Hence also, CC' .

Applications.

CC" 242.06 239.93 2.13 yards,

which is the height of the station A above station B.

PROBLEMS.

1. Wanting to know the distance between two inaccessible objects, which lie in a direct line from the bottom of a tower of 120 feet in height, the angles of depression are measured, and are found to be, of the nearer 57°, of the more remote 25° 30' required the distance between them.

2. In order to find the distance between two trees A and B, which could not be directly measured because of a pool which occupied the intermediate space, the distances of a third point C from each of them were measured, and also the included angle ACB. it was found that

Ans. 173.656 feet.

3. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51°; then measuring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45'; required the height of the tower.

Ans. 83.998 feet.

each other, equal to 200 yards; from the former of these points A could be seen, and from the latter B, and at each of the points C and D a staff was set up. From C a distance CF was measured, not in the direction DC, equal to 200 yards, and from a distance DE equal to 200 yards, and the following angler taken,

OF

GEOMETRY.

MENSURATION

OF SURFACES.

DEFINITIONS.

1 The area of any figure has already been defined to be the measure of its surface (Bk. IV. Def. 7). This measure is merely the number of squares which the figure contains.

A square whose side is one inch, one foot, or one yard, &c., is called the measuring unit; and the area or contents of a figure is expressed by the number of such squares which the figure contains.

2. In the questions involving decimals, the decimals are generally carried to four places, and then taken to the nearest figure. That is, if the fifth decimal figure is 5, or greater than 5, the fourth figure is increased by one.

3. Surveyors, in measuring land, generally use a chain called Gunter's chain. This chain is four rods, or 66 feet in length, and is divided into 100 links.

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

One quarter of an acre, is called a "ood.

Since the chain is 4 rods in length, 1 square cha'n 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.

Mensuration of Surfaces.

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

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

TABLE.

1

square chain=100 x 100=10000 square links.

1 acre 10 square chains=100000 square links

=

1 square mile=6400 square chains 640 acres.

6. Now, when the linear dimensions are links, the area will be expressed in square links, and may be reduced to acres by dividing by 100000, the number of square links in an acro: 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 hand 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 ci phers, and the product reduced to acres and decimals of an acre, by pointing off three decimal places at the right hand.

When both dimensions are in chains, the product is re

Mensuration of Surfaces.

luced to acres by dividing by 10, or pointing off one decimal place.

From which we conclude: that,

I. If links be multiplied by links, the product is reduced to avres by pointing off five decimal places from the right hand.

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

III. If chains be multiplied by chains, the product is reduced to acres by pointing off one decimal place from the right hand. 7. Since there are 16.5 feet in a rod, a square rod is equal 16.5 × 16.5=272.25 square feet.

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.

To find the area of a square, a rectangle, a rhombus, or a parallelogram.

RULE.

Multiply the base by the perpendicular height and the produc will be the area (Bk. IV. Th. viii).

EXAMPLES.

1. Required the area of the square ABCD, each of whose sides is 36 feet

D