the same intervals as the permanent stakes, which may be 50, 66, 100 or 200 feet; but this is comparatively of little use, if the pegs on each curve are started from each tangent point, as the permanent stakes will afterwards require setting out throughout the line, and no saving of labor is effected; it is hence equally usual to range the curve with pegs at any convenient intervals, and to set-out the permanent stakes afterwards by measurement on the alignment thus obtained.

The following are practical methods of ranging simple circular curves of centre lines on canals, railways, and main roads.

I. With one theodolite and a chain.

2. With two theodolites and observers, without a chain.

3. Kröhnke's tangential system, giving equidistant points on the curve.

4. Baker's parabolic system.

5. The author's first polygonal system, or six-point system, giving unequal intervals on the curve.

6. The author's equidistant six-point system, giving equal intervals on the curve.

7. The single chord system.

These methods will here be described separately, illustrated by figures, and accompanied by small tables to facilitate their application in the field.

Ist. The one-theodolite and chain system.

The method of ranging a curve most usually adopted in setting-out with one theodolite is the following: Having obtained the necessary preliminary data, set

a theodolite over one of the tangent points, obtain from the following table (on p. 230) the angle of deflection cor

responding to an arc or chord of 50 feet to the adopted radius, and set it off from the tangent, fix or hold one end of a 50-feet measure, whether chain or metallic tape, to the tangent point, and keeping it stretched, cause it to be moved around that point as a centre until the other end falls in the direction set off by the instrument, mark that end, thus obtaining one point in the curve. Keeping the instrument at the tangent point, set off successively the double, treble, etc., of the angle originally obtained, by continually adding its value to the last amount, and cause measurements of 50 feet to be made from the last point set-out in each case so as to fall in the corresponding directions set-out: the intersections, or points, may be often set-out in this way throughout the whole of the curve, without moving the instrument. On arriving at the direction of the other tangent point, the total angle of deflection should be observed, and its value compared with half the supplement of the angle of intersection originally obtained; thus affording a check on the work.

As the total length of curve is rarely an exact multiple of 50, there is generally a short piece remaining

after setting out a number of these points, between the last point and the tangent point. To verify this, observe the angle of deflection subtended by this piece; also measure the piece, multiply its value by 1000, and divide it by the radius used, and obtain, corresponding to this result, the subtended angle given in the table on p. 232; these values should be nearly equal.

In case obstacles should prevent all the points from being set-out from the one tangent point, move the instrument to the last point set-out, having the last angle clamped on its upper plate; sight with this on the original tangent point, and set off the angle, thus obtaining a new tangential direction from the production of which angles can be set off for more points as at the commencement.

4

In the accompanying figure (27), the instrument is supposed to be placed at A, the angles BAp ̧, BAр2 BAP, being set off from the tangent BA with the distances or chords, Ap1; PP2; P2P3; measured in the manner described. An obstacle prevents the point p from being seen from A; the instrument is then moved to having the angle Bap, clamped on it, the lower circle being moveable, with which sighting A, the angle ApC is set off, thus obtaining the direction of a new tangent CpE: from which the angles Epas; Elapsi etc., ED; are laid off in the same way as the others.

This method of setting-out assumes that the chord of an arc is equal to the arc itself, and hence the check before-mentioned will never be exactly right, although sufficiently so for many practical purposes. The error for 50-feet arcs when the radius is greater than 500 feet and less than 2000 feet, may be neglected-when the radius

is less than 500 feet, 25-feet arcs may be set-out with greater correctness: when the radius is greater than 2000 feet, 100-feet arcs may be considered equal to their chords. In obtaining angles of deflection from the tables under these latter circumstances, it is useful to remember that when both the arc and the radius are doubled or halved, the angle is doubled or halved also.

It is exceedingly rare that greater exactness is required than the foregoing method admits of: but in such cases, the value of the chord corresponding to the arc of 50 should be calculated and used instead of a 50-feet chord; for this purpose Rankine's approximate formula (see p. 214) may be sufficiently precise.

TABLE OF ANGLES OF DEFLECTION SUBTENDED BY USUAL CHORDS FOR VARIOUS RADII.

2nd. The two-theodolite system.

Setting-out a curve with two-theodolites is, when circumstances permit of it, by far the most rapid method

FIGURE 28. THE TWO-THEODOLITE SYSTEM.

The two operators set their instruments on the two tangent points, the one setting off from the one tangent any convenient angles, and their multiples (round numbers are generally chosen), the other simultaneously setting-out angles made with the other tangent equal to the differences between those angles and the total angle. of deflection for the whole curve, mentioned as (y) in the formula; the intersections of all such corresponding pairs of directions are points in the curve.

The small angles should be so chosen with the help of the following table as to subtend arcs, either less than 50 or 100, according to the distances required for points along the curve. To check these points, an instrument is set up at each one of them, and if the angle contained by the directions of the two tangent points amounts exactly to the supplement of the angle of deflection, viz. 180°-y, that point is correct.

For example-(refer to figure 28).

Let R=2000 feet and y= 30° (BAD) it is required to set-out points not more than 50 feet apart.