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75 T

Substituting the above given values, we have sin

D = &-, whence A- C = R sin D, and we have

C=1R sin D. (9O.)

The central angle for the chord B G is 5. The deflection
angle D is, therefore, | = 2 30'. Sin 2 30' = .04362.

Since the deflection angle E B C = 10 for this case, R =
50 -4- sin 10 = 287.94 ft. Hence, chord C = 2 x 287.94 X
.04362 = 25.12 ft.

Accordingly, in measuring the short chords, 25.12 feet
should be used instead of 25 feet.

1251. Tangent Distances. When an intersection
of tangents has been made and the intersection angle meas-
ured, the next question is the degree of curve which is to
unite them, which being decided, the next step in order is
the location of the points on the tangents where the curve


begins and ends. These two points are equally distant from
the point of intersection of the tangents, which is called the
P. I. The point where the curve begins is called the point
of curve, or the P.C. ; the point where the curve terminates
is called the point of tangent, or the P. T. The distance
of the P. C. and P. T, from the P. I. is called the tangent

In Fig. 282, let A B and C D be tangents intersecting at
the point E and forming an angle C E F = 40 00' with each
other. It is decided to unite these tangents by a 10 curve
whose radius is 573.7 feet. Call the angle of intersection /,
the radius B O, R, and the tangent distance B E, T. From
Art. 1247, proposition G, we have B O C = C E F; hence,
the angle B O E = % C E F. From the right triangle E B O

we have tan B O E -TTTY
b O


Substituting the above equivalents we have tan |- 1= - ^-,

whence T R tan \ L (91.)

In our example R = 573. 7 f t. ; \ I = 20 ; tan 20 = . 36397.
573. 7 X . 36397 = 208. 81 ft. Measure back from the point E
on both tangents the distance 208.81 ft. to the points B and
C. Drive plugs flush with the ground at both points and
set accurate center points, marked by tacks, in both. Di-
rectly opposite each of these plugs drive a stake called a
guard stake, because it guards or rather indicates where
the plug is. The stake at B, if the numbering of the stations
runs from B towards C, will be marked P. C., and the stake
at C marked P. T.

1252. To Lay Out a Curve With a Transit.

Having set the tangent points B and C, Fig. 282, set up
the transit at B, the P. C. Set the vernier at zero and
sight to E, the intersection point. Suppose B to be an even
or "full station," say 18, and that it has been decided to
set stakes at each hundred feet. Let the central angle
BOG, measured by the 100-feet chord B G, be 10 ; then,
the deflection angle E B G, whose vertex B is in the circum-


ference and subtended by the same chord B G, will be %
B O G or 5. Turn an angle of 5 from B, which in this
case will be to the right; measure a full chain, 100 feet,
from B and line in the flag at G ; drive a stake at G, which
will be marked 19. Turn off an additional 5 making 10
from zero, and at the end of another chain, at //, set a stake
marked 20. Continue turning deflections of 5 until 20 or
one-half of the intersection angle is reached. This last
deflection, if the work has been correctly done, will bring
the head chainman to the point of tangent C. It is but
rarely that the P. C. comes at a full station. When the
P. C. comes between full stations it is called a sub-
station, and the chord between it and the next full
station is called a sub-chord. Had the P. C. of the curve
come at the sub-station, say 17 + 32, the deflection for the
sub-chord of 100 32 or 68 feet, the distance to the next
station, is found as follows: The deflection for a full station,
i. e., 100 feet, is 5 = 300', and the deflection for 1 foot is


= 3', and for G8 feet the deflection will be 68 X 3 = 204' =

3 24', which is turned off from zero and a stake set on
line, 68 feet from the transit, at Station 18. The length of
a curve uniting two given tangents whose intersection angle
is determined, is found as follows:

Suppose / = 32 40', and that the tangents are to be united
by a 6 curve; 32 40' reduced to the decimal form is 32.666;
as each central angle of 6 will subtend a 100-foot chord, or
one chain, there will be as many such chords or chains as 6
is contained times in 32.666, which is 5.444, that is, there
will be 5.444 chains in the curve, or 544.4 feet, which is the
required length of the curve. The P. C. and P. T. having
been set and the station of the P. C. determined by actual
measurement, say 58 -4- 71, the station of the P. T. is foun$
by adding to 58 -4- 71, the station of the P. C., the calculated
length of the curve, 544.4 feet. 58 + 71 + 544.4 = 64 + 15.4,
the station of the P. T.

