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PRELIMINARY SURVEY
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PRINTED BY
SPOTTISWOODB AND CO., NBW-5TRBBT SQUARE
LONDON
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PRELIMINARY SURVEY
AND
ESTIMATES
BY
THEODORE GRAHAM GRIBBLE
M
CIVIL ENGINEER
LONDON
LONGMANS, GREEN, AND Ca
AND NEW YORK : 15 EAST let** STREET
1891
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INTRODUCTION
The Anglo-Saxon race, in sundry climates and conditions,
and under divers forms of government, is unquestionably
pre-eminent to-day in the civilisation of the world.
It is not alone becab^e they are the greatest traders, but
because they are at the same time the greatest navigators
and engineers of the world that the English-speaking nations
hold the proud primacy of race. Whether it be at the
first appearance of railways upon the Eastern hemisphere, or
the first cable-knot between the old world and the new, or
the development of virgin continents, and the carrying of a
luxurious civilisation into the heart of nature's wilderness,
the Anglo-Saxon is always at the front.
The strengthening of the Anglo-Saxon bond from year
to year is more attributable to improved means of com-
munication than to sentiment. The constantly conflicting
interests of commerce, the intense rivalries of handicrafts,
the minor jealousies of social life, fostered by selfish iso-
lation, produce barriers which would increase and gulfs
which would widen ; but the iron horse, the ocean grey-
hound, and the subtler electric fluid are for ever making
the men who speak the same tongue shake hands again.
To the pioneer surveyor, however, the field available for
, Digitized by v_ _ _ __
Ti Preliminary Survey
new enterprise is rapidly becoming less, whilst the number
of surveyors increases. It becomes, therefore, more and
more important for those who leave our shores to possess
Ae handiest and most efficient instruments, to know the best
and most rapid methods of using them, and to understand
the diverse conditions of the countries to which they go.
The members' list of the British Institution of Civil
Engineers has now reached the colossal total of about six
thousand. These figures alone would serve to show the
extent of the demand for foreign employment, for certainly
there is not enough home work to go round so many ; but,
especially in the department of surveying, the Institution list
gives but little idea of the number of young men who are
issuing year by year from pupilage or college with their eye
on our distant dependencies.
It is furthermore a noteworthy fact that, especially for
surveyors, although the field of engineering enterprise is be-
coming greater and greater, the colonial door is closed to
young Englishmen, just as soon as men can be trained abroad.
Both in Canada and Australia, a diploma is needed to
qualify a man to practise as a land-surveyor ; the studies
for which cannot be easily pursued in England.
In the case of India the Government have met the
difficulty by giving their men the special training needed
for that country at Cooper's Hill College, but the door is
closed to others.
In the Colonies the reason of this is because in the first
place Australians are independent in their ideas, but also
very much because the young English surveyor is too often
an importer of instruments of which he knows little into a
country of which he knows less, so they prefer to educate
their engineers on the spot. ^ ,
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Introduction vii
Things have changed for the better, no doubt, but
about twenty years ago it seemed as if the English engineer
were educated as much as possible in things he cotUd not
use, and as little as possible in things which would be
needed by him in a new country. The writer enjoyed the
last two years of the lectures of one of the most celebrated
professors of engineering of his day, and purchased the whole
of that scientist's textbooks ; but it is a significant fact that
one of the former students earned his living by explaining after
the lecture what the professor had meant to convey to his
hearers. Finding a year or two afterwards that he could
get the kind of information he wanted in a smaller compass
and simpler language the writer parted with his little library
of textbooks.
The same drawbacks attended the pupil in the engineer's
ofl&ce as the student in the university.
Men were not then made to keep their levels in adjust^
ment, but allowed to run to the nearest instrument-maker.
They were never taught the American method of levelling
or ciure-ranging, and the road and railway making which
they learned was that which was suitable to a country
hke England, but of little use for the Colonies. The con-
sequence has been that when they arrived there they were
thrown upon their own ingenuity, and produced a conglome-
rate of different types of construction upon different gauges,
which has been the reverse of profitable to the investors and
without reflecting much credit upon themselves.
On the Canadian Pacific Railway the writer rarely met
a young engineer fresh from England who .could quickly
adjust his level or theodolite or who knew anything of the
American system of curve-ranging or had the least notion
of telemetry. ^ t
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viii Preliminary Survey
It has been the fashion to criticise America for her
cheap railways, her numerous gauges, her erroneous
curve-ranging, and in fact everything that was not the way
we do it in England. This has been the language of those
who have either not been there or who have not understood
the methods adopted there when casually observing them
with a biassed judgment on a passing visit.
The American is just an Anglo-Saxon like ourselves,
only with a little more liberty and a great deal more scope.
He is not at all ashamed to come and learn from the old
country what age and experience have qualified her to teach
him, but in the handling of a virgin colony, with great
undeveloped resources, we may do well to learn of him.
In simplicity of survey practice, uniformity of gauge,
tjrpes of bridges and of rolling stock, the American engi-
neer may be profitably (though not slavishly) imitated in
the work of opening out a new sphere of enterprise such
as our recently acquired colonies, and it is to be hoped
that, profiting by past experience, English engineers will fuse
their ideas into something like uniformity and produce a
harmonious construction.
The methods of surveying considered in the following
pages are by no means exclusively American. In the
class of work formerly called telemetry, but now tacheo-
metry, we have to go to Italians, French, and Germans for
most of the original conceptions and the best modern
developments. Comparatively few English engineers
really practise these methods unless they have learned them
abroad, although some are thoroughly proficient in them.
The title of this book, * Preliminary Survey,' is American,
and answers somewhat to our * Parliamentary Work ; ' but it
covers a wider range, in fact the whole science of surveying
Introduction ix
in condensed form with the exception of those minute
details where very great accuracy is needed.
The object in view has been to present to the young
engineer going abroad a handy vade-mecum which with the
necessary tables will enable him to carry out a survey in a new
country rapidly, correctly, and according to the ideas and
requirements of the people. It has also been sought to furnish
in the first and third chapters an aide-mtmoire to the expe-
rienced surveyor for his assistance in roughly estimating the
cost of the proposed works, and so to guide his decision in
the case of alternative routes and situations.
Considerable use has been made of standard authorities
on both sides of the Atlantic, but the subject matter is in
the main the result of actual experience. The necessary
compactness of such a work has made it eclectic. Some
methods have been passed over with slender comment,
although occupying much space in other textbooks. On
the other hand such subjects as tacheometry, computation
by diagram and slide-rule, signalling, &c., which are as yet
hardly known to the general public except in pamphlet form,
are here treated of at considerable length. An attempt has
been made to explain the elements of astronomy, as far as
they are needed in the simple problems used by the surveyor,
in such a manner as will be understood by those having no
previous knowledge of the subject, and a great many of the
definitions which take up much space in ordinary textbooks
have been placed in a glossary. No tables are given which
are to be found in the Nautical Almanac or in ordinary
mathematical tables, as these have to form part of the
surveyor's impedimenta.
The following extract from the statute book of the
Dominion of Canada will give a fair idea of what the
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X Preliminary Sufvey
pioneer surveyor in any of the colonies should know, both
in theory and practice. Both in Australia and India survey
practice is carried on very much in the American manner.
The subjects enumerated in the Canadian statute are not
treated so much in detail in this work, in order to leave
space for other subjects, such as tacheometry and curve-
ranging, which are equally useful to the railway man.
The author desires to express his acknowledgments for
a great deal of useful material to the following gentlemen
who have kindly given their courteous permission to use
tables, maps, diagrams, and formulae in works of which they
are either the authors or custodians :
James Forrest, Esq., Secretary Inst. C.E. and editor of
* Minutes of Proceedings.'
Captain Wharton, Hydrographer to the Navy, and author
of * Hydrography.'
A. M. Wellington, Esq., C.E., editor of * Engineering
News,' New York, and author of standard works referred to
in the text.
John C. Trautwine, Esq. (jun.), editor of Trautwine's
* Pocket Book.'
Other authorities on different subjects have been also
referred to, and acknowledged in different parts of the book.
The calculations in chapters three and eight have been
very kindly checked by an old friend, Mr. William T.
Olive, Resident Engineer on the Manchester Main Drainage ;
most of the other figures have been checked in one way or
another, but it is possible in a first edition that errors may
still remain undetected, and any information as to mistakes
in the text, figures, or diagrams will be gladly welcomed by
the author.
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Intraduction xi
Qualifications of the Dominion Land and
Topographical Surveyor
Excerpt 49 Victoria^ Chapter 17. Royal assent, May 2 $y
1883. {Dominion of Canada,)
99. No person shall receive a commission from the
Board of Examiners authorising him to practise as a Dominion
land surveyor until he has attained the full age of twenty-
one years, and has passed a satisfactory examination before
the said Board on the following subjects ; that is to say :
Euclid first four books, and propositions first to twenty-first
of the sixth book ; plane trigonometry, so far as it includes
solution of triangles ; the use of logarithms, mensuration of
superficies, including the calculation of the area of right-lined
figures by latitude and departure, and the dividing or laying
off of land ; a knowledge of the rules for the solution of
spherical triangles, and of their use in the application to
surveying of the following elementary problems of practical
astronomy.
1. To ascertain the latitude of a place from an observa-
tion of a meridian altitude of the sun or of a star.
2. To obtain the local time and the azimuth from an
observed altitude of the sun or a star.
From an observed azimuth of a circumpolar star, when
at its greatest elongation from the meridian, to ascertain the
direction of the latter.
He must be practically familiar with surveying opera-
tions, and capable of intelligently reporting thereon, and be
conversant with the keeping of field notes, their plotting
and representation on plans of survey, the describing of
land by metes and bounds for title, and with the adjust-
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xii ' Preliminary Survey
ments and methods of use of ordinary surveying instruments,
and must also be perfectly conversant with the system of
survey as embodied in this Act, and with the manual of
standing instructions and regulations published by the
authority of the Minister of the Interior from time to time
for the guidance of Dominion land surveyors. ^
1 02. Any person entitled to receive, or already possess^
ing a commission as Dominion land surveyor, and having
previously given the notice prescribed in clause 98 of this
Act, may be examined as to the knowledge he may possess
of the following subjects relating to the higher surveying,
qualifying him, in addition to the performance of the duties
declared by this Act to be within the competence of
Dominion land surveyors, for the prosecution of extensive
geodetic or topographic surveys, or those of geographic
exploration, that is to say :
1. Algebra, including "quadratic equations, series, and
calculation of logarithms.
2. The analytic deduction of formulas of plane and
spherical trigonometry.
3. The plane co-ordinate geometry of the point, straight
line, the circle, and ellipse, transformation of co-ordinates,
and the determination, either geometrically or analytically,,
of the radius of curvature at any point in an ellipse.
4. Projections : the theory of those usually employed ia
the delineation of spheric surfaces.
5. Method of trigonometric surveying : of observing
the angles and calculating the sides of large triangles oa
the earth's surface, and of obtaining the differences of
latitude and longitude of points in a series of such triangles^
having regard to the effect of the figure of the earth.
6. The portion of the theory of practical astronomy
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Introduction xiii
relating to the determination of the geographic position of
points on the earth's surface, and the direction of lines on
the same, that is to say :
Methods of determining latitude.
a. By circum-meridian altitudes.
b. By differences of meridional zenith distance (Talcott's
method).
c. By transits across prime vertical.
Determination of azimuth.
a. By extra-meridional observations.
b. By meridian transits.
Determination of time.
a. By equal altitudes.
b. By meridian transits.
Determination of differences of longitude.
a. By electric telegraph.
b. By moon-culminating stars.
7. The theory of the instruments used in connection
with the foregoing, that is to say, the sextant or reflecting
circle, altitude and azimuth instrument, astronomic transit,
zenith telescope, and the management of chronometers \ also
of the ordinary meteorological instruments, barometer
(mercury and aneroid), thermometers, ordinary and self-
registering, anemometer, and rain gauges, and on his know-
ledge of the use of the same.
15 Great George Street, London, S.W., 1890.
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CONTENTS
INTRODUCTION
PAGS
Extent of the demand for surveyors. Necessity for adaptation to
the requirements of a new country. Objects aimed at in the
present work. Statutory qualifications of the land and topo-
graphical surveyor for the Dominion of Canada . . . v
CHAPTER I
GENERAL CONSIDERATIONS
Qualifications of the pioneer surveyor. Subject matter of a pre-
liminary report. Extent to which general considerations a£fect
the location. Tables of resistance to traction from gradients
and curvature. Abt system. Cape railways. Rule for finding
amount of trafiic necessary to pay a given dividend. Maximum
amount of business with a single line. Arrangement of curves
and gradients. Tables of ditto on American and Australian
railways. Rough estimates for railways, highwajrs, and tram-
ways. Cost of plant i
CHAPTER II
ROUTE-SURVEYING OR RECONNAISSANCE
Pioneering for railway location in America. Methods suit-
able to different circumstances. Sketching. Photography.
The plane-table and prismatic compass. The meridian by the
plane-table. Traversing with plane-table and stadia. Traverse
with passometer, aneroid, and pocket altazimuth* Profile and
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xvi Preliminary Survey
PAGE
contours by ditto. Surveying with the sextant. Range-finding
with a two-foot rule. Distance-measurement by time, by
horses' and camels' gait, by patent log, by revolutions of the
propeller. Mapping by plane-construction, by conical projec-
tion, by stereographic projection. Mercator's projection. . 30
CHAPTER III
HYDROGRAPHY AND HYDRAULICS
Coast-lining. Boat-survey. Running survey from the ship.
Distance-measurement by gun-fire. Gnomonic projection.
Symbols in charting. Sun -signalling. Flag-signalling. The
!. Morse code. Tides and currents. Hydraulics. Kutter's
formula and diagram. Discharge from tanks, cisterns, and
weirs. Diagram for ditto. Trautwine's approximate rule for
discharge of pipes under pressure. Ditto for discharge over
weirs. Horse-power of falling water. Efficiency of water-
wheels. Horse-power of a running stream. Hydrostatic pres-
sure and diagram. Notes on dredging, dredging plant, boring,
concrete and dock work 81
CHAPTER IV
GEODETIC ASTRONOMY
Compared with nautical astronomy. General principles. * Gain-
ing or losing a day.' Classification of methods. Observations
for determining the meridian. By equal altitudes. By cir-
cumpolar stars in same vertical. By time interval of culmina-
tion of circumpolar stars. By pole star at elongation. By
solar azimuth. Observations for local mean time and longi-
tude. Table for reducing arc to time, and vice versd. Time
by solar transit. By solar hour-angle. Sidereal hour-angle.
* Absolute methods ' of determining the longitude. Jupiter's
satellites. Lunar occultation. Lunar distance. Terrestrial
difference of longitude. Moon-culminating stars. Observa-
tions for latitude. Rules for different cases in both hemispheres.
By circumpolar stars. By meridional altitude of a fixed star.
By meridional altitude of the sun. By an altitude of the sun
out of the meridian. By two altitudes of the snn or a star.
Gjraphic latitude 116
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Contents xvii
CHAPTER V
TACHEOMETRY
PAGE
Derivation. Definitions. History and first principles of stadia
measurement. Method of putting in stadia-hairs to any
required reading. Anal<^ of tacheometry with celestial
parallax. Limits of error. Different types of telescope.
Different methods of holding the stadia- staff. Comparison of
results. Surveying with the tacheometer. Auxiliary work to
ditto. Description of the author's methods in the Sandwich
Islands. Reduction of the traverse to difference of latitude
and departure. Ditto to longitude and latitude. Levelling
with the tacheometer. Contouring. Plotting the profile.
Sketch of a Hawaiian ravine with part of a horse-shoe curve
and contours. Tacheometric curve -ranging. Mr. Lyman's
conclusions on stadia-telescopes. The plane-table and stadia.
Time occupied in tacheometric survey 158
CHAPTER VI
CHAIN-SURVEYING
Application to preliminary survey. Chaining. Sources of error.
Setting out a square. Deflection distance. Surveying with
chain only. With chain and cross-staff. Fieldbook. Areas.
Traverse with transit and chain. Different methods of fieldbook.
Curve-ranging with the chain only. Krohnke's tangential
system. Jackson's six-point equidistant system. Kennedy and
Hack wood's method. Method by offsets without a table . 194
CHAPTER VII
CURVE-RANGING WITH TRANSIT AND CHAIN
Properties of the circle and general nomenclature. Confusion
arising from diversity of terms. Advantages of decimal gradua-
tion. Fundamental problems. Different methods of keeping
the fieldbook and specimens. Curve-ranging by tangential
angles. Dalrymple-Hay's curve-ranger. Parallel tangents.
"- . ^-
xviii Preliminary Survey
PAGK
Reverse curves. Turn-outs. Transition curves. The * tramway '
spiral. The * horse-shoe ' spiral. The 'mountain ' spiral. The
* trunk-line ' spiral 209
CHAPTER VIII
GRAPHIC CALCULATION FOR PRELIMINARY ESTIMATES
Objecls of preliminary measurement for estimates. Use of
diagrams. The slide-rule for ordinary arithmetic. Proportion.
Multiplication. Division. Involution and evolution. The
slide-rule as a universal decimal scale. Calculation of slopes
and gradients with tables and slide-rule. Table of squares
and square- roots. Table of railway sleepers for all gauges.
Measurement of earthwork by diagram and slide-rule. Measure-
ment of iron biidges by diagram and slide-rule. Stone and
brick bridges by diagram and slide-rule. Service diagram for
railways and tramways. Centrifugal force. Reduction of
thermometer scales. The slide-rule as a universal measurer.
Circles. Areas. Volumes. Weights and measures, with
tables. Tree-timber. Permanent way. Angle and tee iron.
Round and square iron. The slide-rule as a ready reckoner.
Wages table. British and foreign money . . . .241
CHAPTER IX
INSTRUMENTS
Levels and their adjustment. Levelling. Level-staves. Theo-
dolites. The * Ideal * tacheometer. The sketch-board plane-
table. Pocket altazimuths. Passometers and pedometers.
Sextants. The solar compass. The heliostat and heliograph.
Principles of telemeters. Eckhold's omnimeter. The Wagner-
Fennel tacheometer. The Dredge -Steward omni-telemeter.
The Weldon range-finder. The simplex range-finder. The
Bate range-finder. The aneroid barometer. The boiling-point
thermometer. H)rpsometry and diagram. Drawing instru-
ments. The slide-rule. The protracting tee-square. Im-
l^rovised protractors. The eidograph and pantagraph. The
planimeter. Stanley's computing scales. The station-pointer.
Books. Mathematical tables. Field and MS. books.
Stationery . ...... ^ . 286
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Contents xix
APPENDIX
PAGK
Functions of right-angled plane triangles. Functions of oblique
plane triangles. Functions of right-angled spherical triangles.
Functions of oblique spherical triangles. Tables for azimuth
by circumpolar stars. Table for transition curves Tables of
specific gravity 373
Glossary 39*'
Index 416
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LIST OF PLATES
Plate
I. FlG. 26, Kutter's Formula . . . To face page 100
II. Figs. 28, 29, Theoretical Velocity in
Feet per Second, Mii^es per Hour
DUE TO given Head . . • „ ,, 106
III. Fig. 30, Coefficient m in Trautwine's
Formula for Flow in Pipes . . ,, ,, 108
Fig. 31, Coefficient c in Trautwine for
Flow over Weirs . . . . „ ,,108
IV. Fig. 32, Hydrostatic Pressure . • ,, ,, no
V. Figs. 75, 76, Earthwork . . . . ,, ,, 252
VI. Figs. 80, 81, Iron Bridges . . . „ ,, 256
VII. Stone Arches ,, »» 258
VIII. Timber Trestles ,, m 259
Fig. 82, Service Diagram for Tramways On ,, 261
Fig. 83, ,, ,, Railways ,, ,, 262
IX. Figs, no, in, Barometrical Pressure . To face ,, 350
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PRELIMINARY SURVEY
AND
ESTIMATES
CHAPTER I
GENERAL CONSIDERATIONS
The following remarks will be more applicable to railway
reconnaissance, though much of the principle contained in
them is also that which guides the surveyor for trunk roads
for military or commercial transportation.
Qualifications of the Surveyor
The man who is first in the field should be a man of
wide range of experience rather than a minute technologist.
He is usually given much discretionary power as to his
location. He has also advisory powers, or rather duties,
which are great responsibilities. He is called upon to report
upon the scheme from a bare possibility down to a desirable
investment. Before engaging his services, the promoters
have generally made up their minds that there must be
' money in it,' and they want, like most other people, to
obtain a maximum of good showing for a minimum amount
of outlay.
The surveyor is generally disposed to favour a new
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2 Preliminary Survey
undertaking, because, however much or little money there
may be in it for him, there is likely to be * work in it,' and he
has often to resist the natural tendency to make too good
a showing.
There are several considerations which likewise influence
him in this direction. Shareholders always expect a sanguine
report, and take discount off it in any case ; so that a mode-
rate report is to them a bad one. There is a moral certainty
that, however carefully a walk-over survey may be made, a
revised location will show a material improvement in the
line of economy or efficiency, or both, and therefore the
surveyor is tempted to make allowance for this in his trial
profile. He is, perhaps, well aware that the nearer his profile
resembles the surface of a billiard table the better he will
please his employers.
When the country is so rough that chaining is out of the
question — unless he is able to adopt one of the rapid methods
to which these pages are meant to draw attention — a large
element of conjecture enters into his calculations, and he
is naturally disposed to conjecture favourably rather than
critically.
Subject Matter of a Railway Report
The surveyor is generally called upon to advise his
promoters —
1. Whether any kind of line is feasible.
2. Whether it is likely to be profitable.
3. What type of railway would be most suitable, and
what style of rolling-stock.
4. He is to furnish a plan, profile, and estimate of one
or more routes which he considers eligible.
All these points are closely connected. One kind of
line is feasible where another is wholly impracticable ; a light,
cheap railway will often yield a handsome dividend where a
heavy line would never emerge from the hands of a receiver.
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General Considerations 3
On the other hand, a light railway built to carry heavy traffic
will probably be wedged out of existence by a higher-ciass
competing line. The style of rolling-stock procurable to
handle the business often regulates the location as much as
the location rules the rolling-stock. The route is dependent
on the topography to a great extent, but the situation of towns
with which communication is necessary often overrides the
consideration of topography.
It is only experience which can enable an engineer to
form rapidly and correctly the general idea of the class of
line suited to the circumstances. If, as is often the case,
gauge and rolling-stock are fixed factors in the problem,
there remain the questions of grades and curves, which must
be to a large extent dependent upon the topography, and it
is there that the judgment of the surveyor is most needed,
both in the limiting and the arrangement of these vital
elements of a railway.
With regard to the first point of feasibility. This has
almost dropped out of the reckoning of to-day. It may be
taken for granted that a railway can be constructed nearly
anywhere. The only insurmountable difficulties to railway
projects are, first, lack of funds ; second, opposition of
vested interests. It is another thing when we come to the
question of —
Whether it will pay. Here the engineer has to study,
I. What the existing traffic of the district is, and how it
would be likely to be affected by the introduction of a rail-
way. 2. What is the probability of the traffic being handled
by some other means of transit in competition. 3. What rates
can be commanded, and whether it will be in the main a
through or a local traffic. 4. What is the outlook for
development of the business, with any possibly counteracting
causes. 5. Probabilities of another competing railway in
the future.
All these subjects dovetail themselves into the actual
reconnaissance of the route ; engineering difficulties give
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4 Preliminary Survey
way to commercial exigencies and vice versd, until the sur-
veyor has evolved his ideas of a line with maximum efficiency
at minimum cost which will command the maximum amount
of business.
The clifnate in which the undertaking is to be carried on
is a very important consideration. In tropical climates the
use of timber is to be avoided where possible, on account
of predatory insects and the rapid decay produced by alter-
nations of hot sun and heavy rains. The rainy season
regularly changes the trickling rivulet into the mighty river,
and these actions of the weather greatly affect the construction
of culverts and bridges, and therefore, indirectly, the location.
In some places streams are turned into tunnels to save build-
ing culverts, and an overflow channel provided at the junction
of an embankment and side-hill for abnormal freshets.
In cold climates snowsheds are a very costly item, and
the study of the principles of drifting snow will often modify
the location.
The general topography also radically affects the location.
It rules both gradients and curvature and the type both of
gauge and equipment.
If the land falls toward the seaboard, with a heavier export
than import trade — such as a mineral railway which only
takes back lumber, agricultural produce, and so forth — the
gradients can be steeper than would be otherwise permis-
sible ; the rule being then to adopt that which can be sur-
mounted as a contrary grade by the light traffic.
The method of * bunching,' or concentrating the severe
gradients in order to handle them specially, is a very im-
portant one. The best policy for a new country is to carry
long trains as far as possible with one engine^ and then to
divide them on a turn-out and take them over the climb in
sections, or else to provide an assistant engine for the
district.
The following table (No. XXIV. of Mr. Wellington's
standard work on American * Railway Location ') shows the
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General Considerations 5
engine ton-mileage required to move i ton of net load (ex.
engine) 100 miles on a level, except for a rise of 2,400 feet
on different grades, worked with assistant engines : according
to the average daily experience of American railways.
Table I. — Traction on Grades,
Engine ton-mileage per ton of
Rate of
grade on
Length
Length of
level
net load moved loo
miles
incline
incline
track
While on
While on
Total
incline
level track
feet per mile
miles
miles
1
24
100
—
1*056
—
1*056
t.
60
40
0862
0-2I0
1*072
30
70
0760
0369
1*129
100
24
76
0755
0400
1*155
120
20
80
0766
0*421
1*187
150
16
84
0-803
0*442
1-245
200
12
88
0-900
0*463
1 1-363
* It would be seen that the rate of incline had an incon-
siderable influence on the motive power required, for the
reason, largely, that the length of the run on which large
power was required decreased pari passu with the increase
of rate, which was not the case with through grades.
*In this table moreover it was assumed that the total
length of the road remained uniform at 100 miles, whatever
the rate of grade adopted for the high-grade section. This
is ordinarily quite out of the question, the lower grade being
usually attainable only by adding so much further develop-
ment within an approximately uniform air-line distance.
* Assuming, for example, that in the above table eighty
miles of level track was essential in any case, and that in the
remaining air- line distance of twenty miles, any one of the
above rates of pusher-grades from twenty -four to 200 feet
per mile was obtainable, but only by development — a rather
extreme assumption, but sufficient for illustration — the
table would thus read .- ' —
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Preliminary Survey
Table II. — Traction on Grades.
Rate of
grade.
ft. per mile
24
30
80
100
120
150
2(X)
miles
100
60
30
24
20
16
12
Length
Incline ' Level
miles I
80 I
80 I
80 ,
80 i
80
84
Engine ton-mileage per
ton of net
loatd moved between the incline
Total
While on
incline
While on
level track
Total '
miles
180
1-056
0-421
1-477
140
0-862
0-421
1283
IIO
0-760
0-42I
I-181 1
104
0755
0-421
1-176
ICXD
0766
0-421
1-187
100
0-803
0-442
1-245
100
0-900
0-463
1-363
Table III. — Adjustment of Gradients for Assistant Entities y according
to the Average Daily Performatice on American Railivays. {H, M.
Wellington, )
Grade at which the same train can be drawn by the aid of
Ruling
grade
worked by ~ —
one engine |
in feet Of equal
per mile weight on
drivers
One assistant engine
Heavier by
level
24
10
42
20
59
30
76
40
92
50
107
60
122
70
136
80
150
90
164
100
177
IIO
190
120
203
130
215
140
227
150
238
20 per
cent.
40 per
cent.
29 ;
48
66
84 I
loi I
117
133 I
148 '
162 I
176 I
189
202
215 I
227 j
239 I
250 I
Two assistant engines
I Of equal
I weight on
i drivers
Heavier by
20 per
cent.
40 per
cent.
33
46 1
54
62
53
70
80
90
72
92
104
116
91
"3
126
138
109
133
147
160
126
152
167
180
142
169
185
199
158
185
201
216
173
201
217
232
187
216
232
247
201
230
247
261
214
—
—
227
—
—
239
—
—
—
251
—
—
• —
262
—
—
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General Considerations 7
Caution, In calculating the increase of motive power due
to severe gradients, the wear and tear on locomotives, such
as the 'thrashing' of an engine up a steep incline by an
inexperienced driver, is an item which, though difficult to
calculate, should be allowed for by a large margin.
The assumptions in *he above table are that the rolling
friction on the level is 10 lbs. per ton ; for lower frictions
the gradients are proportionally lessened. The gradients
are compensated for curvature.
A good method of overcoming steep gradients is by
the Abt rack system. The special feature about it is that
sections of mountain line can be worked thus without
changing gauge or altering the rolling stock. The loco-
motives do not depend on adhesion, therefore they can be
much lighter just where the construction of bridges is the
most serious item.
The Abt system is also specially adapted to short branch
mountain lines.
The notes and memoranda that the surveyor wants are
to give him a general but accurate idea of the alternative
advantages of the different schemes that arise, and it is
with that object that the chapter on Graphic Calculation
has been added ; condensed to its utmost limit.
In selecting a route and deciding upon the class of line
for a railway scheme it should be considered —
First: If a competing line, how to obtain a pronounced
superiority to the existing one, either on the score of
efficiency or economy.
Second : If the first in the district, how without wasteful
expenditure to secure primacy. That is to obtain a line which
will so handle the existing and prospective business as to
hold its own^ and that the best, of the business.
In order to ensure that his line will be suitable to the
future mode of working it, the surveyor should be acquainted
both with the ordinary and the special types of rolling-stock
that are to be used. In new countries it is essential to
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8 Preliminary Survey
economy that the erijgines and cars should be flexible, not
only as regards side-play, but also, if one may coin the term,
* up and down ' play.
It is furthermore necessary that level crossings should
be permitted over all country highways, and, when on the
level prairie, over existing railways also.
Station buildings should be very primitive, and the
booking performed on the train.
It was stated by Mr. J. C. Mackay at the Institution of
Civil Engineers in 1886, that *the present railways of the
Cape Colony had been constructed on a lavish scale with
rails weighing 60 lbs. per yard, and expensive stations, some .
of them costing over 20,000/., while the railways alone had
cost 8,000/. per mile. This great expenditure had been
incurred y^r the sake of conveying one train per day, in some
cases only one train every other day, and the consequence
was that the revenue did not pay one half the interest on
the loan, after deducting working expenses, and the working
of these lines was obliged to be carried on in such a manner
that the bullock waggon competed successfully with the rail-
ways,
' At Kimberley, with its 1,500 passengers and 350 tons
of goods per mile of railway per annum, a line with rails of
60 lbs. per yard, and expensive rolling-stock and stations,
had been adopted.*
The writer is not in a position to verify at the moment
the accuracy of these figures, nor to state to what extent
they may be modified by subsequent development of
the districts ; but as they stand, reflecting most adversely
upon the judgment of the promoters and the engineers, it
should be added as a qualification that they only serve to
show one side of the question, but that which needs to
be most emphasised for a new country — the danger of put-
ting old-country ideas upon young-country shoulders. The
counter-evil of putting down poor lines where there is busi-
ness for good ones, probably to pad the promoters' pockets.
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General Considerations g
has plenty of illustrations both at home and abroad, and the
engineer is too often compelled against his judgment to
make both his location and construction suit the * spirit of
the times.'
The former evil of wasteful or even premature expendi-
ture is one which greatly checks the inflow of fresh capital
into a country. A receiver is a perfect scarecrow to fresh
enterprise. Purely speculative railway-making is as great a
hindrance to bon^ fide undertakings as jerry-building in its
smaller sphere ; whether it come in the form of too good or
too bad a line of railway.
A single line with properly arranged passing places,
rails 30 to 40 lbs. per yard, engines of 15 to 20 tons, in easy
country, can be built for 4,000/. per mile, including the
equipment, in almost all parts of the globe ; provided that
the line starts from the seaboard, or from a place in rail-
connection with the seaboard.
This line, properly located, is capable of handling 1,000
tons of freight per day, and is, therefore, even with low rates,
in a position to yield a handsome profit to the investors.
Putting net receipts at \d, per ton-mile, it would return 9^ per
cent, on the cost of construction with that volume of business.
Approximate Rule for finding the Amount of
Traffic required to pay five per cent, on the
Cost of Construction of a Railway.
Assumptions, Tariff, 75^/. per ton-mile for passengers
and freight. Passengers* reckoned at two tons each.
Expenses '50^. per ton-mile. Net receipts, -25^^. per ton-
mile. 365 days to the working year.
Traffic in tons per diem = cost in JT^ per mile x *i3i5.
Example : On a road costing 4,000/. per mile, the traffic
needed in tons per diem =4, 000 x '13 15 =526. The amount
of traffic required varies inversely as the net value of the
receipts per ton-mile.
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lo Preliminary Survey
Therefore for any other net value such as '53//. per ton-
mile, the amount of traffic is found by the slide-rule in the
following manner :
Place the given value '53//. on the upper scale of the
slide under the -25 on the rule. Find the required multiplier
0*62 on the rule opposite to the 131 5 on the slide. Leave
the brass marker at 62 on the rule, and make a i of the
slide coincide with it. Then the result, 248 tons, will be
found on the rule opposite to 4,000/. cost, or 186 tons
opposite to 3,000/. cost and so on.
Converse Rule
To find the percentage on cost of construction when
the argument is :
1. The tonnage of freight per day (average of 365 days).
2. The net profit per ton -mile in decimals of a penny.
3. The cost of construction of one mile in ^ sterling.
Multiply the tonnage by the profit. Divide the product
by the cost of construction. Multiply the quotient by
152*1. Result is the percentage.
Example,
Data ate : Daily tonnage .... 542
Net receipts per ton -mile . '47^.
Cost of construction per mile . 4650/.
By Slide-Rule, using lower scales of Slide and
Rule
Place a I of the slide over the 542 (tonnage) on the
rule. Place the brass marker at 47 (receipts) on the slide.
Place 465 (cost) on the slide at the marker. Place the
marker at the right-hand i of the slide. Place the left-hand
I of the slide at the brass marker. Read the percentage
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General Considerations ii
8*34 on the rule opposite to the constant 152 'i on the
slide.* (The operation can be done in about 35 seconds.)
Mr. Robert Gordon in his paper on Economic Construc-
tion of Railways, *Min. Proc. Inst. C.E.,' gives a note on
the problem of the maximum capacity of a single track,
quoting from Mr. Thompson of the New York, Pennsylvania,
and Ohio Railroad (* Railway Gazette,' 1884, 1-43). He
says that for trains running an average of twenty miles per
hour, the most economical speed for freight, the maximum
is reached with stations 37 miles apart, when with an
allowance of six minutes' detention for each train crossed, and
eight minutes extra for each passenger train passed, it is
found that the limit is reached when the time of detention
equals that of running in the 24 hours, which gives sixty
trains per day both ways. For fifty trains and over, the
track should be doubled between the termini and the
next stations. In practice it is found that grades limit the
number of cars run in a train, so that, if forty loaded cars be
the ordinary number on the level, only twenty are taken
over an undulating country by a single engine. Actually,
on Mr. Thompson's division, 98 miles in length, the
Standard engine takes nineteen cars, the Mogul twenty-
three cars, and the Consolidation thirty-three cars each per
train. 2
The type of railway is affected first of all by grade. There
• For the explanation of the principle of the slide-rule see pp.
242, &c., also 361, &c.
^ The total annual expenses on railroads in the United States usu-
ally range between 65 and 130 cents (2j. %\d, and 55. <^d.) per train-
mile, that is, per mile actually run by trains. Also between I and 2
cents {\ and \d.) per ton of freight and per passenger carried one mile.
When a road does a very large business, and of such a character that
the trains may be heavy and the cars full (as in coal-carrying roads), the
expense per train-mile becomes large, but that per ton or passenger
small ; and vice versd, although on coal- roads half the train-miles are
with empty cars. — Trautwine's * Engineer Pocket Book.'
See also, at p. 21, Table of * Earnings of American Railways.'
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12 Preliminary Survey
can be no question that there are many light narrow-gauge
railways which are earning a good return on the capital
which could not have kept their heads above water as
standard- gauge railways. It is true, on the other hand, that
a very large mileage of narrow gauge is every year being
converted into standard gauge, in order to avoid breaking
bulk. Surveyors are not gifted with prophecy to know what
the changes of the next fifty years will be, but a fair amount
of experience will enable them to tell whether the line will
always remain a feeder, or whether it is likely to form part of
a trunk line.
The narrow gauge enables the engineer to adopt curves
of very small radius, but then he must keep to a flexible
type of locomotive and car which involves lighter loads and
less speed ; consequently, less business done for the same
labour and superintendence.
A theoretically perfect railway is an air-line on a dead
level between two points, and yet, apart from its cost, in
nine cases out of ten it would not be the best line.
A great deal of the sinuosity of a well laid-out railway in
a new country is productive. It carries the trains to where
the business is or where it is going to be.
Sometimes the local traffic is the major part of the busi-
ness. It generally commands a higher rate than through
traffic, and it would be a serious mistake to straighten a line
to catch through traffic of small bulk carried at cut-rates and
by so doing to lose a steady monopoly of a lucrative local
business.
Bad gradients are worse than sharp curves ; the latter
can be to a great extent mitigated in their discomfort by
well-made carriages ; in their resistance by flexible locomo-
tive frames ; and in their danger by careful signalling. But
gravity no skill can dispose of, and bad gradients have killed
many a promising line.
Considerable opportunity exists in every line ot railway
for arrangement of the curves and gradients so as to make
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General Considerations
13
them as little objectionable as possible. The first question
for the surveyor is whether or not the construction is to
be homogeneous. If, for instance, he has to traverse level
prairie for fifty miles, then cross a range of hills 3,000 feet
high within an air-distance of another fifty miles, then another
stretch of prairie of fifty miles, he has to consider whether
it would be advisable to develop his mountain section for
better gradients or whether he should arrange the line for
a different class of locomotives in the three sections. "All
this affects the location.
The two main points to be borne in mind with regard to
curvature are the speed at which trains will have to run and
the kind of rolling-stock which will have to be carried. The
following Tables IV. to IX. from Mr. Wellington's book
already referred to will be found useful in fixing the limit
of curvature and gradient to be adopted on a new line. The
American curve-nomenclature is explained in Chapter VII.
Table IV. — Resistance on Curves,
Type of engine
Weight
in lbs.
Length of wheelbase
Resistance on 4*' curve
at 10 miles per hour .
Rigid
Total
Total
lbs. per
ton
lbs. per
degree
American
Ten-wheel .
Consolidation
101,000
123,000
136,000
f 6"
12' 5"
14' 6i
21' io|"
23' 8''
21' l"
1963
1750
1850
39-0
28-4
20 'O
975
7-1
5-0
Note, — Consolidation engines are made to run round a Wye (see
p. 240) with curves of 136 feet radius without any trouble.
Average
degree
per mile
Average Curvature per mile on some of the Railways in the Rocky
Mountains,
Length of
Name of Railway section.
miles
Colorado Central . . . " . .34
Virginia, Truckee 22
Union Pacific 65
Texas Central 143
Southern Pacific 142
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Preliminary Survey
Table V. — Curvature and Grades on Sections of
Eastern Trunk-roads.
^
s,,
^
^
^ V
Name of road
Miles
I1
r u
u b
n
1"^
Ruling
Grades
3^
(S"
021
(^
New York Central .
296*6
0-78
19*8
15*1
^4-8
1-8
rs
21-23
Boston and Albany .
202*0
1*55
56*0
72*5
o'o
9*3
13*7
80
Concord and Ports-
mouth ......
40*0
1-41
37*2
93'o
3*8
19 'o
15*3
80
Ulster and Delaware
Montrose Penna . .
73 '2
3-60
41-0
8o-5
1.V6
24*1
II-6
160-142
27*2
6-3S
490
24*1
3*9
^8-4
9*6
95-74
Summary of N. East-
em States . . .
5,372
I -88
35*5
55*9
22-8
4*7
13*0
The column *Rise per mile* gives the average excess
of rise over the fall in one mile. The next column gives the
feet of rise and fall. Thus, if a road rose 500 feet and fell
200 in 100 miles, it would be given above : Rise per mile
3'o, Rise and fall 2*0. The first quantity is an unavoidable
necessity, due to difference of level at the termini.
From the same work, on the authority of Mr. M. N.
Forney, locomotive expert, it is stated that :
Table VI. — Curve Limits for fixed Wheel Bases.
Feet rad.
Axles 3 feet apart will roll in a curve of 67
4
9ii
5
133
6
174I
r
251
8
337J
9
479
10
643i
Centrifugal Force
On any three curves having radii as i, 2, 3, the centri-
fugal force at any given velocity is as 3, 2, i ; but the
coefficient of safety against overturning or disagreeable
effect is as n/3, n/2, V 1 = 173, 1-41, I'oo.
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General Considerations
^S
Table VII. — Giving for various curves the inferior and superior limits
of speed within which the centrifugal force is more or less objection-
abU or dangerous.
Curve
Maximum and minimum limits of speed
in miles per hour
Minimum. Having
Maximum. On the
Degree
Radius in feet
point of overturning
effect
the vehicle
2
2,865
41*39
130-89
4
1.433
29-27
92-55
6
955
23*90
75-57
8
717
20*70
65*44
10
574
18*51
58-54
12
478
16*90
53*43
14
410
15*64
49*47
i6
359
14-63
46*28
i8
3i9
13-78
43-58
20
288
1309
41*39
22
262
12*48
39-46
24
240
"•95
37-78
26
222
1 1 '61
3672
28
207
11*06
3498
30
193
10*69
33-80
40
146
11
29-27
50
"5
26*18
6o
96
7-56
23*90
^ule. For the centrifugal force in lbs. per ton of 2,000 lbs.
C= -02 3348 V^ D, where C=centrifugal force in lbs. per ton,
V=velocity in miles per hour, and D=degree of curvature
from which the following table is made.
Table Vlll,— Curve Limits at Different Speeds,
Speed in
miles per
hour
Degree of curvature
1°
5°
10°
15°
2e'»
10
20
30
40
60
70
2*33
9-34
21*01
37*36
58-37
84*05
1 14-4
11*67
46-70
105*07
186*78
291*85
42026
572*03
23*35
93*39
210*13
373-57
58373
840-53
1,144-05
35*02
140*09
315*20
875-55
1,260*8
1,706*08
186-78
420-26
747-14
1,167-4
1,681-1
2,288-1
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1 6 Preliminary Survey
The centrifugal force varies directly as the degree of
curvature.
The heavy division lines mark the assumed maximum
limit of speed for safety, when the centrifugal force is=:JW.
On the 4 per cent grades of the Mexican Railway, re-
versed curves of 150 feet radius were worked for a year
with ordinary locomotives.
Narrow-gauge railways have rarely been constructed with
curves sharper than 24° in the United States, but there are
a few as sharp as 30° in Colorado.
In the writer's location of 3 ft. gauge railways in the
Sandwich Islands ; curves of 40° (146-2 ft. radii) were the
minimum, but many of them were more than a semicircle ;
previous railways had been constructed with 75 ft. radii in
that district, but in most cases without necessity. The trains
run at ten miles an hour with perfect safety, though it can
hardly be called comfortable. There is practically no
danger of the trains overturning because the loss of speed
due to curvature keeps them well within the limits of safety.
For rules on the same subject applicable either to feet or
Gunter's chains, see p. 264.
There is chronic trouble to railway managers from curves
of unnecessary sharpness, put in either to save the trouble
of a second revision, or from lack of experience on the part
of the surveyor.
It is true that rolling-stock can be made to go round
almost anything, but not without suffering from it.
It is a curious fact that single-line railways which have
DOWN ^wh-
-<fm UP
Fig. I.
their crossing stations arranged in the usual way, as Fig. i,
wear the locomotive wheels more on one side than the
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General Considerations 17
other. This arises from the trains always entering the
stations faster than they leave them. Probably, if the
crossing stations were made as shown in Fig. 2 this would
not take place. It is a preferable arrangement, except
DOWN mh-
Fig. 2.
-ms UP
where the through express trains run past the station with-
out stopping, in which case the usual arrangement is to
be preferred unless very flat curves are used in the switch
on Fig. 2.
Effect of Increased Distance from Development,
AND OF Curvature upon Working Expenses
The following notes and Table IX. are from Mr. Welling-
ton's book, being deduced from extensive American statistics.
Fractional changes of distance increase or decrease
expenses by only 25 to 40 per cent, of the average cost of
operating an equal distance,
600 degrees of curvature will waste about ^o per cent, as
much fuel as the average burned per mile run.
The lowest probable limit of curve-resistance at ordinary
speeds in ordinary curves is about ^ lb. per ton per degree
of curvature. With worn rails and rough track it may be as
high as § lb. per ton.
Curve-resistance per degree of curve is very much
greater on easy than on sharp curves, so that when, for ex-
ample, the resistance is i lb. per ton on a 1° curve, it may
be 6 lbs. to 8 lbs. per ton on a 10° curve, and not more than
15 to 18 lbs. on a 40° or 50° curve.
The almost uniform increase in cost in the first three
main divisions of the line is principally due to grades and
curves, which get worse as the line stretches inland.
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Preliminary Survey
Table IX. — Running Expenses on Pennsylvania Railroad affected
by Curvature.
Average cost per train
mile in cents
Eastern division .
Middle division .
Western division .
Mountain and
Tyrone . . .
Repairs
6*42
8-87
925
660
Fuel
6-93
7-00
7-59
7-04
Stores
115
1*07
1*39
0-83
Total
14-50
16-94
18-23
14*47
Compensation for Curves
Some of the various rules used in compensating steep
gradients for curvature are given by Mr. Robert Gordon in
' Min. Proc. Inst. C.E.,' vol. Ixxxv. : —
*The best American practice invariably allows com-
pensation when the curve falls on a gradient by lessening
the inclination as the sharpness of the curve increases.
Some difference of opinion exists amongst the authorities as
to the amount of reduction required, but the average given
is 0-05 per TOO ft. per degree of curvature.
' This practice varies, however, and Mr. A. A. Robinson,
who has had great experience on steep gradients, gives as
follows : —
Table '^,— Compensation for Curves.
Per 100 feet
per degree
* Rate of maximum grade ,0 to I in 166 'O compensation o*o6
,, ,, ,, I in i66 to I in 62*5 ,, 0*05
I in 62-5 to I in 33-5 „ 0*04
* Mr. Blinkensdorfer gives 0^03 to o-py in the same limits ;
while Mr. Wellington allows o-o6 on all maximum curves.
The practice also of widening the gauge on curves varies
much. Some engineers allow only the same play of \ inch
that is given on straight hnes ; while others increase it \
inch" and more on curves. But opinion is unanimous in
requiring a tangent between reverse curves, and sharp
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General Consiaerations 19
curves are eased off at both ends. In some cases gradients
also are eased at the approaches.'
The following tables are taken from Mr. Trautwine's
* Pocket- Book : '—
Table YA.— Table of Annual Expenses on iome tfnited <?
Slates Railroads. r ' -.
Name of company
Lehigh Valley, 1872 ....
Pennsylvania Central, 1869
Philadelphia and Reading, 1859 •
)> )) 1868 .
Connecticut, average of all the R. R., »
1861 ... . . ; }
Massachusetts, average of all the»
R. R., 1861 ;
New YoA State, average of all the )
R. R., 1867 }
New York Central, average of all the y
R. R., 1867 . . . . . }
English R. R., averages for 1856-7-8
Scotch „ „
Irish
Per mile
of road
%
32,000
17,200
3,781
3,785
13,856
15,620
mile
'<-l
cents
144
95
85
170
66
56
52
receipfe
^5|
7o|
54i
71
57
60
76
50
44
40
Table ^\\,— Statistics of several United States Narrow-mu^e
Railroads for \%^^ {Poot^s Manual)
Bridgton & Saco )
Riv., Me. . .;
Profile and Fran- )
cooia, N. H. . f
Camden and Mt. i
Ephraim,N.J.j
Bradford and 1
Kinzua, Pa.
Denver and Rio i
Grande . . . )
i6
6
39
i,68s
Rolling-stock
16
6
6q
5,676
a
*3
1-!
^1
%
12,167
IS1430
13,64s
14,922
3S,ooo
"3°
il
o
%
1,112
1.346
2,868
i>793
3i5i9
11
B u
%
834
640
2,642
1,717
2,573
•I- bO
•75
•48
•92
•96
•73
^3-
20
Preliminary Survey
Table XIII. — Items of Total Annual Expenses for Maintenance and
Operation of all the Railroads of the United States in 1880. {Poor's
Manual.)
%
Per cent.
Percent, of
per mile
total
earnings
Repairs of road, bed, and track
451
11-23
6-82
Renewals of rails ....
197
4-89
2-97
Renewals of ties ....
122
3-04
1-85
Repairs of bridges ....
102
2-55
I 55
Repairs of buildings
87
217
1-32
Repairs of fences, crossings, &c.
17
•42
•25
Telegraph expenses
41
i-oi
-62
Taxes
Maintenance of road and real)
estate '
Repairs, &c. of locomotives
152
377
2-29
1,169
29-08
17-67
376
249
6-19
Repairs, &c. of passenger, baggage, »
and mail cars . . . . )
120
299
1-82
Repairs, &c. of freight cars
Repairs, &c. of rolling-stock |
257
6-40
3-89
(including renewals and addi- V
627
15-58
9*47
tions) )
1
Passenger train expenses .
137
3-41
2-07
Freight train expenses
330
8-21
4 99
Fuel for locomotives
374
931
5-66
Water-supply, oil, and waste .
70
1-74
106
Wages of locomotive runners and)
firemen i
310
772
4-69
Agents, and station service and)
supplies i
451
11-23
6-82
Salaries of officers and clerks .
139
3*46
2-10
Advertising, insurance, legal ex-^
penses, stationery and printing . j
123
3-06
1-87
Damages to persons and property .
40
•98
•60
Sundries .....
Running and general expenses
Aggregate annual expenses
250
6-22
378
2,224
55*34
33-64
4,019
100 00
60-78
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General Considerations
21
Table XIV. — Gross Annual Earnings per Mile^ per Passenger Mile^
and per Ton Mile, of some of the Principal United States Railways
in 1880.
Len^h
in miles
From
passen-
gers per
mile of
road
From
passen-
gers per
passen-
mile
From
freight
mile
of
road
From
freight
per
ton
mile
Pennsylvania R. R. .
New York Central
Central Pacific .
Chicago, Burl., and Quincey.
Philadelphia and Reading .
Union Pacific
Atchison, Topeka, and )
Santa F^ . . ,\
Average of United States .
1,806
994
2,447
1,805
780
1,215
1,398
87,801
8
4,700
6,651
2,237
1,532
3»429
2,624
1,144
1,641
$
•0242
•0200
•0303
•0240
•0201
•0320
•0606
•0251
8
15,615
21,794
4,577
7,202
17,200
7,154
3»974
4,740
•0089
•0086
•0249
•01 1 1
•01 6 1
•0199
•0209
•0129
Table XV. — Annual Earnings and Expenses of the
above roads in 1880.
Len^h in
miles
Gross earn-
ings per
mile
Expenses
per mile
Expenses
-i- gross
earnings
Pennsylvania R.R. . . .
New York Central . . .
Central Pacific ....
Chicago, Burlington, & Qu.
Philadelphia and Reading .
Union Pacific
Atchison, Topeka, & Santa )
F^ \
Total United States . . .
1,806
994
2,447
1,805
780
1,215
1,398
87,801
$
20,315
28,445
6,814
8,734
20,629
9,778
5,118
6,611
12,267
17,969
3,340
4,454
11,754
4,507
2,408
4,019
f5
•609
•470
•497
.568
•426
•458
•608
Estimates
A few estimates of roads, railways, and tramways, will
now be given, which will to some extent fix the ideas on
what must necessarily be subject to very great diversity,
according to the circumstances of each case.
The surveyor would do well to make rough shots at his
estimate on his first walk-over, so as to guide him in his
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22 Preliminary Survey
choice of alternative routes, and even the most approximate
* aide-memoires ' are often of great service.
The tendency of even experienced men is to over-esti-
mate rough country and under-estimate easy country. A
big gulch or canon is apt to scare most men, and swaggering
viaducts float across their mind's eye, which often by patient
reconnaissance melt down to one or two reverse curves and
a * bit of a trestle.'
An ordinary American railway of about loo miles in
length in moderately easy country will require about 15,000
cubic yards of earthwork per mile.
Mr, Trautwine's estimate, made quite a number of years
ago, is near enough for a rough shot to-day.
It is as follows for a single line in United States : —
Gauge 4' 8i". Labour $1 75 = 7^. per day.
Grubbing and clearing (average of entire road) 3 acres $
at ^50 150
Grading 20,000 -cubic yards of earth at 35 cents . 7,000
Ditto 2,000 cubic yards of rock at j$'l . . . 2,000
Masonry of culverts, drains, abutments of small
bridges, retaining walls, 400 cubic yards at $^ . 3,200
Ballast 3,000 cubic yards broken stone at ^^i . 3,000
Cross-ties 2,640 at 60 cents delivered . . 1*584
Rails (60 lbs. to a yard) 96 tons at ^30 delivered . 2,880
Spikes 150
Rail-joints 300
Subdelivery of material along the line . . 300
Laying track 600
Fencing (average of entire road) supposing only one
half of its length to be fenced .... 450
Small wooden bridges, trestles, sidings, road -crossings,
cattle-guards, &c 1,000
Land damages 1,000
Engineering, superintendence, officers of Co., sta-
tionery, instruments, rents, printing, law expenses,
and other incidentals 2,386 *
26,000
' This amount is only extended to units to bring the total to a lump
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General Considerations 23
Add for depots, shops, engine-houses, passenger and
freight stations, platforms, wood sheds, water stations
with their tanks and pumps, telegraph, engine-cars,
weigh scales, tools, &c. ; also for large bridges, tunnels,
turnouts, &c. (Trautwine.)
Mr. R. C. Rapier, of the well-known firm of Ransomes
and Rapier, in his ' Remunerative Railways ' gives an
estimate for the equipment of a metre-gauge single- line
railway, 40 miles long, including 40 lb. rails, wooden sleepers,
seven engines weighing 15 tons each on six wheels, turn
tables, tanks, water-cranes, weigh-bridges, sheerlegs, signals,
35 passenger carriages and brake vans, 1 50 freight waggons
of different kinds, workshop-fittings, and stores. The total
for 40 miles 86,708/., or for i mile 2,168/.
The same author gives another estimate for the equip-
ment of a forty-mile 3 ft. 6 inch gauge-railway with 45 lb.
rails, eight engines weighing 18 tons on six wheels, and a
similar list of materials to the metre-gauge, somewhat in-
creased. The total for forty miles 98,840/., or for one mile
2,471/.
Where it is necessary, as in England, to put up gate-
keepers' houses, and sometimes signal-interlocking gates at
level crossings, it is often as cheap, having regard to future
expenses of operating the line, to go over or under the
road.
Gatekeeper^ s house,
£ s. d.
Wages 15J. per week = 5 per cent, on capital ot 780 o o
House 220 o o
;fl,000 o o
Even where there is no help from the ground the earth-
work can generally be got under 20,000 cubic yards.
Alternative bridge.
£ s. d.
20,oc» cubic yards earthwork at 9a.
750
One bridge ..»,..
250
Mh999^(30^^
l^l
^
^4<
^
o ?
J1
i2
f
, jL.^ y
1 r>. .-i
?.£ J ^ (
• X
^1
H c3 Co6 i
qiSuaq
g « s.s
£ 2 o"^
Rinitizprl hv\, lOi
General Considerations 25
A railway 87 miles long was completed for the Nizam of
Hyderabad about the year 1885, for under 6,000/. per mile.
It was promoted by the Indian Government, and built
under the supervision of the Government engineer. The
gauge was 5 ft. 6 inches, the rails were 66^ lbs. per yard, of
steel, and the sleepers steel also. The price included
equipment and fencing.
Roads
The Grand Trunk carriage road of the Bengal Presi-
dency cost approximately 500/. per mile. The work was
done mainly by starving poor during famine time, under the
administration of Lord William Bentinck about 1835. The
width was 40 feet, metalled in the centre 16 feet wide with
either broken stone or a natural concretion of carbonate of
lime, called * Kunkur,' rammed by hand (there being no
steam rollers in those days). The cost of laying the metal
at an average lead of one mile was 162/. ds, od, per mile,
and cost of repairs and maintenance of ditto 33/. 12s, od.
per annum. Total cost of maintenance was 50/. per annum.
(Gen. Tremenheere, * Min. Proc. Inst. C. E.' vol. xvii.)
The main roads of South Australia are described by
Mr. Charles T. Hargrave in * Min. Proc. Inst. C.E.,' vol. 1.
and he gives the average cost with a metalled way 18 feet
wide and a 60 feet right of way as follows : —
Earthwork in cuttings &c., 800 yards at u. . 40 o o
Culverts, nine at 15 cubic yards each ; 135
cubic yds. at \2s 8100
Wheelguards and posts ; 18 sets of 2 posts and
one guard at 40J. per set . . . . 36 o o
Forming the metal bed ; 80 chains, 8j. . . 32 o o
Bottom metal or soling, 4" thick, 15 cubic yards
per chain for 80 chains ; 1,200 cubic yards
at 4f 240 o o
Metal 24" thick, 22 cubic yards per ch. for 80
ch. ; 1,760 cubic yards at 7 J. . . . 616 o o
Blinding [ji,e, thin top dressing of gravel) 7 cubic
yards per ch : for 80 ch. : 560 cubic yards
at If. 28 o o
Rolling eight days at 255 10 o o
Carried forward . . v^\c^^\\v<£l^
26 Prelhni7iary Survey
Brought forward . . 1,083
To this must be added fencing (not always done)
80 ch. at 8 rods = 640 rods at 5j-. . . 160
And if land has to be purchased ; 8 acres per
mile at 5/. . . . . . . 40
;^i,283 o o
For the first six miles out of Adelaide during ten years,
300 to 400 cubic yards of metal were needed per mile per
annum. At fifty miles from the city only slight repairs were
needed.
Annual cost of clearing culverts and weeding 3/. per mile
for each side of road. On 326 miles constructed the cost
of maintaining the metalled portion was 122/. per annum.
The price of labour was 5^. 6^. to 65. per day ; masons
and carpenters %s, to gx.
A temporary road for conveyance of railway materials
into the bush costs from 50/. to 100/. per mile.
Stone-Crushers
' Blake's or Blake-Marsden Stone-Crushers ' vary in size
and cost, from a 10'' x 8'' machine breaking stone of that size
to the extent of 3^ cubic yards per hour, nominal horse-power
3, total weight including screening apparatus 5 tons 6 cwts.,
price 157/., up to a 30" x 13'', breaking stone of that size to
the extent of 14 cubic yards per hour, 16 horse-power, weight
16 tons 2 cwt., price 440/.
ROAD-ROLLERS
A 1 5 -ton * Avehng and Porter Roller ' costs about 650/.
It will roll on an average 1,100 square yards per day. The
cost of rolling, including all charges, is somewhere between
\d, and \d, in England ; and the cost of binding material
about 3^. per square yard.
The two last estimates are gathered from Mr. Boulnois'
Municipal Surveyor's Handbook.'
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General Considerations 27
Tramway Estimates
The following estimates of tramway construction in
the United States are from the report of the committee
of the American Street Railway Association at the Min-
neapolis Convention in October 1889, upon *The Con-
ditions necessary to the Financial Success of Electricity as a
Motive Power.'
They are for a single line ten miles long equipped com-
plete with fifteen cars.
Cable-road
Road-bed^ rails, conduit cable .... 140,000
Power station 25,000
Cars 3,000
Total for ten miles ;^ 168,000
Electrical Overhead-wire Construction
Road-bed and rails 14,000
Wiring 6,000
Cars 12,000
Power station 6,000
Total for ten miles ;^38,ooo
Storage Batteries
{also termed Sitondary Batteries or Accumulators)
£
Road-bed and rails 14,000
Cars 15,000
Power station 6,000
Total for ten miles ;£"35»ooo
' In the above cases of electrical construction the motor
car would be capable of pulling one or two tow-cars if
necessary. These figures your committee has no doubt will
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28
Preliminary Survey
be found to be cal-
culated within a
reasonable limit of
cost/
Figs. 3 and
4, excerpt *Min.
Proc. Inst. C. E.'
vol. Ixxxv., will
give some idea of
the way difficulties
of ascent are over-
come by curve-
development, and
the ground plan of
Burlington shops
will show the way
the lines are laid
out for the engine
depots in America.
The actual
measurement for
the preliminary
estimate is much
facilitated by dia-
grams, some of
which will be found
in the chapter on
'Graphic Calcula-
tion.' These ope-
rations follow upon
the actual survey,
whether it be
merely a route-
survey or a tele-
metric survey, and
are therefore de-
scribed later on^
General Considerations
29
30 Preliminary Survey
CHAPTER II
ROUTE-SURVEYING OR RECONNAISSANCE
Before going into the subject of * Route-survey ' in detail,
the first steps of American pioneer railway-making will be
briefly described.
The Nipissing division of the Canadian Pacific Railway
will furnish a fair illustration of American location. It is on
the north shore of Lake Superior, and it was there that the
writer gained his first experience of that kind of work. A
netwcMrk of lakes and rivers leads the Indian, on his fishing
and hunting expeditions, clear across the watershed of
Northern Ontario down to Hudson's Bay. The first pioneer-
ing was done by Mr. W. R. Ramsey, with an Indian guide
and one or two white men. Taking only the aneroid and
prismatic compass, they followed up the course of several
rivers with a canoe, often having to carry it over long * por-
tages,' and undergoing considerable hardships, until they
emerged at the junction point with another survey, proceed-
ing from Port Arthur on the other side of the great lake. A
sketch-map was made for the approval of the general
manager, showing the topography to a small scale, and the
length of line.
The next party, headed by Mr. Ramsey, consisted of the
usual preliminary location party : —
1 Transit-man 2 Axe-men
2 Levellers 2 Chain-men
2 Rod-men i Slope-man
I Picket-man i Cook
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Route-Surveying 31
The modus operandi is then as follows : —
The leader keeps ahead of the party, revising his previous
survey with aneroid and compass, and keeping the whole of
the operations under control. Every morning he indicates
the direction to the transit-man, who has a line cut through
the bush and chains it, driving stakes at every hundred feet.
The leveller follows on the transit-man, making the profile
(or section as we call it in England), whilst another takes the
side-slopes of the hills with a clinometer. The field-work
is plotted in tent at night. The profile is greatly facilitated
by printed profile paper, which dispenses with all scaling.
Under favourable circumstances a mile or a mile and a
half a day can be surveyed in this manner, but on the division
here referred to only an average of ten miles a month was
possible with one party. At times there were four parties, in
the field to distribute the work. Nearly all of it was gone
over three or even four times, always saving money on the
construction. A hundred pounds extra in survey will often
save a thousand pounds in construction.
The present chapter is devoted to methods which do
not require chaining, but more on the American methods of
location will be found in Chapters VI. and VII.
When the country through which a road or railway is to
be made is not well known— -specially when the inhabitants
are unfriendly, but in almost every case of projected work in a
new country — it is necessary to obtain, cheaply and speedily,
one or more surveys of alternative practicable routes or of
the general topographical features of the area.
A chained survey would be out of the question, and even
a telemetaric traverse would generally be too expensive. A
fairly accurate map is required, which will be made by one
man at the rate of ten to twenty miles per day, and which
will serve the projectors with sufficient information to guide
them in approving the route favoured, in capitalising their
enterprise, and in preparing their prospectus.
The class of work most suitable for this object has been
.^ , Google
32 Preliminary Survey
more thoroughly studied and practised, though with a dif-
ferent object, by military than by civil engineers, and the
best instruments and books extant are from those sources.
Failing time or money for chaining or triangulation, it is
necessary to have recourse for linear measurement to records
of pace of men, gait of horses, or speed of river steamers
taken ^with as much care as possible. Military movements
are made with trained regularity, and no one who has not
studied the subject would believe what great precision is
attained in making maps of marches the distances of which
are laid down simply and solely from either time-records or
pace-measurers.
These maps, which are called reconnaissances, route-
surveys, or flying surveys, vary in accuracy, from a rapid
field-sketch relying upon a trained eye for estimation of dis-
tance and a knowledge of perspective for the filling in of
detail, to a survey correct within a margin of from one to five
percent.; based upon true trigonometrical and traverse prin-
ciples and preserved from cumulative error by astronomical
observations similar to those which determine the position
of a ship at sea.
•The term * a mere sketch ' is often applied rather con-
temptuously to what may be a most valuable and perhaps the
only record of the topography of an important position.
Military engineers know best the value of a good sketch.
They have to do their work under fire or in danger of
surprise, and a man who can dash off a sketch of the enemy's
position, the points of vantage, and the best line of attack,
will by so doing provide in a few moments information which
may decide the final issue of the struggle.
Surveyors have more time at their command, and their
disposition is rather to depreciate methods which overstep
the bounds of rigid mechanical exactness. The standing
orders of our English Parliament for deposited plans of
public works call for a high standard of accuracy, and justly
so in a country like this, possessing already excellent
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Route- Surveying .,33
maps published by the Ordnance Department It was
due to the existing vested interests that they should be
protected from the attempts of speculative promoters
to interfere with their property without fully and clearly
representing in all its bearings what the effect of such
interference would be. The standing orders, however, per-
mit of the framing of memorials of opposition containing
frivolous allegations which as such are frequently overridden
by the Exsuniner.
Surveyors who get their training in England do
not consequently have much opportunity of practising
either route-survey or field-sketching, and so, specially
in regard to the latter, do not know if they have a
talent for it or not. Artists are born, not made, but there
are few engineers who are so wanting in 'eye* that they
cannot greatly increase their efficiency as pioneers -by a
course of study and practice in sketching. Even when the
route survey is carried out with all possible care so as to
approach to the purely mechanical system, a knowledge
of sketching will be most useful in filling in the adjacent
country. Lake, wood, bluff, canon, or river will, by a few
dashes of the pencil, give the promoters at home something
more for their minds to feed upon than a system of straight
lines with the bearings written up. It should of course be
evident from the map, when part is sketched and part scale-
able, which is the reliable portion.
The art of free-hand drawing can hardly be dwelt upon
in a work of this kind. A very few lessons in perspective
combined with a thorough course of practice in the field is
the only way to obtain the needed skill. Even in the course
of telemetric survey, the sketching of detail is a great help
both in the plotting of contours and as an accessory to the
finished map.
Photography is also becoming a useful aid to the sur-
veyor. The little * Detective' camera recently introduced
is only a little larger than a carriage clock ; it is carried in
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34 Preliminary Survey
the hand without needing a tripod or other paraphernalia. It
is directed and focussed by observing a reflected image of the
view on the top of the camera. It has also the advantage
over previous inventions of the kind that the views may
be removed one by one instead of in batches.^
The scales adopted in route-surveying vary from six
inches to the mile, to thirty or more miles to the inch. If
to a small scale, it is usual to plot in miles of latitude and
longitude, so as to correspond with the daily astronomical
observations.
The route survey is essentially a traverse, checked where
possible by triangulation with compass , angles and range-
finder, or simply by rays drawn upon the sketch-board or
plane-table.
Surveying with the Sketch-Board or Plane-Table
Those who have become proficient in the use of the
plane-table are generally enthusiastic in favour of it, and
there can be no doubt that for a topographical survey of
a large area, based upon the known stations of a primary
triangulation, it is both rapid and accurate. During the
season of 1886-7 the U.S. Geological Survey mapped in
this way an area of 56,000 square miles with a staff of 160
men, at an average cost of \2s, per square mile.
* A camera should be preferred which does not require the removal
and changing of the slide every time. Abrahams & Co. make the
* Ideal ' camera, which is quite suitable for the purpose. If the surveyor
is on one side of a wide canon and a train is passing along the other
side, he can get a good representation of the whole hillside by taking
views of the train all along its passage, the steam from the engine will
make a good landmark and especially just as the train goes into tunnel.
The views can afterwards be pieced together ; the length of the train
will serve as a measure of relative distances, and considerable informa-
tion be placed on record by a few seconds* field-work. The slides
can be preserved and developed at leisure or sent home for develop-
ment, but the developers are provided in soluble pellets for foreign
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Route- Surveying 35
The author's practice in surveying for railways has led
him to the persuasion that the plane-table is more useful
as an adjunct than as a universal instrument. He uses the
modification of Captain Verner's military sketch-board plane-
table either buckled on the wrist or mounted on its light
tripod, and in this form finds it one of the most valuable
portions of the surveying outfit.
When the country is very magnetic, the needle becomes
unreliable, and the plane-table may have to be used for the
whole of the field-work.
Under favourable conditions, it is possible to do pretty
much the same class and quantity of work with the
plane-table alone as with the prismatic compass and field
notes.
With the plane-table, the work is all done in the field,
which has the advantage of exhibiting the errors liable to
arise with either class of work, and so to enable them to be
eliminated on the spot. It has the disadvantage of occupy-
ing time in mapping during daylight, which might be wholly
given to fieldwork, and the mapping done either at night
or when the sun is too powerful, or when too wet to go out.
It has the further disadvantage of disclosing to unfriendly
natives the object of the traveller's journey, being much more
conspicuous than a prismatic compass. It has the advantage
of aflfording to the skilful surveyor a complete method of
triangulation, producing results of an accuracy (even over ah
extended area) very little short of that obtained with good
theodolites ; its horizontal angles being determined by rays
aided if desired by the telescopic sight-rule, and indepen-
dent of magnetic variation. It has the disadvantage of being
with difficulty checked when carrying forward a continuous
traverse, especially so when the bases have to be short, as
in a crooked road with high hedges. There are generally
some salient points on which rays can be taken as in-
dependent checks, but without them every mistake in angle
is perpetuated.
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36 Preliminary Survey
For an extended triangulation a measured or calculated,
base is required as in geodetic survey, whether working
with the plane-t.able or compass ; and where the needle
can be relied upon, the angles can be taken with the same
accuracy with either, but in the case of a continuous traverse,
the compass has the advantage of giving the magnetic
bearing every time independently of the previous work.
When the plane-table is used as a universal instrument,
a stadia telescope is attached to the sight-rule, which has
to be transported in a separate case. The class of work
performed by it under these circumstances answers to the
telemetric survey with the transit, but when passing rapidly
through a country, it is much more cumbersome to transport
a heavy plane-table with sight-rule and telescope than to
shoulder a theodolite.
A few remarks will now be made upon the method of
triangulation with a large-sized plane-table over an extended
area, after which the subject of the plane-table will be
only considered in the form suitable to preliminary railway
or road survey with the smaller instrument described in
Chap. IX.
Triangulation with the Plane-table
Where a base is known by the exact geographical posi-
tions of two towns or villages a few miles from one another,
and commanding a good view of surrounding country, an
almost unlimited tract may be correctly surveyed from it.
When this is not the case, the two extremities of the base
have each to be located astronomically, as described on
Chap. IV., and the distance between them calculated by
problem on p. 150. The accuracy of the succeeding
triangulation will depend fundamentally upon this astrono-
mical work. The latitude should be easily obtainable within
a sixth of a mile, and by careful repetitions to about 100
yards of the earth's surface, so that if the watch be not
reliable to give Greenwich time to a second, the more the
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Route- Surveying 37
base approximates to a trae meridional line the better for
the results.
To obtain practice in England the plane-tabler should
locate two suitable points for his base from the six-inch Ord-
nance map of the district, which gives parallels and meridians
to every single second or about thirty yards. They should
be selected as far apart as possible, say from two to seven
miles.
A common military plane-table 30 inches x 24, of one-
inch panelled deal, will map the whole county of Yorkshire
on a scale of four miles to the inch. It can afterwards be
compared with the shilling edition of W. H. Smith & Son's
reduced Ordnance maps on that scale.
The paper should be pasted on the board by its
margins, well damped so as to stretch as tight as a drum
when dry. The parallels and meridians should then be
drawn as described on p. 75, and the base line AB laid
on from the data taken from the six-inch Ordnance map.
The table is then carried to station A, and, with the sight-
rule touching the base line, the board is turned on its axis
until the opposite base-station B is brought into the exact
line of sight. The compass, which should be of the portable
trough-needle type, is then placed in one corner of the
map, and carefully turned to and fro until the needle is at
rest at the centre of its run. A fine pencil line is then drawn
round the compass box with the letter N. (mag.) opposite
the north point of the needle.
With the plane-table firmly clamped, rays are then taken
from A to all prominent points in view ; they should not
be drawn right across the paper, but merely fixed by a fine
pencil line in the margin of the map about half an inch long,
with marks denoting the point of observation and the point
observed, thus A/Ai to correspond with descriptive entries in
the Field Book, shown overleaf.
When all possible rays have been taken from A, the
instrument is shifted to B, and * set in meridian ' by a back
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Preliminary Survey
FlELDBOOK
!
Date Time fta^
' i
Ob-
ject
Latitude
Longitude
Ane*
roid
Varia-
tion of
compass
Remarks
"
/
//
1890 1
Jan. I 3 P.M., A
i
B
A,
A.
^1
640
i9i°W.
Spire of Hockley
Parish Church
Clump of Beeches
near Hockley
ray on A. The compass-box is placed on its delineated
position on the paper, and if the sight-rule has been cor-
rectly adjusted, and there be no local magnetic deviation or
sufficient diurnal variation to account for it, the needle will
still be at rest at the centre of its run. If not, its position
should be measured by the graduation on the compass-box
as an addition to or deduction from the initial variation, and
an entry made of the variation at B.
In selecting the next base C by intersection of two rays
from A and B, a point should be chosen which, in addition
to advantages of commanding position, should form as
nearly as possible with AB an equilateral triangle ; the
reason being that in any triangulation the more acute the
angles, the less reliable must of necessity their intersections
be and an equilateral triangle is the only one in which
no included angle is less than 60°.
When primary points have thus been located all over
the map, the filling in of roads and other detail is done
either with the prismatic compass and passometer, as shown
in the example on p. 54, or, if more accuracy is required,
with compass and tape. This subsidiary survey is plotted
on tracing paper, and pricked through on to the map. This
method is preferable to traversing in the detail with the
plane-table itself as described on p. 41 because it saves
the map from getting soiled, and is an independent check*
(when the tape is used) upon the other work.
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Route- Surveying 39
The main practical points to be observed in using the
table are to set it up firmly upon its tripod ; to level it per-
fectly true with a circular bubble ; to be very exact with
its setting in meridian ; to keep the pencil finely pointed ;
to draw the rays with the utmost nicety, and above all
things not to get hurried.
Road and Railway Work
The work required for preliminary road and railway
survey resolves itself into two classes : the first a reconnais-
sance of such rapidity that even stadia measurements take
too long, and, except at intervals, there has to be as little
time as possible devoted to erecting, levelling, and adjusting
of instruments of any kind ; the second, a telemetric location
survey, which will give the levels sufficiently close to plot a
profile from them, and this is quicker and better done by
the transit-telemeter than by the plane-table.
The *Vemer' sketch-board is described on p. 318, to
which the reader's careful attention is directed; its use will
be first explained when accompanied by a prismatic compass
on reconnaissance, and then some few further illustrations
will be given of its more extended application when en-
larged and developed into the regular surveying plane-table
for the sake of those who may wish to make the most of it.
Variation of the Compass by the Plane-table
Before starting, the variation of the compass should be
ascertained either by an observation of the solar azimuth, as
explained on p. 132, or, failing suitable instruments for that
purpose, by equilinear shadows on the sketch-board, as shown
by Fig. 5. Erect the instrument on its wooden tripod, having
the roll of mapping paper tightened up into its place. Fix the
brass stile in the hole provided for it near the compass-box,
and adjust the board level and stile vertical by the clino-
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Preliminary Survey
meter or otherwise. Draw the centre Une of the table across
the paper from headpiece to tailpiece, and * set ' the table so
that the needle shall be at rest in line with this centre line.
Check once more the adjust-
ments by clinometer, and
the instrument is ready.
About an hour or two
before noon place a mark
with a pencil at the extremity
of the shadow cast by the
stile, and from the base of
the stile as a centre and with
a radius equal to the length
of the shadow describe a
circle right round the board.
When the sun begins to
drop again in the afternoon,
watch the shadow until
it once more touches the
circle. Mark it, and bisect
the chord drawn between
the two shadow-points in
the circle, and draw a line
to the base of the stile from
the centre of the chord.
This will be very nearly the
astronomical meridian, and
the angle between it and the centre-line will be the variation
of the compass. The error, as explained on p. 135, arises from
the altered declination of the sun during the lapsed time,
which varies from o to i angular minute per hour. The
error may at all times be neglected for this class of work.
When the weather is uncertain it is best to take two or three
points, say at 2 hours, i^ hours, and i hour before the
meridional passage, in order to ensure getting the sun in the
afternoon again.
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Route- Surveying 4 1
Sketching and Plane-tabling
The sketch-board is used upon two distinct principles.
When buckled on to the wrist, the correctness of the align-
ment depends on the little compass in the head-piece, checked
as much as possible by bearings taken with the prismatic com-
pass. No back and forward rays can, of course, be taken. The
board has to be ^seV at every point of observation by the
* working meridian,' which is a line drawn across the compass-
box with an index at its end, by which the compass-box can
be turned in any direction. The circle round the compass is
graduated into divisions of ten degrees each, and the index is
placed in such a position that when the needle is under it the
board will be directed in the general line of route. When once
fixed the index is never touched unless the survey begins to run
off the paper. It is a fiducial line by which to adjust or * set '
the board whenever a sight is taken ; and considerable
practice is required to hold the board steady on the wrist
with the needle truly under the index whilst the sight is
being taken. It is also quite an awkward business to keep
the edge of an ordinary sight-rule or ruler at the station-
point whilst it is being rotated to take a sight. It was to
meet this difficulty that the needle-point sight-rule, described
in Chap. IX., was contrived, which has proved a great
convenience, and much better than merely sticking a needle
in the station. Most surveyors use elastic bands for keeping
the ruler in position, but a thin string with a spring under-
neath is neater.
The * working meridian ' is fixed upon before starting
by means of any existing map, or, failing all such data, by
inspection of the ground. For instance, if a route-survey be
required from London to Birmingham we draw a pencil line
between the two cities upon an ordinary atlas, and, running
up the line to the intersection of a parallel, we find the
astronomical bearing to be roughly 50° N.W. Supposing
also that the variation of the compass has been just deter-
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42
Preliminary Survey
mined, as already described, to be 20° W., the magnetic
bearing of the air-line between the two cities will be 30° N. W.
This is called the * Line of Direction,' and is marked as such
on the headpiece, corresponding with the direction in which
the paper has to be fed forward upon the rollers. The words
*Line of correspond with a due northerly direction ; that is
to say, when the index is placed in line with the * Line of
Fig. 6.
Direction,' and the needle is brought under it with its north
pole towards the words * Line of,' the board will be held in
position for running due north.
We want to run a course of 30® N.W., and therefore fix
the index 30°, that is, three divisions to the right of the words
* Line of,' so when the needle is brought to rest under the
index, the * Line of Direction ' will point to Birmingham.
The width of the paper being ten inches, it will take in
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Route- Surveying 43
20 miles on either side of the air-Hne to a scale of four miles
to the inch, and this would take in the whole map if we were
following one of the main highways.
If the hne should run off the paper a * cut-line * must be
drawn and a fresh start made in the middle of the paper with
the same line of direction. This method of using the sketch-
board is illustrated by Fig. 6. The magnetic bearing of the
line of direction shotild be written up before commencing,
so that the index, if accidentally shifted, may be replaced.
