of the lines AB, AC, &c., from A to the several corners of the field; find the areas of the triangles BAC, CAD, &c., as in the last article, and add them together. If the boundary is irregular, as represented in the figure, measure offsets, calculate

the contents of these smaller

portions separately, and add or

E

FIG. 107.

subtract them as may be necessary, to find the true area of the tract.

217.-E X AMPLES.

1. Required the contents and plot of a piece of land, of which the following are the field notes:

2. Required the contents and plot of a piece of land, of which

the following are the field notes:

3. Required the area of a piece of land, of which the follow

Offsets from the line AB were taken as follows:

BOOK V.

LAYING OUT AND DIVIDING LAND.

SECTION I.

OF DIVIDING LAND.

218. The surveyor is often required to lay off a given quantity of land, in such a way that its bounding lines shall form a particular figure, viz., a square, a rectangle, a triangle, &c. He is also often called upon to divide given pieces of land into parts containing given areas, or, into areas bearing certain relations to each other.

The manner of making such divisions must always depend on a skilful and judicious application of the principles of geometry and trigonometry to the particular case.

For example, if it were required to lay out an acre of ground, in a square form, it would be necessary to find, by calculation, the side of such a square, and then trace, on the ground, a figure bounded by four equal sides at right angles to each other.

219. To lay out a given quantity of land in a square form.

RULE.-Reduce the given area to square chains or square rods; then extract the square root, and the result will be the side of the required square. This square being described on the ground, will be the figure required.

220. To lay out a given quantity of land in a rectangular form, when one of the sides of the rectangle is given.

RULE.-Divide the given area, reduced to square chains or square rods, by the given side of the required rectangle, and the quotient will be the other side. Then, trace the rectangle on the ground.

221. To run a line from the vertex of a triangular field which shall divide it into two parts, having to each other the ratio of m to n.

Let ABC be any triangular field.

Divide the side BC into two parts,

such that (Geom., Bk. IV, Prob. I)

For, the two triangles ABD, ADC, having the same altitude, are to each other as their bases (Geom., Bk. IV, P. 6, C.); hence, the triangle is divided into parts having the ratio of m to n.

222. To run a line parallel to one side of a triangular field, that shall form with the parts of the two other sides a triangle equal to the part of the field.

m

n

and on the side BA, take BF equal to BA V

m

n

the line EF is the line required; for, since it divides the sides BC and BA proportionally, it is parallel to the side CA (Geom., Bk. IV, P. XVI); and from the similar triangles, we have (Geom., Bk. IV, P. XXV),

EXAMPLE.-Let it be required to divide the triangular field CAB, in which AC 9 ch.,

=

=

AB 11 ch., and CB = 7 ch., into two such parts that ADE shall be one-fourth of the whole field.

In this case, we have,

1

=

=

4

2

hence,

=

AE4 ch. 50 1., and AD 5 ch. 50 1.

223. To run a line from a given point in the boundary of a piece of land, so as to cut off, on either side of the line, a given portion of the field.

Make a complete survey of the field, by the rules already given. Let us take, as an example, the field whose area is computed in Ex. 1, Art. 140. That field contains 105 A. 2 R. 33 P., and Fig. 112 is a plot of it.

Let it now be required to run a line from station A in such a manner as to cut off, on the left, any part of the field; say, 26 A. 2 R. 31 P.