Another method of calculation is the following: The sum
of all the deflection angles is equal to one-half the intersection


angle. The intersection angle being 32 40', one-half
equals 16 20', which, reduced to minutes, equals 980'. The

deflection for 100 feet is - = 3 = 180', and the deflection

1 RO
for 1 foot is- = 1.8'; then, 980', the total deflection, di-


vided by 1.8', gives 544.4 feet, the required length of the


In the following examples, let /= angle of intersection, T= tan-
gent, and L = length of curve.

1. /. 16 13', degree of curve = 3, required, Tand L.

\T= 272.13 ft.
1S ' ( L = 540.55 ft.

2. /= 59 20', degree of curve = 8 30', required, 2" and L.

A ( T = 384.32 ft.
IS - 1 L = 698.04 ft.

3. /= 21 35', degree of curve = 4 15', required, T and L.

Ans J^=257.03ft.
1S ' 1 L = 507.84 ft.

4. The degree of a curve is 5 30' ; what is the deflection angle for
a chord of 16.2 feet ? Ans. 26.7'.

5. The degree of a curve is 7 15' ; what is the deflection angle for a
chord of 38.4 feet ? Ans. 1 23'.

1253. Obstructions in the Line of Curve. Fre-
quently it happens that the entire curve can not be run in
from the P. C. on account of obstructions. This is especi-
ally the case in either hilly or wooded country, and the
transit has to "move up" to an intermediate point. For
example, in Fig. 282, we will suppose that Station H, 200
feet from B, is the last point which can be set from the
P. C. at B. A plug is driven at H flush with the ground
and carefully centered, and a tack driven at the point. The
deflection angle E B H is 10 to the right. The transit is
set up at //, an angle of 10 to the left is laid off from zero,
and the vernier clamped. The instrument is then sighted
to a flag at B, the spindle clamped, and a close sight to the
flag taken, the lower tangent screw being used to adjust the
sight. The vernier clamp is then loosened and the vernier



set at zero. The line of sight will then be on a tangent to
the curve at //, and the deflection angles to K and C can be
laid off as before and the stations K and C located.

1 254. Tangent and Chord Deflections. Let A B

in Fig. 283 be a tangent, and B C E H a curve commencing
at B. Produce the tangent A to the point D. The line
C D is a tangent deflection, and is the perpendicular
distance from the tangent to the curve. If the chord B C
be produced to the point G, making C G = B C = C E, the
distance G E is a chord deflection and is double the
tangent deflection D C.

1255. Given the radius B O = R, Fig. 283, to find the
chord deflection E G and the tangent deflection C D = F E,

FIG. 283.

The triangles O C E and C E G are similar, since both
are isosceles, and the angle G C E = angle C O E. Hence,


we have O C : C E\:C E \ E G. Denoting the chord C E
by c and the chord deflection E G by d, we have, from the
above proportion, R : c:\c : d. Therefore,

</=-. (92.)

To find the tangent deflection, draw C F to the middle
point of E G. By Art. 1254, F E = D C= the tangent
deflection. Hence, tangent deflection = one-half the chord

deflection, from which

tangent deflection = -^. (93.)

1256. Practical Method of Determining Tan-
gent and Chord Deflections. Let it be remembered for
a basis of calculation that the chord deflection for a one-
degree curve, the chord being 100 feet in length, is 1.745
feet; for a 2 curve, double the deflection for a 1 curve, or
3.49 feet, and so on. The tangent deflection being one-half
the chord deflection, for a 1 curve it will be .873 foot, for
a 2 curve it will be 1.745 feet, etc.

Distances measured either on chords or tangents are
expressed in decimal parts of a station, which is 100 feet, and

is assumed as 1. Thus, the tangent deflection for 75 feet
will be expressed as the tangent deflection for .75 of a
station. This expression is, however, confined entirely to


the calculation, and is spoken of as the tangent deflection for
75 feet. Fig. 284 will be used to demonstrate the principle
upon which tangent deflections are based.

Let A B be a tangent, and B the P. C. of a 2 curve to
the right. We determine the chord deflection for 100 feet
chord of a 2 curve to be 3.49 feet. The tangent deflection
is one-half the chord deflection, or 1.745 feet.

