
wrongly assessed in practice. It is frequently
compared with combinations of other
instruments rather than judged by its own
peculiar merits.
(a) The plane table dispenses with field
notes, and the survey is plotted concurrently
with the field work, thus obviating mistakes
in plotting recorded measurements. Also
the area is in view, and measurements which might otherwise be over-
looked are at once detected. In contour work it is superb under good
climatic conditions, and features that mean little in field notes are seen
as they really are. Even the finest photographic methods are im-
paired by shadows and other defects in the negatives.
But field plotting is disagreeable, if not impossible, under certain
conditions of weather and climate, and the observer's position is
cramped and tiring, being exceedingly trying in the heat of the sun.
Also no notes are available for precise calculations of areas and the like.
FIG. 67
110 ELEMENTARY SURVEYING
(6) Little knowledge is required to use the plane table, but to
manipulate it correctly and effectively demands considerable skill'. It
is cumbersome and awkward to carry and requires several accessories.
(c) The chief use of the instrument is the filling in of details of
surveys where the skeleton has been surveyed with the theodolite.
Used discriminately in this way, it is unsurpassed in certain topo-
graphical work. It is rapid, covering more ground in a given time than
any other instrument, when plotting is also taken into account. In-
accessible points can be plotted without trigonometrical calculations,
and elevations are readily found from the graphical construction of
the tangents of vertical angles observed with a clinometer or the
vertical arc of a complete alidade. Also the facility of three-point
resection permits the occupation of unknown points which give
excellent command without working through obstacles or areas devoid
of detail.
Withal it is not intended for extremely accurate work, yet exception-
ally good results can be obtained if due care and understanding are
exercised.
In conclusion, a few practical hints must suffice, though pages could
be devoted to the technique of plane tabling from the surveyors' point
of view.
(1) Paper. Good quality paper should be used for surveys proper,
the cheap grades being admissible only to rough exercise work Faintly
tinted papers are best in intense sunlight They relieve eye-strain and
obviate the necessity of coloured spectacles in bright sunlight. The
paper should be fastened to the board so that neither the wind nor the
movements of the alidade can disturb it As few drawing-pins as
possible should be used on the drawing surface of the board The best
plan is to cut the sheets barely the width of the board, turn under the
excess length, and fix pins either in the edges or underneath, never
using more than two pins on the upper surface. A waterproof cover
is desirable in the case of sudden showers, and if this is not at hand
the board must be removed and turned upside down.
(2) Plotting. Either a HH or HHH pencil should be used, one end
being sharpened to a chisel point for ruling lines, and the other to a
round point for indexing. Both points should be kept sharp, and for
this purpose a sandpaper block should be suspended from the top of
a leg of the tripod. Lines should be few, fine, and short, and unneces-
sary lines, or parts of lines, should be avoided. There is never any
need for continuous lines, except for clearness in the case of main
survey lines. A line, half an inch or so in length, at the station and a
similar one near the estimated position of the observed point, will
suffice in both radiation and intersection. Some simple system of
referencing and indexing the lines should be adopted, and notes should
never be written in the vicinity of points. A very good plan is to pro-
duce lines, not actually drawing them, except at the margins, where
PLANE TABLING ill
half-inch lengths can be referenced without the possibility of con-
fusiRn. Cleanliness is of highest importance, and the paper can be
messed up with the slightest provocation. Pencils should never be
sharpened over the board, erasures should be as few as possible, the
cleanings flicked off the paper; the base of the rule should be kept
clean, particularly if metal, and heavy alidades should be placed in
position, never slid.
(3) Manipulation. See that the legs of telescopic or folding tripods
are secure, and press these into the ground lengthwise, never crosswise.
Aim at getting a level board, oriented and centred over the station,
with the board a little below the bent elbow. Avoid unnecessary
scruples in centring over a station when working to medium scales.
Remember that eccentricity between point and station varies inversely
as the observed distance, and that 1 inch means 1 minute in error in
280 ft., and 1 minute represents the highest grade work. Level up with
the spirit level central in two positions at right angles near the centre
of the board, and also test near the edges. Avoid undue pressure or
leaning on the board and keep all accessories oil the table when not
in immediate use.
(4) Sighting. Fix a pin in the board at stations and keep the ruling
edge against it when sighting. Fine bead-headed pins are the best.
Always draw a magnetic meridian with the compass in the top left-
hand corner when levelled up and oriented at the first station. It may
prove useful besides giving a necessary detail in the finish of a plan.
Always sight at the lowest possible points of station poles, pickets, etc.
When vertical angles are observed, ensure that the board is level, and
place table pattern clinometers as nearly as may be at the centre of
the board.
Brief as these words of advice may be, they imply that you will aim
at making a proper plan, rather than thinking you know how it is done.
Finally, do not mind if you are corrected for using the original term
"oriented," which is now being ousted by the affectation "orientated."
CLASS EXERCISES
8 (a). An unfinished map is fixed to the board of a plane table at a station A.
The map contains a magnetic north and south line, but only one plotted
station, B y is visible.
Describe how you would (a) Set the map by means of the compass, and
(b) Orient it by use of the sight rule, or alidade.
State clearly, giving reasons, which of these two methods you would use.
(G.S,)
8 (b). Describe with reference to neat sketches the use of the India pattern
clinometer in connection with the plane table.
8 (c). Describe how you would carry out the following surveys with the
plane table, a chain, tape, and pickets being included in your outfit.
(a) A large isolated wood; (b) a flat open field with straight fences; (c) a
crooked boundary with a stream running along the inner side.
8 (d). In a plane table survey it is essential to occupy as a station a point P
112 ELEMENTARY SURVEYING
from which three stations, A, B, and C, are visible and are already plotted
as a, b, and c respectively. +
Describe with reference to a sketch how you would plot the position of P
and orient the board at the point for further plotting. Under what conditions
would your method fail.
8 (e). Describe how you would make a rapid survey of a mountain valley,
determining the heights and positions of mountain peaks en route.
Your outfit consists of a light plane table with sight rule and compass,
clinometer, and a passometer. The proposed scale is 1 in 25,000.
FIELD EXERCISES
Problem 8 (a). Survey the (specified) wood by means of the plane table.
Equipment: Plane table with sight rule and compass, chain arrows, tape,
and a set of pickets.
Problem 8 (b). Survey the (specified) pond (or lake) by means of the
plane table.
Equipment: As in 8 (a).
Problem 8 (c). Survey (specified owner's) field with the plane table.
Equipment: As in 8 (a).
Problem 8 (d). Make a rapid plane table survey of the (specified) lane
between (named points).
Equipment: Plane table with sight rule and compass (passometer), and
three pickets.
Problem 8 (e). The points indicated, A y B, and C, have been plotted as a,
b, and c, on the board of the assigned plane table, which now stands at an
unknown station P. Determine and plot the position of P with the aid of
the tracing-paper supplied.
ORIGINAL PROBLEMS
Survey a part of the (specified) building by means of the plane table.
Select three prominent points on the 6-inch Ordnance sheet attached to
the board of the assigned plane table. Find the horizontal distances and
heights of these with respect to the station at which the table now stands.
CHAPTER IX
CONTOURING
A contour is a line drawn through points of the same elevation on any
portion of the earth's surface as represented on a map.
Contour lines are figured with that elevation above datum as an
integral or whole value, and successive contour lines are inserted at
regular increments from that value, such as 5 ft., 50 ft., or 10 metres.
The difference in elevation, or reduced level, of successive contour
lines is known as the contour interval, or vertical interval (V.I.) in geo-
graphical and military surveying, where the corresponding distance in
plan is called the horizontal equivalent (H.I.), leading to the relation
(H.I.)=(V.I.) cot. a, with a the angle of slope between successive
contours.
Contour intervals vary from 1 ft. to 10 ft. in engineering work, 5 ft.
being the usual interval in English-speaking countries; from 10 ft. to
50 ft. in preliminary and pioneer surveys; and 100 ft. and upwards in
exploratory surveys.
There would have been no reason for the inclusion of this chapter if,
during the Great Flood, the waters had receded from the peak of
Mount Ararat with solar regularity, and at noon each day had left a
permanent watermark on the face of the earth, at intervals of 4 cubits,
which approximate to our fathom units so easily conceived by the
mind of mankind, if not by that of the scientist.
In solid geometry these watermarks would be defined as the traces
of horizontal section planes, and the ground plane, or horizontal plane
of reference, would be the sea-level datum to which the water ultimately
recycled.
'Uses of Contours. The uses to which contours are put may be
summarised concisely as follows:
(1) Giving general information as to the surface characteristics of the
country and showing if points are intervisible, as in military surveying.
(2) Giving data for drawing trial vertical and oblique sections for
the construction of roads, railways, etc., and the layout of engineering
schemes.
(3) Giving data for the calculation of earthwork volumes indirectly,
as in the case of cuttings and embankments, and directly, as in the
case of impounding reservoirs.
Characteristics. Among the various characteristics of contour lines
the following should be noted:
(a) Contour lines close upon themselves somewhere, each to its own
elevation, if not within the limits of the map.
113
114 ELEMENTARY SURVEYING
(b) Contour lines cannot intersect one another, whether they Ije of
the same elevation or not.
(c) Contour lines on the tops of ridges and in the bottoms of valleys
either close or run in pairs within the limits of the map; and no single
line can ever run between two of higher or lower elevation.
(d) Contour lines indicate uniform slopes when they are equally
spaced ; convex slopes when becoming farther apart with increasing
elevations; and concave slopes when becoming closer together with
increasing elevations.
METHODS. Contouring involves both surveying and levelling; in fact
all the first five principles are employed in surveying, or Horizontal
Control, as it may be called, and the two geometrical principles in
levelling, or in Vertical Control, though the use of hypsometrical
levelling is resorted to in the case of great intervals.
Contouring is the prime feature of topographical surveying, and
there is no branch of surveying in which so many combinations of
instruments and methods have been employed.
There are two general methods of Contour Location: (1) Direct, and
(2) Indirect.
(1) Direct Contouring. As the term should imply, points on the
actual contour lines are found on the surface by spirit levelling which,
with one possible exception, is the sole practical method of vertical
control. These contour points are then surveyed in the horizontal plane,
and any of the methods of horizontal control are at the surveyor's
disposal, though only one is selected primarily, another being in
reserve for parts where the primary method would prove inexpedient.
At first sight, the procedure in vertical control appears to be tedious
(as it really is at the outset), but on ground of definite surface character
it is often best for intervals of 5 ft., though it may prove tedious on
intervals of 2 ft., and exhausting in body to the staffman (and in soul
for the levelman) on intervals of 10 ft. The levelman improvises
targets from paper, straps, pocket handkerchiefs, etc., attaches these
to the levelling staff; two and sometimes three. He then directs the
staffman by signals until a point is found at which the (level) line of
collimation strikes the target, which may be as wide as 3 inches in
locating 5 ft. intervals. Unfortunately, among students, signals are
soon evolved into acrobatics and clamours, the latter of the nature:
"Go back," "Come nearer," "Up a bit," "Up yards," and (that gesture
of comfort) "Down a millimetre." Then a lull. The staffman thinks
he is forgotten until the levelman discovers that the line of sight is
really "yards" above the top of the staff on account of a forgotten
bubble. (It might be noted here that a traversing bubble is really
necessary, since the modern tilting level is of little use in trial and
error work.)
But after a few points have been found, the clamours will subside
CONTOURING 115
and the staffman will begin to sense the trend of the contours. And
when direct contouring is done, it is done; which is to say, that hours of
monotonous office work will not follow, as in the case of indirect
methods.
In small parties the levelman signals to the surveying party that the
staffman awaits them at a contour point, which may be located
immediately with some instruments, such as the plane table with a
stadia alidade, or the theodolite provided with stadia lines in the tele-
scope. The work is then in Dual Control Otherwise, or in extensive
surveys, the staffman inserts a short length of coloured lath, selected
from a haversack, which carries white laths for the 50 contour, red
for the 55, black for the 60, green for the 65, blue for the 70, etc., all
prepared by dipping the tops into paint pots. These coloured whites
can then be located in Detached Control, the surveying party working
at their own convenience and collecting the sticks as the points are
located. A convenient plan of booking contour points is to enclose
the interval number in a circle, thus ; and in general, tabular notes
are best, with a page devoted only to one station.
Something sensational, though analogous to the above process,
happens when photographic plate pairs are inserted in a Stereo-
comparator, where a single plastic, or relief model, is seen with a
wandering mark moving robot-like along the contours like a willing
(and silent) staffman.
(2) Indirect Contouring. In this method salient points in the area
are selected as ruling points of elevation representative of the general
surface character; the elevations of these are found and the points are
conveniently recorded by a cross with the reduced level; thus, x (57-6).
