THE SURVEYOR'S COMPANION.
A
~annail for ~radital a;aRregprs,
CONTAINING
METHODS INDISPENSABLY NECESSARY
FOR
ACTUAL FIELD OPERATIONS,
BY E. W. BEANS,
Niorristown, Mlontgomery Co. Pa.
PHILAD)ELPHIIA:
PUBLISHED BY J. W. MOORE,
No. 195 CHESTNUT STREET.
1854.
Entered according to the Act of Congress, in the year 1854, by
JOHN W. MOORE,
in the Office of the Clerk of the District Court of the Eastern
District of Pennsylvania.
PRINTED BY ISAAC ASHMEAD.
PRE FA CE.
THE want of a work on Practical Surveying has been long felt, and is generally acknowledged. The numerous publications on
surveying would seem to preclude the necessity of anything new. But upon exanination we find the wants of the student have
been consulted, rather than those of the practical man. Indeed, many of the publications
in general use appear to have been written
by those who were engaged in the instruction
of youth, and who were unacquainted with
the practical part of surveying, excepting
perhaps so far as may have been requisite
for the information of the classes under their
direction.
I have conversed with many persons who
Vi PREFACE.
have been extensively engaged in land surveying, and I have not in a single instance
met with any one who has not expressed his
unqualified conviction of the want of a work
adapted to the purposes of the practical
surveyor. Within a few years there have
been published several works on surveying:
amongst them may be mentioned that by
John Gummere, a treatise which cannot be
too strongly recommended to those who wish
to become familiar with this subject; and
had it been as well adapted to the wants of
the practitioner, as to those of the student,
no other need have been desired. Professor
Davies' 1"Elements of Surveying" is an excellent performance. Flint's "Surveying"
also contains much useful practical information. But still there seems to be wanted a
more minute detail of expedients employed
in the field.
With a view to supply this defect, the
PREFACE. vii
following pages have been written, designed
as a suitable treatise to be placed in the
hands of those who wish to become familiar
with the practice of surveying. A systematic arrangement has not been followed;
but as my object is to supply the wants of
the practical man, (those of the student having already been supplied by the authors
mentioned,) this will be a matter of minor
importance.
In submitting the work to the public, it is
not pretended to be complete in itself, but
only introductory to a subject of much importance hitherto almost entirely neglected.
If it prove a useful auxiliary to those who
are about assuming the responsible duties of
practical surveyors, the object of the publication will have been accomplished.
E. W. BEANS.
Norristown, AMontyomery county,
_Pa., June 1853.
A MANUAL FOR SURVEYORS.
CHOICE OF INSTRUMENTS.
IT is of the first importance to every young
person, about commencing the practice of Surveying, to furnish himself with suitable instruments.
These should be of the very best character, both as
regards the workmanship and their adaptation to
purposes in which they are intended to be employed. His success may depend upon this choice,
his pretensions to accuracy, in what he may undertake, must rest very much upon it. Unless he is
acquainted with the use of the Instruments in the
field, it would be well to advise with some person
in whose judgment he can confide. It would also
be advisable to visit the shops of the different instrument makers, and examine the various instruments in use. He may be enabled by these means
2
10 A MANUAL FOR SURVEYORS.
to make a more suitable choice than he would
otherwise do. Having the instruments before
him, and their peculiar advantages and uses explained, he would be enabled to select those best
suited to his purpose. He should be provided with
an instrument for measuring angles, and taking
bearings, a chain, measuring-tape, plummet, reading-glass, &c. If he intends laying off town lots, he
should have a leveling instrument and rod, also a
20 feet frame for measuring distances accurately.
The two pole chain is commonly used. It
should be made of heavy wire, the links connected by double rings to prevent the chain kinking, it should be firm in all its parts, and of the
standard length.
The compass or circumferenter is in general use
amongst surveyors in this country. In selecting a compass, particular attention should be paid
to the needle. If it vibrates long and settles with
a gradual diminution in the are of vibration, it
moves with freedom on the centre pin; but if it
settles after a few vibrations or suddenly at any
point, it does not move with sufficient freedom on
A MANUAL FOR SURVEYORS. 11
its centre, or is defective, and should not be used on
valuable lands.
To examine the divisions of the limb of the instrument, set the needle to any degree; if both
ends coincide with the corresponding divisions on
the opposite limbs of the compass, it shows the
adjustment is correct on that course. In the same
manner examine various divisions round the compass box, the coincidence of the opposite ends of
the needle, with the same degree, is a proof of its
being correct. If it be a nonius compass, the
nonius may be moved to the right or left a few
degrees; if the needle move over the same number
of degrees, it is a proof of the correctness of the
graduations. The instrument may be further
tested by laying off, with it, on the ground, a
square, the side of which may be 50 or more
perches. Then if the diagonals be measured and
found equal (or very nearly so, as errors may be
in the measured distances,) it is evidence of the
correctness of the divisions, needle, &c., and that
the instrument is a good one. Any regular figure
may be used for this purpose. A nonius compass
12 A MANUAL FOR SURVEYORS.
is preferable to a plain one in several particularssome of which may be mentioned. Having found
the difference of variation between the present
time, and that at which a survey was formerly
made, set the nonius to this difference, the needle
will show, on the face of the instrument, when the
sights are set to the lines of survey, the same course
as formerly, unless the compasses, by which the
bearings were taken, differ in other particulars.
Another reason for preferring the nonius compass
is, that we may run a line to minutes, by setting
the needle to the whole degrees, and the nonius to
the minutes of the course.
Angles of elevation, when a level is attached to
the compass, (which it should always have,) may
be taken, by having one of the sights graduated
into degrees. This may be easily done by the
surveyor himself calculating the tangents of the
angles, to a radius, equal to the distance of the
sights apart, and setting off those distances, on the
sight. This may be carried to about 200. The
links to be deducted, for every chain length oblique
measure, to reduce it to horizontal measure, may
A MANUAL FOR SURVEYORS. 13
also be marked on the sights, corresponding with
the degrees of elevation. A compass so fitted up
is very useful in hilly grounds; for having an object at the top of the hill, we may easily take the
angle of elevation of the hill, and thence reduce
the oblique line measured, to the horizontal line.
Horizontal angles may be measured with the
nonius compass. For this purpose, procure an
extra pair of sights, made so as to be attached to
the face or movable part of the instrument containing the needle, and at right angles with the
fixed sights. Bring the fixed sights to bear on an
object in one of the lines, including the angle to be
measured, and clamp them in that position. Unclamp the nonius plate, to which the attached sights
are affixed, and bring these sights to bear on an
object in the other line, including the required angle. The number of degrees over which the vernier plate has moved, gives the difference between
the angle measured and a right angle, and must be
added to or subtracted from a right angle, according as the movable sights move from, or towards
the fixed ones.
2*
14 A MANUAL FOR SURVEYORS.
An arm may be also temporarily attached to the
ball or spindle of the staff head, extending to the
edge of the compass box, which should be divided
up to 900. This arm being clamped at 0~ by
means of a thumb screw, connecting the arm to the
spindle or head of the staff, will point out the number of degrees the compass box may move over, as
the motion of the compass is about the immovable
spindle of the staff head or axis of the instrument.
A nonius compass fitted up as here directed, will
be nearly as efficient as a theodolite, in the ordinary cases of farm surveying; and especially
where the needle may be rendered almost useless
from local attraction, or other accidental occurrences.
A nonius compass may be used where there is
local attraction, as follows: Bring the sights to
bear on an object at the back station, then move
the nonius plate until the needle settles at the same
course as at the last station, and clamp the nonius
plate; a course may then be taken in any direction with the needle, as correctly as if free from
the effects of local attraction.
A MANUAL FOR SURVEYORS. 15
The following description of an instrument for
surveying was received from Enoch Lewis, a practical surveyor, and author of several mathematical
works.
"A circumferenter for surveying in the usual
way, and for taking horizontal angles, is formed
with two circular plates, the lower of which is firmly
attached to the stem of the instrument, and may
therefore be clamped to the tripod which supports
it; the upper plate is connected with the sights,
and moves with them. These circles are concentric, and are firmly attached by a screw, or moved
the one on the other by a rack wheel. The circumference of the lower circle is graduated, and
the upper one contains a nonius.
"' To take an angle, place the middle of the nonius
in conjunction with the zero of the lower plate, and
fix them together by the screw; then direct the
sights along one of the lines which include the
angle in question, and clamp the lower plate to the
tripod. Loose the plates from each other, and
applying the eye to the sights, turn them by means
of the rack wheel till they coincide with the other
16 A MANUAL FOR SURVEYORS.
line, including the angle. The angle may then be
read on the fixed plate by means of the nonius.
"An instrument constructed in this manner is
very convenient in laying out the curves of railroads.' A small telescopic tube fixed on the side of the
instrument parallel to the line of sights, and moveable on the centre of a graduated vertical circle,
will serve to take angles of elevation or depression
with great facility.
"' A small table of versed sines to a radius one,
or unity, extending to fifteen or twenty degrees,
inserted on a leaf of the note-book, may answer
the purpose of reduction from oblique to horizontal
mleasure.'"
It is generally recommended, when the compass
is not in use, to place it in its box, where it will not
be disturbed, and let the needle settle, and remain
on its centre pin. However, I believe it preferable
to let the needle settle, and then carefully screw it
up or off the centre pin; it will then be very nearly
in the magnetic meridian, and will have all the
advantages of its remaining on the centre pin,
A MANUAL FOR SURVEYORS. 17
without the danger of blunting it by continual friction on the point.
The cross is a very useful instrument to lay off
perpendiculars when running lines or measuring
irregular pieces of ground. A block of hardgrained wood, three or four inches square, and one
and a half or two inches thick, having two saw
kerfs cut more than half through its thickness, and
intersecting each other at right angles at the centre
of the block, will be sufficiently exact for the purposes above-mentioned; which will also be of convenient size to carry in the pocket.
To use this cross, we have only to lay it on the
face of the compass adjusted for observation, and
directing one of the kerfs to an object; a stake set
in the direction of the other kerf will be at right
angles with a line joining the cross, and the object
observed.
The Graphometer, or Semicircle, is also a very
useful instrument in surveying. It consists of a
graduated semicircle, with a pair of sights in the
direction of the diameter of the instrument, and
having a pair of movable sights moving on the
18 A MANUAL FOR- SURVEYORS.o
centre of the semicircle, to which the verniers are
attached. Underneath the centre of the instrument is a ball and socket to attach it to the stand
when in use.
To measure an angle, turn the fixed sights until
you see an object in the direction of one of the
lines between which the angle is to be measured,
then turn the movable sights until an object is
seen in the direction of the other line, the vernier
will point out the angle between them. The socket
has a notch in one side, so as to enable the suveyor
to take vertical angles or altitudes. It is a very
useful instrument for many purposes in surveying.