Let BC = 100 feet, a full station (which express as 1),
then C L, the tangent deflection at C, will = 1.745 feet, for,
since this is a 2 curve, the chord deflection = 1.745 X 2, and

1 745 X 2
the tangent deflection = = 1.745 ft.

To find the tangent deflection for any intermediate point
G, 75 ft. from B, express the distance as a decimal of the
full station, or, in this case, .75. Square the decimal thus
formed, and multiply by the tangent deflection, in this case,
1.745; the result will be the tangent deflection for the point
considered. Thus, the tangent deflection for the point is
the line G K, and the length of G K= .75' X 1.745 =.5G2 X
1.745 = .981 ft.

For the point D, 125 ft. from B, the tangent deflection is
D M, and the length of D M is found as above. Thus, to
express 125 as a decimal of a full station, divide 125 by
100, obtaining 1.25. Then 1.25" X 1.745 = 1.5G2 X 1.745 =
2.725 ft.

In the above, we have assumed that the chord and the
corresponding tangents were of equal length; i. e., that
B 1= B F, B K = B G, etc. This is not strictly true, but
is near enough for all practical purposes, particularly when
the degree of the curve is small.

1257. Laying Out Curves Without a Transit.

During construction, the engineer is often called upon to
restore center stakes on a curve when the transit is not at
hand. With the aid of a tape and a few stakes for lining in,
a line can be run closely approximating the true one, by
applying the principle demonstrated in Art. 1256.

A practical application of this principle is shown in



Fig. 285, in which A B is a .tangent, B the P. C. of a 4
curve R*. The chord deflection of a 4 curve for 100 feet
chord is G.98 ft. The tangent deflection = \ the chord
deflection, is 3.49 ft. Let B = Sta. 8 -f 25, a stake at each
full station on the curve being required. Restore the
stakes at A and B, which will determine the P. C., and give
the direction of the tangent A B. The distance from the
P. C. to the next full station C is 75 feet, or .75 of a full
station; .75 2 X 3.49 = .562 X 3.49 = 1.96 ft., the tangent
deflection at C. The engineer being without a transit, the
point C is found by measuring 75 feet from B and setting a
stake at C in line with a stake at B, the P. C., and a point


FIG. 285.

on the tangent A B as A. With a tape, measure the dis-
tance 1.96 ft. from C at right angles to B C, and drive a
stake at that point F, which will be Station 9. Measure
100 feet from F and set a point at D in the line B F. By
previous calculation, we know the chord deflection for
100 feet is 6.98ft. Measure the distance 6.98 ft. at right
angles to the line FD and drive a stake at G, which will
be Station 10. In like manner set the remaining Station 11,
which is previously known to be the P. T. Although the
chord deflection D G is not theoretically at right angles to
FD, yet D G is so small compared with FD that for curves
of ordinary degree the offset is made at right angles.



1. The degree of curve is 5, the chord 67 ft. ; what are the tangent
and chord deflections ? . f Tan def. = 1.959 ft.

' \ Chord def. = 3.918 ft.

2. The degree of curve is 7 30', the chord 23.5 ft. ; what are the
tangent and chord deflections ? A -I Tan def ' = -359 ft '

'} Chord def. = .718 ft.

3. The degree of curve is 6 15', the chord 117 ft. ; what are the
tangent and chord deflections ? . ( Tan def. = 7.465ft.

L S ' \ Chord def. = 14.930 ft.

1258. To Determine Degree of Curve by Meas-
uring a Middle Ordinate. In track work, it is often
necessary to know the degree of a curve when no transit is
available for measuring it. The degree can be found by
measuring the middle ordinate of any convenient chord, and
multiplying its length by 8, which will give the chord deflec-
tion for that curve.

Let A B, in Fig. 286, be a 50-foot chord, measured on the
track, and let the middle ordinate a b be .44 ft. .44 X 8 =
3. 52 = chord deflection for ^ .44

50', which, expressed

decimal part of a full sta- 50'

tion, is. 5; .5 2 = .25. The FIG. 286.

chord deflection for 100 feet multiplied by .25 = the chord
deflection for 50 feet, which we know by calculation to be
3.52 feet. Hence, 3.52 -j- .25 = 14.08 ft., the chord deflec-
tion for 100 feet, which, divided by 1.745, the chord deflec-
tion for a 1 curve, gives a quotient of 8.07, nearly. The
inference is that the curve is 8.


