The angle corresponding to this arc will be that given for a radius of 1000 in the Table for an arc of

consequently 40′ subtending less than 50 feet will be the largest convenient angle in round numbers.

The one operator will then set off an angle BAp1= 40′ simultaneously with the other setting off an angle of 30°-40′=29°20′, viz., BDp, from his tangent: the intersection of these two directions is in one point in the curve, p; then the first operator will set off the angle BAp2=1° 20′ while the other sets off the angle BDp2= 28°40′; the one adding and the other subtracting 40' each time until all the points are ranged in.

2

To check the work they would then go to each of the points Pa P3 etc., thus set out, and see that each of the angles ApD, Aƒ‚D, Aƒ‚D, etc., were equal to 180°-7=150°.

The following table, constructed for a radius of 1000, may be used for obtaining either the angle subtended at the tangent-point by any arc or length of curve: for the converse purpose the table of curve-lengths and arcs on p. 223 will give the arc subtended by a given angle, for any radius whatever; as illustrated in the example already given.

TABLE OF ANGLES OF DEFLECTION SUBTENDED BY ANY LENGTH OF ARC WHEN R= 1000.

3rd. Kröhnke's tangential system.

Kröhnke's system consists in setting-out equidistant points on the curve by ordinates from the two tangents.

In doing this the tangent points B, C (figure 29), are taken as the origins of rectangular co-ordinates, the abscissæ being taken along the tangents and measured from them as Be, Be,, Beg, Beg, and Cg, Cg1, Cg2, C831 etc., the ordinates at right angles to them as eƒ, ‹ƒ„ eaf, etc., gh, gh, gh, etc.; thus giving number of points f, ff, etc., h, ha, etc.; the values of ordinates and abscissæ corresponding to equidistant points on the curve may be obtained from the following Table. When the two last points set-out on either branch as ƒ, and h1, are sufficiently near each other, the curve is set-out; and

if for any special reason the middle point of the curve M is wanted, it may be obtained sufficiently closely by bisecting the distance between the two last points.

This setting-out may be checked by comparing the calculated total length of curve which may be obtained from the preceding Table of curve-lengths, as due to the angle of deflection, or (which ought to have been previously found and registered on the plan) with the sum of the distances between the equidistant points, together with the small additional pieces ƒM, h M which must be measured on the ground.

Thus, if L length of curve=5237, and five points have been set-out on either branch at distances of 50' apart, then 5237 — 500=23'7, which ought to be the sum of the additional pieces measured on the ground; and f,M=hM should=1185 feet. When the radius is large and the angle of intersection small, the ordinates near the middle point of the curve become inconveniently long, and if not laid off at very accurate right angles with the abscissa, great error will result. As for instance with a radius of 5000 feet and a central angle of 100°, the last ordinates will be about 1000 feet long, and the serious error from a defective right angle may be imagined.

To avoid these inconveniences it is necessary to make use of an additional tangent at the middle point of the curve, making this a new origin for rectangular co-ordinate, and measuring abscissa both ways from it, the ordinates will be smaller-thus each branch of the curve I will be treated as the whole curve was in the first instance. In this case the tangent QMP must be set-out very accurately, the distances BQ and CP must each= and M should be at the middle of QP; as a

R tan

В

4

further check on the position of the point M, it may be set-out by means of abscissa and ordinate BI, IM from the original tangent, where

being the centre angle or supplement of the angle of intersection (a); this being done, M should fall in the line QP. Should it happen to be more convenient, BK and KM may be measured instead of BI and IM, the distances being identical.

The table herewith given is confined to a radius of 1000; and from this ordinates and abscissæ for equidistant points on curves having other radii may be calculated by either multiplying or dividing the whole of the three corresponding quantities-curve, abscissa, and ordinate-by the ratio that the given radius bears to 1000. In the author's Curve Book, published in 1869, such results were given for sixteen values of radius, and occupied fifteen pages of tabular matter, which were seldom used, and are hence not here reproduced. In Kröhnke's work a comparatively large table, consisting of 47 pages, and containing a great deal of repetition of figure, is given to enable one to set-out curves on this system.

There is a great disadvantage in this system, that of requiring a bisection of the curve, and sometimes even many bisections, when the offsets from the tangent become long; and this necessitates the use of an angular instrument. The presumed advantage of this system, viz., that of setting-out points for permanent stakes that are exactly 50 feet or 100 feet apart on the curve itself is unreal; for as they happen to commence from the tangent point of one curve, they would not suit that purpose on other curves further on, and hence the intention fails.