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To be calculated by either of the preceding Rules.
1. Given the boundaries of a tract of land as follow, viz. 1st. S 350 W, 11.20 ch. 2nd. N 45° W, 24.36 ch. 3rd. N 15°1 E, 10.80 ch. 4th. S 77° E, 16 ch. 5th. N 8701 E, 21.50 ch. 6th S 60° E, 14.80 ch. South 10.91 ch. 8th N 85° W, 29.28 ch. to the place of beginning; required the area. Ans. 85 A. 3 R. 17 P.
2. Given the boundaries of a tract of land as follow, viz. 1st. N 19° E, 27 ch. 2nd. S 77° E, 22.75 ch. 3rd. S 27° E, 28.75 ch. 4th. S 52° W. 14.50 ch. 5th. S 15°1 E. 19 ch. 6th. West, 17.72 ch. 7th N. 36o W, 11.75 ch. 8th North, 16.07 ch. 9th N 62° W, 14.88 ch. to the place of beginning; required the area.
Ans. 152 A. 2 R. 6 P.
3. Required the area of a tract of land bounded as follows: 1st. S 62° W. 7.57 ch. 2nd. N. 43°1 W, 5.89 ch. 3rd. North, 5.82 ch. 4th. N 33°1 W, 8.83 ch. 5th N 48° E, 4.81 ch. 6th. N 12° E, 4.66 ch. 7th. N 6201 E, 5.27 ch. 8th S 601 E, 5.60 ch. 9th S 40° E, 5.87 ch. 10th. East, 6.54 ch. 11th. North, 5.52 ch. 12th. N 68°1 E, 3.10 ch. 13th. S 30° E, 7.90 ch. 14th. S 23° W, 8.80 ch. 15th. S 31°1 E, 6.42 ch. 16th. S 50° W, 8.40 ch. 17th. N 44° W, 6.85 ch. to the place of beginning. Ans. 44 A. 2 R. 18 P.
4. Given the following field-notes to find the area of the survey; also the bearing and distance of the 3rd side, which were omitted to be taken on account of obstacles in the way.
Ans. Area 182 A. 0 R. 21.7 P. and the bearing and distance of the 3d side S. 66° 23′ W. 28.06 ch.
5. Being furnished with the field-notes of a tract of land, and requested to calculate the area, I found, on examining them, that the figures expressing the angles of bearing of the 4th and 5th sides were so defaced as 'to be illegible; but as the remaining data are sufficient, the area is required. The field-notes are as follow.
Off-sets are lines drawn or measured, perpendicularly
from a stationary line, to the angular points of the land
In taking surveys, bounded on some of their sides by streams of water, it is unnecessary to make a station at every bend in the stream, because the field-work can be taken, and the calculations made with more facility, and with equal accuracy, by making use of off-sets.
Directions for taking Off-sets.
Take as many stations in the irregular boundary as may be most convenient. Then take the bearing from the first station to the second; and in measuring the distance stop against each bend in the stream and measure the perpendicular distance from it to the stationary line. Note the distance in the field-book as a right-hand, or left-hand off-set, according as the boundary lies on the right, or left of the stationary line; also note against each off-set, its distance from the beginning of the stationary line. If there be more than two stations, proceed in the same manner with the others.
Note. In calculating by off-sets, the irregular boundary is considered as straight between the ends of each two adjacent off-sets; there should therefore be so many taken that this supposition may be made without any material error in the survey.
To find the area contained between a stationary line and an irregular boundary by means of offsets.
Subtract the stationary distance of each off-set, from that of the one immediately following; the remainders
When the boundary is a brook or rivulet, it is customary to measure to the middle of the channel; but when it is a river in which the tide flows, the mea
will be the distances, intercepted on the stationary line, between each two adjacent off-sets respectively.
Multiply the sum of each two adjacent off-sets by their intercepted distance on the stationary line; half the sum of the products will be the area required*.
Note. The area of the off-sets must be added to, or subtracted from, the area within the stationary lines, according as the stationary lines on which the off-sets are taken are within, or without, the boundary of the survey.
EXAMPLE 1. Fig. 82.
Required the area of a piece of meadow, bounded on one side by a brook, the field-notes being as follow.
The area of the part ABCD within the stationary lines will be found, by either of the rules in the preceding section, to be 18 A. 1 R. 11 P.
* DEMONSTRATION. Considering the boundary as straight between the ends of each two adjacent off-sets, it is plain that the area contained between the stationary line and boundary will be divided by the off-sets into trapezoids