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with the use of these tables, provided the same denomination is taken for the chord as is assumed for the radius. The tangential angle in minutes for 100 ft. chords when the radius is expressed in chains of 66 ft.
Radius of curve in chains.
The tables of multiples give the tangential angle in minutes and decimals for units of radius up to 9, and are intended to facilitate the determination of the tangential angle for fractional chords (page 217). Thus if a curve of 20 chains radius commence at ... miles, ... furlongs, and 37 links from the starting point, the tangential angle for the fraction 63 links will be ascertained thus:
Angle for 63 links minutes.
The length of chord of 63 links is then set out at an angle of 54 minutes with the tangential line, after which the unit angle for chain chords is added to this value at each setting out of points one chain apart. Again, if a curve is to terminate at 63 links beyond a full chain measurement, this value is to be added to the tangential angle taken for the last whole chord of the curve. To return to a straight line, as at D in fig. 1 (pages 218, 219), the theodolite is set up over this point, and any previous point along the curve C, B, or A is selected, the distance of which measured by chords is known. The instrument is clamped to the tangential angle for this distance, and the telescope is directed to this point and the lower plate clamped. The vernier is then unclamped and set back to 360°, when the telescope will be found to be in the direction of the tangent line D K, and when traversed vertically to be in the direction of D H. If, as in fig. 1, the vernier for a radius of 20 units has been successively set to 358° 34' - 357° 8', 355° 42' for pegging out respectively the points B, C, D, when the tangent line is to the right hand of the curve, we must remember that the tangent line D K being to the left of the curve when the instrument is set up at D, the point A must be viewed with the vernier
54 1445121 or 0% of 1° 26' 54
clamped to 4° 17' or the point B with the vernier clamped to 2° 52', and then the direction of a line joining the 360° or zero point with the centre of the instrument will give the direction D K. (Pages 218, 219.)
The poles shown in the direction of the chord lines A B, A C, A D (Case 5) are not generally necessary. The usual method is for one man to hold one end of the chain at the last point determined, taking care, if the curve be flat, to place his body upon the outside of the curve, so as not to impede the line of sight when the theodolite is set for fixing the next point in the curve. The other assistant pulls out the chain or the tape to the given length, and holds up a peg or lath, which he keeps vertical at the correct distance, moving it about as directed by the surveyor, to the right or left hand, until it accurately appears in the required direction. Should any obstacle render it necessary to remove and reset up the theodolite over a new point in the curve, the direction of a new tangent line must be found by the method shown in fig. 1 (Case 5), and the same process of setting out by means of tangential angles re-commenced. The use of the tangential angles, which are calculated from the formula proved by Case 1, enables the curve to be set out to the right-hand side of the tangent line, when the theodolite is placed over the beginning of the curve, as the primary scale of divisions upon the horizontal circle of the instrument is numbered to read in the direction of the hands of a watch; hence when the curve is to be set out to the left-hand side of a tangent line, the column upon the card containing the differences of the tangential angles must be adopted. Thus with a radius of 20 chains, if 1° 25′ 57′′ be the tangential angle for a chord of one chain in length, and an angle of 2° 51′ 54′′ be the tangential angle for the intersection of a second chord of one chain's length round the arc, when the curve is to be set out to the right-hand side of the tangent line; then the tangential angles to be employed for setting out two points at the same distances for a curve to the left-hand side of the tangent line will be 358° 34′ 3′′ and 357° 8′ 6′′ respectively.
Fig. 2 (pages 222, 223) illustrates a method of setting out
RELATIVE VALUE OF USEFUL ANCLES TO ANCLE OF INTERSEC
LET = HALF THE ANCLE
THEN A 90°- X
SETTING OUT WITH TWO THEODOLITES
IF ANY TWO LINES BE SET OUT FROM THE STARTING POINTS OF THE CURVE HAVINC
THEIR TANGENTIAL ANCLES (E+F)=90°-X;THE INTERSECTION OF THESE TWO LINES
POINT IN THE CURVE BY A SERIES OF SUCH POINTS THE WHOLE CURVE MAY BE SET COUNTRY BE SUFFICIENTLY OPEN TO ALLOW OF IT WHEN THE COUNTRY IS UNEVEN SHOWN IN FIC I BY SHIFTING AND RE-SETTING UP THE THEODOLITE WHERE NECE MORE APPLICABLE
D = 90°+ E+F=90 G+ H = 90°· R=TXTAN
DEVIATION OF TANCENT LINE
ANCLE OF DEVIATION
ORICINAL DIRECTION OF TANCENT LINE
curves with the use of two theodolites, each set up respec tively over points at the beginning and the end of a curve, when the angle of intersection between the tangent lines which pass through those points is known.
In-road work, curves may be set out by offsets from lines drawn upon a plan, and afterwards be checked by means of offsets from the tangent lines. When in road-work it is desirable to accurately set out one curve running into another, the formula given in fig. 3 may be employed. The proof is given in figs. 4 and 5 (pages 222, 223).
C B = R sin. A - √ R2 sin.3 A – 4 ( R r sin.
If R = 7 and r = 5 when A = 20°
R sin. A = 7 x 342 = 2'394
= sin. 10° = '173
4 Rr (029929)=4'190060.
The square root of the difference between 5'731376 and 4*190060 equals 1.241.
And (2394-1.241)=1'153=C B, the distance required. The geometrical proof is shown in fig. 4. The algebraical proof is illustrated by fig. 5. (See pages 222, 223).
Fig. 4. In order that the circle with the centre G, in fig. 3, should touch the tangent line E A, its centre must be in a line parallel to it, at a distance equal to its radius (see fig. 4), and in order that it should touch the circle having its centre at K, its centre must be in a circle having a radius equal to the difference of the two given radii, concentrical with the circle D C, which touches the tangent line F H (fig. 4). Therefore it lies in the intersection of the parallel line and the circles with a radius equal to the