1. Length of chord is 50 ft., middle ordinate .35 ft. ; required, de-
gree of curve. Ans. 6 25.08'.

(The original curve probably 6 30'.) ^

2. Length of chord 40 ft., middle ordinate .21 ft. ; required, degree
of curve. Ans. 6 1.02'.

(The original curve probably 6.)

3. Length of chord 25 ft., middle ordinate .22 ft. ; required, degree
of curve. Ans. 16 8.28'.

(The original curve probably 16.)



1259. Field Books. To facilitate the field work of
the engineer, field books have been published. They are
portable, being carried in the pocket, and contain, in con-
densed form, general directions for the conduct of field work,
together with all the necessary data in the form of tables,
for prosecuting such work with accuracy and dispatch.

One of the first published in America is the work of John
B. Henck, to whom most American engineers are under

1260. Note Books. Various styles of note books are
published, the pages being ruled to suit the particular kind
of work being done. They are of three classes, viz., transit,
level, and topography books. The latter are ruled in squares,
which may be given any desired scale and greatly facilitate
the accurate platting of topography in the field.

1261. How to Keep Transit Notes.

A good form for location notes is the following:


Deflection] Tot. Angle: Mag Staring

Dtd. Bearing.

la** *"**"







tf. strife.






%, -r$f*

S '.'.<>














4Curve B*



T.- 188.91 ft.

Def. Angle for OOft-1 Off




Def. Angle for I ftfl.S'








y. wits' g.



In the first column the station numbers are recorded.
In the second column are recorded the deflections with the
abbreviations P. C. and P. T., together with the degree of
curve and the abbreviation R l or Z ( , according as the line
curves to the right or left. At each transit point on the


curve, the total or central angle from the P. C. to that
point is calculated and recorded in the third column. This
total angle is double the deflection angle between the P. C.
and the transit point. In the above notes, there is but one
intermediate transit point between the P. C. and the P. T.
The deflection from the P. C. at Sta. 3 + 20 to the inter-
mediate transit point at Sta. 4 + 50 is 2 30'. The total
angle is double this deflection, or 5 12', which is recorded
on the same line in the third column. The record of total
angles at once indicates the stations at which transit points
are placed. The total angle at the P. T. will be the same
as the angle of intersection., if the work is correct. When
the curve is finished, the transit is set up at the P. T.,
and the bearing of the forward tangent taken, which affords
an additional check upon the previous calculations. The
magnetic bearing is recorded in the fourth column, and the
deduced or calculated bearing is recorded in the fifth

1 262. Preservation of Notes and Records. Notes
should never be erased. If, on account of error or change of
plan, they should cease to be of any value, they are crossed
out, i. e., two diagonals are drawn across the page. All
notes of permanent location should be copied each day into
a separate book for office reference, to prevent confusion,
and for record in case the original notes should be lost.


1263. A Level Surface. A level surface is one
parallel to the surface of standing water. A water surface,
though not theoretically level, owing to the curvature of
the earth's surface, is assumed to be level and perpendicular
to a vertical line, or the line of gravity.

The height of a point is its distance above a given level
surface, measured on a vertical line, and is called its
elevation. The process by which the elevation of a point
is determined is called leveling.


1264. The Three Processes of Determining

They are : 1st. By direct leveling.

2d. By indirect leveling ; and
3d. By barometric leveling.

1265. Direct Leveling. In the process of direct
leveling, a level line either actual or visual is prolonged so
as to pass directly over or under the given point whose eleva-
tion is required. The elevation of any other point being
determined in the same way, the difference in the elevations
of the two points is found by subtracting the elevation of
the lower from the elevation of the higher.

1266. Indirect Leveling. In the process of indirect
leveling, elevations are determined by means of lines and

1267. Barometric Leveling. In barometric level-
ing the elevation of a point is determined by the weight of
the atmosphere at that point as registered by a barometer.
The second and third processes will be explained later.


1 268. General Principles. Direct leveling depends
upon three principles, two of which have already been
stated, viz. : First, that the surface of a liquid in repose is
level; second, that a vertical line is perpendicular to that
surface, and, third, that a bubble of air confined in a vessel
otherwise filled with liquid will rise to the highest point of
that liquid. A common application of the third principle is
seen in the spirit level used by carpenters and the level
board used by masons.

1269. The " Y " Level. There are a great variety of
instruments for determining elevations. The one in most
general use is the " Y " level, shown in Fig. 287.