The prismatic compass is of great use in checking the
angles taken in this manner with the sketching-board. The
bearings of at least the main lines of the traverse can be
taken, and whenever important intersections are wanted upon
salient points they should be likewise checked with the pris-
matic compass. It is quite possible to have too many instru-
ments, but a compass clinometer, as described in Chap. IX.,
is no inconvenience and most valuable. However skilfully
the board is held in line, the slightest jog to the arm may,
imperceptibly to the sketcher, twist the table several degrees.
The value of the sketching-board is not lessened, but on the
contrary much enhanced, by assisting it with the prismatic
compass. The same amount of detail can be filled in, but
with increased accuracy.
Sketching with the aid of Maps
Where existing maps of any kind are available, they
should be made all possible use of. An enlargement from
an atlas, however imperfect, should be laid down to scale, and
unless the map is thoroughly reliable, hke an Ordnance map,
it is best to draw the enlargement in pencil, and to plot in
ink, or vice versA^ so that the divergences may be at once
apparent. To copy detail from the existing map would only
confuse, but a check of alignment and distances is of great
assistance.
It frequently happens that during a rapid traverse a case
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44 Preliminary Survey
arises demanding more careful treatment than can be given
with the board on the wrist. A metallic telescoping tripod
is buckled to the bag of the sketch-board, or, when riding, is
attached to the saddle. It is only sixteen inches long when
shut up, and gives no trouble. By it the sketch-board can be
used as an ordinary plane-table, but being light and vibratory
it is only used in emergencies ; for continuous plane-tabling
the wooden tripod should always be employed.
In this form, the work can be done in fairly easy country
at the rate of twenty miles a day. It is something more
than a sketch and something less than a survey, but seeing
that it takes hardly any more time to a practised man than
if he simply took notes, it has the great advantage of graphic
representation over mere literary description.
Accurate Plane-tabling
The second method of using the sketch-board mounted
on its wooden tripod differs in no way from the ordinary
plane-table except that the instrument is smaller.
The principle of this kind of surveying is the geometrical
law of similar figures, by which when a single side is known
all the rest are determined by their positions in the figure.
It is a graphic triangulation resting on the same mathe-
matical theory as telemetry. The method of setting up the
table has been already explained.
The U.S. Geological and Coast Survey have covered
immense tracts of country with plane-table work and use
instruments of large size — 24 inches by 30 is the maximum.
They are fitted with elaborate joints for levelling them true,
and furnished with sight-rules carrying stadia telescopes
with vertical arcs for measuring angles of inclination. The
figure on p. 42 of a traverse by the first method will also
serve for the second. The preliminaries are all the same,
the only difference being that the board is * set ' at each new
station by a backsight on the previous station with the sight-
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Route- Surveying 45
rule. The precision attainable in this way is very great.
Check angles should still be taken with the prismatic compass,
especially where the bases are short, but the rays are gene-
rally more accurate than the compass-bearings, particularly
so if there is local attraction to the needle. The compass-
bearings, both with the needle in the headpiece (which is
set to the * working meridian ' precisely as before) and those
by the prismatic compass, are simply checks and nothing
more.
In taking a sight, the ray should be projected at the edge
of the paper about half an inch long ; it should be marked
with the signs of the back and fore station thus : A/B if
taken from one main station to another, or B/Bi &c. if taken
from a main station to some outside point. The signs should
Fig. 7.
also be entered in a book of description (see p. 38) specify-
ing what the points represent, and accompanied by little
sketches, especially when the points are not very sharply
defined, such as distant villages, hill-peaks, or river-bends,
so that when arriving at the next station and taking the
backsights of intersection, the memory may be assisted as
to the precise spot viewed on from the last station. These
little sketches are sometimes made upon the map, but it is
not such a good plan, as they are liable to come in the way
of succeeding rays.
The flag on Fig. 6, p. 42, is shown as an out-station, but
the board can be moved to its vicinity and backsights taken
to all the previous stations on flag poles. The coincidence of
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46 Preliminary Survey
these rays with the foresight will prove the accuracy of the
work and close the traverse.
The whole principle of plane-tabling is here embodied.
The stadia work, levelling and contouring, which are added
by means of the accessory instruments to the larger tables,
do not form an essential part of the instrument itself, and
will be treated of under telemetry.
Auxiliary Plane-tabling
The method of using the plane-table as an adjunct to
the tacheometer for filling in detail will be next described.
In Fig. 7 a tacheometer traverse is being carried along a
main road, and a building with enclosure is proposed to
be filled in with the plane-table, being situated upon a
cross road, and it being desired to avoid a deviation with
the tacheometer. A point Bi is fixed in the vicinity of the
building by a sight with the tacheometer, if possible, within
ICO feet of the further corners. The plane-table, worked by
an assistant, is then set up at B,, and aligned by a ray on B.
The line BB^ is then laid down on the paper without any
reference to the compass.
The distance BBj is not absolutely necessary, but, being
known, it is better to plot it, and so locate B on the paper
to enable a checksight to be taken upon it if the table has
to be moved to another sub- station. If all the corners are
within I GO feet, they can be taped without moving the
table and laid off to scale on rays taken with the sight-rule.
If the table has to be moved to another sub-station in order
to command the whole of the detail, it can be triangulated
as already described, but it is generally quicker to locate
several points with the tacheometer as plane-table stations
than to sub-triangulate with the plane-table.
Location of the Instrument
The subject must not be dismissed without touching
upon the well-known three-point problem for locating the
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Route-Surveying
47
instrument, although it should not be used when a more
direct method is possible.
It frequently happens that, when surveying with the aid
of a map, it becomes necessary to determine the position
of the instrument on the ground from one or more points
indicated on the map and transferred from it to the plot.
If only one or two points are known, the problem cannot be
solved without the compass.
Case I. When one point is known. Maps are generally
constructed having the astronomical north at the top of the
sheet. Ordnance maps and ordinary atlases are made thus,
but not so the parish maps. When this is not the case, the
^f^f^. (!^y9^L ^i-t4?5^
I
-J ,
!t"-;5j-
)'
-^
Fig. 8.
position of the astronomical or magnetic meridian or both
must first be determined on the ground from the known
point in the manner already described.
When the geographical alignment of the map is known,
and the variation of the compass has been ascertained, the
magnetic meridian should be marked on the map at the
known point, a bearing taken to it with the prismatic
compass, and plotted backwards from it, i.e. i8o®±bearing.
A base line is then run at right angles if possible, but if the
ground will not admit of it, as nearly square as possible, and
the bearing taken. The base line is measured, and from
its extremity another bearing is taken to the known point.
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4^ Preliminary Survey
Then laying down the two bearings from the point towards
the observer's position on the plot, take the measured
distance by scale and place it with the dividers, in its proper
bearing, where the extremities will coincide with the two rays
from the known point. If the base cannot be run square, the
bearing of the base has to be run out with a parallel ruler.
It is also convenient to lay off the base in an even number
of feet or yards as loo or i,ooo, for then the distance can
be read off from a table of tangents.
Case 2. When two points are known, the prismatic
compass is used in the same way, but a base is dispensed
with* The two known points are plotted from the map in
their proper position upon the plane-table, their bearings
A
Fig. 9.
taken and plotted backwards as before from magnetic
meridians drawn through the known points. Their inter-
section is the locus of the instrument.
Case 3. When three points are known the plane-table
can be located by the sight-rule alone.
Let A, B, C be the three known points and a, b^ c their
position on the plot. It is required to locate upon the
plot the point of observation and to set the table so that
the plot shall have its lines parallel to those in nature, or, as
it is termed, the table be * in meridian.' It is obvious that
rays through A«, B<^, or C^will intersect somewhere, and
by Euc. vi. 2 we know that in any triangle, ^AB, jvBC,
or rAC, when the sides AB, ab ; AC, ac ; BC, be, are
proportional they are parallel. Therefore, if by adjustment
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Route- Surveying
49
of the table we find the position in which the three rays Aa,
^by Cr, intersect in one point, the plot will be parallel and
the table in meridian, and the point of observation correctly
located on the plot.
When the instrument is not * in meridian,' a * triangle of
error ' is formed as shown on the figure, the elimination of
which adjusts the table. There is one exception, when the
^C
^1 — /*^
Fig. io.
point of observation is situated in the circumference of a
circle described about the three known points, the triangle
of error disappears for any position in that arc. This is at
once seen by being able to rotate the table without causing
any error, and some other point must be chosen from which
when located the required point can be determined. See
more on the three-point problem by station-pointer in
Chapter IX.
The Plane-table as a Range-finder
Fig. 8, p. 47, illustrates the principle of range-finding
described on p. 338. The plane-table will, carefully handled,
determine this range or distance of an object with as great
accuracy as with some optical range-finders. Military engi-
neers do not use it for this purpose because the enemy is
fond of making a target of it.
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50 Preliminary Survey
Measurements of Distance
Setting aside the chain, the methods available for deter-
mining the distances upon route survey are of three kinds.
Firsts mere judging by the eye and hand, for which some data
will be presently given of value to the sketcher. Secondly^ the
ancient method of pace-counting, to which may be added
the trocheometer for wheeled vehicles, the time-measure-
ment of horses' gait, and the patent-log records combined
with time measurement when steaming on a river.
Thirdly i the telemetric or optical measurement of distance,
which, including the subject of military range-finding, is
treated of in the chapter on Tacheometry, also in the chapter
on Instruments. It belongs in its practice more to the
route survey, though in principle it is to be classed with
tacheometry.
First Judging by the eye is a faculty which is an im-
portant part of the capabilities of the sketcher. A good
shot will sometimes come pretty near the distance by look-
ing at it as a range for a fowling-piece ; a cricketer by the
distance he can throw a ball. By holding the hand across
the field of view at arm's length, so as to cover the height of
a man, a horse, or a man on horseback either with the palm
or with one or more fingers, we have a measurement some-
what on the stadia principle and of some assistance in
guessing.
The second method, of pace-counting &c., may be
brought to a very fair degree of accuracy for the purpose
of reconnaissance, and we will first touch upon the
passometer or pace-measurer. It is in appearance like a
watch, the mechanism consisting of an escapement carried
by a loaded lever, which is shaken by the shock of the
step and returns by means of a light spring ; the escape-
ment actuates a train of wheelwork, and moves an index on
the dial to record either the number of paces or the actual
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Route- Surveying 5 1
mileage. If the latter, an adjustment is provided to make
the index read correctly to any given length of pace and
instrument is called a pedometer. For long bases, the
mileage indicator is as useful as the other; but for short
distances, and varying inclinations, the counting index is
preferable as no instrumental adjustment is needed, but
scales of paces are drawn upon the board to their ascer-
tained value under different conditions of travel, with their
respective designations. The needle-point sight-rule is
provided with a broad chamfered edge upon which, when
using only one or two scales of paces, they can be
gummed with stamp-margins so as to have the working-
scale at all times on the paper and avoid dividers. This is
dispensed with by using the Mannheim slide-rule as sight-
rule and scale : see p. 245.
The passometer does nothing more than count the
paces \ the accuracy of the measurement depends upon the
regularity with which the walker can pace. A course of
training in this is indispensable, but can be easily gained
during times of recreation. The chief points are to walk
naturally without trying to step yards or any other specified dis-
tance, to hold the body erect, and to maintain the same speed.
The best way to arrive at reliable data of pacing is to
walk over a piece of road where the mile-stones are cor-
rectly indicated. A piece of railway will do very well for
flat walking if the sleepers and ballast are avoided, keeping
to the side of the bank or cutting. A turnpike road is
better. For hill walking a six-inch Ordnance map will give
the contour lines crossing the public roads, from which the
gradients can be calculated.
The time of one's walk has a great deal to do with the
length of step. By educating oneself into the same rate of
time — uphill or downhill, fresh or tired — the length of
pace will be much more uniform than it is ordinarily.
Gradients as steep as i in 40 do not then make any differ-
ence on the average length.
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52
Preliminary Survey
The following trials were made up and down a road
varying from i in 20 to level, but all uphill one way, and
down the other.
Uphill
AtoB .
BtoC .
Cto D.
Total
Downhill
DtoC.
CtoB .
Bto A .
Total
Dis-
tance,
feet
2,900
2,080
820
5,800
820
2,080
2,900
5,800
Number of paces
ist time
2nd time
3rd time
1,091
1,161
1,132
783
705
730
284
285
282
2,158
2,151
2,144
286
296
284
726
725
741
1,156
1,121
1,163
2,168 j 2,142 2,188
Value in feet of
100 paces
xst time and time'srd time
266
266
289
269
287
287
251
267
268
295
287
269-5
277
287
261
270-5
256
285
291
271
288
281
249
265
Average of the three times . . . 2-687 feet per pace
Range of one time from average . • i '4 per cent.
Maximum range of shortest distance) ,^.^
from average .... } ^0-9 per cent.
These trials were under unfavourable conditions as
regards gradient, and are given to show results attainable by
an unpractised walker. See also closing error on p. 55.
They demonstrate very clearly the tendency of pacing
towards uniformity over long distaftces even when there may
be great variations over short lengths.
The distance was only a little over a mile, but this fact
becomes more apparent on daily journeys of ten to twenty
miles, in which the total error can easily be kept within
from I to 2 per cent.
When particular exactness is needed, and an assistant
is present, it is advisable every two or three miles to check
the rate by taping a stretch of 300 paces or so, in order to
detect changes due to fatigue, rough or slippery roads, &c.
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Route- Surveying 5 3
A series of scales may be constructed for use on various
gradients, but it is less confusing to work to one scale and
make a marginal note in the fieldbook to guide in making
the corrections when plotting, or calculating the latitude
an4 departure.
The plotting may be done by the protractor, but the
principle of working to latitude and departure is more exact,
and the calculations are done as quickly with the slide-rule
as if the angles were laid off with the protractor. There is
a harmony, moreover, between this process and the daily
astronomical observations, both being a reference to rect-
angular co-ordinates.
The analogy of traversing with navigation should be
thoroughly studied ; even down to curve-ranging as will be
shown later on. In the traverse for route-survey the work
is nothing more than the dead-reckoning ; only instead of
suffering from liability to error in under-currents, slip of
screw, and what not, it is troubled with magnetic aberra-
tion, irregularities of pace, and * personal error.' The
astronomical observation comes in to help out the land
surveyor with the addition of the frequent sighting of
landmarks whose geographical position is known, and the
number of which becomes every year greater and greater.
The astronomical work is sometimes performed with the
Hadley's sextant, but the surveyor finds a greater range of
usefulness in the transit theodolite.
Surveying with the Sextant
The Hadley's sextant is a favourite instrument with
travellers, who learn its use from one of the ship's officers
when getting to their destination, and then employ it for
traversing on land. It will not take angles any more
correctly than they can be plotted direct upon a plane-
table ; it is necessary to correct the angles when they are
taken between points at considerable difference of level,
whereas the plane-table gives the horizontal projection at
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54
Preliminary Survey
once. It is much slower in sighting, and needs careful
sketches and entries in the fieldbook to avoid mistakes.
Its great advantage is its portability, and an immense deal
of good work may be done with it, but only the principles
of adjusting it are given in the chapter on instruments, its
use being very simple.
Closed Passometer Traverse
In order to exhibit the degree of accuracy attainable
with the passometer and prismatic compass alone, the closed
traverse illustrated by Figs, it, 12, was made in the course of
■so? . . . f , . . , ?
3000 R-
Fig. II.
daily walks and visits to friends, and without making correc-
tions for sloping ground. It has to be remembered that
not only does the pace vary in length according to the slope,
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Route- Surveying
55
but a deduction has to be made for the horizontal projec-
tion of the distance. As going uphill tends to shorten the
step, and increase the number of them to the mile, the
error is aggravated by the projection. Going downhill it is
diminished. The table for deductions due to projection is
[3
Fig. 12.
given in the chapter on chaining, but the pedestrian must
make his own table of pace-variation from actual experi-
ment.
The roads of Sevenoaks are both hilly and tortuous, and
therefore represent an unfavourable case for a route-survey.
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56 Preliminary Survey
Hardly any check angles could be taken on account of
the obstructions to view. It will serve to show a degree
of accuracy which can be at least equalled in more favour-
able situations. The * closing error' of 25 J paces has been
purposely left in the plot, and the process by which it should
be distributed over the bases is placed upon a separate
figure. The closing error does not represent the maximum
error ; it is an average one, arising in large measure from
the sloping ground ; but there is also a slight twist in the
plot, so that in one place where both pace and angle error
assist one another, there is a divergence of fifty paces from
the truth in the neighbourhood of one of the cross roads.
The total closing error amounts to | per cent, of the
periphery, the maximum error to about i^ per cent. On
a day's march of twenty miles, the error at the next solar
observation would be barely detected ; but as each observa-
tion is independent, cumulative error of pace-measurement
is removed within the instrumental limits of about \ mile of
latitude, or, supposing an exact chronometer, of longitude
either, but failing the chronometer, the error in longitude
may be much more, as already explained in Chapter I.
Distribution of Closing Error
This must not be confounded with fudging, which means
correcting by guesswork. Distribution of error is to a great
extent dissipation of its amount all over the plot. The
process is almost self-explanatory on Fig. 12. The total
periphery is laid out on a straight line marking each station,
and an ordinate is laid off at the extremity equal to the
closing error. The extremity of this ordinate being con-
nected with the other end of the line, a triangle is formed the
ordinates to which at every station represent (on the assump-
tion of the error being gradually cumulative) the correction
to be applied at each point in a direction parallel to that of
the line between the two divergent ends of the traverse.
v^ Route- Surveying 57
Where the error is small in proportion to the total periphery
this process reduces it to an unscaleable quantity on any
single base line.
Scale of Paces
The construction of a scale of paces is as follows. Let
us suppose that we wish to produce a map upon a scale of
six inches to the mile. A chained base of i,ooo feet is paced
and repaced until the average has been found to be for
instance 357 paces. We then lay down our scale of miles,
say three inches for half a mile, in furlongs and chains, and
calculate by slide-rule the value of 1,000 paces in furlongs
and chains.^
Taking this amount from the mile scale we lay it down
as our scale of 1,000 paces and subdivide it as follows :
(Fig. 13, p. 59). From one end of it, we erect a perpendicular,
' and selecting some convenient decimal boxwood scale such
as a I oft. to the inch, we adjust it so that one end is at the
extremity of the pace-scale, and some multiple of ten (in this
case the 40) on the perpendicular ; we then draw a line form-
ing the hypotenuse of a triangle and tick off every four
divisions of the boxwood scale so that we have subdivided
our hypotenuse into ten equal parts. All we have to do
' The two proportions are as follows : —
Let X be the number of paces in a mile, and y the number of chains in
1 ,000 paces.
1,000 : 5,280 :: 357 : x-.oxx^ 357 x 5*280
1,000
a: : 80 :: .,000 : y;oty= 8° x '.°°o .
Using the lower scales of rule and slide. Place the right-hand I of
the slide over the 528 of the rule. Place the brass marker at the 357
on the slide. Without displacing the marker, bring the 80 of the slide
to the marker and read off" the result, 42 ch. 44 links, on the slide oppo-
site the left-hand i of the rule.
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Route-Surveying
59
Scale of MileSiOne-Six Inches
iM>le
Nl
Scale ti Paces. i000»43 44 Chain)
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Scales to Fig. 23.
then is to rule down perpendiculars to the pace-scale and it
will be divided into spaces of loo paces each. See also
direct scaling by slide-rule, p. 245.
FlELDBOOK
The fieldbook, on opposite page, requires but a very brief
explanation. The backsights are taken to equalise minute
errors of observation, to locate and remove them when
important, and to detect magnetic deviation.
To reduce the meridional bearings to azimuths from
north or south point, the following table may be used.
Table XVII. — Reduction of Azimtahs
From Qp to 90°, azim. =* bearing unaltered : N. E.
From 90® to 180®, azim. = 180° -bearing: S. E.
From 180° to 270°, azim. = bearing- 180°: S.W.
From 270® to 360®, azim. = bearing— 360®: N. W.
[The author's pocket altazimuth has both graduations. See Chapter IX. ]
The error at closing is seen to be 22 paces to the north
13
and 13 to the west. We will represent it thus \i| „ Then
(i) to find the direction and magnitude of the line itself we
have tangent angle ^ = i^. By slide-rule as before = '591.
22
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6o Preliminary Survey
Keeping the brass marker at the -591 we shift the tangent
scale to its initial position and find under the marker the
angle 30° 35' N. W.
(2) To find the length of the closing error L= 1/22^+13'^
For 22' bring the brass marker to 22 on the lower scale of
this rule, the upper index will then correspond with 484 on
the upper scale. Similarly with the lower index at 13, the
square 169 is found on the upper scale. Adding the two
together=:653 ; direct the upper index to 653 on the upper
scale and the square root 25*5 is read off from the lower.
It will be seen that involution and evolution are performed
by simple inspection without using the slide, and this forms
one of the most important uses of the slide-rule.^
(3) ^^ S^^ latitude and departure. Place the right hand
extremity of the sine-scale of the slide under the distance,
and read off the latitude from the rule above the com-
plement of the angle, and the departure opposite the
angle itself. Thus in the first entry the distance=7o. Place
the extremity of the sine-scale opposite a 7 on the upper
scale of the rule. The reduced bearing is 19^**, of which
the complement is 70^° ; opposite these two angles on the
slide we shall find the departure and latitude respectively.
Check-sights are very useful in such work as this to
correct twists The house shown on the plot was filled in
entirely by angles. When engaged in filling in new roads
to an old but accurately triangulated map such as an old
Ordnance Survey the errors are localised by first plotting the
work in the usual way and then superimposing a tracing of
it upon the Ordnance Map ; the errors of the new work
are thus narrowed within the limits of the nearest reliable
points and the whole made very nearly as correct as the rest
of the Ordnance Map.
» This operation can be also performed by placing the sine-scale
with the angle 30° 35' under the 13 of the rule, and the answer will
be found opposite the right-hand i of the slide.
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Route- Surveying 6 1
Profile
The profile or section is produced from readings of the
aneroid barometer at every station. The distances are laid
off and the heights ruled up as in ordinary levelling. In the
case of a closed traverse, there will generally be a * closing
error' of levels which has to be equahsed or distributed
similarly to the closing error of the traverse. Some of the
sources of error are explained in the chapter on instruments,
but they are usually cumulative and approximately uniform.
They can only be treated as such, and the total periphery of
the base being laid off upon a horizontal line representing
the true datum, the amount of the closing error is laid off
on a perpendicular at its extremity either above or below
according as the last reading is less or greater than it ought
to be. A * false datum ' is then drawn from the starting
point at the end of the perpendicular, and the levels are scaled
from the false datum, which is afterwards erased.
Contours
The contours are laid on by the pocket-altazimuth, check-
angles being taken along the bases and in other directions.
The pocket altazimuth is fully explained in Chapter IX.
When we know the elevation of the point of observation
and the slope of the ground in any direction we can plot the
contours from the cotangent of the angle (that is the tangent
of the complement) which represents the horizontal distance,
corresponding to one foot of difference of level. Thus let
the slope be io° of depression. To find cot lo**. Place
the tangent-scale in its initial position and brass marker at
10°. Reverse the slide and place the right-hand i of the
slide at the index. Read the answer 5*67 on the slide
opposite the left-hand i of the rule. If we want contours
at every 10 feet the horizontal equivalent will be ten times
this, Le. 567 feet. Inasmuch as the aneroid readings are in
62
Preliminary Survey
feet it is best to have a feet scale upon the plot as well as
the two already mentioned for setting out the contours.
This is by no means the only way of contouring. The
contours of the Ordnance Survey are taken with the level,
and dots are placed on the map where the staff was held.
They are either plotted from cross sections or from field
400
300
.4 200
idos laiis tdoo 675 aaoaoo >Feet ]k uUuwWFt.alfovBOrcUIkit>
>Station4 3 2 I For Plan see Fis, II
FttMe^ ofSf^JohFus Road ^^ ^'^ ^
JTor: Scale OMchesmOne-Mle
Vert. Sc4xle lOOFP^OneMUe^
Fig. 14
tracings on which the location of the level points are estab-
lished by tape measurements from hedges, buildings, &c.
The cross section is the most laborious and hardly the more
exact method of the two. If the contours are needed before
the plan is plotted, it is unavoidable, but when the principal
contours are at a hundred or even fifty feet interval it be-
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Route- Surveying 63
comes a very tedious operation in hilly country. By the
second process, the level is kept at nearly the same coUima-
tion going round the hill until it comes round to the same
point again, or leaves the region of the survey. Telemetric
contouring with the level (for which see p. 197) is suffi-
ciently accurate for all ordinary purposes and is indepen-
dent of any plan. Hill-sketching is often done in the
form of contours by the eye, instead of hachures, which
convey but little idea of the topography ; such contours by
being close or far apart show at once the relative steepness
or flatness of the ground.
The examples of profile and contouring given, Figs. 14,
15, and 16, with the fieldbook are, one from the traverse,
Fig. 1 1, the other from a walk through Knole Park. The
instruments used were the aneroid barometer, pocket
altazimuth, and passometer.
The aneroid was first examined between Ordnance bench-
marks, with results as follows : —
Ordnance Aneroid
Ordnance Benchmark at * The Vine ' to
Ordnance Benchmark Railway Bridge
London, Chatham, and Dover Railway 187*8 185 a
Ordnance Benchmark at * The Vine * to v <^
Ordnance Benchmark Railway Tavern 200 '8 apS
Ordnance Benchmark at Railway Bridge ^-.^ %
to Ordnance Benchmark at Railway '^- ^
Tavern 13*8 20
On the profile, the discrepancies between the aneroid
readings and the elevations calculated by the altazimuth
were so small that they were not distributed. In the con-
touring, on the other hand, the aneroid had to be used with
less time allowance for settling, and needed considerable
correction from the altazimuth. The profile was taken
whilst walking up the hill with a friend without detaining
him beyond two or three minutes. The only entries made
in the fieldbook at the time were the station column,
vertical angle, pedometer, aneroid, and remarks.
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64
Preliminary Survey
If a cumulative error of the aneroid is discovered from
benchmarks, or from returning to the starting point on a
closed traverse, it must be eliminated as already described
before entering its readings in the column provided for the
purpose. \Vhen in the field its readings can be entered in
Fig. 15.
»...#
Paces ioo«29fi Feet
. fm?Mm
■ — . — •
Feet Honzontal
3000 m
i4 •...• . .
feet Vertical
.•?•.
. . .>.*«t
4JP fWt
Fig. 16.
the column for remarks. When this is done, the aneroid
levels rule the profile ; the altazimuth angles are only relied
upon for the portion of profile between the aneroid
readings.
The advantage of using a clinometer or altazimuth in
conjunction with thejaneroid, especially in the form recom-
d by Google
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66 Preliminary Survey
mended in Chapter IX. is, first, rapidity, and secondly, accu-
racy of detail. The aneroid coiild be used throughout, but
is not reliable for small differences of level ; for instance, on
a stretch of almost dead level, the aneroid might show jumps
of a few feet here or there which did not exist at all, but the
altazimuth is graduated to show a difference in vertical
angle of lo', which equals i foot in 333. On the other
hand, the clinometer angles on a long stretch will soon
run into a very serious error if unchecked by the aneroid.
For more on the aneroid see Chapter IX.
The use of the slide-rule for the calculations cannot be
over-estimated, it is simply invaluable. To begin with the
scales. Tt is tedious enough to transfer with dividers from
a scale of paces to a scale of feet, but when it comes to
three or more scales of paces for uphill, level, and downhill
at varying inclinations, the work would be interminable,
and recourse has to be had to percentages of addition or
deduction as already described. But with the slide-rule it
simply means a gentle tap upon the slide, and the new scale
is there ; no calculation whatever is needed. Take, for
instance, the scale for the slope of St. John's Road, on which
533 paces measured 1,605 f*set. We use the upper scales of
the slide and rule, and bring the 533 of the slide to the
1,605 of the rule ; the distances in feet corresponding to any
number of paces can then be read off directly from the
slide. If then the rate alters to 543 paces to 1,611 feet, we
have nothing more to do than to slide forward to that ratio.
For rise and fall the operation is almost instantaneous, being
analogous to that for latitude and departure (see Chapter V.
P- I73-)
Contouring with Pocket Altazimuth and Aneroid
The accompanying diagram of contouring (on p. 64) will
illustrate first of all the adjustment of aneroid error already
alluded to. The traverse was made with the passometer
precisely as on Fig. 12, but the correction of the closing error
Digitized by VjOOQ IC
Route- Survey ing 6y
was done by the slide-rule, and as this is by far the most
satisfactory method, as well as much quicker, it will now be
explained.
The northings and southings being added up as before,
the difference of latitude amounted to 1,812 feet south and
1,764 feet north. Placing the diflference of the two, 48, on
the upper scale of the rule opposite to 1,812 on the slide,
the correction for the first of the northings, 141, on the base
9-10 will be given on the scale opposite 141 on the slide,
and equals 37.
.The departures being corrected in the same way, the
plotting of the traverse is proceeded with as before, by
rectangular co-ordinates from a N. and S. line, and an E. and
W. line, the total latitude or departure being scaled for each
station from the starting point afresh, so as to avoid the
errors certain to arise in scaling from point to point.
When the traverse is complete, the total periphery is laid
out on a horizontal base as before described. It was found
in this case that on returning to station 3, where the traverse
closed, there was a discrepancy of fifty feet in the aneroid
readings, arising from change in the weather (see Fig. 16).
As the instrument read too low at closing, the error was
laid off above the true datum, and the aneroid levels scaled
up from the false datum. The profile from the aneroid is
shown by a dotted line. The difference of level by clino-
meter angles was then worked out by slide-rule as before,
and came to a total difference of 14 feet more than the
aneroid. The rises and falls were then reduced in the same
way as the errors of latitude and departure by the slide-rule.
The maximum difference of level existed at the two ex-
tremities of the work, and amounted by the aneroid to 80
feet. This was checked by a second visit ^ith the aneroid
to the two extreme points, and found to be correct. The
rises and falls were then squared with this total by placing
the total error of 14 feet in ratio on the slide-nil^ with 80,
and the error subdivided over the whole traverse propor-
Digitized by VjOQ"^ _
68 Preliminary Survey
tionally to each rise or fall. The full line on the profile
shows the final correction, and the erratic nature of aneroid
readings, which is never got rid of even in the most delicate
instruments. The line lo-ii for instance is shown by the
aneroid as an ascent, whereas it was really a descent, and if
it had not been for the altazimuth would not have been de-
tected. To be perfectly sure of the main difference of level
by the aneroid, the surveyor should repeat his visits to the
two principal positions several times. On extended survey he
should have several aneroids, and despatch an assistant on
horseback with an aneroid to go to and fro between two
objective points until a fair average is obtained. When
long base lines are measured by the passometer, say over
400 paces, the clinometer angles are of little value, and the
aneroid alone has to be followed.
The contours are plotted in two different ways : First,
on lines the extremities of which are known in elevation ;
secondly, on lines of which the slope alone is known by the
clinometer.
Let us take the base-line 3-4, the ends of which were
found to be 393 feet and 404 feet high respectively. The
contours being required at every 5 feet, we want to know
where the 395 and 400 contours cross the line. The distance
was 369 feet, and the difference of level 1 1 feet. Placing the
1 1 on the rule opposite the 369 on the slide, we look for 2
and 7 on the rule, and find opposite to them on the slide their
horizontal equivalents, namely 67 and 235, which we tick off
upon line 3-4, measured from 3, and so on. This plan is
adopted for all the base lines of the traverse, and also for any
points which have been triangulated. In the present case a
series of points «, ^, r, ^, ^, &c., were determined along the
rising ground in the middle of the plot by intersecting com-
pass-rays, and the slope taken with the clinometer. The
distances were then scaled from the plan, and the difference
of level calculated as before with the slide-rule. The points
were all trees of sufficient prominence to be identified from
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Route- Sui'i'eyinj^ 69
successive stations, but these ^re among the least eligible of
points, being badly defined. Low cottages, whose height
above the ground varies but little, are amongst the best
points to choose. The contours were then filled in on the
rays firom the primary stations to these points, and also on
lines connecting the points themselves.
It is convenient to tick off the distance on a slip of
paper, marking each tick with its elevation ; then apply the
distance to the scale, and mark off from it the distances of
the contours without using dividers.
The second method of plotting contours, on lines of
which the direction and slope alone are determined by the
altazimuth, is for the outlying ground in order to show the
trend of the contours after they leave the closed traverse.
The process is as simple with the slide-rule as the former
one. We take a bearing in the direction in which we desire
the slope (sometimes we require several from the one station),
using either a tree or similar mark, which we can describe
for identification in the fieldbook, or else send out a flag.
We then take a vertical angle to the same point and
book it likewise.
When we have plotted the rest of the traverse and con-
tours between fixed points we lay off radial lines from the
respective stations in the directions of the independent com-
pass-shots.
Then with the sHde-rule we place the scale of tangents
under the upper scale of the rule, and note the percentage
of the slope {ue, 100 times the tangent). Thus if the slope
be 3° 25' we find the percentage to be 5*97, which is the
same thing as the tabular tangent of '0597 to a radius i.
It is also the same thing as saying that on that slope every
horizontal stretch of 100 feet has a rise of 5-97 feet. We
then reverse the slide to show the scale of numbers, and,
placing the 5*97 of the rule opposite the 100 on the slide,
we can read off the horizontal equivalents for each difference
of level. Thus, supposing the elevation of the point from
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70 Preliviinary Sun^eyf
which the line was taken was 473, and the slope +, we
should want for 5ft. contours horizontal equivalents for
2ft., 7ft., 12ft., and so on, which we find at 33, 117, and
201. They should be plotted on with the paper-slip from
the scale as before.
When a slide-rule is not available, the foregoing expla-
nation will serve equally well for the use of tables, which is
of course much more lengthy. A set of scales of horizontal
equivalents should be made for every degree of slope. To
save pricking off odd distances, the scales are constructed
from the tabular cotangents of the slopes which are the
horizontal equivalents to a diflference of level of i foot.
They can also be laid off graphically by erecting a perpen-
dicular scale at one end of the profile in the manner shown
on Fig. 16, p. 64, and drawing horizontals cutting the base-
lines at the required elevations.
Distance-measurement other than on Foot.
Hitherto only pedestrian operations have been dwelt
upon, and for several reasons the surveyor should keep to
his feet where he can. He is more independent and can
better give his attention to his work when he has no animal
to look after ; he can use the sketch-board as a plane-table,
and take steadier sights with the prismatic compass. But
it often happens that he has to ride and to depend on his
animal for the measurement of his distances. It is better to
have a horse who walks well than a fast one. A well-trained
cavalry horse will, according to Captain Verner, R.A., walk
with much greater regularity than a man. Horses will fall
into an even pace much better when two are together. The
walking gait is measured over a chained base. The trot is
counted similarly ; each rise in the saddle being a * trot.'
As a rough approximation three yards may be counted to
each ' trot.' A canter requires exceptional horsemanship to
be used to any extent for distance-measurement.
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Route- Surveying 71
When steaming on a river, the speed is measured by
revolutions of the screw or paddle, by the patent log, or by
time only. In any case the speed of the current has first to
be measured before starting, and periodically afterwards.
A good method is by patent log combined with timing for
short distances.
Mr. George Kilgour, M.I.C.E., made a survey for 200
miles in length with a steam launch between the first and
second cataracts of the Nile in five days, for the Soudan
Railway. He adopted time-measurement of speed, for which
he had three scales, full speed, half speed, and dead slow ;
the value of the scales was determined by an accurately
measured base on the centre line of the vessel, at the two
ends of which, near the bow and stern, two lines of sight
were placed square with the centre line, by which the time
was registered in which the vessel passed some well-defined
point on the bank, such as a cocoanut tree, at the three
rates of speed. The survey was made from the deck with
a plane-table kept constantly in the meridian by the com-
pass, and astronomical observations were taken at night.
Distance-measurement by the Range-finder.
The most recent improvements in this class of instru-
ment are described in Chapter IX. but a few further
remarks will here be made upon their use.
There are cases when they can be used for measuring all
the bases of a traverse. All that they require is some well-
defined point to view on and an accurately measured sub-
base at station. When the bases themselves are not measur-
able in this way there are sometimes well-defined points close
to those bases, from which the distances to them can be
accurately determined and so serve equally well. All these
instruments require considerable practice in order to obtain
reliable results every time. Those which have no lens
power require a good and quick eye and a steady hand.
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72 Preliminary Survey
The most important feature about the range-finder is that
it is a rapid means of measuring inaccessible distances ;
and across rough country where pacing is either impossible
or very inaccurate, it is as reliable as on level ground. A
range-finder is always valuable as a check on salient points
off the centre line.
Unless exceptionally proficient in its use it is not advisable
to take long shots except as rough checks. It is better to
keep the bases within i,ooo feet, if possible. An error of
one foot in sub-base means fifty feet in distance. Two Weldon
range-finders are better than an optical square and a range-
finder because then they can be used either in the manner
explained in Chapter IX. or with the two acute angles or
with one right angle and one acute.
Distance-measurement by a Two-foot Rule
The following method needs no instruments beyond a
two-foot rule and a decimal scale. One of the old-fashioned
carpenter's rules, with a brass slide to it, graduated on the
one side with the logarithmic scale and on the other with
inches and tenths, will answer the purpose. Or if a Mann-
heim rule is at hand with its bevelled millimetre scale it will
give a closer reading, or, best of all, a ten-inch slide-rule of
the author's pattern described in Chapter IX. with a fifty scale
on its bevelled edge. This will give an even dividend-number
and as close a reading as it is possible to have.
Dividends are given for all three scales below.
The two-foot rule is held with the eye at the centre of
the joint, and the legs are spread so as exactly to intercept a
known sub-base. It is convenient to measure distances up to
300 feet with a sub-base of ^v^ feet or ten feet in the form of
a pole, with a pair of discs or crossheads painted black and
white, held by an assistant either on foot or on horseback
at the point whose distance is to be measured.
For longer distances, a base of fifty feet should be run
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^outC' Surveying 73
out with a tape by two assistants, who will hold a flag
or a disc at each end of the base. The line of sight to the
centre of the base should be at right angles to it. This is
easily done with a single assistant by a sighting-piece at
the middle of the pole, but when a base is to be taped it
is easier to set it at right angles to a line of sight to one
or other of its extremities. In the latter case, if the angle is
small the error will be hardly scaleable ; but to be exact
the aperture of the legs of the rule should be measured on
the square from one extremity to the line of the other leg
produced.
All that is required is to divide the dividend in the table
according to base and scale by the measurement of the
aperture of the legs of the rule. The only advantage in this
case of a slide-rule is to perform the division, but that is so
simple that it can be done mentally when using an English
scale.
Dividend-Numbers
—
Base 5 feet | Base lo feet
Base 50 feet
Scale of inch and tenths
Scale of millimetres
Scale of inch and fiftieths
600 1 1,200
1,524 ; 3,048
3,000 6,000
6,000
15,240
30,000
1
Example, Viewing on a ten -foot base, measured the
aperture of a two-foot rule twenty- seven fiftieths of an
inch; required the distance.
Ans. -??? = 2 22*2 feet.
27
Good guesses at distance can be made by similar means
without anything more than a base which is itself guessed at.
Perhaps some reader will pooh-pooh such guesswork as
this, but no one who knows what it is to be without any
assistant for measurement and to need some help to the
mere guess at distance in perhaps a very deceptive piece of
country will under-value such a method as this when they
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74 Preliminary Survey
have tried it. In even slightly civilised countries, people
still build with some system, and mud cabins, log huts, or
snake fences go pretty much in sizes.
In countries where they build in brick, one-storey and
two-storey houses bear considerable uniformity of height
per storey. Then a man on horseback, or a cow or any
other animal, may be taken for a guessed base. Even trees
in old wooded countries, though varying individually up to
any size, form as a forest a line of approximately equal
height which can be ascertained for different species and
used as a base. The suggestive fact is that any height of
that kind is much easier guessed at than a distance, because
in the one case there is something to go by, in the other
nothing.
Mapping
We come now to the various contrivances for producing
a correct graphic representation of the fieldwork upon
paper.
The only absolutely true map is a terrestrial globe, but
as we cannot carry globes about vrith us we have recourse
to the principles of projection, which are quite numerous in
variety, but are all of them artificial representations of a
spherical, or more properly spheroidal, surface upon a plane.
If the survey extends over a large area, it becomes neces-
sary to adopt some method of projection by which, in the
first place, the distances are reduced to sea-level, and in the
second place, the meridians converged or distorted so as to
allow for curvature.
When the survey is a continuous traverse of a railway
route this is not necessary. It is not the object of the rail-
way surveyor to know the sea-level dimensions ; he needs
the actual length of his road wherever it may be. The dif-
ference in length between a degree of latitude at sea-level
and at 528ft. (^V mile) elevation is only about nine feet.
At an elevation of 5,280 feet (one mile) it would be about
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Route- Surveying 75
92 feet, and the distance-measurement practicable on route-
survey does not come nearer than that.
Neither does the surveyor want a distorted map, but one
to which he can apply a scale throughout ; he therefore does
not need to take account of the earth's curvature, but plots
his traverse on a horizontal plane. When the area over which
a triangulation extends is not large, the surveyor is still able
to adopt one of two methods of plane construction.
In the first the meridians and parallels of latitude are all
parallel straight lines at right angles to one another.
In the second the parallels of latitude are straight lines,
but the meridians are converging straight lines, or, if great
accuracy is needed, curved lines.
Limits may be given merely to fix the ideas within which
to use the first or second method, of say 1,000 square miles
for the first and 100,000 for the second.
When the survey is in high latitude the spheroidal form
of the earth much more affects the map than near the equator,
in which region a belt could be projected all round the globe
by the first method without sensible error.
Taking as an illustration of the extreme limit given for
the use of the first method, 1,000 square miles ; this area
would be contained in a square of which the side was not
quite half a degree ; let us lay down two plots of squares of
which the side is 2°, which will embrace an area of about
16,000 square miles, or sixteen times the limit given, and
examine from it what the error would amount to in using
the first method, at a mean latitude of 32°.
First, By mean longitude. The length of a degree of
latitude at 32° is 68-90 statute miles (see table, p. 175), and
the length of a degree of longitude at that latitude is = 5870
miles (see table). In the centre of the paper draw a hori-
zontal line to represent the middle parallel of latitude, and
through its centre erect a perpendicular to represent the cen-
tral meridian 5° of longitude, and lay off upon it 68*90 miles
above and below, and through the ends draw parallels to
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76
Preliminary Survey
represent 31° and i'^ of latitude. Lay oft on the middle
parallel 5870 miles on each side of the centre, and for the
first method rule up verticals to represent the meridian of
4° and 6° longitude. The whole figure will then represent
four equal rectangular quadrilaterals.
For the second method lay off" on the 33° parallel two
lengths of 58*09 miles, and on the 31° parallel two lengths
of 59*32 miles, being the length of a degree of longitude at
the respective latitudes (see Table XXV. p. 175), and com-
plete the figure.
The error of contraction at 31° will be seen to be = 1*2
StcULUty.mies
Ja-TP as 70
I
^
V
V>
6870
S870
Q
Q
o»
Oi
•0
*
^
V
68 70
S8 70
• 68 09 S809
fiZ'
1
i
i
88 70
1
<b 1
88 70 1
1
1
«
5
S93Z
5
39 3Z
Fig. 17.
Fig. 18.
miles, and the error of expansion at 33°=! '22 miles by the
first method. No error exists on lines running due north
and south. A diagonal through one of the quadrilaterals is
subject to an error of about half a mile, but on a diagonal
clear through the figure there will be hardly any error,
because the contraction in the upper almost exactly balances
the expansion in the lower.
At the same latitude a similar figure bounded by a ^ degree
of latitude and covering an area of 1,000 square miles would
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Route- Surveying
77
have an error of expansion at the top of the sheet of about fifty
yards and a similar error of contraction at the bottom.
This matter of possible error has been gone into numeri-
cally to hx the ideas before commencing to plot a survey
without wasting time in deciding upon whether to use pro-
jection or not.
The second method, of straight converging meridians,
may be used with quite sufficient accuracy up to a latitude
of 65° for stretches of 100,000 square miles, and there is not
much work done above this latitude anywhere.
It may be said, therefore, that the method of plane con-
struction meets all the ordinary requirements of the surveyor,
but in case he may be called upon to reduce extensive surveys
to atlas scale it may be as well to explain the principles of
Conical Projection
A globe may be conceived to be wholly contamed
inside a cylinder or partly contained inside a hollow cone.
For purposes of projection the cylinder must have a
diameter equal to that of the globe, but the cone must be
Fig. 19.
of such dimensions that its sides will be tangential to the
radius of the sphere at the point of contact. Supposing
the earth to be thus contained ; a belt of a few degrees on
either side of the equator might be conceived to be un-
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78 Preliminary Survey
wrapped or developed on the cylinder without sensible error.
This answers to plane parallel construction.
Similarly a. belt of the cone may be developed as shown
on Fig. iglff with converging meridians and with curved
parallels. Maps of continents are drawn in the atlas upon
this principle, and being of large extent the apex of the cone
is determined, and the radial parallels of latitude drawn
direct from it, with trammels.
Example. It is required to project by conical projection
a belt of io° longitude, say from 20° to 30° east, whose
middle parallel of latitude is 50°. The width to be 10°, /.<?.
from 45" to 55° ; the scale 100 miles to the inch. Draw
the scale of miles at the foot of the paper. Draw a hori-
zontal base-line in the middle of the paper, fix its centre,
and draw a perpendicular through it from top to bottom.
The base will represent the chord of the middle parallel,
and the perpendicular the central meridian 25° longitude E.
Calculate the length of the chord by following formulae.
Radius of Cone^rsid. of earth x cotan lat. = 3,950 x
cot 5o°=3,3i4=R.
Central angle = total longitude x sin latitude = 10° x
sin 5o°=7°-66'=A. '
— — =R sin ^=3,314 X sin 3°-83'=22i miles.
2 2
Versed J///^=R — R cosin =7*4 miles.
2
Lay off ^-?-- on each side of the centre, and 221-
2
less 74 on the perpendicular, and 7*4 below the centre.
Then from the apex draw the middle parallel through the
extremities of the chord and the versed sine, and join the
apex to the two ends of the chord for the two extreme
meridians 20° and 30° east, upon which, on either side of
the middle parallel, lay off distance, = the latitude for each
d^ee (see table on p. 175), and describe the arcs of the
remaining parallels.
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Route- Surveying 79
Then for the meridians draw the bottom chord to the
parallel 55°, subdivide it into ten portions, and draw radial
lines to the apex.
If the radius of the cone be inconveniently long for
plotting, lay off the extreme meridians by protracting an
angle at each end of the chord to middle parallel equal to
. In this case the angle would be 86-17°. Draw
2
top and bottom chords to parallels 45° and 55°, and sub-
divide each into ten equal portions, through which divisions
draw the converging meridians.
Stereographic Projection
is that in which the great circle of a sphere is assumed as
the plane of projection, and one of its poles as the projecting
point. In terrestrial maps it is used for representing a
hemisphere. The plane of projection is termed the primi-
tive. Projections of great circles drawn through the pole of
projection are straight lines, and all others are circles. The
centres of all great circles passing through any point in the
plane of the primitive are situated in a straight line called
the locus.
In the stereographic projection of the eastern hemisphere,
the primitive is usually the great circle of longitude passing
through the 20th western and i6oth eastern meridians, and
the pole of projection is on the equator at 70° east longi-
tude.
The principal use of this projection to the surveyor is
for astronomical problems, such as that of 'graphic lati-
tude,' p. 158, or the chart of circumpolar stars on p. 393.
The explanation of this projection is given in detail in
Chambers's 'Practical Mathematics,' and in Heather's
* Instruments ' in a more general manner.
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8o Preliminary Survey
Mercator's Projection
is a development of the earth's surface by elongation of the
meridian, so that a ship's course will always appear as a
straight line, and is the projection used in the published
Admiralty chart. They are, however, constructed on the
principle of
Gnomonic Projection
for which see p. 89.
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CHAPTER III
HYDROGRAPHY AND HYDRAULICS
When a company is formed to develop the resources of
some new country like Africa, it often falls to its lot
either to make new harbours or improve natural ones, to
canalise rivers or to deepen them ; and the pioneer surveyor
has to be prepared at least to make a hydrographic chart
of a considerable degree of accuracy, to measure the dis-
charge of streams and rivers, or possibly to undertake trial
borings, soundings, or even dredging on a small scale for
the purpose of estimating the cost of a proposed work.
This chapter will be occupied with short descriptions of
methods adopted in carrying out such operations, keeping
strictly to preliminary work.
Hydrography on Land
Some of the naval surveyor's methods differ but very
little from that of the land surveyor. He chains base-lines,
plants permanent trigonometrical stations, triangulates with
the theodolite, and reduces his work to sea-level. It does
not come within our province to follow up such lengthy
methods as these. We will content ourselves with the
subject technically called coast-lining, which signifies the
mapping of the shore-line, either from the ship, aided or
unaided by points on shore, by boats, by traverses on foot,
or by combinations of all the above methods ; together with
a short paragraph on Boat-survey of rivers.
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82 Preliminary Sufn^ey
Coast-lining on Foot
This, where practicable, is the best way of rigorously
determining all the small indentations, creek-mouths,
directions of small streams, &c. Boating tends to over-
sight in these particulars, whilst from the ship itself only
the general outline can be obtained. The first thing is to
obtain fixed points on the shore visible from one another
from which to fill in the intervening detail. These may be
salient points in nature, such as prominent rocks, trees, or
houses, or else they may be beacons, cairns, or flagpoles,
whitewash marks, or any other artificial stations.
Their position is determined by astronomical observa-
tion, and the base is measured by difference of latitude on a
meridional line or by a meridian distance. These problems
are described in Chapter IV.
The plane-table is not much used by naval men, but it
might be made a very useful instrument, both on board
ship and on shore. They do, however, use an ordinary
field-board for plotting in details of coast-line from primary
points, or * fixes' as they call them, plotted previously on
the board. The intervening work between the primary
points is put in by telemetry; they use a ten-foot pole
with a pair of discs as a base, the angle of which is measured
by a sextant or micrometer.
The pole is maintained square with the observer by a
directrix at its middle, or else it is swayed slowly in a hori-
zontal position until the observer has measured the maxi-
mum angle.
Another plan, where the ground is sufficiently open to
admit it, is to use a 500 feet lead-line run out square from
one end of the line to be measured. Fig. 20 is taken from
Captain Wharton's * Hydrographical Surveying,' and repre-
sents a method largely used by Lieutenant W. U. Moore in
the survey of the Fiji Islands, and performed by two officers
only. Starting at A officer No. i moves on to B, leaving
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Hydrography and Hydraulics
83
Fig. 20.
officer No. 2 at A with the lead-line, who fixes a flag at A
and runs out his line either with an optical square or a * 3,
4, and 5 ' line, and plants another flag at its extremity.
Officer No. i then measures the angle between the flags
=500 X cot 0. Each officer then takes sextant angles to
saUent points pre-
viously determined
along the shore, so
as to make intersect-
ing rays for subsidiary
'fixes.' They then
travel to meet one
another, filling in de-
tail by bearings with
the prismatic com-
pass, and distances
with the micrometer
and ten-foot pole or sextant and ditto. The writer never
having done this kind of work cannot speak positively, but
he thinks it probable that the plane-table would be found
preferable for the filling in of detail between the extremities
of the base. The subsidiary fixes form independent checks
to the detail, and if a plane-table were used errors could be
more easily eliminated at the time.
Surveying with Boats
Rock-bound coasts, or rivers with banks covered with
dense jungle, are best surveyed with boats. The use of
boats in coast-lining does not much alter the methods
already mentioned ; we will therefore confine ourselves to
boat-survey of rivers. A steam pinnace is used whenever
one is obtainable, and provided with a compass and patent
log. The latter is attached to the gunwale, and the fan
towed astern. The prismatic compass stands on a tripod
in the stem. The velocity of the current is measured
from time to time by anchoring in midstream. A current-
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84
Preliminary Survey
meter should be added to the equipment as being more
adapted for that purpose than the patent log, especially
with slow currents.
When a large-scale plan of a wide river has to be made,
several boat-parties (Fig. 21) work in concert, four if pos-
sible ; two on either side of the river, triangulating their
way up from point to point. Two starting-points are esta-
blished at the mouth of the river, whose relative position
and distance has been determined with the utmost possible
accuracy. Two of the boats remain at these points, and
the other two jnove up the river to convenient stations, and
each of the four boats takes sextant angles to the other
Fig. 21.
three. Here the point C is fixed from line AB by the
angles BAG and ABC, and the point D by the angles
ABD and BAD; the angles ACB, ACD taken from
C and the angles BDC, BDA taken from D, form
independent checks. The shore-line may be sketched
by the boats A and B when moving up stream to take the
places of C and D. If the sounding has to be done
thoroughly, the boats should return by a diagonal course
AD, BC, or else the boats C, D, when first moving up the
river, can fill in the shore-line with the patent log and
compass angles, and also take soundings along the lines
AC,BD, leaving the boats A, B, only the diagonal sound-
ings AD, BC to take.
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Hydrography and Hydraulics 85
Surveying from the Ship
A great deal of very good charting is done without any
connection with the shore, except the landing of a boat to
make sound signals. This class of work is termed * Running
Survey,' It is the most rapid though the least accurate of
any of the methods which are considered worthy of being
called surveying. The mere sketch independent of any-
thing more than guesses at distance is not now resorted to,
although many islands exist in the Pacific, as well as por-
tions of the mainland of Australasia, and other continents,
where the charts are still, to a large extent, sketchy, and it
is the important duty of the Hydrographical Department of
the British Navy to diminish year by year the * terra incog-*
nita,' which presents its ofttimes hidden dangers to our
world-covering mercantile marine.
Running survey is often assisted by known points on
shore. If there are, for instance, three sharply defined
peaks whose geographical position has been accurately
determined, the work is both facilitated and improved.
For by the station pointer, see p. 367, the ship's place is
located at any time as long as they are kept in view. As
the ship moves forward, her path being thus clearly defined
upon the chart, subsidiary cuts are made by angles with
the sextant to salient points upon the coast and the detail
sketched in. From fifty to a hundred miles of coast-line
can be thus put in in a day.
When no assistance of this kind can be obtained from
the shore, the base is obtained by sound-signals. The
vessel is maintained as nearly as possible in the same
position, whilst a party on shore and a party on board
alternately fire guns within sight of one another, so that
the time between flash and report may be taken by the
chronometer on either side. During this operation angles
are being taken by the sextant to all the important points,
which can be seen both from the shore station and from
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86 Preliminary Survey
the ship. When the length of the base is determined by
the sound signals, the landing party returns on board, the
various peaks are located on the chart by intersections, and
henceforth serve as known points from which to identify the
ship's place at any time as she travels forward.
Sound travels at the rate of about 1,090 feet per second
at a temperature of 32° F. and increases its velocity at the
rate of 115 feet according to some authorities, and 1*25
according to others, for each degree Fahrenheit of rise in
temperature, or vice versd for fall of temperature.
The object of signalling from both ends of the base is
to counteract the effect of the wind, which greatly affects
the accuracy of the result. It is not, however, always re-
sorted to.
The firing is done by prearranged signal, such as the
dipping of a flag, in order that no time may be lyasted in
looking out.
Chronometer watches beat ^vq ticks to two seconds.
English leverwatches nine ticks, and Geneva watches ten ticks.
When watching for the warning signal with a telescope,
the watch should be tied to the ear with a handkerchief,
and the counting commenced thus : nought, nought,
nought, until the flash is seen, then one, two, three, to ten.
At each ten one of the fingers is put down, so that the ten
fingers will represent one hundred beats or forty seconds
with a chronometer=about eight and a quarter miles,
which is longer than bases are usually measured in this way.
The following example is from Captain Wharton's
* Hydrography,' p. 64.
' In meaning the result the arithmetical mean is not strictly
correct, as the acceleration caused by travelling with the
wind is not as great as the retardation caused in the opposite
direction, as in the latter case the disturbing cause has clearly
acted for a longer period. The formula used is
2tt'
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Hydrography and Hydraulics 87
where T is the mean interval required, t the interval
observed one way, t! the interval observed the other way.
The mean interval thus found multiplied into the velocity
of sound for the temperature at the time will give the required
distance.
* As an example let us suppose A and B the two ends of
the base to be measured.
* At A have been observed —
46 beats with watch beating 5 beats to 2 seconds.
47 >j >> » >>
4" » j> )) »
Mean 46*33 beats= 18*532 seconds.
81 beats with watch beating 9 beats to 2 seconds.
^2 ,, „ ,, ,,
^3 j> >> >> »'
Mean 82 beats= 18*222 seconds.
Mean at A= 18*376 seconds.
' At B have been observed —
85 beats with watch beating 9 beats to 2 seconds.
^7 J) n 5> )J
00 ,, ,, ,, ,,
Mean 86*66 beats=i9*258 seconds.
47 beats with watch beating 5 beats to 2 seconds.
47 )' » >? "
4^ )) 5J >> 5)
Mean 47*33 beats= 18*932 seconds.
Mean at B= 19*095 seconds.
2 tf
Then working, T = -, ;= 18*728 seconds.
* Temperature 80° F., at which velocity of sound=
1,145*2 feet per second x 18*728 seconds=2i,448 feet'
The temperature must be taken in the open air with the
thermometer shaded from the direct rays of the sun, but not
in too cool a spot or it will not give the true temperature of
the free air.
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Hydrography and Hydraulics 89
The small island shown on Fig. 22 is a copy from a
recently made chart kindly lent by Lieutenant Vernon
Brooke Webb, one of the hydrographers of H.M.S. * Dart '
on her expedition to New Guinea. It will serve to illustrate
the rapidity of running survey ; the time actually occupied
was nineteen hours, no special hurry having been made
over the work. The distance covered is about sixty
miles. The method was running survey, aided by several
fixed stations on a neighbouring island, from which the
ship's place could be located whilst on one side of the island
by the station-pointer. The vessel steamed at about four
miles an hour, towing the patent log all the time, and taking
soundings where shown. These being pretty deep occupied
about ten minutes each. On the further side of the island
all the distance-measuring was done by patent log. The
hills were put in by sextant angles, both for position and
elevation. The rivers were shot in with compass angles.
The inhabitants being cannibals, the island was not explored
inland. The graduation is a little displaced in order to bring
the margin within the sheet.
The following are the notes appended to this chart : —
Current; ■^->, ebb; "^' , flood; kn, knots; H.W.F. & C ix"*
o"., springs rise 5 ft., neaps, 3^^ f t ; B., bay; C, cape ; Cr., creek ;
D., doubtful; H^, head; H'., harbour; I., island; L., lake; P.,
port ; P^, point ; R., river ; Rk., rock.
bk., black; cl., clay; crl., coral ; f., fine ; g., gravel; h., hard ;
m., mud ; r., rock ; s., sand ; sh., shells ; St., stones.
Figures on the land show the heights in feet.
Bearings to the marks and views are magnetic.
Soundings in fathoms.
Magnetic variation in 1890, nearly stationary.
Gnomonic Projection
The Admiralty charts are now all constructed upon this
projection and published on the Mercator's projection.
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Gnomonic projection is very similar to conical projection,
but the plane of projection is not the development of a
conical zone ; it is a true plane touching the sphere only at
one point, viz., the middle point of the central parallel. The
convergency of the meridians is measured by the difference
of true bearing of one point from the other at the ex-
tremities of the map.
It is the only projection in which all great circles are
represented by straight lines, so that spherical distances from
point to point are scaleable all over the chart when it has
been properly graduated.
This does not mean that one scale may be used through-
out, but, unless in very high latitudes, considerable portions
of the chart are practically to one scale and the graduations at
the side give the scale at any point.
The actual length of any projected line measured on
a meridian from the middle parallel is proportional to the
tangent of the latitude.
It will be seen from the following illustration, that at
a scale of lo miles to the inch in 45° latitude, a stretch of
250 miles square, or 62,500 square miles, would not differ
in measurement from a plane construction with converging
meridians ; because the lengths of the tangents at that scale
are sensibly equal within 2° on either side of the middle
parallel to the developments of the spherical distance. The
difference at 2° between tangent and arc is -0000222, and if
we consider 'oi inch as the limit of scaleable quantity, we
get 37*5 feet as the Umit of radius at which error is appreci-
able, which is equal to about 8f miles to the inch.
The natural scale, or the proportion which the chart
lineally bears to the actual size in nature, is obtained by
dividing the number of inches in the nautical mile at the
latitude by the number of inches corresponding to one mile
on the chart. The result will be the denominator of a fraction
whose numerator is one.
Thus, supposing the scale to be i '8 inches to a mile in
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Hydrography and Hydraulics
9t
latitude 3°, we divide 72,552 (the number of inches in a
mile, see Table XXVI., p. 175) by 1*8 ; this gives :iu^-g^ as
th6 natural scale, which should be noted on all sheets that
are not graduated.
Illustration of Limit up to which Plane Con-
struction DOES not sensibly DIFFER FROM GnO-
monic Projection.
Scale 10 miles per inch=7j^g^j^. At a mean terrestrial
radius of 20,888,628 feet the length of radius to scale=
32*968 feet. The length of 1° from centi'al parallel on
central meridian=-57546 feet. The length of 2° = 1-15127
feet.
But the mean length of a degree of latitude =69 -05
miles, and at 10 miles per inch=6*905 inches or '5754 feet.
Therefore the projection is sensibly equal to the develop-
ment of the spherical distance within at least 3° of latitude.
As it is not Hkely that the surveyor would be called upon,
unless with proper notice for obtaining his special map
equipment, to carry out so extended a survey as to need
the exactitude of the above method, it will not be further
dwelt upon.
He will find a thorough description of the gnomonic
projection in Capt. Wharton's * Hydrography,' Murray's,
Albemarle Street.
Symbols in Charting
The annexed specimen list of symbols, taken by Capt.
Wharton from the Admiralty Manual, shows the authorised
delineation, and is reproduced here almost verbatim.
The Days of the Week.
Sunday . . . Sun's day . . Sun
Monday . . . Moon's day . . Moon }>
Tuesday . . Teut's day . . Mars $
Wednesday . . . Woden's day . . Mercury 5
Thursday . . . Thor's day . . Jupiter %
Friday . . . Friga's day . . Venus ?
Saturday . . . Saturn's day . . Saturn h
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92
Preliminary SurUey
The following Signs are used in the Fteldbooks.
Cutting €«IJIiIi!i,' ,■ I^EBJ'^* Embankment
Objects in line, called transit .
Station where angles are taken
Zero, from which angles are measured
Single altitude sun's lower limb
„ „ „ upper limb .
Double altitude sun's lower limb in artificial horizon
Sun's right limb
Sun's left limb
Sun's centre .
„ upper
Right extreme, or tangent, as of an island
Left
A
01
10
oe
>
<
Zero correct Z K
Windmill ^
Water level w. 1.
Whitewash W. W.
Bridge X"-"^
In Colouring.
Sand ..... Gamboge, dots black.
Low water, sand edge . . Gamboge, dots carmine.
Mud, dry low water . . . Neutral tint, edge offine black dots.
(Burnt sienna and carmine mixed
for wash ; same darker for edg-
ing.
Cliff. . . . . . Dark neutral tint.
Roads ..... Burnt sienna.
-.Either a faint wash of cobalt all
over the area included within
Fathom lines up to five fathoms . J the fathom line, or a narrow
1 edging of the same colour inside
V the dots of the fathom line.
Coral, dry low water or any rocky
ground covering and uncovering
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Hydrography and Hydraulics 93
Signalling with Heliostat or Heliograph
These instruments are described in Chapter IX. Their
use is also sufficiently explained to dispense with anything
further here.
The following description of the methods used is nearly
verbatim from the * Military Handbook of Signals.'
The Morse code is that which both military and naval
officers use. It has nothing but a combination of dots and
dashes for its basis.
The duration of the flash is what constitutes it a dot
or a dash, and practice is required both to give uniform
durations and to read the signals. An unpractised signaller
at the receiving station can take down the flashes by dot
and dash on paper and afterwards read them off. In giving
the signals, the transmitter should count, say, ten for a dash
and three for a dot, or less when in good practice. Counting
the duration of the dot as a unit, the pause between each
letter should be three, and between each word six units.
Table yi^lW,— Alphabet
A .—
N — .
B — • • •
C— • — •
P • •
D — . •
£ •
F • . — .
G •
H • • • •
I • •
Q--.-
R • — .
S • • •
T —
U • .—
V . . .—
J .
K — •-
w .
X — • .—
L • — • •
M
Numbers.
Y — .
Z . .
6 — • • • •
2 • •
7 • • •
3 • • •
8 . .
4 • • • • —
5 • • • • •
9 •
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94 Preliminary^ Sun^ey
Punctuation,
Full stop
Preparative and erasure, a continuous succession of dots.
The preparative sign is to call attention. To answer it
the received either gives the
General Answer
— a single dash, or else he gives the code letter of his name
or station.
When intended as an erasure to signify that a mistake
has been made by the transmitter, it should be answered by
the erasure signal.
The Break signal I I • • • • is used between the
address and text of a message, and after the text if the name
and address of the sender are to be signalled.
The Completion signal V E • • • — • but sent as a
group, not two letters, denotes the completion of a message.
* Repeat ' I M I given continuously • • • •
The Figure signal FI«« — • •• means that figures
are intended.
The Figure Completion signal F F •• — • •• — •
means that figures are done.
Indicator • — • — is sent at commencement and
conclusion of message. It is answered by the same signal.
All right R. T.
Goon G.
Move to your right R.
Move to your left L.
Move higher up or further oft" . . . H.
Move lower down or closer . . O.
Stay where you are . . . . S. R.
Separate your flags S. F.
Use blue flag B.
Use white flag W.
Use large flag L. F.
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Hydrography and Hydraulics
95
Use small flag S.
Your light is bad L. B.
Turn off extra light . . . . . T. O. L.
Wait M. Q.
Say when you are ready . . . . K. Q.
I shall signal without expecting answers . K. K. K. K.
In America the signal for all right is O. K., supposed to
have been invented by a Mr. Joshua Billings.
Some of the above signals refer to flag and light signals.
The lime-light apparatus is not described in this work, but
the flag system will be here explained.
Flag-Signalling
Fig. 25.
Fig. 24.
Fig. 23.
The Morse alphabet is used, but the dots and dashes
are made by movements of the flag and not simply by
exposing it.
Either large or small flags are used. They are both
made of the same material, a sort of muslin, and of two
colours, white with a blue horizontal strip for use with a dark
background and dark blue for use with a light background.
The large flags are three feet square mounted on a pole,
five feet six inches long, one inch diameter at the butt and
tapering to half an inch at the top.
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96 Preliminary Survey
The small flags are two feet square mounted on a pole
three feet six inches long, three-quarter inch diameter at the
butt and tapering to half an inch at the top.
Army Flag Drilu
Large flags. The signaller may work from left to right
or from right to left, or may turn his back to the station to
which he is signalling, according to the direction of the
wind, so that the flag may be waved from the normal posi-
tion against the wind.
To make a dot. Wave the flag from the normal position
«, Fig. 23 to a corresponding position, ^, Fig. 24 on the
opposite side of the body, and without any pause back to ^,
Fig. 23, keeping the left elbow close to the side.
To make a dash wave the flag from ^, Fig. 23, to r. Fig.
25, so that the point of the pole nearly touches the ground,
still keeping the left elbow close to the side and straightening
the right arm ; make a short but distinct pause in this posi-
tion and then return to a. Fig. 23.
When signalling a letter, say R, • — • the flashes repre-
senting it should be made in one continuous wave of the
flag, taking particular care that no pause is made when at
the normal position. Thus to make R wave the flag from a,
Fig. 23, to ^, Fig. 24, back to a^ Fig. 23, and without any
pause down to ^, Fig. 25 ; slight pause at c (see instructions
for making a dash), back to a (Fig. 23) ; then without pause
to b^ Fig. 24, and back to the normal position a. Fig. 23.
A pause equal to the length of a dash should be made at
the normal position a^ Fig. 23, between each letter of a word
or a group of letters. When the word or group is finished
the flag pole is lowered and the flag gathered in with the
left hand.
A slight pause should be made at the normal position
before commencing a word or group. In receiving a
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Hydrography and Hydratdics 97
message the flag should be lowered and gathered in until
required for answering.
In order to keep the flag always exposed while moving
it across the body to form the flashes the point of the pole
should be made to describe an elongated figure of 8 in the
air.
With clear sending and under favourable conditions these
flags can be seen and read with the ordinary service telescope
at distances up to twelve miles.
Uniform unbroken backgrounds are always better than
broken ones. Bare earth rocks and trees form the darkest
backgrounds, and these appear darker or lighter in propor-
tion to their distance in rear of the object projected on them.
If far off" the background will be lighter, if immediately be-
hind the object it will be darker. Sky forms the lightest
background and then water and distant land ; green or
stubble fields form an intermediate class of backgrounds
against which light or dark objects appear almost equally
visible. As dark backgrounds appear darker in proportion
as they are nearer to the object, so white backgrounds, such
as chalk cliffs, whitewashed walls, &c., appear lighter under
the same conditions. The position of the sun should always
be taken into consideration in applying the foregoing obser-
vations. All backgrounds become lighter when the sun is
opposite to them and darker whe;n it is behind them. An
exception to this rule is, however, found in the light mists
which rise from valleys towards evening or the smoke of
habitations, which both form a lighter background than the
surrounding country, whatever be the position of the sun.
The most favourable conditions for flag-signalling are a
clear atmosphere and a clouded sun.
Standard of Efficiency
For the standard of efficiency the minimum rates of
reading correctly from and sending a test message with the
different instruments are as follows.
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9^ Preliminary Smvey
Large flag at the rate of . .9 words per minute
Small flag „ „ . . 12 „
Lamp „ ,, . . 10 ,,
Heliograph ,, ,, . . 10 ,, ,,
The degree of accuracy is tested by the percentage of
letters read correctly. A man is reckoned as —
Very accurate who can read 97 per cent, correctly
Accurate . . . '95 >* »»
Fairly accurate . . '93 »» ,,
Inaccurate . . . .90 ,, ,,
Very inaccurate . below 90 ,, ,,
Tides and Currents
The soundings on charts are given in fathoms of depth
at low water of ordinary spring tides because those are the
times of highest high water and lowest low water. For the
theory of tides the reader must study works of a different
character from the present one. Some definitions of terms are
given in the glossary. No satisfactory data can be obtained
from a casual investigation ; observations should be made
at different times of the day, month, and year, and the results
registered in a systematic manner. The pioneer surveyor is
not supposed to have as much time as this at his command,
and we shall therefore confine ourselves to practical details
for a preliminary examination of tidal phenomena.
Datum
A fiducial point should be chosen on shore as near as
possible to the position where the register is to be kept, but
safe from any encroachment of the sea. A hard rock point
or a benchmark on a house or tree will do. If there is only
a sandy beach sometimes a piece of old iron pipe d" dia-
meter is obtainable, which filled with Portland cement
concrete makes a very durable datum.
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Hydrography and Hydraulics . 99
Tide Gauge
If a pier is available, a planed board is lashed to it
marked in feet and tenths, the former being painted red,
white, and blue alternately so as to be read at a distance.
If there is no pier, a pile must be driven and stayed. To
the side of the pile the planed board is attached, or else a
simple maximum and minimum device is constructed of a
casing made with four boards and containing a float main-
tained in position by guides working in slits and having
indicators one above a guide, the other below the opposite
guide. These indicators have to be set twice in the twenty-
four hours and are then self-registering. Or else if left for
any length of time they will leave on record the maximum
and minimum during that period.
The registration is made from simple inspection of the
tide-board, the zero of which has been previously levelled
with a levelling instrument from the datum point on
shore.
There are many elaborate and expensive self-registering
tide-gauges for use on permanent works of construction or
important hydrographical work.
A very beautiful repertory of tidal instruments was ex-
hibited by Sir William Thomson at the Institution of Civil
Engineers in 1881 in connection with his lecture on that
subject. These comprehended a tidal harmonic analyser,
a tide predicter, and a model of the No. 3 Clyde Tide-gauge.
Wave disturbance is almost entirely annulled in the floater ;
clockwork mechanism records by either an ink or a pencil
marker the movement of the tide upon a revolving diagram
like the steam-engine indicator. The abstruse calculations ot
the harmonic analysis are replaced by those of an automatic
integrator, and the time, not merely of high or low water, but
the position of the water-level at any particular port, is pre-
dicted for any time of day of any future year.
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lOO Preliminary Survey
The velocity of currents at sea are generally taken by a
ship at anchor with a current-meter. They are also ascer-
tinued by observing a drifting boat from two points upon
the shore. When far out at sea they are calculated by
comparing the day's run, ascertained astronomically with the
figures of the patent log, making allowance for instrumental
errors.
An elementary explanation of tidal phenomena, together
with high and low water at the principal ports of the United
Kingdom and tidal constants for minor ports, is given in
Whitaker's Almanack, which should form part of the travelling
library of every surveyor.
The velocity of river currents is best taken by current-
meter, unless sufficient time is available for obtaining the
data required by Kutter's formula.
Floats are either surface- floats, of wood or wax or vertical
tin tubes loaded at bottom. In large rivers the average of
the vertical floats, which gives the approximate mean velocity
at that section, has been found to be about '9 of the surface
velocity.
Hydraulics
The numerous problems of more or less complexity
occurring in hydrostatic and hydraulic science are peculiarly
suitable to treatment by the graphic method, and by the use
of large-scale diagrams may be solved with a greater degree
of mathematical precision than is actually necessary with
any formula, by reason of the element of uncertainty which
in hydraulics must always attach to the resisting power of
the conduit. For the approximate estimates which the pre-
liminary surveyor has to prepare, the small-scale diagrams
on Plate I. Fig. 26, and Figs. 28, 29 will be found amply
sufficient ; any slight inaccuracies due to scale will not pro-
duce variations in the final result as great as those between
two authorities like Beardmore and Kutter as exhibited
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Plate I.
/f—
s«
^ \
ue
//A
SM
#A \
M>0
i/^\ \
IM
£v^i
\ '.
?90
o^^\\\|v
V.
^^v^
1
»0
yS^
9 '
»M
ai3
9/^
IM
IM
^^\\)^^
F
170
ISO
\ ^^^ \?/^
1
tft»
MO
\ \ 4Y )&^
?v.
IM
N. ^5^\ W/v/i
J '
\i?\ v^Sw
•
^
N. XX/^
..o\
^KjC/^
.M \
\
\/t
M
•^^
•0 ^x
\
io
\
«0 ^N
«
»»
»
„ VALUES WR
c
,«< , „?„.,«»ff..s»..»..?f..f.-f.:ra'...'p
Fig. 26.
Kutter's formula for flow in open channels. Diagram
for finding coef. C in formula V=C x ^Rx S-
Nofe.— Thc values of N are from *oi to '04.
,, S ,, '000025 to *oi»
,, R „ o to 60 feet
Intermediate values may be interpolated by eye.
Note I.— The coefficient is to be taken from the thickened, right hand side of
the vertical scale.
Nt?te 2.— If this diagram is in frequent use with dividers, a piece of dull-back
tracing cloth, gummed over it by the four corners, will protect it.
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Hydrography aud Hydraulics loi
later on. The formula of the latter engineer has been
adopted as being the most comprehensive and in accordance
with recent experiments. Messrs Ganguillet and Kutter,
Swiss hydraulic engineers, have of late years developed the
theory of flowing water in open channels, and have furnished
a formula of greater complexity than most of the preceding
•ones, but held by modern authorities to be superior in
accuracy. However perfect the formula may be, an element
of uncertainty must always exist as far as rivers are concerned,
because the coefficient of roughness in the conduit, or n as it
is called by Kutter, has to be obtained, to some extent by ex-
periment but largely by judgment based upon actual study
of the ground or channel. Plate I. Fig. 26 for finding the
coefficient C is constructed for English measurement and
will embrace all ordinary cases, but where much work is
required it will be found advisable to construct a diagram
to a large scale, for which full directions are given in
' Traut wine's Pocket Book.'
It is evident that the varying velocities in any conduit of
water must arise from the force of gravity modified by the
resistances of the surface \ from natural roughness, vege-
table growths, bends, &c. ; the consistency of the fluid, the
wind, and it may be other causes.
Kutter's formula, like those of other good authorities old
and new, is based upon the theory that the resistances to
flow are directly proportional to the area of the surface
exposed to the flow (which would be, in the case of pipes
running full, the entire internal surface) and to the square
of the velocity. The formula stands thus for English
measurement : V=C >/ RS. In which V=velocity in feet
per second ; C a coefficient derived from three independent
variable quantities — the resistance of surface, hydraulic mean
depth, and slope ; R is the hydraulic mean depth ; S is the
slope or sine of the angle of slope.
The velocity V is the mean velocity of the whole stream,
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I02 Preliminary Survey
and is the only reliable means of ascertaining the discharge.
Experimental determinations of surface velocity by floats
are very useful, but are generally too local to give more than
an approximate result. The mean velocity varies in large
rivers from '85 to '95 of the surface velocity found by
floats.
The coefficient C is obtained directly from Fig. 26 by
drawing lines from the horizontal scale of R, that is the
hydraulic mean depth, to the intersection of the hyperbolic
slope curve S with the radial 'roughness' line n. This
line will cut the right-hand side of the vertical scale OC at
the value of C. The dotted line shown on the diagram is
drawn from a hydraulic mean depth of 50 feet to the inter-
section of «= '02 with slope of i in 2,500 ; the coefficient C=
123 is halfway above the first subdivisions from 120, each of
them being 2.
Any intermediate curves or radial lines may be interpo-
lated by the eye with quite sufficient accuracy.
The value of n is obtained from the following table.
Table Y^YA.-- Artificial Chatineis of Uniform Cross- Section.
Coefficient
of roughness,
n
Sides and bottom lined with weU-planed timber . '009
Sides and bottom rendered in cement, glazed ware,
and very smooth iron pipes . . . . 'Oio
I to 3 cement mortar, or smooth iron pipes -oi i
Unplaned timber or ordinary iron pipes . -012
Ashlar or brickwork . . . . '013
Rubble -017
Channels subjtct to Irregularity of Cross-Section.
Canals in very firm gravel *020
Canals and rivers of tolerably uniform cross-section,
slope, and direction in moderately good order
and regimen, and free from stones and weeds '025
Having stones and weeds occasionally . . -030
In bad order and regimen, overgrown with vege-
tation, and strewn with stones and detritus '035
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Hydrography and Hydraulics 103
Messrs. Trautwine and Hering use -015 as a value of n
for ordinary brick sewers instead of -013 given by Kutter, in
consideration of the * usual rough character of sewer brick-
work.' This remark would not apply to the London and
Paris sewers or to those of many other large cities, where
the brickwork is exceedingly good ; but unless properly main-
tained the coefficient would soon increase.
The hydraulic mean depth R is equal to the
area of wet cross-section
length of wet perimeter abco.
In the case of a circular culvert or pipe running full, R=
4
^^ =- or -- where r= radius and ^= diameter. When
half-full R is also = - and the velocity will be the same.
4
At 75 of the diameter, ix. | full, R=D x -3. When \
full R=Dx'i5. To obtain the hydraulic mean depth of
irregular conduits such as rivers, the cross-section must
be taken by soundings and the area measured.
The slope is usually termed the sine of the angle made
by the average bed of the channel with the horizon. At the
small angle of most rivers and sewers the sine is sensibly the
same as the tangent, but why it is called the sine is not clear
to the writer, because slopes are usually from convenience
described in the ratio of base to perpendicular, and not from
hypotenuse to perpendicular.
The length of the conduit is always supposed to be
measured on the level, and it would seem to be proper to
take the ratio from that rather than from the sloping length.