All the methods of vertical control are used in indirect contour loca-
tion. Also, again all the methods of horizontal control are at the
surveyor's disposal, but he chooses one primarily and never makes
his notes an encyclopaedia of methods. More often than not dual
control is kept, though occasions arise when detachment is advisable
for practical reasons.
Contours are inserted between these ruling points (x) by inter-
polation, at best a very monotonous undertaking which is usually
carried out in the office, though occasionally in the field on the plane
table.
Indirect location is the only method that can reasonably be con-
sidered for intervals over 10 ft. to 20 ft. Singularly the method is also
best for very small intervals, 1 ft. or 2 ft., or even 5 ft., when the ground
is devoid of surface character. The question arises, "What is surface
character?" The flattest area certainly has character; flatness which is
nothing in topography; undulatory ridges and furrows, even on hill-
sides are of indefinite character; but hill and valley features are
definitely character, if pronounced in the immediate landscape.
116
ELEMENTARY SURVEYING
Fieldwork. A description of one combination of each of the general
methods will be given with an outline as to how these are varied with
other instruments in particular cases.
(1) Plane Table (H.C.) and Dumpy Level (F.C.). Let A, B y and C in
Fig. 68 represent a traverse which may have been run solely with the
(4-32)BM
plane table and chain or by means of the theodolite and chain, and
afterwards plotted on the board of the plane table. Also let Y 1$ Y 2 ,
etc., represent positions of the level, points on the contours, and the
rectangles the board at the stations.
Control. It will be best to work in detached control, since it is
assumed that only a sight rule or simple alidade is at our disposal; for,
obviously, chaining (or even pacing in rougher work) would hold up
the levelling party. Intersections certainly may be used in horizontal
control wherever possible, though as a rule the use of these in proper
contouring is more restricted than what it may appear. Hence sticks
of conventional colours should be fixed at the contour points, or,
failing paint, cleft twigs may be used with coloured tickets.
(V.C.). A backsight of 4-32 is taken with the dumpy level on a staff
held on B.M. 64-6, giving a height of collimation at Yj of 68-92. Hence
for a reading of 8-92, the foot of the staff will be on the 60 contour,
while readings of 13-92 and 3-92 will likewise give the 55 and 65
contours respectively.
The staffman is ready to move in search of contour points as soon
as he has attached paper targets to the staff at these readings. Assuming
that working uphill is the more convenient, the points on the 55 contour
are found first, then points on the 60 contour, and then on the 65
contour, the sights being up to 500 ft., which length is permissible in
work of this nature.
When the contours are nearly straight or are flat curves, the contour
points may be from 100 ft. to 200 ft. apart, but on sharp curves they
may be as close as 20 ft. or even less.
CONTOURING 117
After a while it will be necessary to move eastwards to locate the
contour points in the vicinity of C. A foresight (of 3-92) is taken on
the contour point (65 C.P.), and with this as a change point the level
is set up at Y 2 , whence a backsight of 2-64 is read, giving a new coliima-
tion height of 67-64. The paper targets will now be shifted to 12-64,
7-64, and 2-64 for the 55, 60, and 65 contour points respectively, which
will be found in the manner described.
(H.C.). Meanwhile the plane tabler draws rays towards the different
contour points from A, represented by a pin at a, and directs the chain-
men to measure the radial distances rapidly to the nearest 2-ft., working
in a manner to reduce walking to a minimum. These distances are
then scaled off along the corresponding rays, a circle and dot is inserted
to represent each contour point, and the contour is inserted, advantage
being taken of the fact that the ground is in view. After all the points
have been surveyed in this manner, the plane table is set up at B, and
is duly oriented by sighting back along ba to A. When the points
commanded from B are surveyed, the table is moved to C, and so on
till the work is completed.
The ideal method of horizontal control is to use a stadia alidade so that
the radial distances can be found by the length of staff seen inteicepted
between the stadia lines of the telescope, or, better, a tacheometer, intro-
ducing the same principle, could be stationed beside the table. Dual control
is then possible in small surveys, the staffman turning the face of the staff
towards the plane table as soon as the levelman has signalled that the staff
is on a contour point.
Similarly the tacheometer, or a theodolite with a stadia telescope, could
replace the plane table and the contour points could be fixed by back angles
or azimuths; but ofcou.se all the plotting would be done indoors.
Dual control is sometimes possible by using the stadia lines in the telescope
of the dumpy level at YI Y 2 , etc., thus obtaining from the staff the distances
from YI, Y 2 , etc., to the staff. The positions of Y! and Y 2 are plotted by
radial distances or by intersections from A to B. Then the rays drawn from
#, 6, etc., arc intersected with arcs centred on YI, Y 2 , etc., the radii being
the stadia distances from these positions of the level.
Sometimes compass-chain traverses are run through the contour
points, and the latter are fixed by offsets, as in the case of boundaries
in land surveying. Occasionally straight lines can be run likewise,
particularly in areas with a general slope in a definite direction. A
special application of direct location is the American method of con-
touring the proposed routes of railways and highways. A reflecting
hand level is strapped to the top of a 5 ft. staff (called a Jacob), and
points on the contours on each side of the centre line are fixed with
great rapidity, the distances right and left being measured with the tape.
(2) (a) Grid Squares (//.C.) and Dumpy Level (F.C.). This is one of
the most effective methods on ground that displays little or no surface
character, and it is applicable to intervals up to 10 ft., but its more
immediate use is for intervals of 1 ft. or 2 ft. in connection with building
or constructional sites, sports grounds, etc. In the latter connection it
118 ELEMENTARY SURVEYING
also provides a ready means of calculating earthwork excavation from
the truncated prisms of which the unit squares are the plans. As the
term implies, the horizontal control consists merely in covering the
area with a network of squares of 50 ft., 66 ft., or 100 ft. side, basing
these on the most convenient side of the survey skeleton, as surveyed
with the chain only, or the theodolite or compass and chain.
(H.C.). In Fig. 69, ABCD is the skeleton of a chain survey. Along
AC at distances of (say) 50 ft., sticks are inserted, and at each of these
points perpendiculars are erected by means of the cross staff or the
optical square, the theodolite being used in highest class work. At
50 ft. distances along each of the
perpendiculars, sticks are inserted
right and left of AC, so that in
effect the entire area is covered with
a grid. It is convenient to number
the perpendiculars to AC as "Line
1," "Line 2," etc., with this and
"O" along AC, and the points to
the right and left "Line 2, 100'
L," "Line 6, 250' R," meaning the
corners of the squares 100 ft. to the
left of AC on Line 2, and 250 ft. to
the right of AC on Line 6. Cards
inserted in the clefts of the sticks
are used.
(V.C.). Starting at a benchmark,
the levelling party take the levels
at the corners of the squares, the
staffman removing the sticks as soon as the levelman has recorded the
staff reading and its position in his notes. A great deal of thought is
necessary in devising a form of notes from which the corner levels can
be found at a glance.
A good plan is to bring the table only of the plane table into the
field and appoint a topographer who will reduce the staff readings to
corner elevations as soon as they are brought to him by someone in
the humbler (though none the less useful) capacity of "runner." Mean-
while the topographer can interpolate the contours in the manner
described at the close of the present chapter.
(2) (b) Plane Table (H.C.) and Dumpy Level (V.C.). Now Method 1 (a)
can be used if the signs used for contour points are replaced by
crosses x denoting points figured with irregular reduced levels, not
55, 60, and 65, but values such as 47-6, 58-4, 66-6, etc., taken along
bottoms of valleys, tops of ridges, or at definite changes in the surface
character. The field work will be more expeditious than in the direct
method, and very often dual control is possible, though in every other
respect there is little difference in the field work, beyond the knowledge
FIG. 69
CONTOURING
119
the laborious process of interpolating contours is yet to come,
This combination is best adapted to intervals up to 10 ft., beyond which
ordinary spirit levelling ceases to be economical. This, the representa-
tive method of low interval contouring, is best carried out with a
tacheomcter in both horizontal and vertical control for intervals of
10 ft. to 20 ft. Also the work with the plane table would be greatly
expedited by the use of a stadia alidade or an independent tacheometer
beside the plane table.
(2) (c) Plane Table (//.C.). When the contour interval exceeds 20 ft.,
the India pattern clinometer is exceedingly useful and the work becomes
more of the nature of an exploratory survey, such as would be carried
out in a valley with eminences of considerable height on cither side
of the traverse. The ruling points for elevations would be conspicuous
points, which would of necessity be fixed by intersections in horizontal
control. The elevations of the ruling points would be found from the
tangents of the vertical angles a observed with the clinometer, with
V=D tan a, where tan a would be read directly and D the horizontal
distance scaled from the map. A similar process is used in ordinary
photographic surveying where, as here, the elevations above the camera
may be found graphically, as described in Chapter VI. In reconnais-
sance work the compass may supersede the plane table, the ruling
points being fixed by bearings observed from the ends of the traverse
lines.
(2) (d) Compass (H.C.) and Clinometer (V.C.). In reconnaissance
and pioneer work it is sometimes possible to run "direction" lines
which radiate from the stations of a compass traverse, the distances
between the traverse stations A, B, C, and D being found by pacing,
riding, or by range-finder. The direction lines, which are fixed by
compass bearings, are chosen along lines in which the ground surface
has a fairly uniform slope; and the slopes are observed with the clino-
meter, as angles 4, 6, etc., or, preferably, as cotangents of a.
Fio. 70
If the reduced levels of A, B, C, etc., are known, it is possible to
interpolate contours in accordance with the relation, D^V cot. a,
where D is the distance between the contours on the direction lines and
Kthe interval, which should not be under 20 ft. in the best applications.
120 ELEMENTARY SURVEYING
For example, if A in Fig. 70 is 404 ft. above datum, and the contour intejval
is 20 ft., then along the N. 32 E. direction line, the horizontal distance from
A for the 420 contour would be 16 cot. 4-= 16 x 14; 3 = 229 ft., after which the
440, 460, etc., contours would follow at even horizontal spacings of 286 ft.,
also to the scale of the map.
Finally, as an idea of the scope of indirect contouring, the use of the
compass and aneroid may be mentioned in regard to intervals of
100 ft. and upwards. The elevations of salient points are found with
the aneroid, and are then fixed in horizontal control either by compass
three-point resection by a lone observer, or by compass or plane
table intersections, made by other observers.
Interpolating Contours. Now that accurate transparent papers, ruled
accurately in tenth-inch or millimetre squares, are readily obtainable,
there is only one method really worth considering; and the work is
very different from the days when the surveyor was compelled to rule
tracing cloth, and, to preserve this, fixed a strip of paper on the left
from time to time for jotting down the values he assigned to the
thicker rulings.
In general, strips of transparent squared paper are cut in widths
from 1 in. to 3 in., 4 in. to 6 in. long, or widths 2 cm. to 6 cm., 10 to
15 cm. long. A system of diagonal decimal division is thus at hand and
the^-in. spaces, or 1 mm. spaces, may represent 0-5 ft., 1 ft., or even
10 ft., according to the range of elevations and the scale of the map.
In Fig. 71 the values 0, 1, 2, and 3, are merely shown for illustrative
purposes, and the elevations actually assigned to the main rulings may
(50) 3
(45) 2
__ y ^
(40) 1
(35)
- ^-^-r^*^0 CONTOUR : T^
X *JF^*1~? 3 1
Fio. 71
have any temporary values, as indicated by the bracketed figures
jotted down on the left.
Let it be required to interpolate 5 ft. contours between two points
x and y of respective elevations 37-3 and 48-7 by means of a strip of
1 inch divided transparent squared paper. (Fifths only are shown for
clearness in Fig. 77.) Assume the zero ruling to represent an elevation
of 35 ft., the next main ruling 40 ft., the next 45 ft., and so on. Each
of the (nine) intermediate lines will then represent 0-5 ft,, and it is
possible to estimate here to 0-1 ft., which is the lowest reading usually
observed in contouring.
Place the strip so that the cross x is between the 4th and 5th lines
from 0, being 0-6 of a small spacing above the 4th for 37*3 ft. Prick
through x with a needle point, and, with this as a pivot, turn the strip
CONTOURING
121
until the cross y Is seen between the 7th and 8th lines above the main
ruling 2, being 0-4 of a small space above the 7th line for 48-7 ft. The
main readings 1 and 2 will intersect the line between x and y for points
respectively on the 40 and 45 contours.
The process is simplified in connection with unit squares, and in
many cases the crossings of contours between x and y are estimated,
sometimes with the aid of a scale.
CLASS EXERCISES
9 (a). As a surveyor with a trained assistant and two men, you are required
to insert the 5-ft. contours on a plane table survey of an area in which it is
advisable to trace the actual contours on the ground.