In surveying valuable lands, or when great accuracy is desired, a theodolite or transit should be
used. A description of the former is deemed unnecessary, as this has been given in some of our
best treatises on surveying, and also in " Simms'
Treatise on the Principal Instruments employed in
Surveying, Leveling, and Astronomy," a work
which every practical mathematician should possess.
The " Transit" may not be so generally known.
A MANUAL FOR SURVEYORS. 19
It consists of two parallel plates attached to each
other in a manner somewhat similar to the circular
plates of a theodolite. To the upper of these circular plates is attached a compass-box of much
larger dimensions than that usually attached to
the theodolite, which enables the observer to read
the bearings from the needle much more correctly.
The circular plates are not graduated at their
outer edges, but so as to be read within and at the
bottom of the compass-box. To the upper plate is
attached the wyes, or vees, which support the axis
of the telescope, that revolves in the same manner
as the astronomical transit. A vertical arc is
sometimes added to measure vertical angles. The
instrument is furnished with clamps, tangents,
screws, and tripod, similar to the theodolite. It
may be used as a circumferenter in taking bearings,
or as a theodolite for measuring angles. The adjustments are few, simple, and readily made. It
is usually provided with only one vernier; two,
however, would be conducive to accuracy, as a
mean of the readings would correct the eccentricity
of the instrument. The transit may be considered,
20 A MANUAL FOR SURVEYORS.
for the general purposes of surveying, superior to
any other instrument in use.
There is one property of the magnetic needle,
(when not disturbed by local attraction, &c.,) which
should not be lost sight of, and that is, that any
error committed in running one line is not communicated to another. But when angles are taken,
the errors may affect or be communicated to others,
even at a remote distance from the line on which
the error is committed. Therefore the accuracy of
the angles must be carefully ascertained by comparison with the courses shown by the needle, otherwise great and perplexing errors may be introduced.
The surveyor should be furnished with tables of
sines, as far as 20~ extending to four decimals at
least; nat. tangents as far as 100, and such others
as may be needful for reference in the field. These,
with the Traverse Table, will be requisite to facilitate many calculations necessary to be made in the
field. Scribner's Engineer's Pocket Table Book
may be particularly recommended. He should
A MANUAL FOR SURVEYORS. 21
also have a reading-glass of considerable power, to
aid him in reading off the bearings, angles, &c.,
with precision.
LAND MARKS.
Judge Wilson gave his decision in regard to lines
and land marks, as follows:
The best evidence is,
1st. Living marks, such as trees, &c., the first
and most substantial land marks; and if marked
trees should not be in a right line, yet the line
must be run from one marked tree to the next,
and thence to the next, and so on.
2d. When there are stones of long standing
along the line in question, the line must be run
from the first to the second; from the 2d to the
3d, &c.
3d. Old residents in the neighborhood, may designate marks or points where the original line
formerly run.
Lastly, the chain and compass.
3
22 A MANUAL FOR SURVEYORS.
It is the practice where there are ditches along
the line, to take for the line the edge of the ditch
lying next the bank of dirt thrown out in digging
it, or if the dirt thrown out in digging be on both
sides, the middle of the ditch must be taken.
A MANUAL FOR SURVEYORS. 23
PRACTICAL SURVEYING.
PROPOSITION i.
To run a line between two points A and B; or
to trace a right line A B on the ground.
Case 1. When the points A and B can
be seen from each other:
Place the instrument over A, and bring
the sights to bear on B; then direct marks
to be set at the required points in', m2,
&c., along the line, and it is done.
N. B. In this and the following propo- B F
sitions, I have used n', n', mn', m2, &c., m,
to denote the number of the stakes set in
the lines, reckoning from the beginning.
Case 2. When obstacles prevent the
points A and B being seen from each
other:
On the same side of A B, and perpendicular to
it set off equal distances A C and B D; place the
24 A MANUAL FOR SURVEYORS.
instrument over one of these points C, and bring
the sights to bear on the other point D: then direct stakes to be set, along the line C D, opposite
the points required at n', n2, &c.; and from these
set off perpendicular to C D, and equal to A C or
B D, the distances n' mn, n 2 m2, &c., towards A ZB,
then A m' m2 B will be the line required.
Case 3. When the points A and B can be seen
from an intermediate point E:
By trials set the instrument at a point E in a
right line between A and B; then the intermediate
points between A E and E B may be found as in
case i.
Case 4. When obstacles prevent A and B being seen from an intermediate point, but C and D,
(case 2d,) may be seen from the intermediate
point F:
Set the instrument at F, the intermediate point
in a right line with C and D, and direct stakes to
be driven in the line C D as in the preceding case;
distances perpendicular to C D, and equal to A C
or B D being set off towards A B will determine
the points Wn', mn, &c., in the line A B.
A MANUAL FOR'SURVEYORS. 25
PROPOSITION 2.
To run a right line on the ground, or to prolong
the line A B to any distance required:
Place the instrument over A, the point from IA
which the line is to be run. Having adjusted
the instrument, bring the sights to bear on B,
a staff placed in the direction of the line to be B
run. Let the needle down on the centre
pin, and take the bearing of the line, screw
up the needle carefully; next remove the instrument to B, set the needle to the course of
A B and run to C, and from thence to D, and
so on as far as required. When it is required D
to run the line accurately, proceed thus: Plant
the instrument at B, and having adjusted it, bring
the sights to bear on the back station A; and direct a stake forward to be set at C; remove to C,
and take a back sight to B, then direct a stake to
be set at a forward station D; and so proceed as
far as required.
This method of running a line by back stakes,
that is by having a stake set up at the last station,
and bringing the sights to bear on it, should always
c),,
26 A MANUAL FOR SURVEYORS.
be adopted, unless the lines to be run are very
short. The needle should not be depended upon,
except where it cannot be dispensed with, as in
taking courses, bearings of objects, &c.
I have used the term "sights," in order that the
methods pointed out may be applicable to the use
either of the circumferenter, theodolite or transit.
Case 2. When there are obstacles which prevent the line being run directly from A to D:
A Run the line from A to B as before directed, where we will suppose an obstacle..p is met with, which prevents the line being
continued.
i.._ Set off at B, perpendicular to A B any
s distance, B n sufficient to pass the obstacle
at B; from a point P between A and B set..... c off another perpendicular P t=B n; prolong t n towards m, until the obstacle at B
is passed. From r and mn in the line t n
produced, set off the perpendiculars thereto, r s=mn C=B n from r to s and m to C
D prolong s C to D, which will be a point in
the line A B produced.
A MANUAL FOR SURVEYORS. 27
Case 3. When it is impracticable on account of
intervening obstacles, to lay off perpendiculars t P
n B, &c., from the line A D:
Run P t any suitable course; then run B n parallel to it, and make B n = P t. Prolong t n to r
and m as in the preceding case; and run r s, m C
parallel to P t and each of them equal to it; produce s C to D as in the last case, so will D be a
point in the line A B produced.
- NOTE. —A line run from a given point which
does not terminate at the point designed, but falls
either to the right or left of it, is called a random
line, or a guide line.
Case 4. Where swampy ground, &c., prevents
measuring, as well as running in the direction A B,
further than to B:
On arriving at B, deflect the line B C with an
angle of 60~, that is, make the angle A B C equal
to 1200, and measure B C. At C deflect C D with
an angle of 600; that is, make the angle B C D
-120~, and measure C D. At D deflect the line
D E with an angle of 60~, or make the angle C D E
=120, and make D E = B C. At E deflect E F
28 A MANUAL FOR SURVEYORS.
with an angle of 60~, or make the
angle D E F = 1200. The points E
and F will be in the line A B produced.
Make B G =BC and join C G,
the triangle B C G is equilateral;
also C G and D E are parallel, there
fore D E G C is a rhomboides, and
E G is equal to C D. Consequently,
ll( A E is equal to the sum of the distances A B, B G, G E, which are
llequal to A B, B C and C D added
together.
PROPOSITION 3.
To measure an angle A B C accurately, where
B C deflects but a few degrees from the line A B:
From a point D opposite to A, a point in one
of the lines, including the angle to be measured,
run the line D E F till we arrive at the point F,
opposite C a point in the other line B C including
the required angle. At E opposite the angular
point B, measure the perpendicular B E, also mea
A MANUAL FOR SURVEYORS. 29
sure the perpendiculars A D and F C;
and also measure D E and E F. A. D
Draw A g and C h parallel to D F.
Then B g=B E-E g=-B E-A D
B h=BE E-E h=B E- CF
Nat. Tang. angle B Ag=B g *. Ay
B g.# D E. Also, nat. tang. angle
BChis equal toBh-. Ch Bh /
EF.* \
The sum of the angles B Ag and
B C h subtracted from 1800, gives -
ABC.
This method of finding the angle
A B C is applicable to measuring little
deviations from right lines in streets,
&c., where the utmost precision is re- c (.
quired that the buildings may be regulated exactly to the line of the street.
A transit or theodolite, should be used in this case
to run out the line D E F, and the distances A D, B
* If C F = AD we may say E F: F D:: Vg AB: v
h C B the angles being very nearly reciprocally proportional.
30 A MANUAL FOR SURVEYORS.
E, C F, &c., should be measured to the hundredth
part of a foot.
It has not been thought proper to introduce methods of measuring angles with the instruments
which have been described, as that has been pretty
fully done in the works to which I have already referred. My object in the present performance is
to supply what has been omitted. In doing this,
I have introduced but little to be found in the authors alluded to. It may not be improper to introduce here the method of verifying the correctness
of an angle by the principle of repetition. Place
the transit over the angular point B (see the preceding figure,) after having adjusted it for observation, set the vernier to zero, and bring the telescope to bear on the staff at A in one of the lines,
including the angle to be measured; clamp the
lower plate to the tripod, unclamp the upper plate,
and bring the telescope to bear on C a staff in the
other line comprehending the required angle; the
vernier will point out the measure of the angle
A B C. In this position clamp the plates together,
unclamp the lower plate from the tripod, and bring
A MANUAL FOR SURVEYORS. 31
the telescope to bear on A, by revolving the instrument bodily on its axis. Now clamp the lower
plate; unclamp the upper plate, and bring the
telescope to bear on C; the vernier will show twice
the angle A B C. We may repeat this operation
at pleasure, the last reading of the vernier being
divided by the number of times the angle had been
measured, will give a mean result more to be relied
on than any single observation, however carefully
made.
It is best to repeat the measure of an angle, that
the vernier may pass over the entire circumference
of the graduated plate, in which case 360~ must be
added to the last reading of the vernier, previous
to dividing by the number of observations.
PROPOSITION 4.
Random Lines.
To run a right line between two given points,
when several intermediate stations have to be taken
on account of intervening obstacles.
