be used the vane should be set to this height; then with the aid of the 50-feet chain, points can be marked or pegged on the ground which will follow the required gradient. This pegging should be connected by cords by a laboring party immediately following, and used as a centre line for a footpath 3 feet wide to be made at once for the convenience of men and animals, as well as for enabling the engineer to go over the whole of this work with a levelling instrument before the bridges and other works are commenced.

On hill-roads the calculation and setting-out the sidewidths is roughly done on the principles already explained, page 204. The final breadth of formation of a hill-road is generally 12, 18, 21, or 24 feet, exclusive of the drain on the inner side in side cutting.

As to gradients, I in 20 is the steepest ascent for wheeled vehicles that can be continued for many miles without causing animals extreme fatigue, it is also the steepest declivity for a horse to canter down for a long distance; a gradient of 1 in 16 should never be continued in exceptional places for more than a quarter of a mile, nor one of 1 in 8 for more than 50 feet; this being the limiting maximum for wheeled vehicles on roads in good order.

For mountain-passes or paths to allow of the passage of beasts of burden only, the gradient generally adopted is I in 10, or I in 8, and the maximum continuous gradient admissible is 1 in 4, although I in 3 is sometimes used for a very short length.

In work of this class extended vertical cliffs often present serious impediment to carrying on the path; if they also happen to be so hard that blasting them away to the required width becomes too expensive, the only

alternative is to adopt a gallery supported on iron cantilevers sunk in the cliff. The setting-out in such work is by pegs at every 25 feet in the mode already described for hill-roads; in the general management much is left to the discretion and skill of the setter-out, who should be conversant with the subject, as well as capable of setting-out pegs.

Section 2.

DETERMINING A CURVE.

Before proceeding to describe practical methods of ranging on the curve, the determination of the curve to be adopted, and the functions and formulæ employed by the setter-out will be given and explained by very short and simple operations.

A curve being, for the purposes under consideration, a convenient method for changing the direction of a route, and a substitute for an angle, the first thing on which it will depend will be the angle formed between the two directions, or angle of intersection as it is termed, as the curve will be tangential to them; the second, the radius of the curve, rendering it sharper or flatter; these being fixed, the curve is determined--and if shown on a plan, the two quantities, designated by the symbols a and R, are noted on it also for convenience of reference.

The former of these two is the result of the previous determination of the centre line of the route with regard to general intention and local circumstances; the second is only approximately so, and generally admits of alteration within certain limits: it should generally be a round number, some multiple or sub-multiple of a hundred being most convenient, as Tables are constructed to suit such numbers in facilitating or saving calculation.

From these two all remaining functions of the curve may be calculated by means of formulæ and Trigonometrical Tables; those, however, that are more practically of use may be easily obtained with the help of the attached Tables by very short and simple operations; while the two pages of curve-formulæ at the end of this section, explained by the figure attached to them, include all such data necessary for the use of the setter-out.

In determining the curve, however, after obtaining the angle of intersection (a) and the radius (R), the two quantities required are the curve-tangent (T), which is the trigonometrical tangent of half the angle, and the total length of the curve (L); values of the former are given in a Table at the end of this section, and values of the latter in another table following it, which is also specially used for purposes of ranging.

The angle of intersection of the tangents is generally observed on the ground with a theodolite, and noted on the plan; in the absence of any instrument the angle may be obtained through calculation in the following way. Let BAC be the angle of intersection whose value is required, measure any two equal distances AB, AC, along the tangents and measure also BC.

Then since BC2=BA2 + AC2-2BA.AC cos BAC and BA =AC

.. BC2=2 AB2 (1-cos BAC)

hence the half-angle of intersection may be found by

P

means of a trigonometrical table of sines, and by doubling this, the whole angle (a) BAC.

Having therefore obtained the angle of intersection (a) and assumed arbitrarily a radius (R), the distances of the tangent points, or beginning and ending of the curve from the point of intersection, must be obtained: these distances are called the curve-tangents, and are given in the Table of curve-tangents at the end of this section, for any angle of intersection and for a radius of 1000: for any other radius they can therefore be obtained by simple proportion.

For example,

If a=102° 40''4 and R=600 feet;

for rad. 1000 and a= 102° 40′ T=800'196

the difference for 1'=2389,

and the difference for '4'

=

*0955

.. for rad. 1000, a= 102° 40′′4 T=800291

It is advisable not only to mark off these tangentpoints on a plan, but also to mark them permanently and accurately on the ground, as from these the settingout or ranging the curve can be done at any time afterwards, either with or without an instrument. The total length of curve should be also recorded on the plan: this may be easily obtained by means of the Table of curve-lengths.

For example.-Let the angle of intersection

a=128°45'1, and the radius=790 feet.

Then the angle of deflection for the whole curve (y), or tangential angle to the other tangent point, will= (180°-128°45′1)=25°37′ 45.

For this angle we get from the Table the curve for a radius 1000,

and the required curve-length

(Z)=894'453 × 790 = 706·618 feet.

1000

In this way the four necessary quantities a, R, T, L, are determined and recorded independently of the subsequent ranging of the curve on the ground.

Systems of nomenclature.—The practice in England formerly was to designate curves by their radii in Gunter's chains of 66 feet each up to 80 chains, and beyond that in miles; this system retains but few advantages, the Gunter's chain being of use chiefly to the land or parish surveyor for calculating acreage rapidly; but since railways have been constructed to a very great extent with such radii, and branches and junctions have to be made into such lines, the old system cannot very well be ignored, and the tables here given are suited to it as well as to the more modern way.

At present the system in vogue in England is to designate curves by their radii expressed in hundreds of feet, or chains of 100 feet, whatever may be their length, thus avoiding both miles and Gunter's chains. As, for instance, a curve of 500 feet radius, or one of 1000 feet radius, is used in preference to an 8 or 16 chain-curve;