2. Required the area in acres of the square whose side is 264 yards.

264

264

1056

1584

528

4840)69696(14a.

4840

21296

19360

1936

4

4840)7744(1r.

4840

2904

40

484,0)11616,0(24p.

968

•1936

1936 Area 14a. Ir. 24p.

3. If the side of a square be 1567 links, what is its area in acres? Ans. 24a. 2r. 9p.

4. If the side of a square be 263 yards, what is its area in acres? Ans. 14a. 1r. 6p.

PROBLEM II.

Rectangular Fields.

When you enter a field which has the appearance of a rectangle, try each angle, and measure each side, as before; and if you find all the angles right-angles, and the opposite sides equal, the figure is a rectangle.

To compute the content.

RULE.-Multiply the length by the breadth, and the product will be the area.

Examples.

1. What is the area of the rectangle ABCD, whose length AB is 1235 links, and breadth AD, 557 links?

2. Required the area of a rectangle, whose length is 235 and

breadth 162 yards.

235

162

470

1410

235

484,0)3807,0(7a.

3388

419

4

484)1676(3r.

1452

224

40

484)8960(18p.

484

4120

3872

248 Ans. 7a. 3r. 18p.

3. The length of a rectangular field is 1225 links, and its breadth

613 links; required the plan and area.

Area 7a. 2r. 1p.

C

4. If the length of a rectangle be 135, and breadth 50 yards; what is its area? Ans. 1a. 1r. 23p.

NOTE. As squares and rectangles seldom occur in surveying, it is more advisable to treat every field of four sides as a trapezium. (See Problem IV.)

PROBLEM III.

Triangular Fields.

When you have to survey a field in the form of a triangle, set up a pole at each corner, when there are no natural marks. Then measure along the base till you come to the point where you think a perpendicular will fall from the opposite angle. There plant your cross, and turn its index till the mark at each end of the base can be seen through one of the grooves. Then apply your eye to the other groove, and if you see the mark at the opposite angle, you are in the right place to measure the perpendicular; if not, move the instrument backward or forward along the line, till you can see the three marks as above directed. Enter in your field-book the distance from the end of the base to the cross, and the length of the perpendicular. Then measure the remainder of the base.

NOTE 1.-Be especially careful that, in measuring the two parts of the base and the perpendicular, no confusion of arrows takes place.

2. In ranging perpendiculars by the cross, you must always proceed as above directed.

Construction.

Having the place of the perpendicular, the figure may be easily constructed, as follows. From any scale of equal parts, lay off the base; erect the perpendicular at its proper point; draw a line from each end of the base to the end of the perpendicular, and the figure will be completed.

NOTE. Having the diagonal, the two perpendiculars, and the place of each perpendicular given, you may construct any trapezium in the same manner.

To compute the Content.

RULE.-Multiply the base and perpendicular together, divide the product by 2, and the quotient will be the area.

Or, multiply half the base by the whole perpendicular, or the whole base by half the perpendicular, and the product will be the

area.

Examples.

1. It is required to survey the triangular field ABC, and to find its

area.

Measure from A toward c, and when you come to m, for instance

at 935 links, try with your cross; and

if this be the point for the perpendicular, measure

MB

= 625 links. Re

turn and measure

3

base = 1563 links; then construct the figure, and find its area.

1563 base.

625 per.

7815

3126

9378

2)976875

4.88437

4

3.53748

40

21-49920 Area 4a. 3r. 21p.

2. The distance between the beginning of the base and the place of the perpendicular is 125, the perpendicular 82, and the whole base 318 yards; what is the area of the triangle?

H

3. Measuring along the base of a triangle 862 links, I found the true place of the perpendicular, and the perpendicular itself 995 links; the remainder of the base measured 1110 links; what is the area of the triangle? Ans. 9a. 3r. 10p.

4. Measuring along the base of a triangle field, I found the perpendicular to range at 865, and its length 645 links; the remainder of the base measured 569 links; required the plan and area.

Area 4a. 2r. 20p.

NOTE.-If the examples in this problem, or any of the following problems, be thought too few, more may easily be supplied by the teacher sketching fields, at pleasure, with his pen, which the learner may measure by a scale. This method will be found very advantageous; as it will give the learner a good idea in what manner he must run his lines, take his dimensions, and enter his notes, when he commences field-practice.

PROBLEM IV.

Fields in the form of a Trapezium.

A quadrilateral field, having unequal sides, may be surveyed by measuring a diagonal. This divides it into two triangles, to each of which it serves as a base.

To compute the Content.

RULE-Multiply the sum of the two perpendiculars by the diagonal, divide the product by 2, and the quotient will be the area.

NOTE 1.-Always make choice of the longer diagonal, because the longer the base line of a triangle, the more obtuse is its subtending angle; and, consequently, there is the less chance to mistake, as the perpendicular will be shorter, and its place more easily and more accurately determined. After finishing the surveying, if you choose, measure the other diagonal, which will enable you to prove your work. (See Problems I. and II., Part IV.)

2. If a field be very long, or elevated in the middle, so that you cannot see from one end to the other, it may be divided into two or more trapeziums; or you may range your lines over the hill, as directed in Part V.

3. When two perpendiculars cannot be taken upon either of the diagonals, such fields must be divided into two triangles by measuring a diagonal for the base of one triangle, and one side of the field for the base of the other. (See Example 6.) 4. Sometimes surveyors affect to reduce trapeziums into squares, or rectangles, by measuring all the sides, adding each two opposite sides together, and taking half their sum respectively for a mean length and breadth; but this method leads to very erroneous results. (See Part IV., Prob. II.)

Examples.

1. It is required to survey the trapezium ABCD, and find its

area.