Case 1. When the line can be run from the
place of beginning:
32 A MANUAL FOR SURVEYORS.
A. Having adjusted the instrument at A,
the point from which the line is to be
run, and directed the sights towards B,.', b i as nearly as can be guessed at, this being the point to which the line is to be
run from A, direct stakes to be set at
ft every 20 perches (or otherwise, as may
~ - suit the nature of ground,) in the di-' —1~ rection of the sights, which let be debteg As signatedl in the order in which they are't \placed by n', n', n', n4, which being
----— " continued until we arrive at C, so that
C B C B may be perpendicular to A C.
The line A C in the direction of the sights, must
be carefully run by Proposition 2d, Case 1st,
Measure C B. Let nt in', mlin n, n3 n3, &c., be
drawn parallel to C B. The triangles A B C,
A mn' n', A ml n', &c., being similar we have,
A C: A n:: B C: n mnl
A C: A nQ:: C B: n2 n2, &c.
A C: A F:: C B:F E.
That is, as the whole distance measured from
A to C, is to any part of the line measured A n',
or A F; so is C B the distance from the termi
A MANUAL FORl SURVEYORS. 33
mination of the randomn line, or line run from the
point at which it should have terminated, to the
distance from (the line measured,) the points,
zl' mn, E, &c., in the true line A 1B.
Having found n' in1, we have n2 mi, equal to
twice n1 ml; also, n mi, equal to three times n' onl,
&c; for as the stakes are equidistant from each other,
the several offsets are some multiple of the first.
1. As an example, suppose it is required to run a
line between two farms having a stone at which to
commence, and also another to which to run. Having run the random line as directed, and leaving
stakes at every 20 perches distance from the beginning of the line, A C was found to be 160 perches,
and B C 2.4 perches; it is required to find the distances to be set off from the random line at every
stake left in the line, as also the distance to be
set off at 96 perches from the place of beginning,
As A C A n:: B C: n' mn
160: 20:: 2.4:.3
n a =.6; n3 n3 —.9; n - 1.2; 2'5 n5
1.5; nr n6 = 1.8; n' m -= 2.1 perches.
4
34 A MANUAL FOR SURVEYORS.
As A C: F A:: C B: F E
160: 96:: 2.4: 1.44 offset at 96 per.
These distances being severally laid off from the
random line, to the right or left, according as B is
to the right or left of C, will give points in the
line A B as required.
2. Being requested to run a line between two
neighbors, I ran a random line 146 perches, and
found I had missed the true corner by 1.23
perches. What perpendicular offsets must be laid
off from this random line, that stakes may be set
at every 10 perches from the beginning of the line.
Reckoning from the beginning the several distances willbe.08;.17;.25;.34;.42;.50;.58;.67;.76;.84;.93; 1.01; 1.09 and 1.18 perches.
In general tenths of a foot will be sufficiently exact.
3. Required the perpendicular distances to be
set off from a random line 647 perches in length,
terminating at the perpendicular distance of 47.3
feet from the point at which the line in question
must end, the stakes in the random line being set
at every 40 perches.
Result, 1st dist. 2.9; 2d, 5.8; 3d, 8.8; 4th, 11.7;
5th, 14.6; 6th, 17.5; 7th, 20.5; 8th, 23.4; 9th,
A MANUAL FOR SURVEYORS. 35
26.3; 10th, 29.2; 11th, 32.2; 12th, 35.1; 13th,
38;0; 14th, 40.9; 15th, 43.9, and 16th, 46.8 feet.
In general it will be most convenient to set the
stakes in the true line in a retrograde order, beginning with the last stake in the random line, and
returning to the first.
To ensure accuracy in our calculations, we must
find the offset corresponding to the distance from
the last stake to the end of the line, which being
applied to the offset at the last stake in the random
line will, if all the calculations are correct, make
up the whole offset C B from the true line. Take
the last example:-say as 647: 7:: 47.3: 5,
which, added to the 16th offset 46.8, gives 47.3 ft.
The following method of finding the offsets may
sometimes be used instead of the preceding, or
may be used to verify their correctness.
Find the proportional part of the offset for the
distance of the last stake from the end of the line,
subtract this distance from the whole offset C B,
and divide the remainder by the number of equidistant stakes in the line, the quotient will be a
number to be continually subtracted from the last
offset to obtain the preceding one.
36 A MANUAL FOR SURVEYORS.
4. Suppose a random line to be run 242 perches,
stakes being set in the line 40 perches apart; what
distance must be set off at each stake so as to give
us the true line, the corner being to the right of
the random line 12.1 feet.
First.-As 242: 2:: 12.1:.1 correction for 2p.
From 12.1 take.1 the remainder 12 being divided
by 6, the number of stakes in the line, gives 2 feet,
the difference of the offsets at each succeeding
stake.
Offset at the end of the line, 12.1 ft.
" to be deducted for 2 per..1
c at the 6th stake, 12.0
difference of offsets, 2.
C at the 5th stakle, 10.
2.
c" 4th " 8.
3d 6.
2.
2d " 4.
A MANUAL FOR SURVEYORS. 37
Offset at the 2d stake, 4.
2.
"c 1st " 2.
2.
at the beginning, 0. proof.
The calculation made by the former method is
as follows:
As 242: 40:: 12.1: 2. the correction from
stake to stake.
Correction at the 1st stake, 2.
cc 2d " 4.
c" 3d " 6.
c" 4th " 8.
5th " 10.
"' 6th " 12.
for 2 per..1
for 242 12.1
It will be sufficiently exact in many cases to
4*
38 A MANUAL FOR SURVEYORS.
omit the fractions of a perch in the first and second
terms of the proportions, the offsets being small,
compared with the line of survey.
5. Run a line 719 perches, when I found I was to
the left of the true line 3.7 per. What distance
must be subtracted from the 3.7 offset to obtain the
offset at 700 perches, and also what number must
b'e-successively subtracted from that offset, to give
the distance to be set off at stakes set at every 100
perches along the line.
Proportional part for 19 per. is.1
Offset at the 700 is 3.6
Number to be deducted for 100 per.
3.6. 7=.514 per.
Offset at the 6th hundred, 3.08 or 3.1
5th " 2.57 " 2.6
"c 4th " 2.06 " 2.1
"c 361d c 1.54 C 1.5'" 2d 2 " 1.03 " 1.0
~c I1st " 52 C.5
The following rule is that generally given in
books on surveying, for determining the true from
A MANUAL FOR SURVEYORS. 39
a random line: From the given point or place of
beginning, run a random line by the given course
of the line, and measure the perpendicular distance
between the line so run and the sought corners
then,
As the length of the line run,
Is to the said perpendicular distance,
So is 57.3 degrees, or 3438 minutes,
To the difference of variation or correction of
the course,
Which, being applied to the given bearing, will
give the present bearing of the line.
Then set the instrument at the place of beginning, and run the line by its present bearing.
This method cannot be relied on, as the shortness of the needle, its aberrations, diurnal variation, &c., may lead to error.
The following method of finding the difference
of variation is convenient and easily remembered:
To the length of the measured line add its half
length, then say,
As that sum is to the perpendicular distance, so
is 860 to the correction of the course.
40 A MANUAL FOR SURVEYORS.
If we have a table of natural tangents, divide
the perpendicular distance aforesaid by the length
of the line, the quotient is the nat. tang. of the
angle of correction; take out the angle corresponding in the table, which will be correct in all cases.
We may use the nat. sines instead of tangs. as
far as 5~.
Example 6th. Suppose a line some years ago
bore N 40~ W 170 per., and that in running by
this course, we came out 1.55 per. to the left hand
of the true corner: what is the present bearing of
the line.
By the first rule:
As 170: 1.55:: 3438': 31' correction.
By the second method:
As 255: 1.55:: 86~: 31'.
By the third:
1.55. 170=.00912, tang. of 31'.
Hence 40 —31'=N 390 29' W, is the present
bearing of the line by which it may be retraced by
the circumferenter.
Or place the transit at the beginning of the line,
and adjust it for observation; set the vernier to
A MANUAL FOR SURVEYORS. 41
zero, and bring the telescope to bear on a stake in
the random line; move the vernier 31' to the right
hand, because the true line is on that side; the
telescope will now be in the direction of the line
to be run out, which may be done as directed in
Proposition 2d.
Rule 4th. Multiply the feet offset by 208, and
divide the product by the length of the line in perches, the result will be the correction in minutes.,Rule 5th. Multiply the feet offset by 100, and
divide the length of the line in 2 pole chains; to the
quotient add its -I for the correction in minutes.
If a nonius compass be used: Set the nonius 31'
to the right hand, the needle being set to N 40~ W,
the sights will give the direction of the required line.
This is a great advantage the nonius compass has
over the common circumferenter; for having set the
nonius, and clamped it to the difference of variation
between the present time, and the time a line was
formerly run; all the lines of survey run at that
time may be retraced, by setting the needle to the
given bearings of the lines. This cuts off the possibility of errors arising from applying the difference of variation by addition or subtraction to the
42 A MANUAL FOR SURVEYORS.
several bearings of the lines; that allowance being
made by the nonius, it being the same on all the
lines of survey, if they have been truly taken.
EExample 7th.-In running a line which, some
years ago, bore N. 220 17' E. 311.7 perches, I
found the true corner 4.5 perches to the right hand,
what is the present bearing of the line?
Ans. N. 230 7' E.
Example 8th.-A line being run by a former
course S. 120 19' E. 128.7 perches, the corner was
found 2.3 perches to the right, what is the present
bearing of the line?
Ans. S. 11~ 18' E.
Example 9th. —Being called upon to run the
line between two townships, the course and distance
of which were given S. 38RF W. 1294 perches, I
found, in running by this bearing, that the true
corner was on my left 7.12 perches, what is the
present bearing of the line between the townships?
Ans. S. 38~ 26' W.
Example 10th.-Wishing to run a line between
two points from one of the points I run a course
N 89 E, and measured the distance with a two
A MANUAL FOR SURVEYORS. 43
pole chain, 129 chains 36 links, when I found the
perpendicular distance from the line run, to the
point designated, was 17 feet 5 inches to the left
hand. Required the course between the required
points and the several offsets to be set off from the
line run to the true line, the stakes being 50 perches asunder.
Ans. The course is N 89~ 31' E.
1st offset 3 feet 41 inches; 2d, 6 feet 81- inches.
3d, " 10 feet 01 inch; 4th, 13 feet 4' "c
5th, " 16 feet9 inches; for the remaining
part of the line (9.44 perches) 8 inches.
11th. What allowance must be made on a course
S 21~ E, distance 727 feet, the perpendicular distance from this line to the point required being 17
links.
Ans. Cor. 53'. The course will be S 20~ 7' E.
Gase 2d. When obstacles prevent running the
line directly from the point A, the beginning of the
line A B:
Choose a point C near A, so that A C may be
perpendicular to A B.
Run the random line C E as before directed,
setting equidistant stakes at n' n2 n3, &c.