Describe concisely, giving sketches, how you would carry out this work
with the following equipment at your disposal: Plane table with sight rule
(or alidade), chain and arrows, tape, dumpy level and staff, range-poles, and
a bundle of laths. (G.S.)
9 (b). The scale of an old map is unknown, but at a place where there is
a regular slope the map shows 5-ft. contours spaced exactly 0-9 inch apart.
The slope of the ground was found by means of a dumpy level, and a fall
of 2-5 ft. was observed in a horizontal distance of 45 ft. Draw a scale for
the map. (100 ft. to 1 inch.)
9 (c). You are required to make a survey of a small lake to show under-
water contours as well as the plan of the lake.
Describe your procedure with the aid of sketches.
*9 (d). The following notes were recorded in a reconnaissance survey in
mountainous country, altitudes being determined with the aneroid baro-
meter and the positions of stations fixed by compass bearings on two known
points P and Q. The magnetic bearing of the line PQ was 82 and its
length 5,500 ft.
Using a scale of 1 inch to 1,000 ft, insert the spot levels and, as far as
possible, interpolate approximate contours at 100-ft. intervals.
Observer's
Bearings to
Altitude
Station
P
Q
Ft.
A
12
58
1,540
B
350
32
2,200
C
314
44
1,160
D
320
15
1,735
E
280
25
1,010
F
310
350
1,950
*9 (e). A, B, C, >, are four points on a straight line in a valley, AB being
1,530 ft., EC, 1,650 ft., and CD, 1,840 ft., and the line has a true bearing of
N. 45 E. The four points are useJ as stations in determining the angles of
uniform ground slope in the area by means of a clinometer and the bearings
of these direction lines by means of the compass. The notes are tabulated
below, the plus and minus signs indicating respectively angles of elevation
and depression along the direction lines.
122
ELEMENTARY SURVEYING
Plot the survey on a scale of 1 inch to 1,000 ft. and insert the 300, 350,
and 400-ft. contours.
Station
Elevation of
Station
(ft.)
Direction Lines
Angle of Slope
True Bearing
A
260
+ 3
+4
355
160
B
290
+ 2J
+ 2
5
140
C
312
+ 2
+ 2
10
85
D
348
Hi
-1
30
220
FIELD EXERCLSFS
Problem 9 (a). Trace the 5-ft. contours within the triangle indicated by
the pickets A, 11, and C, and plotted as a, b, and c on the board of the assigned
plane table.
Equipment: Plane table, sight rule, dumpy level, levelling staff, chain, arrows,
and laths.
Problem 9 (b). The coloured laths inserted by Group ... are at points on
the . . . ft., . . . ft., and ... ft. contours.
Survey the positions of these by means of the compass, chain, and tape,
with reference to the stations A and B, as indicated by flag-poles.
Equipment: Compass, chain, tape, arrows, and a set of pickets.
Problem 9 (c). The flag-poles A and C are at the end stations of the
diagonal of (specified) field, as surveyed by Group ....
Obtain the data for interpolating contours at an interval of 5 ft. by means
of grid squares and spirit levelling.
Equipment: Chain, arrows, cross staff (or optical square), set of pickets,
laths, dumpy level and levelling staff.
Problem 9 (d). The points indicated (on rough ground) are plotted as a
triangle pqr on the board of the assigned plane table. Working in con-
junction with Group II, plot sufficient ruling points in order that 10-ft.
contours may be interpolated within the area pqr.
Equipment: Group I, Plane table with sight rule, chain, arrows; Group II,
dumpy level, levelling staff, and bundle of laths.
(B.M. to be indicated.)
Problem 9 (e). Survey the (specified inaccessible) portion of ... Hills by
means of the plane table, and make sufficient observations of prominent
points so that contours at 20 (50) ft. intervals can be interpolated.
Equipment: Plane table, clinometer, chain, arrows, and set of pickets.
ORIGINAL PROBLEMS
Excursion with compass, aneroid, and 6 inch Ordnance map in very hilly
country.
CHAPTER X
AREAS AND VOLUMES
la a way this chapter brings us back to the seeming drudgery of
arithmetic, and therefore to the things that really count. Those prisms,
pyramids, and cylinders may be something more than geometrical
solids, and little is ever lost of what has been learnt in mensuration.
The calculations that arise from surveying notes require three
things: (a) System in setting out the data simply and methodically,
avoiding unnecessary repetitions and cumbersome arithmetical pro-
cesses; (b) Soundness, selecting the method to meet the particular
requirements of the work in hand; and (c) Certainty in arithmetic,
subjecting the results to checks, preferably by simple processes; for
mistakes, as distinct from errors, can always intrude, and, in a practical
world, a single arithmetical slip can lead to a very material loss, possibly
thousands of pounds (sterling) in a contract. Much more could be
said, but the present would be untimely.
I. AREAS
The British and American unit of square measure in land valuation
is the acre, but the square yard and square foot are used in construc-
tional projects.
1 acre=4 roods^=160 perches 10 square chains=^43,560 square feet.
1 square mile--640acres.
The acre was the estimated amount of land that could be ploughed
by a horse in one day; "by the rod make one rood" It was generally
regarded as an area 10 chains in length, which is 1 furlong, or "furrow
long" with a breadth of 1 chain, which was divided into 72 furrows of
eleven inches.
Simple Plane Figures. The rules for the areas A of simple plane figures
will be summarised only, since the computation of these is common mensura-
tion. Wherefore, plane rectilineal figures will be considered with the letters
A, B, C, (Z>) at the angular points in counter-clockwise order, A being upper-
most and also to the left in the case of quadrilaterals. The altitude, or height
above a horizontal plane or base line, will be denoted by h, and radii by
r, the letter R indicating an outer or larger concentric radius.
Triangles. In the following rules the angles will be expressed in magnitude
by A, B, and C, and the opposite sides by a, b, and c respectively, the semi-
sum of the sides being j=J (a -\-b-\-c).
Altitude h and base a. A=\hb^ ab sin. C=J ac. sin. B=$bc sin. A.
Also the formula attributed to Hero of Alexandria .(120 B.C.):
A= V* (sa) (s-b) (s~c}.
Apart from the square and rectangle, the other (1) Parallelograms include
(2) the Rhombus, with all four sides equal, and Quadrilaterals including
123
124
ELEMENTARY SURVEYING
(3) the Trapezoid, with two sides parallel, and (4) the Trapezium, the general
case of a quadrilateral, no sides being parallel. The Euclidian definiJon
given is not rigidly adhered to, and often the names of the trapezoid and
trapezium are interchanged.
(1) Parallelogram: sides a, b, altitude h. A=bh=ab sin. B.
(2) Rhombus: side a, diagonals AC, BD. A = l (ACxBD)=a* sin. B.
(3) Trapezoid: parallel sides BC, AD, separated by perpendicular distance h.
(4) Trapezium, with perpendiculars hi, h 2 , let fall from A and C on the
diagonal BD. A= BD(hi+h 2 ).
Circle. Sector of radius r, subtending angle of radians at the centre,
A=lr*6.
When 0=2Tc, for the entire circle, A=nr a .
Annulus with outer and inner radii R and r, A=n (# 2 r 2 ).
Approximation to area of a segment intercepted between a chord of
length C and the circle, the perpendicular distance being h at the middle of
the chord. ,4=2/3 hC.
Ellipse, with semi-major and semi-minor axes a and b respectively,
A =s nab.
Sphere: radius r. A=4n:r*.
Zone intercepted between two parallel section planes, distance h apart,
A=2nrh. When /r=2r. 4=4".
METHODS. The areas of surveys may be determined (1) Arithmeti-
cally, (2) Graphically, and (3) Mechanically.
(1) Arithmetical Methods. Occasionally areas are calculated directly
from the field notes, usually as (a) areas of skeletons, and (b) outlying
areas at boundaries. The areas of skeletons are readily found by the
above trigonometrical rules, or by co-ordinates, but the outlying strips
involve tedious calculation by trapezoids between offsets, which also
can be facilitated by a co-ordinate method.
(2) Graphical Methods. Sometimes the area is calculated from (i)
Partial areas, (a) and (b) as above, and sometimes as (ii) Entire areas.
(i) Partial areas, (a) The area of the skeleton is taken off by scale
measurements, usually of the altitudes and bases of the constituent
triangles, which is more accurate and expeditious than the artifice of
reducing polygons to triangles of equal areas.
(6) Although the actual offsets introduced in plotting might be used
as in the foregoing method, the usual plan is to erect false offset
ordinates at regular distances along the survey line.
Fio. 72
The areas of the strips may then be calculated by Trapezoids^ or by
Simpson 's Rule, or by Mid-ordinates, as indicated on the right of Fig. 72.
AREAS AND VOLUMES 125
(1) By Trapezoids. Let y Q and y n be the end ordinates, y l9 j> 2 , >> 3 , etc.,
the Intermediate ones, x the common distance between the ordinates,
and Y the sum of the intermediate ordinates. Then the area A by the
trapezoidal rule:
A=\x 0> +2 Y+y n ).
If the ends converge, as shown dotted at A and B 9 the terms y and y n
disappear, and A=xY.
(2) By Mid-ordinates. A common method, particularly with com-
puting strips, is to insert ordinates midway between the false offset
ordinates. In this case the trapezoidal rule becomes:
where w is the sum of the ordinates m l9 ra 2 , w 3 , etc. (Fig. 72.)
Both the foregoing methods are based upon the assumption that the
several offset figures are trapezoids, and this leads to results that are
sufficiently accurate for most purposes. If, however, the boundaries
are really curved to such an extent that appreciable error is likely to
be introduced, the areas should be calculated by Simpson's parabolic
rule, sometimes called Simpson's First Rule.
(3) By Simpson's Rule. In applying this, the better known of the two
rules, it is necessary to divide the area into an even number of strips
of the same width x 9 the odd number of ordinates again being the
several false offset distances to the boundary. If, as before, y Q and y n
be the end ordinates, y l9 y 2 , y& etc., the intermediate ordinates, and
x the common distance between them, then
A=x/3 (y*+y n +2 (yt+y*+y 9 , etc.)+4 (yi+y*+y 6 , etc.)
Width f Sums of Ordinates
3 LOnce End+Twice Even |- Tour Odd
(ii) Entire Areas. The chief methods applied to whole areas are
(1) By Division into Triangles and (2) By Division into Trapezoids,
introducing the computing scale and
the use of Simpson's rule.
(1) By Division into Triangles. In
this method the resulting outlying
sides of triangles are not wholly
inside the boundaries or identical
with those of the survey skeleton,
but are such that they balance out
the inequalities of the boundaries by
serving as "give and take" lines.
Fig. 73 shows a survey with irregular
boundaries, pencilled into triannlcs //"
for treatment by this method. The Fo 73
resulting triangles ABC, CD A, DEA,
are inserted so that their outlying sides AB 9 BC 9 CD 9 DE 9 and EA
each takes into its own area portions equal to those which it gives
f
\
126 ELEMENTARY SURVEYING
outside. These outlying sides are found by stretching a fine thread
along the boundaries, or, better still, by using a couple of set squares.
After a little adjustment, the lines are drawn, resolving the area
into triangles, the areas of which are found by multiplying half the
respective altitudes by the corresponding bases; thus:
\dD-AC\ IbB-AC, and \eE-AD.
The method is far more accurate than it first appears to be, since
the portions equalised are small in comparison with the areas of the
corresponding triangles.
(2) By Division into Trapezoids. In principle the area is divided into
a number of parallel strips of the same width x, not by ruling equi-
distant parallels across the plan, but preferably on a sheet of tracing-
paper. This tracing-paper is placed over the plan, and is shifted about
so that the area is exactly enclosed
between extreme parallels, thus
avoiding an odd area at one
extremity.
The process of taking out an
area consists in finding the area
of every constituent strip of the
^ . , figure. This is done by measuring
the mean lengths of the strips, as
x indicated by mm 1 in Fig. 74, where
^^^^.^ *_
the dotted line reduces the length
to that of an equivalent rectangle,
the area of which is equal to the width x of the strip multiplied by
the length mm'. Thus the area of the survey,
A^x (aa'+bb'+cc 1 , etc.),
indirectly in square inches or directly in acres, according to the width
x employed.
Square Inches. Commonly strips of convenient width, 1 inch, say,
are used, and the map area is taken out in square inches, which are
afterwards reduced to acres.
This method has the advantage that transparent squared paper can
be used, the small squares serving in obtaining the lengths aa', bb', etc.
In fact, large areas can be dealt with on small sheets of squared paper
if the survey is appropriately divided into parts; four, for instance.
Acres. Even if only an ordinary decimally-divided inch scale is
available, it is possible to rule a sheet of tracing-paper so that the
acreage can be found directly for any given scale. This is done by
making the common width x between the parallels that value in inches
which would be expressed by 10 divided by the square of the number n
of chains to the inch in the scale of the survey. Every inch length of
the strips will then represent an acre and every tenth of an inch one
square chain.
AREAS AND VOLUMES
127
Thus, for 2 chains to an inch, the distance between the parallels
will'be 10/4=2J in.; for 3 chains to an inch, 10/9 = 1-11 in.; and for
4 chains to an inch, 10/16= | in.
Because on a scale of n chs. to 1 in., the width A- will be -, and
10 10 . n "
this will represent -~^.n-- chains on the map, and since an inch length
of the strip represents n chs., the product of these measurements will
10
represent n=lQ sq. chs. = l acre.
FIG. 75
*The best-known device for computing areas in this way is the
computing scale, one form of which is shown in Fig. 75. These instru-
ments can be obtained in various divisions, ranging from two ordinary
scales to universal patterns with six chain scales and two Ordnance
Survey scales. Some patterns are divided for use with J-inch strips and
others for strips representing one chain widths; and therefore each scale
can be applied directly only to maps on the scale for which it is divided.
The use of the computing scale needs little explanation, once the
survey is enclosed between parallels on tracing-paper appropriately
divided. The indicator of the sliding frame is set to zero on the scale,
and the scale is placed parallel to the rulings, with the wire cutting the
beginning of the first strip, "squaring the boundary," as at a in Fig. 74.
The frame is then slid until the wire cuts the end of the strip, squaring
the boundary, as at a', the scale being held firmly in position. The
scale is then lifted and placed parallel for the second strip by moving
it bodily until the wire cuts the beginning of the second strip, as at b.
Then, as before, the scale is held firmly in position while the frame is
moved until the end of the groove in which the frame moves is reached.
A mark is now made under the wire and the scale is inverted and
placed with the wire at the mark, after which the frame is moved as
before, summing up the strips until the other end of the groove is
reached. A mark is then made at the wire, and against it the acreage
of the first double travel is recorded. Next the scale is set right way
up again, and the process is repeated until all the strips have been
measured. The acreage is then cast up for the number of double
travels noted plus the final reading of the scale.
*Simpsori > s Rule. The method described on page 125 is sometimes
used in computing entire areas from figures divided into an even
128 ELEMENTARY SURVEYING
number of strips at regular intervals x inches apart, preferably on a
sheet of tracing-paper. Often this is shifted so as to enclose the 'area
exactly between parallels, and this means that the sums of the first and
last ordinates is zero. Since an open frame is unnecessary when linear
ordinates are measured, a brass frame, or cursor, can be fitted to an
ordinary decimally-divided scale, a pointer at the upper edge serving in
summing up the sets of ordinates, odd and even, as the case may be.
Obviously the area may also be determined by covering the survey,
or a portion of it, with a sheet of transparent paper, or paper may be
ruled so that each square contains so many square chains. Usually it
is quicker to work in square inches on prepared paper and afterwards
reduce the acreage.
*(3) Mechanical Methods. The most popular mechanical method is
by means of an instrument known as the planimeter, which is used less
in surveying than in other connections. This instrument, in its best-
known form, consists of two arms jointed together so as to move
relatively to each other with perfect freedom. Near the joint is
the rolling wheel, and at the extremity of one arm is the fixed
pole P, while at the extremity of the other arm is the tracing point T 9
a tiny handle being provided for guiding it over the plan. Connected
with a gear to the rolling wheel is the index wheel, or dial, which shows
the number of units of area encompassed, fractional parts being read
with a vernier at the edge of the rolling wheel.
In simple patterns one revolution of the indicator corresponds to
ten revolutions of the rolling wheel, which is divided into 100 divisions,
a tenth of each division being read by means of the vernier.
The theory of the planimeter is beyond the scope of this book,
since it involves a knowledge of integration. Also the instrument
is made in many patterns, though the original Amsler instrument
had arms of fixed length, giving areas in square inches or
square centimetres. In later patterns the tracing arm was made
adjustable, being divided to correspond with official scales,
giving areas directly, some designs even allowing for shrinkage of
maps.
Therefore it is always desirable to test an instrument by running it round
a square or circle of known area.
In use the pole P is set outside the area, if possible, the point being
pressed into the drawing board. If P is inside the area, a correction, as
stamped on a weight which fits over P, must be added. The index
wheel is then set to zero by rotating the rolling wheel, stopping with
the zero of this wheel at the vernier index. (Otherwise the initial reading
must be subtracted from the final reading of the instrument.) The
tracing point Tis then guided round the area in the clockwise direction,
following the boundary lightly and carefully, and stopping at the
starting point. The nearest lower value on the index wheel is recorded,
and to this is added the fractional part as read on the rolling wheel and
AREAS AND VOLUMES 129
the vernier. Also the constant value must be added if the point P was
necessarily inside the area during the operation.
Thus with a simple instrument divided for square inches, if 9 is the
reading on the index wheel, and 72 on the rolling wheel, with coinci-
dence at 3 on the vernier, the area is 9-723 sq. in.
It is interesting to note that a planimeter has been improvised with
a jack knife, though with indifferent success on the part of many.
In emergencies areas have been cut out of cardboard of uniform
thickness and compared in weight with a square of the same material
after careful weighings on a chemical balance.
II. VOLUMES
The unit of cubic content, or volume, in earthwork estimates is the
yard cube, which was regarded as the amount that could be hauled in
a one-horse cart. It is also used in the measurement of concrete and
brickwork. In related connections, the cubic foot and the bushel may
be used, while brickwork may be measured in rods and timber in cubic
feet or standards.
The following summary shows the volume content V of certain simple
solids, A being the base area, r the radius generally, and h the altitude as
measured perpendicularly to the base.
Prisms, right or oblique. V^A.h\ square rectangular, trapezoidal, etc.
Cylinders, right or oblique. y=A.h=^nr z .h. Hollow K-TT (R z r 2 ) h.
Frustrum of right cylinder. V=far*(hi + h 2 ), where h t and h* are the
greatest and least heights.
Cones, right or oblique. V=\A.h^\Ti^.h.
Frustrum of cone or pyramid, /* (A f VAa'^a), with h between sectional
areas A and a.
Pyramid, right or oblique. V=^\A.h.
Spheres. V= Jjcr 3 = i (surface) x (radius).
The solids most commonly associated with earthwork calculations
may be defined as (1) Section Prismoids; (2) Truncated Prisms, and
(3) Contour Prisms.
(1) Section Prismoids. A vertical section of the earth's surface as
found by levelling along the centre line of a projected railway or
reservoir is known as a longitudinal section and vertical sections at
right angles to these are known as cross sections, the shapes of which
are also determined by levelling.
(Although the subject of sections is dealt with at the end of this
chapter, it might be consulted during the reading of the present section.)
The solids in the present category are derived from the irregular
cross sections of cuttings and embankments in the construction of
railways and highways, The method consists in finding the areas A l9
A 2 , A 3 , etc., of successive cross sections, usually 1 ch. (or 100 ft.)
apart, and using these in the trapezoidal or prismoidal rules for volume
content. The solids approximate to irregular truncated pyramids, and
may be considered prismoids, which are solids having for their ends
130
ELEMENTARY SURVEYING
any dissimilar parallel plane figures of the same number of sides and
all faces plane figures.
Three standard types of cross sections will be considered. In these
d will be the centre line (C.L.) depth of cutting or banking, w the half
formation width, s the side slope ratio, s horizontally to 1 vertically,
and r the crosswise or lateral slope of the ground, r : 1 likewise. Thus
s and r are the co-tangents of the angles which the side slopes or the
ground surface make with the horizontal, but s is commonly 1, 1 |, or 2,
whereas r can have a wide range of values.
In Cases (a) and (b) of the following treatment, the imaginary
formation triangle OPQ, which is neither excavated nor made, will be
incorporated in order to simplify
the formulae. Its area a is con-
sistently w 2 /s and therefore its
volume v is w 2 /sx I in a length /,
and, being a right triangular prism,
this value will follow from all
rules. Hence, whole areas A' will
be calculated from whole depths
D=d+wls, and the true areas A
will be A' a.
Case (a) Ground Level Across
(Fig. 76). Here the side widths are
the whole area A f = : sD 2 ^s(d+w/s) 2 =(2w+sd)d+w 2 js . . . . (1)
But since the area of the formation triangle a--=w 2 ls 9 the true area
A=(2w+sd)d: a very inconvenient expression.
The values would be precisely the same for a cutting.
Thus for a cutting in which the formation width 2n> is 20 ft., and the centre
line depth of cutting d is 10 ft., and the side slope ratio 1:1, the whole
depth D=d+wls=2Q ft., the whole area A'=sD* = \ x(20) 2 =400 sq. ft.,
and the true area A'=A'-\v*ls=4W- 100-300 sq. ft.
Case (6) Ground Sloping Across (d>wjr) (Fig. 77). Here the side
widths are different, being W\ and W r on the left and right respectively:
W l =CT^W-R f T=W~-s(RT)=W-sWi tan a =W~
Also 0V=
<***
1 l+s/r l+s/r.
= W+S'T' =
W
= W+ W*\r
Or
\-slr~~\-
The whole area A '= J( W t + W r )D = rzrai , (2)
i s / r z
the latter expression being that for ground level across in (1) divided
by 1 2 / r 2, Also the former expression in (2) frequently occurs in "three
level" sections in the American method of cross-sectioning.
AREAS AND VOLUMES ui
Similar expressions would follow if an embankment were considered
witi the higher ground on the left.
Thus for an embankment in which the formation width is 20 ft., the centre
line height of bank 10 ft., the side slope ratio 1J : 1 and the sidelong ground
slope r : 19 : 1, the whole depth D-=IO {- 10/1 i 16-7 ft., and the whole
area 1H16-7)*
A'=- , -430-2 sq. ft.
1 - l ,36
Also the true area ^4 = 363*5 sq. ft., since the area of the formation triangle
w 2 /s=66-7 sq. ft.
Case (c) Hilhide Sections
(d<w!r) (Fig. 78). Cross
sections of this nature are best
treated right away as true areas
since one portion is in 'cut"
and the other in "fill." Also
the side slope s' is necessarily
flatter for the banking than
for the cutting, "made ground"
being less stable than that
under excavated earth. Flo> 78
Here h=(w+x+sh)llr and h f ^(w-x+s'h')l/r
, ^
True area Right-^fir^/y true arca
, IV+A- rs ,
If these areas are equal, ---- ^= y- -7=^;
~
... (3)
r >
The side widths W=h' cot. a x=h'rx
W T =h cot. a +x=hr +x
132 ELEMENTARY SURVEYING
Two rules are used in the calculation of volume content from cross
sectional areas A l9 A 2 , A Zy etc., / units apart, / being usually 66 ft. 'and
100 ft., convenient submultiples being introduced.
(1) Trapezoidal Rule. In this, the average end area rule, the volume
is calculated from the mean of the areas at the ends of horizontal
lengths / along the centre line, A l and A 2 being true or whole areas.
Thus K=J/(^ 1 +^ 2 ).
Hence for a series of areas A ly A 2 , A 3 . . . A n , all / ft. apart, the total
volume will be
V=$l(A 1 +2A 2 +2Az+2A 4 + . . . A J cu. ft.
The well-known earthwork tables of Bidder were based upon this
rule, the centre line distances being in Gunter chains. The trapezoidal
rule is commonly used in preliminary estimates.
(2) Prismoidal Rule. In this rule, which was the basis of Sir John
MacNeill's tables, it is assumed that the surface of the ground between
any two vertical cross sections is such that the volume content is a
prismoid, the end areas not necessarily being similar, but of any shape
whatever, provided the surfaces between their perimeters can be
regarded plane.
|A 3
PIG. 79
According to this rule,
,_>
where A m is the area midway between sections proper, such as A l and
A 2 . Now A m is not the mean or average of the areas A l and A* and
in complex sections it should be determined. If the surface slope is the
same as at A 1 and A 29 the area A m may be calculated from the mean
depth K^i+^a) or \( D \+ D *)> as the rule will apply to both whole and
true areas. But if the end sections have different lateral slopes, the
central depth will still be the mean of the end depths, but the lateral
slope will be the harmonic mean of the end slopes. The prismoidal
rule is used in final estimates, sometimes by applying "prismoidal
corrections" to the trapezoidal rule. It is not strictly confined in
practice to solids with plane faces, but has been used in calculating
curved volumes.
Viewed from this standpoint the prismoidal rule becomes applicable
when the solid is not strictly a prismoid.
Now if the even numbered sections are used as the middle sections,
AREAS AND VOLUMES 133
/ becomes 21 as indicated in Fig. 79, and the rule reduces to that of
Simpson, A x corresponding to y in Fig. 72:
y= 2 ^(A l +4A 2 +2A 3 +4A.+2A, 9 etc.)
