2. Required the plans and areas of two fields from the following dimensions.

A small piece of land, having several sides, may sometimes be most conveniently measured by taking one diagonal, and upon it erecting perpendiculars to all the angles on each side of it. The piece will thus be divided into right angled triangles and trapezoids, the areas of which must be calculated as in the two last Problems.

EXAMPLES.

1. Required the plan and area of a field from the following

NOTE. The method of planning the above field is sufficiently clear, from the preceding field-notes and from what has been already done.

2. Lay down two pieces of ground, and find their areas, from the following dimensions.

PROBLEM V.

FIELDS INCLUDED BY ANY NUMBER OF CROOKED OR CURVED SIDES.

When a field or estate is bounded by crooked fences, a line must be measured as near to each of them, as the angles or bends will admit; in doing which an offset must be taken to each corner or bend in the fence. When the fences are curved, these offsets must be taken so as neither to exclude nor include any of the land belonging to the ground to be measured. The offsets or perpendiculars thus erected, will divide the whole offset space into right angled triangles and trapezoids, the areas of which may be found as already shewn.

NOTE. When the offsets are short, that is, not greatly exceeding a chain in length, their places on the line may be found by laying the offset staff at right angles to the chain, as nearly as can be judged by the eye; but when the offsets are large, and correctness is required, their places must be found by the cross, and measured by the chain.

NOTE 2. The quickest method of laying down offsets, is, by laying the feather edge of the plotting scale against the base or chain line, and sliding the offset scale along the feather edge to the several distances of the offsets and pricking off their lengths, corresponding to their several distances.

NOTE 3. Unskilful surveyors usually add all the offsets taken on one line together and divide the sum by their number for a mean breadth; but this method is very erroneous, especially where the offsets vary greatly in length, and should therefore be avoided where great accuracy is required.

EXAMPLES.

1. Required the plan and content of a right-lined piece of ground by offsets, from the following notes.

270 c d 175 de 120 ef = 170 fg

145

= 124

Calculation by the erroneous method (See Note 3).

955

= AB.

561 = mean breadth.

00

62

84

70

5730

00 0.53957

8)452

561

=

0a. 2r. 6p. Content by this method, which is 17 perches too little. For this method is always erroneous except when the offsets stand at equal distances from one another, and when the first and last offsets are both 0. Some omit all the offsets that are O, dividing the sum of the offsets by the number of real offsets; by this method we shall have 6)452

750

955
75

4775 6685

318

0-719430a. 2r. 85p., which is 114 poles too much.

2. To lay down a crooked piece of land, adjoining a river from the following notes.

The content is found by the same method as in the preceding example.

3. Plan and find the area of a field from the subjoined notes.

Having found the area of the triangle ABC, the areas of the offsets on the line B C must be added thereto, and the sum of the areas of the insets on the line CA must be subtracted from the sum, and the remainder will be the content of the field.

NOTE. The area of the triangle A B C may be found, when the measurement of all three sides are given, (which is the case in the present example,) either by calculation, as shall hereafter be shewn, or by measuring the perpendicular from the plan, which, as already shewn, may be laid down from the three sides of the main triangle.

4. The notes, plan, and content of the following field, the