44 A MANUAL FOR SURVEYORS.
C. —. A Measure E B: from which
deduct B D (=-A C) the remainder will be E D. Then, by, the last case determine the offsets n1 n', n 2 m2n &c., to each of
which add A C, and we obtain
2 the offsets nl p', n2 p2, &c., to
be laid off from the random line
C E to the true A B. If it be
3 required to determine H a point
at a given distance from A, we
may say; As C E: C F:: E D:,].......F G, to which apply A C
(= G H) to obtain F IH: whence
H is determined. As in the
~s | s preceding case we may find the
angle E C D which applied to
D. jjB the bearing of C E, will give the
bearing of C D, that is of A B,
because A B is parallel to C D. When E B is less
than A C, the corrections n'l mn, n2 n2, &c., must
be- subtracted from A C to obtain the corrections
for the points pl p2, &c., in the line A B.
A MANUAL FOR SURVEYORS. 45
Example Ist. Being required to run a line between two points, A and B-I measured a perpendicular A C, equal to 20 feet, and from C run the
line C E, S 290 W 130 per., when I found the distance E B was 27 feet to the right of C E; what
is the bearing of A B; also what distances must be
laid off from C E to determine points in the line
A B at 40 per. apart.
Ans. The bearing is S 29~ 11' W.
Ist offset, 22. 1 ft. 2d, 24.3 ft. 3d, 26.5 ft.
Example 2d. Run the random line C E-S 30~
W 125 per., and measured A C = 20 ft. and E B
= 9 ft., the true line being on the left of the random line. What is the bearing of A B, and what
offsets must be measured off to fix points in A B
40 per. apart.
Ans. The bearing is S 30~ 18' W.
1st offset 17.48; 2d, 12.96; and 3d, 9.44 ft.
Case 3d. When the random line crosses the true
line, between the extreme points.
Let A B be the line required to be run, C D the
random line, C and D being on opposite sides of
5
46 A MANUAL FOR SURVEYORS.
1 Tc X, A the line A B. Suppose AB
or C D measures 165 perches,
A C=15 feet, and B D
= 25 feet. As in the preceding cases, let stakes be
driven into the ground in the
line C D, equidistant from
j each other (say 40 percher.). It is evident that the deviation of the random line C D
from parallelism with A B is' equal to the sum of the distances A C and B D, or
15 +-24 _ 40 feet. Therefore, in this case we must
/;: -7take the sum of the distances
A C and B D to find the corrections.
As C D: C n:: A C + B D: nl = (A C-C F)
165: 40:: 15 +25:9.7 = Af
Or As C D n5 B: A C + B D: B D- n4 m4
165: 5:: 40.: 1.2 =-By
15 AC - 15 feet.
A MANUAL FOR SURVEYORS. 47
AF = 9.7
Tilhe Ist offset n' m' = 5.3
9.7
2d, do. n mn = 4.4
9.7
3d, do. n' m3 = 14.1
9.7
4th, do. n4 m4 = 23.8
Bg 1.2
B D 25 proof.
We cannot subtract 9.7 from 5.3 the first offset,
which shows the random and true line intersect
between the points, the difference 4.4 is laid off
on the other side of the random line, as well as
the rest of the offsets to the end of the line. The
point I, the intersection of the lines may be found
as follows:
As A C + B D: A C:: C D: C I or A I
40: 15:: 165: 61.9
Also A C + B D: B D: C D: D I, or B I
40: 25:: 165: 103.1
48 A MANUAL FOR SURVEYORS.
The correction of the bearing is found as in the
preceding cases.
40 feet = 2.42 perches.
2.42 -. 165 =.01467 nat. tang. of 501'- this applied to the bearing of C D, will give the bearing
of A B. In this case the offsets decrease at the
beginning of the line, the correction for the distance between the equidistant stakes, must be
subtracted, until the remainder is less than the
correction; then subtract this remainder from the
said correction, the last remainder will be the distance of the offset at the next stake, to be laid off
on the opposite side of the random line to a point
in the true one; the random and true line having
intersected each other between the stakes, at one
of which the offset was laid, on a side of the
random line, different from that of the other; after
which the offsets are all laid off on this side to
the end of the line. Those acquainted with the
nature of plus and minus quantities, will readily
perceive the reason of all this.
Example 2. Given A B or C D = 125 perches.
A MANUAL FOR SURVEYORS. 49
A C = 7.7 feet; B D -- 6.2 feet required the offsets, the equidistant stakes being 20 perches apart.
Ans. The 1st offset is 5.5 feet.2d, do. 3.3' To the left.
3d, do. 1.1 J
4th, do. 1.1 1 To the right of
5th, do. 3.4 [ the random
6th, do. 5.6 J line.
The correction of the bearing is 23'.
The lines intersect 69.3 perches from A.
PROPOSITION 5.
Prolonged Lines. A
To determine a point B in the line, A E
produced.
Case I. When the instrument can be placed
at E, from which A is visible. E
Direct the sights to A, and prolong the
line towards B, which continue as far as required.
Case 2. When obstacles prevent A from being
seen from E.
5*
50 A MANUAL FOR SURVEYORS.
Measure off a convenient distance F E,
perpendicular to A B; make A C = E F,
then project C F to D, a point perpendicular to A B from the point B, and make
[ BD=E F; the pointB will be inAE
produced.
Case 3. When obstacles prevent A being seen
from either E or F.
14A Run the line A D by proposition 2d.
At a point F, measure the perpendicular F E; also measure the distances
AE and A B. Then AE:AB::
F - - F E: B D, which being laid off from
the line A D, gives the point B in A E
produced.
---— I- The angle D A B may be found by
the rules given in Proposition 4.
As A E:FE:: 3438': FA E
Or 2 AE: E F:: 86~: 7 E F: F A E
Or F E - A E = Nat. Tang. v F A E.
A MANUAL FOR SURVEYORS. 51
Place the instrument at A and make the / F A E
as found above; then run out A E B.
Case 4. When it is impracticable to run from A,
Set the instrument at C, a point
perpendicular to A B, opposite to A.
Run a line C G II to H, opposite to
B. Measure G E, from which deduct
A C =F E, the remainder is G F. G.. E
Then, as in the preceding case, as
A E: A B:: G F: H D, to which
apply B D =A C, we get H D + i _
AC-=H B.
Case 5. When G E is less than A C.
From A C deduct G E, the remainder
is F G. Then, G C: C H:: F G
D IH and H B =B D-D DH = ACD II. So the point B is determined.
Case 6. When the random line C II crosses the
line A E prolonged.
52 A MANUAL FOR SURVEYORS.
c — ~ In this case, F G=-=A C-G E
CG: C II:: F G: D H.
D HI being greater than A C, their
zr, difference is B H, which must be laid
off to the contrary side of C H, to
which A C was laid; that is, A and
B are on opposite sides of C H, as
D H:A C:: C H: C I. The point
\ of intersection of the random and true
lines may, therefore, be exactly designated.
CYase 7. When the random line crosses the true,
between A and E.
C_ __A In this case, G E will be on the opposite side of the line A E, to which
A C is; therefore,
F G= F E + E G = A C + E G
As C G: CI:: F G: DI) II
H B=D HI-DB=D IH-A C
A and B are on contrary sides of C IIH.
I, the point of intersection is found as
in Case 6. The > D C-H is found by
V W 3 Case 3,
A MANUAL FOR SURVEYORS. 53
PROPOSITION 6.
To retrace the lines of a Survey.
Case 1. When the angular points are established,
run the lines from one angular point to another,
by the methods already given.
Case 2. When the angular point is known to be
in a right line with two other points, and at a given
distance from each of them.
Let A and B be the given points, A
C the point in a right line between
them, which is to be determined in.
order to trace the line D C.
Run the line A B as before directed, from A towards C, measure the distance the point C is
known to be from A; also, from B to C, measure
the distance C is known to be from B, then, if
these distances both terminate at C, the point is
determined. The line C D may then be run out
by the methods already given. But if the measures from A and B do not terminate at the same
point C, which in practice will often be the case,
measure the given distances from A and B towards
C; the distance measured from A, terminating at
54 A MANUAL FOR SURVEYORS.
a; that from B at b. Then it will be, as the given
distance of A from B, is to the given distance
A C; so is the distance a b, to the correction a C,
or as AB: B C ab: b6 C the correction on B b.
The correction a C applied to A a or 6 C applied
to B b, will determine the point C.
This method is also applied, when there is not
full measure.
Example. Given A C = 75 perches; B C = 150
perches, but in measuring these distances, there is
found an excess of measure, a b =.6 perches, required the corrections. As 75 + 150: 75::.6:
aC =.2; or 225: 150::.6: bB =.4.
Therefore, A C - A a + a C = 75 +.2 = 75.2
perches, B C = B b + b C = 150 +.4 = 150.4.
Example 2d. Given A C = 325 perches, B C =
175, and a b = 1.5 perches, excess measure, what
are the true distances?
A C = 325.975 B C = 175.525.
_Example 3d. Given A C = 120 chains; B C =
80 chains, and a deficiency of full measure, a b =
20 links. What is the lengths of A C and B C by
this measure?
A MANUAL FOR SURVEYORS. 55
Ans. A C = 119.88 chains, and B C = 79.92
chains.
Case 3. When the adjacent lands A C D and
B C D have been surveyed at very different periods
of time, measure several lines of the land adjacent
to A C D, and say as the sum of the lines measured, is to the line A C; so is the gain or loss
of measure on the lines measured, to the gain or
loss of measure to be applied to A C. In like
manner find a correction of the measure of B C, by
measuring several lines of the adjacent land B C D.
Use these corrected distances for A a and B b in
the preceding case.
Case 4. When there are line marks in the line
D C at D and E, and also in the line A B at B
and F.
Prolong the line B F towards
D;!c
A, by the method already described; also prolong D E until
it intersects the prolonged line
B F in C, which will be the point sought.
Case 5. When there are marks only at D and B,
runs out ome of the lines on the adjacent lands,
56 A MANUAL FOR SURVEYORS.
which are nearly parallel to the lines to be run, by
which the difference of variation is obtained, which
being applied to the former bearings, gives the
present bearings of the lines. The lines may then
be run out. This may be done with both adjacent
tracts of land, and a mean of the results taken.
A reason for selecting lines nearly parallel to
the line to be run, is that the difference of bearings
of a line as shown by two compasses, will be the
same on lines nearly parallel to it. When this
difference is applied to lines at nearly right angles
with it, a considerable difference will very often be
found, which frequently leads to error, unless
carefully guarded against.
Case 6. When there is given the bearings and
distance of the line A B, running from the point
A, and the year in which it was run.
A The only method likely in this case to approach a satisfactory result, is the following:
If there have been a line or lines in the
neighborhood, run the same year, (or there{B abouts) go to the premises and run them out,
by which you get their present bearing, and therefore the difference of variation between the present
A MANUAL FOR SURVEYORS. 57
time, and the time at which the surveys had been
made; this difference allowed on the bearing of
the line to be run out, will give its present bearing
by which to run it out.
If, however, no survey had been made of any
lands in the neighborhood, by which the difference
of magnetic variation may be found; then, in such
case, if the annual rate of increase or decrease in
the magnetic declination be satisfactorily known,
we may ascertain the change of variation in the
interval of time which applied to the given bearing
of the line to be run, we shall have its present
bearing. The variation at West Chester, in 1845,
is 40 2' W.
The variation of the magnetic needle in declination, is subject to much irregularity, in some instances increasing, in other decreasing, and some
years having scarcely a perceptible motion. The
annual variation at Philadelphia, has been stated
at 1~ in twenty years. In the neighborhood of
West Chester, it is about 1~ in sixteen years, in
Warminster Township, Bucks County, 1~ 3' in
fourteen and one-third years. At any place there
6
58 A MANUAL FOR SURVEYORS.
is much irregularity in a lapse of years. It must,
therefore, be a matter of uncertainty whether we
have the correct bearing of the line, even when the
change for years has been ascertained with the
utmost care.
Another source of error in this case, the diurnal
variation, may be properly mentioned here. If a
survey be commenced early in the morning, which
is not completed until one or two o'clock, P. M., of
a very warm day, it will be found that the bearing
of the first line of survey will vary several minutes,
sometimes a quarter of a degree from its bearing
in the morning. In the winter season, this difference will seldom exceed five minutes of a degree,
but in very warm weather it may amount to fifteen
minutes. There will be little difference in cloudy
weather. Surveys should, therefore, as far as
practicable, be made in the cool part of the day.
A line which is to be established from the course
only, should be re-run at nearly the same season
of the year, a day chosen of much the same temperature, and the same time of day, in order to ensure the nearest approach to accuracy the case will
admit of.
A MANUAL FOR SURVEYORS. 59
Other sources of error are the eccentricity of the
compasses used, the difference of polarity or direction of the needles used, &c., all which should be
carefully guarded against. If the surveyor when
running old lines were to note the difference between the bearing now found and that given, by
applying this difference to the variation of his needle, he may determine very nearly the magnetic
variation at the time of the former survey.
A collection of observations of this kind would
enable him to ascertain the rate of increase or decrease of the variation of the magnetic needle;
and would be highly valued by those who may be
investigating this perplexing subject.
Case 7. When the angular point C is to be determined from the distances A C and B C only,
the points A and B being known.
Measure A C as nearly as may be in
the direction of C, and at the end of
the distance, set two stakes a few feet
apart, so that a line joining them may
be at right angles with A C. Also,
measure B C, and set two stakes a few pi'
feet apart at right angles to B C. Then
60 A MANUAL FOR SURVEYORS.
It line joining the former two stakes will intersect
a line joining the latter two in the point C, the
angular point required.
Case 8. When C B bears but a few degrees
from A C.
A Prolong A C towards D. With the
angle of deflection B C D as a course, and
B C a distance, enter the traverse table,
and take out the latitude C D, and make
D B perpendicular thereto, and equal to
/ the departure; the point C and the line
B C are determined by making D C =
B...... the latitude from the traverse table.
Case 9. When local attraction affects the needle
on the several lines.
A Set the instrument at A, direct a stake
to be set at B, let the needle down on its
centre pin, and direct the sights to B.
B If the needle does not show the same
bearing as formerly, move the vernier
ci plate till it does. Remove to B, direct
the sights to A. If the needle does not
show the same course for B A as at A, there is
local attraction. Move the vernier till the needle
A MANUAL FOR SURVEYORS. 61
shows the reverse of the given bearing of A B.
The needle set to the given bearing of B C will
give its direction. Next remove to C, and take a
sight to B, move the vernier until the needle gives
the reverse bearing of B C. The needle set to the
bearing of C D will give its direction; and so proceed. Otherwise, apply the angle A B C to the
bearing of A B, to get the bearing of B C; apply
the angle B C D to get the bearing of C D, &c.
This may be very accurately done with a theodolite or transit.
PROPOSITION 7.
On -Distances, Bc.
Case 1. Through a given point to run a line
D C at right angles with a given line A B.
In the line A B choose two A
stations, E and F such that the
angle E D F may be less than
100, the point C falling be- DeC
tween E and F.
With an instrument measuring angles to minutes, mea- B
6'
62 A MANUAL FOR SURVEYORS.
sure the angles E P D and F E D, the complements of which give C D E and C D F; also, measure E F; then, as the sum of the angles C D E
and C D F in degrees and minutes, is to either of
them, as C D F in the same measure, so is the
base E F, to the part C F of the base, corresponding to the < F D C, from which the point C is
determined.
If V C D E be used in the above proportion, we
get E C; or correctly, as the sum of the nat. tangs.
of the angles C D E and C D F, is to the nat. tang.
of either of them, so is E F the sum of the segments E C and C F. to the segment corresponding
to the angle used in the second term of the proportion.
Otherwise, subtract the bearing of A B from
90~, the remainder changing N. to S. or S. to N.,
is the bearing of C D; then, by trials make C D
this course; and C will be the point required.
Case 2. From the point C in a right line A B,
to trace a line C D at right angles with it.
This may be readily done, any of the instruments used in taking angles, or with the cross
A MANUAL FOR SURVEYORS. 63
mentioned in the choice of instruments. It may
also be done with the chain as follows:
Make A C = B C = 20, 30, or I
any other number of links less
than a chain, place one end of A I
the chain at A, and with the C
other end, trace an arc on the E
ground; remove the end of the chain from A to B;
with the other describe a second arc, cutting the
former in D. A line joining D C will be at right
angles to A B. In the same manner, another point
E on the other side of A B may be found.
Or thus, set the compass at C, and take the bearing of A B. Subtract this from 90~, changing N
to S or S to N, gives the bearing of C D.
Another method: make A C = 4; with A as a
centre and a radius 5 describe an arc; with the
centre C and radius 3 describe an arc, cutting the
former in D the point required.
Any multiples of 3, 4, 5, as 6, 8, 10, or 30, 40,
50, &c., will form right -d triangles.
Third method: place one end of the chain at C,
the point at which a right angle is to be made, ex
64 A MANUAL FOR SURVEYORS.
tend the other end to any convenient point E; with E as a
centre, move the other end
from C (the chain being radius,) until it crosses the line A B in A; prolong
A E towards D, making D E equal to A E. Join
C D which will be at right angles with A B.
Case 3. To measure an inaccessible distance A B.
A Make B C at right angles to A B,
at C any convenient point. Make
C D at right angles to C B, or paral/ lel to A B. Set a staff at E in the
c B line C B, and in a range with A D.
Measure B E, E C, and C D; then
as E C' B E:C D: A B.
This proportion will hold good, if
B C make any angle whatever with the parallel
lines A B and C D.
Note. —If B C = C E, then A =B - C D.
Case 4. To determine the distance A F, which
cannot be directly measured, on account of the
obstacle between B and E.
A MANUAL FOR SURVEYORS. 65
Trace the line to B, run B C.
From G run C D parallel to A B.
From D run D E parallel to B C,
and make it equal to it. Then run c_,
E F the same course as A B: F will
be a point in the line A B continued.,
The distance A F will be equal to
the sum of the distances A B, C D
and E F. X
Case 5. To find the bearing and distance of A
from B, accessible only at its extremities.
Choose a point O from whichA A
A and B are both accessible. \
Prolong A O to D, making D O i \
equal to A O; also, prolong \
B O to C, making O C=O B. D
Join C D, which will be equal
and parallel to A 13. Its bearing and distance is therefore
determined.
Case 5. When A and B are inaccessible.
Plant a staff at C and find the distance A C and
C B by the preceding methods:
66 A MANUAL FOR SURVEYORS.
-~_~/t Make C D as many parts
/ /D of A C as you do C E of
C B; join D E and measure
ac S it. Then, asC D:CA::
D E: A B, or as C E:
B:: DE: A B D E is
parallel to A B.
Ccase 6. To prolong the line A B across a river,
&c., and determine the distance across to E..A At a point C make the angle B C D a right one, at D
make the angle B D E a right
n- C angle. Measure B C and D B.
Then B C: B::B D:
B E. If D C be measured,
B C: D C:: DC: C E.
Otherwise. Make L C D E
= 450, then C D will be
equal to C E, the breadth of the river.
A MANUAL FOR SURVEYORS. 67
If< C D E-260 34' then C E= CD'<CDE=330 41' " CE=-3CD
"<CDE =450 " CE= CD
"< CDE =560 19' " CE= CD
"< CDE=63026' " CE=-2CD
Or, if C B D = B D E = 60~, the triangle B D E
becomes equilateral, and B D = B E.
fCase 7th. To find the horizontal distance A B
through a precipice or clift too steep to be measured with the chain.
Plant a staff at C on the top of A
the cliff, aligning it with A and B,
both of which may be seen from C.
Run the line A E and measure it.
Also measure the angles C A E =,
D A E; and C EA = D E A(the
point D being directly below C.)o ----- I'
Find, by trigonometry, the dis-
tance A D. In like manner from /
B run B F and measure it also
the angles C B F = D B F; and
B F C= B F D to find B D.
Then A B is the sum of the hori-
68 A MANUAL FOR SURVEYORS.
zontal distances A D and B D. In this solution the
planes A D C, E D C, B D C and F D C are conceived to be at right angles to the plane A E D F B.
By taking the altitude to the top of the staff C from
two stations on each side of the steep, the horizontal distance may be found.
Case 8. To measure an inaccessible distance
AC.
1 A Plant a staff at D in a line with
/ A C. Run B C and measure it;
and make E D parallel thereto, and
e measure E D and also C D. Then
E F (= E D- BC):B B C::CD
to A C. This case is applicable
i/'. when B C and E D differ consider-g ) ID ably, that is, E F bears a due proportion to B F or C D.
Note. — This Case should have followed Case 3.
PROPOSITION 8.
Parallel Lines.
Case 1. From a given point C, to run ql1ine C D
parallel to A B.
A MANUAL FOR SURVEYORS. 69
Run the line A B to find its bearing, c
remove the instrument to C and set the
needle to this bearing, the sights will
then be parallel to A B, so C D may be
run out.
Or set the instrument at B and mea- A
sure the angle A B C. Remove to C, D
making the angle B C D equal to the supplement
of A B C. Then will C D be parallel to A B.
If a transit be used after having removed to C,
reverse the telescope on its axis, bring it to bear
on B (the vernier being at v A B C,) clamp the
lower plate, bring the vernier to zero, the telescope will be parallel to A B; reverse the telescope
on its axis, and set a stake at D.
Case 2. When B is not accessible from C,
Plant stakes at A, B, C, also
at E, in a line with B C. Find, B
by the preceding methods, the
distances A E, E B and E C.
As E B: E C:: A E: ED.
This being laid off in A E to D, a A
line joining C D will be parallel
to A B.
70 A MANUAL FOR SURVEYORS.
Second Method. —Run A C and make. A C D
B A C.
Otherwise. Bisect A C in I. In
J I B I produced, make I D = B I. Join
C D, which will be parallel to A B.
PROPOSITION 9.
To determine a point C in a right line with A B,
the point B being a steeple, or other conspicuous
object, which is visible from A and C, and inaccessible from both.
Measure the angles of deflection B A D d D E,
e E F, D E and F being suitable points for
measuring the angles of deflection. The station
F being selected as near C as can be judged upon
the ground.
The angles of deflection at D and E being to
the left, their sum must be diminished by the
A MANUAL FOR SURVEYORS. 71
angle at A: the remainder is the deviation from
parallelism of the lines A
A B and E F. This remainder subtracted from
900~, or a right angle gives
the angle of deflection
fF G at F, to makeFG
at right angles with A B
prolonged. Run F C G,
measuring the distance
F G. Observe the angles B F G and F G B, the complements of which
are F B C and G B C. The sum of these complements is F B G.
As the angle F B G, in degrees or minutes,
Is to the angle F B C in the same measure,
So is F G
To F C,
Or, as < FB G: < GB C: GF: C G.
From either of these proportions the point C is
determined, which will be in A B produced, F G
is the complementary course of A B. It may be
72 A MANUAL FOR SURVEYORS.
run out where there is no local disturbance of the
needle by setting the instrument at F, and run F G
by this complementary course. The base F G of
the triangle B F G should be of such a length, that
the angle F B G may be only a few degrees; if it
should be too great, the distances F C and C G will
not increase in the same ratio with the angles F B C
and CB G.
N. B.-If F B G is greater than 100, the segments F C and G C should be found by the usual
rules of Trigonometry.
The following proportion is in substance the
same as the preceding. From the external angle
B G H, take the < B F G; then say,
As this remainder is to the difference between
the angle B F G and a right one; so is F G to
F C as before.
It may sometimes be convenient to take several
stations in deflecting from A to F; but in all cases
the angles of deflection to the right hand must be
added together, and also those on the left of the
lines deflected; the difference of these sums will be
A MANUAL FOR SURVEYORS. 73
the deviation, from parallelism, of the last line deflected from the line A B.
If the deflected angles on the right exceed those
to the left, the difference must be laid off to the
left, and vice versa: the telescope will then be
parallel to A B.
If at every station we arrive at, we set the vernier to the same degrees as at the last station, reversing the telescope on its axis or in its wyes, and
bringing it to bear on the last station point; then,
having clamped the lower plate to the tripod, bring
the telescope in its direct position on the next station, the vernier will perform the additions and
subtractions of the angles of deflection; consequently, when we arrive at any station when the
instrument is adjusted by the last station point,
bring the vernier to zero, the telescope will be
parallel to A B-but if set to 90~, it will be at
right angles with it. We may make any angle
whatever with A B, by setting the vernier by
means of the tangent screw to that angle.
Given < F B C = 2 5'; GB -2~0 55' and
74 A MANUAL FOR SURVEYORS.
F G - 12 ft., to find F C (the points F and G
being determined as above).
As F B G (300'): F B C (125'):: F G (12 ft.):
F C = 5 ft.
F B G (300'): C B G (175'):: F G (12 ft.):
GC =7 ft.
Case 2d. When C falls without the triangle
FB G.
Find, as before, the angle F B G,
the angle G B C, which is equal to
the difference between B GU C and n
a right angle, and the distance
F G.
As F B G: <, G B C:: F G:
G C, which, being laid off on F G
produced, determines C, a point in
A B prolonged. The contents of
this proposition and Case 1st of 7
Prop. 7th, are believed to be new; a
nothing of a similar character is to be found in any
publication with which I am acquainted.
Given FB C -= 50'; G B C-10' and F G = 8 ft.
A MANUAL FOR SURVEYORS. 75
to find G C; F G being determined as above directed.
As < F B G (= 50' — 10'=40'): G B C (10')::F G= (8 ft.): G C =2 ft.
Given F B C = 83'; G B C = 17' and F G=
33 ft. to find G C. Ans. G C - 8.5 ft.
if G is between F and C; but if not; G C = 5.61 ft.
PROPOSITION 10.
Dividing Land.
An easy rule for finding the angles of a right
angled triangle, the sides being given. To the
hypothenuse add half the longer leg. Then, as
that sum is to the shorter leg, so is 86~ to the
angle opposite the shorter leg. This rule, which
is easily remembered, is very useful in many calculations in the field, where tables cannot be conveniently used. The greatest error does not exceed-4 minutes. The rule is therefore sufficiently
exact for most purposes in surveying to which it
may be applied.
76 A MANUAL FOR SURVEYORS.
Example. Given the 3 sides of a right angled
triangle, 30, 40 and 50, to find the angles:
(50 + 20): 30:: 860-36~ 53' the less angle.
Example 2d. Given the hyp. and greater leg of
a right angled triangle 50 and 40 angle opposite
less leg 366~, to find the less leg.
As 860: 36-: 70: 30, as required.
This rule may be readily applied in cutting off a
trapezoid from a given tract of land that shall contain a given number of acres, the angles being
nearly right angles.
X \-I A Let there be given A B
south, B C west 40 per. and
C D N. 40 W., to cut off a
a trapezoid A B C D containing
5 acres, by a line A D parallel to B C.
First, 800 _. 40 = 20 = C E approximate.
In this case the leg and hypothenuse are nearly
equal, we may use one for the other, therefore,
As 860: 4:: (20 + 10): 1.4 = E D approximate.
A D = A E + E D = B C + E D =40 + 1.4
= 41.4 approximate.
A MANUAL FOR SURVEYORS. 77
Twice the area of a trapezoid, divided by the
sum of the parallel sides, gives the perpendicular
distance between them.
1600 -- (40 + 41.4) = 1600 _. 81.4 = 19.66 -
CE =A B.
As 86~: 40:: (19.66 + 9.83): 1.37 = D E.
A D = 40 + 1.37 = 41.37 sufficiently correct.
1600 - (40 + 41.37) = 19.66 as before (the
proof.)
The angle B being a right angle, no correction
is required on that side of the trapezoid.
Another method is to find the area of the triangle D C E, and cut off a small trapezoid equal to
it, either within or without, as the case may require. In the preceding example E C + I- D E =
area of C D E = 20 +-.7 = 14., this, divided by
A D 41.4, gives.34, and this subtracted from E C,
20, because A D is greater than C B, gives the
correct value of E C = 20 -.34 = 19.66, as before.
In most cases the use of the traverse table is
more expeditious to find D E. Taking the approximate value of C E in a lat. column, under 4~,
78 A MANUAL FOR SURVEYORS.
gives, in a departure column, 1.4 = D E, from
which a correct value of C E is found. With
19.66 in a lat. column, under 40 in a distance
column, is found 19.,71 = C D.
Example 2d. Given A B south, B C west 40
per., and C D N. 40 E. to find A B, C D and A D,
when the trapezoid A B C D contains 5 acres.
Ans. A B = 20.36 perches.
C D = 20.41 "
A D - 38.58
When it is required to cut off a trapezoid from a trapezoidal piece of
land, it may sometimes be done in the
following manner:
Given A B N. 40~ E. 44.4 perches,
B C S. 501 E. 60.8 perches, C D S.
40~ W. 46 perches, and D A N. 490 W. 60.9 perches, to cut off 5 acres by a line E F parallel to
A B.
First.-800 -- 44.4 = 18 = E A approximate.
Dg = D C- AB = 46- 44.4 = 1.6.
A D: A E:::D E h or 60.9: 18:: 1.6:.47.
E F = F h + h E = A B + E h = 44.4 +.47
- 44.87.
A MANUAL FOR SURVEYORS. 79
A = F B = 1600. (44.4 + 44.87) = 17.93
the perpendicular corrected.
In a lat. column with 17.93 under 1~ ~- g A D
in a distance column, is foundf17.94 A E.
When calculations are made by J. Gummere's
rule, (the sides A D and B C, as in the above example, being nearly parallel,) great care must be
used in extending the decimals to 3 places to ensure accuracy, as two decimals only may throw the
line E F a pole from its true position in many
cases that occur in practice.
PROPOSITION 11.
To determine the correct bear- X
ings of the lines of survey where
local attraction deflects the needle A
from its usual direction or magnetic position.
Let A B C D be several sides of B
a survey on which the needle is
disturbed by some extraneous mat- a
ter. \
80 A MANUAL FOR SURVEYORS.
With the compass or circumferenter: Place the
the compass at A, take a back sight to X, the last
station, and note the bearing, then sight to B and
note its bearing. Having the bearing of A X and
A B, both from the same station A, we can find
the angle X A B as correctly as if the needle settled in its true position, for the needle must be
equally affected when the bearings were noted.
Remove to B, and take a back sight to A, noting
its bearing, then direct the sights to C, and note
its bearing, from which the angle A B C will be
correctly obtained. Thus proceed until all the
angles are taken. If the entire survey has been
made as above directed, the sum of all the internal
angles will be equal to twice as many right angles
as the figure has sides, diminished by four right
angles. If this sum, as in practice will be likely
to be the case, should differ a few minutes from
what it should be, the minutes of error may be distributed among the angles by addition or subtraction, according as there is defect or excess in the
sum of the observed angles.
Now, having all the correct angles, assume some
A MANUAL FOR SURVEYORS. 81
side of survey as X A to be correct, being least
affected by local attraction; then applying the angles severally as they come in order of survey, we
will have the bearings of the sides as correct, relatively, as if no local attraction existed.
If a nonius compass be used, place it at A, take
the bearing of A B, remove to B, take a back
sight to A and clamp the sights upon it. Unclamp the nonius plate, and with the pinion and
rack move the nonius plate until the needle gives
the reverse bearing of A B, which it had at A.
Unclamp the sights and bring them to bear on C,
the needle will show the correct relative bearing of
B C, which note. Remove to C, take a back sight
to B, and clamp the sights upon it; move the
nonius plate by the rack until the needle shows
the reverse bearing of B C; unclamp the sights
and take the bearing to the next station, and so
proceed till the survey is completed. The relative
bearings thus obtained will be as correct as if no
local attraction influenced the needle.
If a theodolite or transit be used, the internal
angles may all be measured by the limb of the in8
82 A MANUAL FOR SURVEYORS.
strument, without regard to the needle. From
which, having also the bearing of one line, the
bearings of all the lines may be found.
The external angles, or angles of deflection, may
also be taken as follows:
Place the instrument over A, reverse the telescope on its axis or in its wyes, set the vernier to
zero, and bring the sight to bear on X; then clamp
the lower or graduated plate in this position, reverse the telescope to bring it again in its direct
position, bring telescope to bear on B, (by means
of the tangent screw or rack,) the index or vernier
will, being read, give the angle of deflection of the
lines X A and A B. Remove to B3 and take the
angle of deflection to C in the same manner as
from A to B; proceed thus the entire circuit of the
survey. If all the angles of deflection have been
outward, their sum must, if correctly taken, be
equal to four right angles, or 360~. If any of the
angles are re-entering, the sum of the external
diminished by the sum of the re-entering angles,
will be equal to 360~.
If the telescope be reversed, the vernier being at
A MANUAL FOR SURVEYORS. 83
the same division as at the last station, the hair of
the telescope cutting the last station point, then
clamping the lower plate to the tripod, reverse the
telescope to bring it in a direct position, bring it
to bear with the tangent screw on next forward station, the vernier will show the angle of deflection.
Proceeding thus from station to station, the vernier
will give the sum of the angles of deflection, and
hence, when we arrive at the first station, the vernier will be at zero, or the point at which it was
placed when the survey was begun. Its distance
from this point is the sum of the errors in observing the angles which may be distributed amongst
them, so as to make the proper sum or quantity
for the angles.
If there should be a considerable difference or
error it would be advisable to retrace some of the
lines until the error be discovered.
CALCULATIONS.
It is the practice with some surveyors to read
the bearings of lines to quarter degrees, and note
84 A MANUAL FOR SURVEYORS.
the distances in chains and links, which, in calculations, they reduce to perches and hundredths.
In measuring distances, where the line to be
measured is one hundred perches or upwards in
length, it must be evident to those who may be
acquainted with the ordinary mode of measuring,
that that distance cannot be measured to the hundredth of a perch, and frequently not even to the
tenth.
Again, when courses are read to the nearest
quarter degree, there is a probability of an error
which may reach to half that quantity or 7~',
which, in the distance of 100 perches, gives 2
tenths of a perch departure from the point of termination. Therefore, when bearings are read to
quarters of a degree, and distances measured to
tenths of a perch, it will not conduce to accuracy
to extend the calculations for the area to hundredths or another decimal figure.
The bearings of lines should always be read to
the nearest five minutes, and distances over 100
perches need not be more exactly noted thanito
tenths of a perch.
A MANUAL FOR SURVEYORS. 85
In laying out town lots, or where the utmost
precision is desired, angles or bearings should be
measured to the nearest minute, and distances to
hundredths of a perch, or tenths, or hundredths of
a foot.
For this purpose a theodolite or transit should
be used to measure the angles, and a twenty feet
frame, with a level or plummet attached, having a
slider affixed at either end of the frame, to adjust
it to horizontal admeasurement. To measure a
line with the frame, in the first place, the line
should be "boned," as it is technically termed,
that is, pegs or short stakes, at the distance of 20
feet, should be driven in the line nearly even with
the surface of the ground; then placing one end
of the frame at the beginning of the line, the other
on the first stake, after having adjusted the frame
to a level, make a fine scribe or mark on the top of
the stake, precisely at the end of the franze; next,
bring the frame forward, adjust the hind end to
the scribe on the stake, bring it to a level and
scribe the second stake at the end of the frame;
and so proceed to the end of the line.
8*t
86 A MANUAL FOR SURVEYORS.
A line, several hundred feet in length, if measured as above directed, will be within an inch or
two of the truth.
METHOD OF OBTAINING THE FIELD NOTES OF A
TRACT OF LAND ACCURATELY.
Having given directions in the preceding part
of this work, best suited to determine correctly the
position of any line of a survey which may be desired to be run out or retraced, it may be proper
here to give an example embracing a number of
those cases, which occur in practice, so as to exhibit the application of the rules which have been
given.
Let us suppose it be required to survey a tract
of land A B C D E A with a transit and common
two pole chain.
Place the instrument, by means of a plummet,
exactly over the point A of beginning. After
having adjusted it for observation, set the vernier
to zero, and clamp it there; next, let the needle
down upon the centre-pin, revolve the instrument
A MANUAL FOR SURVEYORS. 87
I g'
1/A /
so as to bring the needle to the north and south
points of the compass box, clamp the lower plate to
the staff head, unclamp the vernier plate and bring
the telescope to bear on a staff set at B, the second
station; the course may be read from either the vernier or needle, or both, which is preferable. Suppose the vernier reads 50~ 21' and the needle N.
88 A MANUAL FOR SURVEYORS.
50~ 201 E., also A B measures 68 chains and 471
links. The course and distance may be set down
A B N. 500 21' E. 137.9 perches, using the course
shown by the vernier in preference to that shown
by the needle. The observations at A having been
finished, remove the instrument to B, adjust it for
observation; reverse the telescope, the vernier
being at 50~ 21', the bearing at the last station
A B, bring the telescope to bear on a pole at A,
clamp the instrument to the tripod, the zero points
or north and south points of the graduated plate
and the compass box will be in the magnetic meridian, and consequently parallel to its position at
A; the vernier will therefore show the bearing of
a line, the same as the needle. Unclamp the vernier plate and bring the telescope to bear on a
third station C. Let the needle settle-suppose it
reads N. 790 55' E; at the same time the vernier
reads 790 54g. In measuring this line 13 C, an
obstacle, a clump of bushes and swanmp prevents its
being directly measured further than b, 20 chains
from B. An offset, b a, being measured 12 feet at
right angles with B C, also at c, a point between
A MANUAL FOR SURVEYORS. 89
B and 6, a perpendicular offset of 12 feet was made
from c to d. Prolong the line d a, to e, where set
off an offset e f = 12 feet; continue the line a e to
g, make g h = 12 feet, perpendicular to a e, continue f A to C. It is found a e measured 14 chains
25 links, and f c = 10 chains 21 links, so the
whole line B C measures 44 chains 46 links; the
bearing and distance of ]B C is N. 79~ 54' E. 89.8
perches.
Place the instrument at C, take a back sight to
B, the vernier being at 790 54', clamp the lower
plate to the tripod, release the upper plate. The
station D being out of view, run a random line in
that direction, as nearly as may be, setting the
vernier to 130~; the needle reading S. 50~ E. at
the same time, run the line 47 chains to the bank
of a deep creek at nm, on the opposite side of which
set a stake at p, a point in the continuation of the
random line C q. Measure a perpendicular mi n
= 6 chains, and make the angle m n p =- 26 34';
consequently m p is equal the half of m n n= 3
chains. From p continue the random line to q,
19 chains further; when we arrive opposite to D
90 A MANUAL FOR SURVEYORS.
on the left of the random line, D C is 69 chains
138 perches. The corner D from q is 9.25 feet
56 perches; C D = 207:.56:: 86~: 14/ the
correction which apply to C q, gives the bearing of
C D 129~ 46' or S. 500 14' E. 138 perches.
Stakes being set in the random line C q at every
20 perches, the calculation for the offsets is as follows: 138: 20:: 9.25 ft.:1.35 ft. The distances
to be laid off at the several stakes will be 1.35;
2.7; 4.05; 5.4; 6.75 and 8.1 feet, these distances
having been laid off, establishes the line C D.
At a point in the random line C q, near q, and
in the line D E let the instrument be set for observing the course of D E. The telescope being
reversed and brought to bear on a back stake p,
in the random line C q, the vernier at the same
time reading 1300, let the lower plate be clamped
to the tripod, bring the telescope to its forward
direction, and by means of the tangent screw,
make the hair cut E or a stake in a range with
D E, the vernier -gives for the bearing of D E
2700; the needle due west measuring 7 chains 25
links from D towards E, a high and steep rock was
A MANUAL FOR SURVEYORS. 91
encountered, on the top of which a stake was set
at r in a right line with D E. At the termination
s of the 7 chains 25 links, a right angle s t was set
off from the line D E, equal to 5 chains, making
the angle s t r = 63~ 26', the distance r s must be
10 chains. From r to E the ground was a regular
slope. I set my instrument at r, and took the
depression to E 11~ measuring the oblique line r E
17 chains 25 links. With 11~ as a course and 17-1
in a distance, the latitude is 16.68 or 16 chains 34
links, equal to 33.36 perches for the horizontal
distance; therefore D E measures, horizontally, 34
chains 9 links or 68.36 perches. The instrument
being removed to E and adjusted as at the other
stations, the telescope being directed to A, the vernier will read 266~ 12'; the needle S. 86~ 10' W.,
and measuring the distance A E, it will be found
to be 116 chains 20 links. The instrument next
placed at A, reverse the telescope, and bring it to
bear on E, the vernier being at 266~ 12', and clamp
the lower plate to the tripod. The telescope being
in its direct position, and brought to bear on B,
the vernier, if the work is correctly done, will read
92 A MIANUAL FOR SURVEYORS.
50~ 21'; this being the point at which the vernier
was set in first setting out, is the proof that the
angles have been correctly measured. The courses
and distances in this example will be as follows:
A B N. 0~0 21t E., distance 137.9 perches.
B C N. 79~ 541 E., " 89.8
C D S. 500 14' E., " 138.0 "
D E West, " 68.36
E A S. 86~ 12' W.,' 232.8 "
The outward angles by the vernier will be at
A 50~ 21', or 144~ 09' < of deflection.
B 790 54', or 29~ 33'
C 129~ 26', or 49~ 32'
D 270. or 1400 341/
E 2660 12', or 3~0 48'
Sum of right hand <'s 3630 48' positive <'s.
"4 left " " 30 48' negative "
Proof, 360~ 00'
Note.-It is sometimes conducive to accuracy to
measure diagonal lines, or lines to opposite corners
of the tract surveyed.
A MANUAL FOR SURVEYORS. 93
N. B. Angles of a survey may be measured
with the chain as follows:
Let A be the angular point, A -
A B the direction of one of
the lines, and A C the line of
direction of the other line.
Measure A B and make A C equal to it, and join
B C and measure it. The < A in the isosceles
triangle A B C is readily found.
Or, A B, B C and A C may all be unequal in
which case the angles must be found by the rules
for solving oblique angled triangles.
It sometimes happens that an old road is required to be straightened between two given points, in
which case it may be desirable to know approximately the course between the given points. This,
if the courses do not differ much amongst themselves on the old road, may be ascertained as in the
equation of payments, using the courses and distances as the payments and time are used in the
ordinary rules of arithmetic.
9
94 A MANUAL FOR SURVEYORS.
Given N. 41~ E. 20 per. 41~ x 20 - 820
N. 430 E. 30 430 x 30 = 1290
N. 420 E. 80 420~ x 80 3360
N. 440~ E. 120 " 440 x 120 = 5280
N. 400 E. 200 " 40~ x 200= 8000
What is the course of -
the straight line joining 450) 18750
the extreme points?
41-~
The course is N.
41a~ E., nearly.
Or thus, taking the several courses from 41~.
0~ x 20= 0
2~ x 30 = 60
10 x 80= 80
30 x 120 - 360
1~ x 200 = 200
450) 300
20
41
N. 412 E. as before.
By the traverse N. 41~ 35' E.
