plan from a scale of 30 yards to an inch to one of 40 yards to an inch, and it were decided to make the squares on the original 1 inch square, the squares for the new plan should be drawn threequarters of an inch square. Or, to put it in another way, one side of an inch square on the original would represent 30 yards, and to make one on a 40-yard scale to represent a like amount, it would necessarily be three-fourths of an inch. filled in so as to correspond with the original, measurements being made from the corners of the squares to the points where the lines of the plan cross the sides of the square, and to all

Each square is then

bends.

one.

2

3

4 5 6 7 8 9
Scale of Chains

Of course one scale is used for taking the measurements from the original plan, and another for plotting on the new Proportional compasses save much labour in such work, as they can be set to any two scales, so as to give the correct proportion for every measurement. After having been adjusted for the two scales required, if the compasses are opened to a definite width at one end to get off the length of a line, the opening at the other end represents the correct length of the line to the other scale. It is found convenient to number the intersecting lines forming the squares correspondingly on each plan, thus enabling any particular point to be found more quickly.

To set out a Right Angle with the Chain.-Along the line on which it is required to erect the perpendicular, measure 40 links; for the perpendicular take 30 links on the

chain, hold one end at the main line at the beginning of the 40 measurement, and with the chain stretched taut draw an arc of a circle with the radius of 30 links. The arc can be marked on the ground with a piece of chalk, or an arrow held at the end of the chain. Now take 50 links on the chain, hold one end at the further limit of the 40 measurement, and find the point where the 50 measurement will just cut the arc drawn on the ground. A line drawn from this point to the beginning of the 40 measurement is at right angles to the main line.

Any multiples of 3, 4, and 5 may be used to form a rightangled triangle. Thus the lengths 9, 12, 15, or 12, 16, 20 may be employed. That these lengths will correctly form a rightangled triangle can be proved by the proposition that the hypothenuse squared equals the squares on the base and perpendicular. Thus

Setting out Railway Curves.-Curves are usually set out in radii of chains, though, of course, any radius may be used. If the curve is not a very sharp one, that is, if it is of large radius, the simplest method of ranging it out upon the

H

FIG. 123.

ground is to produce the lines representing the straight portions of the railway to meet in a point, and to range out offsets at definite distances along each of these lines. For example, let AB and JK (Fig. 123) be the straight portions of the railway which it is proposed to connect by a curve. Produce AB and

JK to meet in I; along BI measure off a number of equal parts BC, CE, EG, and in a similar manner measure off a number of equal parts along JI. Then range out offsets at right angles to AI from the points C, E, and G, and similar offsets at right angles to JI from the points of division. The length of each offset may be ascertained by plotting the whole on paper, and by applying the scale to each of the lines CD, EF, GH, etc. For sharp curves a better method is to divide the curve into an equal number of arcs, and to range out the offsets from the

D

E

F

H

FIG. 124.

chords of the arcs produced. For example, AB and KL (Fig. 124) represent the straight portions of a railway which are to be connected by a curve of a definite radius. Divide the curve into an equal number of arcs BD, DF, etc., and draw a chord to each arc. Produce AB to C, and make BC

equal to the chord BD, and join CD; produce BD to E, make DE=BD or DF, and join EF, and proceed with the remainder in the like manner. The length of the offsets CD, EF, etc., can be found by measuring on the plan, or can be calculated as follows:

Let R represent the radius of the curve.

The first and final offsets are exactly half of the intermediate offsets, all of which are equal. To state the formula generally—

Ex. What lengths of offsets are required to set out a curve with 10 chains (220 yards) radius with offsets every ten yards.

It will be noticed that the offsets are not right-angle offsets,

but that they form the base of an isosceles triangle.

CHAPTER XVI

LEVELLING

LEVELLING is the finding of a line parallel to the horizon at one or more stations, in order to determine the height or depth of one place in relation to another. There are two methods of performing this, viz. the trigonometrical and the geometrical.

The Trigonometrical Method. By this method either the length of the line joining the two points of observation, or the horizontal distance between them is measured, and the angle between the two lines is taken with an instrument. The difference in height of the two objects can then be found by trigonometry, or by plotting the angle and measurement. For levelling by this method a clinometer, dial, theodolite, or, in fact, any instrument having a vertical graduated arc may be used. The general principles of the trigonometrical method have been described in Chapter VI on Inaccessible Heights and Distances.

The Clinometer.-Where great accuracy is not required, observations may be made with a clinometer, as this is a very portable instrument, and is easy to manipulate.

In its simplest form (Fig. 125) it consists of a square piece of wood, to which is attached a graduated arc, which has for its centre a corner of the square. From this corner is suspended a light plummet, which being free to swing, will always assume a vertical position. If the lower edge of the square be placed so as to correspond with the inclination of the object, the degrees of dip will be shown by the cord of the plummet crossing the graduations of the arc, thus the reading in the illustration gives an inclination of 28 degrees.