=- (End Sum+4 Even Sum+2 Odd Sum)
As in the case of areas, it is inadvisable to say dogmatically that this
method overestimates or that underestimates the content. Such statements
usually refer to areas with straight boundaries between ordinates or plane
surfaces between sections, and not as would really be given by field measure-
ments or levels. Also the comparisons are not always consistent, one often
being in fact an approximation.
Example. The following notes refer to actual sections taken on level across
ground at 50 ft. intervals for an embankment with a formation width of
20 ft. and side slopes 2 horizontally to 1 vertically. In order to show the
discrepancies that can arise from calculations alone, the volumes are calcu-
lated by the Trapezoidal Rule (a) for sections 100 ft. apart, (b) for the 50 ft.
sections, and by the Prismoidal Ride applied as Simpson's rule, (c) with
extreme sections 200 ft. apart and (d) with these 100 ft. apart, the 50 ft.
sections serving as the middle area^.
Distance 50 100 150 200ft.
Centre line depth d 6-2 6-9 8-4 10-0 9-8 ft.
Whole depth />=</+ w/j 11-2 11-9 13-4 15-0 14-8 ft.
Whole area A'=sD* 250-9 283-2 359-1 450-0 438-1 sq. ft.
The volume of the formation prism, 200 ft. in length, = 10,000 sq. ft.
= 370 cu. yds. is to be deducted.
Trapezoidal Volumes:
(a) 100 (250-9+2(359-l)+438-l) = 70,360 cu. ft. =2,606 Cu. yds.
- 2 Net. 2,236 cu. yds.
(b) 50 (250-9 h2(283-2)+2(359-l + 2(450) + 438-l)=71,840 cu. ft.=
~2 2,661 cu. yds. Net 2,291 cu. yds.
Prismoidal Volumes:
(c) 200 (250-9+4(359-l)-f438-l)=70,847 cu. ft. = 2,661 cu. yds.
-~ Net 2,254 cu. yds.
(d) 100 (250-9+ 4(283-2)+2(359-l)+4(450)+ 438-l) = 72,333 cu. ft.-
~6~ 2,079 cu. yds. Net 2,309 cu. yds
Since both (b) and (d) introduce two more measurements, they will more
closely represent the true content, which possibly lies between the given
values, since (d) is not strictly, if practically, applied to the prismoid, though
its use is justified by the assumption common to Simpson's rule for volumes.
Although a single comparison is poor evidence, it shows that a small dis-
crepancy of 0-8 per cent occurs between (b) and (d) against 2-4 per cent
between the two applications of the prismoidal rule in (c) and (d).
(2) Truncated Prisms. Whenever a considerable width of surface is
to be excavated, as in the case or rectangular reservoirs, building, or
other sites, the most accurate method is that of taking out volumes
from a series of vertical truncated prisms, squares being laid out and
levels taken at the corners, as described in Chapter IX with reference
134
ELEMENTARY SURVEYING
/I
FIG. 80
to Fig. 69 in the method of contouring with grid squares in horizontal
control.
Fig. 80 shows a portion of an area marked out in unit squares of
(say) 50 ft. or 100 ft. side, the comer reduced levels having been taken
by means of a dumpy level. Now if
the reduced level of the formation, or
finished surface, is fixed, the difference
of reduced level will be the cut or fill at
each corner, and the total volume of
excavation or filling will be the sum of
the volumes of the constituent truncated
prisms, the right section of which is a
unit square. Also the volume of any
right truncated prism is the area of its
right section multiplied by the distance
between the centres of its bases, and, in
Fig. 80, is the area of a unit square
multiplied by the mean difference of
reduced level of the four corners, which
for a level formation, is the average
reduced level of the corners less the reduced level of formation.
Hence, if the corners abed of the squares indicated are 51-8 ft.,
53-9 ft., 52-7 ft., and 54-8 ft. above datum respectively, and the forma-
tion level is 40 ft., the mean height of the truncated prism will be
KH-8+13-9+12-7+ 14-8) = 13-3, while if the square is of 50-ft. side, the
volume of the prism will be 1 3-3 x 50 x 50 =33,250 cu. ft. 1,321 cu. yds.
If the finished surface is to be inclined or graded, the calculations
will be the same, but the mean height is taken as the differences between
the reduced levels of the corners in the surface and in the formation.
Obviously the area to be levelled or graded may not be exactly
rectangular, and in this case, a number of irregular solids will occur
at the boundaries. These will be mainly trapezoidal in section, triangles
occurring now and then; but the same rule applies as to the height of
the truncated prisms, being the average difference of the reduced levels
at the corners, while the areas will be merely those of trapezoids
or triangles.
In practice, there will usually be parts of the area which are to be
excavated and parts which are to be banked. Hence it is convenient
to prefix the values at the corners + or , signifying cuts and fills
respectively. The cuts will be separated from the fills at formation
level by irregular lines which are actually contours in the case of level
formations. Sometimes these areas are given a light wash of colour
to distinguish them from each other, areas at formation level being
left white.
After taking out the content by the foregoing rules, account must
be taken of the fact that earth expands on excavation and shrinks to
AREAS AND VOLUMES
135
some extent after being placed as filling, the allowance varying with
difterent earths and materials.
The foregoing methods are mainly arithmetical since the calculations
are made directly from the field notes.
Graphical Methods. Sometimes, however, wide vertical cross-sections
are plotted, their areas found graphically or by means of the planimeter,
and the corresponding volumes arc calculated by the average end area
rule, or even the prismoidal method.
(3) Contour Prisms. Also, estimates are sometimes taken from the
horizontal sections given by contour lines, as in the case of the water con-
tent of the impounding reservoir
shown in Fig. 81. Here a dam AB
with a vertical water face is shown,
the top water level (T.W.L.) being
80 ft., as indicated near the
contour. The successive areas at
the different contour elevations
are found graphically or mechani-
cally as >4 80 , ^ 70 , A^ and A 6Qr
(Fig. 81.)
The volume is then calculated
for the several layers, or laminae,
10 ft. in depth from Y=\Q(\A 80 +
A 7o~l~^fio+^5o) cu - ft-, which may be expressed in millions of gallons,
with 6-24 gallons to the cubic foot.
Another of the various methods consists in covering the contour
area with a grid, and reversing the process by finding the corner
elevations by interpolating between the contours on the map.
III. LONGITUDINAL SECTIONS
Longitudinal sections, called "profiles" in the U.S.A., are an impor-
tant feature in engineering plans. They are false sections because the
vertical scale which shows reduced levels is larger than the horizontal
section which shows the corresponding horizontal distances along the
centre line of the proposed railway, road, or sewer. The vertical scale
is roughly 8 to 10 times that of the horizontal scale taken to the nearest
convenient figure. Thus, with a horizontal scale of 200 ft. to 1 in.,
the vertical scale could be 20 ft. to 1 in.; with a horizontal scale of
50 ft. to 1 in., 5 ft. to 1 in. If the horizontal base were the actual
datum of the survey this would often lead to an unsightly section, and
waste of paper. Hence, in order to obtain a neat section, it is usual
to raise the datum, stating the fact thus along the base A B 09 "50 ft.
above datum," as in Fig. 82. Also, the sections are opened out like
a screen unfolded, so that points in elevation are not vertically above
the same points in plan, as in geometrical projection. The fact that
136 ELEMENTARY SURVEYING
the section is longer than the plan is evident in Fig. 82, which shows
the traverse and section of a portion of a proposed railway, the distances
"running through" in chains continuously from the beginning of the
line. The section shows the reduced levels at 1 chain intervals along
the centre line, with additional values at the points at which the direction
of the traverse changes. These points and the beginning and the end
of the section are shown with thicker lines than the rest of the ordinates,
and it is a rule in plotting sections to join the tops of the ordinates with
straight lines, never with a free curve as in the case of a graph.
V
- rm,^
a
1
1
1
i
i
f^i^T^-
i i i
b
21 |22
fc
|24 )25 |2t
27 [ 28 1 2* 1 30
31
50
FT. ABOVE DATUM
(a) VERTICAL SECTION
23-6
(b) PLAN
Fio. 82
The straight line AB in the section is drawn at formation level, and
is known as the gradient, which is expressed by the tangent of the
vertical angle, as 1 in x horizontally, or 1 /80, or as a percentage, 1-25 per
cent, being sometimes prefixed with the plus or minus sign, according
as it is rising (upgrade) or falling (downgrade).
Incidentally, the gradient of pipes is taken at the "invert," which
is the lowest point on the interior.
In Fig. 82 (a) it will be seen that a cutting will occur between 21-0
and 23-3 chs., a bank between 23-3 chs. and 26-6 chs., and a cutting
between 26-6 and 30-4 chs. Also, the ordinates above or below AB are
the centre line cuts and fills, which in the days of more elaborate plans
were often tinted red and blue respectively, a neat array of information
being tabulated along the base A^Q. Now if the gradient is settled
upon from the section, a rapid means of estimating the earthwork
volume is at the surveyor's disposal. Thus, if lines ab and a'b' are
drawn parallel to AB at depths w/s above or below formation, these
will be whole depths Z>, and if the ground surface is level across the
cross-sectional areas will be sD* t while for ground with a lateral slope
5.3
of r to 1, they will have this value divided by 1- -^ as explained in
(2) of page 130.
AREAS AND VOLUMES 137
Cross-sections are true sections, and when drawn, arc plotted on
a (Jommon horizontal and vertical scale.
It is a difficult matter to write a conclusion to a chapter of this
character, since old heads cannot be put on young shoulders, and ex-
perience is something that cannot be imparted by words. In the preamble
to this chapter, the term "correctness" was used to imply arithmetic
devoid of mistakes, since the word "accuracy" alone might suggest
the use of an approximation that would fully satisfy practical require-
ments. Briefly, when information is required the methods should be
adequately accurate and the calculations arithmetically correct. Now
there are not only arithmetical approximations in calculations, but
also visual approximations in the field, in that a feature which appears
even marked to the eye may be trifling as a part of the whole. Thus,
level ground should not suggest a bowling green, but anything up to
a general slope of 3, or a surface warped to a series of slight irregu-
larities. Thus, it often happens that elaborate rules are really ineffec-
tive, and this is frequently the case in ascertaining the cross-sectional
areas of rivers. Of course, there may be some satisfaction in using
the complex. Earlier engineers and surveyors held the mathematician
in awe, and misapplied his teachings, reverently, at least, little realising
that the natural errors of their work overwhelmed any refinements
these rules might otherwise have introduced. Simplicity is the surest
path until experience proves its limitations. Apart from these there
are economic factors that demand rapid or good estimates, each of
which has its place; the former utilising graphs, charts, and other
artifices, and the latter drcreet and careful calculations with appropriate
checking. It has been said that there are computations, estimates,
guesses, and back answers, the last suggesting an absurd response to a
ridiculous request for a statement in an unreasonable period of time.
But this presupposes that the reader will proceed further with the
subject and will learn that much truth is said in jest. Hence, the work
should be kept to the first two categories; and this means no juggling
with rules or scrupling with trivialities, but getting ahead confidently,
obtaining the right data and applying it with reason.
CLASS EXERCISESAREAS
10 (a). Draw an irregular figure about 4J in. x 3 in. to represent a survey
on a scale of 5 chains to 1 inch, and describe with reference to this area two
ways in which you would determine its acreage. (G.S.)
10 (Z>). Draw an irregular closed figure about 3i in.x2i in. to represent
a pond on a scale of 1 in 2,500, and describe with reference to this figure two
methods of determining its area olhe r than by the use of squared paper
10 (c). A race track is to consist of two straight portions and two semi-
circular ends, the width of the track being 29 ft. and the length i mile,
measured around the inner edge of the track.
A rectangular plot which exactly encloses the track is to be purchased for
the purpose at 400 per acre.
138 ELEMENTARY SURVEYING
The committee suggest (a) that the straight portions should oe equal in
length to the outer diameters of the ends, while the surveyor recommends
(6) that the outer radii should be 110 ft.
Calculate the saving that would result by taking the surveyor's advice.
(G.S.)
(245 65. &/.)
10 (d). Sketch an irregular figure approximating to a rhombus of about
4 in. side to represent an area on a scale of 2 chains to 1 inch.
Determine its area by the following methods:
(a) Give and take lines; (b) Division into trapezoids; (c) Simpson's rule.
10 (e). You have a computing scale divided into inches and decimals, and
you are required to find acreages directly on the following scales:
4 chs. to 1 in.; 5 chs. to 1 in.; 1 in 2,500.
State the spacings of the parallel rulings on tracing-paper for use with the
scale. (0-625 in.. 0-40 in., 1-03 in.)
CLASS EXERCISES VOLUMES
10 (A). A straight and level roadway, 20 ft. wide, is being cut through a
plane hillside which slopes 1 vertically in 9 horizontally at right angles to the
road although it is level in the direction of the road.