The table of tangents corresponding to some leading slopes
of feet per mile given on p. 248 will be found sufficient by
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I04 Preliminary Survey ^
the aid of the slide-rule to obtain any required tangent, and
the method of interpolating is there explained. Both the
hydraulic mean depth and the tangent of slope are often
very small decimal quantities of which the square root has
to be extracted. Another short table is given, on p. 249, of
leading values of squares of decimals by aid of which the
operation is greatly facilitated. The diagram only gives
slopes up to 'CI, that is, i in 100. Above that the coefficient
C remains the same. The formula upon which the diagram
for finding the coefficient C is constructed is for English
measure —
^__ slope n
1+— L=
slope /
\/mean radius in feet
When the velocity in feet per second has been obtained
from the diagram and formula V= C >/RS, the discharge, Q,
is obtained in cubic feet per minute by the simple equation
Q=V X4712 D^for circular sewers where D=diam. in feet.
In irregular channels and generally Q=area in square
feet X vel. in feet per second x 60.
Example i. What is the velocity in feet per second and
discharge in cubic feet per minute in a circular brick sewer
running full ; 2 feet diameter ; slope S=io feet per mile=
•00189; R=-25;D='5?
RS= '000945. v/RS= '0307 ; C by diagram 4 is found
to be (using a coefficient of -015 for «)=85, and V=85 x
•0307=2*6 feet per second.
Q=2*6 X 47*12 X 4=490 cubic feet per minute.
Example 2. What will be the velocity in feet per
second and discharge in cubic feet per minute of a V- flume
of unplaned timber, sides sloping 60° and bottom board 10"
wide in the clear, the slope being i in 50 and the depth of
water 9 inches ?
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Hydrography and Hydraulics
los
The whole calculation can be done in a few minutes by
the slide-rule and diagram. The angle of slope of sides gives
the horizontal spread from the line of tangents=5'22 inches,
from which we find area=7o8 square feet and wet perimeter
2-5 feet; R=-^-^=-283 feet; n from Table XIX.=-oi2 ;
S =*02. C is found from Fig. 26 = 98 ; RS = -283 x '02
= •00566; n/RS=-o75 ; and V=98x •075=7-35 feet per
second; Q=7o8+ 7*35 x 60=313 cubic feet per minute.
The following comparison of the flow of circular sewers
running full, calculated from Beardmore's and Kutter^s
formulae, has been made with assistance as regards the former
from the well-executed diagrams of Mr. W. T. Olive
Resident Engineer on the Manchester Main Drainage,
published by the Inst. C. E. in their minutes of
Proceedings, vol. xciii.
It will be seen that the older formula produces results
nearly midway between those by Kutter's rule with the
coefficient '015 and 'oi for roughness, but that in the larger
culverts the discharges by Kutter's formula gradually gain
upon those of Beardmore, until they are considerably ahead
even with the coefficient '015 for roughness.
Tabi.e XX. — Comparison of Beardmore^ s and Kutter's formuUc.
r
"
Discharge in
Diameter
Slope in
Discharge in c. ft.
Discharge in c. ft.
^ per min. for
brickwork; «=*oi5
(Kutter)
for glazed-
in inches
feet per mile
per nun.
(Beardmore)
ware or iron ;
«=*OI
5
(Kutter)
6
'2^
93
155
6
50
38
29-4
49
6
150
66
43*4
727
24
10
540
490
—
24
50
1,210
1,112
—
96
10
17,100
20,850
—
96
15
21,000
25,633
—
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io6 Preliminary Survey
Discharge from Tanks, Pipes, Cisterns, and Weirs.
The diagrams, Plate II., of theoretical velocity due to
different heads will need no explanation.
They are furnished for use when no slide-rule is at hand ;
otherwise it is quicker to work out the velocity by the slide-
rule than to scale it on the diagram, and the velocities for
fractional values are given with the same rapidity as those
of integral values, more accurately than can be scaled from
a diagram, and without the interpolation needed with a
table.
Example. What is the theoretical velocity due to a head
of 15*65 feet? Place the right hand i of the slide under
the upper 15 feet of rule and read the velocity 31*8 feet per
second on the lower scale of the rule opposite to 8*03 on
the lower scale of the slide.
Example 2. What is the theoretical head due to a
velocity of 135 feet per second? Place the 8*03 on the
lower scale of the slide over the 135 of the rule and read the
head 282*5 feet on the upper scale of the rule opposite to
the right-hand i of the slide. ^
Rule I. For tanks and cisterns flowing into the open air
Let V = the theoretical velocity in feet per second due to
head, given on Fig. 28.
A=area of aperture in square feet.
D= diameter of circular aperture in square feet.
C= coefficient of friction due to nozzle or opening
given by table.
VV=actual velocity of discharge in feet per second.
Q= discharge in cubic feet per minute.
ThenVV=VxC
andQ=VVx6oAorVVx47i2D2
^ If the velocity is wanted in miles per hour instead of in feet per
second, use the number 5*475 instead of 8*03. For heads from i to 10
feet use the left-hand half of the rule, and from 10 to 100 the right-
hand half.
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Plate II.
^Sici^^SwE
e
B
■^1
iT (0 o
I -I"
'Z -c
Fig. 28*
Fof'/M't/a. -V =8'o3 ^W.
M = 5-475 VH^
Where H = head in feet.
,, V=vel. in feet per second. ^ i
„ M=vel. in miles per houiQinitiypd h\ V tOOQIP
Noff.— The vel. in feet per second is the only one which can be obtained from
inspection of the diagram : the vel. in miles per hour must be taken with dividers
from the outer scale.
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Hydrography and Hydraulics 107
Values of C for
different orifices
Circular or rectilinear openings in thin iron plate in
bottom or sides of either vertical or inclined tanks = . -62
Short tube projecting outwards, bore \ the length . . 'So
Ditto bore -^^ of the length 70
Short tube projecting inwards '68
Converging tube, outside of tank ; angle 13^° ; velocity
at narrow end '94
Diverging ditto, outside of tank ; angle 5° ; velocity
at narrow end '92
Example, What are the velocity and discharge in feet
per second and cubic feet per minute respectively in the
following case ?
A tank is maintained by a ball cock with a depth of
water of 7^ feet. A short cylindrical tube projects from the
bottom of 2\ inches inside diameter and 2 feet i inch long.
The velocity due to 7*5 feet head is found by Fig. 28 to
be 22 feet per second. C is an intermediate between -So
bore
and 70, - — of orifice being equal to ■^, Interpolating
with the slide-rule we find C=7i and VV=22 x 71 = 15*6
feet per second. The diameter 2\ inches=*2o8 feet and
discharge Q=VVx 47*12 D^ or= 15 •6x47*12 x*2o82.
Ans. W=i5*6 feet per second.
Q=3i'8 cubic feet per minute.
Rule 2. For iron pipes under pressure approx. mean vel.
m feet per second =
coeff. m (Fig. 30) x ^^jeHp^in^^
This approximate rule of the late Mr. Trautwine takes into
account velocity head, entry head, and friction head. It
was not used by him for very long pipes with low heads.
The limit given by him was 1,000 diameters, beyond which
he neglects entry head and treats the flow by the formula
for an open channel. It is not applicable to very high
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io8 Preliminary Survey
pressures either, but is suitable to ordinary reservoirs having
straight iron pipe conduits.
The diagram Fig. 30 gives values of coefficient m for various
diameters of pipes in feet. The horizontal scale gives values
of the A / ^ ^"^ for which the ordinates are values of m,
^ L + 54D
D being diameter in feet, H= total head in feet, L=length
in feet. The value for '200 serves for all ratios above it.
Intermediate values are to be taken by interpolation. The
results are always in excess of those where there is no entry
head and all the fall is in the pipe itself.
Rule 3. Discharge over weirs (Eytelwein).
Q=discharge in cubic feet per minute.
L=length of overfall in feet.
H=head in feet.
Q=204 L n/IP
Remark. The head is measured by ascertaining the
difference of level between the crest of the weir and the
surface of the water before it commences its chute- curve.
The formula is for discharge over a thin plate or a weir
with a sharp edge. A slight current towards the weir makes
hardly any difference in the result.
Example. Required the discharge over a weir in thin
plate length 200 feet. H=i'5 feet.
Q=204 X 200 's/f^= 74,954 cubic feet per minute.
Rule 4. — (Approximate — John C. Trautwine) see Dia-
gram, Fig. 31, for C.
Q=CxLxVxH.
Where Q= discharge in cubic feet per minute.
C= coefficient per diagram.
L= length of weir in feet.
V= theoretical velocity due to H in feet per
second from Fig. 28.
H=head in feet measured as in preceding rule.
This formula gives results somewhat less than the pre-
ceding one for sharp-edged weirs, and is therefore on the
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Plate III.
'200
Values
Fig. 30.
HIAD IN mCMES
S5 ?D I& 11} 5 D
Dischai^ge ovei weirs.
Fig. 3x.
A - crest 2 inches thick.
A. A. = crest sharp edge.
A.A.A.=crest 3 feet thick, smooth ; sloping outward and downward from 1 in 12 Ui
I in 18.
A.A.A.A.=crest 3 feet thick, smooth and level. r^^^^I^
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Hydrography and Hydraulics 109
safe side ; it also meets all the ordinary cases by means of
the coefficient C with quite a sufficient degree of accuracy for
preliminary work.
Example, What would be the discharge over the same
weir described in the preceding example only with a crest
3 feet thick, smooth and level ?
Q=i8*4 X 200 X 9*8 X I '5=54,096 cubic feet per minute.
Over a sharp-edged weir it would be 67,000 instead of
the 74,954 of Eytelwein's formula, but with a crest 2'' thick
it would be 80,000 by Trautwine.
In the case of reservoirs, it is generally easier to deter-
mine the discharge over the weir by that of the supply con-
duit by Kutter's formula.
Horse-Power of Falling Water
HP=^^XQl=.ooi894 QH.
33000 ^ ^
Where Q=discharge in cubic feet per minute, and H=
height of fall or head over a turbine or other motor.
Example, Over a fall 16 feet in vertical height, 800 cubic
feet of water are discharged per minute.
^p^8ooxi6x62'5^ HP
33000
Efficiency of Water-Wheels
For undershot wheels x result found as above by -25 to -33
„ breast wheels „ „ „ by '5.
„ overshot „ „ „ by -5 to 75.
„ turbines „ „ „ by 75 to -85.
Horse-Power of a Running Stream
j^p^QxHx62:5=.ooi894QH.
33000
Where Q= discharge in cubic feet per minute actually
impinging upon the float or bucket ; H= theoretic head, due
to velocity of stream found by Fig. 28.
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no Preliminary Survey
Thus, if the floats of an undershot wheel, driven by
current alone, be 5 feet x i foot, and the velocity of stream
=210 feet per minute, or 3^ feet per second, of which the
theoretical head is '19 feet. Q=5 square feet x 210=1,050
cubic feet per minute ; H=-i9 ;
and HP = ^^50 X •19x62-5^ 8.
33000
The wheels only realise about -4 of this power, from
friction and slip.
The last three rules are from Trautwine's * Pocket Book.'
Hydrostatics
Water pressure is always normal to the surface pressed,
i,e. in flat surfaces perpendicular to them, and in curved
surfaces in line with the radius vector, or, which is the same
thing, perpendicular to the tangent at that point.
Whatever the inclination of the surface or the extent of
its immersion or distance of its submersion, the following
rule holds good.
Rule I. Pressure in pounds =62 -5 AIX
Where 62 -5= the assumed weight of a cubic foot of water.
„ A=area of surface pressed in square feet.
„ D = depth of centre of gravity of surface pressed
below the surface of the water.
Rule 2 For pressure in pounds per square inch, multiply
the depth in feet by -434.
Rule 3. For tons per square foot, multiply the depth in
feet by '0279.
Rule 4. Total pressure from surface in tons on a section
I foot wide=D2 x '0139.
For the depth in feet, at which any given pressure exists :
Divide pounds per square inch by '434.
„ pounds per square foot by 62*5.
„ tons per square foot by -0279.
Plate IV., Fig. 32, gives the normal pressure per square
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Plate IV".
PRESSURE Pcn SQUARE FOOT
TOTAL PRESSURE from SURFACE.
Fig. 32.
Note z. — The left- and right-hand scales have no connection with one another.
The lbs. on the left-hand scale refer solely to pressures per square foot. The tons on
the right hand refer solely to total pressure from surface.
Note a.-— If thb diagram is in frequent use with dividers, a piece of dulUback
tracing-cloth, gummed over it by the four comers, will protect it.
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Hydrography and Hydraulics in
foot of any immersed surface due to any given depth
of its centre of gravity below the surface of the water
thus :
.mprszrjisiemr-.
o e
Fig. 33.
It also gives the total pressure upon a plane standing
upright in the water, such as a sluice-gate, the plane being
one foot wide, thus :
=
'K"~
DEPTH
...,77/^^^'''^
yk^^m^///y
Fig. 34.
In order to combine two diagrams in one, the left-hand
vertical scale gives normal pressure per square foot in
divisions of 100 pounds each, and measures the ordinates to
the straight line. The right-hand vertical scale gives tons
total pressure with each subdivision=-i ton, and measures
the ordinates to the curved line.
Example i. What is the pressure per square inch on a
water-main where the head is 300 feet ? Ans. 300 x '434=
130 pounds per square inch.
Example 2. What is the total pressure on a lock-gate
20 feet wide, with a head of 8 feet ? Ans. 20 x 8 x 8 x 'oi 395
= 17-85 tons, or by Plate IV.=2ox*9=i8 tons.
To divide a vertical surface under hydrostatic pressure.
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112 Preliminary Survey
such as a sluice gate, into sections of equal pressure by
horizontal lines.
Let N= number of sections, then there will be N — i
lines, the distance of which y^^w the top will be D,, Dg, D3,
Djf.i- Let H= height of water producing the pressure.
Then
Di=J|^N; I>2=|n/2N;D3=5^3N
Dn-i=Jn/N(N-i).
Example. luQt H=2o feet and N=4.
I^i=5V4 ; D2=5n/8 ; D3=5v/ 12.
= 10; =i4'i ; =17*3-
Cost of Dredging
Price of labour 45. per day, no allowance for interest or
depreciation of plant.
In from 5 to 10 feet of water, ^d, to 6d, per cubic yard.
In from 10 to 20 feet of water, 6d. to gd. per cubic yard.
These prices include the removal of the material ^ to i
a mile.
Dredging Plant
The dredging plant used at I^eith Docks which remove
about 110,000 tons of silt every year cost 13,000/.
The dredging plant used at St. Nazaire, Loire, France, to
remove 400,000 cubic yards of silt per annum cost 29,800/.
The dredging of soft material in small quantity can be
done with a bag-spoon (Fig. 35). This is simply a bag, b,
of canvas or leather with an iron ring at its mouth. It has
a fixed handle, k^ by which it is thrust down into the mud ;
another man draws it along by therope g, and a third hauls
it up when full by the rope e (Trautwine).
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Hydrography and Hydraulics 1 1 3
Fig. 35.
Borings
in common soils or clay may be made 100 feet deep in a
day or two by a common wood-auger i^ inches in diameter
turned by two to four men with 3-feet levers. This will
bring up samples (Trautwine). Weale's series, * Well Sink-
ing and Boring/ contains a detailed description of boring
tackle.
Concrete
At Aberdeen harbour 16,000 cubic yards of concrete
were deposited in jute bags at a cost, including plant, super-
intendence, and maintenance, of i/. 9^. \od, per cubic yard.
13,000 cubic yards of concrete were deposited in mass at
a cost of i/. 9^. 6^. per cubic yard. The concrete consisted
of I part of Portland cement, 3 of sand, and 4 of shingle.
At Port Said breakwater, concrete in blocks weighing
about 20 tons was deposited, forming two breakwaters, one
9,800 feet long, the other 6, 233 feet long, at a cost of i/. 5^. \d,
per cubic yard. The concrete was composed of 3^ of sand
and gravel to i of shell lime, made in boxes and left two
months to set before being used.
Dock Walls
The cost of the Hull dock- wall, 43 feet deep to bottom
of foundations, cost 19/. 9^. per lineal foot; the South West
India docks, 41 feet deep, 12/. \os. ; the Penarth Extension,
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114 Preliminary Survey
52 feet deep without excavation, 26/. is. ; the cost of the
Dublin quay-wall, 47 feet deep, 40/. per lineal foot.
The cost of the West India Dock wall per cubic yard
was 12^. 6^., of the Penarth Extension wall 17^., of the
Portsmouth wall i/. 2j., of the Chatham wall 7^. \o\d.
Cost of Docks per Acre
The Antwerp docks, of about 50 acres, having about
3,700 lineal yards of quay and a basin lock with entrances
59 feet wide, depth of sills at entrance 23 feet, cost about
4,800/. per acre of water area.
The Joliette basin at Marseilles, of 54 acres contained by
2,300 Hneal yards of quay, cost 640,000/. or nearly 12,000/.
per acre of water area.
At Leith, the old docks cost 28,500/. per acre, the
Victoria Dock 27,000/., the Albert dock 20,400/., and the
Edinburgh dock 24,000/. per acre.
The Chicago breakwaters of cribwork filled with rubble,
30 feet wide and 30 feet deep to bottom of foundations, cost
22/ 18^. per lineal foot.
The New York quay walls of pilework hearting filled
with rubble, cut stone face-wall protected at toe by riprap
and riprap backing, 43 feet deep to bottom of excavation,
cost 49/. 165. per lineal foot.
The New York jetties or landing stages for ocean-going
steamers of timber un-creosoted cost about 55. 6^/. per square
foot of area.
Graving Docks
Twenty-five of the Liverpool Graving Docks, having an
aggregate length of 12,490 feet, cost 940,000/ =about 75/.
per lineal foot of floor length.
Graving docks have been built for vessels of 2,000 tons
for under 20,000/.
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Hydrography and Hydraulics 1 1 S
The Somerset Dock at Malta, 468 feet long, cost
1 50,000/., =about 320/. per foot of floor length.
At per cubic yard of capacity they vary from i/. to 4/.,
the most recent docks at Portsmouth averaging about 2/.
The sluicing basin at Honfleur, having an area of 143
acres, cost 200,000/.
Warehouses of brick with iron columns and fireproof
doors cost from 4^/. to 8^. per cubic foot of total capacity.
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1 1 6 Preliminary Survey
CHAPTER IV
GEODETIC ASTRONOMY
The title of this chapter might convey the idea of a much
more comprehensive treatment of the subject than would
be possible or desirable in a book of the kind. The rough
astronomical calculations which lie within the average
surveyor's attainable knowledge or within the power of his
instruments can hardly be justly termed astronomy ; never-
theless in its most elementary form the subject matter of this
chapter will be what it is stated, and an attempt has been
made in the glossary and also in this chapter to describe the
principles in a manner which will be intelligible to anyone
possessing but very little previous knowledge of the subject.
Surveyors who desire to pursue the study of astronomy
further will find all they seek in Chauvenet's * Spherical and
Practical Astronomy,' clearly and simply explained.
If a work of a more elementary, and less expensive cha-
racter is desired, the volume on * Practical Mathematics ' in
Chambers's Educational Course, will be found both useful
and handy.
Some of the most commonly used formulae of plane and
spherical trigonometry are given in the appendix to the pre-
sent volume, but the demonstration of them is necessarily
omitted; they will nearly all be found, however, in Chambers's
* Mathematics.'
Geodetic astronomy differs but little from nautical astro-
nomy. It is the science of determining by observation and
calculation of celestial phenomena the position and course
on land. It differs from nautical astronomy only in affording
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Geodetic Astronomy 1 17
the facilities of * terra firma ' for some operations which are
not practicable at sea, and in being generally debarred from
the use of those which depend upon the sight of the natural
horizon.
In a treatise on Preliminary Survey the science of geodesy
cannot with propriety be treated fully, neither can the
analysis of the problems of plane and spherical trigonometry
involved therein be closely pursued ; but the object of this
chapter will be to explain general principles sufficiently to
make the formulae intelligible, and to furnish examples of
every problem Hkely to be useful to the pioneer. A great
deal of explanation has been placed in the glossary, which
has been arranged alphabetically to be easily referred to, and
should be studied before commencing this chapter.
The position of any point on the earth's surface is deter-
mined astronomically by finding its relation to the pole and
to some arbitrarily chosen meridian, and these two relations
are termed latitude and longitude.
Latitude
Let us first take the latitude. Everyone has learnt how
to find the pole star by the * pointers ' ; they perhaps also
know that the altitude of the pole star represents roughly
the latitude of the place. We can find our place on a map
by measuring the distances of latitude and longitude on the
circles described upon it for that purpose, but the only fidu-
cial point given us by nature as a starting-point from which
to map the world is the north pole. The map of the earth
is made from the map of the heavens, but the map of the
heavens is first made from the axis of the earth's rotation.
It is the discovery of a stationary point in the heavens
called the celestial pole which determines the position of the
terrestrial pole and terrestrial equator ; and conversely the
position of the observer relatively to the celestial pole deter-
mines the position relatively to the terrestrial equator.
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1 1 8 Preliminary Survey
A moment's thought will serve to grasp this. If I were
standing at the north pole, the celestial pole would be over-
head, because I should be standing in the axis of the earth's
rotation. If I were at the equator, it would appear on the
horizon ; consequently if I measured the altitude of the
pole star it would be in the first case 90**, in the second case
0°. (see Altitude, Glossary). And these are the respective
latitudes, measured from the terrestrial equator. It is evi-
dent, then, that an altitude of a celestial body taken when
passing the meridian will, if we know its distance from the
pole, enable us to determine the latitude by a simple sub-
traction sum.
And now, leaving for the present the subject of latitude,
let us touch upon the principles on which the calculations
of terrestrial longitude are based.
Longitude ^
Unlike latitude, which as we have seen can be deter-
mined independently of almanacs or chronometers, longi-
tude depends upon an arbitrary fiducial point. Both in the
heavens and earth there is nothing fixed by which to deter-
mine the easterly or westerly positions without either a time-
table or a watch or both.
The earth performs a rotation in twenty-four hours of
sidereal time (see * Sidereal Time,' Glossary). A star
which crossed the meridian of Greenwich to-day at mid-
night will cross the i8oth meridian, that is over the Fiji
Islands, at twelve sidereal hours later, and will recross
Greenwich meridian twelve hours later still. Any celestial
phenomenon happening, independently of our meridian,
to the star, such as an occultation of it by the moon, would,
although in itself an instantaneous phenomenon, be seen at
different local times in different places, and the difference of
local time corresponds exactly with that arc of the earth's
rotation which would bring successively the two points of
• See Glossary.
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Geodetic Astronomy 119
observation under the same celestial meridian ; in other
words, is equal to the difference of terrestrial longitude.
Phenomena such as those just described are very useful
in their place, but the chief basis of calculation is the sun,
whose transit across the meridian can be observed with the
utmost exactness, and having by chronometer the time when
he crossed the meridian at some known place, we can at
once, without any mathematics, determine the difference of
longitude by observing his transit and reducing the differ-
ence of time to difference of arc. We use the meridian of
Greenwich as our fiducial point, and perhaps the time is
not far distant when all the nations will agree upon one
common meridian.
Reserving further explanation of the methods available
to the surveyor for ascertaining his longitude, we will add to
these prefatory remarks an attempt to explain the curious
fact which travellers puzzle over when they first cross the
Pacific Ocean ; viz., that of 'losing' a day in going one way
and gaining it when going the other way.
As the sun appears to go round from east to west, a
place east of Greenwich, say Paris, will have sunrise a little
before Greenwich, or west, as Dublin, a little afterwards. In
other words Paris time is ahead of, and Dublin time behind,
Greenwich. If we could travel round the world across
America as quick as lightning, and left Greenwich at mid-
day of January i, 1890, when passing New York we should
find the early risers at breakfast on New Year's morning.
At San Francisco and Honolulu they would still be all in
bed, and at Fiji if we kept to the same calendar we should
find them at midnight of December 31. Fiji is just half
way round, being on the i8oth meridian. Instead of its
being midnight of New Year's eve, however, it would be
styled in that place midnight of January i, and the reason
for this we will explain by supposing that instead of going
vid, America, we had gone vid. Calcutta in our lightning
flight. Leaving Greenwich at the same time, viz. midday
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1 20 Preliminary Survey
of January i, passing Calcutta we should find them
already at their New Year's evening dinner. At Melbourne
they would be just going to bed, and at Fiji it would be
midnight of January i. It will be seen therefore that the
Fijians might keep their time twelve hours back or twelve
hours ahead, according as they chose to consider them-
selves in the Western or Eastern Hemisphere. Conse-
quently they have the option of one day's date. The writer
has been informed by a missionary that the keeping of two
Sundays running has been of frequent occurrence in the
palmy days of evangelisation upon that group, but they have
finally elected to consider themselves in the Eastern Hemi-
sphere in order to be in the same calendar as Australasia.
Now to take the passage of a vessel circumnavigating
from Greenwich eastwards ; she would keep putting on her
time every day at noon until her time was twelve hours
ahead of Greenwich at Fiji, and correct with local time and
date there. A vessel circumnavigating westwards from
Greenwich would have been putting back her time all the
while, until at Fiji she was twelve hours behind Greenwich,
and consequently correct with local time there, but one day
behind in date.
If they both happened to arrive at Fiji at midnight of
January i, 1890, the eastward navigator would be correct
to time and date, but the westward navigator's date would
be midnight of December 31, consequently when he passes
over into the Eastern Hemisphere he will have to skip a day
and call it January i. The eastward navigator passing
into the Western Hemisphere will have to put back a day
and call it December 31.
Classification of Methods
There are three different circumstances in which a surveyor
is ordinarily placed as regards his astronomical work, modi-
fying the methods which he can use and the degrees of
accuracy which he can obtain.
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Geodetic A stronomy 1 2 1
1. When starting from the coast and proceeding with a
carefully measured telemetric survey inwards.
2. When starting from the coast and making a route-
survey inwards.
3. When carrying on an inland survey and having to
locate his starting-point.
The most disadvantageous circumstances, short of having
no instruments to all, in which a surveyor can be placed are
to be —
1. Far inland and out of telegraphic communication.
2. Possessed of a poor watch.
3. Unable to revisit his points of observation. >
4. Compelled to move forward rapidly in an easterly or
westerly direction.
As will be explained presently, the surveyor is very
dependent upon Greenwich time for his longitude. This
is preserved upon important geodetic work with the utmost
care. A number of chronometers in padded cases, swing-
ing on patent joints, wound every day at the same time
by the same individual with the same number of turns,
and constantly compared with one another, enable the
observer to obtain his longitude with the same accuracy as
his latitude. Very different is the case with the preliminary
surveyor. He is rarely able to transport more than one
chronometer, he often does not possess even a semi-chrono-
meter watch, and so when out of range of the telegraph has
to resort to what are termed * absolute methods' of deter-
mining his longitude. Apart from the great inconvenience of
transporting high-class chronometers whilst on rapid tran-
sit, their very delicacy renders them less useful under
rough usage than semi-chronometer watches, two or more
of which in a surveying party are the best instruments
available. 'Absolute methods ' of finding the longitude are
either performed with the sextant by Munar distance' or
by * lunar altitudes ; ' with the transit as in the method
termed * moon-culminating stars'; or with the telescope
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122 Preliminary Survey
alone as by eclipses of Jupiter's satellites or the occultation
of a fixed star by the moon. Sextant observations will
only be explained briefly, in principle ; the surveyor rarely
possesses anything of that description larger than a pocket
sextant, which is of no use for such purposes.
In the first case, when starting from the coast on a
telemetric survey, the daily traverse is reduced to latitude
and departure by slide-rule or by a table, such as that in
Chambers's mathematical tables. A very close check is thus
kept on the astronomical work, and the longitude and
latitude by account is used like the dead reckoning at sea
for determining the argument of the daily observation. The
daily observations should be if possible four.
One for time to correct the watch's rate.
One for longitude by watch.
One for latitude by fixed star.
One for azimuth by sun or star.
If any celestial phenomena are available for * absolute
longitude' they should not be missed.
If only one observation can be taken it should be that
for azimuth, and it can be made en route from the sun
without interrupting the work for more than five or ten
minutes. No continuous telemetric traverse should be
carried forward into an unknown country without a daily
check upon the direction. A slip of one degree which may
be set down to change of magnetic deviation may produce
an error of a thousand feet in a single day's work.
In the second case, when starting on a route survey
from a known point, the observations should be the same
but more strictly kept up every day, because the survey
covers so much more ground, and the check by latitude
and departure is much rougher ; otherwise the principle is
the same.
In the third case, when the location of the starting-point
has to be determined independently, the surveyor must
take time to get reliable data to begin with ; he should stay
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Geodetic A stronomy 1 2 3
two or three days in the same place to take and reduce a
series of observations. The first thing is to determine the
meridian, which is best done by equal altitudes of a fixed
star. From this, local mean time is obtained by observing
the transit of a known star. Latitude is then had from the
pole star or a circumpolar star such as one in the Great Bear
or Cassiopea. Longitude is determined by the successive
transits of the moon and a fixed star close together, or else
by an eclipse of Jupiter's satellites or the occultation of a
star by the moon. This is supposing he has not got Green-
wich time by chronometer to start with.
Three days of favourable weather and careful work will
give the surveyor who has nothing more than a six-inch
transit to work with, a very fair approximation to his true
geographical position.
The starting-point, once fixed, should remain as the
basis of the whole survey. It is probable that errors will be
found later on by bringing greater precision to bear upon it,
but in order to preserve S3niimetry in the work the starting-
point should be assumed as correct for graduating the sheets,
so that any error found afterwards will be a constant through-
out the work. The local mean time and Greenwich date
at starting-point once determined, can be recovered at any
time by returning to the spot and observing a star's transit.
The constant check upon the watch by means of
sidereal time cannot be too strongly recommended. No
mathematics are needed, and no time wasted in waiting.
By it the poorest time-keeper can be made valuable, and the
best chronometers are none the worse for being checked by it.
Observations for determining the True Meridian.
First and foremost amongst the uses of the stars to the
surveyor is the determination of the true north and south
line or meridian of the place. He needs it for running his
survey line true, and he needs it above all things for his
other astronomical calculations.
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124 Preliminary Survey
There are still old-fashioned people who think the
magnetic compass is quite near enough for a railway, and
star-gazing a superfluous luxury ; they are generally of the
same sort who think the chain is the only practical way of
surveying, and a transit instrument only a means of saving
lazy engineers their due amount of manual labour. We will
not enter into controversy with them for fear of being hit
too hard, but proceed to touch upon the various ways of
determining the meridian.
A rough and ready method has been explained in the
chapter on route-surveying at p. 40.
The solar compass as described on p. 328 gives the true
bearing of any points observed, and the true north and
south line.
Some surveyors have found very good results attainable
with simple sun-shadows reduced to true north and south
line by Davis's tables from the time of day recorded opposite
to each shadow.
All such operations, though useful in their place, are
very approximate.
A watch keeping apparent time, and held so that the
hour hand points towards the sun, will roughly give the
meridian, midway between the sun and the xii of the
watch.
Meridian by equal Altitudes of a Star
The simplest of the really accurate methods is by obser-
vation of a fixed star at the same altitude, east and west of
the meridian. The movement being perfectly uniform,
the successive bearings are taken from any clearly defined
terrestrial point, and the mean bearing corresponds exactly
with the meridian.
A good plan is to adjust the transit for coUimation and
bubble in daylight upon a good solid position, then to ad-
just the horizontal limb so that the vernier and compass
needle both stand at zero. Then, clamping the external
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Geodetic Astronomy 125
axis and releasing the parallel plates, take the exact bearing
on the same vernier of some well-defined terrestrial point at
a considerable distance, which will be its magnetic bearing.
On no account use the compass of the instrument for this
bearing.
Cover over the instrument until the stars begin to appear.
Choose a bright star as early as possible and about 30° to
45** from the meridian either north or south and about 20°
to 40° above the horizon.
If it is a circumpolar star taken to the north^ be careful
to note from its position relatively to the pole whether it is
on its way to upper or lower culmination.
If on its upper path, the first altitude must be observed
to the east like the southern stars, but if on its lower path
it must first be observed to the
westward as shown on Fig. 36.
This method requires no
mathematics whatever. Book
the altitude of the star from the
vertical limb in its first position,
and opposite to it the magnetic
bearing from the horizontal limb ;
not from the compass. Unclamp
the vertical arc and bring the
telescope bubble to the centre of
its run, and book any index error which may have crept in
since the first adjustment. Set the vertical arc back to the
altitude of the star, unclamp the horizontal limb and direct
the telescope to the approximate second position of the star,
that is to say, about the same angle from the meridian, only
on the opposite side. The variation of the compass is
known roughly all over the world from a map of equal
variations such as that in * Hints to Travellers ' referred
to on p. 369, so that the telescope can be placed within
a degree or so of its proper place. When the star begins
to get near the field, test the vertical arc with the bubble
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126 Preliminary Survey
as before, and if any index error be found, correct it, and
clamp the vertical arc again at the same altitude as the first
position and free from index error.
Release the horizontal limb, but on no account touch the
vertical limb. Bring the telescope right under or over the
star, and when it enters the field clamp the horizontal limb
with the star on the central vertical hair, and keep it in that
position with the tangent screw of the horizontal limb until
it reaches the horizontal axial hair.
Book the second bearing from the vernier of the hori-
zontal limb, and the mean of the two bearings is the mag-
netic bearing of the meridian.
If the observation has been taken to the north, and the
bearing of the meridian is greater than zero, or if taken to
the south and it is greater than i8o°, the variation of the
compass is west ; and if less than 360° or 180°, as the case
may be, the variation is east.
Example, Observed Spica at same altitude to the south.
Mag. bearing in first position
,, „ second „ .
Mag. bearing of meridian
Variation of compass \V. . 17® 24' 10"
The directrix chosen was a hghtning conductor on a
large building about 500 feet away, whose magnetic bearing
was o , //
233° 52' o"
Applying the variation found . . 17® 24' 10'
155°
239°
08'
39'
40"
40'
2)394°
48'
20
197°
180°
24'
0'
10"
0"
True bearing of directrix -~ . .216° 27' 50"
Thus when we wish to place the transit in the true
meridian we place it in adjustment for collimation and
bubble, and clamp the horizontal limb at 216° 27' 50". We
then release the external axis and adjust the intersection of
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Geodetic Astronomy 127
the cross hairs to the directrix and clamp the external axis.
We then release the parallel plate and set the vernier at 0°
or 180°, when the telescope will be in the true meridian of
the place.
The disadvantage of this method is that it has to be
done at night time. In high latitudes there is twilight all
through the night during summer, and the stars are hard to
see. It takes a long while also, although plotting and other
kinds of work may be done during the interval between the
two stellar positions.
Meridian by Circumpolar Stars
Another method, which may be applied to rough approxi-
mations with a plummet string, but which is a very accurate
and convenient observation when performed with a transit,
is by watching for two circumpolar stars in the same vertical.
It requires the knowledge of the latitude, which can be
Fig. 37.
a is Dubh6 \ p is Merak ; y is Phad ; 8 is Megrez Al Dub ;
e is Alioth ; ^ is Mizar al Inak ; 17 is Al Kaid.
obtained at the same time from Polaris with the transit (see
p. 153), or from two pairs of stars graphically with a plummet
as described on p. 154.
If the azimuth is taken with a plummet-string, the
point of observation is first fixed by a peg with a nail in it,
over which the observer stands with his plummet pointing
to the nail. An assistant has another stake in hand con-
sisting of a piece of 9" x i' board pointed at the end. When
the stars are very nearly in the same vertical the wide stake
Digitized by VjOO . _ _
128 Preliminary Survey
is driven, and at the precise moment the assistant puts in a
nail to the directions of the observer ; a bullseye lantern is
needed to guide the alignment of the nail. The angle of
the line connecting the two nails may then be determined
by a very simple logarithmic sum, as will be presently
described for the accurate method when taken with the
transit.
Adjust the transit as already described (p. 125) with the
magnetic north as a starting-point, and when the two stars
are in the same vertical book their magnetic bearing. This
is all that has to be done with the instrument.
The azimuth of the stars or angle from the pole is then
determined in the following manner for two stars in Ursa
Major.
In Fig. 37 Z represents the zenith, P the pole, HZPR
the meridian of the place, and BC the stars ; ZBCD is the
great circle passing through the zenith and the stars when
the latter are in the same vertical. ZP = 90° — PR, />.
the co-latitude ; hence sin ZP = cos lat. Angle Z in triangle
PZC is the azimuth ; PBC is a spherical triangle which
alters its form very slowly because the fixed stars BC only
move a few seconds a year. In the spherical triangle ZPC
we have ^ Sin Z (azimuth) : sin 3 : : sin C : sin ZP ; or in other
words sin azim x cos lat = sin 3 x sin C. This product
sin 3 X sin C is a constant for the whole year within a
few seconds of arc. Any two stars can be chosen in the
Great Bear, Cassiopea, Draco, or other suitable constella-
tion, and the angle C worked out by the equation given in
the Appendix.
The constants for the two well-known stars /3 and c
Ursae Majoris are given for years 1890 to 1900 at p. 378
of Appendix. This familiar constellation is shown on Fig.
37, B as i3 or Merak. It is the farther one of the two
* pointers,' and Alioth is the third star from it in the train.
Alioth is useful for knowing the position of the pole
■ See Appendix. Formula 78 Rem.
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Geodetic A stronomy 129
star, see * Alioth/ Glossary. The reader should also study
Fig. 122 on p. 395, showing the relative position of all the
important circumpolar stars.
Example. In latitude 51° 16' 45'' observed /3 and c
Urs3e Majoris in the same vertical and west of the meridian,
the magnetic bearing being 318® 27'.
Tabular constant for 1890 + 10 . 19 72910
Cos latitude 979625
Sin azimuth 58° 57" . . . . 9*93285
3600
58° 57'
True bearing 301® 03'
Magnetic 318® 27'
Variation of compass . . . 17° 24' W.
The meridian is found from a previously observed terres-
trial object as already described, p. 126.
The constants for two other stars, not so well known but
easily identified from the sidereal chart on p. 393 are given also
for years 1890 to 1900 on p. 378 of Appendix. As there is
a considerable time interval between their right ascension
and that of the Great Bear, they will serve when the other
is not obtainable.
Meridian by Time Interval of Circumpolar Stars
A very delicate adjustment of the instrument in the
meridian, particularly suitable for a test when it has been
previously adjusted as close as possible by the foregoing, is
to watch the culmination of two circumpolar stars differing
nearly twelve hours in right ascension, such as /3 Cassiopeae
in R. A. 3 min. 2177 sec, and y Ursae Majoris in R. A.
IT hours 48 min. 05*84 sec.
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I30
Preliminary Survey
If the instrument is set to the true meridian and moving
in a true vertical plane, these stars will transit at a sidereal
interval of 15m. 15 •93s. sidereal or 15m. 15*685. mean time,
but if the line of sight be to one side,
as shown by the dotted line, the in-
terval will be less or more. In the
case shown the error would cause
the culmination to be nearly simul-
taneous. In the tacheometer de-
scribed on p. 307, the micrometer
enables" the rate of travel in azimuth
to be accurately measured, and by
a simple proportion the error is
eliminated.
Meridian by Pole Star
Another and extremely useful method is by an observa-
tion of the pole star at elongation. This luminary per-
forms an apparent orbit of an extremely small radius round
the pole, but having twenty-four hours to do it in like the
rest of the stars. He appears to move straight up and down
for a considerable time when at elongation, and to move
horizontally when at his upper and lower culmination.
He is consequently very suitable for observation at elonga-
tion for azimuth, and at culmination for latitude. There is
a table given in the N. Aim. for finding the latitude by an
observed altitude of Polaris, out of the meridian. It requires
the local mean time to be known at least with some approach
to accuracy ; the maximum error produced by an error of
a minute of time is about half a minute of latitude. By
using this method the latitude and meridian can be deter-
mined simultaneously without any mathematics.
The polar distance of the pole star for January i, 1891,
will be I® 16' 23-1'' with an annual variation of— 18-9". Its
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Geodetic A stronoiny 131
azimuth for any latitude when at elongation is given by
formula 59, p. 376.
sin azimuth=^^" P^^ distance x rad
sm co-latitude
Example, In latitude 51° 16' 45'' N. observed Polaris
at eastern elongation. Required the azimuth.
Log sin polar dist 8*3467853
Rad 10^
18-3467853
Log sin co-latitude 97962456
laOg sin azim 2® 2' 9*21" . . . 8*5505397
To find the mean time at place when the elongations will
take place is a time problem, which subject we must not
forestall. To be done exactly, the hour angle corresponding
to the azimuth should be calculated by formula 70, p. 376 : —
Sin hour angle=cos azim -7- cos polar dist.
The polar distance being so small, the hour angle is
within an angular minute or two of the complement of the
azimuth. Thus in the preceding example the hour angle
would be 87° 57' 52-9", which only differs by 2*1" from the
complement of the azimuth.
This is a sidereal interval which reduced to mean time
by rule on p. 412 gives the mean time from the culmination.
The time of culmination is found from Whitaker or N. A. by
rule on p. 412.
The time of elongation ranges according to latitude from
5 hours 49 minutes to 5 hours 54 minutes from the cul-
mination, which is near enough for ordinary purposes.
The following short table of logarithmic values of sin
polar distance of Polaris will reduce the calculation to the
finding and subtracting the log sin co-latitude.
Table XXL— Z<?§ Sine of Polar Distance of Polaris
1891
1892
1893
1894
1895
8-34678
8-34496
8-34313
834130
8*33947
1896 . . 8-33763
^1^7 . . 8-33579
18^8 . . 8-33395
"899 . . 8-33211
*900 -DigitizeabyC^f^Ie
132 Preliminary Survey
Generally speaking the position of the meridian is only
required to the nearest minute of azimuth. This can be
done in one minute of time by the slide-rule.
Example as above. By slide-rule with the sine-scale in its
initial position we find opposite to sin 38° 44' (the co-
latitude) the value -626. Extend the sine-scale to the right
until the sine polar distance 1° 16' is under the -626.
Then the right hand i of the rule will be found over the
required angle of 2° 2'.
The approximate time of Polaris culmination and other
stars is given in * Hints to Travellers/ p. 153.
All the foregoing nocturnal observations possess the
great advantage of regularity of movement in the celestial
bodies observed ; it is often necessary, however, to obtain the
true meridian en route, and the writer has found himself
compelled to take observations for azimuth more than once
in the day, in order to produce anything like accuracy.
The case was one of peculiarly sharp ravines in a densely
wooded country, so that the bases were necessarily very short.
The sun is the great stand-by for such operations as
these. There is no mistaking him, and if he were only a
little more regular in his movements, he would be all that
could be desired. Nevertheless, with a little calculation,
the
Solar Azimuth
is the handiest and best observation, all things considered,
that the surveyor has at his command. The instrumental
work is done in a very few minutes ; the calculation takes
about half an hour. The formula is applicable also to the
pole star or any other celestial body.
On Fig. 127 of Glossary the angle at Z in the triangle
SPZ is the supplement of the azimuth SZS' taken from
the south point.
Having the three sides, we can by formula 75 at p. 377 of
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Geodetic Astronomy 133
Appendix (expressed logarithmically, and remembering that
Jt- =cosec) find the angle Z.
sin
Our data are : Firsts the latitude of the place approxi-
mately. This is usually computed from the latitude and
departure of the day's run as a correction to the observed
latitude of the day before (see p. 173) ; or if in possession
of previously made maps of sufficient accuracy we can scale
it from them.
Secondly^ the declination, which we have from the
almanack for Greenwich, and which we reduce to local time
from the longitude by account.
Thirdly^ the altitude of the sun. The three sides of the
triangle SPZ, Fig. 127, are all of them complements of these
data.
PZ=co-latitude
PS=polar distance
SZ=co-altitude
whence by No. 75, p. 377
2 log cos \ Z=log sin S+log sin (S— SP) + log cosec
ZP + logcosec SZ-20
Example, At Sevenoaks, April 30, 1890, in latitude by
account 51** 16' 45'' N., observed the altitude of and
magnetic bearing of )0 with transit theodolite. Required
the true azimuth, astron. bearing from N. point, and varia-
tion of compass, Greenwich time being known by chrono-
meter.
hrs. min. sec.
Times by watch
Watch slow on Greenwich Qiean time
Greenwich time of observation .
4
2
20
4
4
45
Os"
7
5
4
3
32-5
2
19
4
S
51 5
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134 Preliminary Surv^
To FIND THE Co-latitude PZ
Latitude by account . . 51° i6' 45"
90
PZ 380 43' 15'
To FIND THE CO-DECLINATION OR POLAR DISTANCE
Decl. at noon by * N. A.* . . 14° 50' 42" N.
Diflf. for 4 hours S minutes . . + o" 3' 5-5"
Decl. at time of observation 14° S3' 47 'S"
900
SP 75° 06' i2'S"
Note, If the declination had been south, SP would
have been =90® + declination.
To FIND THE CO-ALTITUDE OR ZeNITH DISTANCE
Altitudes {fi-T%'fo"
2)56<» 58' 50"
28° 29' 25"
Refraction (see Glossary) . . — o^ i' 44"
Semidiameter(N. A.) . .+ 0° i5' 55"
Contraction „ . . - 0° o' i"
Parallax (see Glossary and N. A. ) . . + 0° o' 9"
True altitude of centre . . .28° 43' 44"
Whence SZ= 90°-28° 43' 44" = 6i° 16' 16"
and S-?5±5£±^ = 87*' 32' 5i7".
2
Log sin S . = 9-9996021
+ logsin(S-SP) . . = 9*3334351
+ log cosec ZP . . = 10 2037544
+ logcosecSZ =iox)S7048o
395938396
20
2)19-5938396
Log cos I Z . . . = 97969198= 5 1<* 12' 28"
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Geodetic A stronomy 135
Whence Z . . . = 102° 24' 5^"
i8o«
Azim. from south point . « 77° 35' 04'
180°
Astron. bearing from north point «■ 257° 35' 04"
The magnetic bearings of the sun's |0 at the two altitudes
were taken at the same time, as described on p. 126.
1st observation .
2nd observation
274"
274°
33'
54'
2)549°
27'
274°
43'
15'
30"
55"
274°
257°
59'
35'
25"
04"
Semidiameter .
■ Mag. bearing (J)
Astron. bearing from N.
Variation of compass . 17® 24' 21" W.
NoTA Bene
When the magnetic bearing is in excess of the true
bearing reckoned clear round from the north point, the
variation is that much west ; and vice versd east.
The meridian can be also obtained by equal altitudes of
the sun, similarly to the method by a fixed star. Owing,
however, to the sun's position in the heavens having made
a sensible change during the period of observation — i.e, that
which is due in reality to the earth's orbit, not its rotation —
a correction termed the equation of equal altitudes has to
be calculated at some little length.
Since the methods given are all of them handier and
practicable at any time when the solar equal altitude could
be taken, this latter will not be gone into for want of space.
The rough and ready method described on pp. 39, 40 with
the plane-table is practically this problem, without, however,
applying the correction.
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^
136 Preliminary Survey
For nocturnal observations a good directrix is made
from a piece of two-inch board with a circular hole three
inches diameter in it, and a wire tacked across
it up and down. It is convenient for attach-
ing a lantern behind. It should be driven
^^^^.^^ firmly into the ground as far away from the
Fic. 39. instrument as possible.
Method of determining Local Mean Time and
Longitude
The reader should first peruse the following definitions in
the Glossary — Apparent time, Astronomical time, Civil time,
Mean time (which is the same as mean solar time). Sidereal
time, Equation of time, Hour angle, and Longitude.
The subjects of time and longitude are very closely
allied.
If we have Greenwich time by chronometer or other-
wise, the difference between it and the local mean time is
the longitude in time. This will have become plain from
the preliminary remarks on p. 118 of this chapter.
The tables for converting angles of the earth's rotation
into their time-equivalents are all of course based upon
twenty-four hours being equal to 360°, unless the centesimal
system of 400° is used, which facilitates calculation, but is
not adopted in this work on account of its novelty.
The tables are given in Chambers's * Mathematical
Tables ' and Raper.
By the Slide-rule to reduce Arc to Time
Arc Time
/ Degrees give . . . minutes
Multiplying by 4
Minutes „ . . . seconds
Seconds „ . . . thirds
Example, Reduce 157° 25' 32'' to its time equivalent.
Write down the angle on one side as follows, and placing
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Geodetic Astronomy 1 37
a I of the slide over a 4 of the rule write the equivalents
down opposite.
Arc
Time
min.
hrs.
min. sec.
157°.
. «628.
seconds
. =10
28
25'.
. =100.
thirds
. =
I 40
32". .
. -128.
Answer .
. =
2-13
10
29 42-13
The hours could have been obtained by dividing the
degrees by fifteen. The seconds of arc are generally ex-
pressed in seconds and decimals of time, and are therefore
also more quickly reduced by dividing by fifteen.
To REDUCE Time to Arc
Time Arc
[Hours give . . degrees
] Minutes ,, minutes
I Seconds ,, . . seconds
[Thirds „ . . . thirds
Multiplying by 15
Example, Reduce 10 hrs. 29 mins. 42*13 sec. to its
arc-equivalent.
Time Arc
10 hours X 15 150°
29 minutes x 15 = 435' . . . = 7° 15'
40 seconds x 15 = 600" . . . = 10'
2'i33 seconds x 15 = 32" . . = 0° o' 32"
157° 25' 32"
When we have ascertained the true meridian, the
simplest method of determining the local mean time is by
the
Culmination of a Fixed Star
Any star within the range of the vertical limb of the
instrument will do. Stars of small declination, like those in
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138 Preliminary Survey
Orion, appear to pass the field of view much more quickly
than those near the pole. When culminating at Greenwich
an error of one minute of arc in determining the meridian
would correspond with a time error of three to four seconds,
whereas a star close to the pole would take twelve to fifteen
seconds to make the same change of azimuth ; the former
consequently yields the greater precision. An error of four
seconds of time corresponds with one angular minute of
longitude.
Rule, Find the star's right ascension and add twenty-
four hours- to it, if necessary. Find the sidereal time at
Greenwich mean noon either from Whitaker's Almanack or
N. A. ; in the Nautical Almanac it is called the sun's mean
right ascension at Greenwich mean noon.^ Correct the
sidereal time thus found for the longitude by account,
by rule on p. 142. Subtract this result from the star's
right ascension. Remainder will be the interval in sidereal
time between mean noon and the culmination. Reduce
this interval to mean time by rule p. 412. Final result is
local mean time of culmination.
Example, On December 31, 1890, in longitude by
account 157° 25' W., find the local mean time of the culmi-
nation of Polaris.
hrs. min. sec.
Sid. time M. N. Greenwich, N. Aim. 18 39 27
Diff. for I hr. 9*86" x Ion. in time 10*49 l^rs. = — o i 43*4
Sid. time M. N. at place . 18 37 43-6
R. A. of Polaris i 18 52*05
Add 24 o o
25 18 52-05
Sid. lime M. N. at place . • 18 37 43*6
Sid. interval from mean noon . . 6 41 08*05
Correction subtractive, see p. 412 . o i 5*7
Mean time of culmination . . . 64002*75
' See * Sidereal Time,' Glossary.
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Geodetic Astronomy
139
Local Mean Time by Solar Transit
The actual culmination of the sun can be observed
when not too near the equator by means of the diagonal eye-
piece. The meridian having been determined by a solar
observation for azimuth on the same day or by the stars
the night before, local mean time can be obtained in a
similar way to that just described for a star-transit. If the
sun souths too near the zenith we have to take an obser-
vation for hour angle in apparent time, as will be presently
described, but if we can actually observe the culmination
we can save mathematics by merely applying the equation
of time as explained on p. 399. The equation of time
cannot be perfectly exact unless we know the longitude in
time, which is just what we want to know, because the cul-
mination is only given in the ephemeris for Greenwich noon ;
but its maximum variation is i^ seconds per hour and is
sometimes a small fraction of a second per hour. If there-
fore we know our position within one hour, that is 15° of
longitude, which is over a thousand miles at the equator, it
could not make a greater error than i\ seconds of time.
The equation of time approximately follows the time of
year, and the following rough guide may be useful to fix upon
the memory the positions of maximum and minimum varia-
tion of equation of time.
Table XXJI. — Approximate Equation of Time
January Sun after clock
minutes
•31
minutes
to i3i
February I to 1 1 „
>>
>3i
to I4i
February 11 to 28 „
j> •
14
to I2J
March ,,
,, .
'2j
to 4
April I to 15 ,,
>>
4
to
April 15 to 30 Sun before clock
to 3
May I to 15 „
,, .
3
to 3 50 sec.
May 15 to 31
>» •
3 SO to 2^
June I to 15 „
>>
^\
to
June 15 to 30 Sun after cfock
to 3J
July I to 26 „
>> • • ^
■ ^\
to 6A
July 26 to 31
>» • •
Digitized b\
C^oo^gle
I40 Preliminary Survey
minutes minutes
August Sun before clock . . 6 ' to o
September ,, ,, . . . o to loj
October ,,„... \o\ to 16^
November ,, „ . 16^ to 11
December i to 25 „ ,, . .11 to o
December 25 to 31 Sun after clock . . o to 3 J
The surveyor can hardly be anywhere on land nowa-
days where he cannot tell his position within 100 miles from
some point given on an ordinary atlas.
The maximum error in equation of time at say a latitude
of 70° (which is about as far north as he can go) for a
distance error of 100 miles would only be '33 of a second ;
we may therefore safely say that the surveyor can always
obtain, for the purposes of his calculation, the equation of
time, and when he observes apparent noon, by applying the
equation of time he has the mean time at place.
Example. On December 31, 1890, in longitude by
account 157° W., observed the culmination of sun's western
Hmbs at iih. 58m. 12s. and eastern limb i2h. om. 34s. ; what
was the error of the watch on local mean time ?
Longitude in time 10*43 hours behind Greenwich.
hrs. min. sec.
Equation of time at Greenwich, December j. q , ,5
31, mean noon ^
Diff. for I hour i '19 sec. x 10-43 hrs.. .-00 12*4
Equation of time to be deducted from ap-
parent time o 3 3*6
Apparent noon is 1200
Deduct equation at place . . . . o 3 3*6
Mean time of transit, sun's centre . . 11 56 56*4
hrs. min. sec.
By watch, western limb . . 11 58 12
„ eastern limb . . 12 o 34
„ transit of centre • • • 11 59 23*0
Watch fast on local mean time . . . o 2 '26*6
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Geodetic A stronomy 1 4 1
When the sun souths too near the zenith, we can obtain
the local mean time by a
Solar Hour-angle
y
This observation may be the same as the one for azi-
muth. If the reader will refer back to that problem on
p. 132 it will save space to use that description up to the
finding the three sides of the triangle SPZ.
PZ=co-latitude.
PS = co-declination.
SZ=co-altitude.
Angle SPZ, or for shortness P, is the hour angle because
it is the angle of rotation at the pole between the position S
where the sun is observed and the position S' which it will
have on the meridian, and by formula 75 at p. 377 of App.
we have,
2 log cos \ P=log sin S-hlog sin (S— SZ)-hlog cosec ZP
+log cosec SP— 20.
As on p. 138 SZ=6i^ 16' 16" ; SP=75° 06' 12-5''
ZP=38° 43' 15''
. S, the half sum,=87'' 32' 517''
andS-SZ=26° 16' 357''
Log sin S 9*9996021
Logsin(S — SZ) 9*6461139
Log cosec SP 10*0148468
Log cosec ZP 10*2037544
39*8643172
20
2)19*8643172
Log cos \ P, 31° II' 54*2" . . . 9*9321586
Whence P = 62*' 23' 48*4"
This hour angle of apparent time is first reduced to its
equivaleht in apparent time as follows : —
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142 Preliminary Survey
(i2.^ .... 248 min.
23' . . . .92 sec.
48-4"
hrs.
min. sec.
4
8
I 32
31
4 9 35*1
It is then reduced to mean time by applying the equa-
tion of time in N. A. for Greenwich corrected for the assumed
longitude. The longitude in time '0135 hr. is applied here
merely to show that an error in assumption of 12 miles would
not make an appreciable difference in the result.
min. sec.
Local apparent time as above . . • 4 9 35'i
Equation of time at Greenwich, )
mean noon, April 30 . . '
sec hr.
Long, in time 48*5 x '0135. . o 0*004
o 2 54 '034
Mean time at place 46 41*096
Greenwich mean time by chronometer . 4 5 $2*8
Difference of time o o 48*3
= 12' 5" east longitude.
hrs. min. sec.
Time by watch, see p. 137 . .43 32*5
Local mean time . . . 4 6 41*1
Watch slow on local mean time . . . o 3 8*6
A sidereal hour-angle is obtained in precisely similar
manner from a fixed star, only the arc of rotation, reduced
to its time equivalent, is an interval of sidereal time and has
to be reduced to its mean time equivalent by rule on p. 412.
The time of apparent noon is also obtained by equal
altitudes of the sun by applying the correction of equal
altitudes already referred to, or else it may be obtained from
the equal altitudes of a fixed star described on p. 124 with-
out any correction.
All the previous methods entail the use of a chrono-
meter keeping Greenwich time, which, if done properly, is
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Geodetic Astronomy 143
by far the most satisfactory, and indeed the simplest and
easiest plan.
The surveyor cannot, however, as a rule place very much
dependence upon his semi-chronometer watch. Where he is
able to return to a place at which he has previously fixed the
direction of the true meridian and the geographical position,
he can after any lapse of time recover the true time.
This is a most important matter to bear in mind. It
will often pay to take several days' journey out of the course
to pick up a former station and so readjust the watch,
because with a good timekeeper, although it may suddenly
change its rate from rough usage, it is generally a gradual
acceleration or retardation, and when a cumulative error
has been discovered, it may be distributed proportionally
over the unchecked back work with very close results.
The process is simply to set up the transit in the meri-
dian, and finding the time at place from a stellar culmi-
nation as described on p. 137 determine the error of the
watch over a certain interval of time.
The error of the watch on local mean time can be found
anywhere ; but that does not give Greenwich time. The
object of going back to the known station is to recover
Greenwich time..
We now come to observations for longitude by what
are termed Absolute Methods.
Eclipses of Jupiter's Satellites
These phenomena, visible through a forty-power tele-
scope of one-inch aperture, are timed in the Nautical
Almanac for Greenwich mean time. All the observer has
to do is to watch both the disappearance and reappearance
and reduce the difference of time between the local mean
time of its occurrence and its timed occurrence at Green-
wich into difference of longitude.
The configuration of Jupiter's satellites is given for every
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144 Preliminary Survey
day in the Nautical Almanac for north latitude, and must
be reversed for south latitude. They are also given for an
inverting eyepiece. The first satellite is most rapid in his
orbit, and is therefore to be preferred.
The eclipse being caused by the shadow of the planet
falling upon the satellite, the latter may be at some distance
from the planet when it takes place, so the observer should
be ready a little beforehand. Raper says of this method that
though easy and convenient, it is not very accurate ; the
eclipse is not instantaneous, and the clearness of the air, and
the power of telescope employed, affect considerably the time
of the phenomenon. Observers have been found to differ as
much as 40 to 50 seconds in the same eclipse.
* The observation can only be considered complete when
both immersion and emersion of the same satellite have
been observed on the same evening, and as nearly as pos-
sible under the same circumstances. Thus if the satellite
disappear a little sooner than if the air had been clearer, it
will emerge a little later from the same cause, and the mean
of the two results may be near the truth.'
Lunar Observations
The moon changes her apparent place in the heavens so
rapidly that her R. A. and Decl. are given in the Nautical
Almanac ephemeris for every hour at Greenwich. Her
spherical distances are also given from the sun, certain well-
known fixed stars, and any of the planets to which she may
be near, for every three hours at Greenwich. Consequently
if we observe some recorded phenomena in connection with
the moon, or if we time her culmination relatively to that
of a fixed star, or measure her distance from one of the
registered stars, we have a ready means of finding Green-
wich time. If the ephemeris were absolutely correct, these
methods would be more valuable than they are, but, unfor-
tunately, the moon's apparent motion is so eccentric that it
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Geodetic Astronomy 145
is impossible to register her movements perfectly. In addi-
tiqn to this, the interpolation between the registered times
is far from a matter of simple rule of three. To obtain any
close results, a lengthy process called Bessel's equation for
fourth differences is required, and it is but rarely that the
surveyor is able to afford the time for it. There are, how-
ever, occasions in which even the approximate computation
of the longitude from these lunar phenomena are the only
means at the surveyor's disposal.
Longitude by a Lunar Occultation ^
Occultations of fixed stars by the moon are given in the
Nautical Almanac with Greenwich mean time of immersion
or disappearance and emersion or reappearance. The longi-
tude by account gives a correction for the approximate time
of occurrence at place, and the instrument should be set
up some time beforehand, as the stars are often of small
magnitude and need steady watching.
At the instant of occultation the apparent R. A. of the
moon's limb is the same as that of the star. The calculation
is somewhat lengthy, to remove the effect of the moon's
parallax ; which being done the true R. A. is deduced and
the G. M. T. found.
If the surveyor only has an ordinary six-inch or five-inch
transit-theodolite he will have some difficulty in obtaining a
good observation, but with the * ideal ' tacheometer described
at p. 307 he will see it very well
Any good three-draw telescope attached to a post
will do.
The local mean time has to be first determined by one
or other of the methods given on pp. 136-140.
* A good illustration of this method is given in Raper's * Naviga-
tion,' p. 305.
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146
Preliminary Survey
The Lunar Distance *
This observation for longitude has to be taken with the
sextant, and will therefore only be explained in principle.
Let Z be the zenith and M'S' be
the observed lunar distance from a
star, M'Z and S'Z being the zenith
distances or complements of the
observed altitudes. If M'S' were
the true distance, we could find
Greenwich time by interpolating
between the registered lunar dis-
tances in the Nautical Almanac.
But inasmuch as M' and S' are not the true places on
account of refraction, parallax, and semi-diameter, the dis-
tance is really MS. We can make the corrections MM' and
SS' and then making S equal the sum of the true altitudes
H + H', and s equal the sum of the apparent altitudes >^-|-^',
we have by spher. trig. (Chambers's *Pract. Math.' 751^),
sin2 \ MS=cos2 ^ S -
cos H . cos H' . cos 1 (j + M^S Q. cos ^ (x-M^S^)
cos h . cos h'
Terrestrial Difference of Longitude
"When two points are correctly determined as to latitude
but the longitude is doubtful, their difference of longitude
can be computed as follows,
providing that they are visible
from one another.
Let A, B be the stations,
AP, BP their co-latitudes, the
angles A and B their recipro-
cal true azimuths, and APB,
or P, the required angular
YiG,^x, difference of longitude. As-
' Examples of this method are given in Raper's * Navigation,* p. 283,
Chambers's Math. p. 449, and agraphia method in * Hints to Travellers.'
Geodetic Astronomy 147
sumine the earth to be a sphere we have (by formula 81,
p. 378)-
cot \ P =sin^(a+g ^^ i (A ^B)
Longitude by Moon-culminating Stars
The Nautical Almanac furnishes the R. A. of the moon's
bright limb for the lower as well as the upper culmination,
marked L. C. and U. C. respectively.
It also gives the variation in R. A. in one hour of longi-
tude, that is the variation in her transit over two meridians
equidistant from Greenwich and one hour distant from one
another. The figures are calculated from the right ascen-
sion of the bright limb and include the effect produced by
the change of the semi-diameter.
If we can determine exacdy what the right ascension
of the moon is at any time and place, we can find, by
interpolating between the values for the two nearest hours
in the Nautical Almanac, the Greenwich date corresponding
to our local time of observation.
The method adopted is to watch the successive culmina-
tions of the moon's bright limb and some fixed star close to
her whose right ascension is known. The Nautical Almanac
gives a list of such stars which are peculiarly suitable from
their position for this purpose. They are generally small
however, and need some little practice in star-gazing to
identify them. They are chosen with as short a time
interval as possible and nearly on the same parallel.
Very correct results are obtained by this method when
the distance between the meridians is not great.
The best plan is by corresponding observations at two
places on the same night.
Such a case is the following example taken from Pro-
fessor Loomis' 'Astronomy :* Let the right ascension of the
moon at the two meridians be A and A', from which we
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148 Preliminary Survey
know the moon's motion in R. A. during the interval of the
two transits A' A.
The almanac furnishes the variation of the moon's right
ascension corresponding to one hour, which we will repre-
sent by V.
We shall therefore have the proportion —
V : A'— A:: i hr. : the difference of longitude.
Example i. The R. A. of the moon's first limb Sep-
tember 6, 1840, was observed at Washington to be 19 h. 21 m.
29-90S. \ and on the same day at Hudson, Ohio, i9h. 22m.
972 s. Required the difference of longitude of the two
places.
Here A' — A=39*82sec.
That value of V must be taken which corresponds to
tjie middle of the interval between the observations, which
is found by interpolation to be 135*55 sec.
i35'55 sec. : 39*82 sec.:: i hr. : 17 m. 37-56 sec. .
An accurate method of determining the longitude upon
this principle, both for distant as well as near meridians,
involving lengthy calculation, may be found in Professor
Chauvenet's * Spherical Astronomy.'
Methods of determining the Latitude
The simplest, most accurate, and most rapid of all the
astronomical calculations at the disposal of the surveyor are
those for finding the latitude. The observer does not require
to know his meridian or his local time, all he needs is a
sextant or a transit. In fact, as shown later on, he can even
obtain an approximation to the latitude with a plummet-
string.
In the prefatory remarks to this chapter it was shown
that an. altitude of the true pole is equal to the latitude of
the place. If the pole star were exactly at the pole, we
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Geodetic A stronomy 1 49
should merely have to take its altitude in order to ,get our
latitude. The pole star is not exactly at the celestial pole,
so it has to make its own little circle of apparent rotation
round the true celestial pole. It will be seen on examining
Fig. 127, Glossary, that no star whose polar distance is less
than the latitude can set, neither can any star whose polar
distance is greater than 90*^ + co-latitude ever become visible
above the horizon.
The pole star crosses the meridian in the Northern
Hemisphere twice in the 24 hours like all the rest of the
stars. It is in fact a circumpolar star. When at its upper
passage, its altitude will be 1° 16' 42" greater than, and
when on its lower passage as much less than, the latitude.
Similarly when we know the declination of any other
star we can tell the latitude from a meridional altitude.
All we want besides is to know what side of the globe we
are on, and in which direction, north or south, the body is
culminating.
In north latitude when the polar distance is equal to the
complement of the latitude, the body culminates at the
zenith (Z, Fig. 127) because then the altitude=ZP-i-PR.
When the polar distance is greater than the co-latitude,
it culminates to the south, and when less to the north ; and
when equal to the supplement of the latitude, the body
only touches the horizon when making its meridional pas-
sage. Its diurnal path is represented by HT (Fig. 127).*
When the polar distances are as above in south latitudes
the culminations are in the opposite direction.
When the polar distance is 90°, or in other words when
the body is on the celestial equator, its meridional altitude
EH (Fig. 127) will be the complement of the latitude, because
HEPR=i8o° and EP=9o°, therefore HE + PR=9o°.
The polar distance is either the complement of the
decHnation S'P or 90°+ the declination UT, and the
following rules are deducible from Fig. 127, for obtaining
> The paths of the celestial bodies are shown diagrammatically.
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1 50 Preliminary Survey
the latitude in either hemisphere, when we know the decli-
nation and altitude on the meridian.
Rule I. In the Northern Hemisphere.
(A) When a body culminates to the south of the
observer.
(i) When the declination is north. Subtract the declina-
tion from the altitude, this will be the co-latitude:
HE=S'H-S'E (Fig. 127, Glossary)
(2) When the declination is south. Add the declination
to the altitude ; this will be the co-latitude :
HE=EU'-hU'H
(-B) When a body culminates to the north,
(i) When it culminates above the pole. Subtract the
co-declination from the altitude. This will be the latitude :
PR=VR-VP
(2) When it culminates below the pole. Add the
co-declination to the altitude ; result is the latitude :
PR=V'R+VT
Rule 2. In the Southern Hemisphere.
(A) When the body culminates to the north of the
observer.
(i) When the declination is south. Subtract the decli-
nation from the altitude ; this will be the co-latitude :
QR=:S''R-QS'' (Fig. 128, Glossary)
(2) When the declination is north. Add the declina- .
tion to the altitude ; result will be the co-latitude :
QR=U/'R-hU"Q
(B) When the body culminates to the south of the
observer.
(i) When it culminates above the south pole. Subtract
the co-declin. from the altitude ; result will be the latitude :
OH=V'H-V'0
(2) When it culminates below the south pole. Add the
co-declination to the altitude ; result will be the latitude :
OH=OV-fVH
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Geodetic Astronomy 151
At the equator, all bodies having north declination cul-
minate to the north and all bodies having south declina-
tion culminate to the south, so that when near * the line '
there can be no mistake as to which rule to adopt. The
meridional altitude of a star near either north or south pole,
such as a Urs3e Majoris or a Crucis, will at once show what
side of the line we are on.
The most accurate method of obtaining the latitude is
by a fixed star, but, all things considered, the most useful
celestial body is the sun. The moon or one of the planets
will all serve the purpose as well as a star ; the only diffe-
rence is that the declinations of the bodies of the solar
system vary from day to day and we have to know, approxi-
mately at least, what the Greenwich time of our observation
is* in order to get the correct declination from the almanac.
The declination of the fixed stars is only very slightly
variable from year to year, so that one of them will serve
our purpose just as well if we have not the most remote idea
of our longitude. But the great value of the sun arises from
its being a daylight observation, and to any one not familiar
with the stars an unmistakable object. There are but few
cases in which we cannot obtain the longitude by account
near enough for determining the decHnation.
The formidable array of logarithmic figures, which are
so associated with astronomical problems, convey the impres-
sion of a complexity which does not in fact exist, at least as
far as latitude is concerned. There are no trigonometrical
calculations at all about obtaining the latitude. The first
operation is to free the observed altitude from errors due to
dip (if the visible horizon is used), parallax, and refraction,
and to correct it for semi-diameter, for all of which terms
see explanation in Glossary. The second operation is to
reduce the recorded time to Greenwich time from the longi-
tude, as far as it is known by account. The sun's maximum
variation of declination is only about an angular minute per
hour, and an hour corresponds with 15° longitude, so that
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152 Preliminary Survey
we must be a very long way out in our reckoning for longi-
tude for it to make any serious difference in the calculation.
The third operation is to calculate by a simple rule of
three sum the declination at the time of observation as re-
duced to Greenwich mean time, and finally to apply one of
the foregoing rules to the particular case.
Latitude by Circumpolar Stars
An exceedingly exact though lengthy method of obtaining
the latitude is by observing the upper and lower culminations
of a circumpolar star. This can be done without knowing
the meridian, the declination, or the time^ because it is suffi-
cient to watch for the maximum and minimum altitudes of the
same star. Wfien both observations are corrected for refrac-
tion the mean of the two altitudes is the altitude of the
VR-hV'R
celestial pole, that is the latitude of the place : — =
PR (Fig. 127, Gl.).
Generally the time is known and the true meridian can
be readily ascertained. In this case the operation is very
speedy, for the altitude is measured at either of the culmina-
tions and the declination applied as already explained.
Latitude by Meridional Altitude of a Fixed Star
On February 5, 1889, observed the meridional altitude
of Sirius in Honolulu, north latitude, star culminating to the
south.
Observed altitude 52° 08' 2.0"
Refraction 0° o' 45
S. Declination (N. A. ) .
By Rule i. A. 2, Co-latitude .
- Latitude 21^ 18' 33" N.
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52°
07'
35'
16°
33'
52"
68°
41'
27"
90°
0'
0"
Geodetic Astronomy
153
The latitude by a meridional altitude of the sun, which
is the sailor's stand-by, is identical in principle, only subject
to correction for parallax, which is insensible in the case of
a fixed star.
The following example is taken from Raper's 'Navigation,'
illustrating the calculation made on board ship with the
sextant. May 3, 1878, longitude 38** W., obs. mer. alt.^
56° 10' to the southward ; ind. corr. A- 2' ; height of eye 20
feet ; required the latitude.
Decl. 3rd Table 60 or N. A. . .15° 43' N.
Corr. for 38*^ W + 0° 2'
Obs. alt. Qi
Ind. corr. + 2' )
Dip. -4'f
App. alt.
Refr. - V
Semidiam. + 16
True alt. .
Zen. dist. .
56*^ 10'
- 0° 2'
IS°45'N.
S6« 8'
+ 0° 15'
56° 23
33^ 37' 33° 37'
Latitude 49® 22' N.
This is a case of Rule i. A., but in Raper the zenith
distance is added to the declination for the latitude. An
inspection of Fig. 127, Glossary, will show at once that ZE=
PR, because PE=9o*', and ZR=9o°, and ZP is common.
There are a great variety of methods of determining the
latitude. The foregoing are the most common, and as they
are always available when the heavens are suitable for
obseiTation, we will not take up space with more than an
allusion to two others.
I St. Latitude by an altitude of the sun out of the meri-
dian. In the triangle PZS (Fig. 127, Glossary), if we know
the time, we have the hour angle SPZ. The co-declination
PS we also know, and ZS is the co-altitude. From these
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154 Preliminary Survey
data we can obtain the side PZ, which is the co-latitude.
The demonstration of this is given in Chambers's * Mathe-
matics/ art. 887. The bare formula is No. 78 in the
Appendix.
2. Latitude by two altitudes of the sun or of a star,
and the interval of time between the observations, or the
altitudes of two known stars, taken at the same instant, to
find the latitude of the place. To those fond of mathe-
matical gymnastics of high order this method is recom-
mended. It involves a maximum amount of labour with a
minimum chance of coming out right at the end.
Demonstration in Raper, Chambers's * Mathematics,' &c.
In conclusion, as to observations of meridional altitude,
the pole star and circumpolar stars have this great
advantage that they are longest at culmination, giving good
time for a series of altitudes, and the vital necessity of
repeating the observation face right and face left, as
explained in tacheometry, cannot be too strongly urged.
The subject of latitude will be concluded with a method
of obtaining it without any instrument at all.
Graphic Latitude
The following simple method of obtaining an approxi-
mate latitude is described by Mr. Coles of the Geographical
Society, and is given here somewhat in his words with a
few additional explanations and practical cautions, but the
diagram and example are independently worked out.
By observing the times when two pairs of stars are
vertically above one another the latitude can be obtained
by graphic construction in a very short time. All one
requires is a plummet-string, and a table of mean positions
of fixed stars, such as is given in Whitaker's Almanack.
The operation should be repeated once or twice, and the
results meaned. It requires a time interval of from four to
eight hours, and a timepiece which is at least accurate to a
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Geodetic Astronomy
155
few seconds for that interval, although it may be quite wrong
on local mean time. The positions of the stars are pro-
jected stereographically upon the plane of the celestial
equator ; the circle marked primitive being the equator to a
radius i. P, the centre of the circle, is the projection of the
pole. The first point of Aries is marked Tj and the other
quadrants of right ascension 6 hrs., 12 hrs., and 18 hrs.,
indicated by their numbers. The projections of the vertical
circles through the stars are obtained as will be explained,
Fig. 42.
and the intersection of these circles being the zenith, the
arc ZP, which (Fig. 127, Gl.) is the co-latitude, gives us the
latitude of the place.
It will be seen that the circles through the stars are
much flatter than the primitive, and it is important to have
them cut one another as nearly at right angles as possible,
so the first pair of stars should be as nearly east, and the
second pair as nearly west, as can be conveniently chosen,
with as long a time interval as possible.
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1 56 Preliminary Survey
One of the stars may belong to both pairs as in the
example, but not necessarily so.
We will explain the process direct from the example.
Betelgeuse and Sirius were observed with the plummet-
string when bearing about S.E. by S. ; and after nearly six
hours, when Sirius was nearly setting, it was observed plumb
under Regulus, bearing about W.S.W. The four positions
of the stars are marked S and B for the first pair, and S'
and R for the second pair. The projections of right
ascension are measured by the chords of the angles on the
primitive from ^ . The projections of polar distance are
measured by the tangents of half those arcs from P upon the
radial lines drawn to the positions of the stars in R. A. Thus
by slide-rule the chord of R.A. of Betelgeuse=2 sin \ angle,
and the arc corresponding to 5hrs. 49m. 10 sec. =87** 12'
30''. Placing the extremity of the sine-scale under the 2,
we read for 43® 36', 1*379, which is laid off from nr. The
declination of Betelgeuse is 7® 23' N., therefore polar
distance=82° 37^ The tangent slide in its initial position
gives for 4 1*' 18^' -878, and this we lay off from P. The scale
of the plot was radius = i decimetre, so that the bevelled
edge of the slide-rule served for all the scaling. The dia-
gram as printed has been reduced to one-fifth its original
size. The first pair of stars being projected, the second
pair are treated similarly, except that their right ascension
is decreased by the amount of the sidereal time interval
between the observations. The projections of the vertical
circles are shown on the diagram for both pairs, and their
intersection Z, but only the locus C of the second pair is
shown, and the auxiliary dotted lines by which it is deter-
mined in order to avoid confusion.
Through R draw Rpg, and lay off the perpendicular
HPE. Draw eRd and dpf. Draw efg' cutting Rpg pro-
duced in g'. Bisect Rg' and draw the locus perpendicularly
through its centre. Bisect RS' and draw the perpendicular
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Geodetic A stronotny 1 5 7
from it to c in the locus, which will be the centre of the
projection of the vertical circle RS'.
Draw circle S'R, and similarly, from a fresh locus, the
circle SB cutting S'R in z the zenith, zp measured by
same scale gives the semi-tangent of the co-latitude ; being
the projection of a great circle passing through the pole,
similarly to the circles of polar distance pS', pR, &c.
At Kukuihaele, Hawaii, in latitude 20° 8' 9" by account,
hrs. min. sec.
Observed Betelgeuse and Sinus plumb at . 7 39 15
„ Regulus and Sirius plumb . . 13 26 15
Mean time interval . . 5 47 o
Correction by slide-rule (see Sid. Time — Glossary) + o 57
Sidereal time interval • 5 47 57
hrs. min. sec
R. A. of Betelgeuse 5 49 10
„ ,, Sirius 6 40 15
,, ,, Regulus, less sid. time interval . 4 14 31
„ ,, Sirius, less sid. time interval . o 52 18
Declination of Betelgeuse 7° 23' N. and P. D. =82® 37'
,, „ Sirius . 16® 34' S. and P. D. = 106*^ 34'
„ „ Regulus 12° 30 34" N.; P. D. = 77'* 29' 26"
ZP = semi-tangent of co-latitude = by scale
of diagram '698 .... =tan. 34*=* 55'
2
Co-latitude 69° 50'
90° o'
Latitude 20® 10'
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158 Preliminary Survey
CHAPTER V
TACHEOMETRY
Tacheometry, or rapid measurement, is a term which
signifies more than one kind of measurement. It may be
defined as telescopic surveying, and has for its aim the pro-
duction of a correct map with the least possible assistance and
in the least possible time. It performs at one operation the
measurement of distance formerly only possible with the
chain, and the measurement of elevation formerly made
with the level and staff.
The word is sometimes written tachymetry or takimetry.
The French also have a word * takitechnie ' or * rapid art,'
which includes the use of the slide-rule. The principal dif-
ference in the methods of tacheometry is in the mode of
mapping. If the plane-table and stadia are used, the map-
ping is done in the field, but if the transit and stadia are
used, the mapping is done at home from field notes.
The instruments used are very numerous, some of them
highly complicated, and many of them exceedingly ingenious.