A MANUAL FOR SURVEYORS. 95
Given N. 2~ E. 60 perches.
N. 30 W. 90 "
N. 10 W. 80 "
N. 1~ E. 120 "
to find the course of the strait line joining the extreme points.
E. W.
N. 20 E. x 60 -120N. 3~ W. x 90 — 270
N. 10 W. x 80- - 80
N. 1~ E. x 120 120 -
350 240 350
240
350) 110(N. 0o 19' W.
By the traverse we get N. 0~ 19' W.
There is too much uncertainty in the use of the
above method, except in particular cases, to make
it generally useful. The traverse table should
always be used if at hand.
It is sometimes required that stakes should be
set off from the respective angular points, to the
96 A MANUAL FOR SURVEYORS.
line joining the extreme points, either to save
trouble of running a random line or for proving
the truth of the operations.
In such case take the difference between each
given course and the course of the closing line,
noting whether the given line is to the left or right
of the closing line, with this difference as a course,
and the given distance- of the line, take the departures from the traverse table, then the first departure will be the distance of the closing line from
the angular point at end of the first distance. The
sum or difference of this and the next departure,
according as they are both to the same hand or to
different hands, will be the distance of the closing
line from the end of the second distance, and so
proceed to the last point where the closing line
and last distance will come together, if rightly
done.
-Example. Given the bearings and distances of
several lines as follows, viz: N. 400 E. 50 perches,
N. 38~ E. 24 perches, N. 450 E. 40 perches, N.
39~ E.. 100 perches, required the distance of the
closing line from each angular point.
A MANUAL FOR SURVEYORS. 97
The bearing and distance of the closing line will
be found to be N. 400 15' E. 213.88 perches.
Hence (marking right hand + and left hand -)
N. E. N. E. p. Lat. Dep. p.
400 O 400 15 -_ 15' 50 50. -.22 -.22 B b
38 " - 20.151 24 23.98 -.94 - 1.16 C c
45o " q 4e.45' 40 39.86 3.31 2.15 D d
39 " - 10.15/ 100 99.98 - 2.18 -0.03 E e
213.82
E e should, when correctly done, be 3 0
N. B. The closing line is not exactly N. 400 15'
E.; but offsets computed from this line (N. 40{~
E.) determine B, C, D, &c., as correctly as if the
closing line had been used, and vice versa.
Given A B, N. 100 E. 15 perches, B1 C, N. 150
E. 19 perches, C D, N. 170 E. 40 perches and
D E, N. 120 E. 35 perches, to find the several
offsets from the closing line to the angular points
A, B3 &c.
Ans. The bearing of the closing line running
from A, is N. 14~ 5' E. 108.85 perches, the offsets
are
9,
A MANUAL FOR SURVEYORS.
B b = 1.07 perches.
C c =.77 44
D d = 1.26'"
E e =.00 6C
Note.-The reverse of the
above operation; that is, running a straight line from one
extreme point to another
along an irregular boundary,
and by calculation, finding
how far each angular point is
from the line so run, may be
employed to determine the
angular points in the irregular boundary, especially
where obstructions render the
running on the true line difficult. If the irregular boundary deviates but little from
a right line, this method is
the most accurate that can be
employed.
A MANUAL FOR SURVEYORS. 99
To straighten a crooked boundary between two
estates, so that each estate may have the same
quantity of lando
Let A B C D be a crooked boundary. It is required to run a straight line from A to a point P,
in the line passing through D, that shall equalize
in the line passing through D, that shall equalize
the quantity of land as before. Run any line A E,
near the boundary, and measure perpendicular
offsets from this line, to the several bends in the
crooked line, and find the areas of the several trapezoids, the sums of which areas will be the area
of the irregular figure A B C D E A, which being
divided by the half of A E, will give E P, the point
P being that to which a line drawn from A will
equalize the areas as required.
If the point A should be at a short distance from
the boundary, its distance must be taken from the
above quotient or added thereto, as the case may
require, to obtain E P as before.
100 A MANUAL FOR SURVEYORS.
If ecquidistant ordinates or offsets be taken; add
together half the sum of the extreme offsets, and
the sum of all the intermediate breadths or offsets,
which, being multiplied by the equal distance between the ordinates, the product will be the area
of the irregular figure as before. Whence the distance E F is found as in the former case.
Example. Let the ordinates or offsets, at six
equidistant places be 4, 6, 2, 3, 5 and 8, the equidistance apart being 50.
Here 4x8 12 6 the 1 sum of extremes.
Then (6 + 2 + 3 + 5) =22 and 22 x 50
= 1100 area.
Whence 1100 * (5 x 50.- 2) = 1100. 125
8.8, and 8.8- 4 = 4.8 = E P, therefore, a
line joining the beginning of the boundary near A,
with the point P, will fulfill the conditions proposed.
VARIATION OF THE COMPASS.
The irregularity of the variation or declination
of the magnetic needle in causing uncertainty in
retracing old lines of survey, is well known. A
A MANUAL FOR SURVEYORS. 101
step towards obviating the errors attending the old
method of finding the difference of variation or declination, and thereby obtaining the present magnetic bearing of the lines of survey, has been taken
by the Legislature of Pennsylvania, by enacting,
that meridian lines should be established in the different counties of the commonwealth, and that surveys hereafter made should be returned according
to the true, and not the magnetic bearings of the
lines. Every person will at once perceive that
much uncertainty in retracing lines will hereafter
be removed. Had surveys heretofore been made
according to the true bearings, the surveyor would,
at the present day, have merely to set the vernier
or nonius of his compass to the present variation;
then the needle would point out on the face of the
instrument the true courses of the lines of survey,
by setting it to the proper degree.
All that surveyors would then have to do, would
be to go upon the premises to a known corner, and
run out the true bearings as given in the title, the
bearings shown on the face of the instrument corresponding thereto.
102 A MANUAL FOR SURVEYORS.
We may deduce the true bearing of a former
survey by the following table or accompanying
curve, if we know the year the survey was made or
lines run. This table was formed by a comparison
of the bearings of lines taken at different periods
of time. Much difficulty is found in ascertaining
the date of survey, formerly made.
Of course this table is given only as an approximation merely. It will serve for places north or
south of Philadelphia, and a few miles east or west
of the meridian of that place. The table and diagram are sufficiently plain without explanation.
The application is as follows. Suppose the magnetic bearing of a line run in 1720 to be N. 45~ E.,
what is the true bearing? We find by the table
or diagram the variation in 1720 to be 68 W.,
the true bearing of the line is therefore N. 380~ E.
A nonius compass being set as above mentioned
to the present variation, and a course run N. 38k8
E., will run out the original line.
Again: suppose a line in 1810 bore N. 50~ W.,
what is the true bearing of the line? The variation in 1810 is found to be 2~ W., the true bearing
A MANUAL FOR SURVEYORS. 103
is N. 52~ W. The needle, as in the former case,
being set to N. 520 W., will run out the original
line.
Any person who is at all acquainted with farm
surveying, will at once perceive the advantage of
surveys being made according to the true courses,
and not the magnetic bearings. Ignorant and inexperienced persons will of course object, as more
skill and knowledge will be brought into requisition, and therefore their incompetence will be manifested.
Years. Variation.
1682, 8~0 west.
1690, 8g~0 "
1700, 8~ to 8~0 west.
1710, 710 to 7i~.
1720, 6~.
1730, 6-1 to 6-0.
1740, 55o to 5+0.
1750, 4~0 to 40~.
1760, 41~ to 3~.
1770, 2~0 to 20~.
1780, 20~.
104 A MANUAL FOR SURVEYORS.
Years. Variation.
1790, 18
1800, 1 70
1810, 2~.
1820, 28-.
1830, 3~.
1840, 330 to 380
1850, 48~0
1852, 438
1853, 4~0 west.
N. B. If the surveyor find the variation at his
place for any year, the difference between that variation, and the variation found on the chart or
diagram, will be a correction which may be applied
to variations on the chart, to find the variation,
nearly, at his place for any given time.
/6LIT u'nululu N'I VT(IIJafK aInu
l 11i!11 Z11ill
0
Magnetic eirve for lat. 400 7 N.-Long. from Philadelphia, 3 E. On
Pe' > {Minutes. 10'| 20' 30' 40' 50' Degrees.:,E3 o.0003. 0003.0061.0090.0119.0148 1.0175'
Co i 2.0006.0035.0064.0093.0122.0151 2.0349 o
CD p _ _ _ -
C' ota ~ 3.0009.0038.0067.0096.0125.0154 3.0524 Z'
4.0012.0041.0070.0099.0128.0157 4.0699 d
w. CD ~ ~,
o ~ ~.. 7o 5.0014.0,044.0073.0102.0131.0160 5.0875.
e, ___ o___ __.,__ o.._.
6.0017.0046.0076.0105.0134.0163 6.1051.
0CD C.... CD t,_-.
"'>; 7'.0020.0049.0078.0108.0137.0166 7.1228 vi
p tg c8.0023.0052.0081.0110.0140.0169 8.1405
9.0026.0055.0084.0113.0142.0172 9.1584
&Q, 10.0029.0058.0087.0116.0145;0175 10.1763
A MANUAL FOR SURVEYORS. 107
VERSED SINES, Radius 1~.
00 10~ 20~0
10.00015 -.01837 =.06642
2 /.00061 - - --.02185..07282
3.00137 7- 1.02563.07950
4.00244 - 50.02970 = yl.08642
5.00381 = 3.03407 = 09368
6.00548 7-.03874 = 8.10121
7.00745 - %.04369 = 4.10899
8.00973= To.04894 = -2.11705
9 i.01231 =.05448.12538
10.01519 =.06031 = 3.13397
Multiply the versed sine of the elevation of the
hill by the distance of the slope or surface measure,
which, being deducted from the slope or oblique
measure, gives the horizontal distance.
Example. The oblique distance of a hill of 5~
elevation is 40 perches. What is the horizontal
measure?
108 A MANUAL FOR SURVEYORS.
Here.00381 x 40 -.1524, this deducted from
40, gives 39.85 (40-.15) per. the distance required.
Or, because.0038 = -13 nearly, we have 40
Xo- = 40 -.15 = 39.85 per. as before.
Note. The numbers in this table might have
been expressed by vulgar fractions as in the foregoing example, which would, in the field, be preferable to the decimal form, in many instances
abridging the calculations, and yet be sufficiently
correct. This the surveyor can readily perform
for his own use.
The oblique measure of any line may be reduced
to horizontal measure by the traverse table, in the
following manner: under the degrees of elevation
or depression, and opposite the oblique distance in
the A4istance column, a number will be found in the
lat. column, which is the horizontal distance; the
number in the departure column being the vertical
altitude of the hill. or slope. Taking the preceding
question, we have, under 5~ and distance 40, the
number 39.85 in a lat. column, which is the horizontal distance as before.