The side slopes of the cutting will be 1 vertically in 1 horizontally and
the depth of the cutting will be 10 ft. on the centre line of the road.
Calculate the volume of excavation in a horizontal length of 500 ft. (G.S.)
(5,648 cu. yds.)
10 (B). A reservoir is to be constructed with a flat rectangular bottom in
which the length is 1 times the breadth. It is to hold one million gallons of
water with a depth of 15 ft. Calculate the dimensions of the surface and
bottom rectangles, given that the side slopes are 3 horizontally to 1 vertically.
1 cu. ft. of water = 6 gallons.
(95-44'. x 63-63'., 185-44'. x 153-63'.)
10 (C). The following distances and reduced levels were taken in con-
nection with a drain:
Distance 25 50 75 100 125 150 175 200ft.
Red. level 91-8 92-0 92*4 93-6 94-2 95-2 96-4 95-7 94-1
The invert level of the drain is 88-6 at the beginning and falls 1 in 100.
If the trench is rectangular, 2 ft. 6 in. wide, calculate the cost of excavation
at Is. \d. per cu. yd.
Plot a section of the ground surface and the bottom of the trench on a
horizontal scale of 25 ft. to 1 inch and a vertical scale of 10 ft. to 1 inch.
(4 llj. 4rf.)
10 (D). The following sectional areas were taken at 50 ft. intervals in a
straight trench:
32-5 33-0 35-0 36-0 38-0 sq. ft.
In calculating, the prismoidal rule was used with only the end and middle
areas. Determine the error in cu. yds. due to this misapplication of the rule.
(G.S.)
(75 cu. ft. overestimate. 7,016-7; 6,491-7.)
10 (E). The following heights of embankment were reduced at 100 ft.
sections on a proposed railway, the ground being level across. The formation
width is to be 30 ft. and the side slopes 2 horizontally to 1 vertically. Calculate
the volume of the embankment by Simpson's rule.
7 12 14 13 9 8 4 Oft. (12,720 cu. yds.)
AREAS AND VOLUMES 139
FIELD AND PLOTTING EXERCISES
10(F). Morning:
(A) The range-poles indicate a line AB which is to be levelled with stuff
readings at intervals of 50 ft., starting from an imaginary benchmark 50-0
(chalked A)- Take staff readings at the 50 ft. points and reduce the level
on a form you have prepared in the Answer Book.
Afternoon:
(B) Using your level notes, plot the corresponding vertical section with a
horizontal scale of 25 ft. to 1 in. and a vertical scale of 20 ft. to 1 in. Finish
the section neatly in pencil and insert the horizontal scale.
Imagine that a trench for a drain is to be dug with its bottom" 2 ft. below
ground level at the lower point A, and 3 ft. below ground level at B.
(a) Insert the line of the bottom of the trench on your section and find
its gradient.
(b) Calculate the volume of excavation in cubic yards, given that the trench
is uniformly 3 ft. wide. (G.S.)
ORIGINAL PROBLEMS
Calculate the areas of the surveys in Problems . . .
Find the subaqueous contours of a pond, and from these and the survey
estimate the water content.
Calculate the volume of earth in a knoll from contours directly located.
Estimate the earthwork in levelling a plot for a tennis-court with 15 ft.
level margins around.
CHAPTER XI
"THEODOLITE SURVEYING
It would seem unfair to the reader if his curiosity were not appeased
by some mention of that instrument which has come to be regarded
as the embodiment of surveying: the Theodolite, the most perfect of all
goniometers, or angle-measuring instruments. Thus, opportunely, this
chapter may ease the passage to the more advanced branches of
surveying.
The first mention of the rudimentary form of the instrument in
English literature concerns the "theodolitus" of Thomas Digge's
"Pantometria," 1571. The name, derived from theodicoea, was, in that
old writer's sense of perfection, the most perfect of known surveying
instruments. Also, there are grounds to believe that an equivalent
Arabic root has given us the word "Alidade," which is associated with
the plane table or the upper part of a modern theodolite.
The theodolitus consisted merely of a,horizontal circle divided and
figured up to 360, and fitted with a centred, sighted alidade, the
entire instrument being mounted upon a stand. The nearest instrument
of this form is the almost-extinct "circumfercntor" of about seventy
years ago, some patterns of which very closely resembled the American
Surveyors' Compass. It was not until the close of the eighteenth
century that the theodolite assumed its present form, largely at the
hands of Jonathan Sissons, the inventor of the Y-level; and in the
early years of the nineteenth century Ramsden added substantial
improvements; in particular, the transit principle, by which the telescope
could be rotated in the vertical plane. Quite a romance could be
written about the evolution of the theodolite, introducing its various
forms, ranging from the Great Theodolite of the Ordnance Survey
and Borda's Repeating Circle to the modern geodetic and engineering
models. It is gratifying to know that English makers have been fore-
most in the design and construction of surveying instruments; and
some of the pioneers of American instrument-making received their
early training in this country.
As may have already been concluded, the primary function of the
theodolite is the accurate measurement of horizontal and vertical
angles, i.e. angles respectively in the horizontal and vertical planes.
Apart from special designs, the modern theodolite is made in sizes
ranging from 3,in. to 12 in., the size being specified by the diameter
of the horizontal graduated circle.
In an elementary text-book it is impossible to describe theodolites
in general, though it is desirable that any description should refer to
an actual instrument rather than to an improvised model. For this
140
THEODOLITE SURVEYING 141
reason, a vernier pattern of a general purpose transit theodolite will be
considered as the representative instrument. This is shown dissected
in Fig. 83, in order that the essentials of theodolites may be explained.
The theodolite consists of the following four primary portions, which
are shown separated, the reference letters corresponding to those on
the diagram.
I. The Vertical', II. The Plate-Standards', III. The Limb; and IV. The
Levelling Head. I and II together form the "Alidade" of the instrument.
I. Vertical. This comprises (1) the telescope with its eyepiece E
and ray shade 7?, the azimuthai level B, and the horizontal or transverse
axis o\ (2) the vertical circle and (behind) its two verniers A/, the
magnifiers m being omitted; (3) the clipping frame with its clipping
screws 77, and (behind) the clamp Kand tangent screw or slow-motion
v to the vertical circle; hereafter called the "Vertical Motion"
Sometimes the level B is fitted as an altitude level on the top of the
clipping or vernier frame.
Frequently the verniers M of the vertical circle are stamped C and D,
but usually the former is understood as the vernier.
The horizontal axis fits into the bearings at the tops of the standards
(4) and is secured with little straps and a screw.
II. Plate-Standards. These consist of (4) the standards (here A
frames) which at the tops provide a trunnion bearing O for the hori-
zontal axis 0, and also carry the plate levels, />, for levelling the instru-
ment; (5) the upper horizontal plate, which carries the two verniers TV
(their magnifiers n being omitted), and the clamp U to the upper plate
with its tangent screw or slow motion u, hereafter called the ''Upper
Motion"
Frequently the verniers N are stamped A and B 9 the former being
understood unless qualified.
Centrally at the bottom of the upper plate is the solid inner spindle
(ii) which fits into the outer hollow spindle (iii) of the limb.
III. Limb. This simple component consists of the horizontal circle
divided and figured in degrees on silver, and the outer hollow spindle
(iii) which fits into the bearing afforded inside the levelling head.
IV. Levelling Head. Here the older pattern four-screw device is
shown, the lower parallel plate being bored and threaded in order that
the entire instrument may be screwed to the top of its tripod. F, Fare
the plate screws with which the instrument is levelled after the tripod
has been planted, the levelling being regulated by the position of the
bubbles of the plate levels p, p, as described on page 68. A small
hook is inserted in the nut which secures the spindles (ii) and (iii) in
position in the levelling head, and from this hook a plumb-bob is
suspended. Some levelling heads are fitted with a centring stage or
shifting plates so that the plumb-bob can be set exactly to a cross OD
a peg at the station beneath the instrument.
In the model shown the levelling head carries the clamp L of the
to
Fro. 83
THE THEODOLITE
THEODOLITE SURVEYING 143
limb and its tangent screw, or slow motion /, hereafter called the
"Eower Motion"
Manipulation. When the instrument has been re-assembled and
levelled up at a station O, say, the outer spindle can be clamped to
the levelling head by means of L, while if the inner spindle is undamped,
U being slack, the vernier N can be moved relatively to the divisions
on the horizontal circle, or limb. Hence, in setting the A vernier to
zero (i.e. 360), the upper plate and superstructure are turned until
the vernier index is at 360, as nearly as may be; the upper plate is
then clamped by means of /, and the index is set exactly at 360 by
means of the tangent screw u, the vernier being viewed through its
magnifier n. With the upper motion thus clamped, a station P, say
(normally the one to the left) can be sighted by slackening the clamp
L, and turning the entire superstructure about the outer spindle, as
an axis, until the foot of the station pole is seen inverted near the
intersection of the cross-wires of the telescope; the lower motion is
then clamped by means of/,, and the image of the foot of the pole is
exactly bisected by the vertical wire by turning the tangent screw /.
If now the upper motion is undamped by slackening T7, the telescope
can be directed towards the station Q, say (normally to the right of P),
the inner spindle moving inside the clamped outer spindle; and after
a near sight at the foot of the station pole, the upper plate can be
clamped by means of U, and the inverted image of the foot of
the pole exactly bisected by the vertical wire by turning the tangent
screw u, the upper plate thus moving relatively to the horizontal
circle. The magnitude of the angle POQ is then read on the A
vernier (Fig. 85).
Circles and Verniers. Circles of British and American instruments
are divided into the Sexagesimal division of 1 degree (1)=60 minute?
(60'); l'=60 seconds (60"). This system actually follows from the
ancient nomenclature of the first and second subdivisions of the degree:
'"pars minuta prima" and "pars minuta seconda" although Ptolemy
(A.D. 85-165) actually worked in arcs, not angles, dividing the circum-
ference of the circle into 360 equal arcs. The Continental anguhu
measure is the circle of 400 grades, lOOg. being equal to 90. Whole
Circle Clockwise (0 to 360) is the division of horizontal circles used
exclusively in this country; and the most rational system for vertical
circles is the Quadrant, or Quarter circle division (0-90 -0 -90 -0 ),
the zeros being in a horizontal line. This quadrant division is favoured
by surveyors who prefer to observe bearings directly, particularly in
North America, where the Half Circle (0 to 180 in both directions)
is also used, in each case subsidiary to the whole circle division.
Simple as it sounds, some confusion usually arises as to what the
whole circle division should read when the upper, or vernier, plate is
turned in the counter-clockwise direction. There seems no better
answer to this than to say that if a clock stops at 20 minutes to 5,
144 ELEMENTARY SURVEYING
then, on re-winding at 11.20, it will read 20 minutes past 11 whether
the hands be turned backwards or forwards on re-winding, apart,' of
course, from the fact that it would be indiscreet to turn the hands of
a striking movement in the retrograde direction.
Most of the smaller patterns of theodolites are fitted with verniers,
the simplest and most reliable mechanical contrivance for reading exact
subdivisions of a main division; 1, , or J, in the case of the circles
of theodolites. Named after its inventor, the vernier is a small sliding
scale on which n divisions of length v are equal to n 1 scale divisions
of length c. Thus if the scale divisions c are^fe- in. and 9 of these are
equal 10 divisions v on the vernier, then cv=-fcfo(Yo) ^-^ T^ * n -
which is the least count, signifying that the vernier will read to T ^ in.,
which would require a diagonal scale 1 in. in width.
No difficulty need ever arise with surveying instruments. Merely
divide the angular value of c, the smallest division on the circle, by
the number n of corresponding divisions on the vernier. Now n is not
necessarily the number stamped on the vernier, this often corresponding
to even minutes only.
This simple rule merely follows from:
^ A
(nl)c=nv; v=i ---- Jc; and x=c v=c f - \c=-
Thus, if c=i on the circle and =30 on the vernier, then x=fo =l'.
Verniers of vertical circles are often read upwards and downwards,
and are frequently figured in both directions. Much trouble would be
saved if these were marked plus and minus. Anyway, always read the
vernier with its numbers counted in the same direction as the figures
on the circle.
Since the graduations are finely etched on the circles, it is necessary
to take the readings of the verniers through a magnifier or reader,
attached near the vernier. This device must not be confused with the
micrometer microscopes which are fitted instead of verniers on the
more elaborate instruments. In more accurate work it is usual to read
and take the mean value from both verniers; the A and the B on the
horizontal circle and the C and D on the vertical circle. This is a
precaution against "eccentricity," which is seldom encountered to any
appreciable extent except in old or damaged instruments.