Their names often affect an originality of principle or an ex-
tent of accomplishment which does not really belong to them,
whilst they fail to classify them according to their organic
type.
The measurement of distance performed optically is the
essential principle of the stadia, and places it in the front
rank of all tacheometers.
There should be a means of distinguishing those instru-
ments which measure angles for the calculation of distance
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Tacheometry 159
— such as the Hadley sextant when used in hydrography ;
those which demand the measurement of a base by tape,
chain, or pacing, Hke telemeters and range-finders ; or those
which require the use of a rule of three sum or reference to
a table, like Eckhold's * omnimeter,' or the micrometer tele-
scope — from those in which the distance is practically seen^ or
at least read at a glance from a graduated rod. This latter
achievement, though not by any means new in principle,
deserves a special and typical designation in its modern form.
The word metroscope, forming a good third with tele-
scope and microscope, from metron a measurement and
skopeo I behold, would express more satisfactorily the stadia
principle.
Mr. B. H. Brough, in his paper in the * Min. Proc. Inst.
C. E.' vol. xci., attributes the invention of the stadia to
Mr. William Green in 1778. It was used with a simple tube
. having in its field three horizontal wires, and was used upon
the principle that since, rays of light travel in straight lines
except as diverted by refraction, the distance of an object is
sensibly measured by the extent of the optic retina which is
covered by it.
Thus, supposing we see a man and a boy standing
together 400 yards away from us, and the image of the boy
upon the eye occupies three-fourths of that of the man ;
then, if the boy comes forward so as to be 300 yards from
us, he will appear to be exactly the same height as the man.
If we arrange the two extreme horizontal wires in a plain
tube so that 20 feet away from us a graduated staff will show
2 feet subtended by the wires, then at a distance of 40 feet
they will subtend 4 feet and so on.
If we term the distance of the hairs from the eye * hair
distance,' and the distance of the staff * staff distance,' also
the distance apart of the hairs * hair height ' and the space
subtended by them on the staff * staff height ,' we have by
simple proportion —
Staff distance : staff height :: hair distance : hair height
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i6o Preliminary Survey '
from which we can calculate the distance of the staff when
we know the other three factors. If, however, we graduate
the staff so that one subdivision represents lo or loo of hori-
zontal measurement, and if we then direct the lower hair to
an even figure on the staff, we can simply read the distance
from the staff.
When telescopes are used (and the lens-power is the
backbone of stadia measurements) the factor of *hair
distance ' varies with the focus, so that we have no longer a
single ratio as a multiplier for determining the distance, and
to avoid making a calculation every time, we have to reduce
the varying ratio to a single ratio plus a constant, or else an
Fig. 43.
optical device is resorted to in the instrument itself to
produce the effect described on p. 392.
Fig. 43 applies to a horizontal sight when the staff is
vertical, or to any sight when the staff is held at right
angles to the line of sight.
Let s be the intercepted height on the staff ; /, the height
of its image ; a, the distance of the staff from the object-
glass ; X, the distance of the image from the object-glass ;
f, the focal length ; d, the distance of the axis of the instru-
ment in rear of the object-lens ; A, the distance of the staff
from the axis of the instrument.
Then by simple proportion, as before, we have
^=^ (I)
X t ,
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Tacheometry 1 6 1
But the general formula for the foci of lenses is
^ X a
This multiplied by ^, becomes
--+ I =-^ (2)
and substituting for - on the one side of the equation its
value from (i) we have
a^Mf\
or,
xS
h^a-\-d^f-A-f^d . . :' -. . (3)
/-; is a simple ratio, and /+ ^ is a constant, which ranges in
different instruments from 12 to 24 inches. < is generally
constructed=ioo, that is to say for a fourteen-inch telescope
the hairs are placed one seven-hundredths of an inch above
and the other seven-hundredths below the axial hair. An
amateur will find the best way to do this, to mark the lines
on the brass diaphragm with a needle point, as explained
on p. 312, then to fix one hair permanently with shellac
but to fasten only the extremities of the other hair at the
outer edges of the diaphragm ; this will permit of a slight
adjustment of the hair to fix it finally by drawing it up or
down at the inner edges of the diaphragm with a quill
tipped with shellac. If the measurement of the spaces is
carefully made, it will only require to be moved by about
its own thickness, if at all. This may not leave the hairs
perfectly equidistant from the axial hair, but that is of
small importance when the two distances are known. Most
surveyors who put in their stadia hairs put them in at
haphazard, and find their value by experiment, reducing
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1 62 Preliminary Stin^ey
the distance by slide-rule, but by so doing they sacrifice for
the sake of a very little extra trouble the most valuable
feature of stadia measurement, and open to themselves a
fertile source of error. The diaphragm will hardly ever
have to be taken out more than once to finally fix the
hairs.
The constant of /+^ with a fourteen-inch telescope in
which the vertical axis was eight inches from the object-
glass would be 22 inches or i-8 feet ; but inasmuch as the
readings are only taken to even feet, the constant to be
added would be 2 feet, or with a ten-inch telescope hung
concentrically it would be 1*25 feet ; but unless at very close
distances, only i foot would be deducted. This requires
no entry in the book ; it is usually added systematically
down the whole row of figures as they are entered by the
recorder.
The principle of all telemetry or tacheometry is triangu-
lation, but it may be otherwise styled the determination of
parallax. This is a great word amongst astronomers, for by
it the whole solar system is plotted to scale, and it serves as
a complete illustration of the value of tacheometry. Parallax
is the angle at a distant object which subtends a given base.
The greater the base, and the more exact the angular
measurement, the more correct will the measurement of the
distance be. It is possible to obtain a greater degree oj
accuracy with a small base and high powers of angular
measurement, than with a long base but inaccurate
angles.
For instance, with a range-finder, a base is run out as
explained on p. 338 ; it may be 100 or 200 feet, but if the
angle is taken with a low power and the base roughly
measured, it will not produce results nearly as correct as
those of a stadia telescope of high power when forming upon
a distant graduated staff a base of only a few feet.
The base which the astronomers have for determining
the distance of the sun is vOnly the diameter of the earth, and
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Tacheoinetry 163
that is to the distance itself something Hke a hair's breadth
to a distance of six feet.
Another analogy that should be borne in mind is that
exactness is always proportionate to magnitude. It is
supposed that the error in the sun's distance does not exceed
125,000 miles; this in itself is a large amount, but in
93,000,000 it only represents y^, which, considering the
size of the base, is marvellously accurate. Porro, who in
1823 introduced the anallatic stadia telescope of high
power, claimed never to have exceeded an error of ^^^^ up
to 660 feet, or yxnny ^^ distances up to 1,320 feet. This
would mean that at a distance of 660 feet, he could read to
one-third of a hundredth of a foot on a staff graduated in
our way. His telescope must have been an uncommonly
good one.
If a preliminary survey is correct to the limits of the
scale, it is all that can be demanded of it, and 100 feet to
the inch is about the largest scale used for that purpose, at
which less than i foot would be practically unscaleable.
Sights should not be taken at distances too great for the
instrument to read the graduation distinctly, and this
limit varies from 300 feet in small to 1,000 feet in large
instruments, beyond which only check sights, or those upon
the accuracy of which the survey does not depend, should
be taken.
Even with large telescopes, 300 feet is about the best
working limit for the length of sights. In the first place, the
reliability of the levels rapidly diminishes beyond that
distance just as it does in ordinary levelling, and in the
second place, detail is sure to be missed.
The value of long shots, either within the range of the
stadia or beyond it, with the micrometer is very great, both
as a check, and also for connecting the survey with useful
outlying points, of which the detail is not required, and
which cannot be got by intersections.
M 2
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1 64 Preliminary Survey
Methods of Taking the Sights
In level country it is convenient to set the telescope
horizontal like an ordinary level and read off all three hairs,
booking the stadia hairs in their column and the axial hair
in the column for foresight or intermediate ; there will then
be no entry under the vertical limb column, and the level
will be reduced by the collimation method. This is only
possible where the staff does not travel outside the field.
When it becomes necessary to use the vertical arc, the
lower hair should preferably be directed to an even foot on
the staff, and that one which will bring the axial hair as near
as possible to the same height above ground at the staff as
the line of sight is above ground at the instrument. For
instance, if the telescope is 5 feet above ground and the
staff is 350 feet away, the stadia being arranged at i in 100,
the lower hair should be at 3*00, then the upper will be at
6-50, or, if there is an instrumental constant of i foot at
6*49, the axial hair being at 4745.
There are two methods in vogue for holding the staff. In
Germany the practice is to hold it at right angles to the line
of sight, in America to hold it vertically. It can be easily
shown that in both figures, 44 and 45, angle Jt:=angle X ; and
calling the difference of the two stadia hair readings S and
S', and the reading of the axial hair H, we have for Fig. 44,
A=Sx -C+^+Z
or for the case of an instrument whose stadia value is i per
100, and constant 1*5 feet,
A=iooS+i-5
B=Axsin X
C=A. cos X + ^=A. cos X + H. sin X
In Fig. 45, S=S'. cos X
.•.A = S'.cosXx4+^+/
Or for similar case.
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Tacheometry
165
A=ioo S'. cos X+ 1*5
B=A. sin X=ioo S'. cos X. sin X
C=A. cos X=ioo S'. cos* X+ 15
The writer prefers the first method for the following
reasons :
I. When the staff is held vertically, the accuracy is
dependent upon the staff-holder, who has to watch a circular
Fig. 44.
bubble or plumb-bob, but the observer has no means of
telling whether the staff is correctly held or not. When the
staff is held at right angles, the staff can be furnished with a
Fig. 45.
piece of wood whose two faces are each three or lour inches
long, one painted white and the other black ; this serves the
twofold purpose of enabling the staff-holder to sight square,
and as a tell-tale to the observer, who by a wave of the
signal-flag makes the staff-holder adjust it with perfect
correctness.
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1 66 Preliminary Survey
2, In the first method, the only correction for steep
angles in the method of reduction is to add to C the distance
/, which, as will be seen, is seldom a scaleable quantity,
and from the short table on p. 167 the surveyor will be able
to calculate mentally whether the correction will be appre-
ciable or not. There is, strictly speaking, a second correc-
tion for B, due to H being shorter in Fig. 44 than Fig. 45
by H— H cos X; but this rarely enters into the calculations.
In the second method the use of cos ^ X is required
which necessitates a specially made slide-rule, or failing a
slide-rule a double calculation. For B also a double mul-
tiplier is introduced in the shape of sin X x cos X. These
formulae have to be applied to all vertical angles.
The metallic slide-rule made by Mr. J. Kern, of Aarau,
Switzerland, is a topographical speciality, which will greatly
assist those who prefer to have the staff held vertically, as
it gives the horizontal and vertical components for that
method. The values of cos^ are given upon a short
slide and sin x cos upon the usual long slide. The
metallic division is not more accurate than that of either the
boxwood or celluloid rules, and the fit of the slide upon the
one examined by the writer was not equal to them. It is
more trying to the eye to read, and it is of course much
more expensive. It has, on the other hand, the advantage
of being more durable and less susceptible to humidity.
3. If the staff is intended to be kept always in a vertical
position, an error on the part of the assistant is much more
fatal to accuracy than when it is kept at right angles to the
line of sight.
For instance, let the stadia read one per cent. ; let the
line of sight be at an elevation of 45°, and let the direct
distance be 300 feet. An error of 3° from the vertical
position of the staff will produce an error of 10*3 feet in
horizontal distance ; whereas the same error from a normal
position of the staff will only produce an error of 0*3 foot in
horizontal distance.
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Tacheometry
167
Table XXIII. — Fumiicnis of Angles in per ceniage of perpendicular
to base.
-o
«S 1
i
-o
*>5
13
»>5
'So bo
Angle in
egrees an
minutes
:entag
ndicul
base
glein
grees
.£§8
111
fii
|l
.5 §2
^■s
g 0)
l-s
1^
< M g
% \^
-o
^s.
-o
^^
•0
^s.
I'OO
1° 0'
i'7
6'oo
6° 0'
10-5
10-77
10° 46'
19*0
1*15
1° 9'
2*0
6-28
6° 17'
u*o
, ll'CXJ
16° 0'
19*4
1*73
'°44
3*o
6-85
6«5i
12*0
11-32
tl° 19'
20*0
2'00
2° 0'
3 5
7*oo
7° °
12-3
1 11*87
' I2"00
11° 52'
21 'o
2-30
2° 18'
4*0
7-42
^n^'S
I3-0
12° 0'
21*2
2-87
^\^^'.
S'o
7-97
2o58
140
12-42
12° 25'
22 'O
1 3*oo
3o i,
1'^
1 8*CX5
8° 0'
14*05
12*95
"0 57;
230
3 '43
3° 26'
6-0
8-53
^o3»;
15*0
13*00
^3° °
23*1
4-00
'^o °;
7*o
9'cx>
9° °
15*8
13*50
13° 30
24 "o
4-58
*o35
B-o
9"io
9o ^
i6*o
14*00
^< °
24*9
5-00
So °
8-7
965
9° 39
17-0
14*03
14*58
'*o ^
25-0
5"i5
5o 9
9-0
lo'oo
10° 0'
17-6
14° 35'
26*0
572
5° 43'
lO'O
' 10*20
10° 12'
i8*o
15 'CO
15° 0'
26-8
Example 1. Required by first method, B and C. —.
being 100 ; f-\-d=r^ ; H=5-o ; 8=3*00 ; and X=572°.
This angle corresponds with a slope of i in 10, which is a
steeper ascent than is ordinarily met with in roads.
A=ioo 8+1*5=301*5
B=AxsinX=3oi*5 x*o996=3o*o3
/=io per cent, of 5 feet=*5o feet
C=Aycos X+/=3oi-5 XO-995 + -5
=299*99 feet +*5
=300-49 feet
Note. B' is the height that is actually needed for the re-
duction of the levels, so that B would be, strictly speaking,
subject to an addition of H— H. cos X=o*oi feet, making
B'=3o*04.
Example 2. Required A', B', and C in Fig. 45 ; the
data being the same, except that H will become 5*02 nearly
and 8' = 3*02 nearly. At that distance only the nearest
hundredth can be read to, so the results cannot be made
exactly to correspond with example i ; the difference of 0*1
in 8' makes a difference of -i ft. in B'.
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1 68 Preliminary Survey
A'=ioo S^ cos X+i-5
= 100x0*995x3 '02 + 1 '5 =302*00
(which is practically A + /)
B'=(ioo S'. cos X+i-5) X sin X
=302 X o*995 X 0*0996
=30*08
C=iooS'. cos2X+i*5
=302 X 0*995 X o*995 -I" I '5
=300*5
It is true that by a specially constructed slide-rule, the
multiplication by cos* and by sin x cos is as simple as
that by cos and sin ; but beyond the limits of 300 feet with
angles as steep as that in the example, the slide-rule will
not give the result to a tenth of a foot in elevation, and
therefore most of the turning sights and many others also
where special accuracy is needed have to be reduced
numerically, and then the extra labour of the second method
is felt.
Usually a limit of accuracy is required of about one foot
of elevation and ten feet of distance. This upon a scale of 4
miles horizontal and 100 feet vertical per inch would not be
scaleable, whilst on scales of 400 feet horizontal and 40 feet
vertical per inch they would represent '025 inch, which is not a
large amount, and this accuracy can be attained by beginners.
The field work performed with the tacheometer alone,
apart from the auxiliary work of contouring and plane-tabling
detail, consists of survepng and levelling, and though they
are performed simultaneously we will consider them sepa-
rately ; and, first — ^
Surveying
The greater part of the measurements are independent
rays whose bearings are read from the horizontal limb. Great
care is needed to avoid misreadiiigs, as there is but little
opportunity for checking them. Some useful checks are
practicable which will be presently described, but the chief
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Taclieornetry 169
means of keeping the general direction true is the observation
for azimuth described on p. 136, and a frequent comparison
with the compass-needle.
When the distance of any point is measured by the stadia
and its bearing also taken it can be plotted either by a pro-
tractor or by latitude and departure. Before commencing a
survey the true meridian is acertained and some convenient
reference-mark or 'directrix' chosen. Then the stafF is
carried fb all the points in succession which are to be shown
on the plan, and finally to the new instrument station, the
observations to which are taken with extra care, and a check-
sight taken on the directrix before shifting the instrument.
It is advisable, if the instrument is fitted with two verniers
to the horizontal limb, to remove one of the microscope
* readers ' during all the observations so as to avoid getting
on to the wrong vernier.
When adjusted at the new station, the horizontal limb is
clamped to the same bearing, entered in the book from the
first to the second station, and the telescope is directed by
means of the external axis to the staff held at the first station.
The reader is then transferred to the opposite vernier so as
to give the bearing ± 180° and so replace the zero of the
horizontal limb in correspondence with the meridian.
It may be mentioned that in some countries the south
point is called 360° and the north point 180°.
If the bases are short it will be necessary to align the
telescope from station to station by a plummet, because the
staff being three or four inches wide will give a margin of
error.
At the new station the staff is read a second time by the
stadia as a backsight, so that all the primary lines of the
traverse are measured twice over. // must be borne in mind
that the instrument cannot be carried forward as in levelling
ahead of the staff to take a backsight. At every fresh instru-
mental station the staff and instrument must change places.
The station must be clearly marked. A good stout peg should
Digitized by v__ _ _ __
i
I/O Preliminary Survey
be driven down within an inch of the ground and a reference
stake about three feet long driven a couple of yards away,
having the number or letter of the station chalked on it.
Where the intermediate staff stations are at defined points,
such as comers offences, it will be sufficient to describe them
in the column for remark, but if they are undefined they
should be marked by smaller stakes or builder's laths. A
convenient notation on a continuous traverse is to name the
instrument stations by the capital letters of the alpfiabet to
the end, and then the small letters, after which it will
be safe to begin again with the capitals. The staff stations
can then be marked with the letter of the station from which
their position has been first determined, and their number
as Ai, B26 and so on, a separate column being given to
each. When a new instrument station is observed from an
old one, or a backsight from a new one upon an old one,
the reading of the staff is booked thus :
. station
Staff station
Bearing
A
B
175°
B
A
355°
The Stadia Hairs
As has been already stated, when the vertical arc is used
the lower stadia hair should be directed to an even foot so
that the eye will not have the effort of endeavouring to read
simultaneously three heights upon the staff. There is gene-
rally a slight movement of the staff which makes it very
difficult to read unless the telescope is adjusted in this man
ner ; but by so doing the stadia are read at a glance, and
for the intermediate stations it is not necessary to read the
axial hair at all ; since the stadia hairs are equidistant from
it, the mean of the two readings will be the reading of the
axial hair. It is advisable, however, at the turning points to
read the axial hair as a check.
Some writers advise directing the axial hair to the same
Digitized by VjOOQ IC
Tacheometry 171
height on the staff as the height of the instrument above the
ground or peg ; the value of B or B' will then be the differ-
ence of level of the two stations. This plan loses more time
in observation than it saves in calculation.
Auxiliary Work
Especially with beginners there is a liability to make a
misreading of the horizontal limb with the intermediate
sights, and where the work is very particular it is advisable to
let an assistant tape from point to point. Mistakes are
generally in even degrees, which would at once be shown
by the check lines. Taping also comes in very handily
where instrumental sights are only practicable at long
intervals.
In a topographical survey of a portion of Windward Hawaii?
Sandwich Islands, the author adopted the following plan for
the ravines. These great gorges, locally termed gulches,
were sometimes a quarter of a mile wide, and the side slopes
varied in height from 100 to 400 feet and in angle from a
precipice to 20°. The growth of running shrubs, matted
ferns, and ironwood was such that in one place it took two
hours to obtain a single sight. Half an hour was a very
frequent delay.
The instrument was set up on one side of the ravine and
the staff-party remained on the other. They consisted of a
staff-holder, clinometer man, and two axemen. All four
had axes, and the first thing was to clear for a sight. When
the first point was fixed, the slope was measured and a sub-
sidiary tape-traverse was run just wherever the jungle was
penetrable. The lengths varied from 10 to 50 feet and the
total tape traverse sometimes extended to several hundred
feet. Each tape measurement was aligned by the prismatic
compass, levelled by the Abney level, and the slope taken
up and down with the clinometer. As soon as the staff-
party reached a place practicable for a clearing, all four
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172 Preliminary Survey
would fall to with the axes again and a fresh point would be
thus determined from the instrument. The established points
were first plotted, and then the tape traverse plotted on
tracing paper and squared in with them ; the coincidence
was generally very close, except where the magnetic devi-
ation was great.
When the staff-party reached the limit of instrumental
range, a fresh station was selected on the same side of the
gulch by means of a backsight with a second staff taken to the
station just left, and also check sights to any of the stations
on the opposite side where the clearing had been sufficient
to put in a flagpole visible from the new station. There
was in this way a subsidiary triangulation by which to
eliminate error, and great accuracy was obtained.
When one side of the gulch had been surveyed, the
staff and instrument changed sides and the latter occupied
the stations in the clearings, the staff party repeating their
modus operandi on the other side of the gulch.
When the surveyor is in good practice with the tacheo-
meter, he will very rarely make a misreading, but it is always
a very useful check to let the staff-holder carry a passometer
{not a pedometer) and book it at every station ; this he can do
while the sight is being taken without impeding the progress.
Reduction of the Traverse
The method of latitude and departure has this advantage
that protractor work is replaced by square scaling, and the
position can be determined in the field at any time whether
for the purpose of a check measurement to some known
object or for the assistance of the observation for azimuth.
When no tables or slide-rule are at hand it becomes a very
tedious process and is not often resorted to.
Chambers's Mathematical Tables give difference of lati-
tude and departure for every single degree and for each foot
of distance up to 300. The actual distances of a traverse
Digitized by VjOOQ IC
Tacheometry 173
rarely exceed 600 feet, so that the latitude and departure
can be very readily booked from these tables, if close accu-
racy is not required, especially as it is generally possible to
arrange the base lines so that they come for the most part
on even degrees, by waving the staff-holder to the right or
left until he comes to the nearest degree.
With the slide-rule it is done in the following manner :
First, Reduce the bearing to its equivalent azimuth by
Table XVII. (which should also be pasted on to the sUde-
rule).
Second, Find the difference of the angle thus found from
90°, i.e, the complement.
Third, Place the initial point of the sine-scale under the
distance on the rule, and read off the latitude opposite to
the complementary angle and the departure opposite to the
angle. With small angles it will give the nearest tenth, but
with larger angles the nearest foot.
Example, Find latitude and departure of base line
A/f\in fieldbook, p. 182. Bearing 347*03°, dist. 258.
(The distance must of course be taken as reduced to the
horizontal, which is equal to dist. x cos vert, angle or else
by Table XXIV.)
The azimuth will be 13*97° N.W. and its complement
76*03°. Place the left hand i of the sine-scale under the
258 of the rule, and opposite to angle 13*97° will be found
62*4 feet and opposite to 76°, 250 feet.
To reduce the difference of longitude and latitude thus
found to true longitude and latitude from Greenwich by
Tables XXV. and XXVI.
This is useful for astronomical observations and check
points.
Example
Started on January i in long. 157*582° W.
lat. 21-45° N.
Total northings for the day's run 33,425 feet.
„ westings „ „ 1 7)295 »
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174
Preliminary Survey
Northings in miles by slide-rule 6*33, multiplied by ^
by slide-mk— 552' of latitude N,
Westings in miles by slide inle=3'27; multiplied by-.^^
=3'oi' of longitude W,
Position on January 2 :
Long. 157° 3i'3'H-3Qi'=i57''34'3i'^V^^
Lat 21** 27' + 5-52'^2i^' 32*52' N.
As the maritime and geographical positions are ^iven in
degrees, minutus, and seconds it is more convenient to
retain the old Jiotation for that purpose-
TArtLP XXIV'J —Diffsrrnif tek^mn hypotepmse andimst infitiperhfin-
d red feet // hypotemi.^f\ for sloping grotmd. (Wi?/<?.— This should
he marked on the vtirtic;d limb of ihe transit ; on Y theodolites it
usually is-)
Slope th^ Deduct
defrees > feet
?i[(jpe It
Deduct.
Sbpc ihi
Dndutit.
deRTCi?»
fflftt
desTecsf
reet
'QOI
'J
'420
■■
^004
■460
1
-oog
1
SOJ
I
-015
6
■54^^
4
024
1
i ;
594
1
^034
'^3
'4
•i
047
3 ]
'693
2
■061
7^
745
1
-1
■077
Sao
I
<^S
^856
i
'TIS
n
■913
3
■'37
s'
■973
1
i
^161
i
I
i'03S
1
>iS7
fogii
'214
1 :
K164
4
*244
9
I '231
1
'375
1 I
1-300
'30K
'17'
1
343
I '444
5
3Sj
10
TSi?
13
It 14
1 t
Cfipieil from Tmutwke's * Pnc)tct Botik/
Digitized by VjOOQ IC
Taclteometry
175
Table XXV. — Length oj a minute of longitude in different latitudes at
the level of the sea in statute miles of ^^1^ feet.
Degrees
of
latitude
O
2
4
6
8
10
12
14
16
I minute Degrees
longitude || of
in miles ' latitude
I 153
1*152
1-150
1-146
I-I42
II35
I -128
I-II9
I -108
18
20
22
24
26
28
30
32
34
I minute
[Degrees
X minute
Degrees
I minute
longitude
,• of
longitude
of
longitude
in miles
1 latitude
in miles
0-933
latitude
54
in miles
I -097
36
0-679
1-084
3«
56
0-646
1-069
40
0-886
5«
o-6i2
1-053
42
0-858 '
60
0-578
I -037
44
0-831
62
0-542
I 019
1 46
o-8o2
64
0-507
0-999
i 48
0773
66
0-470
0979
50
0743
68
0-433
0-957
52
0-711
70
0-395
Table XXVI. — Length of a minute of latitude in different latitudes
at the level of the sea in statute miles of $,2'^ feet.
Degrees
pi
latitude
I minute
latitude
in miles
O
10
20
I -145
I -145
I -146
Degrees
of
latitude
30
40
50
I minute
latitude
in miles
1-148
I 150
1-151
Degrees
of
latitude
60
70
80
I minute
latitude
in miles
1*152
I -154
I-I55
Levelling
The elevations of the staff-stations are obtained by first
determining the elevation of the optical axis of the instru-
ment from some bench-mark ; in other words the height
of the centre of the trunnion of the telescope above any-
known or assumed datum. From this elevation the eleva-
tions of all succeeding intermediate staff-stations and the
next instrument station are determined by calculating the
vertical component of the direct distance, and adding to or
deducting from it the height of the staff from the ground to
where it is intercepted by the axial hair of the telescope.
Whether at the commencement or at a turning-point,
the elevation of the optical axis is obtained by a backsight,
and all other sights are foresights. The elevation of optical
axis is henceforth termed O. A. to distinguish it from the
Digitized by VjOOQ IC
176
Preliminary Survey
height of the instrument itself above the p^ over which it
stands, which will be called H. I.
This latter needs a separate column because it is an
independent check. The elevation of instrumental station
will be signified by E.I.S., and that of the staff station by
E.S.S. The direct distance will be marked D., the hori-
zontal component H. C, and the vertical component V. CJ
The Backsight
The elevation of optical axis, O. A., is obtained by the
following formulae :
a. When the vertical angle is one of elevation, which is
termed //«5, O.A.=E.S.S.-|-B.S.-V.C.
Fig. 46.
Fig. 47.
b. When the vertical angle is minus^ O.A.=E.S.S. + B.S.
+V.C.
The elevation of instrumental station E.I.S. is obtained
in either case by deducting H.I. from O.A. It is needed as
a check from the next station.
The Foresight
The elevation of any intermediate staff-station, or of
the next instrumental station, is obtained by the following
formulae :
• See also remarks on adjustment of axial hair to same height as
that of instrument, p. 174 at foot.
Digitized
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Tacheotnetry
177
(a) When the vertical angle is plus, E.S.S.=O.A.+V.C.
- F.S.
{b) When the vertical angle is minus, E.S.S.=O.A.—
(V.C. + F.S.).
In these figures the staff is shown vertical, and the instru-
ment not anallatic, but the rules apply to whatever method
is adopted of obtaining V.C.
Fig. 48. Fig. 49.
The first operation backsight A to /,\is reduced thus
E. S. S. -122-12
+ B.S. 5-28
-V.C.
127 '40
22-34
O. A. = 105-06
The second intermediate sight A to A, thus :
O. A. = 105-06 1-63 V.C.
-(V.C.+F.S.) 4*39 2-76 + F.S.
E. S. S. 100-67
The foresight A to B, O. A. = 105*06 11-96 V. C.
~(V.C. + F.S.) 17-48 5-52-fF. S.
E.S.S. 87-58
It should be remembered that there may be a fall with
a plus angle when the staff-reading is greater than the V.C.,
but never a rise with a minus angle.
The last backsight here agrees exactly with the foresight,
Digitized by VjOCtL _
178
Preliminary Sur-cey
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Tactieonietry 179
but if there is a discrepancy exceeding the working limit of
accuracy the sights should be repeated ; if they only differ
by a small amount, such as one or two hundredths, for close
levelling the mean should be taken for the O.A. of the new
station. The accuracy of the levelling never can equal that
of an ordinary levelling instrument, inasmuch as there is
always a liability to error in the measurement of the distance,
otherwise the accuracy would be the same. Error in the
vertical angle is more or less present with the intermediate
sights ; but, in the back- and foresights, it is ehminated by
the precaution of making each observation in duplicate,
face right and face left, and taking the mean angle. This
is done by releasing the parallel plate, and rotating the
telescope 180^ ; then, revolving it in its trunnions, the
measurement of the vertical angle in the reversed position
will reverse the index error if any. This should be very
little in a day's use of the instrument, if it has been properly
adjusted in the morning as explained on p. 313.
The importance of this operation for eliminating index
error cannot be overstated. It should never be neglected^ how-
ever rapid the rate of march.
Thus, if the vertical angle from A to B was 2° 13' 20"
when the vertical circle faced to the right, and in the
reversed position 2° 14' 40'', the true vertical angle would
be 2° 14'.
The value of the tacheometer for long sights and great
differences of level is a cardinal point, but it must be re-
membered that the accuracy is in opposite ratio both to
the distance and the height. The writer has obtained a
difference of elevation exceeding a hundred feet in one shot
exactly^ the oblique distance being between 300 and 400 feet,
but that cannot be relied upon with a tacheometer any more
than long shots with the level. The surveyor should there-
fore keep to short sights for the centre-line of his survey if
running a railway traverse, but the outlying points, such as
fences, &c., can be put in with longer sights.
„. zedbyGOOgtp
i8o
Preliminary Survey
In working out intermediate sights by slide-rule, or in
checking the back- and foresights, the following short table
will serve to show where the decimal point will come in the
result; for instance, in the backsight A 7?\ the angle is
nearly 5°, and the distance nearly 250, so it will be at once
seen that the result on the slide-rule is 22*34, and not 2*23
or 223*4. The list of multipliers and other factors on the
back of the Mannheim rule is not of much use to surveyors,
so it is best to paste them over and write a table such as
this one ; the rules for back- and foresights on pp. 180, 181 ;
the table of square-roots of small decimals given on p. 250 ;
and some others in this book, in the place of them.
Table X>
CVIL— .
^ines multiplied by various distances.
Degrees
Degrees,
minutes, and
seconds
0° 0^36"
Sine angle
raultipled by 50
Sine angle
multiplied by 250
t
1 Sine angle
multiplied by 750
•01
•0087
! -043
1 -'3
•05
0°
3'o"
-0436
•22
' -6.5
•10
0°
6'o''
•0872
1 '44
1*31
•20
0°
12' 0"
-174
1 -87
2-62
•30
0°
18' 0"
-262
1*31
3*93
•SO
0°
30' 0"
•436
1 2-iS
6-55
75
0°
45' 0"
'%^
1 3-27
9-82 1
i-oo
1°
o'o"
•87
1 4-36
13-09
3'oo
3°
o'o"
2-61
13-07
39-22 1
5-00
t
o'o"
436
21 -80
65*37
7-00
r
o'o"
6 -09
30-47
91*40
10 -oo
10°
o'o"
8-68
1 43*41
130-23 1
15-00
15°
0' 0"
12-94
64-71
194*13
20 -oo
20°
o'o"
17-10
85-50
256-51
25-00
25-
o'o"
21-13
1 105-65
30*00
30-
o'o"
25-00
125-00
—
35 -oo
35°
o'o'-
28-68
143-39
—
40-00
40-
o'o"
32-14
160-69
—
45-00
45°
o'o"
35*36
176-78
[
50-00
50°
o'o"
38-28
191-41
55-00
55°
o'o"
40-96
204-79
—
60 -oo
60°
o'o"
43-30
216-50
—
Referring back to the two first sights in fieldbook, p. 182,
to multiply sin 4° 57' by 259. Place a i of the sine-scale
d by Google
Digitized b
Tacheometry 1 8 1
under 259 of the rule, then opposite to 4° 57' will be found
223, which by reference to Table XXVII. is seen to mean
22*3. The slide-rule will only give the nearest tenth, but as
it is a backsight it ought to be worked out from a table of
sines and tangents. If in the office, Crelle's tables, which
give at a single inspection the products of three figures by
three figures will be found of great assistance, and this is as
close as it is necessary to go. For intermediate sight AAi, to
multiply sin 22' by 254. Place the fiducial mark for minutes
on the number scale of the slide, under the 22 of the rule.
Then opposite to 254 on the slide will be found 163, which
by inspection of the table will be at once identified as 1*63
feet.
If the elevations are required in metres, the staff must be
graduated also in metres, but the operation is the same.
Contouring
The elevations of suitable points having been determined
in the foregoing manner, the intervening ground may be
topographically represented by dotted lines of equal ele-
vation called contours, in precisely the same manner as
described on p. 61. It is well to distinguish contours which
are drawn between two fixed points from those which only
depend upon a slope taken from one point, the former being
naturally more reliable when the distances are not great.
Profile
The profile, as it is called in America, or section, as
termed in England, is produced from the contours, which
are drawn sufficiently close to enable the elevation of each
100 feet station to be judged to the nearest foot by the eye.
The gradients are first fixed approximately. In Fig. 50 for
instance, a suitable place was found about a mile up the
gulch for a horseshoe curve. Crossing the river at a
suitable height would leave a gradient of about i per cent.
Digitized by VjOOQ IC
i
1 82 Preliminary Survey
falling on both sides, so as to meet the rising bottom oi the
gulch. The next operation was then to mark off at say
every 500 feet the position where the gradient intersects the
surface. Then with railway sweeps a trial line is drawn to
pass as nearly as possible through these points, having due
regard to curvature. In a case like this, where the line is
located on the side of a hill, the whole of the line must be
thrown into cutting, except where the slope is flat enough to
admit of embankment.
It is nearly always cheaper to take out a cutting, even in
rock, than to build up a retaining wall. Where this latter
course becomes necessary, it is usual to bench the hillside
and build up cross-loggingi jointed at the ends with axe-
cuts, termed in America cribwork. In other places it is
found cheaper and otherwise preferable to build a wall
of dry-rubble, but, apart from the question of expense,
these erections are liable to destruction from wash-outs
and decay, and a little extra expense in cutting is more
economical in the long run.
When the location is on tolerably even country, the trial
line should aim at equalising the cuttings and embankments,
making them all as short as possible. When the line is on
level ground subject to floods, it should be placed entirely
in embankment, so as to keep it above highest known flood-
level, with frequent openings for letting off" the flood waters.
The details of curvature should all be written up on the
plan, for which see p. 217.
In plotting the profile, one man should take the plan,
with contours and trial line drawn on, and another man the
ruled profile-paper, which dispenses with all scaling. The
first then reads out the hundred-foot stations, and if neces-
sary, intermediate points with their elevations, and the other
plots them. Thus: station 200, elevation 354; station 201,
elevation 353; station 202, elevation 350^; station 202 + 10,
350, and so on.
Fig. 50 is a sketch from memory of one of the Hawaiian
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Tacheomeir)f
183
gulches, and is merely intended as an illustration of the
method. The scales are sufficiently given by the centre line
of the location and the profile. The sharp curve of 40°
Fig. 50.
(146*2' rad.) on the right of the figure was frequently resorted
to either to turn the head of a gulch, or to cut through a
rock-spur as shown.
Tacheometric Curve-Ranging
Curve-ranging is performed by 'tangential angles with
the tacheometer on the same principle as by the transit and
Fig. si.
chain, with this exception. Measurements by the chain are
necessarily repetitions of a short chord round the arc of a
circle, whereas measurements with the tacheometer are each
of them an independent chord, as shown on Fig. 51, where
Digitized by LjOOQ iC
1 84
Preliminary Survey
A is the B.C. (beginning of curve) and AB the tangent
produced; Ajo A20 A30) &c., are each of them rays from the
instrument, which may be either chords forming equidistant
points upon the curve, in which case every radius demands
a different series of tangential angles, or else they may be in
decimal parts of the radius, which is the method shown on
Fig. 51. Whatever the curve may be, points upon it can
be at once determined from the following table.
Table XXVIII. — Tangential angles for chords with radius— 100.
For tacheometric curve-ranging.
Tan-
Tan-
Tan-
1
Tan-
Tan- 1
"S
gential
g gential
"S
gential
gential
"H
gential
jS
angle in
f. angle in
ji
angle in
61
angle in
X,
angle in
^
degrees
^ 1 degrees
21 603
u
41
degrees
11-83
degrees
81
degrees
I
0-28
17-76
23-89
2
0-86
22
632
42
12-12
62
18-06
82
24-21
1 3
23
6-6o
43
12-42
63
18-36
83
24-52
i 4
I-I5
24
6-89
44
I27I
64
18-67
84
24-83
5
1*35
25
7-i8
45
13-00
65
18-97
85
25-15
6
172
26
7*47
46
13-30
66
19-27
86
25-47
7
2*OI
27
776
47
13-59
(^1
19-57
87
25-78
8
2*29
28
8-05
48
13-89
68
19-87
88
2613
9
2*58
29
8-33
49
1428
69
20-18
89
26-42
10
2-87
30
8-62
50
14-48
70
20-49
90
2674
II
315
31
8-92
51
1477
71
20-79
91
27-07
12
3*44
32
9'2I
52
15-07
72
21-10
92
27*39
13
373
33
9-50
53
15*37
73
21-41
93
2771 i
, H
4-02
34
978
54
15-67
74
21-72
94
28-03 i
i ^5
4*30
35
10-07
55
15-96
75
22-02
95
28-36
1 16
4*59
36
10-37
56
16-26
76
22-33
96
28-68
, 17
4-88
37
10 '66
57
1656
77
22-64
97
29-02
; 18
5-17
38
10-95
58
16-86
78
22 96
98
29-33
19
5*45
39
1 1 -24
59
17-16
79
23-27
99
29-67
20
574
40
11-53'
60
17-46
80
23-58
100
30-00 .
Let us suppose that the bearing of AB is 360®, and we wish
to set out a curve of 5°, that is of 1,146ft. radius in chords
of about looft. If ii4-6ft. is near enough, we will take the
tens and begin with 2° -8 7, but if it be necessary to come
closer than that, we should take 9, 18, 27, &c., and begin
with 2-58°, which would give us for a first chord 103 •14ft.
The one is as simple as the other by the slide -rule.
Digitized
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Tacheometry 185
Let us suppose that we take the 9. Place the right
hand i of the sHde under the 1,146 of the rule, and read off
the first dist. 103ft. under the 9, and set the instrument at
2° -58. Put the staff-holder in line, and make him take 37
ordinary steps forward. Keeping him in line wave him back
or forward by the Morse ' B ' or * F ' signal until his staff re-
cords the distance by the stadia-hairs, and then put a stake
in. Clamp the instrument at 5°*i7, and read off the distance
opposite to 18 on the slide-rule=2o6ft. The sub-chords
A- 10, 10-20, 20-30, 30-40, &c., lengthen as the curve is
ranged, but that is quite immaterial for a preliminary
survey. If A- 10 were 103, 70-80 would be 109ft., and
60-70 would be 107^.
Any labourer of average sense will learn to step each
distance within a yard, and the time occupied in ranging
him in line and moving him back or forward is no longer
than that of the ordinary curve-ranging.
This method is quite suitable for ranging the permanent
centre-line of a railway where great precision is not
required.
By leaving a mark at A the instrument can be shifted
to any one of the points on the curve, and directed to the
tangent. For instance, if it were desired to set up the instru-
ment on the tangent at point 80, of which the tangential angle
is 23°-58, we should clamp the parallel plates at 23°'58, and,
reversing the telescope, direct it by means of the external
axis on A. Then, reversing it back again with the external
axis clamped, release the parallel plates, and set the vernier
at 47° -16, which is twice the tangential angle, or in other
words the angle of deflection, and the instrument would
then be on the tangent.
It is even more advantageous to keep the curve-ranging
on true astronomical bearings when on preliminary survey
than when on construction, but the manner of doing so is
considered in the chapter on curve-ranging.
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1 86 Preliminary Sun^ey
Method of Ranging a Curve to Finish at a given
Alignment without Shifting the Instrument ^
It frequently happens that the chaining of a trial line
in order to obtain the topography sufficiently to locate the
best curve would involve a great deal of labour, and
would run a very small chance of being the final one. It
is possible without shifting the instrument to locate a curve
upon the ground which will pass through any given points
and terminate on a given alignment without chaining. Let
us suppose that a tangent has ended upon the edge of a
ravine, and it is desired to lay out a horse-shoe curve which
will follow the slope of the hill ; turn the head of the valley
with a trestle, and, skirting the opposite hillside, terminate
in a parallel to a boundary of some property which could
not be interfered with except at considerable cost. Any
other circumstances may ^y^ the direction of the final
tangent. If there are no obstructions such as the one in
the figure, the tangent will be chosen to suit the ground,
and in the nearest direction practicable to the objective
point of the railway.
The surveyor-in-chief will leave an assistant at the
transit, and proceed to the neighbourhood of B. He will
then fix upon a suitable direction for the final tangent BY,
and measure its magnetic bearing with the pocket altazimuth
described on p. 321, or by an ordinary prismatic compass.
Knowing the variation of the compass by previous observa-
tions, he then reduces the bearing to an astronomical
bearing, and telegraphs it to the transit man by means of a
flag. See Flag-signals (p. 93). The transit-man then works
out the deflection angle D, and tangential angle T, which
is equal to the angle XAB, and setting his instrument to
AB, gives the surveyor-in-chief the E.G. point B. It is
' Before perusing the following, the student should read the chapter
on curve-ranging.
Digitized
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Tacheometry
187
possible that B may fall considerably more to one side than
was intended, so that it would be necessary to prolong
or shorten the tangent at A. It is possible also that, after
locating D, it may not be the best place for the trestle
Fig. 52.
Preliminary Calculations and Fomiulce.
AB= 700 feet.
D =149-48= AC B.
T = 74-74=BAX.
Rad=AC= -4^^=362-8 feet
2 sin T
X=XAD=37-37°
2
AD=— ^?-;=440-4feet
2 cos —
2
AB=2R sin T
Neither of these eventualities need necessitate a change in
the direction of BY, but merely in the position of A and
B, and the radius of the curve. Assuming that B is a
suitable point for the E.G., the staff is held there, and the
Digitized by VjOOQ IC
1 88 Preliminary Survey
transit-man measures the length of AB by the stadia, and
telegraphs it. He then works out the radius and AD by
above formulae and telegraphs them. The surveyor-in-
chief then considers whether the radius is a suitable one,
or whether, by moving his final tangent a little, he can
obtain either a longer radius, or one* which corresponds to a
more simple degree of curve, or one which is on better
ground, and, if necessary, telegraphs back the radius which
he desires. In the illustration, the radius being 362*8 feet,
and a 16° curve having a radius of 359*3 feet, he would
telegraph back simply 16, and place his staff for fresh
alignment about 7 feet nearer to the transit. The transit-
man will then work out AB afresh=2 X359*3 xsin T=
693*3 feet, and AD=436*2. The chief surveyor will then
first drive a stake at the E.G., and then, proceeding to D, will
get his alignment and distance from the transit-man, and, if
suitable, drive another stake at D marked apex.
It may be advisable to know the difference of level
between A and D, to determine the height of the trestle
before finally selecting D. This the transit-man will do by
a foresight, as described on p. 176, and if necessary the
chief surveyor will take one or more trial points above or
below D. Since he is possessed of all the data of the curve,
he can with the slide-rule plot the curve on a sketchboard
and alternative curves above or below D, if he pleases.
When the E.G. and apex points have been satisfactorily
fixed, and stakes driven, the chief surveyor returns to the
transit, during which time the transit man has worked out
the data of the curve, and is ready to range it in the method
described on p. 183.
The particulars of this fieldbook will be better under-
stood after reading the chapter on curve-ranging. The
column ' chainage of,' is so marked for simplicity, although
the chain is not actually used. The columns of subtangent
and apex distance are not filled in because, the curve not
being run to intersection, it is needless to calculate them.
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Tacheometry
189
Description of Curve.
Total 1
angle of i Astronomical bearing
deflection
1
16
feet
359'3
feet
!
feet
934*3
i
feet
Chainage of
(2
c
deg. 1 deg.
149-48' 17*32
P 1 1%
deg. deg. deg.
1
54*69 92*06 166*80
pq
1 ^
1 W
deg.
"3
+24
117 122
+91*1 +58*3
But little is said in Chapter IX. about the various types
of tacheometer which are before the public, many of them
well worth description, and but little has been said here
about the method of using the stadia-rod vertically. The
formulae for cos^ and cos x sin are only tabulated for the
centesimal degree in a somewhat bulky form, published by
Cuartero of Madrid ; they can be obtained through Messrs.
Troughton & Sims. The method given in these pages
requires nothing more than an ordinary table of sines and
cosines, and the writer has proved its accuracy by consider-
able use of it. Upon the Hawaiian survey, a section of
seventy miles of railway included a little over one hundred
ravines of sizes varying from a hundred feet wide by fifty
deep, to a quarter of a mile wide by four hundred feet
deep. The intermediate survey on level ground between
the ravines was made with the transit, chain, and level, by
a separate party in the usual manner. The large ravines
were all surveyed with the tacheometer, and frequently
where two ravines were close together the intervening
ground was also surveyed and afterwards checked with the
results obtained by the chain party. In many cases the
one party used the stakes of the other as reference
marks both for distance and elevation, and the accuracy
of the stadia work was on the whole decidedly greater
than that of the chaining. The space of this work does
not admit of dwelling exhaustively upon many different
190 Preliminary Survey
methods of doing the same thing ; the one given here is the
simplest.
The following are the conclusions of Mr. Lyman, in his
paper, read before the Franklin Institute in 1868, on the
experience gained by him from the Schuylkill topographical
survey, United States of America.
1. That the additional lenses and complications adopted
by Porro, to cause the centre of anallatism to fall in the
exact vertical axis of the instrument, were needless, as the
inconvenience from adding to every distance observed a con-
stant quantity equal to the sum of the focal distance of the
object-glass, and the distance of that glass in front of the
vertical axis, rarely exceeding i foot, is comparatively trifling.
2. That three horizontal hairs or wires are sufficient for
all purposes, as the reading of the middle wire affords suffi-
cient check on the other two ; that fixed are preferable to
movable wires with protected adjusting screws ; and that
the fixed wires should be so set that the visible height on
the staff intercepted between the middle and either outer
wire should bear some exact ratio to the distance such as i
foot to 100, thus avoiding calculation.
3. That the staff should be graduated to hundredths of
a foot.
4. That in combination with the above arrangement, a
telescope magnifying only twenty times and reading to the
200th of a foot at a distance of 660 feet, will produce results
as correct as those of Porro's larger and more complicated
instrument.
5. That the errors arising from spherical aberration may
under these circumstances be neglected in angles of less
than 10° on either side of the focal axis.
The only point in which the writer's views differ from
those of Mr. Lyman is in the power of the telescope
necessary to produce the degree of accuracy named : ^ of
a foot at 660 feet distance. That he maintains is only pos-
sible with double the power specified by Mr. Lyman.
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Tacheometry 191
The Plane-Table and Stadia
The plane-table itself is described at p. 319 ^/ seq,^ but
when used in combination with a distance-measuring tele-
scope it becomes an instrument of much greater value.
The simple sight-rule is developed into a heavy brass
ruler termed an alidade, to which is attached a telescope
usually of 10 inches focal length, furnished with stadia hairs.
Instead of having to fill in detail with taping, pacing, and
sketching, it is all done with the stadia telescope. The
main triangulation still forms the basis of the work, and
occasionally the stadia work itself is checked by supple-
mentary triangulation.
The levelling is done in the same way as with the
tacheometer. The essential difference between the two
methods consists in the absence of a graduated horizontal
limb in the alidade, so that when traversing, the accuracy
depends upon the back and forward ray with the alidade,
which are transferred directly to the map. The transit and
stadia are superseding the plane-table and stadia in America.
The reasons for the writer's preference for the plane-table as
an auxiliary and not as a universal instrument have been
stated on p. 35, but it may be added here —
1. The plane-table is not equal to the tacheometer in
wooded country, because triangulation is often imprac-
ticable.
2. It is not so suitable where only short bases are
possible.
3. It is not suitable to rainy country.
4. It is not suitable for putting in railway curves.
5. It is more awkward to handle.
6. It is not so well adapted to astronomical obser-
vations.
An engineer does not want to over-burden himself with
instruments, but seeks a maximum of efficiency ^ portability^
and economy. The tacheometer will do all thaMhe plane-
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192 Preliminary Stingy
table used as a universal instrument will do ; it will do
most things better, and the peculiar cases in which the
plane-table is superior are not of sufficiently frequent occur-
rence to a man in general practice to make him choose the
instrument.
Rate of Execution
The Hawaiian survey previously alluded to was an
example of as rough a piece of country as could be found
anywhere. The average rate of progress was under half a mile
per diem by two parties, one with the tacheometer and the
other with transit and chain. The work included a belt of
from 500 to 1,500 feet wide, a hundred ravines of more or
less size, and seventeen plantation villages ; the whole being
mapped to a scale of 100 feet per inch in order to show the
contours at every five feet on the steep side-hill. The
ravines were densely wooded, and most of the cultivated
land was covered with sugar-cane standing eight to ten feet
high.
The greatest amount of work done in one day with the
tacheometer alone was two miles. In that case the country
was cleared, the cane was short, there were only two short
ravines and one small settlement of about a dozen cottages.
The little survey of Greatness Mill is intended to show
the convenient manner of doing an awkward piece of work
by the tacheometer. Everything except building detail was
put in from the three stations A, B, and C, and in a few hours.
The buildings were afterwards plotted on a tracing with the
sketchboard plane-table. Every check-line taken tallied
exactly. The profile of the road was obtained from the
sights taken to survey it. A photograph was taken at the
same time somewhat in the direction of AB. In the case
of a railway being made through such a spot as that, the
information obtained would show —
I. From the survey a depression of ground calling for
a crossing by bridge or viaduct.
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Tacheometry
193
2. The narrowest part of the water for a crossing.
3. By the photograph the nature of the property to be
interfered with.
Mr. William Bell Dawson surveyed with the tacheometer
and one assistant an area of 180 square miles, including one
hundred lakes from seven miles long downwards, in
five months. The object was to produce a map of the gold
Scales.
^nani^e D atuny
Ffg.53-
fields on the Atlantic coast of Nova Scotia. The traverses
generally closed within an area of 20 to 30 feet radius, but
closing errors were further eliminated by independent
checks. The map was on a scale of two miles to the inch.
He used a 6 -inch theodolite with stadia hairs and a
Rochon micrometer. The total cost was about 3/. \\s, 6d.
per square mile.
A level should always be furnished with stadia hairs,
and if it has a compass also contouring and small surveys
can be often done with it of sufficient accuracy for the
purpose, and so save bringing out the larger instrument.
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194 Preliminary Survey
CHAPTER VI
CHAINS UR VE YING
This department of survey practice will be only treated in a
condensed form to serve as a reference for the preliminary
surveyor. In exceptional cases it becomes expedient to
make a survey, measure an acreage, or range a curve with
the chain or tape alone, or at most with the assistance of an
optical square or cross-staff. The most frequent of such
cases is where the time of the tacheometer party cannot be
profitably employed on a solitary piece of detail, and one or
two hands are told off to fill it in with the chain only.
A short description will also be given of the methods in
vogue in America for running trial lines for railway location
with the transit and chain, together with a few notes on
curve-ranging with the chain only, reserving for the next
chapter the subject of curve-ranging with the theodolite.
Chaining
If chaining could be carried out with rigorous exactness,
it would be possible to perform the whole survey with no
instruments whatever, although it would be a lengthy affair,
especially when the afterwork of levelling over the staked
survey is considered. But there are also many essential
difficulties which tend to produce error, and render chain
surveys unsatisfactory.
It is just because chaining seems so very simple that it
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Chain-Surveying 195
is often relegated to inferior assistants, who do not study a
systematic method of performing their work, and errors
frequently arise from the following causes :
First The liability of a poor chainman to 'drop a chain,'
that is, to make a mis-count with the arrows.
Secondly. The tendency to measure on sloping ground
without making the proper deductions.
Thirdly, The liability of the chain to being stretched by
tension, or to expansion and contraction by variations of
temperature, all of which require regular attention, not always
devoted to them.
Fourthly. From twists in the map, due to incorrect tie-
lines at the angles of the base lines.
Fifthly. From confusion in the fieldbook.
On a level piece of ground, where the plot is drawn to a
large scale, such as 20 or 30 feet to the inch, the survey
should be made with the transit and chain, as the maximum
error should not exceed three inches. The tacheometer will
not give this exactitude, and the chain alone is almost sure
to develop twists. A survey of this kind, such as a town-
district or a railway terminal depot, is out of our province to
describe. The main distinguishing features about it are :
First, a network of triangulation wherever it is possible to
chain bases, or, combined with triangulation, a closed
traverse. Second, a system of ordinates or offsets from the
bases thus fixed to all the points required to be plotted.
The same operation can be performed with the surveyor's
compass, but the theodolite is much more precise in the
angular measurement.
There are two or three methods of counting the chainage
and it is not of much consequence which system is used,
but it is most important that the same system should be
rigidly adhered to. The following method is recommended.
The leader carries a ranging rod, and the ten arrows, with
the forward end of the chain. The follower carries the
hinder end of the chain and a clinometer; he also has a tape
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196 Prclinwiar)' Sun^eyf
attached to his button-hole. The ranging rod has a hand-
kerchief or piece of paper attached to it at the height of the
follower's eye.
The leader drags forward the chain to the follower's
direction, and hauls it taut.
The follower shakes it up to get rid of kinks and
curvature.
The leader holds up the ranging rod, to get the final
direction, and when fixed, puts the iron point through the
handle of the chain to give the last stretch, and then puts in
an arrow, and goes ahead. When he has put in his last
arrow he runs out the next chain without an arrow, aligns
it by the rod, and, leaving the iron point through the handle
of the chain, signals to the follower to come forward with
the ten arrows.
The follower first ties a knot in his tape to mark the ten
chains, and then comes on, leaving the chain to be dragged
forward by the leader when he has received his arrows. The
leader then puts in one of the ten arrows at the rod, and pro-
ceeds as before.
When the slope exceeds a degree (or, if the greatest
accuracy is required, at every chain) the follower books the
slope, and at the end of the line sums up the corrections
and deducts the result from the chainage. He stands erect
and directs the clinometer to the mark on the ranging rod.
Measuring by short lengths on a side-slope is very un-
satisfactory ; it is difficult to keep the chain level, and still
more difficult to plumb down exactly from the high end of
the chain when rapidity is an object.
A table is given at p. 174 from which the deductions
from the chainage on sloping ground can be readily made
for any given angle.
Setting out a Square
Let it be intended to lay off BD perpendicular to AC
with the chain alone. Measure off AB=3o feet Take the
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Cltain- Surveying
197
10 mark on the chain to A and the opposite end of the chain
to B. Put an arrow through the 40 mark nearest to B.
Draw out the chain and shake it taut, so that the BD line
which is equal to 40 feet, and the AD which is 50 feet, shall
each be perfectly straight ; then because 3^-1-4^=5^ ABD is
a right angle. Links of a Gunter's chain will of course do
just as well, and any multiples of 3, 4, and 5, such as, 6, 8,
and 10 ; 9, 12, and 15 ; or 45, 60, and 75.
To Measure the Deflection Distance of a New
Base Line
This is a problem of curve-ranging, but is needed also
in all cases where closed triangles cannot be formed.
Let it be required to find the deflection distance of BD
from BC. Measure as long a base B^ as possible, align b
correctly with AB, set off bd square and measure it. If
possible choose 100 feet or 200 feet for B^, put in a peg
Fig. 55.
with a nail at ^, and measure ^t/ with a steel tape. BD can
then be plotted direct from these data, or else the angle can
The entry in the
be calculated thus : tan CBD =
B^
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198
Preliminary Survey
fieldbook is marked R or L, according as the new base line
turns right or left.
Another Method
When the angle of deflection is great, it is better to make
the tie-line by the chord of the angle. Measure B^ as long
as possible and also B^ equal to it. The fieldbook should
Fig. 56.
show by a sketch how the tie-line has been taken. The
distance bd should be measured with a steel tape, and the
base can be plotted from the data, or else
sinACBD= — ^,
In making a triangulation survey the aim should be to
get as long bases as possible, and the triangles * well-con-
FiG. 57.
ditioned,' that is approximating to equilaterals. When the
interior of the survey is partly inaccessible this is impossible,
Chain-Surveying
199
and recourse has to be had to external ties Hke the above.
When the triangulation can be made internally, a cross-staff
is of great assistance. The handiest form is a brass octagon
fitted with hair sights in slits. It stands on a short pole
with an iron shoe, and only costs from 5^. to los. The
Fig. 58.
advantage of the- cross-staff will be seen by comparing Figs.
57 and 58. It enables the offsets to be made up to 100 feet
with the tape, and so diminishes the base lines.
The fieldbook is arranged with a central column for
distances measured along the base lines, and on the right
and left are the offsets opposite to the distances at which
they are taken, with remarks to identify them. The field-
book is commenced at the bottom of the page.
Left offsets
Base
B
18
716
49
655
- 68
587
80
505
98
392
382 1
98
312 .
los
190
8S
80
river 17 From
A
Right offsets
330 to C.
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200 Preliminary Survey
Acreage
The area is divided up into triangles and trapezoids.
The area of the triangles is obtained by either of the two
following formulae :
Rule I. To find the area of a triangle from the lengths
of the three sides, a useful method when no perpendicular
can be measured across the triangle.
Let ^, b^ c be the sides, and i= their half sum ; A the
Fig. 59-
area in square measure of the unit adopted, such as square
feet area for lineal feet measurement of th^ sides.
A= \^ sx {s—a) X (s—b) X {s—c)
or logarithmically
A=i[log x+log(j-a) + log(x-^) + (log s-e)l
Rule 2. To find the area of a triangle when any one of
the sides and the perpendicular from it to the opposite angle
are given.
}3 y p
I^t B=base, and P = perpendicular ; area= .
2
The area of a trapezoid is obtained as follows :
o, , .c
Fig. 6o.
Rule 3. Multiply the sum of the parallel sides by the
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Chain- Surveying 201
perpendicular distance between them, and half the product
is the area.
2
or logarithmically
Log 2 A=log(AB + CD) + log h
To reduce areas in Gunter^s chains to acres.
10 square chains =1 acre
100,000 square links =1 acre
To reduce square feet to acreage.
43,560 square feet=i acre
I acre =4 roods
I rood =40 perches
Preliminary Work with Transit and Chain
In running trial lines with the transit and chain, the
curves, unless exceptionally sharp, are not put in. The base
lines are measured, and staked from intersection to intersec-
tion ; the angles measured, and the stakes afterwards levelled
over. The vertical arc of the transit is not used unless
when on a steep gradient it is desired to follow the required
inclination, in which case the vertical arc is clamped to the
corresponding angle, and the line run as nearly as possible
to it.
In America the levelling staff is called a rod, and the
ranging rod is called a picket. In running a line there are
generally two picket men, sometimes only one. The transit-
man sends out a picket-man in the direction which he
intends to take, aligns the picket, and takes a reference sight
to some good fiducial point.
There are two chain -men, leader and follower, and one or
more axe-men, who prepare stakes about three feet long,
pointed at one end, and shaped on the other for a chalk mark.
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202
Preliminary Survey
The transit-man aligns the chaining by the leader's
picket, and the axe-men put in a stake at every hundred
feet, or at shorter distances where necessary.
The chief engineer generally accompanies the chaining
party in order to select the ground.
When the end of the base is reached, the picket-man,
leaving a peg at the end of the base, returns to the instru-
ment, and changes places with the transit-man to enable
the latter to range out a new base.
There are three methods in use for registering the
deflection of the one base from the other.
The first is the most common — see Fig. 6i. It takes no
^>-
• 1 sc
-Z'7
Fig. 6i.
account of the astronomical meridian and only uses the
compass-needle as an occasional check.
The field notes will explain themselves with scarcely
any description. The instrument is set back to zero ever>'
time ; the slopes of the ground, right or left, are either taken
with the vertical arc of the transit or by the clinometer.
Method 2 — see Fig. 62. The second method is by
working out the magnetic or astronomical azimuths. If the
former, it is sufficient to commence by making the zero of
the horizontal limb coincide with the position of the needle
when at rest. If the latter, the meridian must first be deter-
mined by one of the methods mentioned in Chapter IV.
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CliaiU'Surveying
203
Almost all transits are graduated from o to 360, so that
this method entails the reduction of the angle as explained
on p. 59 ; it is, however, very well suited to plotting by
latitude and departure.
Method I. By bearings from o to 360 — see Fig. 63. These
are now also commonly termed azimuths, though not, strictly
speaking, azimuths unless taken from south as well as north
point. Some distinctive term is necessary for these bearings.
* Northerly bearing ' would express the first idea, but then it
needs to be defined as to whether it is astronomical or
SUu6-t60
Fig. 62.
Sea^7^^06
Jia^6^
60
magnetic bearing, which would be expressed by N. Ast.
Bearing, or N. Mag. Bearing.
If we are furthermore to be visited with a centesimal
graduation in competition with the old graduation, we should
require another letter or two of the alphabet to define our
modus operandi. If the term * course ' could by universal
consent be relegated to the magnetic, and 'bearing'
reserved for the astronomical direction, both being kept
solely for graduation from o to 360, whilst the term
* azimuth * was confined to what it strictly means, it would
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204
Preliminary Survey
simplify nomenclature ; but it is to be feared that the sailors
would not fall in with the proposal.
It is not necessary either in Methods 2 or 3 to measure
the angles of deflection separately. They are only inserted
here in order to compare one method with the other. The
transit is fitted with two verniers, and when making a devia-
tion the instrument is reversed and directed to the previous
station, one of the verniers being set at the bearing of the
line just run. This is called the backsight. The instrument
is then again reversed so as to point forwards, and, unclamp-
•i-^
stw e^6o
Six.0
Sixt/7Z^05
6tay6^eO
Fig. 63.
ing the parallel plate, directed along the new base-line to
the picket fixed at the end of it. The bearing is then read
by the other vernier, and booked.
When two * readers ' are provided, it is advisable to use
only one, and at each change of direction shift it over to the
other vernier. Before commencing work, it is of course
necessary to see that the line of sight is in true corre-
spondence with the zero of the horizontal limb as ex-
plained in Chapter IX. It is also necessary to have an
instrument in which the graduation is correct, so that the
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Chai7t-Su7'v eying
205
readings of the two verniers differ by 180°, or in the
centesimal instrument by 200. When this is so, using
the two verniers forms a check upon the readings.
The slopes are put in by all three methods as shown on
Fig. 61.
Circular Curve-ranging with the Chain
Four methods will be briefly dwelt upon.
Method I. Krohnke's tangential system.
This is a scheme for obtaining equidistant points upon
a curve by laying out abscissae BB', BB'', &c., along the
subtangent of the curve and setting off ordinatcs IV b\ B" y\
<S:c., to the required points. When the offsets become long
the curve is bisected, each section treated in a similar
manner. Krohnke's tables of forty-seven pages are published
for setting out curves of any, radius by this method.
Method 2. Jackson's six-point equidistant system. As
in Krohnke's method, he obtains equidistant points from
unequal abscissae, but by reducing the number of points to
six he obtains a manageable curve and much shorter tables.
They are given in his ' Aids to Survey Practice,' Crosby
Lockwood & Co.
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2o6
Preliminary Survey
Method 3. The method given in Kennedy and Hack-
wood's * Curve-tables ' (E. & F. N. Spon) of equal abscissae
and unequal distances on the curve. This is much more
simple, and although, for purposes of construction, it is a dis-
advantage to have points that are not equidistant, for prelimi-
nary survey it makes no difference. The deflection distances
are given for Gunter's chains in the end column of each
table.
It may be here as well to explain the principle of laying
off deflection distances and to show how points may be set
off from a tangent to any required equidistant points upon
a circular curve.
One of the fundamental properties of the circle is that
equal chords subtend equal angles at the centre. It requires
no demonstration that in Fig. 63, BB* is = sin D x AB, or
BB2 = sin (D + D') x AB, or that B^^'= AB - AB. cos D,
or B*^'" = AB - AB. cos (D to D"').
It will be presently shown in the chapter on curve-ranging
that the angles D, D + D', &c., are each of them double the
corresponding tangential angle given in the tables.
Thus, supposing the radius 40 chains, we have a tan-
gential angle for i chain of 0° 42' 58'' = a deflection or
central angle of 1° 25' 56".
If we multiply the radius, 2,640 feet, by sin 1^25' 56'',
we shall get 65*985 feet as the co-ordinate corresponding to
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Chain-Surveying 207
a chord of i Gunter's chain, and by cos 1° 25' 56" we shall
get 2,639-17 feet, which deducted from the radius gives us
the offset '83 feet or 9*96 inches. The table enables us to
find the tangential angle due to length of curve of any radius,
and from this we can obtain the deflection distance in the
manner just described. This will be better understood after
reading the next chapter. In order to throw off a new
tangent when the offsets become too long, we have only to
go back two, three, or four stations and measure off from it
the deflection distance due to its number from the last peg.
For instance, if we wished to lay off a tangent at b"" we
should go back say to b'" and lay off the deflection distance
for I, or to b'\ and lay off the deflection distance for 2 chains.
We should also lay off from b^'^' the abscissae = BB^ or BB^
as the case may be, and this would be a true tangent at b'"\
which we could produce and treat in the same manner as
BB^.
Example, The total curvature \' of the crowning curve
on p. 230 is 63°'6, and the radius 36*93 feet; it is required
to put in equidistant points b\ b", b'" upon the curve. Sub-
divide A' into a suitable number of parts, say 10, then D
(Fig. 65) = 6°-36; (D + D')= i2°72 &c. BBi=36-93 x
sin G'^'^G = 4-09 feet. BB^ = 3693 x sin i2°72 = 8*13
feet. B^ ^'=36*93 (i-cos 6°-36)=o-23 feet. B^/^^ss 36-93
(i— cos i2°*72) = o-9i feet, &c.
Method 4. The oldest and simplest plan for setting out
curves with the chain is the only one which can be done
without tables (see Fig. 66).
The first point in the curve is found similarly to the last
method, but the succeeding points are fixed by ranging lines
B^^fl^; b^b'^a^, and so on, from which the offsets a^b'^^ aH^
will be each of them double the first offset a>b^. The first
offset d^b^ is termed the tangential distance, because it is
the chord subtending the tangential angle to a radius of i
chain. The succeeding offsets a^b^^ a^ H^, &c., are termed
the deflection distances, because they are equal to the chord
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208
Preliviinary Survey
subtending the angle made by the intersection of two tan-
gents to the curve at the ends of a chain and to a radius of
I chain (see more on nomenclature of curve-ranging at
p. 214).
Fig. 66.
The formula used in the field from which to find the
offsets by this method is
for tangential dist. offset =
chord 2
2R
„ deflection dist. offset=— ^^
R
The chord may be i Gunter's chain in links, in which
case the radius will be expressed also in links and the
offset will be in links, or it may be in feet or metres or any
other measure.
This formula is not quite exact for small radii.
On a curve of 100 feet radius, the tangential distance for
100 feet chord would be 5176 feet against 50 feet by the
formula. At 500 the formula would be but -013 feet too
little. So that for curves of 10 chains and upwards, set out
with I chain chords, the formula is practically correct
The subject of curve-ranging with transit and chain is
reserved for another chapter.
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209
CHAPTER VII
Curve-Ranging with Transit and Chain
The subject of curve-ranging has been briefly touched
upon in the previous chapter as far as it can be done with
the chain only, but it was deemed appropriate to devote a
whole chapter to the subject of curve-ranging with the in-
strument and chain as being par excellence curve-ranging.
There are still those who prefer the chain only to any
instrument for the purpose of curve-ranging. In the hands
of experts, very correct results hiay be obtained in level
country, but it would be interesting to know how the chain-
curve-ranger would put in track centres on a 40° curve on the
top of an embankment 30 feet high, the total deflection angle
being 180®.
To take it all round, curve-ranging with the chain only
is, as compared with work by better methods, poor, fudgy,
and muddling.
Nomenclature
The principles of curve-ranging are best understood by
keeping in one's mind the idea of a ship's course at sea.
In the previous chapter, it will have probably been
noticed how the methods of running base lines with the
transit and chain, especially Method 2 , resemble the deter-
mination of a ship's course and position by dead reckoning.
The angular changes of direction 33° 15' and 44"* i' in
Fig. 62 are the angles of deflection^ and this is. the .very, root-
idea of curve-ranging. The internal angle between the tan-
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2IO Preliminary Survey
gents of a curve, commonly termed I in English textbooks,
forms no part whatever of the theory to be described here.
It conveys no idea of the curve beyond the mere fact that it
is the supplement of the angle of deflection.
This angle of deflection may be either represented by its
angular value or else by the difference of bearing of the base-
lines, as in navigation.
Curve-ranging may be defined as deflecting by degrees.
It is for this reason that American surveyors have adopted
the curve-nomenclature by which the quickness or slowness
of a curve is expressed by the number of degrees of a circle
which the curve^ not the chords deflects at each chain length.
Thus a curve in which a i co-foot chord subtends i° of
curvature is termed a i -degree curve.
The reason for this is obvious. When the total number of
degrees contained in the deflection angle have been turned,
the new base line or tangent will have been reached, and
the number of chain lengths contained in the curve will be
given by simply dividing the total angle of deflection by the
* degree of curve.'
For instance, if the total angle of deflection be 17*22'
and it is desired to put in a 2° curve, that is to say, a curve
in which, if a chain length is measured along it, the tangents
to the curve at either end of the chain length will deflect
from one another by 2*, the total length of the curve will
bei^-^ X 100=861 feet if a 100 feet chain is used, or 861
2
links if a Gunter's chain.
Contrast this simple computation with that of English
curve-ranging, where it is expressed as follows :
Let X be half the angle of intersection, and R the radius.
Length of the curve = '000582 R (5400—^). Note ^ x
must be expressed in ;///;z«/^x.
The writer has npt taken the trouble-to ascertain whether
this formula gives the true circular measure of the curve, or
the measure in chains and parts of a chain. If, as is pro-
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Curve-Ranging with Transit and Chain 21 1
bable, it is the former, it has not even the advantage of being
correct, because curves are not set out by true circular
segments but by short chords of chains or parts of a chain,
and this is the principle of the American formula.
The English formula is adapted to logarithmic calcula-
tion, whereas the American is performed either by slide-rule
or even in the head.
The ordinary graduation of instruments into degrees and
minutes necessitates the reduction of the deflection angle to
decimals of a degree,^as for instance where the angle is
29° 47' 30" and the degree of curve is 6^ 35' it requires some
?j / ^ . -
•r^; :^
Fig. 67.
little calculation; but by the author's system of decimal
graduation of the ordinary degree, it is done by the slide-
rule instantaneously.
In Fig. 67, it will be observed that there are two angles
marked A, and two marked d. The upper A is the angle
of deflection of the curve and is equal to the angle at
the centre subtending the curve, which is therefore also
marked A. Similarly the angle of deflection of a portion
of the curve of which the chord is 100 is=</,and is equal
to the central angle also marked d. There has been, and is
still, considerable confusion of nomenclature in both England
and America, from the fact that some writers insist on term-
r 2
212 Preliminary Survey
ing the tangential angle © the angle of deflection because it
is the angle by which the chord deflects from the tangent.
This has been termed the tangential angle, at least from the
first of Rankine's books on the subject, and is still so termed
by the best authorities. There might have been some room
for choice in naming this angle, but it could not with propriety
be termed the angle of deflection.
The only apology for spending so much time in this
demonstration is that, unfortunately, writers of high standing
have taken up this perplexing nomenclature. To put it con-
cisely, the deflection of a curve is its actual change of direc-
tion^ or curvature^ and those who adopt the wrong designation
would have to describe a curve which had turned round
a quarter of a circle as having deflected forty-five degrees,
because the angle between the chord and the tangent (the
tangential angle) would be 45°. The term * tangential angle '
is used either for the angle for i chain of 66 feet or of 100
feet or for the total angle, which is vague. The term ' total
tangential angle' should be given to that for the whole curve,
and * single tangential angle ' for that for the unit of measure-
ment. The term * total deflection angle,' and * single deflec-
tion angle,' will similarly apply to those angles, the latter being
the * degree ' of curve.
One English writer, imperfectly acquainted with American
curve-ranging principles, describes their nomenclature as * a
confusion between the angle at the centre and the angle of
deflection,' from which it would appear that the writer was
not himself aware that these angles were equal to one
another.
The tangential angle T, or ©, is equal to half the angle
of deflection in circular curves.
In America the straight is termed the tangent, and the
prolongation from the springing to the intersection is termed
the subtangent. The point of intersection is so- named, or
else the- Vertex, 'and the midway point of the curve is called
the * apex,' or * crowning point,' or * summit.' The curve is
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Curve- Ranging with Transit and Chain 213
marked B.C. at beginning of curve, and E.G. at end of
curve. Wherever a change of position occurs with the
transit, a *hub,' or short stump, is driven in with a nail in
it, and a reference stake about three feet long close to it. All
the other points are marked by long stakes having their dis-
tance chalked on them in stations of 100 feet and their
excess. Thus 1,143 f'^et would be marked 11 + 43.
Only short curves are run to intersection point. The
method of keeping the fieldbook herein described enables
the surveyor to range a curve of any radius, alter it, compound
it, reverse it, and calculate his position with reference to his
starting point without running the curve to intersection. The
only difference in this system from the ordinary practice in
America is that the lines are kept on their astronomical
bearing. It is a little more trouble when using the gradua-
tion of minutes and seconds, but no more trouble with the
decimal subdivision, and has the advantage of always
affording a ready means of defining the position and checking
the work.
General Properties of Circular Curves
The following demonstrations are amongst the most
useful problems occurring in practice (see Fig. 68).
Let the deflection or total curvature A be represented by
DIE, the radius BO or OE, IB the subtangent, and I A
the apex distance.
I. Prove that DIE=BOE.
In the quadrilateral IBOE angles IBO and lEO are
right angles.
But the internal angles of any quadrilateral are together
equal to four right angles.
Therefore angle BIE+ angle BOE are equal to two
right angles.
But angles DIE-fBIE are also equal to two right
angles ; hence, equating and eliminating BIE, angle DIE
= angle BOE.
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214
Preliminary Survey
2. Prove that angle I BE =
DIE
-or =
BOE
or=BOL
2 2
BOI=EOI because A is at the middle of the curve.
IBE=IEB because IBE is an isosceles triangle.
The three angles of any triangle being equal to two right
angles, and it having been already shown that BIE+BOE
=two right angles, therefore in the triangle BIE angles
BIE, IBE, BEI are equal to the angles BIE+BOE.
Equating, eliminating BIE, and setting IBE + BEI=2lBE;
2 IBE=BOE, or, which is the same thing, IBE=^DIE.
.0
3. Prove that the subtangent BI=the radius BO mul-
tiplied by the tab. tan. of the tangential angle IBE. IB is
evidently proportional to the tangent of angle BOI, which
angle we have shown to be equal to IBE.
4. Prove that the apex distance
IA =
radius
- — radius
IA =
cos IBE
10 — AO, but by No. 5, p. 373
cos BOI
10
BO
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Curve-Ranging with Transit and C/iain 215
... 10 = ^9^, and BOI = IBE
cos BOI
whence lA = ''^^^"^^ - radius
cos IBE
General FoRMULiE
Let A be the total deflection=2®, the tangential angle ;
d = the degree of curve or curvature in 100 feet chord ; R
=the radius ; C = the chord subtending the whole curve ;
S.T. =the subtangent ; AI = the apex distance ; L =
length of curve in feet.
C _ C _ 50
2 sin—
2
sin ©
sin
d
2
. A
2
. d
50
100 sin -
2
100 sin
10
2
-R -
C
■"
C
S.T.=
= R. tan :
AI =
R
_
R
cos®
L=-
100 A
d '
A
_I^
100
We have now sufficient data for ranging a curve.
Example i. The intersection point occurred at station
753 + 34.
The angle of deflection was 37®'9i6. It is intended to
put in a 5° curve. Required, the length of the subtangent,
the apex distance, the length of the curve, and the chainage
of the B. C. and E. C. points and the apex.
The radius of the curve is found from the table p. 379
= 1,146 feet.
Subtan = 1,146 x tan y_-2L = 393-7
2
Apex dist. = — ^lf'^16 " ^'^^^ = ^57 feet
2
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2i6 Prdiminary Survey
Length of curve = ^-^—^^ x loo = 758*3 feet
B. C. point = 753 + 34 - 3937 feet = 749 -f 4o*3
E. C. point = 749 -f 40-3 + 758*3 feet = 756 + 9^*6
Apex point at 749 + 40*3 + H— 5 feet = 753 + 19*9
2
LayoffangleEIAFig.68= ^ ^ ^ and measure off I A
2
= 657 feet, and put a stake in marked * Apex, 753 -f 19*9.'
Measure the subtangent both ways, and put in pegs
with nails, aligning them from the transit at the intersection
point. Put in reference stakes marked B.C. 749 + 40*3 and
E.G. 756 + 98-6.
Shift the transit to the B. C. point and commence to
range the curve. The first odd distance will be 597 feet ;
the tangential angle for this will be 5- x 5z-Z = i°-49. The
2 100
instrument is set to read this angle and a stake put in
marked 750. The next angle will be 2°-5 (2*30') + 1*49
= 3°*99> which is set off and a stake aligned and driven at
another hundred feet and marked 751, and so on. After
stake 753 is driven, at a tangential angle of 8° -98 the
odd distance, 19*9 to the apex point, is set off, the angle
being 2*5 x ^^ = o°-5o, or angle from the beginning 8°*98
100
4- -50 =•. 9°*48. This will be one-half the total tangential
angle, or one-quarter of the angle of deflection. Both the
odd distances to the apex and to the E.G. point are given to
the chain-men, and if they do not coincide with the stakes
already driven the chain-men either signal the transit-man
the amount of divergence when small, or, if large, they
return and report. If, for instance, they signal, E.G. point
six -tenths to the right, the transit-man will signal back to
them to come back three stations or four stations, and he
will distribute the error over them. If he is constructing
an iron trestle or a brick viaduct he will, of course, repeat
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Cun^e- Ranging with Transit and Chain 2 17
his work until there is no sensible error, but it would be a
waste of time for an embankment or cutting, when the error
is only a few inches. With ordinary care on the part of the
transit-man the error, if any, is nearly always in the chainage.
The following fieldbook is arranged for astronomical
bearings. If the curve is run from zero point, the only dif-
ference will be that the entries in the bearing column will
be dispensed with. The operation at a turning-point is this.
Taking as an example the first shift of the transit at Station
Fig. 69.
755. In the fieldbook on next page a picket-man is sent to
the B.C. The instrument is set up at Station 755 over a
peg or * hub ' with a nail in it, fixed by the transit from the
B.C With one of the verniers clamped at 29031 and the
telescope upside down, a backsight is taken by means of the
external axis upon the B.C.
Description of Curve.
Total
angle of
deflection
Astronomical bearing
5
1,146
1
tn
feet
393*7
!
feet
758-3
8
.a
•A
<
feet
65*7
Chainage of
28,
11
deg.
285*81
1
deg.
314*24
B.C.
749+
40*3
Apex
E.G.
98*6
deg.
37 '92
deg.
276*32
deg.
295-28
753+
19-9
The external axis is then clamped, the telescope is reversed
right side up, the parallel plate released, and the instru-
ment directed to the tangent of the curve at Sta. 755 by
reading with the same vernier the angle 304° '30. Comparing
Fig. 69 with the fieldbook we shall at once see that as the
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2l8
Preliminmy Survey
Length
of chord
FieMbooh,
Chainage
Bear.
ing
t
T
Remarks
749 + 40-3
276-32
277-81
B. C.
750
59-7
1-49
1-49
751
280-31
loo-o
2-5
3-99
■
752
282'8l
loo-o
2-5
6-49
.
753
285-31
ioo*o
2-5
8-99
753 + 19-9
^!5'!^
19-9
0-5
9-49
Apex '
754
287-81
8o-i
2*0
11-49
755
290-31
13-99
100 -0
2-5
13-99
Turning point
755
^t
On tangent at 755
756
100 -0
2-5
2-5
756+ 98 ^a
309-271 98-6
2-47
4-97
E. C.
4-97
1399
314*24
!
4-97
18-96
2
On new tangent
1
3792
Check
tangential angle from the B.C. to 755 is i3°-99, ^^ we add to
this an equal amount we obtain the angle of deflection,
which laid off from Sta. 755 throws us on to the tangent.
The following fieldbook is for the same curve, only
turned to the left instead of to the right.
Description of Curve,
Toul
angle of
deflection
Astronomical bearing
4)
1
5
3
1
1
758-3
I
Chainage
or- 1
1
t
%\
H
266-84
u
B.C.
Apex
E.C.
— 37-92
276-32
257-36
238-40
1,146
393*7
65-7
749+
40*3
753 +
19*9
98-6
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Curve-Ranging with Transit and Chain 219
Fieldbook,
Chainage
Bear-
ing
Length
of chord
t
T
Remarks
749 + 403
276-32
B.C.
750
274-83
59-7
1-49
1-49
751
272-33
loo-o
2-5
3*99
752
26983
loo-o
2-5
6-49
753
267-33
loo-o
2-5
8-99
753+19-9
266-83
19-9
0-5
9-49
Apex
754
264-83
80-1
2*0
11-49
755
262-33
1000
2-5
1399
Turning point
755
13-99
On tangent at 755
248-34
756
245-84
loo-o
2-5
2-5
756 + 98-6
243-37
98-6
2-47
4-97
E. C.
4-97
—
238-40
On new tangent
In practice some of these entries are dispensed with,
especially when in a hurry. // is however very dad policy to
keep a fieldbook that nobody else can understand. If the
columns are headed beforehand, or printed, very little time
is taken up in making all the entries, and a great deal of
time is saved to the reviser when the curve has to be picked
up again at any point.
In England, curve-ranging is not done with a fieldbook.
Tables like those of Kennedy and Hackwood supply the
tangential angles for curves of radii in Gunter's chains.
The B.C. and E.G. points are marked usually with two re-
ference pegs, one on either side of the centre peg and close
to it. Some surveyors also drive two pegs at every ten chains.
Long stakes like those driven in America, with chainage
chalked on them, would be welcomed for firewood by the
villagers in England.
The American system of curve-nomenclature by degrees
of deflection angle could be also used for Gunter's chains.
The chord being expressed as 100 links, and the radius also
Digitized by LjOOQ IC
220 Preliminaiy Survey
in links, the degrees could be used just as they are in the
tables, i.e, a. curve of i° deflection angle per chord of loo
links has a radius of 57*30 chains. Compare Table LIV.
Curve-ranging is nearly always performed with a single
theodolite. There is another method with two theodolites
which dispenses with chaining ; it is rarely resorted to, and
will not be further described than by the remark that the
location of points upon the curve is determined by the inter-
section of two tangential angles from the two instruments.
Thus if the two instruments were set up one at the B.C. and
the other at the E.C. point in the fieldbook on p. 218 the
first point at 750 would be formed by the intersection of an
angle of i°'49 from the B.C. with one of i7°*47 from the
E.C, &c. The points are fixed by a picket held by an
assistant, first in line with one theodolite and then retreating
and advancing along that line until in the line of the other
theodolite. If the chain is not used at all, the curves had
best be put in telemetrically as described on p. 183; but if it is
used at all there is no time saved by dispensing with it on
the curves, unless in the case of a trestle across a steep ravine
on a very sharp curve, especially if the measurement is im-
peded by undergrowth. As such cases are exceptional it is
not advisable to have the time of two transit-men taken up
at the one spot on that account.
An ingenious instrument has been lately invented by
Mr. Dalrymple-Hay for ranging curves of radii expressed in
Gunter's chains, but it could be easily modified for curves of
any other radii. The principle is the adoption of an open
and clear graduation in terms of tangential angles, by means
of an extended horizontal arc. The index of the limb for
a curve of say 20 chains is set at i for the tangential angle of
I chain, at 2 for 2 chains, &c., and thereby simplicity of read-
ing is obtained.
This device can be fitted to any transit theodoHte for
eight to twelve guineas.
The objections to it are :
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Curve- Romging with Transit and Chain 221
Firsts that in ordinary practice a transit is wanted alter-
nately for running tangents and curves, and when on the
tangent the curve apparatus is only an awkward appendage.
Secondly^ it tends to make curve-ranging more mechanical
This cannot be called an advantage, because there is hardly
anything that calls for more intelligent skill in surveying
than this branch of it, and a system which enables a man to
do work without knowing why he does it will not be likely
to produce a good workman.
Reverse Curves
It > is always preferable, having regard to the running
gear of the rolling stock, to put in a piece of tangent between
the ends of a reverse curve, and when this is done the
problem is in nothing different from ranging two distinct
curves, one to the right and the other to the left, in the
manner just described.
When the main tangents are parallel it is impossible to
unite them by any other than a reverse curve ; but when
they are inclined to one another the only reason for pre-
ferring a reverse to a plain connecting curve is to obtain a
more rapid transit, in which case the reverse curve will
either intersect or lie wholly on the further side of one of
the main tangents. A sub-tangent at the point of contrary
flexure can be chosen, which will make any desired angle
with the main tangents ; and an endless number of cases
can be formed with a pair of curves of equal deflection and
unequal radii, or unequal deflection and equal radii. If
curves are wanted which will meet without any intervening
tangent, the lengths of sub-tangents are the fixed data, and
from the formula on p. 215 transposed, we have R=S.T.
cot ©, from which we can obtain the radius and other
elements.
The following problem is the commonest amongst true
i:everse curves, and suitable to tun>outs and cross-over
roads between parallel tangents.
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222
Preliminary Survey
Whatever the radii, the deflections A of the two curves
will each of them be equal to twice the inclination © of the
common tangent to either of the main tangents. When the
radii are equal the point of contrary flexure I bisects
the common tangent. When unequal they will be propor-
tional to the abscissae of the common tangent BI and IE.
Reverse curve of unequal radii between parallel tangents.
Fig. 70.
If the distance D on the square between the tangents
be given, and either L the straight reach, or BE the common
tangent be given, the other elements can be found from the
following expressions:
tan(H)=5; cos =^.
R^ BI .^^ IE
2 sin ^ 2 sin %
L=D cot © ; D=L tan
or if one of the radii R be given, and the distances L and
D or BE be given, to find the other radius.
^ BE - B I_ BE (BE - BI)
2 sin % 2D
Turn-outs are now generally curved from the commence-
ment of the switch, when the switch is of the split type,
such as is always used on passenger raihvays in England.
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I
Curve- Ranging with Transit and Chain 223
In America, the old-fashioned stub-switch, having its heel
at the point of curvature, is still largely used. The switch,
being common to both tracks, is straight, and the curvature
to which it is a tangent commences from its further end.
To lay out a turn-out for a stub-switch, instead of the
main tangent, take the line of direction of the switch when
directed towards the curve and which forms an angle / with
the main tangent of which
. . throw of switch
sm /=, , — ^ — .— ,
length of switch
Treat the end of the switch as the point of curvature,
and calculate from it as if it were B. The two tangents
will then not be quite parallel, but the curve can either be
run out to a parallel with the switch, by foregoing formula,
and produced to a parallel with the main tangent, or else a
separate calculation can be made, which want of space
precludes our giving here.
To Divert a Curve to a Parallel Tangent
Very frequently the end of a curve comes too near
some fixed point, and it is desired to throw it to one side
Fig. 71.
ot the other by advancing or receding the B.C. point upon
the old tangent
Let the distance between the two parallel tangents be>',
224 Preliminary Survey
and the length of advance or retreat of the B.C. point be x,
the deflection angle being A.
Then jif = .--^ — or y^=^x x sin A
sin A
To FIND THE Linear Advance of the Inner Rail
ON Circular Curves
In consequence of the outer rail being longer than the
inner, the joints of the latter gradually forge ahead unless
a special rail is put in occasionally, or the rails cut. Let
LA=difference of length between outer and inner
rails.
A=deflection angle or total curvature.
G=gauge.
General Rule
LA=Gx Ax -0x745 (log 8-24188)1
Or if LA be in inches and G in feet —
L A=G X A X -20944 (log 9-32 106)
Rule for Standard Gauge 0/4 feet 8| inches
LA in inches=A x -98611 (log 9*99392)
Rule for 3 feet 6 inch Gauge
Lx\ in inches=A x -73304 (log 9-86513)
Rule for 3 feet Gauge
LA in inches=Ax -62832 (log 9*79818)
Rule for Metre Gauge
LA in millimetres=A x 17-4533 (log 1*2418774)
Generally a consignment of rails includes some
* specials,' six inches or twelve inches shorter than the
usual length ; in which case we can find the. extent of curva-
ture "'needed for putting in a * special 'by substitiittng for
* Ten added to index No. of unity as with log. sines.
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Curve- Ranging with Transit and Chain 225
LA the decrement /, and inverting the equation the general
rule becomes A =^ x 57*296, and A has following values in
G
the several cases : *
A
Standard gauge, 6 inch decrement .... 6° "084
3 feet 6 inch gauge ,, ,,.... 8'i85
3 feet gauge ,, ,,.... 9-549
Metre gauge ; decrement of 2 decimetres . .11 -459
If the radius and length of the curve are given, find from
Table LIV. p. 379 the degree of curve, and A=_JL. Thus
100
if the curve were 1,146 feet radius, and 2,762 feet long,
from Table LIV. ^=:5°-oo,and A=?^?^^^^=i38°-i.
Transition Curves
The transition or parabolic curve is a means of toning
down the abrupt passage from a tangent into a circular curve.
It is called the *Uebergang' by the Germans and constructed
as a true parabola.
The objection to it in this form is that if a curve becomes
deformed or requires renewal the tracklayers are apt to put
in what one of them termed to a friend of mine * somethink
of a paraboler.' As the man literally knew as much as
the proverbial cow about conic sections, the curve was
unique.
Mr. Searles in his little book entitled *The Railroad
Spiral' has placed the matter on a much more practical basis,
using a combination of short circular curves, forming a close
approximation to the parabola, in which, with equal chords,
the curvature increases by an equal amount at the end of
each chord. He adopts ten angular minutes as the basis,
and uses chords of from ten to one hundred feet. Whatever
its length, the curvature of the first chord is 10, the second
20, the third 30, the fourth 40 minutes, and so on.
Digitized by VjOOQ IC
226 Prelinmiary Survey
Mr. Searles gives exhaustive and lengthy tables and
formulae for putting in any kind of spiral or for substituting a
spiral for an existing circular curve whilst retaining as nearly
as possible the same ground an^ the same length of line.
The object of the parabolic curve is not to dispense en-
tirely with the circular arc, but to attain gradually any suit-
able radius with which to form a circular connecting or
* crowning curve ' and leave it again by the same gradual
transition to join the further tangent.
The advantages are chiefly :
First, Less wear upon the running-gear.
Second, Less discomfort to the passengers.
Sharp curves and high speed are the factors that make
the demand for the transition -curve. The explanations
herein will deal firstly with a case suitable to a tramway or
workshop siding of about 40 feet radius, because a better
illustration can be given where the whole curve is shown.
Spirals for street-railways are becoming every year more
important. The radii cannot be increased, but the speed is
constantly being accelerated up to the utmost Hmit in
response to the public demand for rapid transit. Motors of
one kind and another have been invented which meet the
requirements of speed combined with safety. The company
protects its cars from derailment upon sharp curves by
means of guard rails, but cannot prevent its passengers from
the unpleasant swing and jolt when the cars turn a square
corner, nor can it avoid the wear and tear to the car wheels
and axles from the sudden cross-strain. * C^est le premier
pas qui coiite.'
As pointed out at p. 17 in Chapter L, the wear of loco-
motive tyres proves itself to depend largely upon the degree
of shock which is imparted. Once the running gear has
accommodated itself by its structural flexibility to a sharp
curve, the extra wear due to pressure against the outer rail
is comparatively small. The super-elevation of the outer
rail relieves it, and by the transition curve this elevation is
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Curve- Ranging zvitk Transit and Chain 227
commenced and gradually increased with the curve, so that
the unpleasant sensation of being tilted up in one's carriage
whilst still on the tangent is avoided.
The spiral, even on the sharpest curves, can be made to
follow very closely the same ground as a circular arc. It will
be seen from Fig. 72 how very slightly the two curves differ,
and yet the spiral commences with a radius of 573 feet
instead of one of 40 feet.
The spiral adopted in the following pages differs from
that of Mr. Searles in having for its base '2 of a degree
instead of ten minutes. For the sharper spirals the radii
are given in feet, for the others the degree of curvature per
100 feet chord. Sharp short curves are best put in by offsets
as shown in Fig. 73; flatter and longer ones by tangential
angles as explained at p. 215 &c.
Table LV. of Appendix gives the general elements of the
decimal spiral which are common to all the other tables. The
first column, n, gives the number of the chord from the
commencement. The second, n. c, gives the curvature in
that chord, which is the same whatever its length may be.
The length of the short chord c in feet defines the spiral :
thus No. 2 spiral is one in which ^ is 2 feet.
The third column, s, gives the total curvature of the
spiral from the commencement, in other words the angle of
deflection formed by a tangent at any point n with the main
tangent.^ The fourth column, k, is the inclination of any
chord to the main tangent, in other words the total curvature
opposite the middle point of said chord ; it is the basis of
the computation of the ordinates x and y.
The fifth column, /, is the tangential angle formed by the
long chord C at the point of spiral S with the main tangent.
It is needed for setting out the curve by tangential angles
similarly to a circular curve. In the other tables the column r
is the radius of curvature ; d the degree or deflection of 100
feet chord. The column x is the ordinate to the main tan-
* It is called by Mr. Searles the spiral angle,
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228 Preliminary Stirvey
gent from any point ;/, and y is the corresponding abscissa.
The column L gives the length of the spiral. The letter R
denoting the radius of the crowning curve does not occur in
the tables. C is a long chord ; c the short chord.
General formulae of the spiral :
j--= ^— ^ — ^ ; >e = — : tan i-=z-
lo lo y
^=S*. chord X sin k ; y=2, chord X cos k
c=-;
X ^ y . ^^ c
sm t cos / . n,c,
2 sm
2
sub-tangent along 1
main - tangent for > =zy—x, cot s
any value of 7i)
corresponding sub-tangent to ;?= . ^
sm J
It will be seen, on comparing these columns with the
corresponding ones in Mr. Searles' tables, how much simpler
they become by the use of the decimal degree.
The angles \ and /i indicated in Fig. 72, are formed by
the semichord of the crowning curve with a normal to the
main tangent and the central radius of the curve respectively.
They are found as follows (see Fig. 73) :
and as a check X -f a* + ® = i ^o°-
The point to be aimed at with spirals, as with ordinary
curve-ranging, is to obtain the easiest curve possible within
the limits prescribed by the situation. With tramway curves,
and in many other cases, it is often the crown of the curve
or apex which fixes its other elements. The formulae for
putting in a spiral to conform to a pre- determined apex
distance is somewhat longer thah the other, and requires
the finding of angles \ and /i. A piece of spiral is first fixed
♦ The symbol 2 is used for the summation up to any point of the
product of each chord by the corresponding value of k.
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Curve-Ranging with Transit and Chain 229
upon, and the distance of the point of spiral S from the
point of intersection I, and the radius R of the crowning
curve, are found by the following formulae :
^ AI. cos®— Jt /K
j<-= .^ v^;
2 cos \. cos /Li
and IS==y-|- tan A [(AI. cos 0)-^]-|-AI. sin (a) . . (2)
As will be presently shown, these formulae are very
useful for 'putting in a spiral upon a tramway ; but it
is not always practicable to fix the apex distance in this
way, neither is it always important to do so. The data for
a crowning curve of any required radius can be found by
the following formulae for a spiral of any chord-length and
any curvature.
Let R be the radius of the crowning curve.
Let % be the half total curvature.
Let s be the total curvature of one spiral.
Let X be the ordinate to end of spiral.
Let y be the abscissa to end of spiral.
Let IS be the distance of point of spiral from point of
intersection.
Let AI be the apex distance.
Then IS=y+ tan ® (R. cos j + a:)— R. sin i- . . (3)
and AI=^:^?ii±^-R (4)
cos ^^^
The radius of an approximately corresponding circular
arc, or trial curve, should be first ascertained by rule on
p. 215 for the whole curvature, remembering that the
spiral will always somewhat sharpen the rate of curvature
at the crown. The next operation is to select from one
of the tables a portion of spiral which will lead in a con-
venient manner into a crowning curve of not much less
radius than the trial curve just found. The third step is
to work out the data, either from a fixed apex distance by
(i) and (2), or from the fixed radius of crowning curve by
(3) and (4).
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230 Preliminary Survey
If not limited by a fixed apex distance, a radius of
crowning curve will naturally be chosen which forms a
harmonious transition from the spiral, but any radius may
be put in to join any spiral, provided s is less than ®. When
j=©, the two spirals meet, and form a continuous parabola,
or nearly so.
It is also possible to have two spirals of different length
and different curvature at the ends of a crowning curve, but
the discussion of such peculiar cases as that would be out of
place in this work.
Tramway Spiral
(Nos. 2 and 5 Spiral) Tables LVI. and LVII.
Fig. 72 represents the very common case of a square
street-turning, the streets being only 30 feet wide, with 5 feet
footpaths, and the centre-line is placed somewhat to one
side in order to give equal clearance at the comer, and on
the further side of the street. This, of course, is only a
secondary matter, and has nothing to do with the principle
of the spiral ; but the figure will serve to show how, when
the Umit of apex distance has been fixed by a trial curve, a
spiral can be put in which will differ very slightly from it,
and have more clearance at the tight corner.
The data are as follows.
A =90° ; ©=45° ; from which a 40 feet radius is selected
for a trial curve, and the subtangent ID, and apex distance
AT found by rule on p. 215. It is presumed that the limit
of curve-radius is 35 feet. AT is found to be equal to
1 6 5 7 feet. In order that the spiral may approach co-
incidence with the trial curve, the crowning curve must lie
outside, and the spiral inside of it. It is found that an
approximately constant relation exists between the ordinate
X of the end of the spiral, and the distance AA' between
the trial curve and the crowning curve at the crown, and for
curves of this character it will do to diminish AT by one-
tenth of X.
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Curve- Ranging ivith Transit and Chain 231
To select a spiral, on examination of Table LVI. it will be
seen that any of the points 11 to 14 are of a curvature
greater than 40 feet, so that in using them there will be no
danger of the crowning curve flattening the spiral. Which-
ever point we choose, we first reduce the apex distance by
one-tenth of .r. Supposing we use point 11. Our data are,
Fig. 72.
^=5209 feet ; 5=i3°-2 ; x^=-\'*j^ feet ; j=2i*88 feet. De-
ducting from A'l one-tenth of x we have AI= 16*40 feet.
\=9o°-^ (®4-j)=6o°-9
,i=9o°-^(0-j)=74°-i
and by (I) R= ^^-40 xcos 45°-r76 ^35.93 feet
2 cos 6o°-9Xcos 74°'i
and by (2) IS=2i-88 feet + 16-40 x sin 45°+tan 6o°-9
X (16-40 cos 45°— i-76)=5i-i6 feet
The transition at end of spiral will be from ^=52-09, to
R= 39*93, or a change of 15-16 feet.
If we were to end the spiral at point 13, we should have
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232 Preliminary Survey
AI= 1 6*5 7 — •28=16-29, ^^d adapting X and/i to the altered
data, R would become 3574 feet, and 18=51-36 feet. The
transition would then be from ^=44 -08 to R=35-74=
8*34 feet. The change would be more gradual, but the
crowning curve would be 1-19 feet sharper. In either case
the travelling public would be none the wiser, the rolling
stock would not be affected, and it would be difficult to say
which of the points of 11, 12, and 13, would be preferable.
If it were not necessary to fix the apex distance absolutely
and it was desired to have a crowning curve with a perfectly
harmonious transition, we should choose for a value of R
that of r next in order to the point we selected as the end of
spiral. Thus with «= 1 3 we should take R=4o-93 feet:
which is the value of r when ;^=i4. We do not then need
X or /i ; and by (3) —
18=2573 + tan 45° (40-93 x cos 18^-2+2-84)
—40-93 sin i8°-2
and by (4) —
Ai= 4o-93 cos I8--2 + 2-84 _
cos 45°
or 18 =54-67 feet and AI=i8-o7 feet.
The point of spiral would not be at an inconvenient
length, but the apex would be 1-67 feet nearer to the foot-
path.
When finally selected and calculated the curve should
be tabulated for reference and a working drawing made to
a large scale in the form of Fig. 73. Any practical track-
layer can then put it in or replace it with nothing more than
a chalk-line^ a set-square^ and a steel tape^ graduated to feet
and hundredths.
When the survey has been made previous to construction,
as it always should be if possible, the data of each curve
should be worked out and the rails bent at the rolling mills
to the required spirals and crowning curves, painted and
stamped so as to identify them. When there is no survey
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Curve- Ranging with Transit and Chain 2^^
and where no facilities for accurate bending are available at
the job, a number of spiral rails should be included in the
shipment, from 12 to 30 feet long, having holes for fish-
bolts at every two feet in all lengths exceeding 12 feet, so that
they will only need cutting at the most suitable value of n
for each particular curve.
<- -Alecs i
Fig. 73.
A template will be made for the centre line of the spiral
track, and each of the rails adjusted to it ; they will need a
little humouring with the * Jini Crow' to bring them to gauge,
not being perfectly concentric circular arcs.
Another and more exact method is to calculate the
spiral for the outer rail, and make a second working drawing
from it by laying off ordinates at every two feet equal to the
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234 Preliminary Survey
gauge. The ordinates for the crowning curve can be put in
from the tangent by formula on p. 207, or from the chord as
shown on Fig. 73 by formula in Trautwine's * Pocket Book/
The Horse-Shoe Spiral
(Nos. 5, ID, 15, and 25) Tables LVII. to LX.
When the total curvature approaches a semicircle (see
Fig. 74) it is impossible to run the tangents to an intersec-
tion. Long before so great a curvature is reached, as shown
in the example, it becomes very inconvenient to do so, and
recourse should be had to the apex tangent HAK
The deflection A of the main tangents can be measured by
Fig. 74.
running an auxiliary base-line DG from one tangent to the
other. The sum of the interior angles IDG, IGD is equal
to A, but if the instrument is kept to the astronomical bear-
ing as explained on p. 203, the difference of bearing
between the two tangents will give their deflection, even
though two or three auxiliary base-lines have been run
between them.
The point A is generally fixed by inspection of the
ground and measured ; HAK being set out with angle
IHK=0.
In exceptional cases, such as when A falls in the middle
of a torrent, H and K being on the sides of a precipitous
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Curve- Ranging tvith Transit and Chain 235
ravine, it is inconvenient to measure HK with the chain,
and telemetric measurement may be also difficult on account
of heavy brushwood ; we can then measure the distance HG
from the end of the auxiliary base to the point opposite
the intended apex, and calculate AH by the following
formula :
AH=^^^,^^- HG. cos®- DG. sin (©- 1//). cot
2 sin
(5)
Although lengthy, the only angles involved are and
the t\to interior angles IDG, IGD, so that the formula is
less tedious than it looks.
When the distance AH has been obtained, the radius
of a circular curve is calculated which will join the tangents
and pass through A by the formula
R'=AH. cot ® (6)
R' should be from 5 to 10 per cent, greater than the
curve-limit of the survey, in order to allow for the diminu-
tion due to the spiral. If R' is too small, a fresh point A'
must be chosen somewhat further from the intersection
point. It is generally possible, even in very rough country,
to obtain a sufficient approximation to AH by stepping or
even guessing in order to be within the limit of radius.
For instance, supposing to be 76*25°, and the curve-limit
250 feet. AH must not be less than 250 x tan ~=
2
198*8 feet, therefore before selecting A, HK is roughly
measured to be sure that it is over 400 feet. If the object
be the staking out of the curve for actual construction, it
will not do to put in A with the tacheometer ; it must be
fixed to a tenth of a foot by the chain, but for the first
approximation to HK it can be measured by the tacheometer,
by pacing, or by the aperture of the two-foot rule, explained
on p. 72. Flags are fixed at two points on the first main
tangent, such as D and S, and the assistant, keeping himself
236 Preliminary Survey
in line with them, moves forward along the tangent until
he is in the line of HK, the direction of which is given by
the surveyor at H with either the theodolite or prismatic
compass.
When a point A has been found which appears to suit
the ground best and also to be within the curve- limit, it is
finally determined either by the exact measurement of AH
(=HK) or by calculation with formula (5).
To select a spiral, examine any of the tables suitable to
the class of curve intended and find one which attains
a radius r= or somewhat > R' within a suitable length of
spiral. For instance, although it is possible to apply No. 2
spiral to a 1° curve on a trunk railroad, it would be no
use whatever, because it attains a radius of that degree at a
distance of eight feet.
No. 50 would attain that radius in a length of 200 feet,
or No. 100 in 400 feet, so that one of the last three spirals
would be chosen.
At the transition point T or end of spiral, s should be
somewhere between one-fourth and one-half of % ; one-third
is best.
When the spiral is chosen, angles X and ft are calculated
(see p. 228) and the data of point of spiral KS and radius of
crowning curve R are given by the following formulae :
j^^ AH. sin®--^' .
2 cos X. cos /Lt
KS = y-^. cot© + ^^ COSMIC (8)
Example (see Fig. 74). Let the curve limit be 250 feet.
0=76° 2 5. AH is measured=2i7 feet.
Selecting No. 15 spiral at point 15, we have further data,
^=32'-o6 ; j^=22o'-96 ; j=24°'oo. From these we obtain
X = 39°-875, /i = 63^-875, R=2i7 x sin 76°-25- 32'*o6^
2 cos 39°-875. cos 63°-875"=264'-43 ; KS=22o'-96— 32'-o6.
cot 76^-25 + ?x 264^^-43 cos « 63°-875 =318' -69.
' ^ sm 76°*25 ^ /o o y
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Curve-Ranging with Transit and Chain 237
This will give a perfect transition ; r being 286*5 ^^ ^^^
transition point. The length of spiral, 225 feet, will be
ample for a curve of^ this character, being as long as many
narrow-gauge trains.
If we had chosen No. 10 spiral at point 10, we should
have had a crowning curve of 2 74' -32 radius, the last value
of r being 2 86' '5 2. The transition would be equally good,
but the length of spiral, 100 feet, would not be sufficient to
be thoroughly effective.
Spirals up to No. 15 inclusive are more suitable for
setting out by the ordinates of x and y^ although values of /,
the tangential angle, are given from point 10 onwards in
No. 15 spiral. Su{)posing a horseshoe of total curvature 170
degrees were required with a curve limit of 200 feet radius,
it would be' advisable to adopt. No. 15 spiral at point 21,
rather than No. 10 spiral at point 13, and the ordinate x of
8274 would be inconveniently large for tape measurement.
The Mountain Spiral
Tables LX. and LXI.
This term has been chosen to distinguish spirals Nos. 25
and 50 as suitable for sharp curves in a standard-gauge line
or ordinary curves in a narrow-gauge line.
The ordinates of x and y will not be used in the field,
as they become inconveniently large. The curve will be
ranged by the tangential angle /, similarly to the ranging of
circular curves.
Example, Having a total curve deflection of 30°, it
is desired to put in a 5° crowning curve with a uniform
transition from a spiral.
Choosing a No. 25 spiral we find the transition point is
at «=6 and the data are :
©=15-0; 5'=4°-2 ; :v: = 3-97 ; y^\^^'^\.
r=:i, 193-6; R= 1, 146 feet.
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238 Preliminary Survey
By (3)IS ~ 149*91 + tan i5°-o (1,146 cos 4^-2 + 3*97)
— 1,146 sin 4*'*2 = 373*28 feet
By (4) AI = ''^^'-^^Jf°^. + 3'97 - M46 = 41-33 feet
The apex distance will here only differ by 0*9 foot from
that of a 5° curve throughout.
The Trunk Line Spiral
Tables LXI. to LXIII.
This term has been chosen to distinguish spirals 75 and
100, but No. 50 is also applicable in many cases.
Example. Having a total curve deflection, of 45° be-
tween the two main tangents, it is desired to put in a 1°
crowning curve, so as to form a uniform transition from the
spiral.
Choosing a No. 100 spiral, we find the transition point
for a 1° curve to be where «=4, and the data are :
(H)=»=22°-5; j=2°; x= 523 feet; j=399-94; R=5,73o feet
By (3) IS=;399-94 + tan 22°-5 (5,730 cos 2° + 5-23)-
5,730 + sin 2°=2,574-i9 feet.
By (4) AI=5.73o cos 2^-5-23 .5,730=474.00 feet
^ ^^' cos 22*5 ^''^ ^'^
The apex distance will here only differ by two feet from
that of a 1° curve throughout. For practice the reader
might take the apex distance, 474 feet, as the fixed quantity
and find the radius and distance IS by formulae (i) and
(2). The results will agree with the foregoing assumptions
within a small fraction of a foot, but inasmuch as AI is
small as compared with R, in order to get R correct to two
places of decimals, AI should be given to three or four
places, which is not necessary for practical curve-ranging.
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Curve- Ranging ivith Transit and Chain 239
Ranging the Spiral from an Intermediate Point
Hitherto the spiral has been supposed to be visible from
end to end. Such is generally the case, because when the
first trial lines are run, the bush is cleared, and it is only
when unusual obstructions, such as sharp rock-points, inter-
fere with the view that the spiral cannot be ranged con-
tinuously. When, however, from any cause, a break or
turning point has to be made, the operation is analogous to
that described in circular curves on p. 21 7, except that the
tangential angle / is not half the deflection or 'spiral
angle s.
Supposing a No. 25 spiral is being ranged from a main
tangent whose bearing is 33° '34, and it is desired to make a
turning point, where «=i6, the curve being to the right.
Point 16 will be ranged from j", with a tangential angle /*=
9-33; consequently on a bearing of 33*34 + 9'33=42'67,
on shifting the instrument to n^ we require for the tangent
at that pointabearingof 33°'344-J', that is 6o°-54; we there-
fore clamp the vernier at 42° -6 7 for the back sight, clamp
the external axis, reverse the telescope to the forward
position, and set the vernier to 60° -54, which will bring the
line of sight on to the tangent.
In ranging the remainder of the curve either by ordinates
or tangential angles, a separate set of values of k'^ od^ y,
and /' have to be used as given in Table LXIV. If the
curve is ranged by ordinates, the values of x' and y have
to be calculated by simple proportion from those of No.
100 spiral given in the table; for instance a No. 25 spiral
will have values of x and ^=25 per cent, of those in the
table. They are found by the same equation as those for
the primary tables, viz. jc'=S. chord, sin k', y=^, chord, cos
k\ If the curve is ranged by tangential angles /', no cal-
culation is required, as /' is the same for all spirals.
By keeping the instrument on the astronomical or any
continuous bearing, the transition point or end of spiral can
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240 Preliminary Survey
be very conveniently checked, however many turning points
may have been taken. For instance, supposing a curve to
the right, and calHng the bearing of the main tangent B, the
bearing of the transition point when viewed from S will be
B + /. With one turning point it will be B + / + /' and so
on. When the transition point is reached, the direction of
the tangent to the crowning curve is found by taking a
back sight to the last turning point 'with the bearing B + z
+/' &c., and reversing the telescope, with the external axis
clamped, and the vernier released, the line of sight is then
set to the bearing B+x. It can easily be seen on the
ground whether this is a tangent to the last chord of the
spiral, and so check the calculation.
It is also useful as an independent check to lay off and
measure the subtangent from the transition point, or if
necessary from any turning point, to the main tangent,
which is done by the formula on p. 228. *
Wyes and Loops
When a substitute for a turn-table is needed, it is usual
in America to put in a pair of curves turning to the right
and left on the same side of the main track, and tenninating
in a common tangent at right angles to the main track ; the
engine runs round one curve into the common tangent, and
backshunts on to the main track through the other curve,
so coming out end for end. As a substitute for a cross-
over road on a double track a loop is sometimes made, by
which the train, after describing a complete circle, occupies
the other track, but in a reverse position.
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241
CHAPTER VIII
GRAPHIC CALCULATION FOR PRELIMINARY
ESTIMATES
The surveyor is compelled to form his first estimates of cost
without detailed measurements with chain, tape, and rule,
but as little as possible by guess-work.
If the estimate is based upon a walk-over survey and
sketches, he must rely upon his experience of similarly con-
structed works, and he will judge the cost mile by mile,
according to its general character.
When sufficient time is allowed to produce a topo-
graphical map of more or less accuracy, the operations of
the preliminary surveyor, although more rapid, are analogous
in principle to those of the executive engineer, who succeeds
him with more time at his command'.
The earthwork is measured on a profile plotted from the
contours ; the trestles scaled for distance and heights from
the same profile, as also the bridges and'culverts. The
quantities in all cases are usually taken from tables.
The use of quantity diagrams combined with the slide-
rule is much more suitable to this class of work than long
tables of figures and elaborate formulae.
Calculation by Slide-rule
Before entering upon the subject of estimates, it will be
necessary to describe somewhat fully the use of the slide-rule.
The instrument itself is described on p. 361, but it is
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242 Preliminary Survey
desired to give a detailed description of its uses as applied
to preliminary survey.
The printed explanations sold with the slide-rule con-
tain directions for its use, but it is preferred here to arrange
them in a different form.
The easiest way to become familiar with the instrument
is to look upon it as a —
I. Geometrical proportion or rule of three sum. Place a i
of the slide opposite to a 9 of the rule, and give the figures
on the right-hand side of the rule ten times their indicated
value. Then, beginning with the central i of the slide, it
will be found under 9 of the rule, 2 under 18, 3 under 27,
4 under 2i^^ i*i under 9-9, 2*3 under 207, 37 under 33*3,
and so on, every value of the rule corresponding with the
slide X 9. We may put it in the form of multiplication as
9x3=27x1, or in the form of division as t= V> ^^ ^^ ^^
ordinary rule of three form 9 : i :: 27 : 3. In any case, the
figures on the rule are opposite to their proportionals on the
slide.
We may give the figures on the slide ten times their in-
dicated value, and those on the rule 100 times, but the pro-
portion remains the same. Thus, in the above illustration
37 would become 37 on the slide, and it would be under
333 on the rule.
One of the most useful cases of proportion by the slide -
rule is the finding of the angular value of odd distances in
railway curves by tangential angles, or in getting the loga-
rithms of intermediate numbers or angles by interpolation.
Example, What is the cosecant of 35° 11' i3"'3?
Log. cosec. 35° ii'by table . . . = 10*2394308
Tab. difference for 60" = - 1,791 ;
i^^x 1,791 by slide-rule . .= ~397
60
Log. cosec. 35° II' i3"-3 . , = 10-2393911
This is done by placing the 1,791 of the slide opposite to
the 60 on the rule, and looking for the value on the slide
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Graphic Calculation for Preliminary Estimates 243
corresponding with 13*3 on the rule. It is more exactly
done by the lower scales of the rule, although it involves
two operations. If the upper scales be used we can only
read 1,790 instead of 1,791, but there will still be no mis-
take about the 397 to an eye which has had a little practice
in estimating the value of the subdivisions. We will, how-
ever, take the lower scales.
Giving the slide 1,000 times its indicated value, we place
the 1,791 opposite the 6 of the rule, which we call 60,
and we see that the 133 on the rule has been overshot
altogether, the left hand i of the slide being opposite to
the 3*35 on the rule. We slide back until the right hand i of
the slide corresponds with 3-35 of the rule, and we shall be
still in adjustment for giving the proportion of ^\%^ as
before. Opposite to the i '33 of the rule (which we give the
value of 133) we find the exact figure of 397 on the sHde.
This operation is again much simplified by decimal
graduation. z \
The whole of the succeeding examples o£ sc^l^ rechi^-
tion and plotting, weights and measures and^oinage, a^e
based upon this principle of proportion or g^oAi^ricar
ratio. ^.
2. Multiplication. When, as described on last page, we-
put the I of the slide opposite the 9 of the rule, we mul-
tiply it by 9, and any other figure on the slide is likewise
multiplied by 9 on the rule opposite to it ; therefore, if we
want to multiply by any number, we place a i of the slide
opposite that number, and the slide and rule will be in adjust-
ment to read like a column of a multiplication table headed
by that number. Thus, if we wish to multiply 57 X35, we
place a I of the lower slide- scale over the latter number.
Every figure on the slide will then be opposite to 35 times
its value on the rule, and 57 will be found opposite to 1,995
exactly.
3. Division, This is the converse of multiplication
contained in the same principle of proportion.
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244 Preliminary Survey
Thus, in the preceding example, to divide 1,995 t>y 57
we adjust the instrument in that ratio by placing the 57 of
the slide over the 1,995 of the rule, or vice versa, and read
off the quotient 35 under the corresponding i.
4. Involution and Evolution. This is done by simple
inspection without using the slide. Any of the figures on
the lower scale of the rule are the square roots of those on
the upper scale, and vice versa. They are made to * teach '
with one another by the brass marker. Thus the 3 on
the lower scale is under the 9 of the upper.
Example a. Find the square root of 718. Direct the
brass index to that figure on the upper scale of the rule,
and on the lower scale will be found 26-8.
Example h. Find the square of 718. The answer will
obviously have six figures. Placing the marker at 718 on
the lower scale, the upper scale is, as near as one can read,
516 — that is to say, 516,000. The exact answer is 515,524.
If we wanted it exactly we should have to make a double
inspection with the aid of the lower slide.
Thus, 718 X 700 = 502,600
718 X 18 = 12,924
515,524
and in so doing we should scarcely save any time, because
directly we begin to have to put down figures we might as
well work it out.
The Slide-Rule as a Universal Decimal Scale
Nothing can compare with the slide-rule for plotting
in the field. In the Mannheim rufe one of the edges is
bevelled so as to be used on the plot, and is graduated to
millimetres. If we set the slide to read with the rule the
proportion of millimetres to the given scale of feet, chains,
miles, or what not, we can then apply the millimetre scale
directly to the paper without any further calculation, and this
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Graphic Calculation for Preliminary Estimates ^45
with the most awkward scales imaginable. Example (a).
Let it be desired to plot with the slide-rule from a scale which
was intended to be 6 inches to the mile, but which by con-
traction of the paper has shrunk to 5*95 inches to the mile.
On the back of the rule in the table of useful memoranda
will be found under mesuresanglaises^^itd (foot)=:M. 0*3048,
t,e, 12 inches= 304-8 millimetres. If we then make 304*8
on the slide correspond with 12 on the rule, we shall find
opposite to 5*95 on the rule 1 5 1 on the slide. This being the
scale value of a mile in millimetres, we can make 151 corre-
spond with 5,280, to get the number of millimetres correspon-
ding with feet, or 151 with 8,000, to find the value for links.
Thus, adjusting the 151 of the rule opposite the 80 of the
slide, we have for, say, 23*1 1 chains on slide 43*62 on the rule.
The second place of decimals, that is, single links, can scarcely
be relied upon at this scale with a small Mannheim rule,
but as it cannot be estimated by the eye on the millimetre
scale, the rule will give the measurement as closely as it can
be scaled. To use the lower scale we must first reduce the
proportion of -^^^ to its equivalent of ^V ^"^ order to keep
within the range of the rule. See p. 243.
The way in which odd scales of paces can be figured off
at a glance in feet or links is explained on p. 57 of chapter
on Route Surveying, and they can be plotted on the plan
without any further reduction.
If the scale of miles, chains, or feet, to which it is
intended to plot with the slide-rule, be given on the plan,
the first process of determining the value of a mile, or other
English measure of distance, in millimetres to scale, is dis-
pensed with by merely applying the millimetre scale to the
paper, and then adjusting the slide-rule to the proportion.
Railway Gradients
The nomenclature of gradients on English parliamentary
maps for roads or railways is the ratio of perpendicular to
base, and is expressed as inclination i in —
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246 Preliminary Suruey
The nomenclature of railway promoters abroad and
ordinary business men is the same ratio expressed in feet
per mile.
Engineers have adopted percentage, such as i per cent,
instead of i in 100, 2\ per cent, instead of i in 40 &c., as
being the most convenient, both for levelling and contouring.
Side slopes are named by the ratio of base to perpendicular,
as 2 to I, &c., or sometimes in degrees of slope from the
horizontal.
To Reduce Percentage to Feet per Mile by the
Slide-rule
As I per cent. =52 -8 feet per mile, place a i of the lower
scale of the slide over the 5 2 8 of the lower scale of the
rule, then 5 of the slide will be over 26*4, that is to say, '5
per cent, is equal to 26*4 feet per mile, or 5 per cent, equal
to 264 feet per mile.
Example i. What percentage will be a grade of 324
feet per mile ? Opposite the 324 on the rule we find 6*13 on
the slide.
The left hand i of the slide gives grades of i to 1*9 per
cent., or 10 to 19 per cent.
The right hand i gives grades of 1*9 to 10 per cent., and
19 to 100 per cent.
To Reduce the Inclination, such as i in 20, i in
30, &c., to Feet per Mile
The result is obviously 5,280 divided by the ratio.
Place a I of the lower slide-scale over the ratio on the rule,
and read off the feet per mile on the slide opposite 5,280 on
the rule.
Example 2. How many feet per mile are there in a
grade of i in i8-i ?
Place the left hand i of the slide opposite the i8-i on
the rule, and opposite 5,280 on the rule will be found 292*1
on the slide.
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Graphic Calculation for Preliminary Estimates 247
To Reduce the Inclination as above to the
Percentage
Place the slide as above, but read the percentage on the
slide opposite the right hand of the rule. Thus, in the
preceding example the 5*52 is found opposite to the right
hand i of the rule.
To Find the Angle of Slope corresponding to any
Gradient
1. Reduce the grade in whatever form it is given to its
equivalent percentage as just explained.
2. Find the angle from the line of tangents by placing
it in its initial position, and reading off the angle on the slide
opposite to the percentage on the rule.
Example 3. What is the angle corresponding t® 270
feet per mile ?
Placing the 270 on the slide above the 5,280 on the rule,
we have opposite to the i of the rule 5-11.
Reversing the slide to line of tangents, we have opposite
to 5'ii the angle 2° 55', or 2^-92.
Example 4. What is the angle corresponding to 27 feet
per mile ?
Similarly to the preceding example we find the percent-
age=*5i, and this is less than the line of tangents will give.
Without reversing the slide we bring the mark for single
minutes situated at 3*44 on the slide, and indicated by a
single stroke i, opposite to the i on the rule, then the slide
will give tangents or sines of small angles which are alike
proportional to those angles. The following table shows
that '51 per cent, lies somewhere between 6' 30" and
34' 18" of angular value, and we find the exact angle 17*5'
under the 5 1 of the rule.
Example 5. Suitable for flow of rivers. What is the
angle of slope corresponding to a fall of six inches to the
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248
Preliminary Surt'cy
mile ? We see from Table XXIX. that the percentage is
somewhere between -002 and 'oi, and the angle between
2I' and 20''. Placing 5 over 5,280 we find the percentage
•00948, and by the double stroke mark for seconds the
angle i9*5''.
Example 6. What is the angle of a i in 7*32 slope?
With a I of the slide over 7-32 we find the percentage to be
13*66, and from the tangent scale in its initial position angle
=7« 46'.
-Of leading values of slopes in percentage and angle
for checking the slide-rule.
Table XXIX.-
Feet per
mile
S = tan
of slope
Percentage
Angle
(sexagesimal)
Angle
(decimal)
0*1
•0000189
•00189
0° 0' 3-9" :
•001 1
0-528
•oooiooo
•oiooo
0° 0' 20-6"
-0057
i*o
•0001894
•01894
0° 0' 39^o"
•oio8
5-28
•ooiooo
•lOOOO
0° 3' 25-8"
•0571
lo-o
•001894
•1894
0° 6'30^o"
•1083
52-8
•0X0000
I^OOOO
0° 34' x8^o" 1
•5717
100 -o
•01894
1-894
t 1° 5' I4-0" 1
10833
5280
•lOOOO
lO^OOO
50 42' 38-0" ^
57106
1000 'O
•18940
18^940
10^ 43' 45 'o" 1
107292
All the values of S are also equal to the sines except the
last three, which are somewhat larger.
To interpolate between the values in this table.
Example 7. Find all particulars as in the table for a
grade of 7 feet per mile. 8 = 7 x •000189 = '00134.
Percentage=7 x '0189=' 134. Angle=7 X39"=4' 2>2l' *
Squares and Square-roots of Small Decimals
Table XXX. will facilitate the use of the slide-rule in the
involution and evolution of small decimal numbers, specially
of those used in Kutter's formula.
Example 8. Find the square of -0029. From the table
we see that this must be between '00000625 and 'ooooi, and
by inspection of the rule find it to be '00000841.
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Graphic Calculation for Preliminary Estimates 249
Table XXX. — Of squares and cubes for checking the slide-rule.
Number
Square
•00000625
Cube \
Number
*0025
1*414
•00316
•ooooi
1*50
•005
"000025
1*732
•0075
•00005625
2'0
•01
•0001
•oooooio 1
2*1544
•025
•000625
•0000156
3*1623
•0316
"OOI
•0000316
4^64i6
*o5
•0025
•000125
lo^o !
•07s
•005625
•000422 j
•001 '
21-5443
*!
•01
31-6228
*25
•0625
'0156
46-416
•316
•l
•0316 i
lOO^O
*5
*25
•125 !
215*4435
'75
•5625
•422
316^227
I'O
I'O
i-o
464*1587
1*25
1*563
1*953 1
lOOO'O
Square
2^oo
2^25
3*oo
4'oo
4'64i6
lO'OOOO
21*5443
lOO^O
464*1587
lOOO'O
2154*4
TOOOO'O
46415*87
lOOOOO^O
215443*5
lOOOOOO^O
Cube
2-8i8
3*375
4-196
5*359
lO'O
3 1 •6228
lOO^O
lOOO^O
toooo^o
31622^8
lOOOOO^O
lOOOOOO'O
lOOOOOOO^O
3i622777^o
lOOOOOOOO^O
lOOOOOOOOO^O
Example 9. Find the square root of -00095. From the
table we see that this must be between -025 and •0316.
From inspection of the rule it is determined as -0308.
If the student will take the trouble to work out one or
two of these sums in the ordinary way, and then endeavour
to obtain one or two decimal square roots by the slide-rule
alone without the table, he will at once see what an assist-
ance it is.
Cubes and Cube-roots of Numbers
Invert the slide, keeping the numerical scale upwards.
Adjust it so that what was the left-hand upper scale of the
slide becomes the right-hand lower scale, with the figures
upside down. Now place the number to be cubed on the
slide over the same number on the rule, and read off the
cube on the slide opposite to the left hand i of the rule^
Example 10. Find the cube of -373. From the table we
see it will be somewhere between '0316 ^nd '125. Placing
the 373 of the slide over the 373 of the rule, we find above
the left hand i of the rule 519, which we read '0519.
To extract the cube-root with the rule as above, place
the number on the slide over the left hand i of the rule,
and search for a number on the slide which is opposite
to the same number on the rule, that is the cube root.
• Digitized by VjOOQ IC
250 Preliminary Survey
Example 11. Extract the cube root of 1,575. We see
from the table that it will be somewhere between 21 and 10.
Place the left hand i of the rule under the 1,575 of the slide,
and it will be found that the coincident number is 11-63.
Example 12. Extract the cube root of '0954. From the
table we see that it will be between '5 and '316, and placing
the left hand i of the rule under the 954 of the slide, we find
the coincident number to be 457, which we write as -457.
, Railway Sleepers
Method of obtaining by a single adjustment of the rule,
when the dimensions and pitch are given, the number,
quantity of cubic feet, quantity of cubic yards, ^ weight in
tons of 2,240 lbs., and price in sterling or dollars. Place
the pitch in feet and decimals upon the slide opposite 5,280
on the rule.
Read the number per mile on the rule in thousands
opposite the i of the slide.
Read the quantity of cubic feet per mile on the rule in
thousands opposite the value of A in Table XXXI.
Read the quantity of cubic yards per mile on the rule in
hundreds opposite the value of B in the table. Read the
quantity of tons per mile on the rule in tens or hundreds, op-
posite the value of D in the table. Read the price per mile
in sterling or dollars, in hundreds or thousands, from E or F.
Example i. Find the above desiderata for standard-gauge
sleepers 9 feet x 10'' x6", pitched 2' 9'', at 2s, apiece.
Place the pitch 275 on the slide opposite to 5,280 on
the rule.
Then the i of the slide is opposite to 1,920 No. on the rule.
375
»
(A)„
»
7,200 c. ft.
1*39
»>
(B),,
J>
267 c. yds.
5-86
»>
(D)„
J»
1 1 2*1 tons
100
9»
(E)„
»
192/.
* The quantity of cubic yards is required as a deduction from the
ballast.
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Graphic Calculation for Preliminary Estimates 251
• Example 2. Find the above particulars for narrow-gauge
sleepers, 6' 6" x;" X4'', pitched 2' 3", at 35 cents apiece.
Place the pitch 2 2 5 on the slide opposite to 5,280 on
the rule.
Then the i of the slide is opposite to 2,350 No. on the rule.
1-2 7 „ (A) „ „ 2,985 c. ft.
„ 4-69 „ (B) „ „ iio-ic. yds. „
1-98 „ (D) „ „ 46-5 tons
350
(F)„
522
Table XXXI. — Quantity, weight and cost table of railway
sleepers for standard and narroiv gauge.
C.
D.
A.
Cub. feet
B.
Cub. yds.
Wt. in
Wt. in
Dimeosion
lbs. of
tons of
Ft. )
<in. xin.
in one
sleeper
in ten
sleepers
one
sleeper at
35 lbs. per
sleepers
at 35 lbs.
eft.
per c. ft.
4
6
4
0-667
0-247
23*3
1-04
5
7
4
0-972
0-360
34 -o
1-52
6
7
4
1-167
0-432
40*6
1-82
6i
7
4
1-267
0-469
44*3
1-98
7
7
5
I -70s
0632
59*7
2-67
8
8
6
2-667
0-988
93*5
4-17
8
9
6
3-000
I-IIO
105-0
4*69
8
9
7
3-500
1-296
122-5
5*47
8
10
6
3*333
1*233
116-5
5-20
8
10
7
3-889
1*441
136-0
6-07
8
10
8
4*444
1-645
155*3
6-93
8
12
8
5*333
1976
i86-8
8-33
^
8
6
2833
1-048
99*3
4*42
8
9
6
3'i88
I -181
II I -6
4*98
8
9
7
3*719
1*377
130-1
5*8i
8|
10
6
3*542
1-312
1240
5*53
8r
10
7
4-132
1-530
144*8
6-45
8
10
8
4*722
1-750
1650
7*39
8
12
8
5-667
2-095
198-0
8-85
9
8
6
3-000
I-IIO
105-0
4*69
9
9
6
3*375
1-250
118-0
5-27
9
9
7
3*938
1-456
138-0
^•Jl
9
10
6
3*750
1-388
131-1
5-86
9
10
7
4*375
1-619
152-8
6-84
9
10
8
5-000
I -851
175-0
7-81
9
12
8
6*ooo
2*221
210-1
9*37
E, cost of 1,000
sleepers in £ sterling
at 1/6, 75; 1/9, 87-5;
2/0, 100; 2/3, 1 12-5 ; I
2/6, 125; 2/9, 137*5; ,
3/0, 150; 4/0, 200; 1
5/0, 250. I
F, cost of 1,000
sleepers in $ at 35c. ,
350; 50c., 500; 70c.,
700 ; looc., 1000.
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252 Preliminary Survey
Earthwork
It is becoming very customary to use diagrams for
measuring earthwork for preliminary estimates. They are
usually drawn to large scale, and are correct to a fraction of
a yard for level cuttings. To avoid folds, they are in a thin
quarto or folio book, which becomes quite unwieldy. The
little diagram given on Plate V. is quite close enough for
a preliminary estimate. It has the advantage over tables —
first, that it is all on one page ; secondly, that it is applicable
to any base and any slope. It will be observed that for
any additional depth of cutting, the quantities for varying
width of base increase as the ordinates of a triangle, but the
quantities in the side- slopes increase as ordinates to a
curve. The line marked base=o is a datum line for slopes
only ; ordinates measured upwards from it to the curve give
quantities in the two side- slopes for loo feet length.
It is also a datum for the central portion alone ; ordinates
measured downwards from it to the line marked with the
given width of base give the quantities in the central
portion.
When the total quantity, central and sides, is required,
the ordinate is measured clear through from the line marked
with the given wjdth of base upwards to the curve. This
has to be done with the dividers, or with a slip of paper.
If with the former, and much work has to be done, a piece
of -tracing-cloth should be gummed over the diagram to
preserve it. The length of the ordinate is then applied to
the vertical scale of cubic yards. When considerable work
is to be done from the same width of base a piece of paper
can be gummed down so as to cover all below the datum
which is being used.
Although the side-slopes do not vary directly as the depth,
they do vary for the same depth directly as the slope. For
instance, a 2 to i slope contains twice as much as a i to i
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Graphic Calculation for Preliminary Estimates 253
for the same depth. We can therefore obtain other values
by a simple proportion.
The diagram is calculated from the prismoidal formula,
which is a modification of Simpson's rules for land areas
and marine displacements, and may be relied upon within
o*5 per cent. If used with a paper slip instead of dividers,
it will stand a considerable amount of usage.
The smaller diagram, Fig. 76, is only an enlargement of
that portion of the larger one between o and 25 feet of
vertical depth, in order to obtain a clearer reading.
The diagram gives the contents in cubic yards in 100
feet length of level cuttings of any base from 6 to 28 feet,
and having side-slopes of i to i or i^ to i. For any
less distance than 100 feet, such as Gunter's chains of
66 feet, or odd distances, the quantities given by scale are
obtained by simple proportion. With the slide-rule, by
placing the i of the slide opposite the 66 or other fraction
of 100 feet on the rule, and reading opposite the full
quantity on the slide, the odd quantity on the rule.
Example, To find the cubic contents of 100 feet length
of embankment, base 18 feet, side-slopes i to i, and 27 feet
deep. On the larger diagram, tracing the base line marked
18 to where it intersects the vertical 27, we scale from said
intersection up the vertical to the lower curve, and applying
the quantity to the end vertical, we find it measures exactly
4,500 cubic yards, which is the required quantity in 100 feet
length.
The cubic content of a level cutting of any length and
any slope may be obtained from this diagram by a simple
rule of three sum, or by direct scaling with proportional
compasses.
Rule. Multiply the quantity in i to i slopes by the
given ratio, and add to the product the quantity for the given
base. If the slope is given in degrees from the vertical,
multiply by the tabular tangent ; if from the horizontal, by
the cotangent of the slope.
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254 Preliminary Survey
Example, Required the cubic content of a piece of
level cutting. Base 24 feet, height 27 feet, slopes | to i,
length 13 feet. By diagram, the quantities are for the
base 2,400, and for the side slopes 2,700.
(2400 + ^*^^^^^ ') X-i^=575 cubic yards.
\ 4 y 100 ^
When the cuttings are in side-hill they sometimes have
to be equalised, even on preliminary survey, for which a
graphic method is given on Fig. 77, which represents a section
of * hog-backed ' cutting ABCDHG, in which the crooked
portion ABC is replaced by the straight line CFE ; BE is
drawn parallel to AC ; to find E.
Fig. 77.
The triangles EBC, BEA are equal, being on the same
base and between the same parallels ; deducting the common
portion EFB, the remainder EFA, which is added by the
equalising line, is equal to the remainder FBC, which is
subtracted. The crooked portion DCE is replaced by
another equalising line in a similar manner. Finally the
one sloping surface line is replaced by a horizontal equivalent
as follows :
Let DE, Fig. 78, be the final equalising slope line.
Assume a point a in the centre line of the section above
DE, and mark ^, b' by a parallel run up from HG. Find b''
by a parallel run up from ^E to D ; halve the error b'b^' in b"\
and b^'" a' b'" will be the horizontal surface line of an equiva-
lent level cutting. This should be checked by a parallel
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Graphic Calculation for Preliminary Estimates 255
run up from b^'" E to D, and if not quite correct, the error
bisected a second time.
^^^^^^^S
Fig. 78.
The diagram, Fig. 79, is a rough approximation curve
constructed from a few actually measured areas, and may be
used only within the limits given.
Fig. 79.
Example, What would be the equivalent level cutting
when the slope of the surface of the ground (or its equiva-
lent DE, Fig. 78) is 25% and the central depth 18 feet, base
21 feet ? The percentage here is 22, therefore the equivalent
level cutting=i8x'22 + i8=2i-96 deep.
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2S6 Preliminary Survey
Iron Bridges
The diagram on Plate VI. has been prepared from the
empirical formula of the late Mr. Trautwine, after checking
several spans both of plate-girders and trusses. Designs
vary greatly, especially in British construction, sometimes
from unavoidable circumstances, more often from a desire
for novelty on the part of the designer. It is therefore im-
possible to give any precise formula which will cover the
engineer's * personal equation ; ' but it is perfectly possible
to give a formula and a diagram which will represent a weight
which a bridge of moderate span need not exceed^ in order to
be safe within a certain limit under given permanent and
rolling loads, and that condition is fulfilled by the diagram.
The standard gauge covers a range from 4 feet 8^ inches to
5 feet 3 inches, and the narrow gauge 3 feet or metre.
The formulae are as follows :
For plate-girders up to 75 feet.
Weight in pounds per foot run of girders only =5 xspan
in feet + 50 \/span in feet.
For open trusses up to 250 feet.
Weight in pounds per foot run of trusses only=4-5 x
span in feet +22 \/span in feet.
The weights of the complete bridge are all scaled for
either gauge from the bottom to the curve bearing the proper
designation ; the weights of the trusses alone are scaled from
the line marked platform only ; the weights of the platform
only are scaled from the bottom.
The following comparisons with the weights of actual
structures will show to some extent the divergences to be
expected from the formula.
Oak Orchard Viaduct, New York State, 23 spans of 30 feet
each, by Mr. Chas. Macdonald, engineer. They are plate-
girders 20" deep, trussed by a centrepost and eyebars carry-
ing a standard-gauge single-line railway. Actual weight 3*1 2
tons. Weight by diagram 5*00 tons.
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Plate VI.
PLATE CIRDEPS
CiiiversaJ Iron UnMi^fi: Dl;^ roiri.
.Voie T. I'he standard -sauE^e curve is
4 ^", bulWllldoal^io for 5 '3". The nai row-
g^uge curve is 3' ^'\ but will do ;lI^ for
TTit'tre, Cnlculaied fora rol ling load of r j ions
per foot nin. Ironta bear 5 tam per square
inch in tenAioiu
^ JVtf/j' 3, - Ifthi^duifframift In Trequcnt use
With li Holders, SI pietx of diiH-biicJc trpting
Ll;Jth, gL3inmed nvcr it by ihe fonr mrciers
wuJ i)Tote«:i it.
Frr;. Bt
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Graphic Calculation for Preliminary Estimates 257
Railway bridge over the Ohio at Cincinnati; Mr. I. H.
Linville engineer. Deck span of no feet carrying standard-
gauge single line. Braced on the Pratt system ; actual
weight 467 tons, weight by diagram 48 tons.
Double line, standard-gauge railway truss bridge at
Harrisburg, Pa. 21 spans of 156' 6" each. Weight of a
span by actual measurement 129 tons. Weight by diagram
119 tons.
Jhelum bridge, British India. Mr. Lee Smith engineer-in-
chief Metre gauge. Single line and two footways. Deck
system with lattice bracing of single intersection, spans fifty
in number, 97' 6'' long. Actual weight of one span 42*4
tons ; weight by diagram for single line 29 tons, for double
line 50 tons. Mean 39*5 tons.
Iron Trestle Piers
Such wide differences of design exist in iron trestles that
it is almost impossible to form either a table or a diagram
of their weights.
One of the largest of such structures, the New Portage
Viaduct, of America, has towers of maximum height 203 feet.
It is built for double-line standard gauge, and carries spans
on either side of 118 feet. The weight of the trestle, con-
sisting of four columns, and bracing weighs about 1,400 lbs.
per foot of vertical height.
Another celebrated trestle, the Kinzua Viaduct in
Pennsylvania, is built for single-line standard gauge. The
maximum height of any tower is 278 feet. The tower is
capped by girders of 38 feet span, and supports adjacent
spans of 61 feet. The weights vary from 500 to 700 lbs. per
foot of vertical height.
On the Oak Orchard Viaduct of twenty-three 30-feet
spans on single trestles, called * bents,' consisting of a pair
of raking posts and bracing, carrying a single-line standard-
gauge railway, the weight of the bents per foot of vertical
height up to 75 feet was 160 lbs.
Digitized by VjOO^IC
2S8 Preliminary Sun^ey
Stone Bridges of Coursed Rubble
The diagrams on Plate VII. for stone bridges are con-
structed from Mr. Trautwine's formulae. The arches are
semicircular. The depth of keystone is taken from the
formula
Depth of keystoneinfeet=^'^^'^- +>^l^-^" + -2 feet.
The thickness of the abutment is taken from the
formula
rr., . 1 • r i. 4. • • rad. in ft. rise in ft.
Thickness m feet at spnngmg= ±.- -l 2 ft.
5 10
The abutment is plumb on the face, and battered on
the back from the thickness at the springings found by the
formula to a thickness at ground level =§ the vertical height
from ground to springing, which batter is continued down
through the ground to the bottom of the foundations, 3 feet
below ground level, and footings added.
The spandrel walls are y\ of their vertical height where
they join the wing walls and 2 feet 6 inches at cope.
The wing walls are also -^^ of their vertical height at
base, and diminishing to 2 feet 6 inches at cope.
Brick Bridges
The formula for stone bridges will serve for brick bridges
by taking half the quantity for the wing-walls. The quan-
tities in the diagram agree with American practice for rubble
stone, but wing- walls of brick are not nearly so wasteful of
material.
The two following examples are type bridges on the
Eastern and Midlands Railway, England, and agree in the
main as to quantity with those of most other railways in
England for similar height, span, and width.
Example i. 2 5 -foot span over-bridge for double line ;
elliptical arch ; total height zy^i^^iy width 18 feet; one
(::ounterfort to each abutment,
Digitized
by Google
n I ^ ' I ; — f 1 ;T ; I ^ [ ; ; : I ; ri'rTT ;- [ ; 'I ; r- 7jJ^:l:i^^^
•5^
-mj^/ja^n^ ^ '^ % <" J ^ S ->
Digitized
by Google
Digitized
by Google
Digitized
by Google
PlJ^rKVIII
TrMBER TRESTLES
QUANTfTlEIS mONL BEt^T
HT: ^
10 FEET
y<rte. — If thfe diagram is in frequent use with dividers, a piece of dull-back tracing
doth, gummed over it by the four corners, will protect it. ^ ,
Digitized by VjOOQ IC
Graphic Calculation for Pteliminary Estimates 259
Actual Diagram (with half
measurement, quantity for wings),
cubic yards cubic yards
Arch, abuts., cforts., and parapets . 1 96 "5 180
4 wings 147*9 140
Spandrels 9-3 41
3537 361
Example 2. 15 -foot span under-bridge ; segmental arch ;
total height 20 feet ; two counterforts to each abut ; width
23-5 feet.
Actual Diagram (with half
measurement, quantity for wings),
cubic yards cubic yards
Arch, abuts. , cforts. , and ppts. . 200*0 194
4 wings 126-0 123*5
Spandrel walls . . . . 4*0 21
330*0 338*5
Timber Trestles
Plate VIII. gives quantities in one bent. Types i, 2,
and 3 are for a trestle carrying a single-line standard-gauge
railway on a pair of stringers 14'' x 14''. The bents 12 feet
apart centre to centre. The posts, sills, and caps 12'' x 12''.
The sway-braces and wales 12" x 6". The quantities include
the longitudinal bracing but not the sleepers (cross-ties).
Type 4 is for one of a pair of bents forming a pier for a
Howe truss. Thus for a height of 45 feet the diagram
gives 445 cubic feet in one bent, therefore the pier would
contain 890 cubic feet. For narrow-gauge railways three-
quarters of the quantities may be taken.
A Howe truss is a composite bridge commonly used in
America, resembling in appearance the lattice-girder in
England. The top and bottom chords and the diagonal
bracing are of timber, and the stresses are distributed through
the members to the piers by vertical tension-rods.
The following table is an extract from Trautwine's * Pocket
Book,' which, in an excellent article upon trusses, fully
Digitized by VjOOQ IC
26o
Preliminary Survey
Table XXXII. — Hcrwe Trusses of timber and iron. Weights
approximately equal to iron trusses of same span.
All
A Lower
An end
A centre
A
EmJ
Centre
"I'S ' '=ho«i
liract bracie
i^ouiiter
IDll
rod
1
1
1' --"_.,: . _
1
_. .
P5
D 1 U
■si
1
•3S 1
■5J
8
1
%l
1
•SB
1
4J
u
1 -:
,s5 -:
1» a
p,.
□
■5
.3:.^
E
•3
-S'^
"g
y^j u
C/3 (J ( ^
V:''u
m u
"8
tf3^
A
g
1
.-'^
e
d
Iz;
11
1
i
ft. : ft.
ins.
ins-
ins.
ins.
ins-
inl*
ins.
^h\ 6
H 3 SX 6 3
S^ta
'1
S>f B
s
5^ 6
I
55= 6
^
ll
■
r
1 50' ^
g 1 3 Ifix 9 3
6x14 a
fix g
a
SK B
{
5X &
a
I
D
't
7Sf lis
iM \ 1 (SHia 1
6x14
2
Situ
a
6x a
T
fiK S
a
a :
a
4
l\tXl
^5
II
.'
6XJ4
3
6xt6
a
3^1^
^ lexio
1
6 IT to
a
il
3
i|
describes how the truss should be built, and gives the theory
of static equilibrium in so simple a manner that for this, or
for any ordinary structure of the kind, those who have never
studied mathematics can easily work out the proper dimen-
sions for any intermediate size. Composite bridges are also
largely made of the N type, having the uprights of timber
and oblique tension-rods. The total quantity of material is
somewhat less with this form of bridge, but there is more
ironwork in it. High timber viaducts are not much built
now for several reasons, but mainly on account of their
inflammability.
Howe trusses up to 50 feet span can be placed upon
timber trestle piers, but if the latter are higher than 60 feet
they require three bents to a pier, like the old viaduct at
Portage, New York State, which was 234 feet high from bed
of river to rail and contained 133,000 cubic feet of timber
including trusses. The quantity curve increases very rapidly
for trestles over the heights in the diagram. The maximum
length of bents on the Portage Viaduct was 190 feet.
Taking a proportion of \^® x 595 (the quantity in the
diagram for a bent 60 feet high), multiplying by 3, and
allowing for timber in the trusses, the actual quantity would
Digitized
by Google
Graphic Calculation for Preliminary Estimates 26 1
be about 2\ times more than the diagram. It is so very
rare that timber trestles are built above 60 feet nowadays
that it would be needless to extend the diagram.
! ^ i ? g g ^ t t
i 1 g s s s * E g
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tl
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^^ ^^^^ F^k ^
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:^^^ ^^ ^^^ ^.^
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CM
Service Diagrams
The service diagrams Figs. 82 and 83, for tramways and
railways working single line with crossing stations, are
262
Preliminary Survey
WV'Ol
Digitized
by Google
Graphic Calculation for Preliminary Estimates 26^
necessarily to a small scale. They should be drawn for
working purposes on a sheet of double elephant, which will
include the 24 hours for railway and the working hours for
tramways. The value of the diagrams is that they show at a
glance where the trains meet and pass, so that a special train
can be interpolated at any time with no danger of mistaking
its connections.
EngHsh railways worked upon the block system with the
double line, or with the staff or tablet on single line, need
nothing of this kind, but pioneer railways are generally
worked merely by telegraph or even telephone, and the
diagram forms an indispensable adjunct. It also serves the
purpose with tramways of showing the number of cars
required to work a certain service.
In diagram Fig. 82 we will explain the first round trip
of No. I tramway car.
Leave depot 6 a.m. ; run through to terminus with
line clear ; arrive terminus 6.55 ; leave terminus 7.00 ;
cross No. 2 car on No. 1 1 switch ; cross No. 3 car on No.
^ 10 switch. No. 4 car on No. 9 switch, and so on ; arrive
depot 7.55 ; leave again at 8.
In diagram Fig. 83 we will follow No. i freight, whose
waybill will be as follows. Leave N at 3 a.m. ; switch off
at M to allow No. 2 express to pass ; leave M at 4.10 ;
switch off at L for No. 3 ordinary ; leave L at 5.15 ; switch
at K for No. 100 express ; leave K at 6.05, switch at J for
No. X special ; leave J at 6.45 ; switch at I for No. 10 1
freight and No. 102 ordinary ; leave I at 8.10 ; switch at H
for shunting from 8.25 to 8.40 ; ditto at G from 9 to 9.12
and at F from 9.40 to 9.45.
Centrifugal Force
The following rules are prepared on the assumption of
gravity being 32*2 feet per second, and R the radius of
rotation ; they are sufficiently approximate when the thick-
Digitized by VjOO - _ _ '
264 Preliminary Survey
ness of the body such as the rim of a flywheel is not
more than ^ of the radius from out to out ; the radius being
measured to the centre of gravity of the rim.
Let F=centrifugal force in lbs. per lb. weight of rotat-
ing body.
Let F' = centrifugal force in lbs. per ton of rotating
body.
Let R= radius of rotation in feet.
Let N= number of revolutions per minute.
Let M= number of revolutions per second.
F= -00034 RN2 (i)
F=i-224 RM2 (2)
F=o76i6 RN2 * . (3)
Example, What is the centrifugal force in lbs. of a
body making 120 revolutions per minute, at a radius of
5 feet, and weighing 3 lbs. ?
F= I -224x5 X 22=24-49 lbs. per lb.
or for 3 lbs.=73-47 lbs.
Formulae suitable for side-stresses on viaducts due to
centrifugal force :
Let V=velocity in feet per second.
Let VV=velocity in miles per hour.
Let F=centrifugal force in lbs. per ton of 2,240 lbs.
Let FF= centrifugal force in lbs. per ton of 2,000 lbs.
Let R=radius in feet.
Let RR=radius in Gunter's chains of 66 feet.
F=69-S-R- (4)
FF=62i-^- (S)
F=2-268 ^' (6)
F=i49-7 -^ (7)
Digitized
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Graphic Calculation for Preliminary Estimates 265
FF=2-o25 ^ (8)
FF=i33-6 1± (9)
Example i. What is the centrifugal force in lbs. per ton
of 2,240 lbs. on a curve of 23*5 chains at a velocity of 31*3
miles per hour ?
By slide-rule. Place the brass marker opposite 31*3 on the
lower scale of the rule, and bring a 23-5 of the upper scale
of the slide to coincide with the brass marker so placed.
Read off the result 94*5 feet per ton on the upper scale
of the rule opposite to the 2*268 of the slide.
Example 2. What is the centrifugal force in lbs. per ton
of 2,000 lbs. at 60 miles per hour on a curve of 4 chains, 33
links ?
Place the brass marker opposite 60 on the lower scale of
the rule. Bring a 4*33 of the upper scale of the slide to
coincide with the brass marker so placed. Read off the
result, 1,681, on the upper scale of the rule opposite to the
2-025 of the slide.
Example 3. What is the centrifugal force in lbs. per ton
of 2,240 lbs. at 40 miles per hour on a curve of 5** (1,146 feet
radius)?
By No. 7, F=i497 x -3_^ = 209 lbs.
1 146
For table of certain values of F at different curvatures
and speeds, see Tables VII. and VIII., p. 15.
To Reduce Thermometer Scales
Fahrenheit and Centigrade,
Boiling-point is at 100° Cent., and 212° Fahr. Zero
Cent, corresponds with 32° Fahr., whence 180° Fahr.=
100° Cent.
Place the left hand i of the upper scale of the slide
Digitized by VjOOQ IC
266
Preliminary Survey
under the I'S of the rule ; then the slide will give Centigrade
and the rule Fahrenheit, when the constant 32 is added to
the latter.
Thus -1° Cent. = •18 + 32=32-18 Fahr.
i°Cent.= i-8 + 32=33-8
5° Cent. = 9*o + 32=41*0
25^Cent.=45-o+32=77
100° Cent.=i8o + 32 =212
250° Cent. = 450 + 32 =482
35° Fahr. (32 + 3) =r66 Cent.
300° Fahr. (32 + 268)= 149 Cent.
or by rule C=5 (^~32)
9
Below freezing-point, the values Fahr. read on the rule
are deducted from 32.
Thus —1° Cent. = 32 — I -8= 30*2 Fahr.
», -5° » =32-9 =23° ,,
Fahrenheit and Reaumur.
Boiling-point is at 80° R^. and 212° Fahr.
Zero Re. corresponds with 32° Fahr.
Place the left hand 8 of the upper scale of the slide
under the i'8 of the rule, and read Re. on the slide and
Fahr. on the rule, adding the constant 32.
1° R^.=32-l- 2-25=34-25 Fahr.
5°Re.=32 + ii-25=43*25 ,,
25°Re.=32 + 56-25=88-25 „
or by rule F=-5 — + 32.
Centigrade and Reaumur.
Zero coincides in these scales, so there is no constant.
Place the 8 of the slide under a i of the rule, and read
Re. on the slide, and Cent, on the rule.
Digitized by VjOOQ IC
Graphic Calculation for Preliminary Estimates 267
Thus 1° Re. =1-2 5 Cent.
2-4° Re. =3-00 „
100° Re.=i25° „
or by rule C= ^R.
4
The Slide-rule as a Universal Measurer
Circular Measure of Angles,
Two movements of the slide are all that is needed for
the length of any circular arc to any radius.
An arc of 1° to a radius of 100=1745, that is to say is
1745 per cent, of the radius whatever that radius may be.
An arc of 10° is 17 '45 per cent., and so on.
(a) To obtain the value of any arc in percentage of the
radius. Place the left hand i of the slide opposite to the
1,745 of the rule ; then opposite to the angle in degrees and
decimals on the slide will be given the percentage on the rule.
Example, What is the percentage of an arc of 22° 30',
i,e, 22°-5 ? Placing the slide as directed we find opposite
the 22*5 upon it 39*2, which is the required percentage.
(b) To obtain the linear value of the same arc to any
given radius. Leave the brass marker at the percentage
found on the rule so as not to lose it, and move the slide
until a I upon it coincides with the given radius. Then
move the marker to the percentage on the slide and read off
the linear value upon the rule.
Example, With the 39*2 per cent, found as above, re-
quired the linear value of the said arc to a radius of
234 feet.
Placing the slide as directed with the left hand i oppo-
site to 234 on the rule, move the marker to 39*2 on the slide
and read off the result, 917, on the rule.
Circumferences of circles. Multiply 3 • 1 4 1 6 by the diameter,
or 6-2832 by the radius. When the argument is given in
twelfths or sixteenths or sixtieths reduce them to decimals by
268 Preliminary Survey
Tables XXXIV., XXXIX., and XL., respectively, or by
the slide-rule as explained further on.
(d) Areas of circles. Multiply the square of the diameter
by 7854, or the square of the radius by 3*142.
(e) Surface of cylinders. Multiply the circumference
found as above by the length.
(f) Volume of cylinders. Multiply the area found as
above by the length.
(g) Surface of spheres. Multiply the square of the dia-
meter by 31416.
Volume of spheres=^^\zxs\eiQx ^ x •5236.
(h) To reduce inches to decimals of a foot. Place 1 2 on
the slide opposite to 10 on the rule, and read the decimals
on the rule opposite the inches on the slide.
(i) To reduce fractions of inches to decimals of a foot
Place the denominator on the slide opposite the 833 on the
rule, which will be regarded as -0833, ^'^^ then read off the
decimals on the rule opposite to the numerator on the
slide. Thus, to express f ^ of an inch in decimals of a foot,
place the 32 of the slide opposite thie 833 on the rule, which
represents the decimal for one inch '0833 ; then 2 1 on the
slide will be found opposite to '0547 on the rule.
(j ) When the fraction is a mixed one of inches and fractions
of inches. Find the decimal of the integer by the first pro-
cess, and keeping the result on record with the brass marker
find the decimal of the fraction by the second process, and
add the two results together.
(k) To reduce fractions of inches to decimals of inches.
Place the denominator on the slide opposite a i of the rule
and read the decimals on the rule opposite the numerators
on the slide.
(1) To rediue feet to metres or vice versa. Place a i of
the slide, representing 100 feet, under the 30*48 of the
rule and read off feet on the slide and metres on the rule.
(m) To reduce yards into metres or vice versi. Place a
I of the slide, representing 100 yards, opposite to 91*44 on
Digitized by VjOOQ IC
Graphic Calculatipn for Pr-eliminary Estimates 269
the rule ; read off yards on the slide and metres on the
rule.
(n) To reduce inches and decimals into millimetres or vice
vers^. Place a i of the slide to represent one inch opposite
to 25*4 on the rule ; read inches on the slide and milli-
metres on the rule.
(o) To reduce kilometres into statute miles and vice versa.
Place a I of the slide, to represent one statute mile, opposite
1*609 on the rule, and read miles on the slide and kilo-
metres on the rule.
(p) To reduce kilometres into geographical miles and vice
versi. Place a i of the slide, to represent one geographical
mile, opposite 1*853 on the rule, and read geographical miles
on the slide and kilometres on the rule.
(q) To reduce statute miles to geographical miles and vice
versi. Place a i of the slide, to represent one geographical
mile, opposite to 1-151 on the rule, and read geographical
miles on the slide and statute miles on the rule.
Metric Square Measu7'e.
I square centimetre
I ,, metre .
I „ „ .
I square kilometre
= 0*155 square inch
= 107641 „ feet
= 1-19601 ,, yard
= 247*11 acres
= 0*3861 1 square miles
Metric Weight s^
I centigramme
I gramme .
