CHAPTER XVI.

SETTING-OUT.

Ranging Straight Lines.-Setting-out, or the location of predetermined points, is defined as that branch of geodetic operations which is the converse of surveying and levelling, the latter consisting in discovering the position of a series of actuallyexisting points.

In ranging and setting-out a base-line for a surface-survey, ranging rods, 5 to 7 feet in length, are used. They are usually circular in section, and painted in lengths of 1 foot or 1 link, black, white, and red alternately. When one colour cannot be clearly seen, one of the other coloured portions can generally be distinguished.

The rods are planted vertically in the ground, the verticality being judged by the eye. When great accuracy is required, a plumb-bob must be used. Its string is turned over the first and second fingers of the hand, so that when it hangs vertically, the rod may be placed parallel to it. The distance apart of the rods varies from 66 feet to 300 feet.

For ranging straight lines of moderate length, the most convenient instrument is the transit-theodolite, because the telescope may be turned completely over about its horizontal axis, so as to range one straight line in two opposite directions from one station. The error with this instrument should not exceed 10 seconds in angular direction—that is, about 3 inches in a distance of a mile.

For straight lines of very great length, the theodolite is not sufficiently exact. It is then advisable to use a transit-instrument. In order that a vertical circle may be correctly described by that instrument, it is necessary that the line of collimation shall be precisely at right angles to the horizontal axis about which it revolves, and that the pivots of that axis shall be pre

cisely level with each other when they rest in their Y's on the iron stand.

Plotting the Underground Traverse on the Surface.—If it is required to plot the traverse on the surface of the earth, a process which in former times was in general use for determining the position of the boundary of the mine, the first course, from which azimuths were measured, must first be laid down in horizontal length and direction, and its ends marked with stakes. The position of the first station being thus determined, the second station may be found by laying off from the first station, at the proper angle, a horizontal distance equal to the length of the course. In the same way, all the successive underground stations may be marked out on the surface.

This process is tedious, and liable to error, and should consequently only be employed when absolutely necessary. Instead of repeating the traverse on the surface, the required distance and its bearing should be calculated trigonometrically, and marked out on the surface.

Setting-Out Railways to Mines.-Railways to mines may easily be set-out when the ground presents no great irregularities, the best line for the railway being determined by levelling from the starting-point to the mine. The line, of which a trial section shows the fewest difficulties of construction, having been selected, it is roughly marked out on the ground by strong pegs. entire line is then carefully levelled and an accurate section drawn. From this the amount of cutting and embankment necessary may be determined. The entire line is then set out on the ground.

The

Two stakes are driven into the ground with their heads at the intended formation level, at distances of about 50 feet apart, near the commencement of a proposed cutting. The excavators are then able to carry on the cutting at the proper rate of inclination by a process called boning. This consists in ranging a line, of uniform inclination, from two points in it with T-shaped instruments, called boning rods. Boning rods of the same height are held vertically upon the two stakes driven into the ground, and a third rod is held at some point along the intended slope; then, if the inclination is correct, the tops of the three rods will be in line. If the third rod is too low or too high, it must be raised or lowered until it is in line with the tops of the other rods.

Ranging Curves.-Railway curves are of frequent occurrence, and even branch railways to mines, which are usually not of great length, can rarely be made without them.

A common method of setting-out a railway curve of a given radius

on the ground is by means of offsets. In Fig. 78, A and G are

D

B

E

Fig 78.

F

G

the ends of the straight portions of the line to be connected by a curve, being the two points at which the curve falls into the straight lines. Let A C, CE, E G be the distances which it is desired that the points found in the curve shall

be apart. Then measure, upon the

straight line AC produced to D, the distance C D equal to C E, and join D E. This distance D E is called the offset, and gives a point E in the curve. Range a straight line through the points C E, and upon it lay off the distance EF equal to EG, and join F G. The point G will be the next point in the curve. Proceed in the same way until the whole extent of the curve has been set out. Let r be the radius of the curve, and d the distance A C, C E, or EG, which it is desired that the points found in the curve shall be apart, then the value of the offset is

If CE and E G are two equal chords, the offset is

A B being a tangent to the curve at A, the value of the offset from the tangent is

The values of C D and E F will be found from the following equations :

662

d = 1 chain or 66 feet, the length of the first offset is

2 × 990

= 2.2 feet.

The distance to be laid off upon the line A C

produced to give the place for this offset is

65.963 feet. Again the length of the second offset is

662 2.22 =

990

66 x 65.963

=

4.397 feet, and the distance to be laid off upon the chord produced to give the place for this offset is

66 × (990 2.2)

990

A rough method of setting out curves is to extend a line from the tangential portion of the railway and measure an offset at the end of each chain. The length of the offset is found from the formula

Length in inches =

792

radius of curve in chains'

in which 792 represents the number of inches in a chain.

For example.-If the curve has the radius of 40 chains, the length, in inches, of the offset at the end of the first chain is 792

40

=

19.8.

A more rapid and more accurate method of setting-out circular curves is by means of angles at the circumference, a method first described by Rankine in 1843. It is based on the theorem (Euclid, III., 20) that the angle subtended by any arc of a circle at the centre of the circle is double

the angle subtended by the same arc at any point in the circumference of the circle. In Fig. 79, A B is an arc of a circle, C is a point in its circumference lying beyond the arc. The angle ACB is half the angle subtended by A B at the centre of the circle. When the point at which the angle is measured lies upon the arc, as at E, it is the angle BEF = AEG

Fig. 79.

that is equal to half the angle at the centre of the circle. When the point at which the angle is measured is one of the ends of the arc, as at A, it is the angle D A B that is equal to half the angle at the centre of the circle, expressed by a formula, the angle at the circumference in minutes = ACB FEB =

=

=

in which formula, the coefficient is the value in minutes of one half of the arc that is equal to the radius.

The English practice of designating curves by their radii in chains has but few advantages, as there are no acreage calculations involved. It is preferable to express the radii in hundreds of feet, or chains of 100 feet. This method is now coming into general use. The American practice is to designate curves by the number of degrees in the angle subtended at the centre by an arc 100 feet in length. Thus, curves are named one-degree curves, two-degree curves, &c., when the central angle subtended by 100 feet is one degree, two degrees, &c. this method, which is one of great convenience, confusion is created by terming the centre angle "the angle of deflection.' The value of this angle in degrees is 5729-6 divided by the radius in feet.

In

In setting out curves by means of angles at the circumference, a 6-inch transit-theodolite is set and adjusted at a tangent point, as A, and directed along the tangent to D. An angle equal to half the degree of curvature is deflected from A D towards the side on which the curve is to run. Of the two chainmen, the follower holds his end of the chain at A, and the leader, keeping the chain stretched, is directed by the observer at the instrument into line with the axis of the telescope. In this way, the position of the point E on the curve is fixed. From the line A E, the same angle is set off, the instrument remaining at the tangent point. The chainmen move forward; the follower stopping at E, and the leader moving the stretched chain around that point as centre until the other end comes into line with the axis of the telescope. A second point B on the curve is thus obtained. By continuing the process of setting off angles equal to half the degree of curvature, and causing them to subtend distances of 100 feet each, the entire curve is set-out. It is necessary that the angle formed by producing the straight portions of the line, should be known, in order to find the place on the ground from which the curve is to start. As a rule, this may be taken at once with the theodolite. If an obstacle x intervenes, a pointy is selected on one tangent, from which the distance y z to the other tangent may be measured. The lengths a y and xz are then found by solving the triangle xyz.

When obstacles prevent all the points from being set-out from one tangent point, the theodolite must be moved to the last point set-out, having the last angle clamped on its upper plate.