in the triangle ABC, we have two angles and the included side, to find the side BA.

IV To find the altitude of an object, when the distance to the vertical line passing through the top of it is known.

37. Let CD be the altitude required, and AC the known distance.

From A, measure on the line AC, any convenient distance AB, and place a staff vertically at B. Then placing the eye at A, sight to the object D, and let the

E

B

point, at which the line AD cuts the staff BE, be marked. Measure the distance BE on the staff; then,

AB : BE :: AC: CD,

whence CD becomes known.

If the line AC cannot be measured, on account of intervening objects, it may be determined by calculation, as in the last problem, and then, having found the horizontal

SECTION II.

AREA OR CONTENTS OF GROUND.-LAY ING OUT LAND.

1. We come next to the determination of the area or superficial contents of ground.

The surface of the ground being, in general, broken and uneven, it is impossible, without great trouble and expense, to ascertain its exact area or contents. To avoid this inconvenience, it has been agreed to refer every surface to a horizontal plane: that is, to regard all its bounding lines as horizontal, and its area as measured by that portion of the horizontal plane which the boundary lines enclose.

For example, if ABCD were a piece of ground having an uneven surface, we should refer the whole to a horizontal plane, and take for the measure of the area that part of the plane which is included between the bounding horizontal lines AB, BC, CD, DA.

In estimating land in this manner, the sum of the areas of all the parts into which a tract may be divided, is equal to the area, estimating it as an entire piece: but this would not be the case if the areas of the parts had reference to the actual surface, and the area of the whole were calcu lated from its bounding lines.

2. The unit of measure of a quantity is a quantity of the same kind regarded as a standard, and with which all quantities of that kind may be compared.

unit is a right line of a known length, as 1 chain, or any other fixed distance.

For lines, the foot, 1 link, 1

It has been already observed (Bk. II., Sec. I., Art. 16), that Gunter's chain of four rods or 66 feet in length, and

use among surveyors. In measuring land, the length of this chain is generally taken for the unit of linear measure,

3. The unit of measure for surfaces is a square de scribed on the unit of linear measure.

When, therefore, the linear measures of ground are feet, Jards, rods, or chains, the superficial measures are square feet, square yards, square rods, or square chains; and the numerical expression for the area is the number of times which the unit of superficial measure is contained in the land measured.

4. An acre is a surface equivalent in extent to 10 square chains; that is, equivalent to a rectangle of which one side. is ten chains, and the other side one chain.

One quarter of an acre is called a rood.

Since the chain is 4 rods in length, 1 square chain contains 16 square rods; and therefore, an acre, which is 10 square chains, contains 160 square rods, and a rood contains 40 square rods. The square rods are called perches.

5. Land is generally computed in acres, roods, and perches, which are respectively designated by the letters

When the linear dimensions of a survey are chains or links, the area will be expressed in square chains or square links, and it is necessary to form a rule for reducing this area to acres, roods, and perches. For this purpose, let us form the following

Now, when the

=

will be expressed in acres by dividing by

linear dimensions are links, the area square links, and may be reduced to 100000, the number of square links in an acre: that is, by pointing off five decimal places from the right hand.

If the decimal part be then multiplied by 4, and five places of decimals pointed off from the right hand, the figures to the left will express the roods.

If the decimal part of this result be now multiplied by 40, and five places for decimals pointed off, as before, the figures to the left will express the perches.

If one of the dimensions be in links, and the other in chains, the chains may be reduced to links by annexing two ciphers: or, the multiplication may be made without annexing the ciphers, and the product reduced to acres and decimals of an acre, by pointing off three decimal places from the right hand.

When both the dimensions are in chains, the product is reduced to acres by dividing by 10, or pointing off one decimal place.

From which we conclude; that,

1st. If links be multiplied by links, the product is reduced to

2d. If chains be multiplied by links, the product is reduced to acres by pointing off three decimal places from the right hand.

3d. If chains be multiplied by chains, the product is reduced to acres by pointing off one decimal place from the right hand. feet in a rod, a square rod is 272.25 square feet.

6. Since there are 16.5 16.5 x 16.5

equal to

If the last number be multiplied by 160, we shall have, 272.25 × 160=43560 = the square feet in an acre. Since there are 9 square feet in a square yard, if the last number be divided by 9, we obtain,

=

4840 the number of square yards in an acre.

PROBLEM I.

7. To find the area of a piece of ground in the for of a square, rectangle, or parallelogram.

Multiply the base by the altitude, and the product will express the area (Geom., Bk. IV., Prop. IV. and V.)

1. To find the area of the rectangular

field ABCD.

D

Measure the two sides AB, BC: let us suppose that we have found AB= 14 chains 27 links, and BC= 9 chains 75 links. Then, A

B

2. What is the area of a square field, of which the

sides are each 33 ch. 8 1.?