I kilogramme
I tonne
0*15432 gram
- 15-432
■a 2*2046 pounds
= 2204*6 ,,
^ 0*9842 ton
Metric Cubic Measure,
I decalitre = 0*35316 c. ft = 2 *2009 British gals.
,, =0*28378 U. S. struck bushel -2 -64 1 79 U. S. liquid gals.
' See also Specific Gravity, pp. 389, 390.
Digitized by VjOOQ iC
Table XXXlll.— For converting Inches into Decimals of a Foot,
In.
Feet
In.
Feet
jin.
Feet
lln.
Feet
•oooo
8
•2500
6
•5000
9
7500
•0052
•2552
•5052
•7552
16
•0104
tt
•2604
^ A
•5104
1 4
7604
H
•0156
•2656
1
•5156
1
7656
A
•0208
\
•2708
' 1
4
•5208
i \
7708
4
•0260
•2760
•5260
•7760
1
•0313
\
•2813
1
•5313
\ ^
•7813
•0365
•2865
•5365
1 ,
7865
\
•0417
\
•2917
\
•5417
5
•7917
•0469
•2969
•5469
7969
1
•0521
1
•3021
h
•5521
s
•8021
•0573
•3073
•5573
•8073
J
•0625
1
•3125
f
•5625
^
•8125
•0677
•3177
•5677
1
•8177
1
•0729
i
•3229
i 1
•5729
i
•8229
•0781
•3281
1
•5781
•8281
1
•0833
4
•3333
7
•5833
10
1^33
•0885
•3385
•5885
1
•8385
i
•0938
*
•3438
*
•5937
. i
•8437
o
•0990
•3490
•5990
•8490
\
•1042
\
•3542
k
•6042
; \
•8542
•1094
•3594
•6094
•8594
f
•I 146
1
•3646
t
•6146
, f
•8646
•I 198
•3698
•6198
•8698
1
•1250
1
2
•3750
! i
•6250
1 \
•8750
1
•1302
•3802
•6302
•8802
s
•1354 1
1
•3854
1
•6354
8
•8854
•1406 I
•3906
•6406
•8906
1
•1458
*
•3958
1-
•6458
\
•8958
•151O
•4010
•6510
•9010
J
•1563
*
•4063
I
•6563
1 «
•9063
8
•1615
•41 15
•6615
1
•91 15
2
•1667
6
•4167
. 8
•6667
11
•9167
•1719
•4219
•6719
•9219
*
•1771
i
•4271
*
•6771
•9271
•1823 j
•4323
•6823
1 ,
•9323
i
•1875
i
•4375
, i
•6875
\
•9375
•1927
•4427
1
•6927
•9427
i
•1979
f
•4479
I *
•6979
i
•9479
•2031
•4531
•7031
•9531
1
•2083
}
•4583
\
7083
1
•9583
•2135
•4635
•7135
•9635
1
•2188 ;
%
.4688
; *
•7187
1
•9688
•2240 :
•4740
•7240
•9740
3
•2292
3
4
•4792
4
7292
f
•9792
4
•2344
•4844
1
•7344
•9844
J
•2396
I
•4896
1 \
•7396
*
•9896
•2448 ;
•4948
\
7448
•9948
Digitized
by Google
Graphic Calculation for Preliminary Estimates 271
Table XXXIV. — Timey Coinage, and Measurement in Decimals,
Pence
Shillings
Months
Years
Yards
Pounds sterling
Inches
Feet
I
•0833
•0277
'00466
2
•1667
•0554
•00833
3
•2500
•0833
•01250
4
•3300
•nil
•01666
5
•4167
•1389
•02083
6
•5000
•1666
•02500
7
•5833
•1944
•02916
8
•6667
•2222
•03333
9
•7500
•2500
•03750
10
•8333
•2777
•04166
II
•9167
•3056
•04583
Table XXXV. — Shillings in Decimals of a Pound.
Shil-
Pounds
Shil-
Pounds ]
Shil-
Pounds
Shil-
Pounds
lings
sterling
lings
sterling '
lings
sterling
lings
sterling
I
•05
6
•30 I
II
•55
16
•80
2
•10
7
•35 '
12
•60
17
•85
3
•15
8
•40
13
•65
18
•90
4
•20
9
•45
14
•70
19
•95
5
•25
JO
•50 1
15
•75
20
I^OO
Table XXXVI. —Days, Hotirs, Minutes, and Seconds.
Minutes
Seconds
Hours
Minutes
Days
I
2
3
4
5
•01667
•03333
•05000
•06667
•08333
•000694
•001389
•002083
•002777
•003472 ,
Minutes
Hours
Days
Seconds
Minutes
6
•lOOOO
•004166
7
•I 1667
•004861
8
•13333
•005555
9
•15000
•006250
10
•16667
•006940
Table y.y.yM\\. — Weeks, Months, and Years.
Weeks
Months
Years
Weeks
Months
Years
I
•23077
•01923
6
1-3846
•I 1538
2
•46154
•03846
7
1^6154
•I3461
3
•69231
•05769 ;
8
I ^8462
•15384
4
•92308
•07692 '
9
2*0769
•17307
5
1-15380
•09615
10
2-3077
•19230
Digitized
by Google
272
Prelimifiary Sun^ey
Table XXXVIII. —Z>flyj, Weeks, Months, and Years,
Days
I
Weeks
•1429
Months
Years
Days
Weeks
Months
Years
•0329
•00274 1
6
•8572
•1973
•01644
2
•2857
•0657
•00548
7
i-oooo
•2302
•OI918
3
•4286
•0986
•00822 1
8
1-1429
•2630
02192 '
1 4
•5714
•1315
•01096
' 9
I -2857
•2959
•02466 !
5
7143
•1644
•01370
i ^^
1-4290
•3290
•02740
Table XXXIX. — For converting Fractions into Decimals,
h
•0156 1
11
1
t
•0312
^
•0469
el
A
•0625
A
h
•0781
64
•0937
P
•1094
el
¥
•1250
i
ii
•1406
85
64
•1562
•I7I9
i
t\
•1875
-j2g-
13
•2031
29
u
64
•2187
•2344
if
i
•2500
i
•7656
•7812
•7969
•8125
•8281
•8437
•8594
•8750
•8906
•9062
•9219
•9375
•9531
•9687
•9844
Table Xh. — For converting Minutes into Decimals of a
Degree, or
Seconds into Decimals of
a Minute (^J.
Min.
Degree
Min.
Degree
Min.
Degree
Min.
Degree
Sec.
I
Minute
•0167
Sec.
Minute j
Sec.
31
Minute
Sec.
46
Minute
I 16
•2667 ,
•5167 !
•7667
2
•0333
' 17
•2833 :
32
•5333 1
47
7833
3
•0500
1 18
•3000
33
•5500
48
•8000
4
•0667
1 19
•3167
34
•5667
49
•8167
5
•0833
1 20
•3333
35
•5833
50
•8333
6
•1000
1 21
•3500 !
36
•6000
51
•8500
7
•II67
' 22
•3667 i
37
•6167
52
•8667
8
•1333
23
•3833 1
38
•6333
53
•8833
1 9
•1500
24
•4000 i
39
•6500
54
•9000
10
•1667
25
•4167
40
•6667
55
•9167
II
•1833
26
•4333
41
•6833
56
•9333
12
•2000
27
•4500
42
•7000
57
•9500
13
•2167
28
•4667
43
•7167
58
•9667
14
•2333
29
•4833
44
•7333
59
•9833
15
•2500
30
•5000
45
•7500
60
I'OOOO
Graphic Calculation for Prelimin6;^ry Estimates 273
Table XLI. — For cotiverting Seconds into Decimals of a Degreey or
Thirds into Decimals of a Minute (gego)*
Sec.
Thds.
Deg.
Min.
Sec.
Thds.
Deg.
Min.
, Sec.
Thds.
Mm.
Sec.
Thds.
Deg.
Min.
I
•ocx)3
II
•0030
21
1
•00581
31
•0086
2
3
■0005
•cx»8 '
12
13
•0033
•0036
22
23
•0061
"0064 !
32
33
•0089 '
•0092 .
4
•001 1 1
14
•0039
24
•00671
34
•0094
5
•0014
15
•0042
^1
•0069 ,
35
•0097 ,
6
•0017
16
•0044
26
•0072 1
3b
•0100
7
8
•cx)i9
•0022
17
18
•0047
•0050
27
28
•0075 1
•0078 i
%
•0103
*oio6
9
10
•0025
•0028
19
20
•0053
•0055
29
30
•0080
•0083
39
40
•0108
"on I
Sec.
Deg.
Thds.
Min.
1 5'
•0142
' 52
•0144
53
•0147
54
•0150
i 55
•0153
i 56
•0156
57
•0158
58
•0161
59
•0164
60
•0167
Example, What is 17° 11' 29" 47'" in decimals of a degree.^
^f" = '0130' = •CXX)2°
29" = 'OoSo
II' - -1833
17
I7°-I9I5
Table XLI I. — Decimals of a Degree in Minutes and Sec of ids.
•01
o'36;;i
*2I
12' 36"
•41
24; 36'' 1
•61
36 36
•81
48' 36'-
•02
i' 12"
•22
13' 12"
•42
25' 12"
•62
37' 12"
•82
49' i2"
•03
"^.K
•23
13' 48''
•43
25' 48" ,
•63
37 48
•83
49' 48"
•04
2' 24"
•24
^«
•44
26' 24"
•b4
3S><
•84
50' 24"
•05
3' 0" ;
•25
15' 0"
•45
2/ 0"
•65
39 '
•8s
51' 0"
•06
336';
•26
15; 36;;
•46
27' 36"
•66
39 36
•86
5\36;;
•07
4' 12"
■27
16' 12"
•47
28' 12"
•67
40' 12"
•87
52' 12"
•08
iK
•28
16; 48;;
•48
28-48;;
•68
40' 48';
•88
52; 48"
•09
i K
•29
^t<
•49
29' 24"
•69
'*^ ^*//
•89
53, 24"
•10
6' 0" 1
•30
18' 0"
•50
30 0" ]
•70
42' 0"
•90
54' 0"
•11
6' 36"
■31
18' 36';
•51
30 36 1
'71
42' 36"
'91
54; 36"
*I2
7! ^il, 1
•32
19' 12"
•52
3^ <
'7^
43, 12 '
•92
55' 12"
•13
i^K\
•33
19' 48;
•53
3^48
•73
43 48
•93
55 48"
•14
^ <
•34
20 24"
•54
3^<
•74
44, 24
•94
56 24''
•15
9' 0" 1
•35
21' 0"
'55
33
•75
45' 0'
•95
57' 0"
•16
i^K\
•36
21' 36"
•56
3336 1
•76
45 36'
•96
5/36"
•17
10' 12" 1
22' 12"
•57
34' 12"
•77
46 12"
•97
58' 12"
•18
10' 48''
'38
22' 48"
•58
3<<
•78
46; 48';
•98
58; 48"
•19
"! ^^!!
•39
23' 24 '
•59
35 24
•79
47 24
•99
59' 24"
•20
12' 0"
•40
24' 0"
•60
36' 0" 1
•80
48' 0"
I'OO
i°o'o"
Thousandths of a Degree in Secottds and Decimals of a Second,
•001 I 3''-6
•002 I 7" '2
•003 10" -8
•004 14" "4
•005 i8"'o
*oo6 2i"-6
*oo7 25 '"2
•008 I 28'-8
009
'oio
32 '4
36 o
Digitized
by Google
274
Ptelhninary Survey
Table XLIII. — Of Cotangents of a few leading Angles with their
corresponding Tangents for checking the Slide-rule,
Tan of
Cotan of
Tan of
Cotan of
5
0-0874
!
60
1-732
30
lO
0-1763
80
65
2-144
25
15
0-2679
75
70
2747
20
20
0-364
70
75
3732
15
25
0-466
65
80
5-671
10
30
0-577
60
!5
"•430
5
35
0-700
55
86
14-300
4
40
0-839
50
!7
19-081
3
45
I -000
45
88
28-636
2
50
I -192
40
89
5729
I
55
1-428
35
Rule, To find a tangent of a higher angle than is given
by the slide.
tan ®=_i— = —L. _
cot® tan (90°-©)
Example, Find tan 73° 20'.
The table shows that the decimal punctuation will be
between 27 and 37.
tan (9o°-73° 2o')=tan 16° 40'
Place the scale of tangents in its initial position, and find
tan i6°4o'='299.
Reverse the slide and adjust the scale of numbers with
299 over the i of the rule, and opposite to the other i of
the slide will be found 3-34 the required tangent.
Table XLIV. — Ofsome Higher Sines than are clearly
given on the Slide-rule,
Sine of
—
1
Cosine of
Sine of
Cosine of
-9998
I
85
-9962
5
88
•9994
2
80
-9848
10
87
-9986
3
75
•9659
15
86
•9976
4
70
•9397
20
Digitized
by Google
Graphic Calculation for Preliminary Estimates 275
Measurement of Tree Timber
In measuring felled logs, allowance is made first for
squaring, secondly for bark. As to the first, instead of multi-
plying the area of cross section by the length, \ of the girth
squared is taken as the area, and this is the quantity given
in square feet in Hurst's * Pocket Book.' It is about 28 per
cent, less than the actual cross section. When the tree
tapers considerably the two ends and middle are girthed
and the average taken. A tree is not called timber unless
the stem measures 24 inches in circumference.
Rule. Marketable area=(|^ girth)^ ; marketable cubic
contents= length x (\ girth)^.
By slide-rule. Find the \ girth in decimals of a foot by
2^6 Preliminary Survey
Table XXXIII. (unless a decimal tape is used), square by
Rule 4, p. 244, and multiply by Rule 2.
If the bark is on the tree, deduct as follows :
For oak, old and thick barked . . . . ^^ of girth.
For oak, young and thin barked . . • 13 »»
For elm, pine, and fir ^ f>
For ash and beech A* >>
Comparison of various beams cut from a log 2 feet in diameter.
sq. ft.
fl, the most serviceable beam . . x = \ diam., areas 1*92
^, the stiffest beam . . . jc « J diam., area « 175
<:, square timber .... jr = ^ diam., area = 2*oo
Marketable measure of log \ girth square . . area = 2 '47
Gross area of cross section of log . . . '=3*14
To obtain a 12'^ piece of square timber, the tree must
be, allowing for bark, 1 5'' diameter, or 4 feet in circumference,
at its smallest end.
Railway Track
Weight per mile of single tracks consisting of two rails
and fastenings. For rails only : Weight in tons per mile of
single track=area of rail in square inches x 157 143 ; the
weight in pounds per yard being ten times the area of cross-
section in inches.
Molesworth's tables of Indian State railways produce re-
sults as follows, including allowance for waste in fastenings :
^ft, 6 in. gauge.
Rails only .... weight in tons -area x 15*65
Rails, fishplates, fishbolts, and
spikes .... ,, ,, X i6'65
Rails &c. as above, and bearing
plates .... ,, ,, X 17*86
Metre gauge.
Rails only .... weight in tons = area x 15*65
Rails, fishplates, fishbolts, and
spikes .... ,, ,, X 16*65
Rails &c. as above, and bearing
plates .... ,, ,, X 17*48
^ Hurst.
Digitized by VjOOQ IC
Graphic Calculation for Preliminary Estimates 277
Example i. What will be the weight of iron in a mile
of single track of narrow-gauge railway with 40 pound
rails and fastenings, but no bearing plates ?
Place the i of the slide opposite the 4 of the rule, and
opposite to 16-65 (the weight per mile corresponding to an
inch of section) we find 66*6 tons.
Example 2. What section must a rail have so that the
rails only, without fastenings, will amount to 100 tons per
mile? Adopting the factor 15*7143, place the 1571 of the
slide opposite the i of the rule ; then opposite the i of the
slide will be found 6*36, the required sectional area ; the
weight per )^ard would be 63-6 pounds.
Note, The weights of the fastenings have not been
given in detail ; they cover the weight of ordinary fishplates,
but not angled or sleeve fishplates.
Weight of Angle and Tee Iron,
W=(B + D-/)w.
Where W=weight in pounds per foot run.
B=breadth of one flange of angle, or clear
breadth of head of Tee in inches.
D=breadth of other flange of angle, or extreme
depth of Tee in inches.
/= thickness of iron in inches.
ze/= coefficient in table.
Table YAN .—Multipliers for Weights of Structural Iron,
t
W
/
W
\ t \ ..
t
W
1
If
•308
Vk
1-041
4
1-874
13
?
8
2-708
•416
1
1-25
2-082
2-916
•62s
''11?
2-29
tl
3-124
4
•833
1-666
2-50
. I
3-333
Intermediate values of w^ e,g, for 32nds of an inch, or
decimals, can be interpolated by slide-rule.
Example i. What is the weight of a lineal foot of Tee
iron3|x4ixfJ?
d by Google
Digitized b
278 Preliminary Survey
4i+3|— H=7'32 inches, and w=2'2()
Answer. 7*32 x 2*29=1675 pounds per foot.
Weight of channel and H iron may be found similarly,
when the web and flanges are of the same mean thickness,
by the formula W=[DH-2 (B— /)]ze;.
n FT
■X-
B
6
U..
- Fig. 85.
Example 2. What is the weight of channel iron in pounds
per foot run, size 8'' x d" x ^'' ?
8 + 2 (6~^)=i9;a;=r666.
Answer. 19x1 '666 = 3 1 -66 pounds.
Weight of Round and Square Iron,
Rule I. — Round iron. Place the 2*61 of the upper scale
of the slide under the middle i of the rule. Read the
weight in pounds per lineal foot on the upper slide-scale,
opposite the square of the diameter in inches on the rule.
Example. What is the weight of 2^' round iron per lineal
foot? Adjusting the rule as described, we first find in
Table XXXIX. the diameter in decimals 2*375, ^^^ by brass
marker we find its square 5*64 on the upper scale of the
rule. Under this latter figure we find the result 14*8 lbs.
on the slide.
Rule 2. — Square iron. Instead of the number of 2*61
use 3*33, and proceed as before. For flat iron, use the pro-
duct of the breadth by thickness instead of the square of the
side.
For round and square cast iron use 2*43 and 3*097
respectively.
For steel, 2*66 and 3*397.
„ copper, 3*00 and 3*83.
„ brass, 2*84 and 3*63.
„ lead, 3*84 and 4*89.
, zinc, 2*40 and 3*06.
Digitized by VjOOQ IC
Graphic Calculation for Preliminary Estimates 279
The Slide-rule as a 'Ready Reckoner'
Table XLVL— Wages and Salaries,
Multiplier
Pence per hour into shillings per day of 11 hours 0*917
„ „ 10
0-833
9
0750
8
0-667
,, week of 66
5-5
60
5-0
54
4-5
48
4-0
£ per mo. of 26 d. of 1 1 hours
I '192
10
1-083
9
8
0-975
0-867
„an. of 313 d. at II
14-34
10
13-04
9
"73
8
10 -43
Shillings per day into £ per week of 6 days
0-3
>»
,, month of 4 weeks
I -20
>»
>> »> 4i »
1-299
,,
„ ann. of 313 days
15-65
week into £ per month of 4 weeks
0-2
>>
>» >> 45 »>
0-216
}>
„ ann. of 52 weeks
2-6
;f per
week into £ per month of 4 weeks
4-0
>i
4j ,,
4-33
>)
ann. of 52 weeks
52-0
month (lunar)
into £ per annum
13-0
(calend
ar) into £ per annum
I2'0
See also tables,
pp. 271,
272.
Example i. How much per annum is an hourly wage of
3|df. worth, at the rate of 9 hours per day, working every
day except Sundays? Place the left hand i of the slide
opposite the multiplier 11-73 on the rule, and opposite to
the wage 375^. on the slide will be found 44/.
Example 2. A man has made 178/. 15X. in the year.
What is the equivalent wage in shillings per day, if he had
d by Google
Digitized b
28o
Preliminary Survey
worked steadily every day but Sundays? Place the left
hand i of the slide opposite the multiplier 15 '65 on the
rule, and opposite to 17875 on the rule will be found 11-42
==iix. s^r.
Example 3. A servant gets 1 7/. 3^. 5^. per annum ; what
are his wages for 5 months 2 weeks and 5 days ?
Find from the tables the decimal equivalents of the time
and money as follows :
Years
5 months (Table XXXIV.) -417
2 weeks (Table XXXVII. ) -038
5 days (Table XXXVIII. ) -014
Rate in decimals . 1 7 'ooo
3 shillings (Table XLVII. ) -150
5 pence (Table XLVII.) -021
17-171
•469
By slide-rule. Place the left hand i of the slide over
the 469 on the rule, and read the answer 8*07/. opposite to
1,717 on the slide.
By table, 8-o7/.=8/. \s. 4^d.
From Table XLVII. read off -05 as is. and divide the
balance '02 by the multiplier for one penny '00417 ; the
equivalent fraction would be -^~. For division by slide-
417
rule see p. 245. The result will be 4*8^.
English Money
Table XhVll.— Decimal Multipliers,
One farthmg
One penny .
One shilling
One pound
£
s.
d.
/
•00104
•0208
•25
I
•00417
•0833
I
4
•05
I-o
12
48
i-o
20
240
960
Example. To reduce 131/. 13^. g\d. to pounds and deci-
mals by the slide-rule.
Digitized
by Google
Graphic Calculation for Preliminary Estimates 281 •
ith the multiplier '00104 we find f^. .
. « '0031 ]
„ -00417 „ 90^.
. = -0374
„ -05 „ 13^- •
. -es
131
Answer. . . ;f 131 '6905
Colonial and Foreign
Where the value is not at par under the gold standard, or where
silver is not at 44d. per ounce, the slide-rule will give the current value
by rule I, /. 242.
Argentine Republic— see Chili.
Austria. (Par value.)
I/.
. = 10-215 fl- •
. =
1,021 '5 kreuzer
\s.
= '5107,, .
. =
51-07 ,,
\d
= -0425 „ .
Brazil.
"
4-2 „
I/..
. = 8*925 milreis
. = 8,925 reis
\s, .
. = -446 „
. =. 446 „
\d.
. = -037 „
. = 37 „
Canada and United States of America.
Par Value in Sterling,
I £ « 4*87 dollars
I shilling = 24*35 cents
I penny = 2*03 ,,
I farthing- 0*51 ,,
Chili, Colombia, and Uruguay. (Par value.)
1/. . . . -5*340 peso . . = 534 centavos
is. . . = -267 ,, . . =r 26*7 ,,
id. . . = -022 „ . . = 2-2 ,,
China.
Intrinsic value with silver at 44^. per oz, troy.
il, . . = 4*28 taels . . = 428 conderin
IS. . . ^ -214 „ . . = 21*4 ,»
id. . . = -oiS „ . . - 1-8 ,,
Digitized by VjOOQ iC
282 Preliminary Survey
France. (Par
value. )
IS.
25*220 francs
I '261 ,,
•105 „ .
126*1 centimes
io*5 „
Empire of Germany.
(Par value. )
I/.
IS.
Id.
. = 20*420 marks
. = I '021 .,
. = -085 „
. = 102*1 pfennig
8-5 „
Table XLYlll.— Indian Money at par (i Rupee -2s.).
Rupees
Annas
16
I
•0833
Pie
Sterling
I
•0625
•005208
192
12
I
2 shillings
\\ pence
0*125 penny
I lakh = 100,000 rupees, i crore = 10,000,000 rupees. I pice = 3
pie or \ anna. In Ceylon the rupee is divided into 100 cents.
Example a. What is the value of 473/. sterling in rupees
at the exchange of \s. i\d. per rupee ? Using the multiplier
in Table XLIX.,
10 : 4,730:: 12*15 : ^
whence ^^=5,747 rupees. For rule of three sum by slide-
rule see p. 243.
Example b. What is the value of 72 pie at the exchange
of \s. %\d. sterling per rupee ?
First operation : To obtain the par value. By Table
XLVIIL,
I pie : '125^/. :: 72 pie : x
whence ^=9^.
Second operation : To obtain the current value. By
Table XLIX.,
11*56 : 10:: 9^. \y
whence j'= 7 -8^/., nearly.
Digitized by VjOOQ IC
Graphic Calculation for Preliminary Estimates 283
Ed
u
P<
2
5
to
M
(A
^
s?
>
^
^ 1
>
p:;
s?
II
"
II
"
II
II
«^
m
Q
1
^
•%
^
CO
fO
;^
CO
g
ON
00
fO
CO
vo
to
&
1 1
*o
J;
?
t^
CO
P
^
ro
1^
<^
^3
*o
1
1
vo
Op
vo
1
to
1
CO
1
i-i
1
:i
t^
vo
:i
CO
;i
■8
a
CO
H|e»
»o
CO
t^
>
<-»
^
00
R
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s;
^
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284 Preliminary Survey
Example c. To reduce 15 annas 7 pie to cents (looths
of a rupee).
16 annas : 100 cents:: 15 annas : x cents
whence ^=937 ;
192 pie : 100 cents:: 7 pie \y cents
whence j= 3 7 cents,
and answer, 937 + 37=97*4 cents.
Japan.
Intrinsic value with silver at 44^. per oz. troy.
i/. . . . 6*500 yen . . . 650 sen
IS, . . . -325 „ . . . 32*5 »»
i^. • . . -027 „ . . . 27,,
Mexico. (Par value.)
i/. . . « 6 -160 dollars . . = 616 'O cents
IS. . . = -308 „ . . . = 30-8 „
\d. . . « -025 „ . . . = 2*5 „
Netherlands.
i/. . . = 1 2 'oo florins . . =1,200 cents
\s. . . - '60 ,, . . = 60 ,,
i^. . . = -05 » • . = 5 ».
The par value according to United States Treasury circular is about
I per cent. more.
Persia.
I/.
= 2 '800 thomans . « 28*0 banabats
. =«28o shahis
IS.
Id.
. = 140
= 012
. = 1-4
. = -128 „
Portugal. ( Par value. )
- 14
= 1*28 „
I/. .
IS. .
Id..
•
= 4*500 milreis
= 225 „
= -019 ,,
= 4,500 reis
= 225 „
= 19 „
Russia. (Paper currency.)
Current value in Whitaket^s Almanack for 1890.
(/. . - 6*3000 roubles . . = 630 kopek
u. . . = *3i5o „ . . = 31-5 „
\d. . . — -0265 ,, . . = 2*65 ,,
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Graphic Calculation for Preliminary Estimates 285
Spain. (Par value.)
i/. » 97000 scudo = 25*200 peseta = 2,520 centimos
IS. - '4850 ,, = I •260 „ = 1,260 ,,
id. = -0404 „ = -105 „ = 105
Sweden. ( Par value. )
i/. . . =18 '200 crowns . . — 1,820 ore
IS. . . = -910 „ . . = 91 „
let. . . = -076 „ . . = 7-6„
Tu RKEY. ( Par value. )
i/. . . = 1 10 'O piastres
is. . . = 5*5 ,, . . = 220 paras
Id. . . = -46 „ . . = 18-3 „
Venezuela.
i/. . . = 25*220 bolivars . .= 50*44 decimos
ij. . . = 1*261 ,, . . s= 2*522 ,,
id, . . - *io5 ,, . . » '210 ,,
Example d. How many Chinese taels are contained in
10 Mexican dollars, the former at intrinsic value of 42^.
per oz., the latter at par value ?
First operation : To obtain the current value in Chinese
taels of i/. sterling.
42 : 44:: 4*28 : :i:=4*48
Second operation : To obtain the value of 10 Mexican
dollars in taels.
6*16 : 4*48:: 10 : 7*11 taels
Example e. How many Japanese yen at 44^. per oz.
should one receive for 105 U.S. dollars at par value ?
4*87 : 6*50 : : 105 : ^
x-=-\\o yen 44 sen
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286 Preliminary Survey
CHAPTER IX
INSTRUMENTS
Levels and Levelling
The necessity for condensation has led to the insertion
here of all that can be said within our limits upon the theory
of levelling instead of giving it a separate chapter. It will be
attempted to place the fundamental principles of levelling
and the adjustments of different kinds of levels in a practical
and simple light. The abstruse disquisitions upon possible
sources of error in levelling to be found in proceedings of
learned societies have no place here. The surveyor ought
to be able to put his instrument in adjustment every morning
without referring to a book, and considerable space has
been required to make the reason of each adjustment plain.
Secondly, he should be alive to the limits of error arising
from taking long or unequal sights, curvature of the earth,
&c., so that on the one hand he should be prepared to
adopt extra precautions for special cases, and on the other
hand not to waste time upon refinements which are not
essential to his object.
The adjustment of the level is also the foundation of the
adjustment of the theodolite and tacheometer. This is
shown in Fig. 88, where the error in the line of sight is
treated as a vertical angle which has to be eliminated
similarly to the index error of a theodolite.
For these reasons, the lion's share of this chapter has
fallen to the subject of levels.
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Instruments 287
Theory
Two points are said to be upon the same level when
they are equidistant from the earth's centre : conseqently, a
level line cannot be strictly speaking a straight line. It is
a parallel to the curvature of the sea. A horizontal line is
a parallel to a tangent to the earth's circumference, and
therefore a straight line. In common parlance, the words
level and horizontal are synonymous, and it would be pedantic
to endeavour to keep their application wholly distinct, but
the difference is mentioned because it comes into the
question of adjustment for ' coUimation.'
A level adjusted horizontally will not represent an object
as high as it really is. This is proved by looking at the top
of a vessel's masts when just appearing above the horizon.
A level would report the mast-head to be at the same
elevation as the observer's eye, whereas it would be perhaps
fifty or 100 feet further from the earth's centre.
The object of the leveller is to determine the relative
elevations, or heights above sea-level, of points upon the
earth's surface, which is the same thing as their difference
of distance from the earth's centre.
Light always travels in a straight line^ unless diverted
from its path by the medium traversed. The bending of
light by the atmosphere is called refraction : see Glossary.
Levels are either adjusted with the line of sight, hori-
zontal, and consequently subject to correction for both
curvature and refraction, or else they are adjusted with the
line of sight parallel to a chord of the earth's circumference
to allow for curvature and refraction, as by Mr. Gravatt's
method, described in * Heather on Instruments.'
Refraction makes objects appear too high, consequently
it counteracts to a small extent the effect of curvature. It
varies with the state of the weather, but as regards levelling
it is quite near enough to assume it at its average amount
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288 Preliminary Survey '
at sea-level. That is '095 of a foot at a distance of a mile,
and varying directly as the square of the distance.
The Curvature of the earth at a mean radius is==*667
foot at a distance of a mile, and also varies (within a limit of
many miles) directly as the square of the distance. The
allowance to be made for refraction and curvature at a distance
of a mile is therefore '572 foot, or for any distance in miles
up to 100, more exactly, the correction /(?/^^adfd?l?df,is '5717 D*^.
Table L. — Correction for Curvature of the Earth.
Distance in feet
Correction +
Distance in miles |
Correction +
250
•00149
-^ - j
•0356 i
500
•005 1 1
•50
•143
750
•01 16
•75
•321
1,000
•0205
i*o
•572
1,250
•0320
125
•893 1
1,500
•0460
I '50
1-286
1,750
•0620
175
1750
2,000 '
•0820
2-00 ,
2-287
1
Whether the line of sight is adjusted tangentially or to
a terrestrial chord, if the back-sights and fore-sights are of
equal length, there will be no error in the result on account
of curvature. The correction to be added in each case is
entirely dependent upon the excess in distance of the one
sight over the other. Thus with a back-sight of 300 feet, and
a fore-sight of 900 feet, the correction of '002 foot for the
back-sight would have to be taken from the correction -017
for the fore-sight, leaving an additive correction of '015 foot.
But if the back-sight is longer than the fore-sight the correc-
tion will be subtractive. Now with the ordinary 14'' level, at a
distance of 500 feet an almost imperceptible movement of the
bubble from the centre of its run will produce a difference
in the reading of '03 foot, amounting as it does to about one
second of vertical arc, or the twentieth part of one of the
small subdivisions usually . marked upon the level bubble.
There are not many men who would have the hardi-
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Instruments
289
hood to swear to their levels to one-hundredth at the distance
of 500 feet with a 14-inch level ; but it will be seen from
the table that both instrument and man must be true to half
a hundredth every time at a distance of five hundred feet,
or else they may as well leave curvature and refraction out
of their calculations. It was partly (i) to obtain this minute
accuracy, and partly (2) to set the line of sight in the exact
optical axis of the telescope, that Mr. Gravatt invented his
elaborate ' three peg ' adjustment. To go through a tedious
process to obtain the first condition is a waste of time when
it can be added for special exactitude from the table, but
besides that, the condition is only really fulfilled for the
C
B- —
A- —
d
^ ^"" "" n- ^
-f^
q=p
[[^
-D
-C
-A
Fig. 86.— Dumpy Level.
A. A. Plane of rotation ; a. a. adjusting screws to same.
B. B. Horizontal bar ; b. adjusting screw to same.
C. C. Line of sight ; c. c. adjusting screws to same.
D. D. Bubble tube ; d. d. adjusting screws to same.
length of base used in the adjustment ; it produces an equal
and opposite error at a midway point to what is produced
by a horizontal adjustment at the further point. This is evi-
dently the case from the fact that no line of sight can be
made to follow the curvature of the earth. The second
condition is not essential to correct levelling, but it can be
fulfilled by a perfectly simple and rapid process, as will be
presently shown, but a few words are first needed on the
organic principle of the level whether Y or dumpy.
The spirit-level, like the plummet, is a device for utilising
the law of gravity to establish a horizontal or perpendicular
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290 Preliminary Survey
line. Either of them can, by means of a square, do the
work of the other.
If we fill a bottle nearly full of water and cork it, the air-
space is always at the top because the water is heavier. If
we turn it on its side on a level table the air-space will if
very small form a bubble which will stop in any position
along the side, because, the sides of the bottle being parallel,
no one part is higher than the other.
The level- tubes of good spirit-levels are very carefully
ground to a true curve, so that a movement of the tube in a
vertical plane is equal to the right or left for equal vertical
angles. The worst kind of level is that which has hardly
any curvature and is everlastingly getting off the centre with
inappreciable vertical movements. A bubble-glass which is
not perfectly uniform in curvature requires more care than
a perfect one, but correct levelling can be done with it
also.
Suppose now a bubble-tube with legs like the striding
bubble of a theodolite standing upon a table. If the bubble
be at the middle it does not prove the table level. It only
proves that either the table is level or else it has a slope
which is equal to a corresponding inequality in the length
of the legs. For if we reverse the bubble-tube end for end,
the bubble may be displaced from the centre. If the bubble
remains in the centre when reversed it proves yfrj-/^ that the
table is horizontal ; secondly^ and in consequence, that the
legs of the tube are equal. We cannot reverse a bubble upon
a sloping plane. The first thing therefore in the adjustment
of the level is to make the plane of rotation AA (Fig. 86)
horizontal, which we can do whether the line of sight is
horizontal or not, and it is therefore here mentioned
first.
The correctness of the bubble is the basis of all the
adjustments. The common expression, * correcting' or
'adjusting' the bubble, is a misleading one. The only
correction suitable to a defective bubble -tube is to break it
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Instruments 291
up, because nothing can correct an imperfect grinding.
When the bubble-tube is truly ground, the bubble is always
at the top of the curved surface, and just as a plummet
gives the vertical, so the bubble gives a true horizontal
line. It is the different parts of the instrument, such as
the plane of rotation and line of sight, which have to be
adjusted to the bubble, and not the bubble to them. The
test of accurate grinding is by marking and measuring the
travel of the bubble within equal angular movements in
opposite directions. When the bubble-tube may be turned
completely round without disturbing the bubble from the centre
of its run it proves that the plane of rotation is horizontal.
It is not necessary for practical levelling that the line of sight
should coincide with the optical or focal axis of the telescope
tube ; it is sufficient for it to be sensibly horizontal within
the range of the focussing screw, and therefore parallel to the
plane of rotation.
When both the plane of rotation and line of sight are
horizontal, if the telescope tube is not parallel to them, the
line of sight cannot be in the optical axis and will be
theoretically thrown out of its horizontality by actuating the
focussing screw. This fact is met by another, which is that
the adjustment is made for the longest distance that the
telescope will read correctly, and in that position the move-
ment of the focussing screw is not sufficient to produce
sensible error. It is only when the distances are very short
that the, effect would be appreciable, and then the divergence
of the line of sight has not sufficient distance in which to
accumulate sensible error.
The travel of the diaphragm in a vertical plane is so small
compared with the field that if the hairs are displaced as far
as they will go and the instrument adjusted to horizontahty
by the method described on p. 296, it will give the same
difference of level when set up midway between the stakes
or close to either of them.
The method given here will, however, include an almost
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292
Preliminary Survey
perfect coincidence of the line of sight with the optical axis.
It makes all four lines, AA, BB, CC, DD, in Fig. 86
perfectly horizontal. It takes a quarter of an hour the
first time, and five minutes when the pegs are driven and
their difference of level known.
The adjustment of the plane of rotation is analogous to
STRIDING BUBBLE
ADJUSTMENTS
■y rrrrrr>7 , - / 7 rry rrr-fr.
in rrrrrr/yrftrr-i
I
f / f r/ /\^>// rf////^r .
Fig. 87.
that of the striding bubble on a table alluded to on p. 290,
and will be therefore illustrated in that manner. Referring
to Figs. ^, b, c, 4 ^) the line with hatching represents
the plane upon which the level stands. In the case of a
theodolite it would be the plane passing through the trunnions.
In the case of the Y or dumpy it is A A, Fig. 86, the plane of
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Instruments 293
rotation. It is supposed to be inclined to the horizontal by
an angle =6. The bubble-legs are drawn of unequal length
to represent in an exaggerated manner the error of adjust-
ment which may be in these legs or in the little legs d^ d,
Fig. 86, and for adjustment of which capstan -headed screws
are provided. The angle of error by which the feet are
supposed to be out of parallel with the axis of the bubble
(that is with a tangent to the upper curved surface of the
bubble-tube) is L It does not matter what proportion each
error bears to the other ; they can at once be removed.
In Fig. a the striding bubble is placed so that the two
angles augment the divergency of the bubble from the centre
of its run, which becomes=0-f^. In Fig. b the level is
turned end for end and the error becomes l—i^.
Now if we bring the bubble from its position in Fig. a
to the centre of its run, as shown in Fig. r, correcting the
total error h-\-Q, half by the screws on the bubble, and half
by altering the inclination of the plane (in the transit by the
capstan-headed screws under the trunnion), we shall obtain
the following equation.
Placing the striding bubble in the position of augmented
error. Fig. a, and deducting -^^ from I gives ^ (^-~^)-
Deducting ^±^ from B gives ^ (6-^) or-^ (a-^),
.■2
that is to say, the error of the bubble and the plane in Fig. c
will each then be equal and opposite.
If, now, we reverse the striding bubble into the position
Fig. d, the reduced error of the bubble will be ^— (^, and if
we bring it to the centre of its run by equally dividing the
error as before, the error will be eliminated, as in Fig. e.
Example, The error of the plarie was -f5°, and the
bubble 7°. Placing the latter so as to augment the error,
2
7°-6°= + I°=rc'
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294 Preliminary Sun^ey
Reversing, as in Fig. d^ combined error= —2°, and again
levelling with halved correction,
-i°-h?=o
2
+ i°-~=o
2
Probably the correction will not be accurately halved
the first time, and if very much out of adjustment it will
take three or four trials.
The ordinary tradesman's level is planed or ground on
its base to a true parallelism with the axis of the bubble,
and can only be adjusted, if out of order, by re^planing.
1. To adjust the plane of rotation for either Y or dumpy
levels. Bring the bubble to the centre of its run over a pair
of parallel-plate screws. Turn the telescope 90°, and repeat
over the opposite pair of screws. Bring it back to its first
position, and retouch the parallel-plate screws. Turn the
telescope 180°, so as to be over the first pair of screws, but
end for end. If the bubble is off the centre, correct half
the error by the screws d^ d, and half by a, a, Fig. 86. Repeat
until the bubble remains central in each position. Turn
the telescope 90°, and retouch the parallel-plate screws so
as to bring the bubble to the centre, after which it ought
to revolve completely in a central position. If it will not,
it proves that the plane of rotation is not accurately ground,
or that it has become worn by sand or what not, so that it
is impossible to produce horizontality in all directions over
its surface. This, of course, could only be remedied by a
maker, or the instrument could only be used by adjusting
the parallel-plate screws at every sight.
2. To remove parallax in Y or dumpy levels. This is an
adjustment of the eyepiece to bring the cross hairs into the
common focus of the eyepiece and object-glass. Some-
times, with very short-sighted people, the little brass tube
into which the eyepiece fits has to be ground down,
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Instruments
295
The adjustment is performed by directing the telescope
upon a distant object, and first focussing the object with
the focussing screw, then the cross hairs with the eyepiece,
until the object is seen perfectly distinctly, and the hairs
are clear and do not appear to shift when the eye is
moved.
3. To place the line of sight approximately in the optical
axis of the telescope. Dumpy levels. Hold the staff about
thirty feet off, with a sheet of white paper against its back.
DUMPY LEVEL
ADJUSTMENTS
'^^':^'77^'?7>^
^^^^77777^.
Fig. 88.
Direct the staff-holder to mark with a pencil the top and
bottom of the circular field made upon the paper. If
possible, let the staff be held against a wall to steady it.
Bisect the space between the pencil marks, and mark the
centre. Bring the axial hair to coincidence with this
mark by the screws c, r. Fig. 86. The hair will be then in the
centre of the shutter, and if the shutter is concentric with the
tube, which it is in all properly constructed instruments, and
if the glass is perfectly ground, it will also be in the optical
axis. In any case it will be quite near enough to avoid all
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296 Preliminary Survey
error, since it is not from this adjustment that the line of
sight is made horizontal. When this is done, the screws
c^ c need never be touched unless the hairs are broken. An
ivory scale fastened against a wall will enable the foregoing
to be done without assistance.
4. To make the line of sight horizontal. See Fig. 88.
Drive two pegs on nearly level ground about a hundred
paces apart. Set up the level so close to one of them that,
when the levelling staff is held upon it, the eyepiece will be
about half an inch away from it.
Level the instrument very carefully, and look through
the object-glass at the staff, swaying the latter gently until
it comes into the focus of the eyepiece about half an inch
from the face of the staff. Mark the staff at the centre of
the field, which will be about one-eighth of an inch in
diameter. Book the height, which we will call instrument-
height A. Then remove the staff to the distant peg, and
with the bubble in the middle, read the height there in the
usual way, and call it staff-reading B.
Next carry the instrument to the distant peg, and set it
up, carefully levelled, close to the staff, taking a similar
reading through the object-glass, which call instrument-
height A'. Remove the staff to the first peg and read it,
calling it staff-reading B'. Now referring to Fig. 88, if the
line of sight had been parallel to the bubble, it would be
represented by the two horizontal lines in the figure, and the
difference of the readings in each position would be the
same, that is to say A— B'=B— A'.
When, however, the line of sight is inclined to the axis
of the bubble, as shown on the figure, it makes a vertical
angle of elevation or depression =(^. In the case illustrated,
being an angle of elevation, the first difference of readings
is augmented, and the second difference of readings is
diminished by that angle.
When, therefore, the difference of readings at A, that is
A— B', is less than the difference of readings at B, that is
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Instruments 297
B — A', the line of sight * throws' upwards, and when more
it * throws ' downwards.
We can express this more simply by the following :
Rule, When the sum of the instrument-heights A + A' is
less than ^^^^ ^^^ ^^ ^^^ staff-readings B + B', the line of
more than
sight throws -~^ r-, and the half difference is to be
downwards
deducted from ^^^ ^^^^^^ staff-reading,
added to
This last clause needs a little explanation. The dif-
ference of the two differences (B— A') — (A— B') clearly
measures double the angle 6/. The above may be written
BH-B'—(A' + A), and this is twice the error. In the illus-
tration, as the line of sight threw upwards, it has to be
deducted.
The crucial test of this method is that wherever the
level is placed when the adjustment is complete, whether
touching either of the staves or midway between them, the
difference of level recorded is precisely the same.
The angle 6 is caused by the line of sight not being
parallel to the plane of rotation, and having determined its
amount, we eliminate the whok of it by the screw ^, Fig. 86,
by screwing it up or down until the reading on the staff is
corrected by the amount which measures the angle ^ on the
staff. This will of course disturb the bubble, and we bring
the bubble back to the centre of its run entirely by the
screws d^ d. It will be noticed that we have kept the plane
of rotation horizontal all the time^ and we have now all four
lines horizontal.
The correctness of this adjustment can be seen by
treating the Y level in the same manner, for it will be found
when completed that the Y's can be thrown open, and the
telescope reversed end for end without disturbing the
centrality of the bubble.
We could have produced horizontaUty in the line of
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298 Preliminary Survey
sight by correcting the final error by the screws r, r, and
neglecting the third adjustment altogether, but the method
as given is the best.
When, however, there is no screw b^ Fig. 86, under the
horizontal bar, as is frequently the case with the smaller
dumpy levels, the parallelism of BB with AA can only be
corrected by an instrument-maker. We cannot then use the
third adjustment on p. 295. We make the first and second
adjustments and then proceed directly to the fourth, making
the final correction with the coUimation screws r, r, Fig. 86.
If the level is properly made, the parallelism of BB with
AA will be quite near enough to bring the line of sight
sufficiently close to the focal axis for all practical
purposes.
Example i. (Line of sight throwing upwards.)
Readings at Station A.
Ht. of inst. at A. . , . (A) 5-23
Staff-read, at B 5*15 (B)
Readings at Station B.
Ht. of inst. at B. . . . (A') 4-63
Staff-read, at A 5*03 (B')
Sum of staff readings ... io*i8(B4-B')
Sum of inst. heights .... .^^^9-86 (A + A')
Difference ..... '32
Half difference . . . . '16
Reading of true horizontal line from B=5*o3 — '16=
4*87. To which reading the cross hairs are to be brought
by the screw b.
Example 2. (Line of sight throwing downwards.)
Readings at Station A.
Ht. of inst. at A. . . . 5-08 (A)
Staff- reading at B 5*l6 (B)
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Instruments 299
Readings at Station B.
Ht. ofinst. at B.
Staff-reading at A.
4-95 (A')
4-55 (B')
Sum of inst. heights .
Sum of staff-readings .
Difference
Half difference
Reading of the true horizontal Hne of sight from B=
4*55 ^•16=471.
The difference being the same in these two examples is
a mere coincidence.
The difference of level between the two pegs might have
been determined by setting up the instrument midway
between them, and taking readings alternately on one and
the other ; but the method as given takes no longer, and
needs no measurement.
When^ however^ the difference of level is known, the
instrument need only be set up beside one of them.
When the adjustments are completed, particular care
should be given that all the screws are tight ; if not the
adjustment will last but a very short time, but if carefully
made it will probably not need touching after a month's
steady work. Every day before starting, five or ten
minutes at the peg will suffice to show that all the adjust-
ments are in order.
Adjustments of the Y Level
1. To adjust the plane of rotation horizontal, see
p. 294.
2. To remove parallax, see p. 294.
3. To place the line of sight in the optical axis.
Direct the telescope on some clearly defined point, and
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300 Preliminary Sun^ey
intersect it with the cross-hairs, revolve the telescope half
round in its Y's (being careful not to rotate it on its plane).
If the object is not still intersected, correct one half by the
screws r, <:, Fig. 86, and the other half by the parallel-plate
screws (the horizontality of the plane of rotation has nothing
to do with this adjustment). Repeat until the intersection
is the same in all positions.
4. To make the line of sight horizontal repeat No. i
adjustment and open the Y's. Reverse the telescope end
for end. If the bubble is not still central, correct half the
error by the screws d^ d^ and half by b. Repeat until the
telescope can be reversed without disturbing the bubble.
DD and BB will then be both horizontal.
5. To place the bubble axis in the same vertical plane
with the axis of the telescope.
An error in this respect is detected by the fact of the
bubble not retaining its central position when the telescope
is turned a little way in the Y's. The two ends of the
bubble are not quite in line, consequently as the telescope
is turned, one end rises a little before the other, and this
error is corrected by the capstan-headed screw at the side
of the screws which correspond to d^ but are only found in
Y levels and not shown on the figure.
The Y level has the advantage of requiring no peg-
adjustment for coUimation. The dumpy level is handier
for small sizes. English surveyors prefer it on the ground
of its supposed superiority to the Y level in retaining its
adjustment. This of course is a consideration when they
are in the habit of sending their level to the makers for
adjustment. The writer does not hold that view, but
believes the Y level will retain its adjustment just as well,
last longer without re-grinding of the axis, and is adjusted
in the field in a few minutes. The peg adjustment of the
dumpy in the usual form of the text-books is a great bug-
bear to young engineers, so it brings much grist to the
instrument-makers' mill.
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Instruments 301
The Y level is almost exclusively the type adopted in
America.
A modification of Y level has been recently introduced
by Messrs. Cooke and Sons, of York. The principal dif-
. ference is that the telescope is contained in a shorter ex-
ternal tube terminating in sockets into which the telescope
fits very exactly ; it is reversed by withdrawing it most care-
fully, and inserting it end for end to obtain the adjustment
No. 4, on p. 296. Never having used this instrument, the
writer does not wish to speak decidedly about it. Coming
from that firm, the workmanship would no doubt be excel-
lent, and that always means an instrument which will retain
its adjustment well. On the one hand it must be less liable
to wear in the bearings, and on the other hand, more
awkward to reverse it when it needs adjusting.
The wear of ordinary Y's can be corrected by the screw
b. Fig. 86.
- Level- STAVES
Two types of these are used : those in which the sight is
taken on a sliding vane or target, painted black and white
so as to obtain very precise intersection with the cross hairs.
On the target is a vernier which reads with graduations on
the staff. This type is still somewhat used in America, but
hardly at all in England. It can be read at a greater
distance than the graduated rod, but requires an assistant
who can be depended upon to book the readings correctly.
Graduated rods are usually marked by lines across at
every hundredth of a foot, the spaces being alternately
black and white.
When the staff is used for telemetry, it should have a
device for ensuring that it is held at right angles to the line
of sight, or else plumb according to the manner of working.
For colliery work, an illuminated staff has been successfully
employed having the figures painted on glass, and a lamp
carried in a thin casing at the back of them.
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Preliminary Survey
Different Methods of Keeping the Fieldbook
The 'rise andfalV Method
This is the most rigorous plan. The reduction of inter-
mediate sights forms a check upon the turning-points. The
form of fieldbook is as under :
•1
4-65
a
•1
•a
1
Rise
Fall
II
8 ■ Red. level of i
Q turning-points
.2
Q
Remarks
B. M. /^ A on
mile- 1 /N 1 ston®
2-23
2-42
9*5t
102*42 j
11-74
92-91
9'37
2*37
95-28
8-43
11-13
1-76
93*52
Turning-point on
peg
5'22
3*21
I
96-73 1
1
1*13
4-09
IOO'82
7*84
6-71
94-11
i3'o8
i8-97
\
i3'o8
5-89
1
TOO'OO
1
Check
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Instruments
303
By keeping the reduced levels of the intermediate
separated from the turning-points, the independent check
of back-sights and fore-sights is more conveniently applied
at any time, and it should be done at the end of every
page.
The Collimation Method
This is almost exclusively used in North America, and
to a large extent in the Colonies. It is much quicker, and
will receive a little more explanation because it is the
foundation of the tacheometer fieldbook. It has the dis-
advantage of greater liability to error in the reduction of the
intermediate sights, each calculation being independent ;
but as it enables the railway surveyor to put in his gradient
stakes without first sitting down to reduce his levels, its
advantages are deemed to outweigh the one disadvantage.
The back-sight by this method becomes a plus sight, and
the only one. All the intermediate sights and the fore-sight
are minus sights.
In Fig. 89 the operation commences with a back-sight
to determine the elevation of the line of sight, commonly
called 'collimation;' the staff-reading of 9*00 feet being
added to the known or assumed elevation of the fiducial
point on which the staff is held at starting. The collimation
being known, the elevation of any other point which can be
seen is equal to the collimation height less the reading on
the staff, and the reduction of the levels is relieved of the
column of rise and fall.
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304
Preliminary Survey
Form
of Fieldbook
9-00
-T. p.
l
109*00
1 Red. level of
1 intermediate
8 Red. level of
g turning-points
Remarks
B.M. on mile-
stone
4-10
105*00
lO'OO
iiS'oo
99 -oo
1
II 'oo
2"00
107*00
\
7-00
III'OO
[
20 'oo
9'oo
i
} II'OO
9'oo ^^^
^
' lOO'OO
Check
1 1 "oo _ --
-^^ 1
The best rules for precise levelling are —
First. To adjust the line of sight truly horizontal, /. e,
tangential to the earth's circumference.
Second, Not to allow any sight to exceed 250 feet
distance with a 14-inch level or less with a shorter one.
Third, To keep the back-sights and fore-sights as nearly
as possible at equal distances.
Fourth. When back-sight and fore-sight are at unequal
distances to correct by table — by adding to every sight the
tabular amount where it is not less than -ooi. It will be quite
near enough to measure the distances with a passometer.
Theodolites
The plain theodolite for ranging base lines and curves
is the foundation of most tacheometers. The two types in
use are the Y and the transit. For ordinary work the
transit is very much to be preferred on account of the
liabihty of the line of sight to be shaken off its position when
Instruments 305
reversing the telescope in its Y's ; the time taken in rever-
sing ; and the danger of leaving the clips loose and dropping
the telescope out when shouldering the instrument.
When telemetric work is the chief use of the instrument
the Y type has two advantages which bring it more into
competition with the transit.
I St. Within a considerable range of vertical arc, a tele-
scope of twice the focal length usually supplied can be
safely and steadily carried in specially constructed Y's. '
2nd. The adjustment for coUimation is made more
rapidly.
On the Hawaiian survey, the author used a seven -inch
Y theodolite by Elliott Brothers, carrying an eighteen -inch
telescope with eyepiece magnifying forty diameters. It was
furnished with stadia-hairs reading i per 100, and a
movable micrometer hair for long distances. It was a
heavy instrument, but, being of such long range, did not
want so much shifting. It was carried by one man, over
very bad country, and was quite satisfactory.
The adjustment of the Y theodolite is the same as the
Y level for parallax coUimation and bubble. The zero is
then brought to coincide with the zero of the vernier of the
vertical arc when the bubble is at the centre of its run. This
is done by means of a small screw fastening the vernier of
the vertical limb to the vernier plate over the compass-box.
The adjustment of the horizontal limb is the same as
that for the transit theodolite.
Stanley's New Patent Telemetrical Theodolite
Stanley's New Telemetrical Theodolite, Fig. 90, has the
following improvements : A tribrach stage instead of four
parallel screws, which permits the instrument being used on
a wall without the legs, and prevents all possibility of strain-
ing the centre.
The instrument has a mechanical stage for shifting the
centre with exactness over the desired spot.
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Preliminary Survey
The telescope supports and the centre are in one casting,
also the size of the bearing on the centre has been increased.
Instead of webs, which are a constant anxiety to the
engineer or surveyor, platinum-iridium points are substi-
FiG. 90
tuted. These neither rust nor break, and allow of any dust
being brushed off with a camel-hair pencil without inter-
fering with the adjustment.
The eyepiece is also fitted with two vertical adjustable
points for measuring distances.
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Instruments 307
A long trough needle is supplied with each instrument
in place of the usual small circular compass.
The head of the tripod is of an improved form, giving
greater rigidity.
The transit-theodolite is the basis of the author's tacheo-
meter, and the following description will embrace all that is
required for the use and adjustment of the ordinary transit,
together with the special points of difference in his parti-
cular design.
Gribble's * Ideal' Tacheometer
The author has been fortunate enough to be able to ex-
periment a good deal in instruments without so much out-
lay as most inventors in that line. He has been able to
dispose of his theodolites in foreign countries and then get
new ones. The * ideal' is the fifth instrument specially
designed by him for preliminary survey. There is really
nothing original about its principle, and yet in its actual
form and combination of parts it is unlike any other instru-
ment he has met with.
The principal features of novelty are :
First, A decimal subdivision of the ordinary degree of
90 to the quadrant.
By this means all the advantages of the centesimal
graduation are obtained whilst retaining correspondence
with the published astronomical ephemeris.
Tables to five places of decimals for the trigonometrical
functions, together with logarithms of numbers to fiv^ places,
can be procured from Messrs. Ascher & Co., Bedford Street,
London. They are compiled by Dr. C. Bremiker, and cost
IS, (yd, bound in cloth. They are the only tables extant for
this graduation. Transits can be adapted by merely changing
the vernier.^
' It is the original graduation of Briggs in 1633, followed by Roe
and Oughtred, in which they endeavoured to get rid of the senseless
sexagesimal subdivision, which survives in spite of them.
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Preliminary Sun^ey
A vemiir plate
Ph lower plate
C adju!}t[nj£ screw to horizontal
Limb
D compass^boji:
E clamping screw It) exi^rnal axis
F telescope
parallel-plate screws
N fraincs
R vertical arc
VV tangent screw
a. striding bnbble
b riicrcmieter head
c diagopal eyepiece
d levels af horizontal I imli
£ lantern
f rack-screw
^ adjustment screws to level
A steel tape
1 plummet
z clip**crtw and lacknut
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Instruments 309
Slide-rules graduated decimally can also be obtained, at
the same price as for the sexagesimal subdivision, from either
Messrs. Tavernier & Gravet at Paris or Messrs. Davis & Sons,
Derby.
Curve-ranging and tacheometry are especially facilitated
by this graduation : see Chapter VII.
Secondly. The advantages of a glass diaphragm are
obtained without the usual drawbacks. The loss of light is
reduced to a minimum, and the diaphragm can be removed
without disturbing the micrometer diaphragm, if it should
be found necessary to clean it. Its adjustment does not
require the delicate handling which is needed for the micro-
meter hair, so that any one who can adjust a transit can
adjust this instrument with greater ease and in less time.
Thirdly, The instrument is a combination of the stadia
principle within the limits of legibility of the figures upon
the staff, together with the micrometer principle for sights at
longer range ; the level staff being adapted for use either
on the one principle or the other. »
Fourthly, The telescope is of unusual power ; probably
no instrument has yet been constructed of the same lightness
and portability with a magnifying power of fifty diameters.
It is consequently equally well adapted to astronomical
observations and long-range sights of the staff.
Fifthly, It combines minuteness of levelling power with
lightness of construction by making the vertical arc 6 inches,
and the horizontal arc 5 inches diameter. Both arcs read
with the vernier to -oi degree, which is equal to 36 seconds,
but the vertical arc can be estimated by the eye to 18 seconds.
In tacheometry it is generally sufficiently accurate to read
the horizontal line to the nearest minute, which is more than
obtained by the five-inch circle, and the saving in weight is
considerable. The trunnion standards also are stayed.
Sixthly, It has a wide range of efficiency.
As an astronomical telescope, its power enables a good
observation to be taken of the phenomena of Jupiter's
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3IO Preliminary Survey
satellites, or the culmination of one of the small stars in the
British Association Catalogue, used in conjunction with a
lunar transit for determination of the longitude.
As a micrometer telescope of long range it will determine
distances with considerable accuracy at several miles. As a
tacheometer it is of superior power to any the writer has yet
met with.
As a handy theodolite for ranging base-lines and railway
curves, it has none of the complications of the Wagner-
Fennel Tacheometer, the Eckhold Omnimeter, or the Porro
type of telemeter. It has the appearance of a medium sized
simple transit theodolite and can be
more easily adjusted.
The appearance of the diaphragm is
not so complicated as the usual tacheo-
meter. Only the axial lines are drawn
across the glass, the rest are replaced
by ticks which cannot possibly lead to
Fig. 92. confusion.
The ticks a and d are the stadia
lines. The vertical ticks, or comb as they are called, are
spaces of 100 micrometer units, or 1,000 units on the
micrometer vernier.
A three-screw or * tribrach ' stage has been adopted to
avoid strain on the pivot and expedite the adjustment. A
centering arrangement was at first included, but has been
left out of the last instrument made. Where the chief use
is for short bases with the chain, or curve-ranging, the
centering clamp is a convenience, but it weighs from 4 to
5 pounds, and in tacheometry the bases are long, and the
instrument is not so frequently shifted. Any small errors
from the width of the staff as a picket, or from inaccurate
centering, are eliminated by the observation for azimuth.
The addition of another lens as first introduced by Porro
to make the centre of anallatism (see Glossary) coincide
with the vertical axis has been abandoned. The constant
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Instruments 311
of the instrument is from 175 to 2*0 feet, according to size.
Practice has proved that no time need be lost nor any
mistake made in adding this constant. It is not worth any
extra expense or loss of light in order to eliminate it.
Instead of the ordinary compass-box, a trough-needle is
substituted, which saves the headway under the telescope,
and answers the purpose of obtaining the magnetic variation
at the commencement of the work equally well.
A word of caution is needed about the use of the tribrach
stage. A very grave defect exists in the form frequently
supplied to theodolites, namely, that the clamping plate is
not left attached to the parallel-plate screws when the
instrument is put away in the box. It forms a separate
piece. In addition to the extra trouble in packing away, it
is a great danger, inasmuch as there is no warning to the
surveyor, who may chance one day to leave his clamp open.
With the old-fashioned screw head, his instrument may be
loose and turn once or twice round without falling, but one
slip of memory with the clamping plate and it would be
sudden death to his instrument the moment he shifted it.
To remove this danger, a screw is put into the clamping-
plate, which has to be removed if the transit has to be set
on a wall. Comparatively few surveyors use it thus, and it is
generally preferable when taking observations extending over
several days to drive very solid pegs ^' x \" into the ground
and cut notches for the feet of the tripod. At any rate, it
is so very seldom that the instrument requires to be taken
out of the clamping-plate, that it is a most mistaken plan
to risk the safety of the instrument in order to make it
easily removed. In the * Ideal ' tacheometer, the instrument
packs into its case in two pieces.
For astronomical observation, the usual diagonal eye-
piece and lamp for illuminating the axis are provided.
The plummet is suspended from a short chain, which is
a fixture with the instrument. It is preferable to a rigid
hook, which is liable to get bent. The hook of the chain
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312
Preliminary Survey
is strong enough to carry a heavy weight for steadying the
instrument in a high wind. The distance of the hook from
the centre of the trunnion forms with that from the initial
point of a 5 feet steel spring tape 1*50 feet, so as readily
to take the height of instrument as an independent check.
The box of the steel tape forms for ordinary use the
plummet, but for special accuracy a pointed plummet is
also provided.
Adjustments of the * Ideal' Tacheometer
Supposing that the micrometer hair has broken, we will
commence with
I. Putting in a spider hair. Prepare the rectangular
frame shown in Fig. 93 of copper wire, yV diameter or
thereabout, soldered together. Catch a field spider and let
Fig. 93.
him drop by his web from one end, then wind round the frame
till full. Preserve it at the bottom of the level or transit case
in a little casing with blocking strips to keep the web from
rubbing off. It will then be ready for use at any time. The
points where the web touches the frame should be tipped
with shellac to fix them.
Place the diaphragm on the table with a strong light
bearing upon it. Tip with shellac the faint lines cut on
it to mark where the web should go. Superimpose the
web by delicately turning the frame until the web is in its
position. Hold it there till the shellac cakes, very gently
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Instruments 313
pressing the web, so that it shall be quite taut. Remove
the frame, and the web will be left in position.
2. To replace the diaphragms. To put in the glass
diaphragm requires no description. To replace the mov-
able diaphragm, hold it by a pair of tweezers while insert-
ing the capstan-headed screws. Turn the screws until the
hair is as nearly vertical and as near the middle of the tube
as can be estimated by the eye. Leave the micrometer
screw till the last thing. Slide the movable hair to one side
so as to be out of the way.
Insert the eyepiece.
3. To remove parallax. See p. 294.
4. To make the line of sight correspond with the zero of the
horizontal limb. Set the instrument as nearly level as can be
done with the eye, then clamp the lower plate B, and, having
undamped the vernier-plate A, direct the telescope on some
well-defined object, and bring it into coincidence with the
point of intersection of the axial lines on the diaphragm.
Take the reading on the horizontal limb AB ; suppose it to
be 20° '00 ; then move the vernier-plate A half-round, turn
the telescope over, and again intersect the object, taking the
reading on the horizontal limb AB — suppose it 2 00° -04 ; take
the difference between this and the first reading +180°
(which in the present case would be 200°, and the differ-
ence -04°) ; halve this difference, and subtract it from the
second reading when it is greater than the first reading -f
180°, and add it when it is less ; this is the mean reading
(=2 00° -02). Set and clamp the instrument to this mean
reading, and intersect the object by means of the screw
which moves the glass diaphragm, sideways. Repeat this
operation until the readings taken with the instrument in
these two different positions, face right and face left, differ
from one another by 180°.
5. To make the line of sight correspond with the zero of
the vertical limb. It is not necessary that the line of sight
should also be identical with the optical axis. See remarks
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314 Preliminary Survey
on levels, p. 291. Parallelism is sufficient. Level the
instrument carefully on the external axis, by means of the
levels d^ d on the horizontal limb AB ; next take a pair of
verticals — ix. on faces right and left to any well-defined
terrestrial object ; set the vertical circle R to the mean
of these readings, and clamp it ; now intersect the object,
using the two screws z^ z which clip the vertical circle
R to the stud in the telescope frames N, N, and not
the tangent screw W. When the readings on the right
face agree with the left face, the index error will be o.
The cHp-screws 2;, z are provided with capstan-headed
locknuts, which should then be screwed home, as these
screws are not used except for adjustment. The leather
eaps on the nuts of the clip-screws are to prevent their
being moved by mistake instead of the tangent screw.
One of them need never be touched. It can be marked,
and only the other one used for releasing the telescope, so
that the locknut on the untouched clip-screw will bring
the vertical limb to its proper position when it is replaced
without fresh adjustment.
Example. With the telescope in its normal position
observed the top of church spire. Vertical angle + ii°-i3.
Rotating the horizontal limb and reversing the telescope,
vertical angle=4-ii°*i9. Set the vertical limb at ii°*i6
by the tangent screw W, and make the line of sight intersect
the object by means of the clip-screw, z. Then reverse the
telescope, and rotate the horizontal limb to their original
positions, and the vertical arc will still read ii°'i6. If not
quite exact repeat the operation.
6. To make the plane of rotation of the vernier plate
horizontal Clamp the vertical limb R at 0°. Tighten
the clamp-screw E, unclamp the vernier-plate A, and turn
it round until the telescope is immediately over two of
the parallel-plate screws I, I ; bring the bubble in the
telescope-level P to the middle of its run by the screws
I, I ; turn the vernier-plate 180° so as to bring the telescope
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Instruments ' 315
again over the isame screws, but with its ends in a reverse
position. If the bubble of the telescope level does not
remain in the middle of its run, bring it back to that
position, half by the parallel-plate screws I, I, and half by
the screws g, g. This operation must be repeated until the
bubble remains accurately in the centre of its run in
both positions of the telescope ; now turn the vernier-
plate A until the telescope is directly over the third parallel-
plate screw, and bring the bubble to the middle of its run
by turning that screw. The bubble .should now retain its
position while the vernier-plate is turned completely round,
showing that the plane of rotation is truly horizontal or,
parallel to the axis of the bubble.
7. To test the accuracy of the external axis. Clamp the
vernier-plate to the lower plate by turning the clamp-screw
C, and loosen the clamp -screw E ; move the instrument
round its external axis, and if the bubble retains its central
position during a complete revolution there is no error. If
one exists it is only to be remedied by the maker. If, how-
ever, adjustment 6 is correct, the work will be correct.
8. To adjust the levels on the vernier plate. These are
only guides for approximate adjustment of the instrument
when it is being set up. They are set true when the other
adjustments are completed.
9. Horizontality of the axis of the telescope. This is done
with the striding bubble. See full description under levels,
p. 292.
10. To test whether th£ vertical axial hair has been placed
in a vertical line. This adjustment is not required in the
* Ideal ' tacheometer, but is added for the use of those who
have ordinary transits. It is not essential to correct work if
the intersection of the hairs is always observed, but it is
convenient to be able to intersect an object with any part of
the vertical line, and it is also useful for telling if a staff &c.
is held vertical.
Fix the intersection of the axial lines on some sharply
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Preliminary Survey
defined terrestrial object, such as the lightning conductor of
a building, or, failing any similar point, drive a stake loo
paces off and put a nail in. Move the vertical arc up and
down and watch whether the intersection point remains on
the vertical line ; if not correct it by the diaphragm screws,
and check over the former adjustments in case they may
have become deranged.
Other and very delicate adjustments are needed for
special purposes, but the above are sufficient to put the
instrument in ordinary working order.
11. To place the micrometer hair in position. Turn the
spindle of the screw with the micrometer head removed
until the hair coincides with the vertical axial line ; put on
the micrometer head very carefully so that the zero is at the
zero of the vernier. This needs practice and generally
takes a few trials.
12. To check the stadia lines. From the vertical axis ot
the instrument determined by the point of the plummet
y
-^— d^ — ->
Fig. 94.
measure out on the ground the position of the anterior focus,
that is a distance composed of the focal length of the object
glass 4- the length from the vertical axis to the object glass.
This is the centre of anallatism or point from which heights
subtended by the stadia are proportional to the distances of
the objects. (See Chapter V .) In the * Ideal ' /= 1 2 inches
and ^=8 inches, therefore /-l-^= 1*67 feet. Hold up an
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Instruments 317
ivory scale against a wall or house 80 or 90 feet away,
and direct the stadia upon it. Suppose it is a sixty
scale and the stadia subtend 662 divisions. Place a 720
on the rule of the slide-rule, representing the value of
a foot in divisions of a sixty scale opposite a i of the slide,
and find the value in feet '9195 opposite to the 662 on the
rule. Then with the steel tape measure from the centre of
anallatism to the wall. If the stadia are set to i per 100,
this ought to measure 91*95 feet.
Suppose it measures 92*05 feet. Set the 91*95 feet op-
posite the 92*05 feet on the shde-rule, and any distance can
then be read off direct, by looking from one scale to the
other and adding the constant. Practically the stadia lines
are correct, and as it is not required to plot closer than a foot
the constant of 2 feet is added to the number of hundreds
thus subtended on the staff as the distance from the centre
of the instrument.
Another method when plotting to a large scale is to draw
a circle with a radius=/H-^/ round the trigonometrical station,
and plot the distances (ex. the constant) from the circle.
A check should be made on the foregoing computation
of stadia by measuring a base of about 600 feet on level
ground and reading the stadia there also.
13. To form a table of micrometer values with a ten-foot
base. For this a long base on a tangent is needed ; a piece
of level railway track is the best. The instrument can be
set up a little beyond the B. C. point of the curve so as to be
well off the track and out of the vibration of passing trains
and yet in line with the tangent.
Beginning at 500 feet the staflf is held in position
with the vanes up, and the micrometer value is measured
two or three times over.
At each 100 feet up to about 6,000 feet, fresh readings
are taken and booked. They are then plotted as follows .
A differential curve is prepared by laying off on a sheet
of drawing paper a horizontal base equal by scale to the
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31 8 . Preliminary Survey
total measured length, and at every loo feet measuring up
ordinates equal to the micrometer values. Between the
hundreds, the values are interpolated by ordinates to the
curve down to the tens, and the units are then interpolated
by simple proportion with the slide-rule.
The curves are put in with ordinary railway sweeps.
This will be found quite as accurate as more frequent
observations in the field ; as a matter of fact, the curves will
soften several asperities due to eccentricities of observation.
Printed profile paper with faint scale lines is convenient
for this purpose.
The * Ideal ' reads the level staff with the stadia up to
I, GOO feet ; beyond that the micrometer is used.
A close approximation to the actual magnifying power
of the telescope may be very simply obtained in the follow-
ing manner. Set it up at a distance of 20 feet from the
anterior focus and cause a levelling-staif to be held there
perfectly steady. The stadia will then subtend '2 upon the
staff. Direct the lower stadia hair to some convenient
figure such as 5*00 feet, then the upper hair will be at 5*20.
Open the other eye and look with the one eye through the
telescope, and the other unassisted at the staff, until the
actual staff appears to be superimposed upon the magnified
portion and you seem to see the one through the other.
Book the two positions of the stadia hairs on the actual staff.
For instance, if the eyepiece magnified 20 diameters, the
stadia hairs would appear to cover 20 x '2=4 feet of the
natural staff. The lower would then appear to be at 3'io
and the upper at 7'io feet.
If there are no stadia hairs, take the top and bottom of
the field and proceed similarly.
The Sketch-board Plane-table
Fig. 95 represents the present improved type of cavalry
sketching-board as modified from Col. Richard's original
design by Capt. Willoughby Verner, whose field-sketching
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K -. ..7orB ^
M-
Lzne^ of
CE
m rni
Fig. 95.
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320 * Preliminary Survey
and reconnaissance in Egjrpt form an interesting feature of
the Soudan Campaign.
The paper is in a continuous roll, wound round the
brass rollers, and turned down under them so as to be
always quite tight. It is to a large extent waterproof, being
specially prepared, and graduated in quarter-inch squares.
The line marked * line of direction ' is that in which the
paper is unrolled so that when the line of march does not
deviate very much from it, the work will not run off the
board.
The thin line drawn across the compass-box is a cut in
the glass, the object of which is to indicate, by turning the
compass-box round, the position of the needle, when the
board is in the line of direction. The needle, being a loose
one, is first observed before starting, the board being set to
the line of direction. The compass-box is then turned until
the cut in the glass coincides with the needle, and serves
ever after to bring back the board into the line of direction.
This is further explained on p. 41.
The heads to the brass rollers fit tight in the wood and
keep the paper stretched by their friction. In the cross
section the buckle is shown attached to a dovetailed slide
and pivot. When used on horseback, the strap fastens the
board to the arm, and when fastened to a tripod, the dove-
tailed plate runs into a corresponding slide on the tripod.
The seven-inch by nine-inch size is more of a sketchboard
and the nine-inch by thirteen-inch size is more of a plane-
table, but both sizes are used in either way effectually.
Pocket Altazimuths and Compass-Clinometers.
Strictly speaking an altazimuth is an instrument which
gives the altitude and azimuth from one adjustment of its
line of sight.
The first altazimuth was a stationary transit instrument ;
what is commonly called now a wall transit.
Three-screw transit theodolites are now adapted to being
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placed on walls ; but the term altazimuth has also become
applicable to pocket instruments, in which the azimuth is
found by the magnetic bearing and the angle of elevation or
depression is read by means of a plummet or a level.
Another variety might be designed combining the pocket
sextant with the measurement of altitudes. Possibly there
may be such already.
Messrs. Casella make an altazimuth in which the mag-
netic bearing and altitude by plummet are both read by a
microscope and the observation is assisted by a telescope.
Fig. 96.
Messrs. Elliott make Colonel O'Grady Haly's compass-
clinometer in which the same operations are performed
without any lens power.
All these instruments are practically two instruments in
one, to save the time and trouble of taking out first one and
then another from separate sling-cases.
Hand instruments are better without telescopes. Work
requiring telescopic accuracy is better done with a tripod.
A combination of the box sextant with the altitude
measurer would have the advantage of greater precision in
the horizontal angles and fully as much despatch. On the
other hand it would be subject to cumulative error, which
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322 Preliminary Surz>ey
the compass is not. It would be free, however, from magnetic
deviation, which sometimes renders the compass wholly
unreliable.
The plummet makes a slow clinometer and not a sure
one either. If the instrument is not held in a vertical
plane, the plummet is apt to stick against the side.
The writer has used nothing of the kind of late years,
but confines himself to an, arrangement of his own, con-
sisting of a combination of Captain Abney's reflecting level
with a prismatic compass.
This is illustrated in Fig. 96. The Abney level is now^
so well known that it only needs a passing description (see Fig.
Fig. 97.
97). Its principle is somewhat that of the vertical arc of a
theodolite, only it is in miniature and without lens power. A
small telescope has been tried but not with success. A re-
flector occupies half the field of the sighting-tube, adjusted
at an angle so that when the bubble is at the centre of its
run, its reflection is seen in the centre of the tube.
Whatever angle is given to the line of sight, the bubble can
be brought to the centre of its run without removing the eye
from the tube, and the little handle which moves the bubble
tube is furnished with an arm and vernier, which indicates
the angle of altitude upon a graduated semicircle. The
range of view of the bubble through the tube is only up to
60°, but the author uses his instrument also up to 90° for
getting the batter of a wall or the side slope of a steep
cutting by placing it on a straight edge, and then bringing
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the bubble to the centre of its run without looking through
the tube. He therefore has the circle graduated up to 90°,
and often uses it as a 'plumb -level.'
The prismatic compass forms the handle, which turns
the bubble, in place of the usual little brass wheel. Its
disc is only i^ inch, but it has a graduation from o to 360°
and also a quadrantal graduation for working by latitude and
departure. The prism magnifies so well that the graduation
of single degrees can be easily subdivided by estimation
to quarter degrees. It is made by Messrs. Elliot Bros., and
does them credit.
The further use of this instrument is explained in ' Route
Surveying,' pp. 61, 64, &c.
Passometers
The passometer is a register of the number of paces
taken at any time by a walker ; its use has been explained
on p. 54, &c. The graduation of passometers is often very
clumsy. The arrangement in Fig. 98 is recommended as
being easily readable. The right-hand small dial reads up
to 20,000, which is over ten miles. The large hand gives
the fifties up to 2,000, and the left-hand small dial the units
between the fifties.
The instrument in Fig. 98 is reading 0,558 paces. The
hands of passometers are generally set loose, so that they
can be adjusted to zero. For surveying this is a mistake,
because they soon get out of teaching with one another.
A permanent fit is best — that is to say, with square bearings
instead of conical, like the hands of a watch.
The passometer should be attached to the centre of the
person ; if placed on one leg it will only count half paces. The
best place is hanging from a waistcoat button, with the hook
well buttoned in to be secure, and just kept from shaking up
and down by the edge of the waistcoat. If too free it will
occasionally made a double count. When the counting
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324 Preliminary Survey
has to be suspended for a while, the passometer can be turned
upside down.
The pedometer and passometer can be made to work
together, by adjusting the former until it records a quarter
Fig. 98.
mile against the average number of paces to that distance
by the latter. The ordinary adjustment of the pedometer to
length of pace in inches is not always exact.
Sextants
Both the Hadley sextant and the box sextant are so
well known that only the adjustments will be here given,
which are taken from Mr. Heather's excellent little work on
* Instruments.'
To examine the error arising from the imperfection of the
dark glasses. View the sun through the dark glass at the
end of the telescope, removing the shades ; make a contact
with the reflected image of the sun and its direct image
seen through the unsilvered part of the horizon glass. Then,
removing the dark glass, set up one shade glass after the
other, and book any alterations of angle due to each succes-
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Instru7nents 325
sive combination. No adjustment can be made for this error ;
when registered it has to be applied at every observation.
The adjustments consist in setting the horizon glass
perpendicular to the plane of the instrument, and in setting
the line of collimation of the telescope parallel to the plane
of the instrument.
To adjust the horizon glass. While looking steadily at
any convenient object, sweep the index slowly along the limb,
and, if the reflected image do not pass exactly over the direct
image, but one projects laterally beyond the other, then the
reflectors are not both perpendicular to the face of the limb.
Now the index glass is fixed in its place by the maker, and
generally remains perpendicular to the plane of the instru-
ment ; and, if it be correctly so, the horizon glass is adjusted
by turning a small screw at the bottom of the frame in
which it is set, till the reflected image passes exactly over
the direct image.
