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A Treatise on Engineering Field-Work by Peter Bruff. 1838


[Front Free Fly Leaf]


[Half Title]

A TREATISE
ON
ENGINEERING FIELD-WORK.


[Title Page]

A TREATISE
ON
ENGINEERING FIELD-WORK:

CONTAINING
PRACTICAL LAND SURVEYING FOR
RAILWAYS, &c.
WITH THE THEORY, PRINCIPLES, AND
PRACTICE OF LEVELLING,
AND THEIR
APPLICATION TO THE PURPOSES OF CIVIL ENGINEERING.
ALSO
PARISH AND SUBTERRANEAN SURVEYING,
WITH SECTIO-PLANOGRAPHY,
AND
EVERY INFORMATION NECESSARY TO BE KNOWN IN THE
ELEMENTARY PARTS OF CIVIL ENGINEERING;
WITH DESCRIPTIONS OF THE BEST INSTRUMENTS EMPLOYED IN SURVEYING AND
LEVELLING, THEIR ADJUSTMENTS AND METHODS OF USING IN THE FIELD.

Illustrated by numerous Plates and Diagrams.

BY
PETER BRUFF,
SURVEYOR. &c.
LONDON:
SIMPKIN, MARSHALL, & CO., STATIONERS' HALL COURT
SOLD BY
HEBERT, CHEAPSIDE; TAYLOR, WELLINGTON STREET, STRAND; WEALE HIGH
HOLBORN; AND WILLIAMS, GREAT RUSSELL STREET, BLOOMSBURY
1838.
[Printer's Imprint]


Imprinted by EDWARD RAVENSCROFT of London,
18, Tooks Court, Chancery Lane.
Udført af hærens bogbestand
den 19/7 1995
Det Kgl. Garnisonsbibliotek

Bibliotekar

[Preface]

PREFACE

THE idea of writing this Treatise suggested itself
to me a considerable time since, from knowing
the repeated inquiries that had been made for such
a book; and after perusing, I believe, every work
that has been published on the subject, I was fully
convinced of the necessity of such a work, adapted
to the present improved practice. Of the works
on Surveying, I may say, they are all elementary,
and of ante-date, no treatise, that I am aware of,
having been published since the fine mathematical
instruments at present in use have been considered
a necessary adjunct to the successful prosecution of
land surveying. Of the treatises published on
levelling, there is only one of recent date, by Mr.
Sims
, that can be referred to, and I consider that
does not supply the wants of persons seeking in-
formation
on the subject.

The few rules and suggestions that I have
thrown in at the end of the volume, I hope will be
found of service.


[page vi]

Many facts and suggestions that I wished to
embody in these pages, I have been compelled to
omit, from not having time to put them in an
intelligible shape. I have transcribed but little
from other works, although, where the subjects
are the same, there is necessarily a similarity.

In the Theory of Levelling, I have extracted
portions from Playfair's Philosophy, and Hutton
and Barlow's Philosophical Dictionaries; the article
on Refraction, from Robson's Marine Surveying;
some facts connected with the tides, from the En-
cyclopædia
Britannica; a portion of Subterranean
Surveying, from Fenning's work on that subject.
I have also carefully looked over Mr. Sims's work
on Levelling, and his Treatise on Mathematical
Instruments, to which the reader, requiring more
detailed accounts of, is referred.

The method of laying out Curves is adapted
from the Railway Magazine, to which publication
I feel indebted. I would have added further exam-
ples
in Surveying, but for want of time, am unable
to prepare them for the press.

P.B.
22, Charlotte Street, Bloomsbury,
April, 1838.


[Table of Contents]

CONTENTS.
SURVEYING.
Page
Introduction 1
General Observations and Rules to be observed 3
Method of taking Offsets 9
To Survey a Single Field with the Chain only, and method of Plotting
  the same
9
To Survey a Single Field without a Diagonal, by means of Chain Angles,
  and method of Plotting ditto
13
To Survey a Field with the Theodolite or Sextant 15
To Survey a Road with the Chain only 16
To Survey a Coppice of Wood with the Chain only 17
To take any inaccessible distances with the Chain only 19
To Measure over a Steep Hill with the Chain, so as to reduce it to Hori-
  zontal
Measurement
20
Example in Surveying, with different methods of keeping the Field-book,
  (Plates 1, 2, and 5)
21
Example in Railway Surveying with Field-book, (Plates 3, 4, and 7) 24
Of Parish Surveying 31
Practical Directions for ditto 37
Method of Keeping a Base Line straight 39
Method of Passing Obstructions, as Buildings, Woods, &c., without
  breaking your line, and continuing it onward without interrup-
  tion
39
On Subterranean Surveying, with the methods adopted in Surveying
  Colliery and other Subterranean Works
46
On the Protracting and Plotting of Surveys 50
On the Reducing of Figures, and Equalizing of Boundaries 53
On the method of Computing Areas 57
SURVEYING INSTRUMENTS
The Chain and Offsett Staff 63
Of the Circumferenter, and method of Observing with it 65
Of the Prismatic Compass, and ditto 67
Of the Pocket Sextant, its adjustments and use 70
Of the Theodolite, its adjustments and use 72
Method of Observing with the Theodolite 75
Theodolite Improved by Captain Everest, description of, its adjustments,
  and use
78
Instrumental Parallax 81
Cross-hairs of the Diaphragm 81
Theodolite Stand 81
THEORY OF LEVELLING.
The Figure of the Earth, with the Theory and Principles of Levelling 83
Method of Correcting for Curvature 85
Ditto for Refraction 87
Of the Causes which produce Atmospherical Refraction 88
Formula for Computing the Correction for Curvature 91


[page ]

CONTENTS.
PRACTICE OF LEVELLING
Page
Example in Levelling 94
Field-book for ditto and Reduction of Levels 97
Plotted Section of ditto 98
Scales (different) used in Plotting a Section 98
Example in Levelling for a Contract Section (Plate 7) with the Field-
  book
, and explanation of ditto
99
Curvature, Correction for, not applied in Practice; with the method
  adopted in Levelling Operations, whereby the Correction be-
  comes
unnecessary
103
Bench Marks, explanation and use of 104
Trial Sections, directions for taking 107
On the Assuming of a Datum Line 107
Example of Trial Sections, (Plate 8) with directions for Plotting ditto for
  the purposes of Comparison
108
Observations to be made on taking Trial Sections 110
Check Levels, explanation of, with directions for taking 110
Field-book for Check Levels 112
Example in Levelling with Cross Sections (Plate 8) 113
On Chaining, and overcoming difficulties that occur in Levelling 115
Levelling with the Theodolite 118
Formula for calculating the differences of Level 120
Field-book for Levelling with the Theodolite, with an example, (Plate 8) 122
On the method of taking Cross Sections with the Theodolite 124
Sectio-Planography, or New Method of laying down Railway and other
  Sections (Plate 8)
126
On the Choosing of a Datum Line, the differences of level of high water
  at various places, and how to obtain the mean level of the Sea
128
LEVELLING INSTRUMENTS.
Observations on Levelling Instruments 131
The Y Level, description of, its adjustments, and use 132
Troughton's Improved Level, ditto 135
Improved Dumpy Level, by Mr. Gravatt, ditto 137
On Instrumental Parallax, the cause of, and remedy 141
Of the Diaphragm, and bow to repair the Cross-Wires when broken 143
Levelling Staves 143
The Vane-Staff 143
The Sliding Staff 144
Improved Staff by Mr. Gravatt 144
Ditto, by the Author 144
Observations on the Formation of Railways.
On the Setting out of Railways 148
Method of Laying out Curves 150
Manner of Putting in the Widths 154
Of Gradients, or Rates of Clivity, showing the method pursued in forming
  an Embankment, or cutting to any rate of clivity, &c.
154
The Forming of Slopes 156
The Method of Calculating Earth-work 157
Tables of Slopes, Inclines, Curvature, &c 157-8
ADDENDA 161


[B]

THE
PRACTICE OF SURVEYING.

SURVEYING in a general sense, denotes the art of
measuring the angular and linear distances of objects,
so as to be able to delineate their several positions
on paper, and to ascertain the superficial area, or
space between them. It is a branch of applied ma-
thematics
, and supposes a good knowledge of arith-
metic
and geometry. In the erection of extensive
buildings; the forming of new roads, or altering of
old ones; in the sale or purchase of property; or, in
fact, in any alteration of, or change in, landed pro-
perty
, a survey is required.

It is most essential that the survey when made
should not only contain the correct area, but also
that every part should occupy its proper and true
position, otherwise serious evils will result. The
importance then of a survey being admitted, it
becomes the student's duty to make inquiry for
the most accurate method. It is admitted, when a
survey is made with the chain only, that numerous


[page 2]

lines are measured only for the purpose of fixing
points by which to determine the positions of the
objects to be delineated. Now by the use of angular
instruments these points are determined at once, and
far more correctly, without scarcely any other lines
being measured than are used in the detail of the
work; thus time is saved by the use of the instrument.
Many people object to the use of angular instruments
in a survey, alleging that far more correct results
may be obtained with the chain only, which is a
complete delusion--position, not distances, being
determined by the instrument (at least in land
surveying); and where the angles of a trapezium or
triangle, into which all figures should be resolved,
are taken with a good instrument, and the sides of
these figures accurately measured--we say, when
the work thus taken is laid down on paper, if the
measured distances of the lines coincide with those
protracted on paper, the position of such lines must
be correct, consequently such points, as it was the
object to delineate by means of the lines, must also
occupy their true position. It will be our object,
in the following pages, to direct the student how to
accomplish his object in the best manner, under a
variety of circumstances.


[B2]

GENERAL OBSERVATIONS,
WITH DIRECTIONS TO BE OBSERVED ON COMMENCING
A SURVEY.

SURVEYING may be performed in various ways with
the chain only, or by means of angular instruments
with the chain. In the first place, it should be
observed, that the base or principal line of a survey,
from which all the other lines diverge, should,
if practicable, be carried through the greatest extent
of property to be surveyed, so as to intersect the
principal or most intricate parts of the work; it is
also as well to carry your base line near midway
through the property, so as to leave nearly the same
quantity of work on the one side the base, as on the
other. Then your tye-lines in filling in, crossing
your base, and tying into the opposite line, will be
a satisfactory test of the accuracy of the work, and
will ensure the exact positions of the different objects
to be delineated in the survey. Chain surveying is
much more limited in its capabilities than survey-
ing
with an instrument, and certainly not so
correct. In surveying with a chain you are, in
every case, limited to one figure, a triangle; and
the correctness of the survey, and relative position
of all objects to be delineated, entirely depend on


[page 4]

the extreme accuracy with which the sides of this
figure are respectively measured. Precipitous or
enclosed ground with strong fences, render it almost
impossible to measure the distances with that de-
gree
of accuracy requisite; an error of only a few
links in the side of a triangle, only determined
by admeasurement, is not confined to that side on
which the error is made, but extends to the whole
figure; altering the position of every object enclosed
within it. Not so, if the positions of these sides
are determined with a theodolite, or other angular
instrument; for, in any case, if the angles are cor-
rectly
taken, the sides will be placed in their true
position, and the admeasurement with the chain of
these sides will determine the correctness of the
angles, so that the angles and measurement are
mutually checks on each other. Too much care
cannot be bestowed on the measurement of a base
line, for on the correctness of a fundamental base,
every part of the survey depends--whether trigono-
metrical
, or plain surveying with the theodolite and
chain, or chain only. Another important point to
be attended to is, always previous to commencing a
survey to accurately measure your chain. To do
this, tighten it on a level piece of ground, and with
a ten foot rod (correctly marked off from a two foot
carpenter's rule, or a plotting scale) carefully mea-
sure
the chain, and, if in error, you must remove a
few rings, or shorten some of the links, observing
that you correct the error equally on each side the
[page 5]

mark denoting 50 links; if the error be consider-
able
, you must distribute it equally, as near as
possible, over the ten divisions of the, chain; but
if only half a link or so, you might shorten the
first link from each end of the chain; the centre
division will then always be in the right place,
and the other divisions will be so triflingly in
error, as not to be worthy of notice. It is not the
error existing in one chain's length that is so dan-
gerous
, but that error occasionally increasing; as for
instance, suppose a chain to have expanded only
one inch in its length, and you measure a base line
of three miles through a parish or estate, the error
in the length would be 20 feet, which would make
a great diminution in the quantity of land surveyed.
We cannot impress too earnestly on the surveyor's
mind the absolute necessity of attending to this
last instruction, and never, in any instance, to com-
mence
a survey without previously testing his chain,
as it is generally, we may almost say universally,
neglected; surveyors being satisfied that their chains
were correct when they purchased them (but which
is not always the case), and that they will continue
so, or perhaps trying them two or three times in a
year. To show the necessity of attending to this
point, in a recent survey in which the Author was
engaged, there was also employed an eminent local
surveyor: at the conclusion of the survey, our
separate portions would not connect, indeed it was
very apparent on inspecting the plans that a serious
[page 6]

error existed; our chains were immediately measured,
when the local surveyor's was found to be nearly
2 feet longer than it ought to have been, immedi-
ately
showing where the error existed. When the
surveyor is: quite satisfied that his chain is correct, he
should set off its length upon some convenient spot,
and mark the extremes in such a manner as to
preserve the means of daily comparing it while in
use; thus the slightest elongation would be imme-
diately
detected, and easily rectified. While speaking
of chains, it would not here be amiss to recommend
a 100 foot chain, or a 132 foot chain, divided into
double links, in every case, in preference to Gunter's
chain of 66 feet; as you can measure a line with
much greater accuracy with a long chain than a
short one, for the same reason that you can measure
a line on paper more correct with a long scale than
a short one; viz., that you have not so many lengths
to measure off, each repetition producing an error,
which, although small in itself, when multiplied
by a large number produces a very sensible quan-
tity
; which any one might prove by measuring off
a few lengths of a short scale on paper, and after-
wards
applying thereto a much longer one. Also
in measuring over a wall or thick hedge, through
which you cannot pass the chain, the angle
formed with the long chain would be more obtuse,
and approach nearer to a horizontal line than if
a short one was used; as in the latter case you
would lose several links, the angle formed with
[page 7]

the horizon being greater than the former, and if
you allow for it by bringing the chain a few links
forward, it would be at random, and consequently,
very uncertain. Rapidity in plotting is also another
advantage attached to the use of the foot or double-
link
chain, and a great saving of time in reading off
the distances in the field, although, for computing
the quantities, you must still come back to the
chain scale of 66 feet for convenience in reducing
to acres. Though it is generally held bad to plot
with one scale, and calculate the contents with
another, yet we are convinced that much more
accurate results would be arrived at if adopted;
always observing, that the scales you plot and cal-
culate
with are both of the same material, either
box or ivory, although we would recommend box,
as being less liable to be acted on by changes in the
atmosphere.--(See article on the Chain.) A custom
is also very general, among some country surveyors,
of only using nine pins or arrows instead of ten,
making a huge mark or hole in the ground in
place of the tenth; a custom which cannot be too
much condemned, as being incompatible with cor-
rect
admeasurement; as, in the next chain forward,
in place of holding to a pin, you hold over a large
hole, generally a link or two broad, which, on a line
of any great extent, would introduce large errors.
All lines measured over steep ground must be
reduced to the horizontal measurement, otherwise
the work will not plot, and there will appear a
greater quantity of land than there really is: this
[page 8]

must always be attended to. Generally where
hedge and ditch divides property, the brow of the
ditch is the boundary; hence the advantage of
the surveyor passing his lines along the ditch side
of fences. But the brow of the ditch is not always
the boundary, it being in some districts the roots
of the quicks, or the foot of the bank; therefore the
necessity of making inquiries as to local custom:
the width taken for a ditch that is partly filled, is
generally about 6 links, and for the bank about 9.
In some places, 3 feet from the roots of the quicks
are for the breadth of the ditch, in some
4, in some 5, and in some 6; but 6 links are
commonly allowed for ditches between neighbour-
ing
estates, and 7 links for ditches adjoining
roads, commons, &c. Where a boarded or post
and rail fence comes in the surveyor's way, coupled
with hedge and ditch, he will often be at a loss to
know the precise boundary of the property--when
this is the case, observe from which side the nails
are driven (it being generally understood that nails
are driven homewards) ; if on the ditch side, the
brow of it will be the boundary, if on the other side,
the fence itself: where a fence changes from one
field to another, correctly mark it on the plan--the
breadth of the hedge and ditch must be shown,
which, as we before observed, is generally taken at
15 links. When plotting a plan, the surveyor should
be careful always to have the North upwards, and
the writing from West to East.


[page 9]

Of the Method of Taking Offsets.

A, c, d, e,f, g, h, being a brook or crooked hedge.

From A measure in a straight direction along the
side of it to B; and in measuring along the line
A B observe when you are opposite any bends or
corners of the hedge, as at i j k, &c., and from thence
measure the perpendicular offsets, i c, j d, &c., with a
tape or offset staff. When these offsets are marked
off from the distances on the line A B, a line drawn
through their extremities will represent the crooked
hedge A c d, &c.

To Survey a single Field with the Chain only.

Having carefully read the general observations,
and the method of taking offsets, directions will now
be given for surveying a single field with the chain
alone. As before observed, in a chain survey you
are confined to one figure, a triangle; and the
correctness of every part depends on the extreme
accuracy with which its relative parts are measured,
as well as the judgment displayed in the arranging
or laying out the sides of this figure on the ground,
which should always, as near as possible, be an


[page 10]

equilateral triangle; for if the angle at the apex be
either very obtuse or acute the most trivial error in
the admeasurement of any one of the sides will
materially alter the figure, and consequently the
area. As it is better to proceed gradually, we will
commence with a single field, as the same system is
pursued throughout, whether it be a small enclosure
or a large estate. The first operation is in the
arranging the ground to be surveyed, either into
one or more triangles: that is to say, you station
yourself at one corner of the field, and having
erected a conspicuous mark at your starting point,
look to the opposite corner, and if no natural mark,
as a tree, house, or any other object, exist in the line
you intend measuring, you must erect one; having'
done which, commence chaining from the first
mark in the direction of the second, always ob-
serving
that you measure in a perfectly straight
line. Leave marks on this diagonal or principal
line for the purpose of measuring tye or check
lines to the apex of the triangles, to ensure the
accurate measurement of the sides. Note in your
field-book at what distance from your station or
starting point you put down these marks, or false
stations as they are termed. When you arrive at
the opposite corner of the field, put down another
mark, and from this station commence measuring
a line by the Side of one of the fences, without
regard to the angle it makes with the preceding
line, taking offsets to all the bends in the hedge
[page 11]

as you proceed; put down a mark at the end of this
line, as before, and commence measuring a new one
to the station first started from, also taking offsets as
you go along.
Now measure the tye line from the apex of the
triangle or junction of the side lines to one of the
previous marks left on the diagonal, which will
ensure the accurate measurement of the sides. The
same operation will be repeated on the other side of
the diagonal, when the survey of the field will be
complete.



The preceding sketch shows at once the method
of procedure in surveying a single field: A B
being the diagonal or base line, E D the false
stations, left when measuring the diagonal; B C one
side of the triangle, commenced from the termina-
tion
of the diagonal, and C A the remaining side of
triangle, which finishes the figure; having arrived
at the point from whence we set out. We have
now to measure the line C D, which verifies the
measurement of the triangle A C B. The same
method of procedure is adopted on the other side of


[page 12]

the line A B, respectively measuring the sides B F
and F A, and the tye line F E; or one false station
may be left on the diagonal, and the continuous
dotted line F C measured. The next operation is
to lay down these lines on paper; to do which, fix
on a scale to which it is intended plotting the field,
as one, two, or three chains to an inch, according
to circumstances. This determined on, draw the
line A B in any position, and measure off the length
with the scale, being careful to mark the position of
the false stations E and D; then take the length
B C with a pair of compasses (beam compasses are
the best) and describe an arc of a circle from B as
a centre; also take the length A C in the same
manner, and describe an arc from A as a centre:
the intersection of these arcs will fix the relative
positions of the lines A C and B C, through which
point draw them from A and B. The same method
of procedure would be observed on the other side of
the diagonal in laying down the lines A F and B F.
Then apply the scale to D, and observe what it
measures to the point C; also from E to F. If
these distances are the same as measured in the
field, it shows that the measurements were correctly
taken; if not, it shows that an error must have been
committed, either in laying off the lines on the plot,
or measuring them in the field; in which case it
must be gone over again until it proves satisfactory.


[page 13]

To Survey the same Field, without a Diagonal,
by means of Chain Angles.

This method is not at all advisable, although
practised to a considerable extent; there is a little
time saved, but the chances of error are con-
siderably
multiplied. Commence, as previously
directed, at one corner of the field; but instead of
measuring to the opposite corner, go down the
longest side, and within 200 links, or some con-
venient
number, of the end of the line leave a
mark. Now commence a new line, as before
observed, without regard to the angle it makes
with the preceding; and at the same distance
on the new line from the station or point at the
end of the last line leave a similar mark. Now
measure the distance between these two marks,
noting exactly the measurement, even to half a link;
or continue the line to the end of the fence, and
measure it afterwards. The same method may be
pursued at the other angles of the field, although
not absolutely necessary.

The following diagram will illustrate the pre-
ceding
method of surveying a field and the manner
of plotting it: --Commence at the angle A, and
measure down one side of the fence towards B, and
at 200 links, or some convenient number, before
arriving at B, leave a mark, a; when arrived at B


[page 14]

commence a new line, B C, and at the exact dis-
tance
from B, as you had previously left the mark a,
put down the mark b, and accurately measure the
line a b. Then continue the line to C, where it

would be advisable to pursue the same method,
although it is not absolutely necessary beyond being
a check on the work.
We think we have now sufficiently shown the
method pursued in surveying an enclosure without
a diagonal; and will proceed to plot these lines.
First: draw the line A B in any position, and with
the scale to which it is intended to plot the en-
closure
, accurately mark off the distance from A to
B; then take the distance from B to a, or B to b,
which is the same, with a pair of compasses, from
a much larger scale, the larger the better, and from
B, as a centre, describe the arc a b; then with the
compasses take the distance a b, and put one foot in
a, (observing also to take this distance from the
large scale) and with the other make a fine puncture
in b, or describe an arc cutting that point, then
with a fine pointed pencil draw in the line B - C,
passing through the point b. Mark off the distance

[page 15]

B C from the same scale A B was marked from;
then take the distance C D with a pair of compasses,
and from C, as a centre, describe an arc, passing
through D; also take the distance A D, and describe
another arc, cutting the former one in the point D;
then, with a fine pointed pencil, draw the lines
C D and A P to the point of intersection: the
figure will then be complete, or you may take an-
other
chain angle at C, or any one of the three
remaining angles, similar to the one at B; which
would be advisable; for the following reason
In measuring the diagonal a b at the angle B, an
error may have been committed in reading the
chain; if so, you have no means of detecting it:
but if a chain angle is also measured at C you
will have a check on the work, as the points P and
A being fixed by the chain angles the measured
distance P A will not at all come in if an error
has been committed, but will either fall short or
beyond those points.

The Method of Surveying a Field with a
Theodolite or Sextant.

Commence on the longest side of the field, and
measure quite round it, as in the preceding example,
taking offsets at all bends in the fence as before.
In the diagram (see last example) commence at A,
and measure to B, leaving a mark at the point A,
whence you started. When arrived at B, with the


[page 16]

instrument look along the line B A, and also to a
mark at C; read off on the instrument the angle
formed by the line B C with the line B A, and
carefully note it in your field-book, measure the
line B C, leaving a mark at B. When arrived
at C, observe the angle formed by the line C P
with C B; measure C D, and when arrived at P
observe the angle formed by P A with P C. This
will finish the figure; and when the last angle
C P A is protracted it will pass through the
point A, whence you started, and the measurement
of the line P A will be the same on the paper as
measured in the field, if the angles and measure-
ments
have been correctly taken.--For the method
of observing with the theodolite and sextant, and
plotting the lines, see the directions given under
those heads.

To Survey a Road with the Chain only.

Suppose A B C D to be a piece of road re-
quiring
to be surveyed, commence at A, and if no


[page 17]
[C]


natural mark exists in the line A B, set up one,
observing to get the longest possible sight along the
line of road, and taking offsets to all bends in the
fences on each side; likewise to all houses and
buildings, and note where a fence runs up from the
road, as at 1, 2, 3, and so on. Before arriving at
B set down a mark, as at C, otherwise continue the
line 50 or 100 links, as may be convenient, beyond
B to a; then commence a new line from B towards
C, and at b, the exact distance from B as you had
previously set down a, leave another mark, and accu-
rately
measure the distance, a b; then continue the
line towards C, pursuing the same method at C and
each subsequent bend in the road. In plotting the
road the same method would he pursued as directed
for plotting a field surveyed without a diagonal.
The above method would not have much pretensions
to accuracy; the angles A B C, B C D, &c., should
be determined with a theodolite, or sextant.

To Survey a Coppice of Wood with the Chain only.

Let A, B, C, D, represent a coppice of wood,
very much overrun with brushwood, so that it cannot
be measured through. In that case take an angle
with the chain by measuring 100 links from A to e,
keeping in a line with A D, and put in a mark at e;
then measure 100 links from A to f, in a line with
A B, and measure the distance from f to e; then


[page 18]


measure, on the outside of the wood, from A to B,
and continue the line beyond B to g, which is 100
links: also measure out 100 links to h, in the line
B C, and measure h g. Now measure B C, C D,
and D A.

To plot the work draw a line at pleasure, to
represent the line e A D, then put one foot of the
compasses in A, and sweep an arch from e to f, after
having taken off 100 links from a large scale. Then
take off the distance, e f, from the same large scale,
put one foot of the compasses in e, and bisect the line
at f; then lay a straight edge on the point of inter-
section
at f and the angle of the wood at A, and
draw the line A B, and lay off the distance A B,
with any scale you like to plot your work with,
which may be much smaller than the scale used for
laying down the angle. Then take the distance
from A to D, which lay off upon the line from
A to D by the same scale you used from A to B.
If the angle is taken at B, which should be done, the


[page 19]
[C2]


same process is repeated in laying off the angle as
at A; if not, take the distance B C, put one foot of
the compasses in B, and describe an arch at C, then
take the distance from D to C, put one foot of the
compasses in D and bisect the arc at C, which will
give the exact shape of the wood. Of course the
necessary offsets must be taken, as each line is
measured. By taking the angle at B a check is ob-
tained
on the work, as it fixes the point C, and if
that point is laid down in any other position than
the right one, the line C D will not come in, but
will be too short or too long, thereby proving that
an error has been committed.

To take any Inaccessible Distance with the Chain,
as the Width of a River, &c.

Suppose in the annexed sketch you brought
your survey up to the river, the width of which


[page 20]

was necessary for you to know:--let D A be your
base, or some line well connected with your survey,
bring it down to the river edge at A, at which leave
a mark, also set up a mark in a line with D A at
E, then measure out any distance, as A B, at which
set up another mark; then retire back to C, and set
up a mark in a line with B and the object on the
opposite side of the river, at E: measure B C, and
also C P; the distance, D A, you had previously
obtained. If you have set out the line D C parallel
with A B you may easily calculate the distance
A E, in the following manner:--say, as the difference
between A B and D C is to D A, so is A B to A E
the distance required.

If you wish to lay these lines on paper so as to
show the method pursued in crossing the river, it
will be necessary to measure either of the diagonals
A C or D B, which will divide the figure into two
triangles; plot these triangles by intersection, as
before directed, then produce D A and C B, until
they intersect at E, apply your scale to A E, and it
will give you the width of the river.

To Measure over a Steep Hill with the Chain, so as
to reduce it to Horizontal Measurement.

The method adopted in measuring over a hill
to reduce the line to horizontal measurement, is by
taking short lengths of the chain. lithe hill is not


[page 21]

very steep, take half a chain's length, or if very steep,
take 25 links, or less. The foremost chainman in
ascending a hill holds the chain quite to the ground,
while the hindmost chainman takes 50 links, or as
much as the ground will permit him, and holds it
up over the mark, and as near level as he can guess.
In descending, the reverse will be the case, the hind-
most
chainman holding to the ground, and the
forward man elevating the chain until he thinks it
level, and then dropping his pin.
Many surveyors who adopt this method in
crossing hilly ground have a plumb line, which the
man elevating the chain holds in his hand. By
this means you can tell exactly, when ascending, if
the chain is over your mark, and, in descending,
where to put in a pin. This method may be
successfully practised where the ground is at a
moderate inclination, but when very steep the ver-
tical
angles should be taken with a theodolite, and
the requisite allowance made.--(See description of
the Theodolite.)

Example in Surveying, with different Methods of
Keeping the Field-book.

To survey the annexed three fields the same
method should be observed, whether an instrument
was used or the chain only. You will commence at
A (see Plate 5 where you must set up a mark, that


[page 22]

being the extreme boundary, and measure a straight
line to some object at B, which is the longest line
that could be obtained through the property, leav-
ing
marks, which are termed false stations, at a
and b. You must then go off with a line to D,
leaving a false station at c. When you arrive at
about D you will be able to see the mark you first
set up at A, to which you must measure a line from
the termination of the last at D, leaving another
false station at d. You will then have enclosed all
the property on one side of your base. You must
now commence again from A, and measure a line
to some object at C. When you come to about e
you must put down the mark e in a line with the
two false stations at b and c; when you arrive at C
you will be able to see your mark at B, up to which
you must measure. This will finish your great
figure, enclosing all the property. You have now
to fill in the detail, to do which measure e, b, c,
which is a straight line, cutting the base at b; when
you arrive at f, which is in a line with the two
false stations at a and d, you must put down a
mark, f; you must also note exactly the distance
from e to b, and continue on the chainage to c.
Then measure f, a, d, correctly noting the chainage
when you arrive at each false station. These in-
ternal
lines, it is evident, will form a check on the
accuracy of the triangles A B D, and A C B.

Of the field-books we need only to refer to them,
and leave the reader to make his choice. The field-


[page 23]

sketch we prefer, as being less liable to error, and all
the minutiæ of the survey being sketched in at the
time, so as greatly to assist the memory in plotting.
It may be thought by many persons, that the sketch
field-book is only applicable to the survey of a few
fields, but with a little practice it may be used in
surveys of any extent. In the common method of
keeping a field-book, it is absolutely necessary to
plot the work as it proceeds, which is advisable in
all cases,
but in the sketch field-book it may remain
for months or years, and then be plotted with as
much facility as at first. The Author has a sketch
field-book of an intricate district of some thousand
acres, comprehending one of the suburban villages,
which was non plotted for some months after being
surveyed, but was then done with the greatest ease
and expedition.

If an instrument had been used in the above
survey, the angles A B D, B D A, would have been
taken; also D A C, or B A C, and A C B. The
angles formed by the internal lines need not be
taken, as their positions would be correctly deter-
mined
by the outer lines. But without an instru-
ment
, if an error of 20 or 30 links was committed
in measuring the line A C it would not be detected,
the point C being determined by intersection, except.
by measuring a line from the apex of the triangle at
C to some part of the base, and the same on the
other side; or a line should be measured from
C to D, (see Parish Surveying) which would occupy


[page 24]

considerable more time than taking the requisite;
angles, neither would it be so correct. The same
liability to error and uncertainty of detection would
exist in the measurement of the other lines: but
where an instrument is used to determine the posi-
tion
of the several lines, and every line measured
and tied to the base, it is impossible but the whole
must be correct.

EXAMPLE IN RAILWAY SURVEYING,
Showing in what manner the Survey for the Rail-
way
was performed, and an Explanation of the
Field-book.

The base-line A B C was first carefully measured,
of which the portion, A B, is only given in the field-
book
. At the commencement of the base, at station
A, was set up a flag, natural mark being visible at
C, in a line with the intended base. Before arriving
at the first fence a false station was put down a, and
the exact chainage (540) was entered in the field-
book
, and also at the crossing of the fence; farther
on another false station b (1,330), was put down, the
crossing of the fence being noted as before; and in
a similar manner were put down the other false
stations c (2,240), d (3,380), e (4,080), and, lastly,
that at B (4,720).


[page 25]

Many surveyors, after measuring out a certain
length of base, as A B, would commence at B, and
work back to A, but as that method would perhaps
perplex the beginner, it will be better to return to A
to commence. The first thing to be done is to take
the angle formed by the line A f with the base; but
you must take care to choose your mark at f, so
that you can produce or back your line to g, so as
to form one straight line g A f, then put down a
false station at g, from which commence your chain-
age
to f, taking the necessary offsets, and accurately
noting the distance g A, at the crossing of your base;
when you arrive at the top of the field you must fix
your station at f in such a position as to get the
most favourable and longest line, by the side of the
next fence or fences, as f, h, i, j; enter the false sta-
tion
at f in your book (580), and continue the line
beyond f, until it cuts the fence; which it does at
600. Then take the angle formed by the line f h i j
with the line you have just measured f a g, which is
108° 10', and measure the distance: on arriving at h
(480), put yourself in the best position for passing a
line down the adjoining fence, as h a l k, take the
angle formed by this line with your object at j,
which is 96°, measure this line, which cuts your base,
at a, the exact chainage of which you must notice
(it is 240), and proceed on towards k; at l you
must put down a false station, which will presently
be of service to form the line l m n, to take up the
adjoining fences. From k you will measure a line

[page 26]

to g, which will finish the first field. It will be at once
seen that there is no occasion to take the angle h k g,
as the two points, k and g, being correctly fixed by
the former angles and measurement: the measured
distance, k g, will not at all come in if the slightest
error has been committed. Now return to h, and
continue the line on towards j; when you come to i
(1,220), take another angle j i m (83° 50'), cutting
the base at b, measure this line, noting the chainage
at b (220), and continue it on to m (615); at which
leave a mark. Return to i, and continue the line on
towards j, until you can fix your false station, as at j,
to command the line j a, cutting the base at c.

Now you may either measure the angle formed
by the line j n, off the line j f, or you may proceed
to c, and take it off the base,--which would, perhaps,
be advisable; then from c (2,200 on the base), take
the angle A c j (79° 40'), extend the line a conve-
nient
distance to o, so as to command the line o p q,
and measure from o to n, noting the chainage at j
and c; when you come to n you will find yourself
in a line with m and l, the two stations you had
previously left. Measure n, m, l, noting the chain-
age
very exactly at m and l. Now return to o, take
the angle n o q (83° 18'), and measure the line o p q,
leaving a mark at p (1,135), proceed on to q (2,535),
and at q take the angle o q a (88° 15'), measure
q s, noting the chainage, cutting the base at B:
leave a mark at r (675), so as to form a line r e t,
and proceed on to s (1,000).


[page 27]

If it had been thought necessary to give the
field-book farther than this road, a false station
would have been left at s, and the work carried on
to x on the base, the line s q being extended to y;
the angle s y z would be taken, and a false station
made at z; the angle y z x would be also taken,
and the line z x extended to u, so as to form a station
in the line u s v, the angle being taken also at u.
But to return to s; take the angle B s v (87° 20'),
and measure s v; but when you arrive near the end
of the line you must fix your false station v, so as to
range with the others previously put down at d and p,
measure v p, and about w you may or may not leave
a mark; enter the chainage (445), cutting your base
at d, put down a false station at t, in a line with
e r, and continue it on to p (915), which is close.
You may now return to e, and measure a line e a,
which will cut the line v p somewhere about w, and
continue the line on to n (1,815), which is also close.
Return to t, and measure t e r, which will finish the
survey up to the road. The other part, on to C,
would be continued in the same manner; the lines
measured in the field being marked on the plan.

When the student becomes conversant with the
above method much of the labour would be abridged,
as he would run his lines backwards and forwards,
in several instances, without walking to a distant
point to resume his work. Neither is it absolutely
necessary
that so many angles should be taken, as
those at h, i, and c, might have been omitted, the


[page 28]

line f j being fixed by the angle taken at f. But the
advantage of taking these angles is very great, as it
ensures the accuracy of the work; as, for instance:
suppose in fixing the false station at i, you made
an error of 10 or 20 links in reading the chain, by
taking the angle at i this would be immediately
detected, otherwise the distance i b, would appear to
plot correctly; for the false station i, being moved
forward or backward 10 or 20 links, would not make
a sensible difference in the length i b: this error in
the distance i b, would therefore not be detected, but
on producing it to m the error would be great.--
(See the Plan.)

Suppose the false station at i moved forwards
20 links to the black line, the distance i b would
appear to be correct, and the line produced through
b to * would be also assumed as correct, in conse-
quence
of the measurement, i b, answering to the
position of i on the plan, on reference to which it
will be seen how much the position of m would be
altered: it is true the line m l would correct this,
but suppose the error of 20 links to be committed at
h, the station at l would then be as much in error as
that at m, and the line m l would plot correctly as
to length; but it would be said the measured
distance, k g, would ensure the accurate position of
k, but in the general method of surveying; the dis-
tance
, k g, would be necessary to fix the point g,
which would consequently be as much out of its
proper position as the other stations, therefore k g


[page 29]

would plot correctly, and the whole survey be twisted
out of its proper position.

We do not mean to assert that a person making
the above survey without an instrument would lay
out his lines in the same manner; therefore the
inference drawn of an error committed at i, extend-
ing
back to g, may by many persons be set down
as erroneous; but, although the same distribution
of lines would not be observed, the consequences
pointed out would ensue wherever an error was
committed, which, if of small amount, it would be
almost impossible to detect.

The field-book is a sketch made as the work
progressed; the several lines being entered as they
were measured, and the offsets in the order they
were taken. The same method is observed, as in
the common system, by commencing at the bottom
and writing upwards.

This survey was made with a sextant--if the
angles had been taken with a theodolite, the first
thing determined would have been the angle formed
by the base line with the magnetic meridian (for the
method of doing which see directions for observing
with the theodolite) which it is always desirable to
ascertain, as your theodolite being set to this read-
ing
at any subsequent station on your base will point
out its direction; also when arrived at the end of
your base, if of any considerable extent, it would be
desirable to bisect the beck station, and note the
bearing, the difference would be exactly 180° if you


[page 30]

have measured in a straight line; it also often
happens when surveying near a town, that you are
compelled to take up a long line of road running
out from your survey with many bends in it, your
bearing will here be of service, as probably you will
not be able to connect the extremity of this road
with any other part of your survey than that you
started from, the position of every part of the road
(which is generally of importance) depending on the
accuracy of your angles. In this case there are
various ways of checking the work, but the most
simple and generally practised, where you are
pretty certain of the correctness of your angles,
is to note the bearing of your last line, and when
plotting to lay off this bearing, and with a parallel
rule bring it down to the bearing taken on your
base, with which it should exactly correspond.

It will also be found of great advantage in
measuring a base line to have its bearing, otherwise
(except in a very open country) it will be almost
impossible to measure it straight, without first
ranging it out with poles, as in passing through
a plantation or orchard you are pretty certain to
lose sight of your marks, you have then nothing
but your bearing to satisfy you that you are in the
right line.--For further directions in measuring a
base, and in general surveying, see observations on
parish surveying.


[page 31]

Of Parish Surveying

Previous to appending any remarks or instruc-
tion
of our own on parish surveying, we think it right
to direct attention to the following extracts on ties
subject from Captain Dawson's invaluable Report
to the Tithe Commissioners: it is necessary, he
observes, to determine the area of the whole parish
by some means, which make the correctness of that
area independent of the result obtained by summing
up the contents of each enclosure, minute errors in
many of which would escape observation, if not
checked by comparison, with the correctly ascer-
tained
whole, It is essential, in fact, to arrive at the
total area of the pariah by direct admeasurement
of the apace included within its external boundary;
and the simplest and cheapest means by which a
survey and plan may be made for effecting this
object appear to me to be as follows:--

1st. To measure two straight lines through the
entire length and breadth of the parish.

2nd. To connect the ends of those lines by
means of other measured lines: and

3rd. From those connecting lines (by measured
triangles and offsets) to determine the entire pariah
boundary.

The true area of the parish may then be ob-
tained
by calculation from the measured distances,


[page 32]

and by the admeasurement of the included space
upon the plan.

Lines of the description herein proposed to be
measured are ordinarily used by surveyors in the
construction of their plan, but are not always shown
on the finished map; I propose to retain them per-
manently
for purposes which will presently appear.

The object and application of these lines will
be better seen by reference to the diagram beneath,
which is a rough sketch of a parish to be surveyed.

The two main lines which I should recommend
to be measured through it are marked A B and
C D; A C, C B, B D, D A are the connecting
lines, a a a, are the offsets, or perpendicular


[page 33]
[D]


distances of the several angular points of the parish
boundary from the measured lines.

Now, if the main lines A B and C D be mea-
sured
accurately, and their true lengths from the
point (O), at which they cross one another, be laid
down upon the plan, it will be seen that the connect-
ing
lines A C, C B, &c. will form an efficient check
on the general direction of the two main lines with
reference to one another. A satisfactory check on
the lengths of the several lines will, by the same
meant be afforded; for as the points A, C, B, D, are
in each case determined by the intersections of three
lines, an error in any one of these lines must im-
mediately
be discovered.

Thus the true relative position of four extreme
points (A, C, B, D), in the parish boundary will be
obtained, and such portions of the boundary as fall
within the ordinary range of offset-distances from
the connecting lines (A C, C B, &c.) will also be
determined, and may be laid down in their true
positions.

The more remote parts of the parish boundary
may be determined by means of the triangles
(T, T, T), the sides of which (E F, G H, K I, &c.)
being prolonged on the ground to intersect the
main lines A B, C D (as they do at M, N, P, &c.),
may be laid down correctly in position and direction
upon the plan. By this simple process the whole
boundary will be determined, and the total area
may then be ascertained.


[page 34]

Among the objects to be particularly attended
to in practice, is that of reducing the lines, mea-
sured
over steep slopes in hilly districts, to the
horizontal plane.

This demands especial mention, because anise
inattention to it is not unusual, though the necessity
for such reduction is well known to practised sur-
veyors
, and all should be alive to the importance of
using a theodolite, spirit-level, or other assured
means, in the measurement of lines over hilly
ground, for determining the exact allowance to
be made. Without this reduction of the lines
they cannot be laid down in plan upon a flat
surface, and distortion of the outline must inevitably
result.

Care, of course, most be taken in all cases to
measure the lines straight to the points desired;
sad this will require more particular care in a
mountainous, rocky, marshy, wooded, or thickly
inhabited country. The expedients in use among
practical surveyors will of course be resorted to for
overcoming any difficulties which may attend the
measurement of these main lines, and the theodolite
offers a never failing resource in all cases where a
departure from the direct line is inevitable.


[page 35]
[D2]


Observations

The lines which have been described as essen-
tial
to be surveyed, should, in all cases, be marked
upon the plans. They should he drawn in red ink,
in order to distinguish them from, and prevent
their interfering with the lines of fences, &c. and
the length of each line in links should be marked
in red figures upon it. Lines measured in the
direction of external objects, should be drawn out
to the margin of the plan, and the name of the
external object should be written upon the line
thus:-


[page 36]

The main lines should be selected, as much as
possible, with reference to permanent well-defined
objects, such as churches, &c. In other cases it
will be desirable that the extremities of the lines (or
of some of them at least) should be marked, and
preserved on the ground by stones or posts, or by
trees, planted there so as to admit of the points
being referred to at a future time.

The parish boundary should be shown, in all
cases, by a dotted line; and when it passes along
the middle of a fence, the dots should be drawn on
both sides of the fence, thus:-

When a road forms part of the boundary of a
parish, both fences of the road should be shown;
and it will be desirable also to mark the abutments
of other fences upon the outer fence of the road.
The same remark will apply to rivers generally; and
in Lincolnshire and other fen districts, to droves
and the drains by which they are bounded, &c.

When a parish boundary passes through a field
or other enclosure, without being defined by a fence,
the whole of such field or enclosure should be shown
on the plan, with the parish boundary, marked by
a dotted line, passing through it. The area of the
included portion only of such field or inclosure will
appear in the schedule; but the area of the excluded


[page 37]

portion may with propriety be given on the plan,
and be marked as belonging to the adjoining
parish.

In all cases of fences, the actual boundary line
of the adjacent properties should be marked upon
the plan, whether it be the central line or the side
of a hedge, ditch, wall, bank, &c.; and when the
fence belongs entirely to one property or the other,
that should be indicated by the proper mark.

The plans are to be drawn to the scale of three
chains to one inch, to admit of the correct com-
putation
of the contents of the several lands. And
the ordinary usage should be observed with regard
to placing the north towards the top of the plan;
writing the name of the parish, as a title, with that
of the county in which it is situated, and adding
the name and address of the surveyor, the date of
performance, the scale, and the total contents.

The extracts we have made from Captain
Dawson
's Report are so much to the point, as ne-
cessarily
greatly to abridge our remarks, which will
be confined simply to the guiding and directing
persons in the measurement of such lines as are
therein recommended. In the first place, then, we
will endeavour to point out the best and most
correct method of measuring the principal base
through the entire length of the parish. Previously
to commencing or arranging the work, the surveyor
should, if possible, procure an old map of the parish,
which, however incorrect it may be, will still serve


[page 38]

generally to point out the best parts of the parish
through which to pass his lines: but whether this is
obtained or not, let him be in no hurry to lay out the
work, but look carefully to the consequences result-
ing
from transverse lines running through various
parts of the parish--whether their extremities can
be easily connected, and if they intersect any par-
ticular
or important points within, or are on, with
any without the parish. This is particularly to be
attended to, as it would greatly facilitate the tracing
of the boundary at any future time.

We will suppose, then, the surveyor to have
decided on the point of commencement and direc-
tion
of the base, which, if possible, should be on,
with some conspicuous permanent mark, without
the bounds of the parish, as a church, windmill,
house, or such like. At this point set up your
theodolite, and ascertain very exactly the angle
formed by this line with the magnetic meridian;
then take angles to several conspicuous objects
around, which would serve hereafter very accu-
rately
to determine the point. At this spot erect a
pole, very perpendicular, and commence the mea-
surement
of the line; but before proceeding further,
it cannot be too strongly enforced on the surveyor's
mind the absolute necessity of extreme exactness in
this part of the operation; for which purpose a
much longer chain is recommended than that
usually adopted.--(See remarks on the chain.)

At about every 5 or 10 chains, it would be


[page 39]

advisable to drive a stake firmly into the ground,
with the chainage inscribed thereon in Roman cha-
racters
: thus, if at every ten chains, call the first
ten 1, at twenty it would be 2; or if left at every
five chains, at five it would be 1, at ten it would be
2, and so on. The reason of this will be presently
apparent.

The roads, rivers, brooks, fences, &c. as they
are crossed, should be very carefully noted; but in
this stage of the proceedings it would be quite
useless putting down false stations at nearly all the
fences, as in common surveying. Offsets, if within
distance, should be taken to all conspicuous objects.
At certain prominent points, as you pass along, set
up poles; these will serve to keep you in a direct
line, even if you entirely lose your forward object.
Your forward chainman must at each chains length
plumb back to those poles you have erected, and by
keeping them exactly in a line, you need not fear of
departing from your true course.

If you come upon a house, or gentleman's
pleasure ground, through which it is impossible to
measure a line (but this always should be avoided,
if possible:) the means of overcoming the difficulty
will be found by referring to a chapter on the sub-
ject
, in the section devoted to levelling; but the
most ready and correct method would be, very
carefully to measure an angle with the theodolite,
either to the right or left of your line, of exactly
60°, and measure out any length until clear of the


[page 40]

obstruction; then take another angle of exactly 60°,
and measure the same distance as the last line.
This will bring you to the exact spot you would
have arrived at, could you have continued your line
onward without interruption. You will thus have
measured two sides and angles of an equilateral
triangle. The remaining angle and side will be
the same; that is, the angle will be 60°, and the dis-
tance
, if it could be measured through the obstruc-
tion
, would be exactly the same as that of either of
the measured sides; or a line forming any angle
with the base (but which must be determined)
being measured clear of the obstruction, and an
angle taken at the extremity so as to cut the base
beyond the obstruction, the length of this side and
of that passing though the obstruction may be
easily calculated by plane trigonometry. This
difficulty overcome, and the continuous distance
entered in your book we will proceed onward; but
the poles you have set up behind are not visible,
neither probably is your forward mark. To extri-
cute
yourself front this dilemma, measure the sup-
plementary
angle of 120° from the last measured
side of the equilateral triangle; this will direct you
in the precise line; but to verify it, ascertain its
bearing, which should be the same as at first: and
in this manner you will be able to overcome all
similar obstructions.

We will now suppose the surveyor arrived at
She extremity of his base, where he must set up his


[page 41]

theodolite, and take the angle of one of the side
lines, which should not be very oblique, but as near
45° as circumstances will permit, and, as directed
for the base, should, if possible, be in a line with
some natural mark. To measure this angle, which
is most important, with the requisite degree of
accuracy, it should be repeated several times, and a
mean taken as the correct angle. Set up a pole at
the extremity of the base, and measure this line in
a similar manner as directed for the base, putting
down stakes at intervals. When arrived at the
boundary of the parish, or so far as may be desira-
ble
, set up the theodolite and measure an angle
from the last line to some object on the opposite side
of the parish, transversely, to your base, and another
angle to the first station at the commencement of
the base line: set up a pole at the exact spot from
whence the angles were taken, and measure the
transverse line, which can be measured in a per-
fectly
straight line, by adopting the Same means as
already directed. When this line is measured up
to the crossing of the principal base, stop, and, from
one of the stakes previously left, measure up to the
exact spot at which you cross, and enter the two
distances in the field-book. Continue the mea-
surement
of the transverse base (driving in stakes at
regular intervals as before), to the extremity of the
parish, or so far beyond it, as by tye-lines, measured
to the extremities of the principal base, the entire
parish can be circumscribed; or leaving out such
[page 42]

small portions only, as may be determined by small
triangles from these principal lines, similar to those
marked T, T, T, in the diagram.

In throwing out triangles to enclose any part
of the parish that may be without these aide lines, if
an instrument be used, there will be no occasion to
extend them back to the base, without the figure
should be very large, or the internal lines can be
used for other purposes, than merely to verify the
position of the figure; but where a chain only is
used, it is indispensable to the correct fixing of
the figure that those lines should be so extended
to one of the bases. From the extremity of the
transverse base very accurately observe the angles
of the tye-lines to the extremities of the principal
base, and measure these tye-lines in the same
accurate manner as the bases, leaving stakes at
intervals, and taking offsets to the parish boundary
and conspicuous objects wherever within distance.
When these angles and tye-line are measured and
protracted, there will be four principal stations in
the parish very accurately determined; and by
these stations being correctly fixed, each stake on
the lines connecting the stations, may be consi-
dered
as a correctly determined station, and used as
such.

It would be advisable, before filling in any por-
tion
of the work, to get the boundary of the parish,
and all the work laying outside the lines; but if not
all the boundary, at least the part on that side from


[page 43]

whence it is intended to commence filling in. In-
ternal
lines may now be used wherever it is thought
necessary, the surveyor confining himself to one
portion of the surrey only, and entirely filling it up
before any other part is commenced; his work will
then never get confused. With regard to the
direction of such lines as it may be necessary to
measure within the principal ones, circumstances
must alone direct; but lines may be measured in
any direction within this boundary, without regard
to poles or false stations that may have been erected
during the measurement of the base or tye-lines;
for, having stakes at regular intervals of 5 or 10
chains, the distance from any one of them to the
point at which an internal line crosses can be mea-
sured
, and the point determined as correctly as if
that spot had been fixed on for a station, when
measuring these principal lines; and thus can lines
be measured in any direction, always observing
that from one point to another must be perfectly
straight. The angles of the first few internal lines
should be very carefully taken, which will fix their
position without regard to their measurements; and
on the scale being applied thereto, the distance at
which any one of them bisects either of the
principal lines will be the same as measured
in the field, and the point bisected will be at the
same chainage as determined by reference to one
of the stakes. If on protraction of the angle it
should not pass exactly through the point as deter-
[page 44]

mined
by measurement, it should be made to do so;
more dependence having to be placed on the dis-
tances
than the angle in this case; but by taking the
angle, any error committed in putting down or mea-
suring
from any one of the stakes will be imme-
diately
detected: points thus determined must be
correct.

If the surveyor is expert in the use of the
sextant, it would be very desirable to have the
angles taken of all the lines, except where well tied;
but where only determined by their extremities, the
angle should in every case be taken. Particular
care is necessary in reducing lines measured over
steep ground to the horizontal plane; for the method
of doing which see description and use of theodolite,
also the method of correction with the chain only.
The surveyor is advised to lay down his work as he
proceeds, if done every day it would be best; he
will then, in the event of committing an error be
able immediately to rectify it. The sextant may be
used with advantage in filling in, but on no account
should any other instrument than the sextant and
theodolite be employed.

With regard to computing the aggregate quan-
tity
of land in the parish, the principal measured
lines, as suggested by Captain Dawson, may be used
for that purpose, equalizing and arranging into tri-
angles
what may be without; and for the separate
enclosures within they may be equalized and arranged
in a similar manner. But it would appear to us


[page 45]

the most correct method to form parallelograms, or
squares, as usually done in large surveys, of about
2 chains, by which the quantities in each enclosure
would be very correctly ascertained; and for the
aggregate, every fifth or tenth parallelogram might
be distinguished by a thicker line; the aggregate
could then be easily calculated, there being so many
parallelograms of 10 or 20 chains square, or as
much larger as pleased; the broken parts of the
parallelograms would be calculated as directed in
another part of this volume. By this method the
contents of all the enclosures added together, and
the computed whole would be found (if carefully
done) to be so nearly the same, that the difference
would be beneath notice.

The contents of the whole, computed by the
measured lines, might be used as a check on the
preceding method. The lines forming the parallelo-
grams
should be permanent; either in faint red or
blue; but probably blue would be the best, so as to
be distinguished from the measured lines, which
Captain Dawson desires to have retained. All the
entire squares must be numbered consecutively, and
the broken figures (as where a fence crosses), calcu-
lated
separately; and it will be evident that, in this
separate calculation, the ascertained contents must
be correct; for, having the contents of the whole
square, its parts added together must of course be.
the same.


[page 46]

On Subterranean Surveying, with Directions for
Procedure in Surveying Coal Pits, Mines, &c.

The instrument usually employed in all subter-
ranean
surveys is the circumferenter, or a modified
instrument, which is half circumferenter, half theo-
dolite
, having the large compass of the farmer, with
the limb and vernier of the latter; by which means
the bearing can be obtained with much greater
accuracy than with the common circumferenter.
The method of procedure is to plant your instru-
ment
where the survey is intended to commence,
and take the bearing to a lighted candle placed at
as great a distance in the required direction as can
be seen; the distance must then be measured; to do
which, remove the instrument, and let a person
stand on the exact spot where it stood, holding in
his hand one end of the chain, while another person
takes the other end with a lighted candle in the
same hand, being directed by the former until that
hand which holds the candle and the chain is in a
direct line with the light whose bearing was taken;
there mark the first chain, to which mark the hind-
most
man comes, the other advancing another chain
forward as before; this is repeated until the dis-
tance of the light to which the bearing was taken is
determined. The instrument is now fixed where
the light stood as an object, or at the termination of


[page 47]

the preceding bearing and distance, and a second
bearing taken, and the distance measured as before:
this is repeated until the whole is completed.

As surveys of pits or mines are generally made
for the purpose of ascertaining if the workings ex-
tend
into adjoining property, or for sinking shafts,
either for ventilation, or convenience for mining the
produce--it is essential, in either case, to mark on
the surface the extent of the workings, or the exact
spot at which the shaft is to be sunk, so as to open
on a precise spot in the pit previously determined
on; or it is often the case, that you may have to di-
rect
the miner as to the bearing and distance from
the extremity of some working to another pit: in
either case it is essential to trace the survey on the
surface, to do which, plant the instrument as near
the pit as convenience will allow, so that when the
foresight is put in the direction of the first bearing
you may, by looking backwards, cut exactly the
centre of the pit; if it does not do so, the instrument
is not placed in a proper position, which must be
obtained by shifting the instrument to the right or
left, until it is in the situation beforementioned; after
this is found, measure out the distance of the first
bearing from the centre of the pit, remove the in-
strument
to the end of the line, and take the various
bearings, and measure the distances, the same as
below; or, instead of the above manner, choose any
spot near the pit, as the point of commencement,
and from this point take the first bearing and dis-


[page 48]

tance as to the pit. Before the instrument is removed,
take the bearing and distance of the centre of the
pit; then remove the instrument to the end of the
measured distance of the first bearing, and set the in-
struments
to the same bearing as the centre of the pit
had; likewise, measure off the same distance as was
the centre of the pit, from the spot commenced on;
then a line from the centre of the pit to this spot will
be the proper bearing and distance.

It will be frequently found necessary, where
only particular points are to be determined so the
surface, to reduce the intermediate bearings and dis-
tances
into one bearing and distance; the most easy
and practicable method of doing whish, into apply a
good protracter to the meridian line, with its centre
no the angular point--accurately noting the angle
subtended, and measure the distance with the scale
by which the survey was plotted; during the time
of making the survey, care should be taken not to
admit any iron within three or four feet of the in-
strument
, for fear of attracting the needle, although
a large mass of iron will attract it at a much greater
distance. The tram plates or rails generally laid
down in pits do not seem to attract the needle, if
elevated from two to three feet above them; but we
have ourselves found the needle very sensibly affected
by a large clasp-knife in our breast packet. As
these surveys from records which are being continu-
ally
referred to from time to time, and additions
made thereto, it is necessary to lay down the work


[page 49]
[E]


in such a manner, that additions at any future time
can be correctly attached: this can only be done by
referring all the angles to the true meridian, as it
must be well known to the most common informed
that the magnetic meridian has been continually, and
is at present varying--the present variation of the
needle is about 27° westward of North; the method
adopted by practical miners for ascertaining the
variation (for it is not the same in all places), is,
to erect a pole exactly perpendicular--its shadow at
12 o'clock will be due North and South, or in the
direction of the true meridian; besides the above
variation of the needle from the true meridian it has
a diurnal variation, which has been often observed to
amount to one degree and a half, which may account
for inaccuracies that have occurred where the greatest
care has been observed in the use of the instrument.

A different method of using the circumferenter,
from what we have just now described, is sometimes
adopted: it is to plant the instrument at alternate
angles, and take back and foresights; thus, in place
of setting up the instrument at the commencement
of the survey, it is done at the extremity of the first
line, and its bearing taken, but with the instrument
reversed, that is, you apply your eye to the opposite
sight vane to that when a forward bearing is taken:
it should be noticed in this case, that the angle is
read off from the South end of the needle, which
angle is easily reversed and laid off, as if taken from
the North end--the forward bearings are taken in


[page 50]

the - usual manner. A back observation is taken
thus, S. 10° 30' West; reversed, it is N. 10° 30' East;
by this method the instrument is setup only half the
number of times, as by the former. We would,
however, recommend the theodolite in this as in all
other kinds of surveying: in the commencement it
would be necessary to determine the bearing, but in
the remainder of the survey, it may be used in pre-
cisely
the same manner as already described for
common land surveying: the bearing might also
be noted at each station, which would be an effectual
check on the correctness of the angles.

ON THE PROTRACTING AND PLOTTING OF SUR-
VEYS
; THE REDUCING AND COMPUTING OF
AREAS, &c.

It is usual with practical surveyors to plot their
work daily, which, if possible, should always be
done; otherwise, if left alone for a few days, and an
error should have been committed in the first day's
work, it will be very troublesome to correct, besides
a great loss of time: but by plotting daily, any
mistake that may have occurred can be easily
rectified. There is also another advantage in so
doing, which is, that you are enabled to lay down
your work with the most scrupulous exactness, every
part being fresh in your memory; and in all sur-
veys
there are particular parts which can only be laid


[page 51]
[E2]


down on the plan from memory. If it should be
inconvenient to plot daily, your lines of construc-
tion
should certainly be laid down; the method of
doing which, where a chain only is used, is pointed
out in the commencement of this treatise; and
where an instrument is used, and the angles taken
from the meridian, directions will be found in the
description and use of the circumferenter.

The method of laying down a survey made
with a theodolite or sextant, will, therefore, only be
given here; to do which, with any degree of ac-
curacy
, a circular metallic protracter is indispensable.
This instrument, for the general purposes of survey-
ing
, should be of about 5 or 6 inches diameter,
divided on silver in a similar manner to a theodolite,
with two projecting arms carrying verniers, and a
third by which the other two are moved round the
circle, either with a rack and pinion, or clamp and
tangent-screws; but where great accuracy is re-
quired
, the latter is preferable. The projecting
arms carrying the verniers have each a branch, with
a fine pricker at its extremity. The inner part of
the circle is chamfered off at each quadrant to an
edge, and the divisions brought down to it. A
small circular space of metal in the centre of the
instrument is removed, and a circular disc of glass
inserted in its place, on which are drawn lines cross-
ing
each other at right angles, and dividing the
small circle into four quadrants, the intersection of
the lines denoting the centre of the protracter.


[page 52]

When this instrument is used for laying down
an angle, it must be so placed on the paper,
that its centre exactly coincides with, or covers,
the angular point, which may easily be done,
as the paper can be seen through the glass cen-
tre-piece
. The divisions at 360° and 180°, which
are brought down on the internal chamfered edge,
must be on the line passing through the exact spot
over which is the centre of the instrument. When
the protracter is thus placed, it is prevented from
moving by four small studs, which take sufficient
hold of the paper without damaging it; then, by
means of the rack and pinion, or clamp and tan-
gent-screws
, the vernier may be set to the required
angle. A slight downward pressure on the ex-
tremities
of the branches will make two small
punctures in the paper, a line passing through one
of them, and the angular point or centre will be the
required angle. The use of the second vernier is,
that often in setting the instrument to the required
angle, the protracter is stirred, and its centre is no
longer over the angular point. When such is the
case, a line drawn from their punctures will not
pass through the centre: the branches will also
sometimes get deranged, and the same consequences
ensue. To correct this, the branches must be al-
tered
by means of two small screws, on which they
play, until a line will pass through the three points:
this should be attended to before the instrument is
used in laying down the angles. When the angles
[page 53]

on a survey are taken with a sextant, they are often
laid down with a semicircular protracter, without a
vernier; which may be also used with advantage
when plotting a survey made with a. theodolite,
except for the principal angles, which must be laid
down with the greatest possible accuracy. It should
also be an invariable custom with a surveyor to
protract all his lines before commencing to plot the
fences, &c., as it often happens on closing a day's
work, that the last line will not protract, which
arises either from some slight error in laying down
the previous angles or distances, or in noting them
in the field: if this is the case, and the early part
of the day's work should be plotted, it is so much.
waste of time.

ON THE REDUCING OF FIGURES AND EQUALIZING
OF BOUNDARIES.
To Reduce a Parallelogram to a Triangle of
equal Area.

Suppose the parallelogram A B C D, is to be
reduced to a triangle, whose area will be the same.

Produce or extend one of the sides, suppose A B,
then lay a parallel ruler on the diagonal D B, and


[page 54]

move it parallel to C; draw in the line C E, or mark
where it cuts the produced line at E; draw in D E,
and it is done: it will be the same thing if you
prick off the distance A B, on the produced line,
which will reach to E.

To Reduce a Trapezium to a Triangle of equal area.

Let A B C D be the trapezium. Draw the
diagonal A C, and extend the base from A to E;
draw B E parallel to A C; from C draw C E,
and the triangle E C D will be equal to the trape-
zium
A B C D, which may be proved by scaling
the figures.

To Reduce a Figure of Five Sides to a Triangle.

Let A B C D E be the given figure; extend
the base each way; draw C A, and C E, and B G,
and D F, parallel thereto. C F G will be the re-
quired
triangle.


[page 55]

To Reduce or Equalize an Irregular Side or
Boundary to a Mean Line.

Suppose the side of a field to be of the irre-
gular
form below. Draw the line A B, and at A

draw a transverse line, which is usually at right
angles thereto, except when the equalizing line of
the adjoining fence passes through that point. Lay
a parallel ruler from A to the third point at c; slide
the ruler up to b, and draw in the dotted line to the
transverse line, or, without drawing it, mark where
it cuts it; from which point lay the ruler to d, slide
it down to c, and draw in the dotted line as before;
from the point at which it cuts the transverse line
lay the ruler to e, and slide it up to d, and draw in
the dotted line from d; lay the ruler from the point
bisection to f, and slide it down to e, draw in this
line, and, from the point of bisection, lay the ruler
to B, slide it up to f, draw in the dotted line; and
from the point of bisection at C, draw the line C B,


[page 56]

which will equalize the irregular boundary, as much
being cut off as taken in.

This method is rarely adopted in practice; too
much time being taken up in the operation, and
equally as accurate results being arrived at by a
much shorter process, which is to equalize those
irregular boundaries by the eye, and by a little
practice it may be done with the greatest exactness.
A thin piece of transparent horn, or a strip of glass,
is recommended for this purpose, by which means
you can very exactly judge if you include as much
new space as you exclude of the original; or a bow
of whalebone, or any elastic substance, strung with
horse-hair, will suit as well. But the method gene-
rally
adopted is to draw an equalizing line, in pencil,
with a parallel ruler or straight-edge, which, on
being removed, if the line is found to exclude a
greater portion of the original than it includes of
new space, is rubbed out, and fresh lines drawn,
until the eye judges it correct.


[page 57]

ON THE METHOD OF COMPUTING
AREAS.

In computing ,the contents of any piece of land,
whether it be one enclosure or a great number, it is
done quite independent of the several lines mea-
sured
in the field, except in some cases where the
base line and a few others, from their position on
the plan, may be used with advantage; otherwise
new lines are drawn, dividing each separate enclo-
sure
into trapeziums and triangles, the bases and
perpendiculars of which are measured on the plan
by means of the scale from which it was plotted,
and so multiplied, and added together for the total
contents. After all the separate quantities are thus
computed, and added together in one sum, calculate
the whole estate, independent of the fields, by dividing
it into large triangles and trapeziums, and add these
also together. If this sum be equal to the former,
or nearly so, the work may be considered right; but
if the sums have any considerable difference, it is
wrong, and they must be examined and re-computed
until they nearly agree; or the contents may. be
found in a much more correct manner, by dividing
it into a great number of parallelograms, as here-
after
explained; it may then be proved by large
triangles and trapeziums as above directed.


[page 58]

Of the computing of Areas.

The area of any plain figure is the measure
of the space contained within its extremes or bounds.
This area, or the content of the plane figure, is esti-
mated
by the number of squares that may be con-
tained
in it; the side of these squares being an inch,
a foot, a yard, a chain, or any other fixed quantity;
and hence the area or content is said to be so many
square inches, feet, yards, or chains, &c.
Land is estimated in acres, roods, and perches.
An acre is equal to 10 square chains, that is, 10
chains in length and 1 chain in breadth: also an
acre is divided into four parts, called roods; and a
rood into 40 parts, called rods, perches, or poles.
The chain generally used, called Gunter's chain, from
its inventor, the Rev. Edmund Gunter, is 4 poles, or
22 yards, or 66 feet in length. It consists of 100
equal links, and the length of each link is therefore
22/100 of a yard, or 66/100 of a foot, or 7,92 inches.

An acre of land then consists of

  1,000 X 100 =100,000 square links.
660 X 66 = 43,560 " feet.
220 X 22 = 4,840 " yards.
40 X 4 = 160 " rods.

Lines measured with a chain are set down in
links as integers, every chain in length being 100
links; therefore, after the content is found, it will be
in square links; then cut off five of the figures on the
right hand for decimals, and the rest will be acres.
These decimals are then multiplied by 4 for roods,
and the decimals of these again by 40 for perches.


[page 59]

To find the Area of any Parallelogram.

Multiply the length by the perpendicular
breadth or heighth, and the product will be the
area.

To Find the Area of a Triangle.

Multiply the base by the perpendicular height,
and half the product will be the area; or multiply
the one of these dimensions by half the other.
Example.--Required the area of a triangle,
whose base is 1020, and perpendicular height 580
links.

1020   510
580 580
81600 40800
5100 2550
2)591600 2,95800
2,95800 4
4 3,83200
3,83200 40
40 33,28000
33,28000
Equal, 2 acres, 3 roods, 33 rods.


[page 60]

To Find the Area of a Trapezoid

Add together the two parallel sides; then
multiply their sum by the perpendicular breadth
or distance between them, and half the product
will be the area; or multiply one of these dimen-
sions
by half the other.

Example.--In a trapezoid the parallel sides are
750 and 1225, and the perpendicular distance be-
tween
them 1540 links; to find the area
1225
750
1975 x 770 = 1,520,750 square links = 15 acres, 33 rods.

To Find the Area of any Trapezium.

Divide the trapezium into two triangles by a
diagonal, add the perpendicular heights together,
and multiply by the diagonal; half the product will
be the area: or multiply one of these dimensions by
half the other


[page 61]

In place of measuring the two perpendiculars
separately, and adding them together, and then
halving the sum, a practical surveyor would lay a
parallel ruler upon the diagonal A B, and move it
parallel to the angle C or D, and draw the line C e
or D f, and take off the distance with his compasses
from one of the angles to the nearest part of the
dotted line, passing through the opposite angle,
which will give the same distance as the sum of the
two perpendiculars. Apply this extent to a scale
double that to which the figure is laid down by, and
the trouble of adding and dividing will be saved.
This method has also the advantage in point of
correctness, for in measuring the perpendicular
heights separately there will be small fractional parts
that you cannot well estimate.

To Find the Area of an Irregular Polygon.*

Draw diagonals, dividing the proposed polygon
into trapeziums and triangles; find the areas of all
these separately, and add them together for the con-
tents
of the whole figure.

To Find the Area of any Figure by means of
Parallelograms.

This is done by drawing parallel lines, in some
faint colour, all over the plan, forming squares of one

* A poylgon is ANY irregular figure having more than four sides, and,
consequently, more than four right angles.


[page 62]

or two chains each, which are afterwards numbered
consecutively: the contents of all the squares are
then easily ascertained, to which is added the con-
tents
of the broken squares, which are equalized and
calculated as triangles or parallelograms, and the
area of each generally inserted on the plan. An
example of this method of calculating areas is shown
in Plate 5. The squares are of one chain each,
consequently the number (79) divided by 10 (the
number of square chains in an acre), or, what is the
same thing, cut off one figure on the right hand,
which is multiplied for roods and perches--that on
the left is acres, thus
7,9 the number of squares
4
3,6
40
24,0
A. r. p.
Contents of squares = 7 3 24
Ditto broken squares = 1 1 31
9 1 15
It will be perceived on reference to the plan,
that in some of the broken squares the contents of
an entire square (16 perches) is inserted; but it
will be perceived that immediately adjoining these
broken squares, in which it is so inserted, that a
piece equal to the deficiency is omitted in the cal-
culation
. Also where a piece in a broken square is
very small, it is carried into the contents of the
adjoining one.


[page 63]

DESCRIPTION, USES, AND ADJUSTMENTS OF THE
VARIOUS INSTRUMENTS EMPLOYED IN SUR-
VEYING.

Of the Chain and Offset-staff.

THESE are two of the principal instruments employed
in surveying. The chain, as before mentioned, is
66 feet, or 100 links, in length, comprising ten
divisions of ten links each, which are distinguish-
able
by brass marks, cut into fingers, from one to
four, one denoting 10 links, and 4 fingers, 40.--The
chain is marked thus from both ends, the centre or
50 links being denoted by a round brass mark and
swivel. It is, therefore, immaterial which way you
measure with the chain. Four fingers past the
round brass mark denotes 60, 3 fingers 70, 2 fingers
80, and 1 finger 90; the end of the chain is 100.
It will be found rather perplexing at first reckoning
above 50, but a little practice will soon make it
familiar: but the student should be very particular
when reading the chain at about 60 or 70, as
nothing is more common than the mistaking of
these marks for 40 or 30.

The offset-staff is a light rod, generally of 10
links in length; the links being marked by notches
or brass nails. This is carried by the person fol-


[page 64]

lowing the chain, which should always be the
surveyor himself, never trusting to an assistant in
this the most particular part of his work. A tape
is used generally for measuring buildings, and oc-
casionally
for taking offsets; although it is not so
handy as the staff, without an extra assistant is
employed: a tape of 100 feet in length will be
found the most advantageous.

Having, in our general observations at the
commencement of the volume, spoken on the ad-
vantages
of a much longer chain than that of Gun-
ter's
, it will only be necessary here to point out the
method of setting down the chainage and plotting
the work. If a chain of 100 feet in length is used,
the chainage would be entered in the field-book, in
the same manner as if a common link-chain was
used; the only difference would be in the plotting
of the work, thus: If a survey made with a 100-
feet
chain was to be plotted to a scale of 5 chains to
an inch, it must be done with a scale divided into
330 to an inch, there being that number of feet
contained in 5 chains; that is, there would be 33
divisions, each division representing 10. To plot
to a scale of 2½ chains to an inch, there would be
the same number of divisions, each division repre-
senting
5; but there are some scales into which the
100-feet chain cannot be well subdivided. The
most correct and convenient chain for use is one
double the length of Gunter's, divided into 100
links, each link of which will be double the usual


[page 65]
[F]


length. This chain may be used in the same man-
ner
as Gunter's; that is, the links set down as
integers, each link counting but one, although it is
actually two: but in plotting, by using a scale
double the size of that to which the work is laid
down, the results will be the same as if a chain and
scale of the usual kind had been used: thus, if the
survey is to be laid down to a scale of 4 chains to
an inch, by plotting with a two-chain scale the
results will be the same as if the common chain had
been used, and the work plotted to the proper scale.
Also, by having stamped thereon a second division
of figures, denoting the proper scale of 4 chains, the
work may be plotted, and the contents calculated
with the same.

Description of a Gircumferenter, and Method of
Observing with it.

This instrument is principally used in mines
and coal pits, and where a country is thickly over-
grown
with wood. Each angle is taken with the
needle from the meridian, without being at all con-
nected
with the preceding portion of the work. A
large compass fitted with plain sights, mounted on
a ball and socket, and connected with a stand similar
to that of a theodolite, forms the instrument.
The internal part of the compass box is gra-
duated
very distinctly, sometimes into twice 180°, or


[page 66]

four times 90°, as well as quite round the circle to
360°. At zero, or 360°, is fixed a perpendicular
sight vane, with a fine wire strained on its opening:
opposite to it, over the division 180°, is fixed a
similar vane and strained wire; these wires being
in the meridian or N. S. line of the compass box. A
small orifice is also made in each sight vane, in a
line with the strained wire, to which the eye is
applied on taking an observation; the object being
bisected by the perpendicular wire, thus:--at any
station from whence you intend carrying out a line,
set up your instrument, and by means of the ball
and socket and, spirit-bubbles set it level; or, if you.
have no spirit-bubbles attached to your instrument,
notice if the needle plays freely, by which means
you may set it up very nearly level; clamp the ball
and socket tight, and turn the sights which are fixed
to the compass box in the direction of the proposed
line, which, when bisected, will give the desired
angle, by noting exactly what degree the north
point of the needle becomes stationary opposite. As
the correctness of this instrument entirely depends
on the fine performance of the needle; it will be
requisite to have a very sensitive one, and great
care must be taken that the instrument is not stirred
after bisecting the object, and previous to noting
the angle. The needle should always be thrown off
its centre, and not allowed to play except when in.
the act of using it, otherwise the fine point on which
it revolves, and its accuracy depends, will soon be
[page 67]
[F2]


destroyed. The student should also notice that in
this instrument and, generally, in all modern in-
struments
, the compass is marked contrary to
nature; that is, the West is substituted for the East,
the North East for the South West, &c.: by thus
altering the cardinal points the readings of the
needle show the actual direction of the line.
For a further description and method of using
this instrument, and plotting the work, see "Subter-
ranean
Surveying" and "Prismatic Compass."

Description of the Prismatic Compass, and Method
of Observing with it.

This is a very useful little instrument, and is
used in the same manner as the circumferenter, that
is, all the angles are taken from the magnetic meri-
dian
; but, instead of a needle with the graduations
in the compass box, it has a graduated floating card
attached to the needle, similar to a mariner's com-
pass
. This card is usually graduated to 15' of a
degree, but angles cannot be taken with it to less
than 30', or half a degree, which renders it very
unfit to be used, except in filling in the detail of a
survey, where the principal points have been accu-
rately
determined by means of a theodolite, or where
great accuracy is not required. The graduations
on the floating card commence at the North point, or
zero, and are numbered 5°, 10°, &c. round the


[page 68]

circle to 360°. Attached to the instrument is a
fixed perpendicular sight vane, with a fine thread or
hair strained along its opening, opposite to which is
the prism. On applying your eye to the prism, and
bisecting any object with the thread in the sight
vane, the division on the card coinciding with the
thread, and reflected to the eye of the observer, will
be the angle formed by the object with the meridian;
but care should be always taken that the card is
quite stationary at the time you note the angle.
You should also be careful to hold the instrument
in such a position that the card will play freely in
its centre, otherwise the results obtained will be
liable to great inaccuracy.

The angle formed by one object with another
may be easily ascertained with this instrument, first
finding the angle formed by each with the meridian,
the difference will be the angle required; as for
example: suppose you find the bearing of one object
to be 20°, of the other 45°, the angle subtended by
these two objects will be 25°, which is the less sub-
tracted
from the greater; but if the difference be
greater than 180° it must be subtracted from 360°,
as in the following example: the bearing of one
object is found to be 345° of another 30°. The
angle subtended in this case will be the difference
between 30° and 345° = to 315°, subtracted from
360° = to 45°. But the best method of using this
instrument is to keep each angle separate, noting
them all in your book from the meridian. You are


[page 69]

then less liable to error, and the plotting is per-
formed
with great rapidity, by means of a pro-
tracter
formed on the same sheet of paper that you
plot on.

If only filling in the detail of a map, the
magnetic meridian will have been previously deter-
mined
; if not, you must draw one, taking care that
you so place it that the paper will take in the great-
est
extent of survey. A line must then be drawn in
a vacant corner of the map parallel with this meri-
dian
, and from a metallic protracter mark off the
degrees and half degrees, which may be done in a
few minutes; number them as 10°, 20°, &c. to 360°,
commencing from the North point as on the floating
card. Then, to lay off an angle from the meridian,
you have only to apply the edge of a parallel ruler
to the centre of the circle forming the protracter and
the required angle, and slide it parallel to itself to
the point from whence it was taken; then draw in
the line and mark off the distance: proceed in this
way until all the angles taken are laid down on the
plan. If the survey is very extensive you may
describe two or three of these protracters on different
parts of the plan, taking, great care that the meri-
dian
lines are exactly parallel.

But a much more convenient method of laying
down angles taken from the meridian is to make a
protracter on a separate sheet, of paper or card-board
in the manner described above; then cut out the
blank paper withinside the circle on which the


[page 70]

degrees are marked, and it will prove a most con-
venient
protracter, and may be attached to any part
of the map, and the angles laid off as before by
means of the parallel rule; or the meridian line
being drawn through any point on the plan from
whence angles have been taken, can be instantly
marked off by laying this paper protracter on the
meridian line, observing that the N.S. line on the
protracter exactly coincides with that on the plan.
The same method of plotting is also observed when
the circumferenter is used.

Description, Use, and Adjustment of the Box or
Pocket Sextant.

This instrument is the most useful a surveyor
can have, and sufficiently accurate for all purposes,
except in setting out long lines, or laying out large
triangles in a hilly country, where a good theodolite
is indispensable; but for filling in the details of a
survey, it is unrivalled both for accuracy and expe-
dition
. The pocket sextant is usually divided on
silver to 140°, although a greater angle than 110°
should not be taken with it. A small telescope is
sometimes attached to assist the sight, but is very
inconvenient, taking up more time to arrange for,
distinct vision than for taking the angle. There is
a small aperture opposite to the half silvered or
horizon glass to take plain sights with, in place of


[page 71]

the telescope, which is far preferable. The method
of observing with the sextant is to place yourself in
a line at the exact point you intend carrying out
another line, the angle of which with the preceding,
you are desirous of knowing. You must take the
sextant in your right hand, the case of the instru-
ment
forming a handle, and apply your eye to the
small aperture. looking through the ursilvered part
of the horizon glass to some object or station mark
on your line; you will then, wtth your left hand,
turn the milled head screw, which carries the
silvered glass until it reflects an object on the
second line, in the silvered part of the horizon
glass; and when the two objects are in exact con-
junction
(that is, the object viewed direct through
the unsilvered part of the horizon glass, and the
reflected object appearing as one), the desired angle
will be given.

The method of adjusting this instrument is ex-
ceedingly
easy and correct. You must first observe
some well defined object through the unsilvered part
of the horizon glass, and turn the milled head screw
tarrying the silvered glass until the same object is
reflected; the observed and reflected object, which
in this case are the same, appearing as one; the
index or vernier standing at 0° on the limb if the
instrument is correct, if not the reading on the limb
will show the error of the instrument; or the vernier
being set at 0°, the object viewed direct and by
reflection should appear as one if the instrument is


[page 72]

in adjustment, if not they will overlap and appear
unconnected. To remedy this the horizon glass
must be altered by means of two screws (one in the
upper part of the instrument, over the horizon glass,
and the other at the side of it), until the object
viewed direct and reflected appear as they really
are--one, the vernier standing at 0' on the limb.
A key to fit both screws of the horizon glass is fitted
into some spare place in the instrument. The angle
is read off with the assistance of a lens, fitted to the
instrument with a hinge joint, which allows it to be
moved over every part of the limb and vernier.

Description, Uses, and Adjustments of the
Theodolite.

Of all angular instruments for surveying this
is the best; the improvements it has received from
time to time rendering it almost perfect. The the-
odolites
in general use have telescopes mounted in
Y s, similar to that of the Y level, with the spirit-
bubble
attached. A semicircular arc for taking
vertical angles is also fixed to the Y s, the whole
moving in a vertical plane, and from the centre of the
circle describing this arc are projecting arms resting
on standards, which are fixed to the upper plate of the
instrument on which the verniers are marked; the
plates are called the limb: the chamfered edge of
the lower one is divided quite round the circle, as


[page 73]

10°, 20°, &c. to 360°--the intermediates as 5°, 15°,
25°, &c. being represented by longer divisions than
the single degrees, and the half degrees, into which
the limb is generally divided, by shorter lines than
the degrees.

For the purpose of bisecting objects correctly,
a slow motion screw is attached to the upper
plate, the clamp screw securing it when you have
nearly bisected the object; and the slow motion
screw moving the vernier through the least pos-
sible
space, perfecting the bisection. In good in-
struments
, similar clamp and slow motion screws
are attached to the lower plate, by which means
(as will be presently explained) angles may be
repeated any number of times, thereby ensuring the
degree of accuracy required. A vernier is fixed
to the upper plate, through which the semicircular
arc for taking vertical angles passes; two spirit-
bubbles
are also attached to this plate at right
angles to each other, for the purpose of setting the
instrument horizontal. A small compass is also
attached to the upper plate; the N.S. line of the
compass box ranging with the line of sight. The
bearings being noted at the different stations serve
as a check on the angles. The whole is mounted
on parallel plates similar to those of a level.

The principal adjustments requisite to this
instrument are the same as those for the Y level (to
which the reader is particularly referred); viz., that
the line of collimation, or optical axis, of the telescope


[page 74]

must coincide with the cylindrical rings on which
it turns in the Y s, and that the bubble must be
parallel to this optical axis. The method of per-
forming
these adjustments have been already ex-
plained
in the account of the Y level. The next
adjustment is to make the axis of the horizontal
plates truly vertical: to do which, set up the instru-
ment
as near level as you can, the telescope lying
over two of the parallel plate-screws; then, by
means of the clamp and slow-motion screws at-
tached
to the vertical arc, bring the bubble con-
nected
with the telescope into the centre of the
tube; then move the instrument half round, and
the telescope will be over the other pair of plate-
screws
. if the bubble remains in the centre of the
tube, it is right, if not, you must correct it, one half
by the clamp and slow-motion screw attached to the
vertical arc, and the other half by the parallel plate-
screws
. If the bubble will not now remain in the
centre of the tube, while the instrument is turned
quite round, it must be repeated until the result is
satisfactory, when, if the bubbles on the vernier-
plate
are correct, they will stand in the centre of
their tubes; if not, the screws at either end con-
necting
the bubble-tubes with the vernier-plate
must be altered as much as will bring them to the
centre of their runs.

The adjustment necessary to the vertical arc
may now be attended to. If the vernier stands at
zero, when the former adjustments are perfect, it is


[page 75]

correct; if not, you must alter the vernier by means
of the screws attaching it to the plate until it does,
or you may note the error, and allow for it in each
vertical angle. But for the purpose of accurately
noting this error, it is necessary to take the vertical
angle of some conspicuous object with the telescope,
reversed in the Y s; to do which the instrument
must be turned half round, when the telescope will
occupy the same position as at first, but the vertical
arc will be reversed: the mean of these two read-
ings
will be the amount of error; which we should
advise the surveyor to correct, in preference to
allowing for it at each vertical angle, as being less
liable to error.

To Observe with the Theodolite.

The method of observing angles with the theo-
dolite
is to set it up exactly over some station, which
you can easily do with the assistance of a plummet
and line suspended exactly under its centre from a
hook attached to the stand; then set it level by
means of the parallel plate-screws, clamp the upper
and lower plates together, and turn the instrument
towards some station mark; clamp the lower plate
and make the bisection with the slow-motion screw,
observing in every case to bisect the object as near
the ground as possible. Then read off the degrees
and minutes, also the seconds if necessary, taking


[page 76]

the mean of two or three verniers, if you have so
many, but two should always be used. The upper
plate may now be released (the lower one remaining
clamped, care being taken that it has not the least
motion), and turned towards another station, clamp
it, and bisect with its own slow-motion screw, read
the degrees, minutes, and seconds, as before, and
their difference will be the desired angle. But for
general purposes it will be found much easier and
be less liable to error, to set the vernier at 360° and
clamp it; then turn the instrument bodily round in
the direction of the first object, clamp the lower
plate, and bisect with its slow-motion screw; then
release the upper or vernier-plate (the lower plate
remaining clamped), and turn the instrument in the
direction of the second station, clamp and bisect as
before, and the angle read off will be the desired
one. This will be found in practice generally the
most preferable method of taking angles, although
not so correct as the former, it being almost impos-
sible
to set the instrument exactly to 360°: but in
extensive operations, and where many angles are
taken from one station, the former method is greatly
superior. The bearing, which is the angle pointed
out by the compass-needle, may be also noted at
4each principal station, which will be a check on the
accuracy of the angle; but in the usual description
of theodolites, the bearing cannot be read with any
degree of accuracy, except by setting the plates and
needle at zero, and reading the angle on the limb
[page 77]

when the object is bisected, which will be the
bearing.

Vertical angles are taken for the purpose of
reducing lines measured over steep ground to the
horizontal measure. On one side of the vertical arc
are engraved the requisite divisions for determining
the angle, and on the other the number of links to
be deducted frorn teach chain's length; to reduce it
to the horizontal measurement; but for the purpose
of measuring this angle it is necessary to set up a
mark at the exact height of the optical axis of the
telescope, at the extreme point to be measured to,
on which this reduction is to be made, which can
then be allowed for in the field in the follpwing
manner :-suppose the angle of elevation was found
to be 14° 30', on the other side of the arc will be
found the figure 3 and a fractional part, which
signifies that the chain must be lengthened 3 links
and a fraction, or that the chain should be drawn
forward on the ground that quantity to bring it to the
horizontal measurement. In the common operations
of surveying it will be found most advantageous to
make the necessary allowance in the field, especially
where you cross many fences or take a good quan-
tity
of offsets, as it will be found very troublesome
in plotting to make the necessary reduction for each
distance: but where great accuracy is required, the
angle should be noted and the necessary reduction
made when plotting the work. Thus, on reference
to the table for reducing hypothenusal lines to hori-


[page 78]

zontal, it will be found that a reduction of 3,18
links must be made from each chain's length. Now
in the field you could very well allow for the three
links on each chain, but not for the decimal parts,
which, on a line of any great extent, would make
some difference: it should, therefore, depend on the
description of work whether the allowance be made
in the field or in the office, the latter, as we have
observed, being the most correct.

Further information respecting vertical angles
will be found in the account of levelling with the
theodolite.

Captain Everest's Improved Theodolite.

This instrument differs considerably, and has
many decided advantages over those in common
use. It is extensively used in India, and is now
becoming very common in this country. It is not
thought necessary to give a minute description of it,
as five minutes' examination of the instrument would
more fully explain its advantages than the most
lengthened account. This theodolite has three ver-
niers
for the horizontal angles, the mean of which
is taken, as also of the two for the vertical angles.
The whole instrument, in place of being mounted
on parallel plates, is fixed on a tripod stand, three
foot-screws serving to set it level, and is usually
bronzed. The method of adjusting this instrument
is somewhat different from other theodolites.


[page 79]

To adjust the Level, so as to make the Axis of the
Horizontal Limb truly Vertical.

Bring the spirit-bubble attached to the bar
over two of the foot-screws, and by their motion
bring the bubble to the centre; then turn the in-
strument
half round, when, if the bubble remains
in the centre of the tube, it is right; if not, correct
half the error by raising or lowering the tube itself,
and the other half by one of the foot-screws.

For the Line of Collimation:--After bisecting
an object with the vertical wire, the telescope must
be reversed, the bisection remaining perfect; if not,
correct half the difference by the collimating screws,
and the other half by a horizontal motion of the
instrument until perfected.

To Correct for the Vertical Arcs:--Take the
altitude or depression of any object with the telescope
reversed, half the sum will be the true angle to which
the verniers must be set, and bring the level connect-
ing
the verniers to the centre of its run by the ad-
justment
screws at either end.

The Author has a 5-inch theodolite (which
is the most useful size for general purposes) of
the above description, made by TROUGHTON and
SIMMS
, which performs admirably; but he has
found it necessary to adopt parallel plates instead of
the tripod (which had a very sensible lateral motion),
but has retained the tripod on the instrument, by
which means he has the advantages (that of being


[page 80]

able to use it without a stand) of the instrument
without its defects Over the verniers are engraved
the letters A, B, and C, degrees and minutes being
always read from A, and minutes only from B and
C, and a mean taken as follows:--
On the instrument being clamped, the reading at
A was 7° 34' -"
B " - 33 -
C " - 33 -
  7 33 20 mean.
On bisecting the object whose angular distance was
required, the reading at A was 45° 59' -"
B " - 58 -
C " - 59 -
  45 58 40
Subtract 7 33 20
Angle required 38 25 20

The vernier A might be set at 360° in the same
manner as described for the common theodolite, and
the reading, when an object is bisected, will be
the desired angle. Neither is it necessary to read
the three verniers if the description of work does
not require such accuracy. Theodolites of the above
size and description will be found the most useful
for all ordinary purposes of surveying; and, more-
over
, are so light as to be carried about in the field
without inconvenience.


[page 81]
[G]


Instrumental Parallax.

Parallax is often the cause of as much per-
plexity
in surveying instruments as in those for
levelling; for the method of procedure to correct
this the reader is referred to the account of levelling
instruments in another part of this treatise.

Cross Hairs of the Diaphragm.

If the reader should happen to break these
hairs, he will see the method of replacing them in
the account of Levelling Instruments.

Theodolite Stand.

The surveyor is recommended to have a good
solid stand for his instrument, similar to that de-
scribed
for levelling instruments; he will thereby
avoid the tremulous motion and consequent uncer-
tainty
attendant on the use of the round weak legs
generally applied to surveying instruments, which
would appear to be made rather for show and con-
venience
of carriage, than use; a solid, firm, and
immovable mounting for the instrument being a
desideratum.


[page 82]


[page 83]
[G2]


THE
THEORY AND PRINCIPLES OF
LEVELLING.

THE figure of the earth is understood to be deter-
mined
by a surface at every point perpendicular to
the direction of gravity, or to the direction of the
plumb-line. This surface is the same that the sea
would have if continued all round the earth; the
surface of every fluid, when at rest, being horizontal
or perpendicular to the direction of gravity. Now
the visible horizon of an observer at a point on the
surface of the earth is a tangent, or at right angles
to the direction of gravity at that point. Levelling
is, therefore, the finding a line parallel to the horizon,
at one or more stations, to determine the height or
depth of one place with respect to another; its appli-
cation
is in the laying out of roads, the conducting
of water, draining, &c., &c.

Two or more places are on a true level when
they are equally distant from the centre of the earth.
One place is also higher than another when it is
equally distant from the centre of the earth, and a
line equally distant from that centre in all its points


[page 84]

is the line of true level. But the earth being round,
that line must be a curve, and make a part of the
earth's circumference, or, at least, be parallel to it,
or concentrical with it, as the line B C F G, which
has all its points equally distant from A, the centre
of the earth, considering it as a perfect globe.

But the line of sight, B P E, &c., given by the
operation of levelling is a tangent, or a right line,
perpendicular to the semi-diameter of the earth at
the point of contact B, rising always higher above
the true line of level the farther the distance is,
which is ealled the apparent line of level. Thus
C P is the height of the apparent level above the
true level, at the distance B C or B D; also E F is
the excess of height at F; and G H at G, &c. The
difference, it is evident, is always equal to the excess
of the secant of the arc of distance above the radius
of the earth.

Now the difference C D, between the true and
apparent level at any distance, B C or B D, may be
found thus:--by a well known property of the circle


[page 85]

(2 A C + C D): B D :: B D : C D, or because
the diameter of the earth is so great with respect to
the line C D, at all distances to which an operation
of levelling commonly extends, that 2 A C may be
safely taken for 2 A C + C D in that proportion with-
out
any sensible error, it will then be 2 A C : B D ::
B D : C D, which therefore is = BD2/2AC or BC2/2AC nearly;
that is, the difference between the true and apparent
level is equal to the square of the distance between
the places divided by the diameter of the earth, and,
consequently, it is always proportional to the square
of the distance. Now the mean diameter of the
earth being nearly 7,916 miles,* if we first take
B C = 1 mile, then the excess BC2/2AC becomes 1/7916 of
a mile, which is 8,004 inches, or 6,670 decimals of
a foot, for the distance of the apfarent above the
true level at the distance of one mile.

Hence proportioning the excesses in altitude
according to the squares of the distances, the height
of the apparent above the true level can be easily
ascertained, as for example:--a spirit-level planted
on a hill, the horizontal wire of which exactly
coincided with the summit of another hill, distant.
5 miles, required the difference of level; the height
of the levelling telescope above the ground being
4,50 feet and decimals.

*The equatorial diameter of the earth is 7,924 miles, and the polar dia-
meter
7,908 miles; the mean diameter, 7,916 miles, is therefore taken.


[page 86]

Allowance for curvature for 1 mile 6670
Square of distance; 5 miles 25
33350
13340
Amount of curvature for 5 miles 16,6750
Height of instrument to be added
-the observed hill appearing
so much higher than where the
instrument was planted
4,5000
21,1750 difference of level.

From this example, although the apparent differ-
ence
of level is only 4,50 (the height of the instru-
ment
) yet the observed hill is actually 21,1750
higher than where the instrument was planted, but,
on account of the curvature of the earth, it is appa-
rently
depressed to the same level as the centre of
the telescope.

But what has been stated has been said without
any regard to refraction in elevating the apparent
places of objects. But as the operation of refraction,
in incurvating, the rays of light proceeding from
objects near the horizon, is very considerable, it can
by no means be neglected, when the difference be-
tween
the true and apparent level is estimated at
considerable distances. The terrestrial refraction,
when the elevation is not very great, varies from
¼ to 1/24 of the angle subtended by the horizontal
distance of the objects; and the radius of curvature


[page 87]

of the ray, therefore, varies from twice to twelve
times the radius of the earth. In the mean state of
the atmosphere the refraction is about 1/14 of the
horizontal angle, and the radius of curvature of the
ray seven times the radius of the earth. The effect
of refraction may be allowed for by computing the
correction for curvature, and taking one-seventh
of it for the quantity by which the object is ren-
dered
by the refraction higher than it ought to be;
it must be observed that refraction is opposed to
curvature.

The above example will then stand thus :-

Curvature 16,6750
Deduct for refraction 1/7 2,3821
14,2929
Add height of instrument as
before
4,5000
18,7929 true difference of level.

The necessary tables containing the correction
for curvature and refraction, for distances in chains,
feet, and miles, are inserted at the end of this
treatise, to which the reader is referred. The
corrections for refraction are taken, as in the above
example, at 1/7 of the apparent above the true level,
although many professional men allow 1/12; but it
is found to vary with the state of the atmosphere in
regard to heat or cold, and humidity, so that deter-


[page 88]

minations obtained for one state of the atmosphere
will not answer correctly for another.

Of the Causes which Produce Atmospherical
Refraction.

From the nature and progression of light
rays, in passing from any object through the
atmosphere or part of it to the eye, do not proceed
in a right line; the atmosphere being composed of
a great number of thin strata of air of different
densities, the densities being greater the nearer they
are to the surface of the earth, the luminous rays
which pass through it are acted on as if they passed
successively through media of increasing density,
and are therefore inflected more and more towards
the earth as the density augments. To explain the
nature of refraction let A B be a portion of the
earth's surface, G H the upper boundary of the
atmosphere, S a star, P the place of the observer,
and Z his zenith.

Let E F, C P represent the boundaries of these
strata, then a ray of light will proceed from the
star S in a straight line until it arrives at the point
K, where it enters the denser medium; it will then


[page 89]

no longer pursue its direction S K N, but will be
deflected in the direction K L; at the point L it
enters a still denser medium, and will be again
deflected from its direction K L 0, and will now
proceed in direction L M.

Similar effects will be produced whatever be
the media through which the ray passes, and it will
at length reach the point P in the line M P. Hence
the ray instead of being a straight line, is broken
into parts K L, L M, M P; and if we suppose the
media through which it passes to be indefinitely
increased, and their boundaries to approach each
other by spaces extremely small, the parts K L,
L M, M P, may be considered as curvilineal, and
the course of the ray a curved line as S B P, where
P is the place of the observer, H P N his horizon,
Z the zenith, and S a star. In this case the ray
is no longer considered as passing through different
strata of variable density, but through a medium of
continually varying density, such as the atmosphere
of the earth, whose density is greatest at its sur-
face
, and decreases towards the higher regions
In passing through such a medium, a ray of light
will be deflected into a curve line, concave towards
the earth's surface, and will enter the eye of the
observer in the direction of a tangent to that curve.
A ray from the star S will therefore not proceed
in the straight line S P, but in the direction of
the curve line S B P, and to the observer at P
the star S will appear as if at S' in the direction


[page 90]

of a straight line which touches the curve at P.
The angle S' P N is therefore the apparent alti-
tude
of the star, and the angle S P N its true
altitude: their difference, the angle S' P S will
consequently be the refraction.

Similar effects take place with regard to the
rays of light by which terrestrial objects are ren-
dered
visible. In their passage through the at-
mosphere
they are bent out of their rectilineal
direction, and enter the eye of the observer in a
curve line, so that the apparent or observed alti-
tude
of an object is always greater than its true
altitude. The correction for refraction of 1/7 the
amount of curvature as given in the tables, will
be found sufficiently correct for every state of
the atmosphere except in very extensive trigono-
metrical
operations--when it is usual to correct
for refraction according to the state of the atmo-
sphere
at the time the observations were taken--
for the method of doing which the reader whom it
concerns is referred to the account by Colonel Mudge
of the Trigonometrical Survey of England.

The following form may be adopted for making
the corrections for curvature:--reject the decimal
004, and assume the difference between the true and
apparent level for one mile, to be exactly eight inches,
then arises the following form:--

Correction = 2 D²
3

[page 91]

D being the distance in miles, or in other words
two-thirds of the square of the distance in miles
will be the amount of correction in feet.

Another convenient form for making the cor-
rection
for curvature for any distance is--to add
to the arithmetical complement of the logarithm
of the diameter of the earth,* or 2,378861--
double the logarithm of the distance in feet, the
sum will be the logarithm of the correction in
feet and decimals, from which, if 1/7 of itself be
subtracted, the result will be the combined cor-
rection
for curvature and refraction.

Example. Required the amount of curvature
for 1350 feet:-

Log. of 1350 feet 3,130334
2
6,260668
Add arithmetical complement of the
log, of the diameter of the earth
2,378861
Log. of ,04361 8,639529
Correction for curvature for 1350 feet ,04361 decimals of
a foot.

But the most convenient method of making
the correction for curvature, is to divide the square

* The arithmetical complement of a logarithm is what the logarithm
wants of 10,00000, &c., and the easiest way to find it, is beginning at the left
hand, to subtract every figure from 9 and the last from 10. Thus the diameter
of the earth is 41,796480 feet, the logarithm of which is 7,621139 which
subtracted from 10,00000 gives 2,378861 the arithmetical complement.


[page 92]

of the distance in chains by 800; the quotient will
be the depression in inches very nearly.

But previous to introducing the reader to
practical levelling, it will be as well to inform him,
that in the common practice of levelling, neither
curvature or refraction are ever allowed for, as
by the simple expedient of taking observations at
equal distances on each side your instrument, both
stations are equally affected. Suppose your instru-
ment
planted at B (see first Diagram) and your
points of observations to be Z and D, which are
equal distances from B, it is evident both would
be affected alike--or, in other words, the points
would be equally distant from the earth's centre.
Suppose further--the instrument planted at B as
before--one point of observation being Z and the
other E, curvature would be allowed for on D E,
or the excess of distance of the point E above that
at Z. It is only in experiments or very delicate
operations, that its effects can be appreciated, or in
extensive trigonometrical surveys, where the observed
distance is generally several miles; but this can
never be the case iii common levelling operations;
the average distance to which levelling at one sight
extends, seldom exceeding a few chains.


[page 93]

THE
PRACTICE OF LEVELLING.

LEVELLING is an art of great practical utility, and
with the beautiful instruments now in use, the
exact difference of level between any number of
places may be obtained, with a degree of accuracy
surprising to those unacquainted with the delicate
adjustments of those instruments.

It will be at once apparent, that to obtain the
difference of level between any number of places,
that they must all have reference to some fixed
mark, from which an imaginary line called the
datum is drawn, and upon which all the levels
are based; or, in other words, every variation in
the surface of the ground is reckoned from this
assumed line, which is generally either high-water
spring tides (Trinity datum), or low-water spring
tides; or if very remote from the sea, some fixed
mark is chosen at either end of the line of levels
to which they are all referred. If it should be
thought advisable to fix the datum line lower than


[page 94]

either end, it may be done by assuming a line 100 or
any number of feet lower than the fixed mark, on
which assumed datum line the levels will be based.
But reference should be made to the tides if prac-
ticable
where the line of levels are of great extent.
There are two instruments necessary to obtaining
the difference of level between two or more points:
first, the spirit-level, which, by mechanical con-
trivances
hereafter described, is set truly horizontal:
secondly, a graduated rod or staff, which is alter-
nately
advanced with the spirit level, denoting
by the graduations bisected, the rise or fall between
any two points. (See description of the Spirit-
Level
.) An example will immediately illustrate
this:--

Let it be required to know the difference of
level between the points A and I. The spirit level
is set up at a point B, and the staff at A, and when,
by the mechanical means previously spoken of, the
spirit-level is brought truly horizontal, on looking
through the telescope attached to it, the line of
sight will be the dotted line towards the staff at A,


[page 95]

which is graduated sufficiently distinct to be read
off with the aid of the telescope. Suppose the line
of sight cuts the staff at 4 feet; the spirit-level
revolving on its stand is turned in the direction of
the staff at C, and being still perfectly horizontal,
it bisects it at 7 feet 6 inches, the difference then
between the points A and C will be 4 feet, minus
7 feet 6 inches, or equal to a fall from A to C of
3 feet 6 inches. The spirit-level is then removed
from B, and placed at D, the staff remaining at
C, the spirit-level, as before, being set truly hori-
zontal
, is turned in the direction of the staff at C,
which it bisects at 2 feet; it is then turned towards
E, which it bisects at 9 feet, the difference of level
between the points C and E will be 2 feet, minus
9 feet, or equal to a fall of 7 feet from C to D;
which, added to the previous fall of 3 feet 6 inches
from A to C, will, on the whole, be equal to a fall
of 10 feet 6 inches from A to E. The spirit-level
being removed to F, and the staff placed alternately
at E and G, reads at the former 4 feet, and at the
latter 7 feet, which is a further fall of 3 feet from
E to G, making altogether from E to G a fall
of 13 feet 6 inches. We now place the spirit-level
at H, and read on the staff at G 6 feet, and on
the staff at I only 2 feet 6 inches, or equal to a
rise of 3 feet 6 inches from G to I. This, instead
of being added to the fall of 13 feet 6 inches, is
(being a rise) to be deducted from it-then 13 feet
6 inches, minus 3 feet 6 inches, (the rise from G
[page 96]

to I) will be equal to 10 feet, which is the difference
of level between A and I, and this method of
procedure may be continued for any distance,
adding or subtracting, as the ground rises or falls.
If it was only necessary to obtain the difference of
level between the extreme points A and I, it would
not be requisite to reduce the levels of the several
intermediate stations, but arrange the fore and back
sights into separate columns, and the difference of
their sums would be the difference of level-thus
Ft. in. Ft. In.
A 4 . 0 x 7 . 6 C
C 2 . 0 x 9 . 0 E
E 4 . 0 x 7 . 0 G
G 6 . 0 x 2 . 6 I
Sum of back sights 16 . 0 Sum of fore sights 26. 0 I
16 . 0
Difference of level between A and I . 10 . 0

But to be able to draw the section so as to
show the undulation of the ground between the
points A and I, it will be necessary to measure the
distances between A and C, C and E, &c., so that
the staff being set up at each change of inclina-
tion
of the ground, and having the distances from
staff to staff, you can proceed to plot the section.
But in this case the form of field-book would be
similar to the following, the difference between the
back and fore sights being in each case added to,
or subtracted from, the preceding reduced level;


[page 97]
[H]


and to prove the correctness of the castings, it is
necessary to add up the columns of the fore and
back sights (as in the example above), and if their
difference equals the last reduced level, it shows it
to be correct.
Field-Book
B. Sights. F. Sights Red. Levels. Distances. Remarks.
    0,00 0,00 Datum.
4,0 7,6 3,6 2,00
2,0 9,0 10,6 4,00
4,0 7,0 13,6 6,00
,0 2,6 10,0 8,00
16,0 26,0    
16,0
10,0 Difference, the same as last reduced level.

You have thus the materials before you for
drawing the section, as in the column containing
the reduced levels (taking the level of the ground
at A as the datum) you have a fall of 3 feet
6 inches at 2 chains from A; and at 4 chains (the
distances being continuous) you have a fall of 10
feet 6 inches below your starting point. You will
thus continue it until you have plotted the result of
each observation, first marking off the distances
on the datum line, and then the reduced levels
answering to such distances:--the method pursued
in plotting a section will be immediately understood


[page 98]

on reference to the following diagram, which is the
plotted section of the example given, the horizontal
scale being 2 chains and a half to 1 inch, and the
vertical scale 20 feet to 1 inch. We should here
observe that the vertical scale to which a section
is plotted is always different to the horizontal: if
it was not the case, the variations of the ground
would be scarcely perceptible.


Thinking that the student has now received
sufficient insight into the principles of levelling,
we will here give an example, with the forms of
field-book made use of by practical men, which
alone are useful; the description, adjustments, and
method of using the different levelling instruments
being given hereafter.

The more simple the form can be reduced to,
the less liable to errors and confusion. The follow-
ing
is the most simple that we are acquainted with,
in which all the levels may be cast out in the
field, by which means much time is saved in the
office, and little remains to be done, but plot the
section.

The following example is part of a contract
section taken for a railway. The distances were mea-
sure
d with a 100-feet chain, consequently the quan-


[page 99]
[H2]


tities in the distance column are feet. The section
(plate 7) is plotted to a scale of 5 chains to the inch,
for which purpose a scale divided into 330 to the
inch was used, that number of feet being contained
in 5 chains.



[page 100]


The method of procedure in setting down,
casting out, and reducing the levels in the above
form of field-book, will be easily understood on
reference thereto. In the second and third
columns are entered the back and fore sights,
opposite to each other: the first contains the
difference of the two, if a rise; if a fall, it is
entered in the fourth,* the differences in this form

* When the difference of columns 2 and 3 have been cast out, the four
columns should always be added up at each page, previous to reducing or
carrying the difference to column 5: the total rise or fall will correspond, if


[page 101]

being set down in the order they are cast out.
The fifth contains the reduced levels, obtained by
adding or subtracting the differences to the pre-
vious
reduced level; the sixth the distance (which
will be perceived is continued from the commence-
ment
to the end of the section); and the last, which
is the largest, for remarks, to note the crossings
of roads, rivers, brooks, &c., and, if necessary, to
enter the bearing of your line of levels; but in the
above case no bearings were taken, as the ground
had been surveyed, and the line definitely marked
out. In the forms of field-books generally used there
is a separate column for the bearing at each station,
which is quite useless; it is sometimes, indeed,
necessary in running trial levels through a part of
the country you are unacquainted with, to note the
bearing of any object you may be running to; and
when you change the direction of your line, again
note the bearing with the chainage, so that you
may be enabled to lay down on a map. the line you
have been taking; but it will not be necessary for
you to note the bearing any oftener than you
change your general direction.

The sketches of the crossings of roads, rivers,
&c. in the fifth column need no explanation; they
are plotted to a large scale on the section, and will
be found of infinite service to the engineer in
guiding him as to the dimensions of his bridges,
so as to maintain a proper width of roadway, &c.

correct; the reduced level must also correspond by deducting the quantity
brought forward at the commencement of each page.


[page 102]

The above example of a field-book comprises
all the data requisite or necessary for drawing a
contract section, for which purpose it was taken.
The levels are based on Trinity datum, which was
previously ascertained, and bench marks left at
each end. To show with what correctness levelling
operations may be carried on, it will be only neces-
sary
to state, that the difference between the bench
marks at each end, from that previously ascertained,
amounted to 0,02 only, or two-hundredths of a foot,
the staff being graduated decimally.

The plotted section from the above field-book
is given at the end of this treatise, by referring
to which, and carefully looking over the field-
book
, the student will become perfectly acquainted
with every particular necessary for the purpose.
Where the ground slopes transversely, it becomes ne-
cessary
to take cross sections at intervals, but iii the
above, the ground laid so fair it was not at all requi-
site
. It is also necessary to take the section of every
road, lane, or public way over or under which it is
intended to build a bridge, for four or five chains
or more on each side of the point of crossing, as
you will then be able to judge of the best method
of crossing it, and be able to show the requisite
quantity of cutting or embankment, in carrying
the road under or over the railway. Curvature is
never allowed for in practice, except in the most
delicate operations, as although the instrument may
not always be midway between the stations, yet a
mean of a number of observations will give the


[page 103]

true level, provided the inclination of the ground
be not continuous ; but even so, if the operation
of levelling is properly conducted, the errors from
this source would be too trivial to be noticed.
In the method of allowing for curvature, the
student is referred to the introductory chapter.
But it should be observed, that it is always de-
sirable
in levelling operations, to place the staff
as near equal distances (roughly estimating it
with the eye) on each side of the instrument as
can be done conveniently; which (if the instru-
ment
should be out of adjustment) will neutralize
the errors thereby occasioned: although, if the
adjustments are perfect, it is not necessary to ensure
accurate results, that the staff should be placed at
equal distances. In practice, the operator generally
chooses the most commanding spot of ground on
which to plant his instrument, without regard to its
being in line with the staffs, or at equal distances;

although, as we have before observed, it should be
attended to when it can be done conveniently.

We should here remark, that it is always
preferable to use one staff only, on account of the
difference of the graduations, although made by
the same person with ever so much care. We
have frequently, when adjusting our instrument,
found a difference of 1/100 of a foot in the read-
ings
of two staves, although made by the same
person, and no doubt the errors are often much
greater. To avoid this source of error, we would


[page 104]

always advise the use of one staff in preference to
two ; as in the latter case there is very little, if
any, time saved, an additional expense incurred,
and the chances of error greatly multiplied.

Bench Marks.

It will be perceived on reference to the field-
book
; that in the distance column is frequently
inserted B. M. in place of the chainage. These are
bench marks, which consist of fixed objects, as
notches cut in trees, hooks of gates, boundary
stones, &c., which are left for the purpose of re-
ference
at any future time, either for checking,
varying, or continuing the levels. The remarks
necessary to identify them are inserted opposite in
the column under that head. A section is always
commenced and finished at a bench mark; if none
exist, they are made. In the above section, the
levels were commenced from a bench mark which
was a notch cut on the root of an oak tree
near to the line, and which was previously ascer-
tained
to be 8,096 feet above Trinity datum.
The staff being first placed on the bench mark,
and then on the ground at the point of com-
mencement
, gave the difference of level 1,11,
which being a rise, was added to the height of
the bench mark, making the point of commence-
ment
of the section 82,07 above Trinity datum.
The levels were then carried on in the usual


[page 105]

manner to the crossing of the first road, where it
was thought necessary to leave a bench mark.
The method of computing the levels of these
bench marks, is to set down the reading of a staff
placed on any one of them as a forward station;
and on placing the staff at your forward station on
the ground, the reading on the bench mark becomes
the back station to it-which will be best under-
stood
by referring to the readings in the field-book;
but where you finish a section, which must always
be done on a bench mark, the reading (as in the
example) becomes the last forward station; the
reading of the staff planed on the ground, at the
extremity of the line of section, becoming the back
station to it. Of course when the levels are re-
newed
, this bench mark becomes the back station,
in the same manner as at the commencement of
the section given as an example.

At the close of a day's work, if a bench mark
cannot be found conveniently, you must drive a
stake firmly into the ground, until its top is on a
level with the surface, the staff being then placed
on it, answers for your forward station, and becomes
a temporary bench mark, from which you resume
your work the next morning, in the same manner
as if you had not suspended operations, not re-
cording
this temporary B. M. in your field-book,
but the distance in the proper column as usual.


[page 106]

Trial Levels.

The manner of taking trial sections, where
there are several lines to be levelled for the pur-
pose
of ascertaining the best, is so similar to the
example of levelling already given, as to be
scarcely worth introducing; but as many surveyors
or students would, perhaps, from the minute data
given in the example, be apt to think the same
necessary in every case, we have thought it best to
give an account. (with examples) of this kind of
levelling.

The object in taking trial sections, is to ascertain
generally the best of several lines, for which pur-
pose
it is not necessary or requisite to show every
trifling variation in the ground, but oniy the
general features of the line over which you are
passing; as in all cases where trial sections are
taken, the levels are taken anew over the particular
line decided on, but even then it is not at all neces-
sary
to give the minute particulars as contained in
the previous example, the sights being taken as far
as the instrument will command backwards and
forwards within limits; of course sudden rises
or falls must be shown, but the surveyor must
hear in mind that he is not taking a working
section, which can only be done when the line is
definitely marked out on the ground: a lateral
difference of only a few feet from the intended line


[page 107]

when not staked out (which in most cases cannot be
avoided), will generally make the minute details of
a section quite different from that obtained when
the centre line shall be accurately set out. The
surveyor will, therefore, see the folly in wasting his
time in those particulars which must, under the
circumstances, be useless.

The accompanying sections were taken for the
same line of railway, although some by the different
routes taken are much longer than others. The
method of plotting these sections for the purpose of
comparison is obvious; they are all based on the
same datum line, the same bench marks being
referred to at the termination, as well as the com-
mencement
of each line, for the purpose of testing
the accuracy of the levels taken for each section,
which, if practicable, should always be done. It
was not known at the time how much above Trinity
datum the point of commencement was, no bench
mark having been fixed; one was accordingly left,
which was assumed as 100 feet above it, to which
all the levels were reduced and plotted.

It was afterwards ascertained that the height of
the B. M. was 115 feet above T. D., being 15 feet
higher than that assumed; it oniy then required a
line to be drawn on the section 15 feet below that
assumed to give the correct datum; and by adding
the difference of 15 feet to any reduced level in the
field-bo6k, of course the exact height of that point
would be given without reference to the section.


[page 108]

By tinting the surface of each section a dif-
ferent
colour, their merits are at once determined
without confusion. The field-book of the section
tinted brown is only given as being sufficient for
the purpose. The page at whitth you commence
your levels should be headed somewhat in the
following manner:--Trial Levels for the--Railway
or Canal, as it may be, from (the name of the place
you commence at, to the place you are to finish),
with the date, name of the engineer, and any other
particulars that may occur to you.



[page 109]


The section (pl. 8) is plotted from the above field-
book
to a scale of 4 in. to the mile horizontal (which
is 20 chains to the in.), and 50 ft. to the in. vertical.


[page 110]

The roads, &c. written on the section refer to that
tinted brown. If the roads, &c. of those tinted red
and blue had been written in, it would have been
done with the same colours as the respective sections
are tinted with. The distances in miles, and also
the vertical and horizontal scales, should be written
on the section. Many engineers also require the
horizontal distances and vertical heights to be
figured, which is a check on the accuracy of the
plotting.

At crossing all roads, lanes, rivers, brooks, &c.
the local names should be ascertained and written
on the section; also the names of places to and
from which the roads lead; in fact, too much
information cannot be obtained:
what at the time
often appears trifling becomes afterwards a matter
of great importance. It would be as well also to
note in your book the description and quality of land
over which you pass, whether clay, gravel, chalk,
&c., &c., and whether ballasting can be had, as on
these particulars often depends the eligibility of any
particular line.

Check Levels.

When the trial levels are taken, and the line
decided on, it is again levelled, but with more care.
At, various points are left bench marks (generally
about every two miles) which, if not previously deter-


[page 111]

mined on, are very minutely described by the person
leaving them (a distinguishing mark being left to
enable the person coming after to find it with greater
facility). To be certain that no1 errors have been com-
mitted
in taking the section, it is usual for some
other person to check the work by ascertaining the
difference of level between the various bench marks.
If these differences of level prove to be the same as
before, or nearly so, it is fairly to be inferred that
the intermediate levels forming the section are also
correct.

The person taking the check levels does not
pass over the line on which the section was taken,
but proceeds by the nearest and most convenient
route from bench mark to bench mark, generally
by a road if running near; not, noticing the varia-
tion
of the ground, and taking as moderate long
sights as his instrument and the ground will permit.
No chain is required in this kind of levelling, but
the surveyor should be careful to plant his stave
(as near as his eye will direct him) at equal distances
on each side of his instrument : errors arising from
curvature, long sights, &c. will then be nugatory.
The section tinted brown was decided on as the best,
but there was not time to level it over again, and
the trial section was obliged to be deposited with
the Clerk of the Peace, but which should never be
done when by any possibility it can be avoided.
But whatever it wanted in detail it was necessary to
have the general results correct, for which purpose


[page 112]

check levels were taken to various bench marks,
but not to all that were left, which would have
been a waste of time, but only to those that were
conveniently situated, or of material consequence, as
the crossing of a summit, or any particular road or
point on the line, to which reference might be made.

The first bench mark on the line that was levelled to,
was that at the style, which was not at all necessary,
being so short a distance from the commencement,
but as it laid very convenient it was taken.

The check levels up to that point are given,
which will explain the method.


It will be seen on reference to the former field-
book
, in which the height of the B. M. at the style
is entered, that the difference of the two levellings
was but 1/100 of a foot. It was, therefore, assumed


[page 113]
[I]


that the intermediate levels were correct, and in
this manner were the check levels taken over the
whole line. The results may be obtained by casting
out the differences of each observation (but not re-
ducing
them), and adding up the several columns to
prove each other; or, as it is generally done by
simply casting up the back and fore sights, and
subtracting the one from the other, being satisfied
of the correctness of such addition by casting the
columns a second time, commencing from the top.
The same form of field-book is here given as
for the other sections, although but two columns are
required, but it is presumed that the surveyor has
his field-book ruled throughout in the same manner.

Example in Levelling with Plotted Cross Sections.
(Plate 8.)

As we have observed in the directions for taking
a working section, where the ground is at all sideling,
cross sections must be taken at right angles to the
main line.

We need only refer to the plate in this case,
it being so plain as scarce to need a description.
We have omitted the field-book as being unnecessary,
the levels of the main line being taken in the same
manner as already described. The cross sections
were taken, at the same time as the main line, and
in the following manner:--On bringing the levels
up to any point of the line where we thought it was


[page 114]

necessary to take a cross section, we drove in a
stake, level with the ground, and placed the staff on
it; then, without removing the instrument, we read
a staff placed at several points at right angles to the
main line on each side the stake, measuring the dis-
tances
. The method of plotting is obvious, the point
where these cross sections intersect the main line
becomes a temporary B. M., and the rise or fall on
each side is plotted from a datum line drawn through
that point at right angles thereto.

The surveyor will, where the line is staked out,
generally be able to command the ground on each
side as far as required, without removing his instru-
ment
, and can then continue the principal section
as usual. He will thus, by exercising his judgment
in the planting of his instrument, be enabled to
take the cross sections, wherever requisite, without
any additional trouble or loss of time. Some engi-
neers
like their cross sections plotted above the main
line, as at A--this method is perhaps the best.


[page 115]
[I2]


ON CHAINING.
WITH METHODS OF PASSING WOODS, RIVERS, AND OVER
COMING DIFFICULTIES THAT OCCUR IN LEVELLING.

IT often happens that in taking any consider-
able
length of section that difficulties intervene, as
woods, lakes, private grounds, forbidden property,
high walls, &c., in which case the surveyor is often
puzzled how to proceed.

In some recent levelling operations in which
we were engaged, in crossing some private property
a very high wall intervened, which apparently put
a stop to further proceedings, at least in a straight
line. To plant the instrument in any position so as
to look over the wall was impossible; and to level
round to the opposite side would have taken too
much time, the distance being nearly a mile, and
expedition absolutely requisite; it was also necessary
that the levels should be taken with the greatest
accuracy. If an inch or two difference in the levels
had been of no consequence this difficulty would
have been easily overcome by measuring from the
top of the wall, or counting the courses of bricks
cut by the line of sight on each side. We removed
the earth (which was only a few inches), from the


[page 116]

side of the wall, to the upper course of footings, and
placed the staff thereon; we then clambered over
the wall, and, exactly on the opposite side, cleared
the earth away in the same manner to the footings,
which was about a couple of feet deep; placed the
staff thereon, as on the opposite side, calling the
reading the same, and then proceeded in the usual
manner with the section. We subsequently levelled
round to the opposite side of the wall, having left
marks for that purpose, when we found the reading
of the staff placed on the footings to be exactly the
same on both sides.

It will also often happen that in taking a section
you come to woods, or private grounds, through
which you cannot pass; it will then be necessary to
level round to the opposite side, until you come to
the spot you would have arrived at had you con-
tinued
on without any obstruction; but in case you
have not a plan of the ground, and cannot see the
opposite point at which you would have arrived had
you continued on, it will be necessary to measure
round it, in as few lines as possible, taking the
angles with your compass, but if you should not
have one to your instrument you must take the
angles with the chain in the manner described for
surveying a field without a diagonal. When you
think you have measured far enough round, you
must plot these lines; then, by laying down on this
plot the line you were taking previous to such
obstruction, you will see at what point you would


[page 117]

have come out; apply your scale to it, enter the
distance in your book, bring your levels round to
that point, and continue the section. By setting
your instrument to the same bearing at this point as
on the opposite side, your direction will be plainly
pointed out.

If you come to a lake or pond, the operation
will still be more easy, by simply turning at right
angles to the right or left of your line, noting the
exact distance, until you can clear the obstruction;
then turn at another right angle, and measure the dis-
tance
forward (which will be parallel to the line you
would have measured, had not the obstruction ex-
isted
) until you get clear, then by turning at another
right angle, and measuring out the same distance
as at first, it is evident you will come to the same
spot you would have arrived at, had you continued
your line forward unbroken. This method may be
adopted where you come in contact with buildings,
or small pieces of wood-land, and generally where
the object to be avoided is not of great extent--if
so, it is attended with great uncertainty, and should
never be practised, but recourse had to the method
pointed out above. If you avoid the object by
means of right angles, you should not judge of
such angles by your eye, but measure them with a
sextant or an optical square, which is a small
instrument, with the glasses of the sextant re-
flecting
a constant angle of 90°. If you have no
such instrument, nor a compass, you must trust to


[page 118]

your eye, and, for short distances, may generally
set it off correct enough; but for any considerable
distance, accuracy cannot be expected. (See Parish
Surveying).

For the method of taking inaccessible distances
with the chain only, as the crossing of a river, &c.,
the reader is referred to the part of the work on
"Surveying," where he will find every information.

Levelling with the Theodolite.

In levelling operations, where despatch is more
essential than results of extreme accuracy, or to take
cross sections where the ground is precipitous, the
theodolite may be used with great advantage. The
manner of applying the theodolite to the purposes
of levelling is by taking a series of vertical angles;
and in the absence of the spirit level it may be
used in a similar manner to that instrument by
clamping the vertical arc at zero, setting it up at
each change of level, and taking back and fore
sights, exactly in the manner described for using
the spirit-level: but the manner in which it is
principally used in levelling operations is, by taking
a series of vertical angles along the proposed line;
but it is absolutely necessary, in this kind of
levelling, that your instrument be adjusted to the
greatest nicety, as on the extreme accuracy of its
performance, and the great care of the observer,


[page 119]

every thing depends; any inattentiveness on the
part of the observer, or carelessness in adjusting
his instrument, will make the results worthless.

The line being determined on over which you
purpose taking a section, you must set up your
theodolite at the commencement (the adjustment,
as explained in the chapter devoted to that
instrument, having been carefully attended to), and
level it by means of the parallel plate screws.
Then ascertain the exact height of the optical axis
of the telescope, which is best done by measuring
to the centre of the eye-piece; a vane staff must
be used, and the vane set carefully to this height.
An assistant must now pass along the line (not
regarding the intermediate undulatioris of the
surface) until the general inclination of the ground
changes, at which spot he must fix it. The
telescope must now be moved in a vertical plane,
until the bisection of the vane on the staff by the
cross wires is perfect, but to ensure accuracy the
instrument and staff should change places, and
the angle be observed as before; if there is any
difference, a mean must be taken, to which the
instrument must be set, and clamped. An assistant
may now commence chaining the distance, and at
each change of level in the surface of the ground,
as in common levelling operations, erect the staff,
and slide the vane up until bisected by the cross-
wires
of the telescope, when it will be in the ima-
ginary
line connecting the instrument with the


[page 120]

staff first erected. Your assistant must then enter
the distance, and the height of the vane on the staff,
in his book, and continue the chainage onward
until a further change in the ground requires the
staff to be again erected, and so on to the end.

The most simple method of applying the data
thus obtained is to prick off the observed vertical
angle, which is the line of sight, and draw it in;
on this line mark off the measured distances, from
which let fall perpendiculars to the horizon, of the
lengths denoted by the vane at the various inter-
mediate
stations; a line being traced from the
extremities of such perpendiculars will represent
the surface of the ground; and these perpendi-
culars
being continued until met by a horizontal
line drawn from the point at which the instrument
was first set up, or any assumed datum, will show
the changes of level at the various points: but the
most correct method is to calculate the difference
of level between the stations, which may be done
in the following easy manner, by the aid of a table
of logarithms and logarithmic sines-the measured
distance being the hypothenuse of a right-angled
triangle:--To the log. sin, of the observed angle
add the logarithm of the measured distance, and
their sum, deducting 10 from the index, will be the
log, of the difference of level;
as for example,
suppose the angle of elevation to be 2° 40'; and
when the instrument and staff changed places, the
angle of depression (which it would then be) to be


[page 121]

2° 42', the mean 2° 41' would be taken, and the
measured distance to be 7,500 feet.

Log. sin 2° 41' 8,670393
Log 7500 3,875061
Log. of 351,018 2,545454
Showing the difference of level to be 351 feet.

Or the difference of level may be found thus,
which is perhaps the most ready method: from a
table of natural sines, take the sine of the angle, and
multiply the hypothenuse (or measured distance) by
it, which will be the difference required. The base
may be also found by taking from the same table
the cosine of the angle, and multiplying it by the
hypothenuse.
In the above example, the distance
being so great, the necessary correction for curva-
ture
and refraction must be applied, but the angle
being so small, the correction may be applied to the
measured distance without sensible error, the hypo-
thenuse
and base being nearly the same. The
example will then stand thus:--

  Feet.
Observed difference of level 351,018
Correction for Curvature 1,35
" Refraction ,19
1,160
  True difference of level 352,178

In taking a section with a theodolite, it would


[page 122]

be desirable to take only avery few intermediate obser-
vations
between your stations, it would then be best
to calculate the bases and difference of level for each
observation in the manner already described, of
course adding to or subtracting from the calculated
difference, the difference between the ground where
the instrument is planted, and each intermediate
station below the line of sight as given by the
sliding vane.


The plotted section of the above field-book is
contained in plate 8; the scales are 330 feet hori-
zontal
, 100 vertical.

Columns 1 and 2 contain the angles of eleva-
tion and depression; col. 6 the hypothenuse, or
measured distances of the stations; col. 7 the bases
answering to the hypothenuses in 6; cols. 3 and 5
contain the hypothenuses and bases of the inter-


[page 123]

mediate stations; col. 4, the difference of the inter-
mediate
stations below the line of sight; as for
example :-at 5,05 the vane on the staff was at
14 feet when bisected, the instrument being set up
4 feet above the ground, the difference 10 feet is en-
tered
as minus; at 10,20 the, vane read 1,5, the
difference 2,5 is entered as plus. In the angles of
depression, these terms produce opposite effects on
the quantities they are connected with to what they
represent; thus at angle 5° 42' D at 600, a rise or
plus 3, this is to be subtracted from the difference of
level, it being a fall, which difference iias to be sub-
tracted
from the previous reduced level. Columns
8 and 9 contain the elevations and depressions from
the various stations, the differences in column 4 is
added to or subtracted from them as the case might
be. Column 10 contains the reduced levels, the re-
maining
part of the book being left for observations.
By thus calculating and reducing the levels, the
section may be plotted exactly in the same manner
as if taken with the spirit-level.

The above method of levelling with the theo-
dolite
, may be advantageously adopted in taking
trial sections for crossing summits-one section
should be carefully taken with the spirit-level, the
remainder may be taken with the theodolite--ver-
tical
angles being measured to the bench-marks at
the extremities; any errors that may have crept into
the calculations will then be detected.

If this plan was adopted, how much expense


[page 124]

would be saved; comparative sections may then be
taken in every direction without the enormous ex-
penses
incurred by the present method. Such
correct results may not be arrived at certainly as
with the spirit-level, but in these preliminary sec-
tions
, of what consequence is the differing a few
feet, when perhaps the crossing of a summit may be
effected thereby at a less elevation by ten times the
amount of error.
The theodolite is not put in com-
parison
with the spirit-level for accuracy, but for
the above purposes, where a near approximation is
all that is necessary, it is presumed it will be found
a far more advantageous instrument, as, in some
districts, the surveyor might do ten times the
quantity of work as with the spirit, level.

Of the Method of taking Cross Sections with the
Theodolite.

The superiority of the theodolite over the
spirit-level will be here manifest. In many cases
new roads are projected along the side of a ravine,
where it would be scarce possible to plant a spirit-
level
; in this case the cross sections which would
be absolutely necessary, might be taken with the
greatest ease and accuracy with the theodolite. At
the bottom of the valley, or so far below or above
the centre line of the road as you can get a firm
footing, plant your instrument as before directed;
set the vane on the staff at the same height as the


[page 125]

optical axis of the telescope, and set it up exactly
on the centre line of the road where you intend
taking the cross section, take the vertical angle, and
measure the distances; where the ground varies,
set up the vane-staff, and note the height at which
it is bisected-the method of reduction and plotting
will then be the same as in the example given.

The following sketch will illustrate this method
of taking cross sections:--

Suppose B A D to be the cross section of the
ground, and A the centre line of the road. At B plant
your instrument, suppose 4 feet from the ground,
set up a staff at A with the vane at the same height,
and take the vertical angle; where there is a change
in the ground from the general inclination, set up
the staff as at a b D, slide the vane up until bisected,
and measure the distances, the calculations for the
bases and differences of level will be the same as in
the example given; the section can be plotted to
any scale as in common levelling operations.


[page 126]

SECTIO-PLANOGRAPHY
Or New Method of laying down Sections for Rail-
ways
and other Public Works, as recommended by
Mr. Macneill, and required by the Standing
Orders of the House of Commons

IN the previous examples we have given, the plan
and section are. detached, and totally unconnected
with each other. The inconvenience this gave rise
to was strikingly mahifest in the numerous railway
projects that were recently before the legislature.
Very few except professional men perfectly com-
prehend
the vertical section; and still fewer, after
finding their property on the plan, can refer to
the corresponding part on the section,--reference
having to be made to scale and compasses. Even to
professional men this is a work of trouble and time,
and is often inaccurately performed. Was it then
probable that a country gentleman or farmer could,
on inspection of the plans and sections, understand
how their estates or farms would be affected? In
many cases it has been proved that the agents of
railways, presuming on their ignorance of such mat-
ters
, have purposely misinformed them for the pur-
pose
of gaining their acquiescence. To remedy the


[page 127]

inconvenience, uncertainty, and fraud attendant on
drawing the plan and section detached, Mr. Mac-
neill
contrived the admirable method of connecting
them, which we will now describe.

On reference to Plate 6 will be found, in the
upper part, a section drawn in the usual manner,--
the plan appearing beneath. The strong black line
on the plan represents the position of the proposed
railway on the ground, and is also considered to
represent a vertical section of the rail, the undula-
tion
of the ground being marked thereon precisely
in the same manner as in the section above:--in
fact, the line on the plan (whether straight or
curved) is taken as a datum; and after the gradient
or rate of clivity is put on the detached section, the
heights of embankment or depths of cutting--ac-
cording
as the rail line is above or below the surface
--is easily and accurately transferred to, or plotted
on the plan. But as the quantity or position of the
various cuttings and embankments might not be
readily seen on the plan, the cuttings are coloured
red, and the fillings blue: where the rails are in-
tended
to be on a level with the surface of the
ground, no colour is applied; where the plans and
sections are lithographed, to save the trouble of co-
louring
, the cuttings are represented by vertical shade
lines, and the fillings by similar horizontal lines. By
this method any person, however slightly informed
on such subjects, could immediately perceive on
inspection of the plan how any particular property


[page 128]

would be affected, and whether the railway would
pass through it in cutting, embankment, or on a
level.

On the Choosing of a Datum Line.

In the commencement of levelling operations,
it is necessary (as already explained) to fix on some
well-defined mark, as the standard to which the level
of each place is referred, from this an imaginary line
is drawn, called the datum, which generally has re-
ference
to the tides. Some engineers choose for their
datum the level of the highest spring tides; others
take low water springs. Let us enquire which is
the best. The height of high water above the iow
water constitutes what is called the tide. Many cir-
cumstances
render it almost impossible to say what
is the elevation of high water above the natural sur-
face
of the ocean: a strong south-west or north-west
wind raises the tides to an unusual height along the
east coast of England, and in other places the reverse
is the case. In the open ocean, the tides rise to but
very small heights in proportion to what they do on
the coast. It is also well known that the tides rise
in many rivers, channels, bays, and estuaries, to an
elevation far above the average height of the same
tide
on more open parts of the coast. In the estuary
of the Severn at King's Road, near Bristol, the rise is
42 feet; at Chepstow-on-the-Wye, a small river
which opens into the same estuary, about 50 feet;


[page 129]
[K]


at Milford Haven, 36 feet; at London, 18 feet; at
the promontory of Beachy Head in Sussex, 18 feet;
at the Needles in the Isle of Wight, 9 feet; at Wey-
mouth
, 7 feet; at Lowestoff, 5 feet; and at Great
Yarmouth
, 3 or 4 feet only.

In many places the surface at low water is
above the natural surface of the ocean;
this is the
case in rivers, at a great distance from their mouths.
This may appear absurd, and is certainly very para-
doxical
; but it is a fact established on the most
unexceptionable authority. One instance will suf-
fice
: the low water-mark at spring tide, in the har-
bour
of Alloa, was found by accurate levelling to be
three feet higher than the top of the stone pier at
Leith, which is several feet above the high water-
mark
of that harbour. A little attention to the mo-
tion
of running water will explain this: whatever
checks the motion of water in a canal, must raise its
surface; therefore a flood tide, coming to the mouth
of a river, checks the current of its waters, and they
accumulate at the mouth; this checks the current
farther up, and therefore the waters accumulate
there also: and this checking of the stream, and con-
sequent
rising of the waters, is gradually communi-
cated
up the river to a great distance; the water rises
everywhere, though its surface still has a slope.

It is an extraordinary fact, but some profes-
sional
men expect the levels of spring tides to cor-
respond
in every part of the kingdom; and that
the level obtained in a river remote from the sea,
but influenced by the tides, will be found to corres-


[page 130]

pond when brought down to the sea. We know
of instances where engineers have pronounced sec-
tions
to be wrong that were taken to the coast,
because the high water spring tides did not corres-
pond
with that at London. What could be more
fallacious,--the difference of the rise of the tides
in many of these instances exceeding 20 feet. The
level of low water spring tides, which is considered
more equable, is now generally taken by engineers
as their datum line,--which has also this advantage,
that in your section you never run below your
datum, which is very troublesome; whereas, by
taking high water springs, you are doing so con-
tinually
; scarce a low track of land in the neigh-
bourhood
of the sea or a river but what is on a level,
or beneath high water spring tides, and only kept
from being flooded by embankment. We should
observe, that in taking high or low water as a
daturn,--if no natural or artificial conspicuous mark
exist at exactly the height of the tides,--a good
bench-mark must be left in the immediate vicinity;
by which means, if not quite certain as to the tide a
mean can be taken, it will also be found necessary
for future reference. But for extensive operations,
we would advise the mean level of the sea to be
taken; which (according to M. De la Lande's me-
thod
, and adopted in the Trigonometrical Survey of
England) may be obtained by taking the level of low
water, and deducting therefrom one-third of the
height to which the tide rises.


[page 131]
[K2]


ON LEVELLING INSTRUMENTS.

PREVIOUS to giving an account of the various kinds
of spirit-levels in use, we would particularly impress
on the minds of surveyors and persons who use
levelling instruments, oniy to use those of a superior
quality; as an inferior instrument (the correctness of
the work entirely depending on the nicety of the
finish) will be a never-ending source of error. Let
no false economy, for the sake of saving a few pounds,
induce you to purchase a common instrument; but
whatever kind you purchase, let it be the best: then,
with care, your workwill always be a credit to your-
self
, and a satisfaction to your employer.

The staff or stand on which your level is mounted
should also be of good substance: a tripod is the best,
and made of light fir, and not less than 5½ feet in
height when closed. You will then have a good
firm mounting for your instrument; the good effects
of, in the field, and great advantages over, the skele-
ton
mahogany legs that are generally applied to
levels (which you would suppose were made more for
show than use), will be very soon apparent. Also,
when in the field, the surveyor should be careful, in


[page 132]

reversing the instrument, to turn it only from righit
to left;
as it often happens (where the level is
screwed on to the upper plate), after observing the
staff at the back station in attempting to turn it
the reverse way to observe the forward staff, that
the instrument becomes partly unscrewed from the
plate, and consequently its horizontality destroyed.
If the staff at the back station should have been
removed, it will then be necessary to go back to the
last B. M., and commence afresh; but by minding
to turn it always from right to left, this will never
be the case.

A Description of the various Levelling Instruments,
with their Adjustments, and the Method of
Using them in the Field.

The Y Level.-We have commenced with the Y
level (so called from the supports which carry the
telescope resembling the letter Y), as being the
oldest description of instrument at present in use,
but which is now nearly superseded by instruments
of far superior and more correct construction. The
adjustments of this instrument are easily performed,
which may account for the pertinacity with which
some people assert its superiority; but on the other
hand, they are also as easily deranged. The teles-
cope
, in this instrument, is generally made to show
objects erect; it is consequently darker and less dis-


[page 133]

tinct than the improved instruments, which have
telescopes of larger diameter, with fewer glasses,
and therefore of more brilliancy.

The first adjustment in this instrument is the
line of collimation; to do which, set up the instru-
ment
in any position, open the rings which confine
the telescope within the Y s, and after adjusting for
distinct vision, (paying no regard to the spirit-bubble)
bisect some well-defined object with the cross wires
and clamp the instrument firm, at the same time
turning the telescope gently round as it lies on its
supports or Y s, and observe if the bisection continues
during a revolution of the telescope: if so, all is
right; if not, you must alter the screws which carry
the cross wires or diaphragm until the telescope will
revolve in the Y s, the bisection remaining perfect.
After you have adjusted the line of collimation, you
must carefully level the instrument by means of the
parallel plate-screws; and when the spirit-bubble
remains steady in the centre of the tube (the rings
or clips remaining open), you must carefully reverse
the telescope end for end, the eye-piece being in the
place previously occupied by the object-glass; if
the bubble returns to the centre of the tube, it is
correct; if not, you must observe the end it retires
to, and correct half the error by raising or lowering
one end of the bubble-tube, by means of the shrews
by which it is attached to the telescope, and the
other half by the parallel plate-screws which will
bring the bubble to the centre of the tube. You


[page 134]

must now again reverse the telescope, the object-
glass
being in the same position as at first. The
bubble should now, if correct, return to the entre
of the tube; if not, you must alter it, as just di-
rected
, until you can reverse the telescope end for
end,--the bubble, in each case, returning to the
centre of the tube.

The next adjustment is to make the adjusted
telescope perpendicular to the vertical axis; or, in
other words, to make the instrument revolve on its
stand,--the bubble remaining in the centre of the
tube the while; if this is not the case, you must
raise or lower the milled head-screw (which carries
one of the Y s, and consequently the telescope and
bubble tube) one half the observed error, the other
half being corrected as before by means of the pa-
rallel
plate-screws; and if the bubble will not now
remain in the centre of the tube, but retires to either
end, it must be repeated until there is no perceptible
difference. The instrument will then be in a proper
state to observe with; but as the adjustments of the
Y level are.easily deranged, it is absolutely necessary
to examine them frequently,--and as it is easily
done, we should recommend it every morning: and
indeed whatever instrument is used, the surveyor
will find it to his advantage to devote a few minutes,
every morning before proceeding to work, to examine
and rectify if necessary the errors of his instrument.


[page 135]

Troughton's Improved Level.

The improved level is an admirable instrument,
capable of taking levels to the greatest degree of ac-
curacy
, and is generally constructed to show objects
inverted. Its adjustments in the hands of a beginner,
or a person only accustomed to the old Y levels, may
appear troublesome, tedious, and difficult; but in
reality, when fully understood, they are performed
with much greater facility, and with far more satis-
faction
to the operator. The only objection that can
be made to this instrument is, that no adjustment is
applied to the spirit-bubble tube; so that if, by
accident or otherwise, the tube gets deranged or dis-
turbed
in the bed in which it is fixed by the maker,
the line of collimation must be adapted to it, although
no longer remaining in its original situation.

The adjustments necessary for the improved level
are the same as those for the Y level, although from
the different construction of the instrument they are
differently performed. The bubble tube in this in-
strument
has no adjustment, being fixed by the
maker in the cell provided for it, which is firmly
attached to the telescope. The line of collimation
must, therefore, be adjusted to suit the bubble-tube;
the most easy and correct method of doing which is
to set up the instrument on a tolerable level piece of
ground, and set it horizontal by means of the parallel


[page 136]

plate-screws; then, at a distance of three or four
chains on each side of the instrument, drive a stake
firmly into the ground: on these stakes alternately
place your staff, and note the graduations bisected
thereon by the cross hairs of the telescope. You
will thus obtain the true level of two points,--the
graduations bisected on the staff being equidistant
from the earth's centre, however much your instru-
ment
may be out of adjustment.

Then remove your instrument beyond one of
the stakes six or eight feet, but the nearer the
better, and again read the staffs alternately placed
on these stakes, and if the readings give the same
difference of level, it shows that the line of collima-
tion
is correct; if not, you must alter the collima-
ting
screws, raising or lowering, the diaphragm, as
the instrument looks upward or downward, until
the reading on the farther staff (the reading on the
nearest remaining the same) gives the exact differ-
ence
of level, previously ascertained, of the two
stakes. You must now ascertain if the instrument
will revolve on the staff-head, without a sensible
difference in the bubble; if not, you must notice
which end it retires to, (as in the adjustments for
the Y level,) and correct half the error by the cap-
stan-headed
screws at one end of the horizontal
bar, and the other half by the parallel plate-screws.
If it will not then revolve without changing, you
must repeat the process, until it will turn quite
mund without perceptible difference.


[page 137]

Another method of collimating this instrument
is by a pool of still water, in which you must
drive two stakes exactly level with the surface, dis-
tant
two or three chains from each other; then set
up your instrument a few feet beyond one of the
stakes, and read a staff alternately placed on each,
which readings will be exactlf the same, if the
line of collimation is correct; if the readings are
different, you must alter the i~o1limating screws,
until the readings are the same on both stakes; or
you may set up your instrument exactly over one
of the stakes, and measure the height of the teles-
cope
above it, which should also be the reading of
the staff placed on the other stake, if correct. If
the reading is different, you must alter the collimat-
ing
screws, as before directed, until it reads the
same; then adjust the vertical axis to the plane of
the instrument, as before directed.

Description, Use, and Adjusiments of Mr. Gravatt's
Improved Level, commonly called (from its
appearance) the Dumpy Level.

This instrument is by far the most perfect foir
levelling operations; its short length (generally 10
inches only) rendering it less liable to accident, and
its adjustments, when perfected, requiring positive
violence to derange them. We have a 10-inch
level of Mr. Gravatt's construction, which has not


[page 138]

required adjusting for twelve months past, although
it has been nearly in constant use, with the addition
of having been sent per coach, at various times,
several hundred miles. The large object-glass
applied to this instrument gives it great advantages,
as greater brilliancy, and consequently, distinctness
in reading the staff, and a much larger field of
view: this instrument also inverts objects.

The bubble-tube, as in Troughton's improved
level, is above the telescope, but unlike it, has
mechanical means of adjustment; there is also a
cross bubble attached to the telescope, to assist the
operator in setting the instrument up more level by
means of the legs than be would be otherwise able
to do. The large bubble-tube in this instrument is
graduated to tenths of an inch, by which means
you can set it up more truly level, and instantly
detect the slightest change in the bubble. This
instrument may be adjusted in the same manner as
described for Troughton's improved level, but much
more correct by the method adopted by its talented
inventor; as, for example, set up your instrument
on a tolerable level piece of ground, and, as di-
rected
for adjusting the improved level, drive in
two stakes, one on each side of the instrument, at
equal distances of one or two chains; then read a
staff placed alternately on each, you will then
find the true level of two points, however much
your instrument may be out of adjustment; then
remove your instrument the same or double the


[page 139]

distance beyond one of the stakes, and again set it
up, and also measure out the same distance beyond
the instrument as the instrument is beyond the
second stake, and drive in a third. Now read a
staff placed alternately on stakes No. 2 and 3
you will then, by adding or subtracting the differ-
ence
of level between stakes No. 1 and 2, to the
difference of level between stakes No. 2 and 3,
obtain the true level of three points, viz., stakes
1, 2, and 3, which we will call A, B, and C, and
taking stake A as the datum, suppose the difference
of level to be as follows:--
Stake A 0,00 above Datum.
B 3,63
C 2,39

You must now place your instrument a few
feet beyond A, in a line with the three stakes, (but
the nearer the better,) and carefully mark, by means
of the graduations on the tube, the exact position
of the bubble, so that you cannot disturb or alter
the instrument, without detecting it. On looking
through the telescope the staff placed on A reads
4,74; on B, 0,95; and on C, 1,75. Nowhad the
instrument been in proper adjustment when the
reading at A was 4,74, the readings on B and C
should have been respectively 1,11 and 2,35; the
instrument, therefore, points downwards, the error
B being 0,16, and at C, 0,60. Now, was the
bubble only in fault, the error at C should be three
times that at B,

the distance being three times as
[page 140]

great; but 0,16 x 3=0,48 only; there is an error,
therefore, of 0,60--0,48=0,12, not due to the
bubble. To correct this error, you must raise the
cross-wires by means of the collimating screws, and,
neglecting the actual error of level, make the error
at B only one third at C; after a few trials, the
staff at B will read 1,05, and at C, 2,17, the reading
at A remaining the same. Now, 1,11--1,05=,06,
and 2,35--2,17=0,18; and as 3 x ,06 (the error
at B)=0,18 (the error at C), the line of collima-
tion
is in perfect adjustment.

What now remains to be done, is to raise the
object-glass of the telescope by means of the parallel
plate-screws until the reading at C is 2,35, the
reading at B will then be 1,11, that at A remaining
as at first, then by means of the capstan-headed
screws carrying the bubble tube, bring the bubble
into the centre of its run. There is still another
adjustment, that of making the telescope parallel to
its vertical axis, or to make it revolve on the staff-
head, the bubble remaining in the centre the while;
this is performed in the same manner as described
for Troughton's improved level, to which the reader
is referred. The operation of collimating upon levels
On Mr. Gravatt's construction may appear tedious and
complex, but after a few trials it will be easily under-
stood
and performed in a few minutes, but when once
perfected it will scarce ever need be repeated. But we
would nevertheless recommend that the adjustments
be always looked to before commencing operations,


[page 141]

the few minutes spent in so doing will be amply
repaid by the satisfaction produced in knowing that
your work is correct. In some instruments it is
found very difficult to make the bubble retain a cen-
tral
position while being turned round on its axis,
or, as it is generally expressed, to make the instru-
ment
reverse, the operator must in this case repeat-
edly
try to correct it, in the manner previously di-
rected
for making the telescope parallel with the
vertical axis. But if after many trials the bubble
will not remain in the centre of the tube, while the
instrument is reversed, it must be brought to that
point by means of the parallel plates at each reading
of the staff.

On Instrumental Parallax.

Instrumental parallax is often the cause of
great errors being committed, and more especially
in levelling operations-in fact observations of any
kind, whether in surveying or levelling, are worth-
less
if parallax exists. The causes of parallax are
easily explained when the method of remedying it
will immediately suggest itself.

The rays of light moving in straight and
parallel lines (although not actually the case, it may
here be considered so without sensible error) are im-
mediately
on coming in contact with the object-
glass
of a telescope bent on one side and turned


[page 142]

from their previous straight course, converging to
a point which is the focus, an image of the ob-
served
object being there formed; and for the pur-
pose
of distinguishing this object, an eye-glass of
magnifying powers is applied to the telescope.

To obtain a proper view of this image formed
at the focus of the object-glass, the focus of the
eye-glass should be at the same point, the cross-
wires of the telescope appearing at the same time
perfectly clear and sharp; if not it produces paral-
lax
,
which is at once detected on looking through the
telescope and bisecting any object, the cross wires
not remaining in contact with the object, but appa-
rently
moving as you move your eye, up or down,
or on either side, rendering it impossible to ascertain
the correct bisection. To remedy this it is neces-
sary
to move the eye-glass a very little until you
obtain a clear and well-defined view of the cross
wires, then turn the screw attached to the telescope
which communicates motion to the slide carrying
the eye-piece and cross-wires, until you obtain a
distinct view of the object, the focal point of the
eye-piece will now coincide with that of the object-
glass
on whatever part of the optical axis it falls
(the focus of the object-glass varying according
to the distance of the object), and on looking
through the telescope at a staff, or any well-defined
object, the observer will obtain a clear and distinct
view of the object and the cross wires, apparently
attached, and appearing equally distant; The proof


[page 143]

of the parallax no longer existing, will be in the
moving about of the observer's eye and no displace-
ment
taking place. The whole sum, substance, and
correction then, of the perplexing parallax, consists
in a very slight movement of the eye-glass of the
telescope, which must be further continued until the
parallax no longer exists.

Of the Cross Wires of the Diaphragm.

It will often happen from changes in the state
of the atmosphere, that the cross wires will break.
To repair this the surveyor should loose the colli-
mating
screws, and take out the diaphragm, then
draw out the finest film of silk from any garment of
that material you may have, and pass a little gum
water over it, let it dry, and gum or glue it on to
the diaphragm; if you should neither have glue or
gum, you may fix it on the diaphragm by making
a small cut in the sides with a knife just sufficient
to raise the metal, lay the film of silk in the notch,
and close it. Replace the diaphragm within the
telescope, and adjust for use as before directed.

Levelling Staves.

There are various descriptions of staves for
levelling operations, the oldest of which is that with
the sliding vane, moveable by the person holding it
with a cord passing through pulleys at top and


[page 144]

bottom, the staff being in one piece of about 12 feet
in length, graduated on the face into feet and inches,
or feet and decimals. There is also another kind of
vane-staff graduated similar to the above, but the
staff itself, instead of being in one, piece, is di-
vided
into two or three sliding pieces of about five
feet each, the vane in this case being moved by the
hand over the first division. When the observation
requires that the vane should be higher than that,
it is effected by leaving it at the summit of the first
division of the staff, and sliding it up on the second,
thereby reaching 10 feet; if this should not be high
enough you slide up the second in like manner on
the third until you reach the required height. The
reading on this staff is by an index on the side; as
for example, slide the vane to the top of the first
division it will be five feet, then slide the division
up on the second two feet, the bisection being per-
fect
, the reading on the index at the side of the staff
will be two feet, which added to the first division of
five feet, will be equal to 7 feet, the height of the
vane from the ground; in like manner if you slide
the second division up on the third 2 feet, the height
would be 12 feet, but these staves are now rarely
used except in remote districts where improvements
have not penetrated.

The staves now in general use (the inven-
tion
of William Gravatt, Esq.) are without any
vane, the graduations, feet, tenths, and hundreths,
being sufficiently distinct to enable the observer to


[page 145]
[L]


note the reading from the-instrument, and with the.
powerful telescopes now applied to spirit-levels, the
graduations can be distinctly, and accurately noted
at a distance exceeding ten chains, thereby saving
much time, and obtaining more accurate results.
The mechanical arrangements of Mr. Gravatt's
staves are very simple; they are in three pieces,
with joints similar to a fishing-rod, and when put
together for use, form staves 17 feet in length, but
when asunder pack very conveniently for carriage.
There are several modifications of Mr. Gravatt's
stave, differing in their mechanical arrangements
only, all retaining the main object, viz, having the
graduations sufficiently distinct to enable the ob-
server
to read off the quantities himself. The per-
son
who has most improved on Mr. Gravatt's
invention is Mr. Sopwith of Newcastle, whose staves
are very convenient; the graduations are nearly the
same, but the decimal parts of the feet are figured,
the subdivisions also are more minute; when closed
it is only five feet in length, but drawing out similar
to a telescope to 15 feet, a spring catch retaining
each joint in its place.

A staff has been contrived by the author, which
will be found more convenient than any that has
appeared before the public; one great fault with the
improved stave is, that in reading off with an in-
verting
telescope (which nearly all levels have now,
for reasons before explained) you are very liable to
error from the figures appearing upside down, you


[page 146]

are consequently apt to mistake one figure or di-
vision
for another, often leading to serious errors.
To remedy this inconvenience the figures on this
stave are inverted,
whereby, when viewed through
an inverting telescope, they appear in their natural
order, doing away with the confusion and uncer-
tainty
hitherto existing, and enabling the observer
instantly to note the reading with expedition and
accuracy, altogether making a considerable differ-
ence
both as to the quantity and accuracy of the
work. There are also various mechanical arrange-
ments
in this staff differing greatly from the pre-
ceding
; when closed for carriage it is only 5 feet in
length, but opens for use to 15 feet. A peculiar shoe
or foot is attached to this stave, which the author
considers of some importance:--practical men are
aware of the irremediable errors committed through
the carelessness of stave holders; when the face of
the stave is turned from the last forward station to
become the next back, an error of an eighth of an inch
or more is often occasioned through the clumsiness
of the holder in pressing the stave into the ground,
or lifting it up carelessly with clods of soil adhering
to it, and again putting it down with the face re-
versed
; the errors committed in this way are greater
than are generally imagined. Many surveyors
attempt to remedy this by putting a coin or some
flat substance beneath the stave, but generally the
stave holder is too careless or lazy to attend to it, and
the staff is in most instances placed on the ground
[page 147]
[L2]


without anything beneath it. Mr. Simms has con-
trived
an iron tripod for resting the staffs on, which
in a great measure removes this source of error, but
is very troublesome for the irian to carry. The
description of shoe the author has applied to his
staves will allow the face to be turned in any di-
rection
, without in the least disturbing that part
resting on the ground, whereby this evil is effectually
remedied.

The stave should be pressed on the ground at
each station, and on turning the face in any direc-
tion
, not the slightest change in the level of the
stave will take place.


[page 148]

ON THE SETTING OUT OF RAILWAYS.

WE will here give a few directions, which are ne-
cessary
to be observed in the carrying of a line of
railway into execution. Previous to the execution
of the works, it becomes necessary to have the centre
line of the road very accurately determined; and a
stake should be firnily driven into the ground at
about every chain's length, or at each 100 feet as
may be advisable; which centre line should be after-
wards
carefully levelled and a sight taken at each
stake, which then becomes as so many bench-marks.
These stakes should be numbered consecutively; and
when the section is plotted, and the gradient (or rate
of clivity) put on, it will at once show you the height
of embankment or depth of cutting at each stake.
The requisite widths (which will depend on the
slopes and the embankment or cutting) should also
be carefully put in, at one view showing the pro-
prietor
of the ground the quantity of his land that
you require, and the damage he will sustain by
severance or otherwise,--which will be more satis-
factory
to him and all parties concerned than any
number of plans.

To set out a straight line of railway is not a


[page 149]

difficult matter; yet, to do it correctly, a good
theodolite is necessary to ensure a straight line,
and all obstacles impeding a clear view should, if
possible, be removed. You may then, with the
assistance of your theodolite, set out a perfectly
straight line for any distance, by. pursuing the
directions we have previously given for measuring
a base line.--(See Parish Surveying.) If any build-
ings
should chance to be in the way, and you cannot
see over them, you may easily and accurately pass
round to the opposite side by forming an equilateral
triangle, as already pointed out in the passage
referred to above. In a recent extensive survey in
which, we were engaged, while measuring the bases
and tye-lines, we unavoidably came on many build-
ings
; but, by pursuing the above method, we passed
round theni without the least difficulty. One case
in particular may be worth notice. In measuring
the principal base (which passed over two ranges of
hills and the intervening valley), we had a very con-
spicuous
forward object; but in passing through the
valley, our line came directly on a very thick wood
of considerable extent, through which it was quite
useless attempting to pass: we therefore set out an
equilateral triangle, the sides of which were each a
quarter of a mile; and on coming to the opposite
side of the wood, and setting off the supplementary
angle of 120°, the instrument exactly bisected the
first observed object: the maximum error on this
line (which was more than 2½ miles in length.) did
[page 150]

not exceed five links. But to return to the point in
question: when the line of railway changes either
to the right or left, it cannot do so at an angle, but
gradually; forming a curve line which is a part of a
circle, the radius being seldom less than a mile. It
must be evident that this circle cannot be struck
from a centre; means must therefore be adopted for
finding certain points on this curved line, to which
points straight lines may be drawn, which will ap-
proximate
so closely to a regular curve as to be
sufficiently correct for practical purposes: although,
when laying the rails for the permanent way, straight
lines are inadmissible, and fresh points on the curve
between these former ones must be found through
which a regular curve can be drawn, and to which
curves the railway bars are generally brought pre-
vious
to laying, as no combination of straight lines
will produce a curve sufficiently regular for the pur-
pose
intended.

A straight line, connected with a curve, should
be a tangent to it; and two curves joined or con-
nected
together should have a common tangent, but
they may turn different ways at the point of contact,
forming a double curve, or curve of contrary fiexure,
similar to the letter S.

The methods pursued in laying out curves are
various; but the following, as communicated by
Mr. D. J. Henry, C. E., of Dublin, and successfully
practised by him, appears to us to be the most easy
and correct method we are acquainted with:--


[page 151]

When the intermediate ground between the
straight lines is level, you may assume any radius at
pleasure, provided it be sufficiently long to connect the
lines with a convenient degree of curvature; then the
points of contact measured from the angle of meeting
of the two lines both ways will be found thus :-
As sine of half the angle of meeting
Is to cosine of said angle,
So is the assumed radius
To the distance of the required point of contact
from the angle of meeting.

Next, when the nature of the ground is such
that the curve must pass through a certain place,
in which case the point of contact is assumed, and
the radius will be found by the converse of the above
statement, thus:--

As cosine of half the angle of meeting
Is to sine of said angle,
So is its distance from the assumed point of
contact
To the radius required.

Now to lay down the curve, take the nat. sine
of 1° and multiply it by the radius in chains, and
lay off the product from the point of contact as tan-
gent
in continuation; then at the extremity lay off
at right angles to it on the concave side of the curve,
a distance equal to the radius diminished by pro-
duct
of the radius and the nat. cosine of 1°, and that
will determine a point in the curve. Now as the
angle at the circumference standing on a chord of


[page 152]

1° is 30', and the exterior angle made by a second
chord of 1° and the first chord produced will be
double thereof,* or 1°, we have the length of the
next line on the first chord produced, and its corre-
sponding
set-off from the following proportions:--
As radius of the tables
Is to the cosine of 1°,
So is the chord of 1° in the given circle
To the distance to be measured on the first
chord produced. And
As radius of the tables
Is to sine of 1°,
So is the chord of 1° in the given circle
To the set-off.

Thus, let the radius of the circle be 1 mile or 80
chains, take the < a r x (see diagram Plate 8) = 1°,
then the nat. sine thereof will be .017452, which
multiplied by 80 will give x z equal to 1 chain

* We have made some alterations in the above, but fearing the method
might not be quite understood, we add the following Note:--
The principle of the method consists in this, that if successive equal
chords be taken in a circle, the angle made by the produced part of any
chord with the adjacent chord is equal to the angle which the chord sub-
tends
at the centre of the circle. Thus in the diagram, Plate 8, let ax=xn,
then the angle m x n = the angle a r x, for the angle m x n = angle x a n + angle
x n a; but each of them = half the angle a r x, therefore, &c. Now from the
point n, let fall the perpendicular n m, then x m is the cosine of the angle m x n
or a r x to the radius x n or a x, and m n is the sine of the same angle to the
same radius; or conversely, if x m be taken in the chord a x produced, equal to
the cosine of the angle a r x to the radius a x; and if m n be taken perpendicular
to x m equal to the sine of the same angle to the same radius, the point n thus
determined is a point in the circle; the same values applied to x n produced,
determines the next point in the curve.


[page 153]

39.6 links, equal to y a, the distance on the tangent,
and the nat. cosine of 1° is .999848, which taken
from radius of the tables will leave 000152, the
versed sine of 1°, which multiplied by 5280 (the
number of feet in one mile), will give .803560 feet
or 9.6 inches, for the numerical value of a z = y x the
first set-off
at y. The chord of 1° is equal to 2 x nat.
sine 30', and therefore the chord a x or x n = 2 x nat.
sine 30' x 80 in chains = 139,63 links. Then say-
As radius of the table
Is to cosine 1°,
So is 139.63 links
To m x, the distance to be set off in a direct line
with the chord a x. And
As radius of the tables
Is to sine 1°,
So is 139.63 links
To m n.

At the point m lay off m n, already found, and
ii will be another point in the curve; and thus con-
tinue
to lay off the distance x m in a line with the
last chord, with its corresponding set-off at right
angles to it extremity, and you have the curve
complete.


[page 154]

Manner of putting in the Widths.

After the centre-line is staked out, the section
taken, and gradient put on, you can easily deter-
mine
the widths that will be required on every part
of the line; having the embankment or cutting at
regular intervals where the stakes are driven, it will
be only necessary to multiply the height or depth
by double the slope, and add the width of roadway
and ditches, which will be the extreme width. In
plate 7, the first figured height on the section is 17
feet; the width of roadway on the plan, 28 feet,
with 12 feet on each side for bank and ditch, and
the slopes 2 to 1; now, 17 x 4 = 68 + 28 + 24 =
120 feet the width set off that point; and in this
way were the other widths calculated and put in.

Of Gradients, or Rates of Clivity, showing the
method to be pursued in forming an Em-
bankment
or Cutting, so as to produce one
regular Incline.

In the constructing of roads, but more espe-
cially
rail-roads, it is essential, as far as circum-
stances
will permit, that a level be obtained, but for
any considerable distance this is generally imprac-
ticable
; it then becomes necessary to surmount


[page 155]

the inequalities of the ground, by means of regu-
lar
inclines, which are, however, kept as near
a level as possible, due regard being paid to
economy. The method pursued in regulating the
quantity of embankment or cutting, so as to pro-
duce
a regular rate of clivity, is what we shall
here describe:--Divide the difference of level by
the distance in chains: the quotient will be the rise
or fall at each chain's length;
thus, in Plate 7, the
rate of clivity is laid on the section in the first in-
stance
, after the rate of 16 feet per mile; take the
distance, 1 mile, with the rate of clivity as above;
then, 1 mile or 80 chains ÷ 16, (the rate of
dilvity) = ,2, showing that a difference of ,2 at
each chain's length will form a rate of clivity equal
to 16 feet per mile. To apply this to practice, set
up your spirit-level at either end of the cutting or
embankment, and each chain's length therefrom,
must show a difference of ,2 to produce the re-
quisite
rate of inclination. The theodolite might
be employed where'the rate of clivity is very great,
by setting it up at an ascertained height, and
taking the vertical angle as already explained in
our directions for levelling with that instrument.
The angles answering to the various rates of in-
clination
will be found at the end of this treatise.


[page 156]

The Forming of Slopes.

Embankments or cuttings are generally brought
to one uniform slope, except where the strata in
deep cutting is found to vary; when this is the
case, some portion will often stand at a much
greater inclination than others; when so, a ledge,
or benching, as it is termed, is left where the angle
of the slope changes. As already explained in our
directions for setting out a railway, the widths
would be put in when the centre line is set out;
then, after excavating to the required depth, it be-
comes
necessary to bring the slopes to the degree
of inclination previously determined on. Various
kinds of levels are occasionally employed, but that
in most common use is of wood, made to the re-
quired
slope, and set up perpendicular by means of
a plumb line. A convenient and portable instru-
ment
for this purpose is the batter-level, or clino-
meter
, which is a small quadrafit, with an attached
bar, to which a rod is affixed when in use; the
quadrant has an index carrying a spirit-level,
which is moveable round the centre of the instru-
ment
; when in use, the rod is laid on the slope,
and the index moved by the hand, until the bubble
assumes a central station in the tube, the angle
denoted by the index will be the inclination. The
ratios of the slopes for the various degrees of in-
clination
will be found at the end of this treatise.


[page 157]

It should be observed, that when speaking of
slopes, as 2 to 1, 1½ to 1, 1/6 to 1, &c., the base is
always first; two to one signifies, that for each foot
perpendicular, a base of two feet is required, or,
what is the same thing, that it batters two feet;
one sixth to one means, that for each foot perpen-
dicular
, a base of 2 inches will be required, or that
it will batter 2 inches.

On the Method of Calculating Earth-work.

There are various methods for calculating
earth-work, but certainly none so correct as the
prismoidal formula adopted by Mr. Macneill. It
would be useless our demonstrating this, as the
reader will find every information by referring to
Mr. Macneill's Tables; and as we suppose no
one would go through the tedious process of calcu-
lating
by the formula, we will merely direct the
student in what mariner to arrange the quantities on
a section, so as to calculate the contents by the
tables above mentioned.

In Plate 7, which is the contract section, the
quantity of cubic yards in the embankment from
the road at B to the road at C is 128.795,6180;
the method pursued in getting this quantity, is to
take a portion of the section so far as the inclination
of the ground is continuous; slight undulations
of the surface being equalized in the same manner


[page 158]

as directed for calculating superficies; the first por-
tion
taken was 510 feet in length, 17 feet in height
at one end, and on a meafl line being drawn, was
found to be 21 feet at the other end, the base being
28 feet, and the slopes 2 to 1 :-the contents found by
table 29 is 23.737,2360 cubic yards for that portion
of the embankment between the two first figured
heights; the next portion taken was 400 feet in
length, extending from the second to the fourth
figured height, the contents of which is 22.617,3600
cubic yards; in this way were taken the several
portions of 570 feet, 2185 feet, and 1670 feet
which finished up to the road, the respective quan-
tities
for these distances being 24.545,6820 cubic
yards, 39.708,4420 cubic yards, and 18.186,9680
cubic yards, the whole of which being added toge-
ther
, will make the total quantity of embankment
in this portion of the railway 128.795,6880 cubic
yards, as given before.


[page 159]
[TABLES]



[page 160]
[TABLES]



[page 161]
[M]


Addenda

WHEN railway surveys are completed, the contents
of each enclosure through which the line passes
should be written within it, with the description of
land, as arable, pasture, &c., also the owner and
occupier's names, which will be found infinitely su-
perior
to having the usual detached reference-book.
In the reduced plans for depositing, &c., a detached
reference-book is indispensable. The method of
computing the contents is by triangles, in the man-
ner
previously described: that required for the
railway may generally be computed as parallelo-
grams
.

In the description of field-books for levelling,
we should have mentioned that an assistant always
chains the distance, giving to the principal the chain-
age
at each point where the staff is set up. A cus-
tom
is prevalent with some people, in place of
inserting the chainage to attach numbers or letters
to each sight, which refer to another book kept by
the assistant who chains the distance-in which, to
the corresponding number or letter, is inserted the
distance; but any error or omission on the part of
the principal or assistant, involves the whole in
inextricable confusion, and it becomes worthless.
We have seen many instances of ruinous errors
being committed, entirely from this method of
keeping the field-book; and we would urge those
persons who are in the habit of keeping their book in
this manner, to abandon it before they have reason
to be sorry. In the deposit plan for railways, &c.,


[page 162]

although the section is drawn on the plan as ex-
plained
in our description of Sectio-Planography,
yet the usual detached section is always required for
engineering purposes.

In taking horizontal angles with the theodolite,
the object is bisected by the vertical wire; but in
taking vertical angles, the bisection is made with
the horizontal wire: the latter is always the case in
levelling instruments, the former serving to show
the operator when the staff is held perpendicular.
A micrometer scale is sometimes attached to survey-
ing
as well as levelling instruments, in place of the
cross-wires. This is a fine slip of mother-of-pearl,
or wire strained across the diaphragm in regular di-
visions
; the object is to measure distances, which
can be done thus:--level your instrument, and mea-
sure
out various distances, at each of which set up
the staff; note in your book how many of the
divisions on the micrometer scale the staff subtends.
To apply this in practice, on the staff being planted
at any distance from your instrument, refer to the
table, and the corresponding division to that sub-
tended
in the scale will give the required distance.
This method, however, has not much pretensions to
accuracy, and cannot at all be practised in windy
weather; but it might always be advantageously
employed to plant the staff at equal distances.

   

LONDON:
Printed by Edward Ravenscroft, of Tooks Court, Chancery Lane.


[page 163]

ERRATA

PREFACE, 1st line, for "suggested itself" read "occurred."
PAGE 21, last line, for "Plate 1" read "Plate 5."
PAGE 22, last line, the Plates referred to are "1 and 2."
PAGE 24, "Plate 7" is here referred to.
PAGE 62, line 11 from bottom, for "31 perches" read "21 perches ;" and
at line 10 from bottom, for "15 perches," read "5 perches."
PAGE 92, line 8 from bottom, 4th word from left-hand, add "the square of the"
For Plates 1, 2, 3, 4, read Plates 4, 3, 2, 1.
IN Plate 7, for "Datum Line.--High Water, Spring Tides," read "Datum
Line.--20 feet above High Water Spring Tides."


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