The main line should be set off from the tangent point C at every chain.

Although the last two offsets ff may not meet on the curve, it is of no consequence, as the points obtained by the offsets do not give the regular distance of one chain on the curve, which is subsequently done, as will hereafter be shown.

When the curve is longer than the radius the offsets become very long, and subject to great inaccuracy; when that is the case, the following example will be found more practicable.

Note.-A common cross staff-merely a piece of board with two grooves cut at right angles, fixed on a short staff-may be applied with considerable advantage in setting out long offsets.

Problem 25.

Fig. 3, Plate 31. Another method to lay out a curve by offsets from its tangents, the extension of the main line being obstructed by buildings, &c.

The tangents BE and CE were found to be obstructed in their intersections at E by buildings, or otherwise, too great a distance from the curve to set out the offsets within the limits, it was therefore necessary to obviate these difficulties by introducing other tangents, as F G and G H.

The radius of the curve is 40 chains, or 2640 feet, and the angle at the centre 60 degrees.

Having carefully laid down the curve on the plan and drawn the several tangents, the sides or lengths of each were proved by the following trigonometrical calculations. (Case 1.)

Then measure off the tangents B F and C H, and leave flags; range out the line F G H and measure the same, leaving a flag at the centre G; if the whole length measures 1414 feet, or 24

*For convenience, the radius lines are not extended to the centre.

chains 43 links, the position and measure of the tangents are

correct.

The radius being the same as the former example, the offsets also will be the same.

In all cases it is most advisable to set out and prove the position of the tangents before setting out the offsets.

Problem 26.

Fig. 4, Plate 31. To set out a curve by offsets from a chord line. Radius 20 chains.

The obstructions on the convex side of the curve prevent the preceding examples being adopted; therefore, to obviate the difficulty, the two following examples are substituted:

Let B C, Fig. 4, be the curve to be connected to the lines A B and C D. (The tangent lines and offsets shown by dotted lines are merely introduced to show the method of obtaining the offsets from the chord line.)

In this instance, if possible, chain the tangent line out to E equal to 10 chains, and there fix a flag; at right angles to E B A set off 165 feet as at G, or 2.50 chains, and put up a flag.

The method of obtaining the tangent offsets is shown in the first example; the chord offsets are obtained by subtracting each height respectively from the greatest, or 165 feet, thus:

Then at the tangent point B, at right angles to A B, set out the first half chord offset 165 feet to the point F, and proceed on regularly up to G; when the ground will admit of it, the flag at the centre would be desirable; at G set out the angle

F G E, equal to 60 degrees; and again set off 165 feet to E, from which point measure off the half chord to C; if the flag at C cannot be seen from E, set out the angle 90 degrees (see Diagram); if required, the curve may be continued; sometimes half chord lines may be set out with less trouble than the whole chord,

Problem 27.

Fig. 5, Plate 31. The method adopted in this example differs only in setting out the whole chord by an angle either from the tangent point B, equal to 30 degrees, or by the angle CBA, equal to 150 degrees; the offsets are the same as before. The calculations are made from the radius of 20 chains.

The angle between the tangent and chord is equal to half the angle at the centre in all cases.

If the curve has to be continued, plant the instrument at C, and set off the double angle, equal to 120 degrees.

Problem 28.

Fig. 6, Plate 32. To lay out a curve intercepted by a river, and other obstacles preventing the use of the chain, and substituting two theodolites.

This method has many advantages that is not contained in other examples, particularly on hilly or sloping ground, and where a wide stream of water has to be crossed, as in this instance.

The angles E B C and E C B, as before stated, are regulated by the two radii, and the angle at the centre, that being double the angles E B C or E C B. These angles may be divided into any number of equal parts, as Ba, B b, Cd, C c, &c. See the 21st proposition of the third book of Euclid, that all angles contained in the same segment of a circle are equal to one another.

The radius is equal to 20 chains, or 1320 feet.

The angles E B C and ECB are each 30 degrees, which,

for example, is divided into five equal parts, or 6 degrees each.

Plant the theodolites correctly over their respective points B and C; adjust the instruments and set the verniers to zero; then direct both telescopes to E, the point of intersection of the two tangent lines A B and C D; at this point clamp the lower plate, and bring the vernier to the first angle, equal to 6 degrees, whilst the other theodolite will fix the opposite angle at 24 degrees, and so on to the two last points. When the verniers are brought to the respective angles, as at a, b, c, &c., an assistant must be at those points with a pole, waiting the signals from each of the telescopes, shifting the pole until both sights have brought it to the correct point; this operation must be done to each of the points, and a stake driven down.

If it be required to set out each point one chain, or 66 feet apart, the following rule will give the first angle, then the succeeding angles are obtained by multiplying that angle by the number of times, as 1° 26' by 2 is equal to 2° 52', and so on.

Therefore to find the angle so that the points in the curve shall be 66 feet apart (or any other distance), as by the following rule:

Divide the distance by twice the radius, the natural sine of the quotient will be the required angle, or by logarithms, thus:

Fig. 7, Plate 32, is similar in its operation, having only one theodolite, and the angles are calculated in the same manner. The theodolite is planted at B, and set to the first angle; one end of the chain is held fast at B, the other end at 1 is strained out, and moved about, until by signals from the instrument;

the line of vision intersects the pole held at the end of the chain, which will be the first point; the chain is then moved on and held firm at this point, the other end at 2 waiting the signals from the instrument as before. At all these points stakes must be fixed; so proceed to C, at which point place the theodolite if the curve has to be continued.

This method has similar objections to the first example, except that by this the instrument overcomes difficulties which the chain could not.

Problem 30.

Fig. 8, Plate 32. To lay out a compound curve.

This example is introduced to show the method of adopting two curves of different radii, and connect them with the proper tangent points B and C.

The curve B E has a radius of 40 chains, and the angle at the centre 50 degrees; the curve E C has a radius of 80 chains, and the angle at the centre 18 degrees 20 minutes.

As on former examples, the plan should be consulted, and the angles carefully measured, so that the tangent points BE and E C may be minutely defined.

The lengths B a and a C may be correctly obtained by calculation, as shown by Fig. 3; the angle B a O is equal to 64° 30', therefore the angle B a E is equal to 129 degrees, which may be set out by the theodolite, leaving a flag at the point E equal the length of B a.

The offsets are calculated from the tangents as in Fig. 2, and if too long an intermediate tangent should be introduced, as shown in Fig. 3.

The curve EC having a greater radius will require similar calculations, for the tangents Eb and b C, as well as the offsets from them, all of which will come within the limits.

To insure accuracy and prevent extra labour, it is better to fix all the tangents first.

It frequently happens that several curves of different radii are combined, as in the following example.