These cross-sectional lines should be set off at right angles with the centre line by means of the optical square. Fig. 111 shows a plan of the longitudinal section with cross-sections at the required intervals. The longitudinal section is taken along the line in the ordinary way already shown, and cross-sections are taken at the points C, D, and E, the distances at which the levels were taken being shown.

The levels of the cross-section may be taken separately or in conjunction with the longitudinal section.

The following (Fig. 112) is a Form of level book for cross-section levelling :

The level of the ground on the section line at 0.50 is 52.04 at 30 to the left of 0.50 53.02

Each cross-section should be plotted separately. Instead of taking the levels of the cross-section with the level, the angle of slope may be taken with the clinometer as explained before, and the angles plotted on the cross-sections by a protractor. The above form of level book is very useful for taking what are called spot levels-that is, levels at numerous different spots. In this

case a central line is pegged out at convenient intervals and at each of these intervals cross-sections at right angles are also pegged out as required. Levelling is then proceeded with in the ordinary way by reading as many of the pegs as possible with one setting up of the instrument, then moving the instrument and proceeding as before. The position of each peg is marked on the plan with its letter or number, and each peg is marked to correspond.

Fig. 113 shows a section and cross-section plotted as it should be shown.

or

Gradients. A gradient is the ratio of the perpendicular to the horizontal distance. It is generally expressed as a fraction, 1 in 20; meaning that the difference of level in a horizontal distance of 20 feet is 1 foot.

To find the rate of inclination of a gradient.-Find the difference in height above the datum at the beginning and end of the gradient, and divide this into the horizontal length of the gradient.

Example (Fig. 113).

The distance from A to B is 42 chains. 103 and at B 166: difference

horizontal length of gradient.

The H.A.B. at A is

= 63 feet. 42 chains = 2772 feet =

An inclination of 1 in 20 is the steepest for wheeled vehicles drawn by horses that is practicable for any distance. For short distances such as a quarter of a mile 1 in 16 is permissible, and even 1 in 8 in exceptional places for (say) 50 feet.

By the angle of slope is meant the number of degrees formed by the slope and the horizontal.

The gradient may be calculated from the angle of slope, and the angle of slope from the gradient. (See chapter on Trigonometry.)

Computation of Earth-work.-When the gradient of a road or railway has been decided upon, and the formation level' (shown by the thick black line in Fig. 113) has been drawn on the section in such a way that the amount of earth to be removed in the cuttings as nearly as possible equals the amount required for the embankments, the next step is to calculate the cubic contents of the cuttings and embankments. It is of great importance that the required embankments should never exceed in cubic contents the quantity of the cuttings for obvious reasons; rather should it be slightly the other way.

To obtain the cubic contents.-First divide the quantities of the cuttings and embankments, between the formation level and the ground level on the section, into triangles and trapezoids, as dictated by the undulations of the surface line. Next it is necessary to consider the slopes' or ' batters' of the cuttings and embankments1 to 1 for the former (i.e. 13 horizontal to 1 perpendicular) and 2 to 1 for the latter may be considered as an average. Suppose the cubic contents of a triangular portion of cutting, such as ABC in Fig. 114, are required to be worked out. Let the slope of the cutting in this instance be 1 to 1, the height 20 feet, the length 500 feet, and the width 30 feet. The Prismoidal Formula (p. 4, No. 5) must be used.

Boning Rods. These are pieces of wood in the shape of T squares, of exactly the same length, by means of which foremen

of works are enabled to work to levels which have been taken by the surveyor. In Fig. 115 let A, B, C be three pegs which have been put in at a certain level. If the foreman places three boning rods upon these pegs and sights along the top of them, held transversely to a boning rod at D, when the four boning rods are seen exactly level at the top all four spots are on the same level. Or, again,

if the pegs A, B, C are on a certain gradient, and D be sighted as before, then the spot D will also be on this gradient.

The boning rods are usually 3 feet long, and the cross-piece 15 inches. It is a good plan to have the first B.R. as shown in Fig. 116, with the top edge painted white, and the last as shown in Fig. 117 with the two ends painted black. When setting out curved lines, the cross-piece may be made longer. In setting out

Painted
white

Painted
black

TT

FIG. 116.

FIG. 117.

Painted
black

the gradient of drains sight rails' are fixed at each end of the inclination, as shown in Fig. 118. They consist of two uprights driven into the ground on either side of the manhole, with a crosspiece nailed to the uprights at a given level above the invert. The invert is the bottom of the channel pipe in the manhole. To the ordinary boning rod another piece of wood is attached so that it is made the same length as the sight-rail above the invert. There,