the same as the area already found by measuring the perpendicular from the plan.

NOTE. This method of finding the areas of triangles is very little used in practice, on account of its requiring a tedious calculation, which may, however, be more readily performed by logarithms, as shall hereafter be shewn.

2. It is required to lay down a survey and find its content from the following field-notes.

Having drawn the figure, the proof line mn will be found to measure 351 links, as in the field-notes; and the perpendicular Bb to be 1056 links.

Double areas.

2644224 Triangle A B C

653112 Offsets on A B and A C

3297336 Sum

199016 Insets on B C

2)3098320 Difference

15.49160 15a. 1r. 381p. nearly the area required.

=

COMPUTATION OF THE AREA BY CASTING, THAT IS, BY REDUCING THE CROOKED SIDES TO STRAIGHT ONES.

The offsets in the last example have been computed, in manner already shewn in Chap. II. ; but by this method straight lines are drawn on the plotted figure so as to include as much space in the area to be measured, as they exclude, as nearly as can be judged by the eye, the area to be measured is thus reduced to a figure bounded by right lines only, which may thence be much more expeditiously reduced to triangles, trapeziums, &c. The method of drawing these lines is usually by a straight edged ruler of transparent horn, or by a silken thread stretched with a bow; the ruler or thread being moved over the crooked fence, till it appear to the eye to enclose as much of the adjoining ground as is left out, a line is then drawn in this position; and so on for other crooked fences. Thus the trouble of calculating numerous offsets is completely avoided, and with proper care equal accuracy is obtained.

2. We shall adopt the last example for this method of casting, that it may be seen how near the two methods agree.

The figure being constructed, and the boundary drawn carefully with ink, the chain-lines must then be rubbed out, and the three dotted lines A B, B C, CA must now be drawn, in such a manner, that the parts excluded by them may be equal to the parts included, as nearly as can be judged by the eye. The base AC will be found to be 2584 links, and the perpendicular Bb 1200 links. Whence 2584×1200

2

—15·50400=15a. 2r. 1p. (nearly, the area by this method.) If the area found by the true method be taken from the area,

just found by casting, it will be seen that they differ by little more than one pole out of 151 acres, or little more than 1 in 4000: thus

15.50400

1.5.49642

758 square links, or little more than one pole.

NOTE. It will hence be seen at once that a great deal of trouble is saved by this method, which is therefore generally adopted by practical surveyors; although it is certainly less correct than by calculating from the offsets, the former method depending chiefly on the accuracy of the casting lines for the truth of its results; but practice will soon render it easy to draw the lines so as to obtain almost perfect accuracy.

3. Required the plans and contents of two fields by both the methods of calculation, viz., offsets and by castings, from the following field-notes.

NOTE. It will be seen that in the main triangles of these two surveys, the proof lines have been taken from a side of each to its opposite angle; which ie the best method of proof, when convenient to make it; but it may be performed with equal accuracy by taking a proof line from one side to another, at a short distance from one of the angles of the triangle.

PROBLEM VI.

FOUR SIDED FIELDS.

When a field has four sides, straight or crooked, measure the four sides, or lines near them, if crooked, taking the offsets: also measure one or both the diagonals, one of which will serve as a base in plotting the work, and the other for a proof-line; or the proof-line may be measured in any other direction that may be most convenient.

Sometimes the measurement of both the diagonals is prevented by obstructions, in such cases it will be sufficient to measure tie-lines across two of the angles of the trapezium, at the distance of from two to five chains from each angle, according to the size of the field. These tie-lines with their distances from the angles on the main-lines will be found sufficient tor planning the lines and proving them.

A

D

B

EXAMPLES.

1. In the annexed figure the lines A B, BC, CD, DA are measured, marks being left at p, q, and r, and their respective distances on the lines noted in the field-book, thus furnishing the following method of laying down the plan.

On AB, as a base, take Ap = given distance, and with the distances Ar, pr, and centres A and p describe arcs cutting in r; then prolong Ar, and lay off C thereon the given length A D.

In the same manner construct

the triangle p B q, and make B C = its given length. Lastly, join DC, which must be of the length shewn in the field-book, otherwise there has been some mistake either in the measurement, or in laying it down. Should this be the case the whole of the work, firstly on the plan, and secondly in the field, must be gone over again till the error be discovered.

NOTE. When the main lines that include the chief part of the ground to be

measured are of considerable length, as from 30 to 40 chains, it will be necessary to take the tie-lines at least 10 chains from the angles, across which they are measured; for a small error, in laying down the plan with short tie-lines, will cause the main lines to deviate considerably from their true position when prolonged. However, it sometimes happens that long tie-lines cannot be cbtained in consequence of obstructions. In such cases the tie-line must be carefully measured to even one-fourth of a link; the distance of each tie-line from its angle and the tie-line itself must then be all multiplied by 4, thus throwing fractions out of the question, and with the three lines, thus increased, the riangle determining the position of an angle of the trapezium, may be accurately constructed. The proof-line and its distances from its angle must be vimilarly treated, that the accuracy of the work may be fully established.

2. Required the plan and area of a straight-sided field from the following dimensions.

When the figure has been laid down, the

diagonal AC will be found 1926, and the perpendiculars thereon from B and D respectively 632 and 514 links. Whence the area is

11a. Or. 53p.

3. Draw the plans, and find the contents of two enclosures, from the following field-notes, both by calculation from the offsets and by casting.

NOTE. In each of the two following examples, it will be seen that there are two straight sides, and two that require offsets: also, in the former example, one of the crooked sides is crossed by the chain-line, thus producing insets, the content corresponding to which must be subtracted, as in former cases.