MEASUREMENT OF ANGLES. Let us assume that the tripod has been
firmly planted at the station O with the telescope at a convenient
height for sighting, the lower plate of the levelling head being fairly
horizontal. The instrument must now be levelled up in the manner
described on page 68, the bubbles of the plate levels being central.
Next the telescope must be focused, eliminating parallax, in the manner
also described. The cross-wires will appear as in Fig. 84, and the
images of the station poles will appear at these, finally with that of the
pole or point exactly bisected by the vertical wire.
THEODOLITE SURVEYING 145
(Some diaphragms will also be webbed or etched with the stadia lines
shbwn dotted in Fig. 84. The object of these is that of determining horizontal
distances D from the amount of vertical staff seen intercepted between them,
D being 1005, but always subject to corrections for vertical angles above 5,
known as Reductions to Horizontal.)
Horizontal Angles. (1) Clamp the lower motion by means of the
clamp L. Unclamp the upper motion, and set the A vernier at zero;
clamp U, and finally set the vernier index at 360 by means of the
tangent screw u.
(2) Unclamp the lower motion and sight the lowest point of the
pole at the left-hand station P\ clamp L, and obtain an exact bisection
of the image of P by means of the tangent screw /.
(3) Unclamp the upper motion, and sight the pole at the right-hand
station Q\ clamp U, and obtain an exact bisection of the image of Q
by means of the tangent screw u.
(4) Read the A vernier and record this reading as the magnitude
of the angle PO Q (Fig. 85).
Fio. 84 FIG. 85
If it is impossible to sight the lowest points of station poles, these
should be carefully "plumbed."
If the magnetic bearing of a line is required, the A vernier should be
set at zero by means of the upper motion, and the lower motion should
be undamped and the alidade turned until the magnetic needle comes
exactly into its meridian, clamping L, and obtaining exact coincidence
by use of the tangent screw /. Then the station P (or Q) should be sighted
by means of the upper motion, clamping U and obtaining an exact
bisection of the image of the station by means of the tangent screw u.
The bearing of OP (or OQ) is then read on the A vernier.
Vertical Angles. When vertical angles are observed, greater accuracy
will result if the azimuthal or altitude level B is utilised in a more
exact levelling-up of the instrument.
(1) Set the C vernier to zero by means of the vertical motion,
clamping at O by means of V, and setting the vernier index precisely
by means of the tangent screw v.
(2) Set the bubble of the leve^ B central by means of the clipping
screws jy, the process being the same whether the level is on the clipping
frame or on the telescope (Fig. 83).
146 ELEMENTARY SURVEYING
(3) Unclamp the vertical motion, and sight the elevated poi,nt;
clamp F, and obtain exact coincidence of the intersection of the
cross-wires and the image by means of the tangent screw v.
Read the magnitude of the vertical angle on the C vernier, taking
care that the vernier is counted in the proper direction.
Face Left and Right. In the case of transit theodolites, it is possible
to "transit," or rotate the telescope about its horizontal axis 0, which
means that the vertical circle may be either on the right or the left of
the observer's eye. These are known as the Face Left (F.L.) and Face
Right (F.R.) positions; one of which is retained in ordinary usage,
this "normal" position being Face Left preferably. When angles are
observed with both faces thus, the mean horizontal angle will be free
from instrumental errors of adjustment, but this is never the case
with vertical angles. Both faces are used thus when great accuracy is
required, as in triangulation surveys.
Back Angles and Bearings. In British practice, horizontal angles are
usually measured directly, as above, or as Back Angles, which are the
angles measured clockwise from a zero reading on the preceding rear
station, a practice commonly followed in town surveying. Thus, if the
pond in Fig. 20 is traversed with the lines, AB, BC, CD, and DA
running in the counter-clockwise direction, these back angles will be
the interior angles of the skeleton; and this is convenient in applying
the check of the angular sum; (27V 4) 90, where TV is the number of
sides or angles. Any error in the observed sum of the angles may
then be divided equally among the angles, and each part applied
appropriately as a correction to the observed angles, provided each
angle is measured with equal accuracy, or equal weight, as it is called.
If reduced bearings are required for plotting by co-ordinates, as
described hereafter, these must be calculated with the bearing of one
side of the traverse, observed with reference to the magnetic or the
true meridian or assumed with reference to any convenient so-called
north and south line. Most theodolites are provided with a magnetic
compass, sometimes in the trough or the telescopic form, and some-
times in the form of a dial. If then the bearing of one side, AB, say,
is observed, or if AB is assumed to have some bearing, conveniently
N.0 0' E., then the bearings of the remaining sides can be reduced
from the observed interior angles. The characteristic of direct angular
measurement is that all angles are measured separately, and errors are
not carried through to succeeding lines. Its advantage is that angles
may be repeated with alternate faces of the instrument, thus eliminating
the effects of instrument errors.
In North America, azimuths and bearings are observed directly by
sighting on the preceding rear station face right, transiting the
telescope, and then sighting forward face left consistently. This
expedites the work and gives a direct reading of the total angular error
on the horizontal circle, but it confines the work to one vernier, angles
THEODOLITE SURVEYING
147
to one measurement, and also exaggerates the effects of errors of
instrumental adjustment. If the bearing of the first line is observed
the compass need not be consulted again, for, in fact, the survey will
be run "fixed needle."
The angular measurements of the surveys shown in both Figs. 19
and 20 will be more precise than when the compass was used (page 93),
and, strictly, the accuracy of the chaining should be raised, or the
results may appear disappointing, simply because the crude and precise
cannot mix.
II. LATITUDES AND DEPARTURES
Like as the traverse of a polygon should close upon the first station,
so is it fit and proper that this little book should return to the co-
ordinates of the opening chapter.
Latitudes and departures are nothing more or less than Cartesian
co-ordinates, more commonly known as "graphs." The Y-axis of
F g. 1 merely becomes the N.S. axis of Latitude and the X-axis the
W.E. axis of Departure, the origin still remaining at O.
Now if S T be the length of a survey line OA and NfiE its bearng;
then its latitude will be the projection on the N.S. axis, which H
A 1 =,y 1 ujb PJ.
Also, its departure will be the
N.
0^
projection 8 l on the W.E. axis,
^
s
which is ^i ^s l sin p x . Hence, N.w.
A
tan PI^J/XJ. + A
Likewise for the line OC\ of - 6
r
"""7! +A
A / i- 6
length ^ 3 and bearing SpgfK, the
! !
/ !
latitude and departure will be
/ c
rpinpptivpl v y
t J
I/ 1 E
1 V'^^/WV'l.l W 1 j , ir
j^
x 3 -=^ 3 cos p 3 ; 3^3 sin p 3 ; ^1
tan p 3 5 3 /x 3 . ^
.sp>, -*
*-,-*]
Amin flip DDpninf rhvmc nixiv
/ r 3
A
/-Vgtfllll lil^/ U^yV^lllll^ A 11 jr 111C- UlCljr _^
be repeated:
^__1 X 3 ^
|\
+ S
"Positive north and positive ~ ^
S.E .
^5^,
Negative south and negative S
s
FIG. 86
Howare these signs determined?
Simply from the initial and final letters of the bearings; N. and S.
and E. and W. respectively, as given in the rhyme.
North bearings give plus latituc^s, or "Northings"; south bearings
give minus latitudes, or "Southings"; east bearings give plus departures,
or "Eastings"; and west bearings .^ive minus departures, or "Westings."
Thus, Xj and ^ l are both plus, while \ and S 3 are both minus, lines
in other quadrants having signs prefixed to them as in the four quadrants
148
ELEMENTARY SURVEYING
of Fig. 86. Algebraical signs are of utmost importance in all problems
which introduce latitudes and departures.
Now latitudes and departures are used in two forms, which, to avoid
confusion, may be styled (a) Individual Co-ordinates, and (b) Total
Co-ordinates. At all stations the existence of the co-ordinate axes must
be imagined when thinking of individual latitudes and departures,
while in working with total co-ordinates these axes exist in fact, as
with graphs, with the origin at the most westerly station of the survey;
and the total co-ordinates of any point are the individual latitudes and
departures summed algebraically from this origin.
N Consider Fig. 87, which
^ - & I - 6 j is a quadrilateral traversed
in the counter-clockwise
direction, so that the forward
reduced bearing of AB is
S.E.; of BC, N.W.; of CD,
N.W.; and of DA, S.W.
The origin O is taken at A,
which is the most westerly
station, and the individual
latitudes and departures X
and 8 are written appropri-
ately on the diagram.
Now at C, the total plus
departure, or easting, is
x ^=J r (8 l Sj), and the total
P g - plus latitude, or northing, is
On returning to A by way of CD and DA, the total latitude y will
be O, since +(x a x^ (X 4 X 3 )==.-O, while the total departure x will
also be 0, since +(S 1 -S 2 )-(S 3 +8 4 )-0.
This introduces a very important principle, which is the basis of
adjusting traverse surveys arithmetically.
Adjusting Traverses. Now it seldom happens that either the alge-
braical sum of the latitudes or of the departures is exactly zero, but
will be small values which are the total errors in latitude and departure,
EI and E 4 respectively, as indicated by the dotted line AA' in Fig._87.
The true error of closure of the traverse is linear, and is E=\/Ei 2 +Ej 2 ,
while the angular error of closure a is found from the difference of
the observed sum of the interior angles of the figure and the geometrical
sum, as found from 2(N 4)90, where N is the number of sides.
Bowditch's method is easily applied by finding the ratios:
m^ and n=~ where S t y=5 1 +j a +J 8 . etc., or the perimeter of the
traverse.
THEODOLITE SURVEYING 149
The corrections in latitude and departure will be l lt / 2 , etc., d l9 d 2t
etc?, to the sides s l9 s 29 etc., accordingly:
/!=/fWi; / a =tfw a ; etc.; and </ 1 =/w 1 ; di~ns 29 etc.,
which arc prefixed with the sign of the corresponding total error in
latitude and departure. These corrections are then subtracted alge-
braically from the corresponding calculated values for the corrected
latitudes and departures to be used in plotting the survey. Many
practical men do not worry about signs. When, for instance, they
sum up the latitudes and find that E t is negative, they say they have
too much negative latitude, and merely increase the plus latitudes and
decrease the minus latitudes by the values of the corrections /!, / 2 ,
etc. Likewise for the departures.
Plotting Surveys. The method of total co-ordinates provides possibly
the best and most accurate method of plotting surveys. But before
the latitudes and departures arc calculated, it is advisable to consider
how the survey is to be "placed" on the drawing-sheet; as, for example,
with the approach road along the bottom of the sheet, which usually
will mean that the meridian will not run parallel to the vertical edges.
Consequently, it is advisable to plot the traverse roughly with the did
of a protractor. This will not only reveal which is the most westerly
station, but will indicate the angle 6 through which the entire survey
must be twisted so that a meridian will run parallel with the vertical
edges of the sheet. Some even value of is then subtracted from all
the bearings, and the latitudes and departures are calculated with
reference to the resulting "false" meridian. They are then duly cor-
rected, as explained in the preceding paragraph. Otherwise it might
be necessary to re-calculate the entire set of latitudes and departures;
and this is no small undertaking without traverse tables, since each
pair of values requires four to five minutes in reducing with five-figure
mathematical tables. It might happen that an extra-outsize sheet of
paper, known as "antiquarian," might be found, and from this the
modest "imperial" sheet could be cut out after plotting.
The individual latitudes and departures are added algebraically from
the most westerly station adopted as the fixed origin, the total latitude
of a station being either plus or minus, and the total departure always
plus. On reaching the origin again the sum will be zero. The values
thus tabulated are the co-ordinates with which the stations of the survey
are plotted with reference to the origin.
In general, it is best to draw a reference rectangle which will exactly
enclose the skeleton to scale, the most westerly station being on the
left-hand side, while the upper, lower, and right-hand sides pass
through the most northerly, southerly, and easterly stations respectively.
The vertical dimension of this rectangle is given by the arithmetical
sum of the greatest total northerly and southerly latitudes, and the
horizontal dimension is merely the greatest total easterly departure.
150
ELEMENTARY SURVEYING
The work may be expedited in surveys with much detail by covering
the rectangle with a graticule or grid of unit squares, each sid* of
which shows on the scale of the plan a convenient unit of latitude and
departure; 1 chain, 100 ft., etc. Otherwise the stations would have to
be plotted with their total scale distances, north and south and east
of the origin O. It is, of course, possible to plot from the two nearest
sides of the rectangle by subtracting the tabular distances of the stations
or points from the lengths of the sides of the rectangle.
The foregoing are only two of the uses of the method of latitudes
and departures. The principles are also used in (c) calculating areas,
(d) supplying omitted measurements, (e) parting land and rectifying
boundaries, and (/) overcoming obstructions where no other method
would be effective.
* EXAMPLE. The foregoing methods may be illustrated through the medium
of the following closed theodolite and chain traverse, in which back angles
were observed, the magnetic bearing of AB being S. 64 36' E.
Reducing Bearings:
Line
AB BC CD
:hs.) 23-53 7-20 10-66
DE EA
8-89 22-45
EAB
58 42'
ABC
101 06'
BCD
143 30'
CDE
131 18'
DEA
105 24'
Back angle
It will be seen that the back angles sum up exactly to 540 00', which
frequently happens in careful work with theodolites reading to single minutes.
The bearings may now be reduced, and since this survey may be plotted
with the magnetic north at the top of the sheet, the latitudes and departures
may be calculated and tabulated also.
Calculating Latitudes and Departures:
Line
Length
(Iks.)
Bearing
Latitude
(Iks.)
Departure
(Iks.)
AB
BC
CD
DE
EA
2353
720
1066
889
2245
S. 64 36' E.
N. 36 30' E.
N. 00' E.
N. 48 42' W.
S. 56 42' W.
-1009-3
4- 578-8
4-1066-0
+ 586-7
-1231-8
+ 2125-6
+ 428-3
00-0
- 667-9
-1875-6
fl -9-6; E d + 10-4
Adjusting the Traverse. It will be seen that there is an excess of minus
latitude of 9-6 Iks. and of plus departure of 10-4 Iks. in a perimeter of 7273 Iks.
The correction factors m and n can now be calculated, although the
fractions are more conveniently run off on a slide rule:
Ei
-
7273 *
" Zs
7273
THEODOLITE SURVEYING
151
The corrections to the latitude and the departure, / and d, are calculated
by hiultiplying respectively m and n by the lengths of the sides thus:
AB BC CD DE EA
I -3-1 -1-0 -1-4 -1-2 -3-0; sum - 9-7
d+3-4 +1-0 +1'5 +1-3 +3-2 +10-4
These corrections are now subtracted algebraically from the observed
latitudes and departures, giving the corrected values in the following table.
In practice, a Traverse Sheet is drawn up with sufficient columns for the
entire notes; but this would require a folding sheet, which is not desirable
in a book of this nature. Hence the tables ate separated, and so curtailed in
width that it is impossible to show + and latitudes in separate columns
as northings and southings respectively, and + and departures as eastings
and westings respectively a great convenience in summing algebraically.
Plotting the Traverse. Now A also happens to be the most westerly station
of the traverse, and the following total co-ordinates are summed algebraically
from that station.
Station
Line
Cor reeled
Total
Latitude
Departure
Latitude
Departure
(Iks.)
(Iks)
(Iks.)
(Iks.)
A3
-1006-2
-[2122-2
B
-1006-2
+2122-2
BC
+ 579-8
+ 427-3
C
- 426-4
+2549-5
CD
+ 1067-4
- 1-5
D
+ 641-0
+2548-0
DE
+ 587-9
- 669-2
E
-} 1228-9
+ 1878-8
EA
-1228-8
-1878-8
A
+ 0-1
0-0
E/+0-1 E d 0-0
If a boundary rectangle is used in plotting, its horizontal length will be
2549-5 Iks. to scale, and its vertical width will be (1067-4+ 1228-8)=2296-2 Iks.
to scale. On a scale of 1 chain to 1 inch the dimensions would thus be
25-50" x 22-96".
152
ELEMENTARY SURVEYING
CLASS EXERCISES
1 1 (a). The following readings were obtained in a triangle ABC, the nfcan
reading of the two verniers being given in each case. Tabulate the mean
observed value of each angle, and state the corrected values, assuming that
the total error is to be distributed equally among the angles.
Station
Point
Observed
Face Left
Face Right
A
B
C
/
00 00
49 10 30
/ IT
229 10 30
278 20 50
B
C
A
91 39 00
170 43 30
350 53 30
69 58 10
C
A
B
195 56 20
247 41 10
67 13 50
118 58 20
BAC
CBA
ACB
49 10 15
79 04 35
51 44 40
49 10 25
79 04 45
51 44 50
Observed 179 59 30 Corrected 1 80 00 00
11 (b). Discuss the measurement of a horizontal angle with the theodolite
when great accuracy is required. State what errors will be eliminated by the
various steps of your procedure.
11 (c). The following back angles were observed in a traverse survey, the
area being traversed in the counter-clockwise direction: ABC, 172 48';
BCD, 96 50'; CDE 7 148 42'; DBF, 128 43'; EFG, 70 40'; zndFGA, 102 17'.
Reduce these to magnetic bearings, given that the line AB had a forward
bearing of N. 24 12' E.
(N. 24 12'; N. 17 00' E.; N. 66 10' W.; S. 82 32' W.; S. 31 15' W.;
S. 78 05' E.; N. 24 12' E. (check)).
1 1 (d). The following notes show the co-ordinates of a closed traverse
with straight boundaries, the stations running in counter-clockwise order.
Line
Latitude
(Iks.)
Departure
(Iks.)
AB
BC
CD
DE
EA
00-00
+ 404-00
-1-437-50
-445-40
-396-10
+ 1133-90
- 188-80
- 269-10
- 526-50
- 149-50
Plot the survey and calculate the area from the co-ordinates.
(563,126 sq. Iks. -5-63 126 acres).
11 (e). The following notes were recorded in a theodolite and chain
traverse in which the length and bearing of the closing line EA were omitted:
THEODOLITE SURVEYING 153
AB: 2342 ft., S. 84 21' E.; BC: 782 ft., N. 14 44' E.; CD 1510 ft., S. 88
32' W.; DE: 462 ft, S. 38 24' W.
Calculate the latitudes and departures, and hence determine the length
and bearing of the missing closing line, assuming that all the measurements
were made with uniform accuracy.
(EA^= V(X^) : K^ : =744 ft., where X and S are the latitude and the departure
that make the entire sums both algebraically zero.
^
Tan, Bearing EA = , the signs indicating the quadrant, S. 80 20' W.)
FIELD EXERCISES
11 (A). Measure the angles of the triangle ABC, using both faces of the
instrument and taking the mean of both verniers. Record the results in
appropriate form, and adjust the triangle to close.
Equipment: Transit theodolite, and three range-poles.
\ 1 (B). Determine the error in the sum of the interior angles of the polygon
ABCDEby observing the back angles with a theodolite. Observe the magnetic
bearing of AB; reduce the corrected bearings, and record these on an appro-
priate note form.
Equipment: Theodolite and five range-poles.
1 1 (C). Determine the height of the fmial on (specified) Tower above the
ordnance datum, given that the reduced level of the peg A is . . .
Equipment: Theodolite, two pickets, chain or band, and levelling staff.
11 (D). Make a theodolite and chain traverse of the (specified) field,
wood, or pond. Plot the skeleton of the survey by latitudes and departures.
Equipment: Theodolite, chain, arrows, tape, and set of pickets.
1 1 (E). Make a theodolite and chain traverse of (specified) road between
. . . and ... (or ... brook, between . . . and . . .).
Equipment: as in 1 1 (/)).
ORIGINAL PROBLEMS
Determine the height and distance of ... (specified point) on ... Hill
(inaccessible)
Determine the true north from the mean horizontal angle when observing
a circumpolar star at equal altitudes on each side of the pole
154
ELEMENTARY SURVEYING
TRIGONOMETRICAL TABLE
Deg.
Chord
Sine
Tangent
Cotangent
Cosine
Deg.
0"
2
3
4
017
-035
052
070
-0175
0349
0523
0698
0175
0349
0524
0699
CO
57 2900
28 6363
19 0811
14 3006
1
9998
9994
9986
9976
1-414
1 402
1 389
1-377
1-364
90
89
88
87
86
5
087
0872
0875
11 4301
9962
1 351
85
6
7
8
9
105
122
139
157
1045
1219
1392
1564
1051
1228
1405
1584
9 5144
8 1H3
7 1 1 54
6 3138
9945
9925
9903
9877
1-338
1-325
1-312
1-299
84
83
82
81
10
174
1736
1763
56713
9848
1-286
80
11
12
13
14
192
209
226
244
1908
2079
2250
2419
1944
2126
2309
2493
5 1416
470+6
4-3315
40108
9816
9781
9744
9703
1-272
1-259
1 245
1-231
79
78
77
76
15
261
2588
2679
3-7321
9659
1-217
75
16
17
18
19
278
296
313
330
2756
2924
3090
3256
2867
3057
3249
3443
34374
3 2709
3 0777
2 9042
9613
9563
9511
9455
1 204
1 1^0
1 176
1 161
74
73
72
71
20
347
3420
3640
27475
9397
1-147
70
2!
-> >
23
24
364
382
399
416
3584
3746
3907
4067
3839
4040
4245
4452
2 6051
24751
2 3559
22460
9116
9272
9205
9 1 35
1 H3
1 1 18
1 10*
1 089
69
68
67
66
25
433
4226
4663
2 1445
9063
1 075
65
26
27
28
29
450
467
484
501
4384
4540
4695
4848
4877
5095
5317
5543
20503
1 9626
1 8807
1 8040
8988
8910
8329
8746
1 060
1 045
1-030
1 015
64
63
62
61
30
518
5000
5774
17321
8660
1-000
60
31
32
33
34
534
551
568
585
5150
5299
5446
5592
6009
6240
6494
6745
1 6643
1 6003
1 5399
1-4826
8572
8480
8387
8290
985
970
954
939
59
58
57
56
35
601
5736
7002
1-4281
8192
923
55
36
37
38
39
618
635
651
668
5878
6018
6157
6293
7265
7536
7813
8098
1 3764
1 3270
1 2799
1 2349
8090
7986
7880
7771
908
892
877
861
54
53
52
51
40
684
6428
8391
1-1918
7660
845
50
41
42
43
44
700
717
733
749
6561
6691
6820
6947
8693
9004
.9325
.9657
1 1504
1 1106
1 0724
1 0355
7547
7431
7314
7193
829
813
797
781
49
48
47
46
45
765
7071
1 0000
1 0000
7071
765
45
Dcg.
Cosine
Cotangent
Tangent
Sine
Chord
Deg.
INDEX
Abney Level, 81
Angles: construction of, 39; measurement of, 53, 145; angular levelling, 79
Areas, 123
Barometer, aneroid, 83
Bearings, 89, 147; affected, 92
Benchmarks, 70, 75
Boundary lines, 26
Chains, chaining, 9
Chain surveys, 24, 45
Clinometer, 17, 81
Collimation, 65; system of notes, 74
Compass, 86; compass surveying, 93, 99, 119
Computing scale, 127
Conventional signs, 23, 37
Contours, contouring, 113
Co-ordinates, 4. 147
Cross sections, 130
Cross staff, 1 2
Datum, 70, 135
Divided circles, 89, 143
Dumpy level, 64; adjustment of, 65; use of, 68, 70
Elevations: by aneroid, 83; by clinometer, 79; by theodolite, 145
Enlarging and reducing maps, 41
Field-book notes, 22; field code, 18; field geometry, 48
Field-work, 8, 24, 70, 79, 1.3, 116
Horizon, reduction to, 14
Latitudes and departures, 147
Level, 60, 67; adjustments, 65, 68, 144
Level book, 71
Levelling, angular, 79; spirit, 70, 75
Levelling stores, 66
Linear measurement, 17
Longitudinal sections, 75, 135
Magnetic needle, 86; declination and variation, 91; local attraction, 91
Magnetic bearings, 89
Maps, Ordnance, 3; enlarging and reducing, 41; plotting and finishing,
34,110,120,149
Meridian, true and magnetic, 89; needle, 36, 43, 105
Obstacles, obstructed distances, 53
Offsets, 11, 26
Optical, square, 13
Ordnance maps, 3
Plane table, plane tabling, 104; three-point problem, 108
Plumb-line level, 61
Plotting plans and maps, 32, 110, 149
Prismatic compass, 86
Prismoidal rule, 132
155
156 INDEX
Ranging-out lines, 10, 48
Range-poles, 8
Reciprocal levelling, 66; ranging, 48
Resection, 6, 96, 108
Rise and fall systems, 73
Scales, 30
Sections, cross, 130; longitudinal, 135
Signals, signalling, 20
Sloping distances, 14; stepping, 16
Spirit levelling, 64, 139
Staff, levelling, 67; Jacob, 13
Stations, 8, 25, 93
Surveys, chain, 24, 45; compass, 93, 99, 119; theodolite, 140
Table, trigonometrical, 154
Tapes, 9, 50
Telescope, 62, 69, 141
Theodolite, theodolite surveying, 140
Three-point problem, 6, 96, 108
Traverse surveys, 7, 27, 93; adjustment of, 95, 148
Triangulation surveys, 7, 24, 80
Verniers, 143
Vertical angles, 17, 80, 145
Volumes, 129
Water level, 62
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