To examine if the index glass be perpendicular to the
plane of the instrument. Bring the vernier to indicate about
45°, and look obliquely into this mirror, so as to view the sharp
edge of the limb of the instrument by direct vision to the
right hand and by reflection to the left. If, then, the edge
and its image appear as one continued arc of a circle, the
index glass is correctly perpendicular to the plane of the
instrument ; but if the arc appears broken the instrument
must be sent to the maker to have the index glass adjusted.
To adjust the line of collimation, i. Fix the telescope
in its place, and turn the eye-tube round, that the wires in
the focus of the eye-glass may be parallel to the plane of the
instrument. 2. Move the index till two objects, as the sun
and the moon or the moon and a star more than 90° dis-
tant from each other, are brought into contact at the wire of
the diaphragm which is nearest the plane of the instrument.
3. Now fix the index, and altering slightly the position of
the instrument, cause the objects to appear on the other
wire, and if the contact still remain perfect the line of col-
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326 Preliminary Survey
limation is in correct adjustment. If, however, the two
objects appear to separate at the wire that is further from
the plane of the instrument, the object end of the telescope
inclines toward the plane of the instrument ; but if they
overlap, then the object end of the telescope declines
from the plane of the instrument. In either case the correct
adjustment is to be obtained by means of the two screws,
which fasten to the up and down piece the collar holding
the telescope, tightening one screw and slackening the other,
till, after a few trials, the contact remains perfect at both
wires.
The instrument having been found by the preceding
methods to be in perfect adjustment, set the index to zero,
and if the direct and reflected images of any object do not
perfectly coincide, the arc through which the index has to be
moved to bring them into perfect coincidence constitutes
what is called the index error, which must be applied to all
observed angles as a constant correction.
To determine the index error, — The most approved method
is to measure the sun's diameter, both on the arc of the
instrument properly so called, to the left of the zero of the
limb, and on the arc of excess to the right of the zero of the
limb. For this purpose, firstly, clamp the index at about
30' to the left of zero, and, looking at the sun, bring the
reflected image of his upper limb into contact with the direct
image of his lower limb, by turning the tangent screw, and
set down the minutes and seconds denoted by the vernier ;
secondly, clamp the index at about 30' to the right of zero
on the arc of excess, and, looking at the sun, bring the
reflected image of his lower limb into contact with the direct
image of his upper limb by turning the tangent screw, and
set down the minutes and seconds denoted by the vernier
underneath the reading before set down.
Then half the sum of these two readings will be the
correct diameter of the sun, and half their difference will be
the i?idex error. Wh^n the reading on the arc of excess is
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the greater of the two, the index error then found must be
added to all the readings of the instrument, and when the
reading on the arc of excess is the less, the index error must
be subtracted in all cases.
To obtain the index error with the greatest accuracy, it
is best to repeat the above operation several times, obtaining
several readings on the arc of the instrument, and the same
number on the arc of excess, and the difference of the
sums of the readings in the two cases, divided by the whole
number of readings, will be the index error ; while the sum
of all the readings divided by their number will be the sun's
diameter.
Example,
: or excess
Readings on the arc of the instrument
35' 0"
35' 5"
35' 10"
Readings on the ai
29' 25"
29' 35"
29' 20"
105' 15"
88' 20"
88' 20"
105' 15"
•' I
N»°^;°f}6) 16' 55" Difference 6)193^' Sum
32' 15-8" Sun's diameter
2' 49" Index error
The Adjustments of the Box Sextant
On the upper surface of the instrument close to the rack-
screw is a little hole with the square head of a screw inside
it. This is a screw for adjusting the horizon glass in the
plane of the instrument by observing the reflected image of
the sun.
There is also another screw at the side of the instrument
for removing index error by taking readings on the arc of
the instrument, and on the arc of excess as described in the
adjustments of the Hadley sextant, and then by correcting
with the screw so as to make both readings the same.
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328 Preliminary Surve)'
Both screws are turned by a little key, which is removed
for the purpose from its position in the arc-plate opposite
the rack screw.
The Solar Compass
The solar compass is an instrument for determining the
true bearing of any object, instead of the magnetic. It
performs automatically the operation of finding at any time
of day and in any latitude the azimuth of the sun. It can-
not of course be used in cloudy weather, and should not be
used when the sun is less than one hour above the horizon
or less than one hour from the meridian. It stands on a
tripod of the same size, and is itself somewhat larger than
the ordinary surveyor's compass.
It is furnished with vertical arcs to set it to the latitude
of the place, and a pair of sight vanes.
Unless the adjustments are very carefully attended to
errors are likely to arise, greater than those due to local
magnetic deviation.
The price ranges somewhat between that of a surveying
compass and a plain transit.
The Heliostat and Heliograph
These instruments are both sun-signals ; the principle is
the same in both. A mirror reflects a sunbeam from one
point to another. The heliostat is only intended to give a
continuous ray, whereas the heliograph is provided with a
spring to the mirror, by which it is made to give little jumps,
causing it to flash short or long flashes corresponding with
the dots and dashes of the Morse code. The heliostat is
capable of being used for signalling also, by alternately
covering and uncovering the mirror. Both instruments are
furnishad with sighting vanes on jointed arms, commonly
called jiggers, to set the mirror in line with the station to
be signalled. The heliograph is also furnished with a duplex
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Instruments 329
mirror, so that when the sun is behind the instrument it
can still be used.
The best heliographs are Galton's sun signal,, fitted with
a telescope, and Mance's heliograph. Both are expensive
instruments, and need not enter into the outfit of the
preliminary surveyor.
The sketch on next page is taken from Capt. Wharton's
' Hydrography,' ^ and illustrates the principle of all such in-
struments. It is a heliostat made by a ship's blacksmith.
The standard is about 2\ feet high. ' In soft ground the
ends of the legs can be pressed into the earth, and on rocky
ground stones placed against the legs will hold the instru-
ment steady. The arm ///, of light iron, is carried separately,
and slips over the shaft of the standard, clamping when
required with a screw.
* Into a circular socket in head of standard shaft the leg
of the frame holding the mirror is shipped ; this is also to
be tightened by a retaining screw. The mirror, which can
be of any size from 2 10 6 inches or more in diameter,
revolves on its retaining screws as an ordinary toilet-
table glass, and can be held in any position by the
screws.
* The ring, of fiat wood, is made as light as possible, so as
to exert less strain in wind. Across it are nailed crossed
strips of copper with a white cardboard disc, about an inch
in diameter, fastened to their centre.
* The rod that carries this ring slips up and down in a
hole at the end of the arm, and is clamped by a retaining
screw.
* In the centre of the back of the mirror a hole of about
I inch diameter is scraped in the tinfoil, being careful to
leave a sharp edge. A similar hole is cut out of the wooden
back of the glass frame. This we shall call the blind
spot.
' A cheap instrument of this description is made by Potter of
London.
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Prehminaryi Survey
* To direct the flash to an object, bring the mirror vertica),
and looking through the hole in the centre, revolve the arm
until in the direction of the object nearly, clamp it, and
adjust the disc-rod as nearly as may be for elevation or
depression. Then, slightly loosening the screw, clamping
the arm, finally adjust the latter so tliat the object, as
regarded through the hole in the mirror, is obscured by the
Fig. 99.— Captain Wharton's * Hydrography.'
a, sliding collar carrying arm w/, revolving round s ; b^ wooden ring,
painted black, with iron wires and white cardboard centre, sliding
vertically by means of rod through arm tn ; c, iron frame to hold
mirror, fitting into socket in top of standard s ; j, iron standard with
fixed tripod legs ; rf, blind spot in mirror ; r, screw for clamping^ in
iron frame ; f^ screw for clamping arm ; g^ screw for clamping ring
rod.
white cardboard disc in centre of the ring. By turning the
mirror so that the dark shade caused by the blind spot is
thrown on to the disc, the flash will be truly directed, and
must be kept so by slight alterations of the position of the
mirror, which should therefore be clamped only sufficiently
to hold it steady and yet admit of gentle movement. The
shadow of the blind spot should be slightly smaller than
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the disc, so as to ensure having it truly in the centre of the
latter.
' The mirror must be of the best glass, with its faces
parallel, or the shadow of the blind spot will be very indis-
tinct when the mirror is at a large angle, and also the beam
of light will be dispersed before it has traversed many
miles.
' It is well to have the mirror a feir size, say 6 inches
square, as in practice it will be found generally necessary,
in order to save time, after once adjusting the flash, to leave
a man to keep it on while the surveyor is taking his angles ;
and although a man will soon pick up the knack, a larger
mirror will allow for eccentricities on his part, and also, on
a dull day, a faint flash will be detected from a large
mirror, where a small one would not carry any distance. ,
' On a bright day a flash from a 3 inch by 2 inch mirror
has been seen 55 miles and more.
* In hazy weather, angles have been got when the place
from which the flash was sent was entirely invisible ; and
b
€1,
,-'^
Fig. 100.
ff, a, bent wire ; b, b^ brightened bullets ; /, jr, line of sight to the
Tugela.
thus whole days have been saved by this simple contriv-
ance.
*Only those who have spent hours, or even days, in
straining their eyes to see a distant mark can appreciate the
value of a heliostat.'
The following is an extract from an article in * Science
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332 Preliminar)' Sun^ey
for All,' by Major C. Cooper King, of the Royal Military
College, Sandhurst, and explains a handy makeshift shown
on Fig. I GO.
*At Ekowe, in the campaign in Zululand, the whole
apparatus had to be improvised. . . . Two wires with the
upper parts bent into the form of a semicircle, and with
cross wires uniting the bent end with the upright part, were
tried. The sights were composed of brightened bullets in
the centre of the cross wires, and when these rods were set
in the ground and aligned with the flash from a common
looking-glass, rio difficulty was experienced in communi-
cating.'
Sun signalling has the great advantage of being indepen-
dent of background.
Telemeters and Range-finders
The term telemeter, which was introduced by surveyors,
has been appropriated to some extent by electricians and
others ; and it is probable that the better term tacheometer,
which has been quite adopted by the French, will soon
become general.
The simplest form of telemeter is the plane-table, which
is a graphic triangulation to attain the same end as the
optical telemeter.
In every case the telemeter measures by a kind or
triangulation, varying from the long base, bearing a con-
siderable proportion to the distance to be measured, down
to the base of a few inches or at most a few feet, from
which distances of over a mile are measured.
Then, again, telemeters vary in size from the Weldon
locket range-finder, about | inch in diameter, to the 6 feet
and 7 feet 6 inch telescope-telemeters of Clarke and Struve.
Organically they may be divided into three classes.
I. Those in which the measured base forms an integral
part of the instrument itself. Such are Adie's i8 inch and
3 feet telemeters, described in Heather's * Instruments ; '
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Instrtunents 333
Piazzi Smyth's 5 feet telemeter, Colonel Clarke's 6 fe^t,
and Otto Struve's 7^ feet telemeter.
The instrument is held square with the line of sight.
The object whose distance has to be measured must be
sharply defined. One end of the instrument receives a re
fleeted image at right angles, and the other end a reflection
at an angle forming the complement of the angle subtended
by the length of the instrument at the point observed.
With so great a multiplication as this, it is evident that
the base must be very exactly measured in order to graduate
the instrument.
Adie's telemeter is of brass, and covered with leather in
order to diminish changes of length under varying tempera-
tures ; but all this class suffer from this cause. They are
not much used now.
2. Those in which the measured base is at the point
observed, generally consisting either of a graduated staff" or
a pair of discs connected by a rod ; the naval surveyors call
it a 10 feet pole, the Americans a target
In this class there are two subdivisions.
a. Those which have a fixed base and varying angle, as
the Rochon micrometer, furnished with a reflector by which
the images of the discs are made to coincide, as in a sextant,
and the angle subtended is read in terms of distance.
Messrs. Elliott's army distance-measuring telescope is
another of this class. The fixed base is the assumed height
of infantry, 5 feet 11 inches, or cavalry 8 feet 10 inches.
Two micrometer wires are fixed in the diaphragm, and are
actuated by the micrometer head so as to exactly intercept
the object. The micrometer head is graduated differentially
in terms of distance ; one side for the cavalry base, the
other for the infantry. Targets set at those respective dis-
tances on a pole would of course give more exact results.
Eckhold's omnimeter. Fig. loi, also made by Messrs.
Elliott Bros., is a combination of this and the first class
alluded to. The instrument is a transit theodolite in prin-
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Preliminary Survey
ciple, but it is furnished with a powerful telescope and a
long and powerful microscope {a\ reading a graduated
baseplate (b). At one operation, by observing top and
bottom of a lo feet pole, the distance and elevation are
given by a rule-of-three sum. The instrument has given
much satisfaction, both in India and the Colonies ; it is,
however, more complicated than the stadia principle, takes
longer to adjust, and is not more accurate. The author is
Fig. ioi.
informed by Messrs. EUiott that an improvement upon this
instrument is in course of development by Mr. W. N.
Bakewell.
b. Telemeters which have a variable base and a fixed
angle. This is the stadia principle, and the commonest
instruments of this type are the stadia or telemetric theo-
dolite and the tacheometer. The former instrument is
illustrated on p. 307 in a new form by Mr. W. F. Stanley.
Theodolites of all sizes are now frequently fitted with stadia-
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' Instruments 335
lines. The tacheometer of Messrs. Troughton & Simms
is graduated to 400 primary degrees and a decimal sub-
division. Tables of the trigonometrical functions are pub-
lished for this graduation. Any kind of decimal graduation
affords great facility both to astronomical calculations and
telemetry.
For the author's system of graduation, see p. 307.
Another feature is the obtaining of what is termed anal-
lalism (see Glossary), or unchangeableness, at the vertical
axis. There is very little practical advantage from this device,
which involves another lens. A micrometer screw can of
course be added if desired, but it does not usually form an
integral part of the tacheometer.
An instrument of this kind with more novelty about it is
the Wagner-Fennel tacheometer.
All the instruments previously described under this class
require the calculations of vertical height to be made by
tables or by slide-rule, but this instrument gives them direct,
without even the rule-of-three sum required by the Eckhold
omnimeter.
The measurement is by stadia hairs, but the movement
of the telescope in a vertical plane instead of being recorded
upon a graduated circle causes a vernier to slide backwards
or forwards along a scale fixed parallel to the line of
sight.
This vernier actuates a pair of other scales, one hori-
zontal and the other vertical, thus giving directly the
horizontal and vertical components of the inclined distance
measured by the stadia.
The instrument is highly ingenious, but it is not so
suitable for the ordinary work of the railway engineer, such
as setting out curves, &c., and it is not adapted to astro-
nomical observations.
3. The third class of telemeters is that in which the
base is measured on the ground, at the observer's station.
The Hadley sextant, Jhough not, strictly speaking, a tele-
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Preliminary Survey
meter, is greatly used as such by naval hydrographers,
sometimes in conjunction with a Rochon micrometer.
The Dredge-Steward omnitelemeter, made by J. H.
Steward of London, illustrated and described in ' Engineer-
ing,' of August 20, 1 886, is a good instrument of this class. It
is in appearance and in principle very much like a box sextant,
stands on a light tripod, and measures the angle formed at
the distant point by the pair of rays from the ends of a
base run out by steel tape at the observer's station. It has
this special feature, that the base need not be run out
square, which saves time when there are obstructions.
Fig. 102. Fig. 103.
Perhaps the most suitable of any of this class of instru-
ments to the ordinary surveyor is the Weldon range-finder,
the patent of Colonel Weldon, R.A. There is indeed a
new range-finder coming out from Woolwich, but the mili-
tary authorities are not as yet communicative upon the
subject.
Figs. 102, 103 represent the watch-size * Weldon ;' there
is also a * locket ' size. The following description is mainly
from the pamphlet written on the sul)ject by Captain Wil-
loughby Verner, R.A. :
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^0-
Fig. 104.— Range-taking with a
direction point (using first and
second pnsms).
A, position of observer.
O, object of which the range is
required.
D, direction point (as distant as
possible).
B A C
Fig. 105. — Ran^e-taking without
a direction point (using second
prism only).
A, position of observer marked
by first picket.
O, object of which the range is
required.
B, second picket.
C, third picket.
Fig. 106.-
by means of the third prism.
Range AO. BC=i. Base A 8=^15 Range
AO.
-Method of measuring a long base
' Base AB=^
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338 Preliminary Survey
The Weldon range-finder consists of three prisms of
crystal, accurately ground to the following angles :
1. 90°.
2. 88° 51' 15''.
3. 74° 53' 15''.
The range of an object, as at O, is taken by observing the
angles OAD, OBD, at the base of a right-angled triangle
ABO in Fig. 104, the measured base AB of which=3:V of the
distance or range AO. In this case the first prism, 90°, and
second prism, 88° 51' 15'', are used.
A second method, and one equally important, is by
observing the angles at the base of an isosceles triangle as
BCO in Fig. 105, when the measured base BC is ^ of the
distance or range AO. In this instance the second prism
(88° 51' 15'') only is used.
In order to measure the base AB or BC accurately and
rapidly the third prism of 74° 53' 51" is used ; but this is
merely a convenience and not a necessity, except under
very exceptional circumstances.
It will be at once seen by those who have had any
experience of range-finding that there are several objections
to this apparently simple process. They may be summa-
rised as follows :
1. Difficulty of obtaining any definite mark at right
angles to the object to reflect the latter upon, when em-
ploying a base of 3^.
2. Difficulty of always finding ground suitable for
measurement of base as regards view, general configura-
tion, and space, whatever base may be employed.
These difficulties are more or less common to instru-
ments of this class, but the Weldon possesses the two great
advantages over most of them of exceeding portability and
reliable, permanent reflectors.
The writer possesses one of these useful little instru-
ments, but has not obtained results with it equal to
those on record. Considerable practice is needed in
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Instruments 339
using them, both in judging a position for measuring a base
and in taking the observation. The right-angled prism is
practically an optical square. In Fig. 104 the range is
taken thus. Choose a good direction point D or else put in
a ranging rod at D making its reflection coincide with the
object by means of the right-angled prism. Then using the
88° prism retreat along AB leaving a mark at A to keep
yourself in line ; when the 88° prism shows a coincidence of
D with O, B is reached. AB is measured either by paces
or tape, and multiplied by 5o=the range.
In the official trials at Aldershot in 1883, the average
error was only 34 yards for each range, whilst in India in
1885 it was 35 yards.
The chief use of this class of hand instrument is on re-
connaissance, when neither plane-table nor theodolite can be
carried. It is more accurate in the hands of a practised
man than the sketching- board used on the portable stand,
described in * Route Surveying,' p. 47, and occupies no more
space than a watch. The sketching-board, however, has
counter advantages of permitting the surveyor to make
some representation of the country as he goes. Captain
Willoughby Verner advocates the use of the range-finder
in conjunction with the sketching-board. The writer has
not yet been able to get more correct results with it than
with the sketching-board used alone, on its light tripod, as
a triangulation, measuring bases with the latter not less than
■^-^ of the distance, but taking them in any convenient direc-
tion and any convenient length, which cannot be done with
the range-finder. A great drawback to this class of instru-
ment, and one which puts it out of competition with
telemetric theodolites wherever the latter can be used, is the
time occupied in running out the base. For taking ranges
in battle military officers are content to step the base.
Captain Verner mentions an amusing incident connected
with 'field-firing' of a certain corps, in which a portly
sergeant was told off to give the range with the range -
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340 Preliminary Sun^ey
finder, but did not understand it, neither did he wish to
show his ignorance. He was doubling along in rear of his
section, and was seen after each * rush ' to raise his range-
finder to his eye, after the manner of a spy-glass, and cail
out whatever his fancy moved him. He had orders to give
ranges constantly and obeyed them.
A rapid method of using the range-finder with the
accuracy needed on route-survey would be to measure
the two angles simultaneously by two observers, each hold-
ing one end of a loo feet steel tape. With the 90° prism in
the hand of one, and the 88° in the hand of the other,
distances up to 5,000 feet could be measured, or, by each
holding the 88° prism, up to 2,500 feet. It would require
a boy to hold up the sag of the tape. Each observer should
hold his handkerchief under the range-finder, for the other
observer to make the conjunction of image with the distant
object through the prism.
The Weldon is now reduced in price, and hardly more
expensive than an ordinary optical square, the duties of
which are well performed by the 90° prism, and it is not at
all liable to derangement.
Mr. Steward also makes a range-finder called *The
Simplex,' which is similar in principle of action with the
Weldon, only the angle is taken with a glass mirror, and
it requires adjustment each time. It is practically an optical
square with an alternative position of the mirror at a lesser
angle.
The Bate Range-finder
This is a binocular combined with two reflectors. The
modus operandi is the same as with the Weldon and similar
instruments.
It has the decided advantage ot optical power combined
with a wide field, so that for a hand- telemeter, used by one
man, it is one of the best of this class.
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Instruments
341
Like the Dredge-Steward, it allows of considerable lati-
tude in measuring the base, and instead of the very small
Fig. 107. — The Bate Range-finder, closed.
telescope of the latter, is furnished with a fine binocular,
suitable for general use.
The angle subtended by the base measured is expressed
in multiples by putting in a gauge between two graduated
Fig. 108.— The Bate Range-finder, ojDen.
limbs, thus measuring their divergence, or, in other words,
plotting the angle in the instrument itself. There is no
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342 Preliminary Survey
calculation beyond the multiplication of the measured base
by the multiplier indicated by the gauge.
It costs about ten guineas, and, considering what there
is in it, is not an expensive instrument.
Hypsometric Instruments and Hypsometry
TJie Barometer, This instrument has become so familiar
to the public as a weather-glass, and to engineers and
travellers as a height-measurer, that it is hardly necessary
to show an external view of it. Fig. 109 represents the internal
Fig. 109.
mechanism. The vacuum chamber B is of German silver,
firmly attached to the base-plate A. The sides of the
vacuum are compressed with a force equal to the weight of
a three-year-old child when the air is pumped out, and are
kept from collapsing by the powerful spring D. The vary-
ing pressure of the atmosphere produces tiny pulsations in
the vacuum, which are very greatly multiplied by levers G
and J, and chain Q. The lever G acts also as a compensator
for changes of length in the mechanism due to temperature,
but this does not dispense with the corrections to the
hypsometric formula due to difference of temperature, which
are given on p. 350, or to the actually recorded difference
of level by any aneroid. The altitude scale corresponding
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Instruments 343
to equal variations of atmospheric pressure is a differential
one, like water pressure on a lock gate, illustrated at p. iii.
By means of a device termed a worm, similar to that used
for equalising the tension of the spring in watches, the
differential movement of the index is in the surveying
aneroid changed into a uniform pne, so as to admit of the
use of a vernier scale. The 5 inch aneroid is the only
satisfactory type for the surveyor. It reads with the vernier
to feet, not that it is reliable to single feet at one reading
by any means. It needs repetition of readings, compensa-
tion for temperature, and every other precaution, and then
is only reliable in proportion to the number of readings
taken, and the range between highest and lowest
results.
It is clumsy and heavy, and it would be a great boon if a
small aneroid could be obtained, say 2^ or 3 inch, which
would give equally good results.
The reason that this has not been attainable is because
there must be a large vacuum and a small range of altitude
in order to get a sensitive instrument which will not * hang '
at all when rapidly changing from rise to fall and back
again.
Travellers sometimes specify to instrument makers such
absurdities as a 2 inch aneroid reading to single feet, and
having a range up to 15,000 feet, and absolutely compen-
sated.
A surveyor needs a maximum of exactitude over small
differences of level. He should have a 5 inch aneroid with
a range of 3,000 to 5,000 feet for all the ordinary work
of reconnaissance.
For mountain-climbing he should have a 2 inch watch
aneroid and a boiling-point thermometer.
The following results are added to show the range of
discrepancy in two aneroids, one a 3 inch and the other
a 5 inch, both of them eight guinea instruments by the
same makers, of the first rank in London.
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344
Preliminary Survey
Distance, A to B, 0*55 mile. Difference of elevation by
Ordnance benchmarks, 69*0 feet. Mean temperature (ave-
rage of all the times), 68° Fahr.
Three-inch Aneroid,
feet
First time.
Second , ,
Third ,,
P^ourth ,,
Fifth „
Up
• . 72 D
• 77
. 77
• 50
. 66
5)342
Average .
Corr. for temperature
. 68-4
• 23
Down
feet
60
63
57
72
68
5)320
707
Range of error from true level
Range of error between two successive readings
Range of final mean of means of * ups ' from true
elevation .......
Range of final mean of means of * downs ' from
true elevation ......
+ 10-3
-10-3
22
17
27
Five-inch- Aneroid.
First time.
Second ,,
Third ,,
Fourth ,,
Fifth ,,
Up
Corr. for temperature
Range from true level
feet
67
78
66
63
64
5)338~
67-6
2*2
698
Down
Range between two successive readings .
Range of final mean of means of * ups ' from true-
elevation .......
Range of final mean of means of * downs ' from
true elevation ......
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feet
65
68
63
71
56
5)323
64-6
+ II-3
-11*3
10
0-8
2-2
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Instruments
345
Distance A. to D, I'l mile. Difference of elevation by
Ordnance benchmarks 113*2. Mean temperature, 68° Fahr
Three-inch Aneroid.
feet
feet
First time. Up
. 112 Down
. 107
Second , , , ,
• "7
• 97
Third ,,
. 125
• 92
Fourth ,,
• 93
• 123
Fifth „
• 103
. 106
5)550
5)525
no
105
Corr. for temperature
3-6
113-6
Range of error from true level
+ 15-4
-i8-4
Range between two successive readings .
= 33
Range of final mean of
means of * ups ' from true
elevation .
,
o*4
Range of final mean of means of * downs
from
true elevation
4-6
Five-inch Aneroid,
feet
reet
First time. Up
. 103 Down
no
Second,, ,,
. 117
• "3
Third ,,
• 107
100
Fourth ,,
• 107
. 108
Fifth „
• 107
100
5)541
5)531
108 -2
ic6-2
Corr. for temperature
. 3-6
III-8
Range of error from tr
ue level
+ TZ
— IO-2
Range between two su
ccessive readings .
7-0
Range of final mean of
means of * ups ' from true
elevation
.
1*4
Range of final mean ol
^ means of ' downs '
from
true elevation
3-4
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346
Preliminary Survey
Distance, A to C, 0776 mile. Correction for tempera-
ture added to each reading. Difference of elevation from
Ordnance benchmarks, 129*8 feet.
Five-inch Aneroid.
feet
First time.
Down . . 130-2
Second ,,
1311
Third ,,
• 132^3
Fourth ,,
„ . . 128-3
Fifth „
.107-0
Sixth ,,
. 124-8
Seventh,,
„ . . II7-6
Eighth „
. 133-9
8)1,005-2
Up
125-7
Range of error from true difference
Range between two successive readings
Range of final mean of means of * ups ' from true
elevation
Range of final mean of means of * downs ' from
true elevation ......
feet
1344
132-2
121-4
132-4
128-9
124-8
1365
123*6
8)1,034-2
129-3
. + 6-7
-22-8
21-9
05
41
The wide readings will surprise those who are not familiar
with the subject, and who have been led to believe other-
wise by those who are proficient in the sale, less versed in
the construction, and least of all conversant with the use of
aneroids. // will be noticed that the five-inch in the above
tests has from a quarter to one-half the range of error of the
three-inch between any two successive readings. This is a
much more important point to the surveyor than the
accuracy of the final mean of means. In the three-inch
the range of error for the half-mile was two-thirds of that
for the mile, whereas in the five-inch the range of error for
the mile was less than that for the half-mile. It is fi*e-
quently the case that errors of five and ten feet will be regis-
tered in going up and down a hillock whereas the height of a
mountain may be correctly given within two or three feet.
It will also be understood from this why stress is laid in
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Instruments 347
Chapter II. upon using the Abney level with the aneroid for
the smaller rises and falls of route-survey, and making as
many repetitions as possible of the aneroid readings which
determine the maximum and minimum elevations. These
repetitions should all be entered in the fieldbook as care-
fully as the original ones, and a symbol placed beside the
mean of means on the plan to indicate the degree of
accuracy which had been obtained. It will be observed,
further, that in every case the mean of the * ups,' whether
the journey commenced by going up or by going down, was
more correct than the mean of the * downs.' Having used
a number of instruments in different parts of the world, the
writer has always found this the case. The pressure of the
atmosphere puts a strain upon the spring D. When going
uphill, the spring is relieved, and when going downhill it is
again depressed. It is natural that the resilience of the spring
should be freer in action than its compression, on account of
the intermediate mechanism needed to procure the action.
Practical Suggestions in Procuring and Using
Aneroids.
1. Procure a first-class instrument.
2. Learn its peculiarities and eccentricities.
3. Have it examined every year.
4. Register atmospheric changes by a second instrument
at camp.
5. When reading, hold horizontally, and tap several times.
6. Take as many repetitions as possible.
7. Where discrepancy, not due to atmospheric disturb-
ance in the district, exists between *ups ' and * downs,' prefer
the *ups.'
8. Always apply the temperature correction.
9. Three * ups ' will generally gw^ a mean result correct
to 3 feet, or two 'ups ' to 6 feet, in difference of level of 50
to 500 feet.
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348 Preliminary Survey
The Kew test is useful, and should be specified in order-
ing valuable instruments, but even that does not prove the
instrument in every way.
A first-class aneroid is as hard to get as a horse without
a blemish. A seasoned vacuum and perfect mechanism
inside, together with clear open graduation outside, and a
sling-case which will open and shut quickly and safely^ are
the chief points to aim at. Five-inch aneroids are now
made in aluminium to save weight, but the price is very
high.
Hypsometry,
or height-measuring, is a term applied to determination of the
level above the sea, by the barometer or boiling-point thermo-
meter, as distinguished from levelling with the spirit level.
Both operations rest upon the same fundamental prin-
ciples of the weight and pressure of the atmosphere, com-
monly known as Mariotte's and Charles's laws. These are,
firstly, that at a uniform temperature the pressure of any
gas varies inversely as its volume, and secondly, that at a
uniform pressure the expansion produced by a given increase
of temperature is the same for all gases.
The standard mercury barometer is now but little used
for hypsometric purposes on account of the inconvenience
in transporting it. The aneroid barometer just described
has superseded it, first because of its exceeding portability,
secondly because of its superior sensitiveness. It is
subject, however, to derangement of its delicate mecha-
nism, not alone from accident, but even from changes of
climate ; and the boiling-point thermometer, which measures
the atmospheric pressure without any mechanism, is gene-
rally added to the outfit of anyone who wishes to make ex-
tended and reliable hypsometric observations. It must not
be supposed that the use of any of these methods is sufficient
to ascertain elevations with the accuracy of a spirit-level.
They do no more than give the difference of atmospheric
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Instruments 349
pressure at two different situatrons. When the condition of
the atmosphere is steady in the district, that is tb say when
a standard barometer at either of the stations remains sta-
tionary during the period of observation, the difference of
pressure at the two situations forms a means of correctly as-
certaining the difference of elevation. When, however, there
is a disturbance of the air-pressure, no reliance can be
placed upon the results unless by also recording the move-
ments of a stationary barometer at the two stations by inde-
pendent observers.
With all their draw-backs hypsometric instruments are
indispensable, because they are the only means of approxi-
mately determining differences of elevation en route. The
real makers of them are few, and hardly ever put their names
on them. It is not sufficient to order them from a good
instrument-maker or even to have them tested at Kew. The
best proof of their excellence is in their daily use for a few
weeks under conditions of varying temperature, elevation,
and methods of transportation. All aneroids are marked
' compensated,' and instrument makers will often tell their
customers that no further correction is needed on account
of this compensation- -a statement which always shows that
they do not understand the principle of the instrument.
The compensation of the mechanism of the aneroid causes
it to record the correct difference of atmospheric pressure at
different times or in different places, and it is therefore not
subject to the correction which is applied to mercurial
barometers to correct them for the expansion or contraction
of the column of mercury.
But the difference of atmospheric pressure between two
different elevations is less at high temperatures than it is at
low ones. If the barometer recorded 30'' pressure at sea
level when the thermometer stood at 32° Fahr., at 1,000
feet elevation with the same temperature the pressure would
be nearly 28''*88. But if the mean temperature were 72°
Fahr., the pressure would be about 28''*96. In a table
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3 so Preliminary Survey
graduated with altitudes corresponding to differences of
atmospheric pressure at 32° Fahr. mean temperature, the
pressure of 28''-96 would only give 923 feet, and the altitude
in the table would have to be multiplied by 1*083 to ^v^
the true altitude. Hence it is much more convenient to
know the multipliers for temperature applicable to the
aneroids as they are actually constructed, for they are not
graduated from 32° Fahr., but from 50° Fahr., by Airy's
fprmula, or from 53® Fahr. by the author's mean formula
of English and American standards.
The compensation of an aneroid is nothing more than a
device analogous to that in a watch by which the expansion
or contraction of the mechanism of the instrument itself
due to changes of temperature is compensated. Nothing
can however prevent the expansion of the air produced by
any increase of temperature from acting differently upon the
aneroid under its altered condition, and nothing can prevent
the altered specific gravity of the air due to different alti-
tudes and latitudes from likewise differently affecting the
instrument. English aneroids are graduated with an altitude
scale corresponding to a mean temperature of 50° calculated
by Airy's formula or 53° by the author's mean formula ;
consequently, as shown in Plate IX. Fig. iii, the altitudes
indicated by the instrument at lower temperatures are too
great and should be treated with the multipliers given, and
vice versd at higher temperatures. Each of the subdivisions
has a value of '002 and reads thus : i.ooo ; 1*002 ; 1*004, ^J^d
so on. The temperature corrections given in the textbooks
are arranged from 32° Fahr., so that a double calculation
is required. This diagram can be used with the slide-rule.
When the altitudes are calculated from readings of the
barometer, or boiling-point thermometer, formulae are used
based upon the researches of Guyot and others, but varying
considerably in the coefi&cients adopted by different experi-
menters in different countries.
The tables given by Mr. Francis Galton in the textbook
Digitized by VjOO . _ _
Plate IX.
50 ^ !S
„ . TJtrULl'J' mil jij.t I T
BAROMETRIC PRtSSURt w INCHES *ndTENTH5
Krci. III. - Correcijoti fur interrriciliate jiir
for anemids scaled from 53=^ Fahr.
Aafe u llie horizontal lintrs
represent "002'^ Fahr.
i\'i7£e 7. 'If ihiii iliagruiii is i[i frcniient use
with divider Aj a (liec^' of duH-bacU tracing
cloth, gummed over it by ihe four L'ome^^
will jjrotcti It, y^-^ T
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Instruments 351
of the Royal Geographical Society are calculated by Loomis,
and differ both from those of Airy, from which most English
barometers are graduated, and still more from those ot
Col. Williamson, the American authority.
The formula of the latter is as follows :
^__ 3oDxT
B
where H=diflference of height in feet between two stations.
D= difference of barometric pressure between the
two stations.
T= tabular number corresponding with mean tem-
perature.
B=mean barometric pressure.
A short approximate rule is given by Mr. J. H. Belville
of Greenwich Observatory, which is nearly correct between
temperatures of 50® and 60° Fahr.
S : D 1:55,000 : H
where H= difference of height in feet between stations.
S=sum of barometric readings.
D= difference of barometric readings.
Examples will be given t9 show the divergence of the
three authorities, and a series of constants, K, for each degree
of temperature from zero to 102** Fahr., by which, with a
modification of Belville's rule, a mean result will be obtained
between those of the English and American formulae and
of so simple a kind that it can be worked by the slide -rule.
It is as follows :
K D
H (difference of height)=— ^
where K is the coefficient for mean temperature given in
Table LI., D=difference of barometric pressures, and S
their sum.
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352
Preliminary Surt'ey
Table LI
. — Fa/ue of K
in Formula H = *,- - .
,n.
K.
LogK.
! M.T.
K.
LogK.
1 Fahr.
' Fahr.
o
48753
4-68800
52
54785
473866
I
48869
4-68903
53
54901
4'73958
2
48985
4-69006
54
55017
4'74049
3
49101
4-69109
55
55133
4*74141 i
4
49217
4-69211
, 56
55249
55365
4*74232 1
I 5
49333
4*69314
57
4-74323 1
1 ^
4-69416
I 5«
55481
4-74414 ,
1 7
4'695i7
^9
55597
4-74505 !
8
49681
4-69619
' 60
55713
:» !
9
49797
4-69720
61
55829
lO
49913
4-69821
62
55945
4-74776
II
50029
4-69922
i ^3
56061
4-74866
12
50145
4-70023
5^
56177
4-74956 '
13
50261
4-70123
1 S
56293
4-75045 1
14
50377
4-70223
i f
56409
4-75135 (
15
50493
4*70323
^2
56525
4-75224 ;
16
50609
4-70423
68
56641
4-75313 '
17
50725
4-70522
1 69
56757
4-75402
1 ^8
50841
4-70621
1 70
56873
4-75491
! ^9
50957
4-70720
71
56989
4-75579
20
51073
4-70819
i 72
57105
4-75667
21
51189
4-70918
73
57221
4-75755
22
51305
4-71016
74
57337
4-75843
23
5142T
4-71114
75
57453
4-75931
1 24
51537
4-71212
76
57569
4-76019
25
51653
4-71309
77
57685
4-76106
26
51769
4-71407
78
57801
4-76193
4-76281
1 27
51885
4-71504
79
579ir
28
52001
4-71601
1 80
58033
4*76367 1
29
52117
471698
81
58149
4-76454 1
30
52233
4-71794
1 82
58265
4-76541
31
52349
4-71891
4-71987
1 83
58381
4-766-27
32
52465
84
58497
4-76713
33
52581
4-72083
' 85
58613
'^, 1
34
52697
4-72179
' 86
58729
35
52813
4-72274
i ?2
58845
4-76971
36
52929
4-72369
1 88
58961
4-77056
11
53045
4-72464
1 89
59077
4-77142
S3161
4-72559
90
59193
4-77227
39
53277
4-72654
91
59309
4-77312
40
53393
4-72748
92
59425
4-77397
41
53509
4-72843
1 93
59541
4-77482
42
53625
4-72937
' 94
59657
4-77566
43
53741
4-73030
i 95
59773
4-77650
44
53857
4-73124
4-73218
1 96
59889
4*77735
45
53973
97
60005
4-77819
46
54089
4-73311
98
60121
4-77903
47
54205
4-73404
99
60237
l-?82?o
48
54321
4-73497
100
60333
49
54437
4-73589
1 lOI
60469 •
4-78153
50
54553
4-73682
' 102
60585
4*78236
_J\
54669
4-73774
i
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Instruments 353
Comparison of Author's Formula with English
AND American Authorities
Example i.
Reading of barometer at lower station
„ „ upper „
Thermometer at lower station .
inches.
. 26-64
. 20-82
. . 70^
upper „ .
. . 40°
Bar. sum = 26 '64 + 20 '%2 = 47 '46
Difference = 26 -64 ~ 20 '82 = 5 '82
T + /
M. T.
K, see
.= 110°
.= 55"
table 55,133
H = Y • LogK . .
. =474141
+ Log 5-82 .
. =0-76492
-Log 47-46 .
.
5*50633
. =1-67633
Log alt. 6760-9 ft. . =3-83000
By slide-rule. Place the 47*46 on the slide opposite to
the 55,133 on the rule ; find the result 6,760 on the rule
opposite to 5*82 on the slide.
The same by the American rule, from the standard work
by Lieut. -Col. R. S. Williamson of the U. S. Army, as
quoted by Trautwine.
Rule, Height in feet=
difference D of barometer x table No. for MT x constant 30
mean reading of barometer
whence height=6 748*5 feet.
The same by the Geographical Society's rule by Francis
Galton, F.R.S. See * Hints to Travellers,' p. 185.
irari i. lui isu U4 uii;iica.
20-82,,
. 18,066
^'^^•^ X (70° + 40'' -64) .
900
6,440-2
329-1
6,769-3
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354 Preliviinary Survey
Example 2.
Reading of barometer at lower station . 30*646
„ „ upper „ .23-66
Thermometer at lower station . • 77 '5
„ „ upper ,, ... 70-3
Sum. bar. 30 -046 + 23 "66 = 53 706 | T + / = 147 *8
Diff. „ 30-046-23-66= 6-386 M. T. = 73-9
1 K, see table 57.337
H=^:^ . LogK ... 47584350
LogD 6-386 . . 0-8052289
5-5636639
Log S 53-706 . . 1-7300228
Log alt. 6,817-8 . . 3-8336411
By slide-rule as before 6,820.
The same by the American rule is 6,829 ^*^^^ ^^^ by
Galton's rule 6,805*6 feet.
Example 3.
Bar. lower station ...... 28*00
„ upper „ 22 -oo
S 50-00
D 6-00
Temp, lower station .... 60'^ Fahr.
, upper „ . . . . 40 ,,
100
M. T 50 K = 54,553
. -H=^^-i^ = 54^553_^^ 6,546-36
S 50
By the American rule 6,527-5
By Galton's rule 6,552*7
By Airy's tables 6,571-5
Examples i and 2 are taken from ' Trautwine ' and
* Hints to Travellers.' Example 3 is formed of barometric
pressures intermediate between the other examples, and
the result from Airy's tables is taken from a pamphlet on
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Instruments
355
the aneroid by Houlston & Sons, lent to the writer by an
instrument maker as being the standard used in graduating
aneroids. The author's two surveying aneroids graduated
I to 10,000 feet, the other to 5,000, have their zero at
31 inches and are graduated almost exactly in the same way.
The former reads 9,325 feet at 22 inches. Airy's tables
above alluded to show an altitude of 9,347*5 at that pressure
and mean temperature of 50° Fahr.
The American rule would give 9,214*4 feet, Mr. Galton's
rule 9,318*4, the author's rule 9,263*6, an almost exact
mean as before between the English and American autho-
rities.
The usual corrections for temperature of intermediate
air are given in tables having 32° Fahr. M. T. as the starting
point, and consequently are not capable of being applied to
ordinary aneroid readings without a double calculation.
In the diagram Fig. in multipliers are given having
53° Fahr. mean temperature as their starting point, that
Correclion for latitude and
decrease of gravity.
Note.~'Y\it. positive correction for de-
crease of gravity is to be scaled for the
required altitude from the left-hand scale.
Thus, for 10,000 feet of altitude the cor-
rection 29*8 feet is marked so. The cor-
rection for latitude is scaled in the same
way, but positive or negative.
Fig. 112.
being the basis of the graduation of the ordinary aneroid
according to the author's mean formula. By using these
multipliers, the correction can be made with one calculation,
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356 Preliminary Sun^ey
and in fact for ordinary work by the slide-rule in the field
as close as most instruments will read.
Example 4. At a mean temperature of 70® Fahr. the
aneroid showed an altitude of 1,265 feet. The multiplier is
1-037.
Place the left hand i of the slide opposite to i'037 on
the rule and we have opposite to 1,265 ^^ ^he slide 1,310 on
the rule, or more correctly, by figures 1,311.
As a remembrancer for every 10 degrees of temperature
above ^^ , , . add to
bd5"w 53°Fahrenheit ^^^^^^ 22 percent, f-- the regis-
tered difference of level by the aneroid.
The diagrams on Fig. 1 1 2 for change and decrease of
gravity are added more to exhibit at a glance the amount of
such corrections than for actual use, because it is rarely of
any practical importance to know them. The local atmo-
spheric disturbances may be far greater within a small area
than the equivalent of these corrections in barometric pres-
sure, so that to apply such refinements as these would be a
sheer waste of time.
The Boiling-point Thermometer '
' The boiling-point apparatus (Fig. 113) consists of a ther-
mometer A graduated from 180° to 215° ; a spirit-lamp B,
which fits into the bottom of a brass tube C that supports
the boiler D, and a telescopic tube E which fits tightly on
to the top of the boiler. The thermometer is passed down
the tube E from the top until within a short distance from
the water, which it should never touchy and is supported in
that position by an india-rubber washer F. The steam
passes from the boiler up the tube E, and escapes by the
hole G. To pack this instrument for travelling, withdraw
the thermometer and put it into a brass tube, lined with
• From ' Hints to Travellers' by John Coles, Esq., Royal Geo-
gr^iphical Society
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Instruments
357
india-rubber having a pad of cotton wool at each end ; take
off the tube E, shut it up and put the small end into the
boiler D, which it fits, then withdraw the spirit lamp B, screw
the cover over the wick, and replace it in C. The whole of
this apparatus fits into a circular tin ^
case 6 inches long and 2 inches in dia-
meter.
' To use the boiling-point thermometer. ^
Take the apparatus to pieces, pour
some water into the boiler D, the less
the better, as it will boil the quicker
(about one quarter full is quite suffi-
cient) ; then put the instrument together
as shown in the drawing, taking care
that the thermometer is at least half an
inch clear of the water, and light the
spirit-lamp ; as soon as the water boils,
the steam, ascending through the tube
E, will cause the mercury to rise ; wait
until the mercury becomes stationary,
and then read the thermometer ; at the
same time take the temperature of the
air in the shade with an ordinary ther-
mometer.
'When purchasing this instrument
be careful to see that the lamp is large
enough to hold a good supply of spirit ;
it is a common fault to make it too
small. A small screw which may be
made of tin to fold up is most useful ^
to place on the windward side, and at a ^^* "^*
very low temperature is almost indispensable, as the heat is
otherwise carried off too rapidly for the water to boil pro-
perly.'
The following rule for finding the height due to temper-
ature of the boiling point, has been prepared in a similar
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358 Preliminary Survey
way to that for barometric pressure by adopting a coefficient
which produces mean results between the English and
American authorities. It is for a mean temperature of inter-
mediate air=53® Fahr. instead of freezing point, as custom-
ary in the textbooks. This has been done to make the
coefficient agree with the graduation of ordinary aneroids,
and for other temperatures the coefficient 540 has to be
multiplied by the multiplier on diagram Fig. in Plate IX. ;
thus, at 32° the coefficient becomes •954x540=515, or
at 82^=540 X i-o64=574-6.
Rule, Let B= temperature of boiling point in degrees
Fahr. deducted from 212°.
H= height of station above level of sea.
K= 540 for a mean temperature of interme-
diate air of 53** — and varying as ex-
plained above.
H=K.B + B2.
Example.
Boiling points -ii'37
„ „ 210-14
Mean temperature . . . . . . 82° Fahr.
Required the difference of elevation :
H = 540 X 1-064 X 0-63 + 0-63'' . . = 362-37
H' = 540 X 1-064 X 1-86 + 1-86- . . = 1,072-14
Ans. diff. in feet . . . . = 709-77
The value of the boiling-point thermometer consists in
the fact that it is a perfectly simple machine, there is no
wheelwork to get out of order, no vacuum to play off its
caprices. Otherwise it performs the same duty as the
aneroid. Its results depend upon the assumption that
water boils at 212° at sea-level. This is not always
true ; water boils at different temperatures in different
latitudes and also varies under different conditions of the
atmosphere. In fact the boiling-point is nothing more than
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Instruments 359
a register of the atmospheric pressure, like the aneroid.
Water can be put under an air pump at sea level, and made
to boil nearly at freezing-point. The boiling-point thermo-
meter is only suitable for the first pioneer work in very
hilly country, such as finding a pass for a railway through a
chain of mountains. It is not frequently used even for
that purpose.
The surveyor generally contents himself with two
aneroids rather than spend time in boiling-thermometers.
It is not, however, such good work, because the best
aneroids are fickle under varying conditions.
Office Instruments
At a pinch the surveyor can get along with hardly any-
thing more than a pair of compasses and a straight-edge.
For a short survey, where impedimenta are a great objec-
tion, with a pocket case of instruments and a slide-rule he
can do all the protracting, contouring, scale-making, and
gradient-drawing that he wants. What he misses most is a
box of good railway sweeps. They are bulky, and it is
seldom he can afford space for them.
The author has combined in one box, 19 inches long
by II inches wide by 8^ inches deep, a complete repertory
of all the drawing instruments he requires, and though
.quite bulky, he has contrived to make room for it wherever
he has gone. A description may be useful.
At the bottom is a full set of railway sweeps, i to 240
inches radius. A boxwood rolling parallel ruler with the
edges graduated as a protractor and loaded in the middle
with lead to steady it.
On the lower tray is a colour-box and water-dish, recep-
tacle for liquid ink, hquid carmine, and liquid prussian blue
in their own bottles. Small hammer, wire-cutting nippers,
* Yankee notion' screw-driver-bradawl, store of pencils,
rubber, and colour-brushes, needles, copper-wire, brass
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360 Preliminary Survey
screws, paperclips, small sponge, chamois-leather, palette,
lancet, surgical needle.
On the upper tray are, six-inch German silver circular
protractor with short arm and vernier, reading to -oi degree;
three ply magnifiers for ditto. Complete set of drawing
instruments, including proportional compasses and trammels.
In the lid ivallet are one 15 inch, 60° set square, one
1 2 inch, 45° set square ; two French curves, card of crow-
quills, one 9 inch paper protractor.
On the lid-flap are one set of ivory scales 10 to 60, one
universal scale, all 1 2 inch. They are laid out in a row in
elastic loops, so that they can be withdrawn without needing
to search for the right one. Also one small 45° and one
small 60^ set square.
It is not only a great convenience, but the best plan of
keeping one's instruments to have them all together in a box
where each one has its own place, and becomes 'conspi-
cuous by its absence.'
A tee-square is much the best means of protracting angles,
where the sheets are not longer than can be reached by it.
The steel tee-square with bronze head sold by Charles
Churchill & Co. of Finsbury, though an expensive instru-
ment, is well worth its price. It is a protractor with a
36 inch arm, and a tee-head. It has a clamping screw with
a large head to it, so that it is adjusted in a moment to any
angle. It is used with an ordinary set square.
Barnett's diagraph, which only costs lu., is another and
ingenious instrument for rapid plotting, by the same makers.
Where no very great accuracy is aimed at, a paper
protractor pinned down in one corner of the board, an
ordinary tee-squaie, and a single jointed two-foot rule with
a pretty stiff joint form a very cheap and efficient method of
protracting.
The tee-square is run up to the pf otractor, and the 2 -foot
rule is set to the angle. It is then run down to the station
from which it is to be laid off by the tee-square. It is useful
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Instruments 361
for laying off bracing of trestles ; equal and opposite battering
walls &c., because the 2 -foot rule has only to be reversed to
give the opposite angle.
It is also very useful for making strain-diagrams —
although that is outside our province because it is not so
liable to shift as a parallel ruler.
The eidograph and pantagraph are occasionally a great
help to the surveyor, but they can hardly be called satis-
factory instruments. Reduction of plans is much better
done by dividing them into squares, and aided with propor-
tional compasses does not take much longer than by the
eidograph.
The planimeter is a most valuable instrument to those
who have extensive computations of acreage to make. It
\
Fig. 114.- Amsler's Planimeter.
does the whole work by recording, on a pair of indicator
wheels, the travel of a roller round the periphery of any
figure. Stanley*s computing scales, which are much cheaper
than the planimeter, are an efficient substitute for it and do
not take very much longer.
The Slide-rule
This ancient but not antiquated instrument is not nearly
as much appreciated in this country as it ought to be.
Thoroughly scientific as it is in principle it has until recent
years suffered from the imperfection of mechanical science
which prevented the attainment of the same accuracy in
graduated scales which is possible with figures.
The increased skill in making dividing-lathes has enabled
the mechanician to produce an instrument of precise accuracy
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362 Preliminary Sun^ey
only limited by its length and the consequent visibility of
the graduation.
The slide-rule has been formerly known as Coggeshairs
rule or the carpenter's slide-rule, and has been regarded as a
rough though labour-saving makeshift.
Ever since its invention by Oughtred it has been mis-
understood, until recently the French and Italians have
brought it into something of the estimation which it deserves.
Oughtred is said to have kept the instrument by him many
years out of a settled contempt for those who would apply
it without knowledge, having * onely the superficial scumme
and froth of instrumental trickes and practices, and wishing
to encourage the way of rationall scientialists, not of ground-
creeping Methodicks.' *
A glance at the catalogue of rules of all sorts, sizes, and
prices from 6s, up to 10/., manufactured by the firm of
Tavernier-Gravet, will show that the French have appreciated
this instrument more than we.
A very decided improvement on the cheaper form of
boxwood rules has, however, been made by Messrs. Davis
& Son of Derby and London, by overlaying celluloid upon
hard wood. It looks like ivory and costs no more than the
lo-inch Mannheim slide-rule. The graduation is quite
equal to that of the French rule. There are some points of
advantage in the writer's opinion in favour of the latter, but
which might be easily adopted by the English makers if
they were so disposed.
In Preliminary Survey the slide-rule is simply invaluable,
and it is astonishing how quickly its manipulation is acquired,
especially by those accustomed to read graduation of any
kind.
If it were only for the one operation of reducing the
optically measured distance to horizontal and vertical co-
ordinates, and the horizontal distance to latitude and de-
parture by two shifts of the slide, no surveyor should be
» Rev. W. E. Elliott, <The Slide-rule,*
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Instruments 363
without one ; but that is only one of its numerous func-
tions, some of which are described in the chapter on Graphic
Calculation.
To do justice to all the applications of the slide-rule to
the various branches of technology would require a small
volume for that purpose alone.
The printed directions supplied with the slide-rule only
give general principles, and therefore instructions are given
in this manual for a considerable number of those opera-
tions which are most commonly required upon survey.
Repetition is of course unavoidable, but inasmuch as the
use of the instrument may not be continuous, if its special
application to some particular rule is forgotten it will not
be used at all. It was therefore considered advisable to
give numerous examples with the various rules.
The instructions for using the slide-rule are principally
contained in Chapter VIII., but they are also necessarily
interspersed all over the book. They are partly for sexa-
gesimal degrees, although in many places decimal degrees
are given along with minutes and seconds ; Tables XL., XLI.,
and XLII., moreover, give a ready reduction of minutes and
seconds to decimals of a degree. Nothing has been worked
out with the centesimal degree, that being so much of
a novelty that ever}^thing would have had to be in duplicate.
MM. Tavernier & Gravet, and Messrs. Davis & Son,
make slide-rules graduated for the ordinary degree divided
decimally, and they can be used for minutes without any
reduction, because the subdivisions are merely at 3', 6', 9',
12', &c., which are '05°, -i®, -15°, -20°, and may be used
either way. Instead of the mark for single minutes, there
is one for -oi degree.
The slide-rule may be used for long rows of figures by
placing them in batches of threes ; but its chief value is for
all those small calculations up to an accuracy of 7^.1^, which
only require one adjustment of the slide ; for in these the
operation is performed in less time than it would take to
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68 •,
noi
"[ ^ I*'' L I'
!^1
l-lt- ■■«-
LXJlt
.Jk_;tjL^_^.^,..^ .
Preliminary Survey
look up the first factor of the
sum in the table of logarithms, or
to put them down on paper for
ordinary reckoning.
The instrument, in its present
form, will perform accurately, and
tvithout any mental effort ivhat-
ever, a whole mass of tiresome,
small calculations in less than
half the time it would take a very
quick man to work them out by
any method he might choose.
In the field every surveyor
knows how books and tables with
soiled pages and fluttering leaves
become lessons of patience and
self-control to him, but the slide-
rule enables him to reserve these
moral exercises for other occa-
. sions- All graphic operations are
essentially approximate, but it is
possible to arrive at as close an
approximation as may be needed
for the purpose in view when the
object and the principle are both
clearly understood, and so to
give perfectly correct results with-
in the limits prescribed.
The principle of the slide-rule
is that oi graphic logarithms.
The organic formula of loga-
rithms is Log (AxB)=log A +
log B. We can obviously per-
form this operation by scaling as
well as by figures. For if we add
the tabular logarithm of A to
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Instruments 365
that of B by scale or by figures, the sum of the two loga-
rithms will represent the product of the two numbers.
Referring to Fig. 115, the slide has been shown withdrawn
one primary division to the left ; and it will be noticed
that the i of the upper scale of the rule is over the 2 of
the slide, the 2 over the 4, the 3 over the 6, and so on.
If the slide had been shown in its initial position, all the
figures would have been in correspondence, because they are
equal logarithmic scales. That is to say, the instrument
maker has constructed the space i to 2, by any convenient
scale of equal parts, =301 units, because the logarithm of
2 is '301. Similarly the space from i to 3 is made equal to
477 units. The space from i to 6 is 778 units for the same
reason.
By retreating the slide 301 units to the left in the manner
shown we add to the whole scale of the rule an amount=
log 2, and consequently represent graphically the multipli-
cation of every figure on that scale by 2. The 3 of the
rule coincides with the 6 of the slide because on the one
477 units are added to 301 units on the other, making 778
units, which is the log 6.
The intermediate graduation is made in the same man-
ner, each line being ruled off in the instrument maker's
dividing lathe at a distance equal to its tabular logarithm.
The upper scale of the rule is doubled, so that if we give
to the left hand i the value of i, the middle t will be 10
and the right hand i will be 100.
The lower scale is only single in the first place because
with less range it has a more open graduation so that by it
some figuring can be done twice as closely as by the upper.
In the second place it will be observed that all its figures
are under their doubles on the upper scale. As twice the
log means the square of the number all the lower figures
are the square roots of the upper and we can perform invo-
lution and evolution without moving the slide at all.
Where the lower figures are given their indicated value
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366 Preliminary Survey
and are under the right hand half of the upper scale, the
figures of the latter must be given ten times their indicated
value ; thus 6 is under 3*6, which must be styled 36 and
so on.
The scales of sines and tangents are constructed from the
tabulated logarithmic sines and tangents ; but with a radius
of 100, of which the logarithm is 2. Tabular values are
always figured to a radius whose logarithm is 10, so that the
integral number 8 is deducted from, all the tabular values.
The process of calculation is the same as with numbers,
only the sines and tangents cannot be read lower than
34' 23" nor the tangents above 45"^. Smaller angles and higher
tangents can be obtained in another way as explained at
p. 247, Ex. 4, and p. 274.
The use of the brass marker or index is both to retain
a number found by one process whilst the slide is being
shifted to make a second calculation and also to perform
involution and evolution without using the slide.
The Station-pointer
This instrument is a treasure to hydrographers. By
it they locate their position at sea from three fixed points on
shore, very rapidly and exactly. A A, BB, and CC are three
arms having a common centre O round which two of them
can be moved in any direction ; DE is a circle graduated
from zero to 180° on each side of the central fixed arm which
is permanently set to zero. The two other arms are provided
with verniers, and when angles are taken with the sextant to
the three fixed points forming two angles, one on each side
of a central point, these two angles are laid off on each side
of zero on the station pointer, which is then laid down on
the chart so that each of the three arms points to one of the
fixed points. The centre of the instrument will then be at the
position of the point of observation on the chart, which is
then pricked off through the pin-hole left for the purpose.
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Instruments
367
This is the same problem as that described in Route Sur-
veying, p. 48, performed by the plane-table.
The only case when the station-pointer will not locate
the station from three known points
is when the station happens to be
in the circumference of a circle de-
scribed about the three points. If a
tracing is made of the three arms it
will be found that the centre of the
graduated limb can be moved into
any position in the circumference,
and the three arms will still go
through the three points.
It is always practicable to choose
points which will not fall into this
position.
The same thing can be done by
plotting the bearings on tracing
paper and superimposing it upon
the chart, working them round until
all three intersect the points. This
is, of course, not so rapid or cor-
Books
The Nautical Almanac is the most indispensable portion
of the surveyor's library. The chapter on astronomy to-
gether with the glossary will explain all that is needeci for
its use on survey.
Whttaker's Almanack is a substitute which is quite suffi-
cient for the more ordinary calculations of the geographical
position. A couple of the shilling edition should be taken,
so that those leaves can be cut out which supply the data
for observations, and carried about without taking up any
room.
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368 Preliminary Survey
Raper^s Navigation is one of the best worics on nautical
astronomy in the English language. It is bulky, and con-
tains a great many more subjects than the surveyor requires.
It has, however, a full supply of mathematical tables, and
if the surveyor has not a copy of Chambers's or Weale's
mathematical tables this work is the best he can procure.
Chambers's Practical Mathematics is a very suitable book
for those who do not intend to go deeply into the subject
It abounds in examples and illustrations, and would doubt-
less clear up many points that it was impossible in a treatise
like the present one to handle thoroughly on the subject
of geodetic astronomy.
Chambers's Mathematical Tables are more comprehensive
than those of Weale's series. Either will do quite well for
the surveyor's purpose.'
Dr. Crelle^s Calculating Tables are a triumph of German
patience. They are a multiplication table in 500 pages ol
quarto. Three figures by three figures are multiplied by
simple inspection. Larger sums are performed by dividing
them up into threes. Every extensive survey should be pro-
vided with them for office use. Ordinary small surveys need
nothing more than the slide-rule. The place of Crelle's tables
in the economy of calculation comes between the logarithmic
tables and the slide-rule, being quicker than the former for
three or four figure calculations and closer than the slide-
rule. But even when they are well understood they do
not approach the rapidity of the slide-rule for small calcula-
tions.
Trautwine's Pocket Book, This wonderful compendium
of general engineering knowledge had reached its 27th
thousand three years ago. There is no one book which the
writer would place beside this as a vade-mecum for the
engineer who travels abroad either to America or the
Colonies.
' Dr. Bremiker's Mathematical Tables are for ordinary degrees
divided decimally. Ascher and Co., Bedford Street. Price is. 6d,
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Instruments 369
MoIeswortNs Pocket Book is likewise a remarkable collec-
tion of useful tables and formulae. It is not so explanatory
as Trautwine but less bulky and expensive. Engineers going
to India will find in it special information upon engineering
as practised in that country, which Sir Guildford Molesworth
has been in the best possible position to collect.
Spon's Shilling Pocket Book of Engineering Formulce con-
tains, along with other most useful information, a table of
sines and tangents which are closer than Molesworth's and
are sufficient for ordinary curve-ranging.
Hints to Travellers, This handbook of the Royal
Geographical Society is as remarkable for the modesty of
its title as the value of its contents. It is mainly devoted
to geodetic astronomy and route-surveying, but it has
chapters on all the subjects of interest and importance to
travellers, including geology, natural history, anthropology,
&c., &c. It is the cheapest five-shillings' worth that the
traveller can put into his outfit. The portion on surveying
is by Mr. John Coles, the accomplished Instructor to the
Royal Geographical Society and Curator of the Library.
Fieldbooks, The form of fieldbook recommended for
tacheometry has been given on p. 178. The printed
books can be had through Messrs Elliott Bros. They are
marked No. — on the outside, and have an eyelet hole for
holding the pencil.
Hold-alls, The best way to preserve pencil and rubber
is to keep them and a lo-inch slide-rule in a canvas hold-
all, buttoned on to the coat, against which it lies quite flat,
and each article when used is replaced in its own proper
place. A red and a blue pencil should be added for detail-
sketching, or else red-ink and blue-ink fountain-pens.
MS. books. These should contain about 100 pages.
Half a dozen should suffice ; well bound to resist
damp climates.
Stationery and drawing paper, A good supply of ruled
profile paper will be found most useful, not only for cross-
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370 Preliminary Sun^ey
sections and profiles, but also for estimates, diagrams, and
different kinds of scale-drawings.
Tracing paper is not of much use in a tr)ring climate.
Tracing cloth with one dull side is best, as pencilling can
be done upon it as easily as on tracing paper. A tough
writing paper suitable for iforeign postage and for using with
the manifold writer will be found the most generally useful.
Ink -pellets soluble in water make a fair substitute for ordi-
nary writing fluid, and are no dearer. A gold pen saves its
cost over and over again.
The Author's Outfit of Surveying Instruments
The following complete hst of a surveying outfit, such as
is recommended for an extensive survey, costs from loo/. to
120/.
'Ideal' tacheometer. — 5-inch horizontal and 6-inch vertical limb.
Telescope of 12-inch focal length and eyepiece naagnifying 50 dia-
meters, having micrometer screw head fitted on right-hand side of
eyepiece. Fixed stadia hairs 1/ 100, and vertical moving hair worked
by micrometer screw graduated to i/io,cxx> inch. Perforated axis
and lantern. Diagonal eyepiece.
Level staff. — One ordinary 3-draw Sopwith 16 feet staff, with 2 vanes
fixed 10 feet apart for micrometer readings at long distances, and
portable table for same.
Field-books.— One dozen specially printed fieldbooks, arranged for
stadia measurements and space for sketching.
One Vemer's large size military sketch-board-plane-table and metal
tripod-stand, the scales being adapted to railway work.
Six dozen strips of impervious paper for ditto, ruled J -inch with co-
ordinate lines.
One lOO-feet steel chain and 2 sets arrows.
One lOO-feet steel tape, divided into feet and hundredths.
One loo-feet linen ditto, divided into feet and tenths.
One bill hook, with long handle.
One small axe.
Three ash ranging rods.
All the above are in two iron-bound deal cases, forming
one mule's burden.
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Instruments 371
One 12-inch Y level.
One Weldon range-finder.
One Abney level, with prismatic compass.
Two survepng aneroids, one reading to 5,000 feet, the other to 15,000
feet, fixed altitude scales and verniers.
One keyless semi-chronometer watch.
One pedometer and one passometer.
One large case of drawing instruments as described.
One pocket case of drawing instruments.
* Hints to Travellers.'— Stanford.
Bremiker's Mathematical Tables.
Chambers's Practical Mathematics.
Nautical Almanac.
Whitaker's Almanack (2).
Dr. Crelle's Calculating Tables.
MS. Books (2).
Tracing cloth with dull back, 24 yards by 30 inches*
Profile paper, ruled to scale. Drawing pins, pens, pencils, liquid ink,
liquid blue, liquid carmine.
Digitized
by Google
Digitized
by Google
APPENDIX
PLANE TRIGONOMETRY
FUNCTIONS OF RIGHT-ANGLED TRIANGLES
Fig.
117.
sin A =
COS A -=
tan A =
tan A I
sec A cosec A
cot A ^ I
cosec A sec A '
sin A ^ I
cot A
cos A
cot A = -= . ,
a sin A
sec A = ~ =
cosec A
tan A
cosec A ^ I
cot A cos A
c ^ sec A _ I
a tan A sin A
tan A
sin - A + cos ^ A = I
sin A = v^i-cos'^ A and cos A =
sec ^ A = I + tan ^ A and cosec ^ A = i + cot
• (I)
. (2)
• (3)
• (4)
■ (5)
• (6)
^/l-Sln'A(8)
A (9)
:i„
sin A X cosec A =
cos Ax sec A
tan A X cot A =
sin ^ A + cos ^ A = J
sin (A + B) = sin A. cos B + cos A. sin B
sin (A - B) = sin A. cos B - cos A. sin B
cos (A + B) = cos A. cos B - sin A. sin B
cos (A - B) - cos A. cos B + sin A. sin B
Digitized
• Co)
• (II)
• (12)
• (13)
. (14)
by Google
374 Preliminary Survey
Functions of any two angles A and B.
SIX (A + B) + sin (A - B) = 2 sin A. cos B (15)
sin (A + B) - sin (A - B) = 2 cos A. sin B (16)
cos (A - B) + cos (A + B) = 2 cos A. cos B (17)
" "' " ~ . - . -^ (jg)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
cos (A - BJ - cos (A + B) = 2 sin A. sin B
sin 2 A = 2 sin A. cos A
cos 2 A « cos 2 A - sin ^ A .
I + cos 2 A = 2 cos ^ A
I — cos 2 A = 2 sin ' A
sin 3 A = 3 sin A — 4 sin' A
cos 3 A = 4 cos ' A — 3 cos A
tan A + tan B
r^~tanA7tan B
tan A - tan B
tan (A + B) =
tan (A - B) =r
tan 2 A =
tan i A =
I + tan A. tan B
2 tan A
I - tan
sin A
^ , . sin A
^ i cot i A = -
I + cos A I — cos A
tan'iA=i5?llA.eotHA = S?Lf4
^ cot i A ^ tan i A
Relations between the sides and angles of triangles,
B
^ab
2 sin ^ i A = (^ j^ b - c)^a^c^b)
^ ibc
2 cosn A = {^^b^c){b-- cj^
2bc
Digitized
(30)
(31)
(32)
(33)
(34)
by Google
Appendix 375
Right-angled triangles.
Case I. Given the hypotenuse r, and a side b.
sec A -= R X ^ -^ /5 (35)
« = tan A X <J -f- R (36)
C = 9o°-A (yj)
Case 2. Given a side ^r and one of the oblique angles A.
C = 9o°-A (38)
« = sin A X iT -^ R (39)
b = cos A X ^ -7- R (40)
Oblique-angled triangles.
Case I. When two angles and a side opposite are given.
The sides are proportional to the sines of the opposite angles.
Let AB and a be given ; then
C = 180° - (A + B) (41)
» «. sin B , .
b = - .- ^ (42)
sin A ^^ '
«. sin C , V
^=-smA • (43)
Case 2. When two sides and an angle opposite to one of
them are given.
Let «, ^, and A be given.
sinB = ^i£>EA (44)
a
si„C = ^:-«^A (45)
B = 180°- (A + C) (46)
or C = 180° - (A + B) (47)
* = ^-.^i"^ (48)
Sin A ^^ '
Case 3. Given two sides and the included angle.
Let a^ b, and C be given.
A + B = 180° - C ;
tan HA - B) = (^ - ^)- ^-^ > i^ 4_B) ^ ^^^^
a + b
A = i (A + B) + i (A - B) . . (50)
B = i(A + B)-HA-B) . . . (51)
c'^= a^ -^ b'^ ±2 ab. cos C . . . . (52)
When C is obtuse the + sign, and when acute the - sign
to be used.
Digitized
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576
Preliminary Survey
Case 4. When the three sides are given.
Let a^ by and c be given and s = ^ {a -¥ b •¥ c).
sin'iC-(^-^)("-^) . . . .
a, b
SPHERICAL TRIGONOMETRY
Right-angled Spherical Triangles.
(53)
Case I. Given the hypotenuse b and one of the angles C, to
find the other parts.
tan a = cos C x R -j- cot b .
sin iT = sin ^ X sin C -f- R .
cot A = cos ^ X R -4- cot C .
Case 2. Given the hypotenuse b and a side a,
cos C = cot b X tan <2 -i- R .
cos c = cos b X K -r- cos a .
sin A = sin ^ X R -T- sin <^ .
Case 3. Given the two sides a and c.
cos b = cos c X cos a -^ R .
cot A = sin £: X R — tan a ,
cot C = sin a X R -T- tan ^ .
C^j^ 4. Given the two angles A and C.
cos b = cot A X cot C -^ R .
cos ^ = R X cos C -T- sin A .
cos a = R X cos A H- sin C .
Case 5. Given a side a and its adjacent angle
cot ^ = R X cos C -f- tan a .
tan c = R X sin a-T- cot C .
cos A = sin C x cos « -f- R .
Crtj^ 6. Given a side ^ and its opposite angle C.
sin ^ = R X sin ^ -T- sin C .
sin A = R X cos C -^ cos c .
sin a = cot C x tan c -^R .
Digitized b
(54)
(55)
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
by Google
Appendix '377
This last is termed the ' ambiguous ' case. See Chambers's
* Practical Mathematics/ p. 379.
R expressed Ipgarithmically is j 0*000000.
Oblique-angled Spherical Trigonometry,
Fig. 120.
Rule I. Given three sides to find the angles.
Let J = ^ (<2 + ^ + r).
ry,, ^ . , . I sin is — U) X sin is -^) . .
Then sm i A = a / ^^ — -. — 4 .— ^ '- . (yi)
V sm ^ X sm ^ .
sin i B = /^.r(7-<z)xsin(.-r) ^ ^
A^ sm « X sm ^ ^
V sm ^ X sm ^
and for B and C the formulae are exactly analogous.
Rule 2. Given two sides and the included angle to find the
other parts.
Let a and d be the sides and C the given angle.
sin i (a + d) : sin ^ (^ -- 3) : : cot ^ C : tan ^ (A '>- B) ;
cos i {a + d) : cosl (a r^ d) : : cot I C : tan ^ (A + B) ;
whence A = ^ (A + B) + J (A - B) . . . (76)
B = i(A + B)-i(A-B) . . . (77)
To find c.
sin J (A ~ B) : sin i (A + B) : : tan ^ {a ^ d) itain ^ c ;
:. tan i ^ = sin J (A + B) x tan i {a '■^ d) -r- sin ^ (A ^^ B)
In every spherical triangle the sines of the angles are pro-
portional to the sines of the opposite sides.
Digitized by VjOOQ iC
37^ Preliminary Survey
Rule 3. Given two sides and the angle opposite to one of
them.
Let a^ b^ and A be given, then B is found by
sin <2 : sin ^ : : sin A : sill B . . . . (79)
and for c
tan i ^ : tan ^ (/i ^ ^) : : sin J (A + B) : sin i (A *- B)
and for C (80)
cot i C : tan ^ (A *- B) : : sin i (^i + ^) : sin \{a r^ b)
(81)
Table LII.
Values of sin b x sin Cfor Years 1890 io i<)Oofor Azimuth
by fi and 5 Draconis in same Vertical,
log sin Azim = tabular constant - log cos Lat
1890 9-48568
1891 9*48559
1892 9-48549
1893 9-48540
1894 9-48531
1895 ... ... 9-48521
1896 9-48512
1897 9-48503
1898 9-48494
1899 9-48484
1900 9-48475
Table LIII.
Valtus of sin b x sin Cfor Years 1890 to igoo for Azimuth by 3 and
€ Urscs Majoris {Merak and Alioth) when in same Vertical.
log sin Azim =■ tabular constant — log cos Lat.
1890 9-72910
1891 972917
1892 9-72923
1893 9-72930
1894 9-72936
1895 9-72943
1896 •. 9-72950
1897 9-72956
1898 9-72963
1899 9-72969
1900 9-72976
Digitized by VjOOQ IC
Appendix
379
Table LIV. \
Radii correspomiing to Decimals of a Degree oj^i^iHt^fcUure
per Chord of lOO feet. V <^ C ;
Angle of
Radius !
Angle of
Radius
Angle of
Radius
Angle of
Radius
deflection
in feet 1
deflection
in feet |
deflection
in feet 1
deflection
in feet
d
r \
d
r
2204
2122
d
r
d
r
0-05
o-io
I 14592 !
57296 j
2-6o
270
8-20
8-40
699-3,
6827 t
20-00
287-9
21 -OO
274-4 1
015
38197
2-8o
204^
8-60
666-8
22-00
262-0 '
0'20
28648 i
2-90
1976
8'8o
651-7
23-00
250-8 1
0-25
0-30
22918
19098
3-00
I9IO
1848
9-00
9-20
6373
623-4
24-00
25-00
240-5 !
3-IO
231-0
o'35
16370
3-20
179I
9-40
610-2
26-00
222-3
0*40
14324
3-30
1736
9-60
597-5
27-00
214-2
0-45
12732
3-40
1685
9-80
585-4
28-00
206-7
0*50
1 1459
10417
, 3-50
1 3.60
1637
1592
10-00
573-7
562-5 1
29-00
30-00
199-7
1 0-55
10-20
193-2
o-6o
9549
' 370
1549
10-40
551-7
31-00
187-I
0-65
i 3-8o
1508
10-60
541-3
32-00
181 -4 1
070
8185
3*90
1469
I0-80
531-3 1
33-00
176-0 :
1 075
7639
4-00
1433
11-00
5217
34-00
171 -0
o-8o
7162
4 '20
1364
1 11-20
512-4
35 -OO
166-3
0-85
6741
4-40
1302
11-40
503-4
36-00
161 -8 '
0-90
6366
4-60
1246
11-60
494-8
37-00
157-6
0-95
6031
4-80
1 194
1 1 -80
486-4
38-00
153-6
i-oo
5730
5209
1 5-00
1 5 -20
1 146
1 102
12-00
478-3
459-3
39-00
40-00
149-8
. i-io
12-50
146-2
1 I -20
4775
4407
5-40
1 5 -60
I061
1024
13-00
441-7
425-4
42-00
44-00
139-52 1
1-30
13-50
133-47 ,
, 1*40
4093
3820
1 5-8o
6'oo
988-3
955-4
14-00
4IO-3 1
396-2
46-00
48-00
127-97
1 1-50
14-50
122-93 j
i i-6o
3581
3370
6 '20
6*40
924-6
8957
15-00
383-1
370-8
50-00
55-00
II8-3I 1
170
i 15-50
108-28 1
I -So
3183
3016
6 -60
6-8o
868-6
843-1 i
16 -oo
359*3
348-5
60 -oo
65-00
100 -oo ,
I 90
16-50
93-06!
1 2-00
2865
2729
7-00
819-0
796-3
' 17-00
338-3
328-7
70-00
1 75 -oo
87-17
2-IO
7-20
17-50
82-13
2 '20
2-30
2605
2491
7-40
7-60
774-8
754-4
18-00
319-6
3II-I
80-00
85-00
77-78 ,
18-50
74-OI
2-40
2387
2292
7-8o
8-00
735-1
716-8
19-00
302-9
295-3
90-00
7071
2-50
19-50
1
i
Digitized
by Google
$o
Prelwiinary Survey
Table LV.
General Elements of t lie Decimal Spiral.
"To. of point
I
2
3
4
5
6
7
8
9
lO
II
12
13
14
15
i6
17
i8
19
20
21
Curvature in
one chord
n. c.
degrees
0*2
0-4
0-6
0-8
i-o
1*2
1-4
1-6
1-8
2'0
2-2
2-4
2-6
2-8
3-0
3*2
3*4
3-6
3-8
4*o
4*2
Total
curvature
up to n
degrees
0-2
0-6
1*2
2-0
3-0
4-2
5-6
7-2
9-0
II -o
132
iS-6
i8-2
21 'O
24-0
27*2
30-6
34-2
38-0
42*0
46-2
Total curvature
Tangential
up to mid-chord
angle, or ang^le
or deflection of
between long
short chord from
chord and
main tangent
main tangent
i
degrees
degrees
01
o-i
0-4
0-25
09
0-47
1-6
075
2-5
i-io
3-6
1-52
4*9
2XX>
6-4
2*55
8-1
3-17
100
3-85
I2-I
4-6o
14-4
5-41
i6-9
630
196
7*24
22-5
8-26 ,
25-6
9*33
28-9
10-48 (
32-4
11-68
36-1
12-95 i
400
14-29
44-1
15-69
Table LVI.
Elements of No. 2 Spiral for Tramivays.
feet
I
572-96
2
286-48
3
19099
4
143-24
5
114-59
6
1 95*49
7
81-85
8
1 71-62
9
63-66
feet
feet
•003
2-00
-017
400
-049
6 00
-105
8-00
-192
10-00
•318
11-99
•488
1 13-98
-711
1 15-97
•993
' 17-95
feet
2
4
6
8
10
12
14
16
18
Digitized
by Google
Appendix
38'
Table L VI . — Continued,
Elements of No. 2 Spiral for Tramways,
10
II
12
13
14
15
16
17
18
19
20
21
feet
57-30
5209
4775
44 'oS
40-93
38-20
3581
3371
3184
3016
28-65
27-29
feet
I 34
1-76
2-26
2-84
3*51
4-27
514
6-IO
7-i8
8-35
9-64
11-03
feet
19-92
21-88
23*82
2573
27-61
29*46
31 26
33*02
3470
36-32
37-85
39-29
L
feet
20
22
24
26
28
30
32
34
36
38
40
42
Table LVII.