This instrument consists of an erecting telescope A B,
i. e., one which shows the image of the object to which the
telescope is directed in its erect or natural position, resting in
Y-shaped supports C and D, from which it takes its name.



The line of sight, or collimation, is identical to that in the
transit explained in Art. 1225, and is parallel to the level
E F. The tube containing the eyepiece G has an exterior
ring 77, which is milled to assist the hand in turning the
tube. This movement adjusts the eyepiece to the cross-
hairs. The object glass at B is moved in or out by the
milled headed screw A'; L and J/are parallel plates; the bar
O P supports the Y's and revolves on a spindle which is

clamped by the screw N. By means of the tangent screw X,
the telescope can be slowly turned horizontally. The tele-
scope is leveled by means of the leveling screws F, <2, A, and 5.
The level is supported by the tripod T. The cross-hairs are
of either platinum wire or spider threads, and are fastened
to a ring which is held in place by capstan screws shown at
U, and their movements are regulated in the same way as
the movements of the cross-hairs of the transit explained in
Art. 1225.


1270. The Bubble Tube. The bubble tube is of
glass bent upwards and so nearly filled with alcohol that only
a bubble of air remains, which is always at the highest point
in the tube. This tube is protected by a brass case, which
is fastened to the underside of the telescope, and provided
with the means for adjustment. The one end may be raised
or lowered and the other end moved horizontally. Through
a slit in the upper side of the case, the bubble tube is seen.
Directly over it is a scale graduated in both directions from
zero, which is over the center of the tube.

The Y's C and D support the telescope, which is held in
place by hinged clasps, or clips, as they are called, fastened
by carefully turned pins, by means of which the tele-
scope can be firmly held in any desired position. The Y's
rest upon the bar O P, to which they are fastened by lock-
nuts, the one above, the other below, the bar, for raising or
lowering. The bar revolves upon a finely turned steel spindle,
resting in a socket of bell metal. The parallel plates L and M
are united by a ball-and-socket joint, and held in place by the
leveling screws F, Q, R, and vS.

1271. Adjustments. The first thing to do in prep-
aration for actual leveling is to make the adjustments of
the instrument.

There are three adjustments, as follows:

1. To make the line of collimation parallel to the bottoms
of the collars, or rings, on which the telescope rests.

2. To make the plane of the level parallel to the line of
collimation, or to the bottom of the collars.

3. To cause the bubble to remain in the center of the tube
while the telescope is being revolved horizontally.

1 272. First Adjustment. To make the line of colli-
mation parallel with the bottoms of the collars.

Plant the tripod firmly; choose some distant and clearly
defined point, the more distant the better so long as the
sight is distinct. Remove the pins from the clips and clamp
the spindle, bringing the intersection of the cross-hairs to


exactly bear on the point by means of the tangent screw.
Revolve or turn the telescope on its supports through one-
half a revolution, i. e., until it is bottom side up. If the
intersection of the cross-hairs is still on the point of sight, it
proves that the line of collimation is parallel to the bottoms
of the collars. If, however, the line of sight is no longer on
the point, move the cross-hairs by means of the capstan
headed screws over one-half the space which measures the
apparent error, being careful to move them in the opposite
direction to that in which it would appear they should be
moved. The apparent error is double the real error, and is
explained in Fig. 288.

Let the instrument stand at A and sight to the point Z?,
and suppose that when the telescope has been revolved half
way around, the point B appears to be at C, then will the

distance B C be double the real error, and the true line of
sight will be at D, half way between B and C. Sometimes
both cross-hairs are out of adjustment and they must be
moved alternately until the intersection of the cross-hairs,
i. e., the line of collimation, will pass through the same
point throughout a complete revolution of the telescope.

1 273. Second Adjustment. The second adjustment
is to make the plane of the level parallel to the line of colli-
mation, or to the bottoms of the collars, and is made as
follows :

Remove the pins and open the clips; place the telescope
over a pair of leveling screws and clamp the spindle. Bring
the bubble to the middle of the tube by means of the le\4el-
ing screws, and revolve the telescope through an eighth of a
revolution. The bubble tube will stand out at an angle with
the Y's. If the bubble runs it shows that a vertical plane
passed through the longitudinal axis of the bubble tube is
not parallel to a vertical plane passed through the line of


collimation. To correct the error, bring the bubble nearly
back by means of the check nuts which regulate the lat-

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