getting the correct quantity of excavation. When the bridge is on the skew the procedure is the same, the lines AOB, COD being set out at the proper angle with each other.

Setting out Culverts.--Usually only type drawings of culverts are furnished, showing the cross section of the culvert and the plan and elevation of the front and wing walls, and it devolves upon the engineer to calculate the length of the culvert and make a correct plan of it on the basis of the type drawing.

The first thing to do is to take levels in the bed of the stream for which the culvert is intended, and decide upon the invert level at each end. From the invert level we get the level of the top of the outside of the arch (the extrados at the crown), and the difference between this and the formation level of the railway is the height of bank above the culvert. Half the length of the culvert is then equal to half formation width plus the slope as calculated from the height of bank above the culvert, plus any allowance for face walls, &c., according to the type drawing. When the culvert is on the square, a rough sketch may usually be made from these figures sufficient to set it out, but when on the skew it may be necessary to draw out a plan of it. Having got out the plan, the process of setting out is exactly similar to that described for setting out a bridge.

Setting out Bridge Abutments. When the foundations of a bridge have been laid in, the corners of the abutments should be accurately set out with steel tapes. This is done by co-ordinates in the same manner as already described for setting out the foundations. When one abutment has been built or set out the corners of the second abutment should be set out by measuring from one centre line only, the span as measured from the first abutment being used as the other co-ordinate. This will ensure the span being correct.

Other Setting out: Centre Lines and Levels in Cuttings and Banks. During the progress of construction the engineer will continually be called upon to give centre lines and levels in the cuttings and on the banks as the work proceeds. Having all his tangent points transferred, he will usually have no difficulty in doing this, the work simply consisting in setting up the theodolite and ranging in a few pegs on a straight or turning

off a few angles on a curve. The levels required are put in by levelling from the nearest bench mark; see also To Locate a given Level, page 253.

"Boning Rods."-When a cutting is started the foreman is given a note of the depths at each peg, but when the cutting is deep and there are lifts or benches in the face he is unable to measure down these depths after the cutting has gone in a little way. The engineer must therefore give him two pegs, 1 or 2 chains apart, driven in so that their tops are at formation level. The foreman places his boning rods (see page 161) on these pegs, and bones in his levels in front as the work proceeds. Level pegs must be put in as often as necessary to keep them within a chain or two of the face of the excavation.

66 Crossheads."-In the case of embankments the foreman is given a note of the heights at each chain peg, and in this case he puts up what are called "crossheads" to indicate the height of the bank. A "crosshead" is simply a post with a crosspiece about 2 ft. long nailed across the top of it, and at right angles to it. The "crosshead" is erected so that this crosspiece is at the given height above the chain peg plus 3 ft., the height of the boning rod. Having erected two or three crossheads, by using his boning rod, which is 3 ft. long, and sighting on to them, the foreman is enabled to keep his bank to the correct level.

Allowance for Shrinkage and Settlement. In banks tipped with soft material an allowance of about 1 in. per foot of height must be added to the height of the bank to allow for shrinkage and settlement. In high banks this allowance becomes excessive, and a smaller percentage must be added. The allowance to be made of course depends on the nature of the material, and is a matter of experience.

Bench Marks and Checking through Levels. -Before doing any setting out on a railway for construction purposes the levels should be carefully checked through from end to end of the railway, and bench marks established not more than mile apart and at each bridge or other important work.

To Locate a given Level.-In order to put in a peg or mark a point at any given reduced level it is simply necessary to

deduct this reduced level from the height of the instrument, and the difference is the "staff reading." The peg has then to be driven in until this reading is observed, when the staff is held on it. In the case of making a mark at any given reduced level on a wall, &c., the staff is to be placed against the wall and moved up or down until the correct "staff reading" is observed. The mark is then made at the foot of the staff.

CHAPTER VI.

TACHEOMETRY OR STADIA SURVEYING.

Tacheometry.-The term tacheometry is derived from the Greek tacheos (quickly) and metreo (I measure), and therefore signifies the art of rapid measurement. It is, however, now confined to distance and height measurement by telescope.

Principle of the Stadia.-The principle of tacheometry in its simplest form is supposed to have been first used by Mr W. Green in 1778. He used

C

C

B

D

Fig. 178.-Principle of the Stadia.

a simple tube containing three horizontal wires, and on the supposition that rays of light travel. in straight lines from the object observed to the eye, the distance of the object is proportional to the extent of it apparently intercepted between the cross wires. Thus in Fig. 178, let a be the eye, b, d, c the cross wires, BC and B'c' the extent of the observed objects apparently intercepted by the straight lines ab, ac produced. If the tube is held so that the line ad from the eye to the central wire is horizontal, then AD and AD' are the distances of the observed objects from the eye, and BC and B'C' being vertical will be bisected by Ad produced in D and D'. The triangles BDA and B'D'A being similar, we have

and as BC and B'c' are respectively double BD and B'D', also

If now the wires are so arranged that when the distance AD is 100 ft., BC is 1 ft., then if B'c' is 2 ft., the distance AD' will be 200 ft., and so on, or in other words the distances will be 100 times the intercepted amounts. Obviously then by placing a staff at c and c', the distances AD and AD' will be equal to the staff readings with the decimal point removed two places to the right.

The horizontal wires at b, c, d are called stadia wires, and the above is the principle of the stadia. The word stadia is Italian,

and was originally used to mean the staff. another name for tacheometry.

Stadia work is simply

To look at the matter in another way, referring to Fig. 178, we see that the triangles ABC, AB'C' are each similar to the triangle Abc, and we have

Ad
bc

That is, the ratio being made 100 so that ad is 100 be, then

the distances AD and AD' are 100 times the intercepted amounts BC and B'c'.

Tacheometry simply consists in applying this principle to the telescope, by means of which the staff may be accurately read at considerable distances.

Distance and Height Measurement by Simple Vertical Angles.-Before describing the tacheometer proper the following method of distance and height measuring by means of simple vertical angles and staff readings may be described :—

In Fig. 179, let T be the theodolite and ABC the staff held at c. Let the horizontal distance TDH be required. Measure the vertical angles DTA and DTB, and let them be 0 and 4 respectively; also let the distance intercepted on the staff be s, being equal to the difference of the staff readings at A and B, which is equal to b-a in Fig. 179.