Having laid down the figure, you will find the perpendicular Ba, upon the base AC, to measure 587 links.

Double areas

633960 triangle ABC

110800 offsets taken on the line AB

100120 ditto on the line BC

149310 ditto on the line CA

Area 4a. 3r. 351p.

PROBLEM II.

Fields in the form of a Trapezium.

When you have a trapezium to survey, measure each side and both the diagonals, one of which will enable you to construct the figure, and the other will serve as a proof-line; or, you may measure the longer diagonal, and a proof-line in any other direction most convenient.

NOTE 1. From various obstructions it is sometimes impossible to take either of the diagonals; in such cases measure tie lines across the angles of the field at any convenient distance (not less than two chains) from the corners. These you will find sufficient for constructing the figure, and for proofs. Or, you may take an external angle, or angles, as directed in Problem IV.

G

2. When the lines, including the angle you intend to take with the chain, are of a considerable length, it will be necessary to measure more than two chains from the angular point before you take the chord-line, because a small inaccuracy in constructing the figure, when the angular distance is short, will throw the lines, when far produced, considerably out of their true position. It sometimes happens, however, in consequence of obstructions, that it is impossible to measure the chord-line at a greater distance from the angular point than one or two chains. In such cases multiply both the chord-line and angular distance by 2, 3, 4, or any larger number, as circumstances may require, and use the products resulting in laying down the figure.

3. When the measurement of the surface is required, for reaping, &c., you must let the chain touch the sides of the lands in all places where you measure across them. If you do not measure across the lands, but along the headland, then you must add as many links to the length of the chain-line as will make it equal to one measured across the lands, parallel to and near the headland.

You may easily ascertain what number of links you ought to add by stretching the chain across the lands, and putting down an arrow at each end, after which leave hold of one of the ends, and you will observe it recede from the arrow. The number of links by which it falls short of its former position add to each chain. Some lands you will find so low that nothing need be added to the chain-line, and some will require a link to four, three, two, or even (where the lands are very high) a link or more to one chain.

When the lands are high, if the lines measured along the headlands be not lengthened, the perpendiculars will obviously measure less than they ought to do; consequently, the horizontal measure will be returned instead of the measure of the surface.

When the diagonal is measured with a slack chain it will give the measure of the surface; but in this case the perpendiculars will evidently be shorter than they would have been, if the diagonal had been measured with a tense chain; consequently, the measurement will be the same, or very nearly the same, whether the diagonal be measured with a tense or slack chain, unless the headland lines be lengthened.

4. If two or three persons measure the same piece of land separately, or even if one person measure the same piece twice over, there will generally be a difference between the measurements; this difference, however, in small pieces, should scarcely ever exceed four or five perches.

5. When land, crops of corn, &c., are bought and sold, the buyer and seller commonly choose each a surveyor, and in their measurements it occasionally happens that there exists a considerable difference. In this case the best method, perhaps, of adjusting a dispute is, that the two surveyors meet and jointly remeasure the land. If this fail, they or their employers, the buyer and seller, jointly choose an experienced surveyor, as an umpire, and abide by his decision.

Examples.

1. It is required to construct a figure, and find its area from the following notes.

Having constructed the figure, lay your scale from B to D; and if you find it exactly 1400 links, as in the field-book, you may then measure the perpendicular Ba= 468 links, and the perpendicular Da = 432 links; from which you will readily compute the area required.

Here the area is found to be 9a. 2r. 34p., which is too much by la. Or. 15p.; but the more nearly a trapezium approaches to a square or rectangle the less will be the error.

2. Required the area of a field from the following notes.

Having constructed the figure, you will find that, in consequence of the angle ADB being obtuse, a perpendicular from the angle A to the diagonal DB cannot be taken; you must, therefore, let fall the perpendicular Da from the angle D to the side AB, which you will find = 638 links. The perpendicnlar ca will be found = 294 links.

B