413. In surveying a field or a number of fields, we shall have a series of operations and records like those exemplified in Art. 410; namely, one for each chain-line. As an example, we will take the case of a field which approximates to the form of a triangle, so that three chainlines will occur in the survey.

The sides of the triangle ADG are respectively 1540 links, 1430 links, and 1650 links; hence it will be found by Art. 152 that the area of this triangle is 1016400 square links. We proceed to calculate the areas of the small pieces lying between the sides of this triangle and the boundary of the field.

Along AD there are offsets to B and to C; thus we have to estimate a triangle, a trapezoid, and another triangle; the areas are the following in square links:

Along DG there is an offset to E, and an inset to F: thus there are two corresponding triangles, the latter of which is to be subtracted:

the balance is 7850, to be added.

Along GA there are offsets to H and to L, and the boundary meets the chain-line at K; thus there are two corresponding triangles:

1016400+23400+7850+22250=1069900.

Thus the area of the whole field is 10'699 acres.

The perpendicular is measured as a proof-line, and found to be 1232 links, while Gd is 726 links.

414. Instead of the field-book, another method of recording the results of measurement is sometimes adopted. A plan is drawn resembling the field to be surveyed, and the lengths are noted in the plan as they are found against the corresponding parts of the figure,

415. We have hitherto supposed that the boundary of a field which is to be surveyed may be regarded practically as composed of a moderate number of straight lines. But if the boundary is so irregular that this supposition is not admissible, we must employ the principle of adjustment which has been explained in Art. 202: a plan of the field must be drawn and the boundary changed into a rectilineal boundary enclosing an equal area. We will explain a convenient method of applying the principle.

416. Let ABDKEC be the plan of the field. Draw on the plan equidistant parallel straight lines; thus dividing the figure into strips of equal width. Consider one of these strips, as BDEC. Draw the straight line bd at right angles to the parallels, so that the area of

bB

E

d

K

the strip may be the same whether BD or bd be regarded as its end; if BD can be regarded as a straight line, bd will pass through its middle point; if BD be not a straight line the position of bd must be determined as well as possible by the eye. Similarly, draw ce at the other end of the strip, so as to leave the area unchanged. Then the area BDEC is equivalent to the rectangle bdec.

Proceeding in this way we obtain a series of rectangles which are together equivalent to the original figure. The area corresponding to all these rectangles can be easily ascertained, and therefore the area of the original figure. Suppose, for example, that the parallel straight lines are drawn an inch apart; and that the sum of the lengths of all the rectangles is 29 inches: the area of the original figure is 29 square inches. Now suppose that the plan has been drawn on the scale of three chains to an inch; then a square inch of the plan corresponds to nine square chains of the field; therefore the area of the field is 9 × 29 square chains, that is, 261 square chains.

In practice the process of forming the sum of the lengths of the rectangles is performed by an instrument called the computation scale.

EXAMPLES. XLVI.

Draw plans and find the areas of fields from the following notes, in which the lengths are expressed in links: