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Ans. Area 182 A. 0 R. 21.7 P. and the bearing and distance of the 3rd 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.

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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 off-sets.

RULE.

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 offsets 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.

Left-hand off-scts on the 3rd side.

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The area of the part ABCD within the stationary lines will be found, by either of the rules in the preceding section, to be 13 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

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EXAMPLE 2. Fig. 83.

Required the area of a survey from the following field

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The area within the stationary lines, found by either of the rules in the preceding section, is 1152.5381 square chains.

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