Elements of No, 5 Spiral for Tramways and Light /Railways.
n
ds
r
X
feet
y
L
degrees
feet-
feet
feet
I
4-00
1432-4
0-009
5-00
5
2
8-01
716-2
0-044
1000
10
3
12-02
477-5
0-122
15-00
15
4
16-05
358-1
0-262
20-00
20
5
20- 10
286-5
0-480
24-99
25
6
24-18
238-7
0-794
29-98 ^
30
7
28-29
204-6
1-22
34-96
35
8
32-43
179-0
1-78
39-93
40
9
36-62
159-16
2-48
44-88
45
10
40-86
143-24
3-35
49-81
50
II
45-16
130-22
4-40
54-70
55
12
49-52
119-40
5-64
59-54
60
13
53-97
110-20
7-10
64-32
65
14
58-50
102-31
8-77
69-03
70
15
63-14
95-54
10-69
73-65
75
16
67-89
89-53
12-85
78-16
80
17
7279
84-25
15-26
82-54
85
18
77-84
79-58
17-94
86-76
90
19
83-07
75-40
20-89
90-80
95
20
88-53
71-63
24-10
94-63
100
21
94-26
68-21
27-58
98-22
105
Digitized
by Google
382
Preliminary Survey
Table LVIII.
Elements of No. lo Spiral for Light Railways,
n
^j
r
X
y
! L
degrees
feet
feet
feet
1 feet 1
I
2-000
2,865-0
0-017
10-00
10 1
2
4-cx>i
1,432-0
0-087
20-00
20 1
3
6-003
954*9
0-244
30-00
30 I
4
8-006
716-2
0-523
39-99
40 ,
5
IO-OI2
573 -o
0-960
49-98
50 \
6
1 2 -022
477-5
1-588
59-96
60
70
80
7
14*034
409-3
2*442
69-93
8
16-052
358-1
3-556
79-87
9
18-074
318-3
4*96
89-77
90 1
lO
20-102
286-5
6-70
99-61
100
II
22-136
260-4
8-80
109-39
no
12
24-197
238-6
11-28
119-08
120
13
26-226
220-4
14-19
128-65
130
14
28-283
204-6
17-55
138-07
140
15
30-350
191 -0
21-37
147-31
150
i6
32-427
179-1
25-69
156-32
160 .
17
34-514
168-5
30-53
165-08
170
18
36-614
159-2
35-88
173*53
180
19
38726
150-8
41-78
181-61
190 ,
20
40-852
143*3
48-20
189-27
200
21
42-992
136-4
55-16
196-45
210
Table LIX.
Elements of No. 15 Spiral for Light Railways,
n
ds
i
i '■
X
y
L
degrees
feet
feet
feet
feet
I
1-33
4297-1
•02
15-00
15
2
2-67
2148-6
-13
30-00
30
3
4-00
1432-4
"^l
45-00
45
4
5-33
1074-3
•78
59-99
60
5
6-67
\ 859-6
1-44
74-98
75
6
8-OI
1 716-2
2-38
89-95
90
7
9-34
613-9
3-66
104-89
105
8
IO-68
537-2
5-33
119-80
120
9
12-02
477-5
7-45
134-65
135
10
13-36
429-7
10-05
149-42
150
ir
14-71
390-7
13-20
164-09
165
12
16-05
1 358-1
16-93
178-62
180
Digitized
by Google
Appendix
383
Table LIX. — Continued,
Elements of No, 15 Spiral for Light Raihvays,
n
ds
r
X
1 y
L
degrees
feet
feet
feet
feet
13
17-40
3306
21 29
192-97
195
14
1875
307-0
26 32
207-10
210
15
20-I0
286-5
32-06
220/96
225
16
21-45
268-6
38-54
234-49
240
•17
22-81
252-8
45-79
247-62
255
18
24-18
238-8
53-83
260-29
270
19
25-54
226-2
62-66
272-41
285
20
26-91
214-9
72-31
283-90
300
21
28-28
204-7
82-74
; 294-67
315
Table LX.
Elements of No, 25 Spiral for Narrow- Gauge Railways^
n
ds
r
X
y
L
degrees
feet
feet
feet
feet
I
0-80
7161-9
0-043
25-00
25
2
I 60
3580-9
0-218
50-00
50
i 3
2 40
23873
o-6io
75-00
75
1 4
3-20
1790-5
1-31
99.99
100
1 5
4-00
1432-4
2-40
124-96
125
I 6
4-80
1 193-6
3-97
149-91
150
1 7
5 -60
1023 -I
6-IO
174-82
175
8
6-40
895-2
8-89
199-66
200
9
720
795-8
12-41
224-41
225
10
8-01
716-2
16-75
249-03
250
II
8-81
6511
21-99
273-48
275
12
9-61
596-9
28-21
297-69
300
13
10-41
551-0
35-48
321-61
325
H
11-22
511-6
43-86
345-16
350
15
1 2 -02
477-5
53-43
368-26
375
16
12-83
447-7
64-23
390-81
400
17
13-63
421-4
76-31
412-70
425
18
14-44
397*9
89-71
433-81
450
19
15-24
3770
104-44
454-01
475
20
16-05
358-2
120-51
473-16
500
21
16-85
341 -I
137-90
491 -I I
525
Digitized
by Google
384
Preliminary Sun>ey
Table LXI.
Elements of No, 50 Spiral for Narrow- Gauge Railways.
ds
degrees
I
0-4
2
0-8
3
1-2
4
1-6
5
2'0
6
2-4
7
2-8
8
3*2
9
3-6
10
4-0
II
4'4
12
4-8
13
5-2
14
5-6
15
60
16
6-4
17
6-8
18
7*2
19
7-6
20
8-0
21
8-41
I
;c
y
feet
L
feet
feet
0-087
50-00
50
0-436
100 00
100
I -221
14999
150
2-62
199-97
200
4-80
249-92
250 •
7*94
29982
300
12-21
349-64
350
1778
399*33
400
24-83
448-83
450
33-51
498-07
500
43 99
546-96
550
56-42
595-39
600
70-96
64323
1 650
87-73
690-33
; 700
106 -86
736-53
750
128-47
781-62
800
152-63
825-40
1 850
179-42
867-61
900
208-88
908-01
950
241 -02
946-32
1000
275-81
982-22
1050
Table LXII.
Elements of No. 75 Spiral for Trunk Lines.
n
ds
degrees
I
0-27
2
0-53
3
0-80
4
1-07
5
1-33 .
6
1-60
7
1-87
8
2-13
9
2*39
10
2-67
feet
feet
feet
0-13
75-00
75
0-65
149-99
150
1-83
1 224-98
225
3*93
299-95
300
7-20
374-88
375
11-92
i 449*73
450
18-31
; 524*46
525
26-67
1 598-99
600
37-24
673-24
675
50-26
748-16
750
Digitized
by Google
Appendix
385
Table LXII. — Continued.
Elements of No. 75 Spiral for Trunk Lines.
n
ds
;r
y
L
II
2-93
65-99
820-44
825
12
319
84-64
893-08
900
13
346
106-44
964-84
975
H •
372
131-60
1035-49
1050
15
4'CX)
160-30
1104-79
1 125
16
4-26
192-70
1172-43
1200
17
4-52
228-94
1238-08
1275
18
479
269-13
1301*41
1350
19
5-05
31332
1362-01
1425
20
5-33
36153
1419-47
1500
21
5-6o
41372
1473-32
1575
Table LXIII.
Elements of No. 100 Spiral for Trunk Lines.
n
^j
c 1
X
y
^
degrees
feet
feet
feet
feet
I
0-2
100-00
0-17
100 -oo
100
2
0-4
199.99
0-87
199-99
200
3
0-6
299-98 i
2-44
299-98
300
4
0-8
399*97 1
5*23
399*94
400
; 5
10
499*94
9-60
499*85
500
i ^
1-2
599-86
15-88
599*65
600
1 7
1-4
699-70
24*42
699-29
700
1 8
1-6
799-45
35*56
798-66
800
9
1-8
899-04
49*65
897-66
900 1
1 10
2-0
998-39
67-02
996-14
1000 [
! II
2-2
1097-44
87*98
1093-92
1 100
1 12
2-4
1196-11 ,
112-85
1190-78
1200
1 ' 13
2-6
1295-71
141-92
1286-46
1300
^4
2-8
1391*77 1
175-47
1380-67
1400
1 15
3-0
1488-52 1
213-73
1473-06
1500
16
3-2
1584-15 !
256-94
1563-24
1600
17
3*4
1678-85
305-27
1650-79
1700
18
3-6
1772-36
358-85
1735*22
1800
19
3-8
1863-42
417*77
1 8 16 -02
1900
20
4-0
1948-59
482-05
1892-63
2000
21
4-2
2040-48
551*64
1964-44
2100
Digitized
byCfoogk
386
Preliminary Survey
1
Table LXIV.
For Ranging the Spiral from an Intermediate point.
The values of », k^ and x are common to all spirals : x and y are for No. loo
spiral, but for other spirals x and y are obtained by simple percentage. Thus for No. i s
«piral take 15 per cent.
n
/t
Instrument at i
y
i
n
_*
X
y
J
*deg^
feet
feet
deg.
deg.
feet
feet
deg.
I
2
1 2
0'2
0-35
lOO'OO
0-20
3
0-3
0-52
100*00
0-30
; 3
0-7
I '57
i9v"y9
0.45
4
I'O
199*98
0-65
1 4
1*4
^•oi
80:1
29./ u6
0-76
5
1*9
10*82
299*93
1*07
1 5
2'3
39'J-^^3
1-15
6
3*o
399*79
1*55
1 6
3*4
1396
49^/7^3
I -60
I
^^i
18-32
499*51
2*10
; 7
U
22-15
59'^ ■r7
2-12
28-42
fH
2-72
• 8
32-95
69? .73
2-70
9
7*5
41*47
3*40
9
n
46-70
797 -i 3
3*35
10
9*4
57*8i
796-80
4*15
10
61-99
8go-5
4*05
II
"■§
77*74
894*79
'X
II
11-9
82-61
994 i I
4*75
12
13-8
101-60
1087*89
12
16-7
107-14
1091-45
5-61
13
i6-3
129-66
6-8o
n
135-88
1187-23
6-53
H
19-0
162-22
1182*44
7'?^
14
19-4
169-10
1281-56
7*52
15
21-9
199*52
1275-22
8-89
15
22-3
207-04
1374-08
8*57
16
25*0
241-78
i365;85
10*04
16
^s'-*
249-94
297-96
1464-41
9-69
'7
28-3
289-19
1538-89
11*25
^l
28-7
1552-13
10-87
18
31-8
341*88
12-55
18
32*2
35^ "^5
1636-75
I2-II
19
35*5
399*96
1620-30
13*87
19
35-9
409-88
1717*75
13*42
20
39*4
463*43
1697*57
15*27
20
39-8
473-89
1794*58
14*79
21
43*5
532*26
1770-11
16-74
21
43*9
543-23
Instrumen
1866-63
tat 3
16-23
Instrumen
tat 4 ,
3
4
4
0-4
0-70
100-00
0-40
5
o'5
0-87
100*00
0-50
5
1*3
2-97
Zf
0-85
6
1-6
3-66
J99*Sj
'25
6
2*4
7-15
1*37
7
2-9
8-72
1*67
7
3*7
13-61
399*67
1*95
8
J-t
16*40
399 '53
2-35
8
5*2
22-67
499-26
II
2-60
9
27*02
498-97
3*io
9
6*9
34*68
3*32
10
8-0
40*94
597 99
%
10
8-8
'£t
4-10
II
lo-i
58-48
696^44
II
io'9
**2I
12
12-4
79*95
7^4" I r
1*75
12
13-2
91*73
989-?8
5*86
13
14*9
105*66
8go"75
6-76
13
157
118-79
6-85
14
17-6
135*90
rifl6'i:»7
7*85
14
18-4
'im
1084 -07
7-90
15
20-5
170-92
iu7y-74
9-00
1 15
21*3
186-68
1177-24
9 -ox
16
23-6
210*96
"71-37
10-21
, 16
24-4
227-99
1268-31
10-19
»7
26*9
256-20
1260-55
11-49
17
27-7
274-47
1356-85
11-44
18
30*4
306-80
1346-80
12-83
18
31*2
326-27
1442-38
12-75
J9
34*1
362-87
1420-61
. 1508*41
1582*61
14-24
19
34'9
383-49
1524*40
1602-33
14-12
20
38-0
424*43
'5*71
ao
38-8
446'X5
15*56
21
42-1
49«*47
17-25
2r
42-9
5x4*22
1675-59
17-06 1
Digitized
by Google
Appendix
387
Table I.XIV. -Continiud.
For Ranging the Spiral from an Intermediate point.
instrument ai 5
Instrument at 6
—
deg.
feet
s
6
0-6
I '05
7
J '9
4-36
8
3 '4
10*29
19*18
9
5**
10
7-0
31*37
11
9-«
47-18
12
11-4
66*95
13
ii-'i
9o*97
14
"9*54
15
i9'5
152-92
16
22-6
191-35
17
25-9
235*03
18
29-4
284*12
19
33'i
338-73
20
37*0
398-91
21
41*1
464*65
feet
o
99*99
199-94
299*76
399*37
498*62
597*36
695*39
792*46
888*29
982*56
1074*88
1164*83
X25z'96
1335-73
1415*59
1490*95
Instrument at 7
o*8
2-5
4-4
6-5
8*8
11-3
14*0
16*9
20*0
23*3
26*8
30-5
34*4
38-5
I 40
5*76
13*43
24*75
40*05
59*64
83*84
112*91
147*11
186*66
231*75
282*50
339*00
401*25
99*99
199*89
299*60
398-96
497*78
595-84
692*87
882*52
974*37
1063*62
1149*79
1232*30
1310*56
Instrument at 9
1*0
3*1
5*4
7*9
io'6
13*5
16*6
19*9
23*4
27*1
31*0 I
35*1 I
1*74
7*15 I
16*56 I
30*31 1
48*70 I
73*05
ioo'o6 I
134*65 I
174*37
219*92 ,
271*43 I
328-93 I
99-58
199-34
29939
39S "44
49Q^74
593 ■■>8
6fc.r^i
7^} -A
9C4
1050'
1 132'
del?-
D
i^^5
1-97
r^
4'5a
6'55
a'S5
10^09
11-41 1
1279!
I
i
o*ao
I 65
J "57
.155 ;
4 "61
o 90
Bj5
9-46 I
ij'So
i7'o«
a "05
3"i7
4 "35
5^60
6^93
9:75
II nVt
14*49
iq"jo
'3
lb
X-
JT
dr^.
feet
Q
07
T'33
^■j
S'o6
t-?
TiS6
'21-97
7"9
35*/i
10*3
S3>2
13-7
?S*40
tB'3
TOI'^fi
169-^5
31-4
^4 '7
311-63
aBit
3SS-B9
^*'g
3""73
35'li
370*33
3^9
434"37
f«t
pg*99
299-09
3^'ii7
596-64
694 '30
790 -eii
33 5 '55
9?a-66
11:169-51
1157-04
ii+iS3
1 3 13 'fid
*4"=|'
:;;j
Inetniment at 8
1 °
! ??
i'57
6-46
4-9
15*00
r^
27 53
9-7
44'38
tB-4
fisSs
^5":^
92 '24
iH-4
TSl'flo
31-7
160-73
35-^
203-36
38*9
35i"69
.i:^"ii
J05-S6
3&'9
36s '90
99^99
199-87
399*50
3yS"7i
497-26
594*95
691-41
735-29.
fl79*3i
969-69
1057-44
21-36
InstriiTiient at id
1-1
3 "4
5 9
i 8'6
IT*S
1410
''>
t»5i
^ D
1 ga
7*e5
tS'13
330S
53 03
78-23
iDB'96
MS"4S
ifl7-a7
33fi'35
2^3-96
Digitized
99-98
199^80
399-37
39S-15
49&*i4
59' '91
fcSS'07
78113
a 71 -74
959'=o
^ I
0*70
a'37 I
4*io
2{
7 "35
E-57
g-S^
11':^
I2'6l
14 '03
15-63
17*33
0*90
1-a.s
a -a;
3 '95 ,
5'w
6-33'
S-95
10* J^
11*6-1
*3*34
15 "Oil
1 la
3"47
6 10
7 5 3
9-00
la-ict
13 "S4
byGoogk
388
Preliminary Survey
Table \2^\\, -Continued.
For Ranging the Spiral from an Intermediate point.
Instrument at ii
Instrument at X2
13
14
15
16
17
18
13
14
15
16
17
18
19
17
18
19
k
X
deg.
feet
I "2
2*09
3*7
8*56
6-4
19-69
9*3
35-85
12*4
57 '33
15*7
84-39
19*2
117-27
22 '9
26-8
156*19
201-27
30-9
252*63
feet
o
99-98
199*77
299-15
397-83
495-50
591-77
686-21
778*32
867-58
953 '39
Instrument at 13
1*4
2*44
4-3
9*94
7*4
22-82
10*7
41*39
65-92
14*2
17-0
21-8
96-65
133-79
25-9
177-47
99-97
199-69
298-86
397*12
494 -06
589*22
682*07
772*02
1-6
2*79
4*9
"-33
8-4
25-94
12*1
46-90
16*0
74-47
20*1
108-83
Instrument at 15
99*96
199*59
298-52
396-30
492-43
586*34
Instrument at 17
99-95
199*49
298*15
395-38
Instrument at 19
Q
1-8
3-14
5-5
12-73
9-4
29-06
13-5
52-40
deg.
o
I '20
2*45
3*77
5*15
6*60
8-12
9-70,
"•35 '
13*06
14*84
V
1 "
k
X
y
i
deg.
feet
feet
deg.
12
13
I "3
2-27
99*97
1*30
H
1*°
6*9
9*24
199-73
2*65
15
21-26
299*01
4-07
16
lO'O
38-62
397*49
5*55
17
13-3
61-63
494-80
7-10
18
16-8
90*53
w;^
8-72
1 '9
20-5
125-55
10 '40 ,
'1 20
1
166*86
775-27
12*15 '
21
214*58
863*15
13-96
20
6-1
3*49
14-12
99 94
199*37
I -80
3*65
5*57
7*55
2 00 ,
4*05.
Instrument at 14
14
1*40
15
1-5
. 2*62
2-85:
i6
4-6
10*64
4-37
17
7*9
24*38
5-95
18
11*4
44-15
7-bo,
19
15-1
70*20
9-321
20
19*0
102*75
11*10 j
21
23*1
141*99
"•95 1
I
1
I
nstrument
16
1-60
Ii
1*7
3-25
5*2
4-97 •
6*75
19
"*2
20
12-8
8-60 .
21
16-9
10*52 ;
2*97
12*03
27*50
49*66
78-73
99'97
199*64
298*69
396*72
493*27
587*82
679*80
99*96
»99-54
298*34
395*85
491-54
Instrument at 18
i"9
5-8
9-9
331
13*42
30*61
99 94
199*43
297*94
Instrument at 20
3*66
99-93
1-50
3-05 i
8*ro
9*92
11*80
1*70
3*45
5*27 i
7*15
9*io
1*90
3-85,
5-87
Digitized.
by Google
Appendix
389
Table LXV.
Basalt . . . .
Bricks and brick-
work . . . .
Cement (American)
„ (Portland)
Concrete in lime
Port
land cement .
Coal, Newcastle
,, anthracite
„ in bulk for
stowage, 48 cubic
feet per ton
Coke
„ in bulk for
stowage 80 to ICX5
cubic feet per ton
Chalk ....
Earth ....
Flint
Specific Gravity of Stofies, Earths ^ ^c. Water
taken cU 62*3 lbs. per c. ft.
G G
275 to 2*95 Glass 2*50 to 3*00
Gneiss and granite average 2*65
I '60 to 2'Oo Limestone and
O'So to 0*90 marble ... ,, 2*65
1-35 to 1*45 Lime ,, 1-50
average i '9 Masonry Of dressed
ashlar same G as
,, 2 '2 the stone
,, 1*25 Masonry of rough
,, 1*50 rubble. . . . i "80 to 2*20
Mortar . . . . i '40 to i -90
Mud I -25 to 1 75
Peat average 0*40
075 Pitch „ 1-15
Sand I "5 to I '95
Sandstones . . . 2*10 to 2*50
Sulphur .... average 2*00
,, 2*50 Tallow .... ,, 0'90
i-5oto2-oo Tar ,, i*oo
average 2 '60 Traprock . . . 2-80 to 3*00
Table LXYL— Metals and A /toys.
Aluminium
Antimony
Babbett or white
metal
Bismuth
Brass .
Copper
Gold .
Gun metal
average 2*60
„ 670
» 7-30
„ 9'8o
8-40
„ 8-8o
„ 19-00
„ 8-50
Iron (cast) .
„ (wrought)
Lead . .
Mercury .
Platinum
Silver .
Steel . .
Tin . .
Zinc . .
average 7*23
„ 778
„ 11-40
» 13*60
21-50 to 23-00
average 10*50
7-75 to 8-00
average 7*30
7*00
Table LXVll.—Ti/nder.
Acacia .... 070 too 'So
Ash 0-70 to 0-76
Beech .... 0-70 to 0-80
Box average 1*25
Cedar (Lebanon) . ,, 0*49
„ (American) . ,, 0-55
„ (West Indies) „ 0-70
Chestnut. ... „ 0-61
Cork ....
Ebony . . .
Elm (English) .
„ (American)
Fir ....
Hornbeam . .
Ironwood . .
Larch . . .
Digitized
average 0*24 '
I-I9
0-56
0-72
0-51
076
by Google
390
Preliminary Survey
Lignum vitse .
Mahogany (Hon
duras) . .
Mahogany (Spa
nish) . . .
Maple . . .
Oak (American)
„ (English) .
Table LXVIL—Timder.
average 1*33
056
„ 0-85
0-67
080
078 to o*93
{Cotttimi^d. )
Pine (white) . .
„ (yellow) . .
„ (red) . . .
,, heart of long-
leafed southern
yellow .
average 0*40
0-S5
0-57 to 0*65
I 04
Teak 074 to 0-86
Table LXVIII
Acetic acid i -06
Alcohol o*8o
Ether 070
Hydrochloric acid . . . i '20
Nitric acid i -22
Oil (linseed) 0*94
,, (olive) 0*92
„ (petroleum) . . . . o-88
,, (whale) 0-92
Liquids.
Sulphuric acid .
Water, distilled at
62° Fahr.,bar.
30 in., 62-355
lbs. per cubic
foot ....
Water, at 212°
Fahr. . . .
Sea ....
I c
1-84
. . . o'957
I -026 to I -030
Table LXIX. — Multipliers for reducing Specific Gravity to Weight oj
certain Volumes^ Water taken at 62*3 lbs.
Weight of I cubic centimetre in grammes . = G
,, I kilolitre or cubic metre in
tonnes of 1,000 kilogrammes . . . ^ G
Weight of I decalitre in kilogrammes = G x 10
,, I cubic inch in lbs. . . = G x 0-036
,, I cubic foot in lbs. . . . = 0x62-30
,, I cubic yard in tons . . . — G x 0-751
,, I Brit. imp. gallon in lbs. = G x 10
,, I Brit. imp. bushel in lbs. . = G x 80
„ I U. S. liquid gal. in lbs. . . = Ox 8-322
,, I U. S. struck bushel in lbs.
No. of cubic yards in one ton . . •
No. of cubic feet in one ton
No. of Brit. imp. gals, in one ton
,, ,, bushels ,,
U. S. liquid gals.
struck bushels in one ton
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GLOSSARY
Accounty Byy a term used for either longitude or latitude when cal-
culated from other data than observations.
Age of the Tide is the interval between the time of new or full moon
and the time of the next spring tide, and varies from i J to 3 days.
Aliothy a star otherwise called e Ursae Majoris, in the constellation of
the, Great Bear {see Fig. 122).
Altazimulh, an instrument for measuring at one adjustment of the
line of sight the angle of altitude and of azimuth {see pp. 320, 322).
■ Altitude is the angular elevation of a heavenly body, or, in other
words, the arc of a great circle passing through a heavenly body
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39^ Preliminary Survey
measured from it to the true horizon {see Horizon). Let A, Fig. 121,
be the observer's eye, S the object in altitude. SOH is the true
altitude plus refraction, which has to be deducted {see Refraction).
10 is drawn parallel to AS. So SOI represents the correction due
to parallax in altitude [see Parallax). The sensible horizon is drawn
for simplicity at the observer's eye, instead of at the earth's surface,
because it only involves an inappreciable error of parallax, due to
the increased length of radius of the earth at high elevations {see
Parallax, Equatorial).
Amplitude is the spherical angle at the zenith contained between the
plane of the prime vertical at the point of observation and that of
the meridian of any celestial body observed on the horizon, rising
or setting. It is measured by the arc of the horizon from the east
or west point to the body ; it is shown by AW in Fig. 133, cm: as
AC in Fig. 127, and is therefore the complement of the azimuth.
It is a very useful method of finding the variation of the compass
at sea, but is of no use where there is no horizon. When a star
is setting its true place is already under the horizon by about half
a degree {see Refraction) and therefore more to the westward than
it appears to be. If we take an amplitude when the star is 34'
above the horizon we shall be pretty near the truth ; if we take it
when actually setting it will appear too much to the west — that is,
to the right of its true place. The greater the southern declination
in the northern hemisphere — that is, the flatter the arc of the bod/s
trstnsit across the heavens— the greater the error will be. In the
case of bodies with parallax {see Parallax) the error due to that
cause makes them appear too low. The moon's parallax is nearly
double the refraction ; so she is still about half a degree above the
horizon when she appears to set, and her true place is to the left of
her apparent place.
Anallatism (Greek a privative, and alasso^ to alter, unchangeableness).
The centre of anallatism is that point in a distance-measuring tele-
scope from which the distance of any object is proportional to the
height intercepted upon the staff by two horizontal wires in the dia-
phragm. In ordinary telescopes it is situated at the anterior focus.
In some tacheometers it is made to coincide with the vertical axis
by means of an additional lens.
Aneroid (Greek a privative, and neros, wet), an instrument for measur-
ing the pressure of the atmosphere {see p. 342).
Angle (Lsitm an£uluSf a corner) may be plane or spherical {see Spheri-
cal Angle). A plane angle is formed by the inclination of two
straight lines to one another ; it has been reckoned from o to 360
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Glossary ^ 393
degrees all over the world until lately, when the centesimal system
of dividing the circle into 400 degrees, and each degree into 100
minutes, has been coming into use for the purpose of simplifying
calculations. An angle of 90° is called a right angle, one less than
90° an acute, and one greater than 90° an obtuse angle.
Angle Complement, The complement of an angle is the difference
between it and 90®. The cosine is the sine of the complement of
an angle, and the cotangent and cosecant are tangent and secant
of the same complementary angle.
Angle Supplement, the difference between an angle and 180°.
Aphelion (Greek apo, from, and helios, sun), that point in the orbit of a
comet or planet which is farthest from the sun.
Apogee (Greek apOy from, and ge, the earth), that point in the moon*s
orbit which is farthest from the earth.
Apparent (Latin ad, to, and parere, to appear), that which is opposite
to the true or real. The apparent position of a celestial object is
that which it appears to have to the observer with an instrument
before being corrected for refraction, parallax, &c.
Apparent Time is the hour angle of the sun {see Hour Angle) reckoned
westward from the meridian. It is the time shown by the sun-
dial. The sun's apparent place in the heavens is constantly chan-
ging, owing to the earth's orbit, but this being elliptical, the movement
is not uniform and is represented in the almanacs by daily changes
of right ascension, with rate for one hour. A clock keeping appa-
rent time would have to be altered every day, so the expedient of
mean or average time is resorted to {see Mean Time). Apparent
time is first found from observation and then reduced to mean time
by the equation of time {see Equation of Time).
Arc of Excess, in sextants that part of the graduated arc behind the
zero.
Aries, First Point of {see Right Ascension).
Argument (Latin argumentum, a thing taken for granted) means any
mathematical datum or known quantity from which to determine
others.
Ascension {see Right Ascension).
Astrcnomical Time {see Civil Time).
Augmentation of the moon's semi-diameter is the increase in angular
dimension when in altitude above what it appears on the horizon,
owing to its approach towards the observer, until, when in the zenith,
it is closer by the amount of the earth's radius. The table is given in
Chambers's * Mathematical Tables.'
Axis (Greek ctxon, an axle), an imaginary line joining the north and
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south poles of a celestial body upon which it is supposed to rotate.
The imaginary line about which the vertical limb of a theodolite
rotates. Is of very wide application {see also Optical).
Azimuth (Arabic samatha^ to go towards) is the spherical angle at
the zenith contained between the plane of the meridian of the place
and that of the great circle of altitude passing through the object
observed {^see Fig. 133, p. 414). If Z be the zenith and NZS the
meridian, Z )) A a great circle of altitude, NA is the azimuth. It is
measured upon the horizon from the north or south point, which-
ever is nearest. {See also Course and Amplitude, of which the
azimuth is the complement.)
Barometer^ an instrument for measuring the pressure of the atmosphere
{see^. 342 &c.).
Binary Stars are double stars which revolve round one another ; when
the motion appears to be rectilinear they are merely called double
stars.
Circumpolar Stars, those whose polar distance is less than the latitude
of the place, and which therefore do not set but culminate twice.
At the North Pole the whole celestial hemisphere is circumpolar ;
at the Equator none. See Fig. 122.
Civil Time, like astronomical time, is a term having reference more to
date than to time. Both are mean time {see Mean Time), but civil
time begins its day from midnight, and astronomical time from the
succeeding noon, so that January i, 1890, at 6 a.m. by civil time, is
December 31, 1889, 18 hours astronomical time. But January i,
1890, 2 P.M. civil time, is the same date and time astronoinically.
Co-altitude. See Zenith Distance.
Collimation, Line of (Lat. cum, with ; limes, a limit), in telescopes
is the axis of a pencil of light reaching the eye through the tube ; or,
which is the same thing, it is the straight line joining the two foci
of the double-convex lens forming the object-glass and the focus of
the eye-piece. The line of coUimation is defined in levels and
theodolites by two intersecting spider hairs or some such device,
attached to a brass diaphragm which is placed in the optical axis
by adjusting screws.
Colure (Gr. kolouo, I cut in the middle), two celestial meridians {see
Meridian, Celestial) whose planes are at right angles to one another ;
whose line of intersection is terminated by the poles, and which cut
the celestial sphere into quarters. One of these semicircles bisects
the equator at the spring and autumn equinoxes, and the other at
the summer and winter solstices.
Compass, Solar. See p. 328.
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Glossary
395
Cofnpassy Variation of^ is the angle between the astronomical meridian
and the direction of the compass needle when at rest under the in-
fluence of terrestrial, but undisturbed by local, magnetism. It is
Fig. 122.
subject both to a diurnal oscillation and an annual variation as well
as to local disturbances.
Constellation (Lat. cum^ together and stella, a star), a portion of
the heavens marked on globes and maps by dotted boundary lines
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in which the main feature is a well-defined group of stars supposed
to resemble some terrestrial object and accordingly designated.
Contraction of the semi-diameter of sun and moon are sensibly the same
• in amount, and arise from unequal refraction in the upper and lower
limb. The table is given in Chambers's * Mathematical Tables.'
Co-ordinates^ Rectangular, a pair of straight lines locating any point
in a plane by measuring its shortest distance from two fiducial
lines at right angles to one another. Thus the rectangular co-
ordinates X yy x' y, x" y determine the positions of the points a,
b, and c relatively to the lines N.S. and E.W., and if N.S. be the
meridian {see * Meridian, Magnetic,* and * Meridian, Celestial ')^,^',
and y^ are the latitude ; jt, x\ and jt" the departure {see Latitude,
Difference of).
Course in navigation and land-surve)dng is the direction of the line
being travelled or measured with reference to the magnetic meridian
or true meridian. It is the azimuth of the objective point. It is
also called * bearing * in land-survejdng more frequently than course.
It is reckoned in each quadrant separately, from the north east-
wards and westwards, and from the south eastwards and west-
wards. This method is suitable to traversing with a large field
compass, being ready for reduction to latitude and departure.
Another method is to reckon clear round from o° to 360**. Most
countries reckon the 0° from the north point, some firom the south.
Theodolites are arranged to read from 0° to 360°, as the former
method would not be suitable ; but when working to latitude and
departure the angles must first be reduced to azimuthal form. See
Table XVII., p. 59.
Conjunction (Lat. cum^ together ; jungere, to join). Two bodies are
said to be in conjunction when they appear in nearly the same part
of the heavens. It is necessary, therefore, that one of them should
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Glossary 397
have an apparent movement through the heavens, such as the moon,
which is in conjunction with a star when it has the same longitude
or right ascension.
Culmination (Lat. culmen, the top) is the passage of a celestial body
across the meridian of a place. In the northern hemisphere the
sun and most of the stars used in observation culminate southwards ;
consequently the term * to south * is used for southern culmination.
See also Circumpolar Stars.
Declination is the distance of a celestial body from the celestial equator
measured north or south on the arc of a celestial meridian passing
through it. Declination corresponds exactly with terrestrial latitude.
In Fig. 127 SD is the north declination of a body S (see also Polar
Distance). In some almanacs N. and S. declination are marked
+ and — respectively.
Degree (Lat. degredior^ to go down : from de^ down, and gradus, a
step), a division of the circle which in the sexagesimal is ^ or
in the centesimal is -^ of the total circumference.
Departure in a traverse means the easting or westing from a known point
which is taken as the origin of rectangular co-ordinates. It is equal
to the distance run multiplied by the sine of the angle, azimuth,
or course {see Course, Azimuth).
Depression or Dip of the Horizon is the angle of depression of the
apparent horizon, due to elevation of the eye above the level of the
sea. If we direct a levelling instrument in good adjustment towards
the horizon from the top of a high cliff we shall at once perceive that
there is a depression, and with a large-sized transit theodolite we
can measure that angle with sufficient accuracy to know the height
of the cliff within ten to twenty feet. Depression arises from the
curvature of the earth. The values are given for different elevations
of the eye in Table L., p. 288. These answer for correcting an
altitude taken at sea, or for estimating the elevation of a cliff in the
manner just described. The depression is always deducted from
the observed altitude. A simple way to keep it in memory is * Dip
makes me see too much, and therefore I deduct it '
Diameter (Gr. dia^ through, and metrony a measure) in its ordinary
use is limited to the circle and the sphere of which it is double the
distance from centre to circumference. It also means the breadth
of anything.
Diaphragm^ in telescopes is an annular brass plate fixed in the focus by
adjusting screws and forming both a passage to confine the light and
a frame to hold the cross hairs.
Dip of the Horizon. See Depression.
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Diurnal Inequcdity of Heights is, in irregular tides, the difference
between the height of high water of each successive tide.
Diurnal Inequality of Time is, in irregular tides, the difference between
the lunitidal intervals of each successive tide.
Eclipse (Gr. ekleipsis, a disappearance), the phenomenon occurring in
the heavens from the disappearance of one body in the shadow
of another. The following diagram illustrates the theory of the
various forms of eclipses of sun and moon, from which it will be
seen that an eclipse of the sun (which is more correctly an occulta-
tion, see Occultation), can only take place when the sun and moon
[t;
Fig. 124.
7[nmUar£dip5&ifthe'Siin
Fig. 125.
Total£c^zpse oflheMoort
Fig. 126.
are in conjunction, or, in other words, at new moon, and an eclipse
of the moon, which is really an eclipse, can only take place at oppo-
sition or full moon.
An eclipse of a satellite of Jupiter is due to similar causes. When
entering the shadow its immersion is said to take place, and on
leaving it is said to emerge. The idea of obtaining the longitude
by observing eclipses of Jupiter's satellites originated with Galileo.
Ecliptic^ the great circle of the heavens which the sun appears to
describe in the year ; it derives its name from the fact that eclipses
can only take place when the moon is also on the ecliptic. It is
commonly called the sun's annual path, to distinguish it from the
sun's diurnal path, due to the earth's axial rotation.
Ecliptic, Obliquity of, the inclination of the plane of the ecliptic to
that of the celestial equator, producing the phenomena of the
seasons. The angle is about 23° 27 , and is very gradually dimi-
1
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Glossary 399
nishing. It is measured by the decHnation at the solstitial points,
June 21 and December 21.
MlangcUum (Lat. longe^ afar ofif), the angular distances of the pole star
eastward or westward of the true pole ; also the similar distances of
a planet from the sun or a satellite from its primary.
Equation of Time^ the daily correction at mean noon to be added to
or deducted from the apparent time ascertained by observation, in
order to obtain mean time {see Mean Time). It is sometimes ex-
pressed as sun * after clock' or 'before clock,' meaning that the
equation is to be added to or deducted from the apparent time.
Thus : when the sun is on the meridian it is noon of apparent time ;
if in the table the equation is marked * sun after clock ' 5 min. 6 sec.
it will be then o hr. 5 min. 6 sec. astronomical, or 12 hrs. 5 min.
6 sec civil time. If the table had it * sun before clock ' 1 1 min.
3 sec, when the sun culminated it would be 11 hrs. 48 min. 57 sec.
civil time, and 23 hrs. 48 min. 57 sec astronomical time of the
previous day's date {see Civil Time, Astronomical Time ; see also
Sidereal Time).
Equator, Celestial, is the intersection of the plane of the terrestrial
equator with the celestial sphere (EQ in Fig. 127).
EqucUoTy Terrestrial^ is the great circle of the earth's surface whose
plane is midway between the poles and at right angles to the earth's
axis. AD in Fig. 129.
Equatorial instrument, a telescope which is made to move, by hand
or by clockwork, in the equator, or, in other words, in which the axis
of rotation, instead of being set vertical, as in an ordinary transit
theodolite, points to the celestial pole.
Equinox, See Equinoctial.
Equinoctial (Lat. cequus, equal; noxy night), another name for the
celestial equator {see Equator), because when the sun is in it the
nights are equal all over the world. It will be obvious in looking at
a celestial globe that whatever angle the pole makes with the horizon
the equator always intersects with the prime vertical {see Prime
Vertical) at the horizon ; consequently the equator is in every latitude
half above and half below the horizon. If an observer could be sta-
tioned precisely at the north or south pole, the equator would then be
coincident with the horizon, and he would see a half-sun going clear
round the horizon at the equinox. The sun is in the equinoctial on
March 21 and September 21 {see Right Ascension).
Establishment, Vulgar y is the lunitidal interval when the time of moon's
meridian passage is o hr. o min. or 12 hrs. It is termed by Raper
the tide-hour, and defined, as the apparent time of the first high
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water that takes place in the afternoon of the day of full or change.
In German Hafenzeit^ or harbour-time.
Establishment y Mean^ is the mean of all the lunitidal intervals in a
semi-lunation, and is often less by ten to forty minutes than the
vulgar establishment.
Fiducial^ any point, line, or arc which is known, fixed, or may be
otherwise relied upon for locating others, such as the meridian of
Greenwich ; the datum line of a profile or cross section ; an ordnance
benchmark, &c. &c.
Focal Lengthy the distance from the * centre ' of the lens to the focus.
The * centre * of a double-convex lens is that point in the axis which
is midway between the two surfaces.
Focus, the common meeting-point of all the converging rays passing
through a lens. In the double-convex lens of a telescope the focus
inside the tube is termed the focus, that outside the tube is termed
the anterior focus. In German Brennpunkt, or burning-point
Geographical Mile, or Admiralty Knot, is 1*15152 statute mile. It
slightly differs from that of the United States, which is i •! 5157 statute
mile and adopted by the Coast Survey as being the linear distance
in the arc of i minute of a great circle of a true sphere whose sur&ce
area is equal to that of the earth at sea level. It also equals i '85324
kilometres.
Gibbous (Lat. gibbus, convex-bunched), when the moon is rather more
than half but less than full.
Great Circle of a sphere is any circle described about it whose plane
passes through its centre, as HZRN or PSDO in Fig. 127.
Great Circle Sailing is sailing between two points on the earth's surface
upon the arc of a great circle {see Spherical Distance).
Horary Angle, See Hour Angle.
Horizon, The sensible horizoft is a plane parallel to that of the true
horizon, but touching the surface of the earth aj the point of obser-
vation.
The true horizon \% the intersection of the celestial sphere by a
plane GH (Fig. 121) passing through the centre of the earth and
at right angles to a diameter of the earth at the observer's stand-
point.
The visible or apparent horizon is the intersection of a conical
surface, of which the apex is thfe observer's eye, with the sphere
{see EF, Fig. 121). The dip of the horizon is equal to the comple-
ment of half the angle of the cone.
Hour Angle is the angle at the pole contained between the meridian of
the place and the celestial meridian passing through any celestial
^
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Glossary
401
body. SPZ, Fig. 127, is the hour angle ; it is measured by the arc
DE of the celestial equator. The calculation of the hour angle is
reduced from arc to time by the proportion of the total time of a re-
volution to the hour angle ; thus, as 360® : 24 hours:: hour angle in
arc : hour angle in time. The tables of log. sines, cosines, &c., in
Raper give the horary values of all angles, in addition to which
tables for converting arc into time and vice versd are given. For
formulae adapted to slide-rule see p. 136.
The calculation of the hour angle of the sun is the commonest
method of obtaining apparent time at place, from which by equa-
tion of time {see Equation of Time) and a chronometer registering
Greenwich mean time the longitude is easily calculated {see also
Apparent Time, Longitude). The hour angle of a star is sidereal
time, which can be reduced to mean time by rule (p. 410).
Hypsometric f 'height-measuring,' is used for observations with the
aneroid or boiling-point thermometer to determine the approximate
elevation above the sea.
Ifitegral, consisting of entire numbers, as contrasted with fractions.
Kilometre^ a distance of one thousand metres {see Metre).
Kfiot. See Geographical Mile.
AT
.0*
/^^^^-^i^S^V^"
Fig. 127.
Latitudey Difference of, in traversing called latitude for shortness, is the
northing or southing of the base line {see Co-ordinates and Traverse).
Latitude { Terrestrial), the spherical distance {see Spherical Distance)
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Preliminary Sufvey
between the equator and the position of the observer, measured
north and south upon a meridian of longitude. It is represented
by BC, Fig. 129.
^"'-ui^it^^^*
Fig. 128.
Lens, in telescopes a circular glass, truly ground to a surface of that
curvature which causes the rays of light passing through it to be
refracted at angles which meet in a common centre.
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Glossary 403
Linib of a celestial body means the extreme edge of the circumference,
upon which the observation is taken.
Limb of a theodolite is the vertical or horizontal portion cf the instru-
ment [see Chapter IX.).
Longitude ( Terrestrial) is the spherical angle at the pole {see Spherical
Angle) between the plane of the meridian of Greenwich {see Meri-
dian, Terrestrial) and that of the meridian of the place of observation.
Thus the angle ANC in Fig. 129 is the longitude of C west of
Greenwich when AEND is the meridian of Greenwich, and is
measured on any of the parallels of latitude in angular measure.
Notice in the figure that though the small circles of latitude give the
same arc of a circle as the equator, and consequently the difference
of longitude, they do not give the true spherical distance — that is
{see Spherical Distance), the arc of a great circle which is the flattest
circle, and consequently the shortest distance between any two points.
Lunitidal Interval is the time that elapses each day between the
transit of the moon over the meridian and high water.
Mean Distance of a planet from the sun is the mean of the perihelion
and aphelion distances, which see.
Mean Time. Instead of correcting the watch daily to keep apparent
time {see Apparent Time) the average length of a solar day throughout
the year is calculated and termed a mean solar day. Both civil
and astronomical time are kept in this way {see Civil Time and
Astronomical Time). The corrections by which to obtain it from
observed apparent time are given in the almanacs and called
equation of time. Mean time is the only possible method of regulat-
ing time by the sun and yet keeping it uniform {see Standard Time).
Meridian, Celestial, The observer's celestial meridian is the great
circle of the celestial sphere passing through the zenith — the pole,
the north and south points of the horizon, and the south pole. Its
plane is therefore at right angles to that of the prime vertical, and is
shown by ZHNR in Fig. 127.
The celestial meridian of a star is a semicircle of the heavens
which passes through it.
Meridian {Distance), See Hour Angle.
Meridian, Magnetic, is the line of direction of the compass needle {see
Compass) produced by the magnetic polarities of the earth. The
lines of equal magnetic variation do not even approximate to great
circles of the earth's surface, although the magnetic poles approximate
in position to the terrestrial poles.
Meridian, Terrestrial, is a great circle of the earth's surface passing
through the north and south poles and the place of observation.
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Every place has its meridian, but certain ones are chosen by the
several sea-going nations as the basis of their calculations of
longitude, such as the meridian of Greenwich, of Paris, of Washing-
ton, of Lisbon. A conference was held two years ago with a view
to adopting one for the world, but no agreement was come to.
Metre, the French standard measure of length = one ten-millionth of
the distance from the pole to the equator as measured by earlier
astronomers = 39*37079 British and American inches. It is about
\ inch longer than the seconds pendulum.
Alicrometer (Gr. mikros, small ; metron, a measure), an instrument for
measuring small angles or distances {see p. 317, Chapter IX.).
Mile {see Statute, Geographical).
Mooft'Culminaiing Stars are certain stars which lie close to the moon's
path through the heavens, and from this cause furnish a ready
means of obtaining, at their culmination, the difference of local time
at any two places, and hence the longitude. The interval between
the culminations at Greenwich is obtained from the * Nautical
Almanac ' by the difference in right ascension of the moon and star.
The interval at place is found by watch, and the difference between
the two intervals is proportional to the difference of longitude.
Nadir, the point in the celestial sphere at the opposite extreme to the
zenith.
Nautical Mile. See Geographical Mile.
Nodes (Lat. nodus, a knot) are the points of intersection of a planet's
orbit with the sun*s path or ecliptic. The * ascending node* is
where it crosses the ecliptic from south to north, and the descend-
ing node the opposite point.
Obliquity of the Ecliptic, See Ecliptic.
Occultation is, the disappearance or hiding of a celestial body by the
intervention of another. Thus the stars in the moon*s path are
occulted by her, and the satellites of a planet by the body of the
planet.
Optical Axis in instruments is the line joining the centres of the true
spherical surfaces of the lenses.
Oriottation, the general direction of a chain of triangles, or the placing
of a plane-table or similar instrument so that it will preserve the
same line of direction.
Parallax {see Fig. 121, where the parallax in altitude is shown
as angle SOI. It is the difference of altitude which would exist
between two simultaneous observations of the same star by two
observers, one stationed at the earth's surface and the other at its
centre). All calculations of celestial bodies are reduced to the earth's
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Glossary 405
centre, and therefore parallax is always to be added. A way to keep
it in memory is to think of the earth's centre as our true place of
observation, so that being raised up too high by an amount equal
to the earth's radius we see the body too low.
When the body is in the zenith the angle of parallax, as will be
seen by inspection of Fig. 121, is eliminated. It is at a maximum
when the body is on the horizon, where it is termed horizontal
parallax, and the table for its values for different bodies and at
different seasons is given in the almanacs together with a table for
the sun's parallax in altitude.
Parallax i Equatorial, In the * Nautical Almanac ' the equatorial or
longest radius of the earth is used for computing parallax ; so when
great accuracy is sought the equatorial parallax must be again re-
duced by a correction for latitude. Only the sun, moon, and planets
have parallax. The fixed stars, being at too vast a distance, have no
appreciable parallax except in a few instances, where the parallax is
measured by assuming as a base not the earth's radius, but the
diameter of the earth's orbit round the sun.
Parallax in Altitude. Given the horizontal parallax of a celestial body
and its altitude, to find its parallax in altitude. The sun's parallax
in altitude is given in a table, because the variation is sensibly
constant, but the horizontal parallax of the moon is deduced by the
following rule :
Let P = the horizontal parallax
P' = the parallax in altitude
A = the apparent altitude
R : cos A : : sin P : sin P'
• R
Or by logarithms :
Ic^ sin P' = log cos A + log sin P - 10.
Parallax in telescopes, the apparent dancing about of the cross hairs when
the eye is shifted about during observation. It arises from the eye-
piece not being correctly in focus. The cross hairs should be clearly
defined by moving the eyepiece out or in.
Perigee (Gr. peri, near ; ge, the earth), the converse of apogee {see
Apogee).
Perihelion (Gk. peri, near ; helios, the sun), the converse of aphelion
{see Aphelion).
Pointers, the stars o and )8 in the Great Bear, Ursa Major {see Fig. 122),
or familiarly the Waggon and Horses. The former of them is also
called Dubh^. The seven principal stars of this constellation are
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406 Preliminary Survey
all that are commonly known as belonging to it, but there are many
more. These appear to revolve round the pole without in our
latitudes even touching the horizon, the pointers maintaining all the
while their direction towards the pole star.
Polar Distance is the arc of a celestial meridian passing through a
celestial body measured from the pole to the body. In the northern
hemisphere it is the complement of the declination when that is
north, and equals 90° + the declination when it is south. In the
southern hemisphere the polar distance, i.e. distance from the sotith
pole, is vice versd,
Pole (Gr. poled t I turn) {Celestial), the intersection of the earth's axis
when produced to the celestial sphere. The apparent unchange-
ableness of this point renders it the basis of all astronomical mea-
surements. The point does actually change from year to year, but
not with sufficient rapidity to enter into daily calculations of latitude
and longitude.
Pole {Terrestrial), the two points north and south forming the apices
of the axis of the earth's rotation (N, S, Fig. 129).
Pole Star is a star of the second magnitude near the celestial pole in the
end of the tail of the Little Bear ; it is called either a Ursae Minoris
or Polaris ; its right ascension for 1890 is i hr. 18 min. 29*1 sec.,
its declination 88^ 43' 18" N. Fifty years ago it was R.A. i hr.
2 min. 10*683 sec. and declination 88° 27' 2i"*94 N. ; it is therefore
travelling (in appearance) slowly towards the celestial pole.
Precession of the Equinoxes is a slow retrc^ade movement of the equi-
noctial points, due to the attraction of the sun. and planets {see
Aries, First Point of).
Prime Vertical is the great circle of the celestial sphere passing through
the zenith, east and west points of the horizon, and the nadir. Its
plane is therefore at right angles to that of the meridian. It is
shown as ZCN in Fig. 127.
Primitive, the great circle upon the plane of which a stere<^[raphic
projection is made.
Quadrant (Lat. quadrans, a fourth part), the fourth part of a circle.
An instrument so named was used in taking altitudes before the
introduction of the sextant.
Radius Vector (Lat. radius, a sunbeam ; vector, a bearer), the
shortest distance from the centre of the earth to the centre of the
sun at any point of the earth's orbit. Has also the meaning of
radius of curvature to any curve other than a circle at any particular
point in the curve, such as the distance from the centre of the
earth to any point upon its spheroidal surface.
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Glossary
407
Range oj Tide is the difference between the height of high- and low-
water levels of any one tide without any reference to datum. Is
also termed height of tide.
Refraction (Lat. refrangere, to bend) is the bending of the ray of light
proceeding from a celestial body when passing at an angle from the
rarer ether, or whatever the medium may be, into our denser atmo-
sphere. It makes bodies appear higher than they are, and the cor-
rection for it is given in Chambers's * Mathematical Tables,'
Whitaker, &c. It is always to be deducted. In consequence of
this law all bodies appear to rise earlier and to set later than they
really do, with the sole exception of the moon {see Amplitude). If
the ray of light were passing from a dense into a rarer medium it
would be bent the opposite way. A simple but entirely unphilo-
sophical and somewhat grotesque way of remembering the direction in
)^--"
Fig. 130.
which the ray is bent is to imagine the pencil of light as a long thin
wand trending down with the weight of the star at the end of it.
When in a vertical position it ¥rili not deflect either way, but the
more the ang^e of depression the greater the deflection. The
direction of the ray as it reaches the eye, or line of sight, is where
the eye sees the star ; but its true position is below that, at the end
of the curved pencil of light ; so that we must deduct the correction
for refraction from the observed altitude. It is just the same for
all bodies, at whatsoever distance they may be. It depends entirely
upon the angle and density of the medium, and the tables give
corrections for difiference of barometric pressure and temperature
where close calculation is required. It varies from o' when the body
is in the zenith to 34' on the horizon.
Right Ascension of a celestial body is analc^ous to the longitude of a
terrestrial position. It is the arc of the celestial equator measured
from a meridian passing through the first point of Aries to a
meridian passing through the celestial body. The first point of
Aries has nothing in particular to mark it in the heavens ; it is the
vernal e(juinoctial point which in the times of the ancient astronomers
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was actually situated in Aries, but, owing to the precession of
the equinoxes, is now in another constellation altogether. It is
quite an imaginary point upon the celestial equator, chosen, like
the observatory of Greenwich in terrestrial calculations, as i
fiducial point from which to map the stars. Fortunately for astro-
nomers there has not been the same display of national feelisg
in the selection of a celestial meridian as there has been about tbe
choice of a common terrestrial meridian, so there is but one. The
right ascension is reckoned from west to east, and is expressed in
hours and minutes, the 360° of the equator being 24 hours {see
Sidereal Time). The celestial semicircle which crosses the j>oles
and the first point of Aries is called the vernal equinoctial colure,
and its opposite semicircle the autumnal equinoctial colure, because
on March 21 and September 21 or thereabouts the sun*s path isee
Ecliptic) intersects the celestial equator. The sun has then no R. A.
and no Decl. , and day and night are equal all over the world [su
Equinoctial).
Rise of a tide is the height of the high-water level above the low spring
datum.
Satellites (Lat. satelles, a companion), the little bodies which revolve
round the planets.
Sea Mile, See Geographical Mile.
Semi-diameter of sun and moon is half the angle subtended by the
diameter of the visible disc ; it varies according to the bodies'
distance from the earth, and values are given in the almanacs ; by
it observations of the upper or lower limb are reduced to the centre.
Sextant (Lat. sextans^ a sixth part), an angular reflecting instrument for
making celestial observations {see Chapter IX. ).
Sidereal Time, If a star is watched passing any fixed point, such as the
line between two perfectly straight vertical rods, on successive
evenings by a correct watch, it will be seen to pass them 3 min.
56 sec. (more correctly 3 min. 55*91 sec.) earlier each evening.
This movement is perfectly regular, and means simply the time of
one complete revolution of the earth upon its axis. Sidereal time
is needed to find the time when any star will culminate, and to
correct watches or chronometers, which may be done by a transit
instrument to a fraction of a second.
It would be of no use as civil time, because the time would keep
dropping back. Sidereal time commences when the first point of
Aries is on the meridian of the place {see Right Ascension), and is
counted through 24 hours until the same point comes round again.
It is a shorter measure of time than mean time.
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Glossary 409
24 hrs. of sidereal time = 23 hrs. 56 min. 4*0906 sec. of mean
time, and 24 hrs. of mean time = 24 hrs. 3 min. 56*5554 sec. of
sidereal time.
Sidereal time at mean noon is the heading of a column in the
Nautical Almanac Ephemeris, and also in Whitaker.
In the American Nautical Almanac the same thing is termed
sidereal time, or right ascension of mean sun.
In Chambers's * Mathematics * it is called the sun*s mean right
ascension at mean noon.
All these names are sufficiently reasonable, but such a difference
of nomenclature is confusing to the beginner, who has some difficulty
in grasping the thought of a * mean sun.'
We will confine ourselves to the first-mentioned expression and
endeavour to put it in popular language.
Everybody knows that the sun appears to go round the heavens
once a day and once a year, owing to the earth's daily rotation and
annual orbit.
The sun's yearly path is indicated by different stars appearing
at sunset at one time of the year from another. At sunset in March
the brilliant constellation of Orion is nearly overhead. In June it
is the sun's bedfellow, so we do not see it at all.
The starting-point of star-measurement {see Right Ascension) is
an arbitrary point in the heavens called the first point of Aries,
situated in a semicircle which is termed the vernal equinoctial
colure, because the sun is always there at spring-time.
This starting-point is marked 24 hrs. or o on the celestial
globes, and it is also the commencement of sidereal time at any
place.
When the first point of Aries is on the meridian of any place it
is sidereal noon, just as when the sun is on the meridian it is apparent
noon. Hence in the American Nautical Almanac there is a
column headed * Mean Time of Sidereal Noon.' In the British
Nautical Almanac it is termed * Mean Time of Transit of First Point
of Aries.'
Sidereal time is a perfectly regular measure of time like mean time,
but it is not the same measure, since we see it gains about 4 minutes
a day.
Why, then, do we use it at all ? Because the sidereal time at mean
noon given in the almanac enables us to tell when any star will
culminate, as will be presently shown.
Why do we not use it exclusively ? Because it does not keep
with the sun. We should have to put our breakfast hour on
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every morning if we kept sidereal time, and say Monday at 8,
Tuesday at 8*04, &c., or else we should soon be breakfasting at
midnight.
If a star has a R.A. of 24 hours, like the second star of Cassi-
opeia nearly {^see Fig. 122), it will culminate at sidereal noon. If its
R.A. is I hour it will culminate I sidereal hour afterwards, and
so •!!. Hence we can find the mean time of any star's culmination
by adding its right ascension reduced to mean time to the mean time
of sidereal noon given in the almanac {see Figs. 131, 132).
Thus : Find the mean time of the culmination of a star in
4 hrs. R.A. on May 18, 1889. 4 hrs. sidereal = 3 hrs. 59 min.
207 sec. mean time, which is the interval between the passing of
the first point of Aries and the star across the meridian.
But sidereal noon by the almanac was at 20 hrs. 15 min. 10*46 sec
mean time on May 17. Adding the R.A. in mean time, we have
o hr. 14 min. 31*16 sec. as the mean time on the i8th of the star's
culmination.
Or we may do it in another way. The sidereal time at mean
noon on May 18 is given as 3 hrs. 45 min. 26*47 sec. This repre-
sents the interval of sidereal time between the culmination of the
first point of Aries and mean noon. But the sidereal interval
between the first point of Aries and th« star is 4 hours. Therefore,
if we deduct the one from the other we get the sidereal interval
14 min. 33*53 sec. from mean noon. This reduced to mean time is
the same as we had before, 14 min. 31*16 sec.
The two rules are therefore as follows : — To find the mean time
of any starts culmination at any meridian.
Rule I. To the mean time of sidereal noon on the previous day
or the given day add the star's right ascension reduced to mean
time.
(If the two quantities make more than 24 hours take the previous
day, if less than 24 hours take the day itself.)
Rule 2. From the star's right ascension, increased if necessary
by 24 hours, deduct the sidereal time at mean noon : result will be a
sidereal interval which reduced to mean time will be the answer.
When the meridian is not the same as Greenwich the mean
time, apparent time, or sidereal time have all to be corrected
for the difference of longitude reduced to time, as explained on
p. 136.
It is, no doubt, with the object of making the matter clearer that
sidereal time at mean noon is expressed in some books as the mean
right ascension of mean sun. One is told to imagine a sun which
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Glossary
411
keeps mean time in its movements, and whose right ascension will
therefore be the sun's right ascension plus or minus the equation of
time.
Since the right ascension of any body is synonymous with the
sidereal time of its culmination ^ the right ascension of this imaginary
sun when on the meridian — that is, at mean noon— is the same thing
as sidereal time at mean noon.
There is not, however, any such thing really as apparent right
ascension or mean right ascension ; it is only a hyperbole for con-
veying the twofold idea of the real sun keeping apparent time and
an imaginary sun keeping mean time.
It is important not to confound the expression ' the sun's mean
right ascension at mean noon ' with that of * the sun's right ascension
at mean noon.' The latter only differs by one or two seconds from
its right ascension at apparent noon— that is to say, it is the difference
of right ascension which the sun has made during the interval re-
presented by the equation of time — whereas the sun's mean right
ascension, or sidereal time, at mean noon is the position of an
imaginary sun whose right ascension always differs from that of the
true sun by the equation of time itself.
This explanation has only been added because of the difference of
nomenclature. The term * sidereal time at mean noon ' is sufficiently
intelligible for our present purpose without introducing the idea of
two kinds of right ascension.
Fig. 131.— Position of the Celes-
tial Equator at Sider^l Noon,
May 17, SK> hrs. 15 min. 10*46
sec. Mean Time.
Fig. 132. — Position of the Celestia
Equator at Culmination of Star
at o hr. 14 min. 31*16 sec. Mean
Time, May 18, and 4 hrs. Sidereal
Time.
The * mean sun ' of the foregoing example is shown in the figure
close to the star ; but 14 min. 31*6 sec. ahead.
Note, The position of the heavens in these two figs, is as they
would appear in the southern hemisphere, where the sun's culmination
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412 Preliminary Survey
is to the north. The primitive is the ecliptic. The only reason for this
arrangement was that the north point and first point of Aries might
both be at the top of the paper, as being perhaps more readily under-
stood.
When the equation of time at mean noon is marked as sun
aftej^clock _ mean time is that much-f'"*^' ^han ^ ^^^^ jj^^.
before clock slower than
and sidereal time at mean noon i% that much -^^ — ^ the sun's
^ more than
right ascension at mean noon. In the illustration just given the
equation of time was 3 min. 46*69 sec. * before clock,' and the sun's
right ascension at mean noon was 3 hrs. 41 min. 3978' sec., wliich
added together make 3 hrs. 45 min. 26*47 sec, which is the sidereal
time at mean noon.
To reduce sidereal to mean time intervals by slide rule :
Place the right-hand i of the slide over the 9*83 of the rule, and for
hours and decimals of sidereal time on the slide read off seconds
and decimals on the rule, which are to be deducted ftom the sidereal
interval to obtain the mean-time interval ; thus 2*50 hrs. sidereal
time » 2 hrs. 30 min. is opposite to 24*6 seconds to be deducted.
To reduce mean to sidereal time intervals by the slide rule :
Place the right-hand i of the slide over the 9*86 of the rule, and
for hours and decimals of mean time on the slide read off seconds and
decimals on the rule, which are to be added io the mean-time interval ;
thus 6*50 hours mean time are opposite to 64*1 sees., or i min. 4*1
sees, to be added.
To reduce sidereal time at mean noon at Greenwich to sidereal
time at local mean noon by slide rule :
Express the difference of longitude in hours and decimals.
Adjust the rule with a i of the slide over 9*86 on the rule, and
for hours of difference of longitude on the slide read the correction in
seconds and decimals of sidereal time on the rule.
Example. What will be the sidereal time at mean noon in New
York, Ion. 74° W., it being 16 hrs. 41 min. ii sec. at Greenwich?
The Ion. in time is 74 x 4 = 296 min., or 4*93 hrs., which being
W. is the amount behind Greenwich.
Sid. time, mean noon, Greenwich . i6*' 41" ii"
4*93 hrs. x 9-86 sec. . . .— 00 48*6
Sid. time at mean noon, New York . 16 40 22*4
Signs of the Zodiac are twelve symbols denoting the constellations suc-
cessively traversed by the sun in his apparent annual circuit of the
heavens. They are as follows with the sun's position in them ;
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Glossary
413
Aries, the Ram . . t .
Taurus, the Bull . .
Gemini, the Twins . . a .
Cancer, the Crab . . s .
Leo, the Lion . • Q *
Virgo, the Virgin . . tgj .
Libra, the Scales . , ^^ .
Scorpio, the Scorpion . m .
Sagittarius, the Archer . ^ i
Capricomus, the He-Goat, vp •
Aquarius, the Waterman . a» •
Pisces, the Fishes . • X .
March 20 to April 20
April 20 to May 21 ^
May 21 to June 21
June 21 to July 22
July 22 to August 23
August 25 to September 23
September 23 to October 23
October 23 to November 22
November 22 to December 21
December 21 to January 20
January 20 to February 18
February 18 to March 20
The signs of the zodiac are supposed to be Chaldean or Egyptian
hieroglyphics, intended to represent some occurrences peculiar to the
month in which the sun occupied each of the constellations at that
time. Thus the spring signs show productiveness of nature.
When the sun is in Libra the autumnal equinox takes place, whence
the origin of that title is evident. Explanations more or less likely
are given to all the rest of them.
Small Circles of a sphere are those whose planes do not pass through its
centre, such as the parallels of latitude (Fig. 129).
Solstice (Lat. sol^ the sun, and stare ^ to stand), the two periods, June 21
and December 21, when the sun's declination is temporarily con-
stant.
Sottthy To, See Culmination.
Sphere^ Celestial, is the apparent vault of the heavens supposed to
be viewed by an observer at the centre of the earth, in which the
heavenly bodies appear to be situated and upon which their relative
positions or movements are determined by measurements of spheri-
cal distances taken from arbitrary but fiducial circles and points
supposed to be drawn upon the surface of the sphere like the meri-
dians of longitude and parallels of latitude upon the terrestrial maps.
Spherical Angle is that formed at any point upon a sphere by two great
circles intersecting there. It is measured by the inclination of their
planes or by the angle between the tangents to the circles at the
point of intersection. Thus the spherical angle NZA (Fig. 133) may
be measured by the angle NOA between the planes, or by the angle
TZT between the tangents, or by the arc NA of the great circle
whose plane is at right angles to those of the two intersecting planes.
Spherical Distance is the arc of a great circle (see Great Circle) passing
through two points which is intercepted between them, as SD,
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Preliminary Survey
PZ in Fig. 127 or as NA Fig. 133. It is the shortest distance
upon the spherical surface between any two points.
Standard Time on the continent of America is a form of keeping mean
time {su Mean Time) by which there are no variations except those
of an even hour at a time, since 15® of Ion. correspond "with one
hour*s difference of time. The time of every fifteenth meridian, be-
ginning at New York with the 75th and ending with San Francisco on
the 1 20th, rules the belt for about 7 J° on each side of it. The conve-
nience of railway systems causes in some cases the overlapping of the
T_
2:.
T>
=f
(n
^ /
T
W
^
Fig. 133.
times, so they are distinguished by the following terms: Intercolonial
time, Eastern time, Central time, Western or Mountain time, and
Pacific time. The first mentioned is time of the 60th meridian and
is used in Nova Scotia.
Statute MiUy the British and American standard of long measure;
it is equal to 8 furlongs, or 80 chains (Gunter), or 320 rods, or
1,760 yards, or 5,280 feet, or 63,360 inches. It is also equal to
0*868719 of a knot {see Geographical Mile) and 1*609315 kilo-
metre.
* Semi-mensual inequality of heights ' is the difference between the
heights of spring and neap tides above mean water-leveL
* Semi-mensual inequality of time * is the difference between the greatest
and smallest lunitidal interval.
Supplement^ the difference between any angle and a semicircle.
Taclieometer (Gr. tachedSy swiftly, and metreo, I measure), same as tele-
meter, but exclusively applied to instruments furnished with tele-
scopes.
Telemeter (Gr. tele^ far off, and metreo, I measure), an instrument
for measuring distance without chaining.
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Glossary 415
Time, See Apparent, Astronomical, Civil, Mean, Sidereal.
Traverse in land-surveying is used in contradistinction to triangu-
lation {see Triangulation). It is the method of surveying by mea-
suring base lines in length and angular direction continuously for-
ward, whereas in triangulation the lengths are computed by trigo-
nometry or graphic construction.
A closed traverse is one in which the base lines box the compass back
to the starting-point {see Course).
* Working a traverse * is the reduction of the angular base lines to
rectangular co-ordinates of latitude and departure {see Latitude
and Departure). The term is also largely used in navigation.
Triangulation in land-surveying is the determination of points by the
intersection of rays taken from the ends of a base of known length.
It is the foundation of geodetic operations of large extent as well as
the most cursory field-sketching with a sketch-board. It is the root
principle of all range-finders and telemeters and of the whole science
of surve)dng. It is hardly ever used without traversing as well. In
primary triangulation for geodetic survey the detail is filled in by
traversing {see Traverse), and upon a route survey traverse the detail
is sketched from triangulation. So the two principles dovetail into
one another, taking alternative forms according as accuracy or
despatch is the main point aimed at.
Zenith is that point in the heavens which is directly overhead. It
would be the celestial pole if the observer were standing at the earth's
pole, and on the celestial equator if he were crossing the *line.'
Zenith Distance is the coaltitude or complement of the altitude. In
Fig. 127 it is indcated by SZ.
Zodiac (Gr. zone^ a girdle), a belt of the heavens extending 8° on
either side of the ecliptic, within which the sun, moon, and the
major and many of the minor planets perform their annual revolu-
tions {see Signs of the Zodiac).
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INDEX
ABN
Abney's level, 322
Acreage, 200
Adie's telemeter, 332, 333
Altazimuth, pocket, traverse and
profile by, 54. 63
— description of, 320
Aneroid barometer, 342
Angle iron, weig:ht of, 277
Arc reduced to time, 126
Astronomy, Chauvenet's, 116
Barnett's diagrapb, 360
Bate range-finder, 340
Boiling-point thermometers, 350
Borings, 113
Breakwater, Port Said, 113
— Chicago, 114
Bremiker s Mathematical Tables,
368
Bridges, iron, 256
— stone, 258
— brick, 258
Chaining, 194 ; methods of, 195
Chambers's Practical Mathe-
matics, 368
— Mathematical Tables, 368
Channel iron, weight of, 278
Chronometers, 121
Cisterns, discharge from, 106
Clarke's, Col., telemeter, 333
Climate, affects location, 3
Clinometer (compass), 321
Coast lining, 82
Coggeshall's slide-rule, 362
Compass-clinometer, 321
— Casella's 321
— O'Grady Haly's, 321
— solar, 328
Computing scales, Stanley's, 361
Concrete, 113
DRE
Crelle's Tables, 368
Cross-ties, railway, 250
— table of, 251
Cuartero's tables, 189
Curvature of the earth, 288
Curve-ranging, Krohnke's me-
thods, 205
— Jackson's method, 205
— Kennedy and Hackwood's
method, 206
— by chord and offset, 208
— with transit and chain, 209
— nomenclature, 209
— by astronomical bearings, 217
— tacheometric, 183
Curves, resistance of iron, 13
— cost of operating, 17
— reverse, 221
— diversion of, 223
— linear advance of inner rail, 224
— transition, 225
Davis's Rules, 362
Dawson, Wm. Bell, his survey of
Nova Scotia, 193
Deflection distance, 197
Directrix for night observations,
136
Distance, measurement of, 150
— measured by gunfire, 86
Dock, Hull, cost of, 113
— West India, 114
— Antwerp, 114
— Marseilles, 114
— Leith, 114
— Liverpool Graving, 114
— Malta Graving, 115
— Portsmouth, 115
— Honfleur Sluicing Basin. 115
Drawing instruments, 359
Dredge-Steward omnitelemeter,
336
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417
DRE
Dredging, 112
— plant, 112
Earth, curvature of, 288
Earthwork, mensuration, 253
— equalisation, 254
Eidograph, 361
Elliott's army telescope, 333
— omnimeter, 333
Equation of time, 139
FlELDBOOK, 369
Force, centrifugal, 263
Glossary, 391
(iradients, method of overcoming.
4
— cost of traction on, 6
— railway, 245
(iraduation of slide-rule, 362
Gravity, specific, 389
(ireatness Mill, survey of, 192
(iribble, ' Ideal ' Tacheometer,
307
Heliograph, 328
Heliostat, 328
Hints to travellers, 369
Hold-alls, 369
Hydraulics, 100
Hydrostatics, no
Hypsometry, 348
Iron, angle, 277
— tin, 277
— channel, 278
— round, 278
— square, 278
Kutter's formula, 103, 105
Latitude, 117
— general rules, ti8
— by circumpolar stars, 152
— by solar altitude out of meri-
dian, 152
— double altitude, 154
NAU
Latitude by a pair of stars, 154
Level, dumpy, 289
— striding bubble, 294
— plane of rotation, 294
— parallax, 294
— adjustment of, 295
— Y. 299
— staves, 301
— fieldbook, 302
Levelling, 286
— theory of, 287
Levels, 286
Longitude, 118
— by culmination of a fixed star,
137
— by solar transit, 139
— by solar hour angle, 141
— by Jupiter's satellites, 143
— by lunar observations, 144
— by lunar occultation, 145
— by lunar distance, 146
— by reciprocal azimuths, 146
— ^Jby moon-culminating stars, 147
Lyman on tacheometry, 190
Makeshifts for protracting, 360
Mannheim's rules, 362
Manuscript books, 369
Mapping, 74
— by plane construction, 75
— by conical projection, 77
— by stereographic projection, 79
— by Mercator's projection r 80
— by gnomonic projection, 89
'Mathematics,' Chambers's, 116
Mensuration by slide-rule, 267
Meridian, by sun shadows, 124
— by ordinary watch, 124
— by equal altitudes of a star,
124
— by circumpolar stars, 127
— by pole star, 130
— by solar azimuth, 132
Metric measures, 269
Molcsworth's Pocket Book, 369
Money, calculations of, by slide-
rule, 280
Moore, Lieut. W. N., survey by,
82
Morse code, 92
Nautical Almanac, 3
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Preliminary Survey
OFF
Office instruments, 359
Olive, W. I., 105
Otto Struve's telemeter, 333
Oughtred slide-rule, 362
Outfit for taQheometric survey.
370
Pantagraph, 361
Parallax, analogy with tacheome-
try, 162
Passometer, 51, 323
— traverse by, 54
— contours by, 61
— profile by, 61
Pedometer, 324
Permanent way, weight of, 277
Photography, 33
Piazzi Smyth's telemeter, 333
Pipes, discharge from, 106
Plane-table, ^18
— triangulation by, 36
— variation of compass by, 39
— : traversing by, 42
— as range-finder, 49
— compared with sextant, 35
— with stadia, 19
Plane- tabling, accuracy of, 44
— as an auxiliary, 46
Plane trigonometry, 373
Planimeter, 361
Protractor, tee-square, 360
Quays, New Yoric, 114
Railway track, weight of, 276
— American, management and
cost of, II
— estimates for, 22
— Australian, 24
— Indian, 25
— service diagram for, 262
Range-finder, 71
Range-finders, 332
Raper's ' Navigation,' 368
Reconnaissance, 30
Report, subject matter of a, 2
Road-rollers, cost of, 26
Roads, Indian, cost of, 25
— Australian, cost of, 26
Rochon's micrometer, 333
Round iron, weight of, 278
TAC
Scales, thermometer, 265
Searles's • Railroad Spiral,' 227
Sextant, siu^eying by, 53
— adjustment of, 324
— box, adjustment of, 327
Signalling, with heliostat, 93
Signals, flag, 95
Sketchboard, 318
Sketching, 33
— with aid of maps, 43
— with plane-table, 34
Sleepers, railway, 250
Slide-rule, Kern's metallic, 166
— calculation by, 241
— arithmetic by, 242, 243
— as decimal scale, 244
— for gradients, 245
— for centrifugal force, 265
— for mensuration, 267
— as a ready reckoner, 279
— described, 361
Sluicegate, pressure on, m
Solar compass, 328
Spherical trigonometry, 376
Spider hairs, to put in, 313
Spiral tramway, 230
— horse-shoe, 234
— mountain, 237
— trunk-line, 238
Spon's shilling P. B. , 369
Square, setting out a, 196
Square iron, weight of, 278
Stadia, theory of, 168
Stanley's telemetric theodolite, 308
Station pointers, 366^
Stationery, 369
Steward's simplex range-finder,
340
Stone crushers, cost of, 26
Survey, with pole and micrometer,
82
— from the boat, 83
— from the ship, 85
— with transit and chain, 201
Surveying, with chain and cross-
staff, 198, 199
Surveyor, qualifications of, i
Symbols in hydrography, 192
Tables, see 'Index to Tables.
p. 419
Tacheometer, Gribble's, 307
— Troughton & Sims's, 189
Index
419
TAC
Tacheometer, adjustment of, for
line of sight, 314
— of micrometer, 316
— registration of micrometer
values, 313
Tacheometric survey, outfit for,
370
Tachieometry, 158
— auxiliary work, 171
— survey of Hawaii, 171
— traverse by, 172
— levelling by, 175
— fieldbook for, 178
— contouring, 181
— profile, 181
— Hawaiian gulch, 182, 183
— curve-ranging, 183
Tanks, discharge from, 106
Tavemier-Gravet's rules, 362
Tee iron, weight of, 277
Telemeters, 332
Theodolites, 304
Tide gauges, 96
Tides and Currents, 89
Timber, tree, measurement of, 276
Time, reduced to arc, 137
Traffic, estimates of, 9
WYE
Tramways, cost of, 27
— service diagram for, 261
Trautwine's ' Pocket Book,' 368
Ttestles, iron, 257
— timber, 259
Trigonometry, plane, 373
— spherical, 3^
Troughton & Sims's tacheometer,
189
Two-foot rule as a range-finder, 72
Union Pacific Railway, 28
Wages and salaries, calculations
of, by slide-rule, 179
Wagner-Fennel tacheometer, 335
Warehouses of brick, 115
Water (falling), horse-power of.
109
Webb, Lieut V.B., 89
Weirs, discharge over, 106
Weldon range-finder, 336
Whitaker's Almanack, 367
Workshops, for railway, 29
Wyes and loops, 240
V
INDEX TO TABLES
TABLE FAGK
I. Traction on grades • • • 5
II. „ , 6
III. Assistant engines 6
IV. Curve-resistance 13
V. Curvature and grades on American roads . -14
VI. Curve limits for fixed wheel bases 14
VI I. Curve limits at different speeds 15
VIII. „ .. 15
IX. Running expenses affected by curves . . . .18
X. Compensation for curves i8
XI. Running expends of American railroads . 19
XII. Statistics of American railroads 19
XIII. ,. .. , 20
XIV. ,. 21
XV. ,. .. 21
XVI. Statistics of Australian roads 24
XVII. Reduction of azimuths 59
XVIII. Morse alphabet 93
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420 ' Preliminary Svy-i^ey
TABLE r.M.b
XIX. Kutter's coefficient 102
XX. Comparison of Beardmore and Kutier ... 105
XXI. Sine of polar distance of Polaris 131
XXII. Equation of time' (approximate) . . . . 139
XXIII. Functions of angles in percentage 167
XXIV. Difference between hypotenuse and base . . . 174
XXV. Length of a minute of longitude 175
XXVI. Length of a minute of latitude 175
XXVII. Sines multiplied by various distances . . . .180
XXVIII. Angles for tacheometric curve-ranging. . . . 184
XXIX. Leading values of slopes in percentage . . .248
XXX. Squares and cubes ... j ... . 249
XXXI. Railway sleepers (cross ties) . . ' . . 251
XXXIl. Howe trusses 260
XXXI II. Inches into decimals of a foot . . 270
XXXIV. Time, coinage, and linear measurement . . 271
XXXV. Shillings and pounds 271
XXXVI. Days, hours, minutes, and seconds 271
XXXVII. Weeks, months, and years 271
XXXVIII. Days, weeks, months, and years 272
XXXIX. Vulgar fractions into decimals 272
XL. Minutes into decimals of a degree 272
XLI. Seconds into decimals of a degree 273
XLII. Decimals of a degree into minutes and seconds . . . 273
XLIU. Cotangents 274
XLIV. Sines 274
XLV. Multipliers for structural iron .277
XLVI. Wages and salaries 279
XLVII. Decimal multipliers for English money .... 280
XLVIII. Indian money at par 282
XLIX. Sterling currency and rupees at various rates of exchange 283
L. Curvature of the earth 288
LI. Multipliers for hypsometry 352
LI I. Azimuth by /3 and 5 Draconis ... . ■ . - . 378
LI II. Azimuth by /3 and € Ursae majoris . . . . 378
LIV. Radii of curves in feet and degrees per 100 feet . . . 379
LV. General elements of spiral 380
LVl. Elements of No. 2 spiral . 380
LVIL ,, No. 5 , 381
LVIII. „ No. 10 382
LLX. „ No. 15 „ 382
LX. „ No. 25 383
LXI. „ No. 50 ,, 384
LXII. ,, No. 75 , 384
LXin. „ No. 100 ,. 38s
LXIV. Tables for surveying the spiral from an intermediate
point 386
LXV. Specific gravity of stones, earth, and other minerals . . 389
LXVI. ,, ,, metals and alloys 389
LXVIl. „ „ timber . 389
LXVIII. ,. .. liquid . . 390
LXIX. Multipliers for reducing specific gravity to weight of
certain volumes 390
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