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In the triangle CBC", to find CC.

We have, Theorem IV.,

R : tan B BC: CC"; whence,


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which is the height of station A above station B.


1. Wanting to know the distance between two inaccessible objects, which lie in a direct level line from the bottom of a tower 120 feet in height, the angles of depression are measured from the top of the tower, and are found to be, of the nearer 57°, and of the more remote 25° 30': required the distance between the objects. Ans. 173.656 feet.

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

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also the included angle ACB: it was found that,

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3. Being on a horizontal plane, and wanting to ascertain

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.

4. Wanting to know the horizontal distance between two inaccessible objects E and W, the following measurements were made.

AB = 536 yards

BAW = 40° 16'

Ans. 83.998.

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were chosen at a distance from 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 a distance CF was measured, not in the direction DC, equal to 200 yards, and from D a distance DE equal to 200 yards, and the following angles taken,

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6. From a station P there can be seen three objects, A, B, and C, whose distances from each other are known: viz.,

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AB = 800, AC = 600, and BC= 400 yards.

Now, there are measured the horizontal angles,

APC 33° 45', and BPC = 22° 30′ :


it is required to find the three distances, PA, PC, and PB.`


With the three given sides construct the triangle ABC. Then, at A lay off the angle BAD = 22° 30', and at B the angle ABD = 33° 45', and note D, the point at which the two lines intersect.

Through the points A, D, and B describe the circumference of a circle, and through C and D draw the line CDP;


the point P in which it intersects the circumference, will be the position of the station.

By observing the equal angles in the figure, the trigonometrical solution is not difficult. We find,

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NOTE. This problem is much used in maritime surveying, for the purpose of locating buoys and sounding-boats. The trigonometrical solution is somewhat tedious, but the geomet



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

The surface of 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 calculated from its bounding lines.

48. The unit of measure of any quantity is a quantity of the same kind, regarded as a standard. For lines, the unit is a right line of a known length, as 1 foot, 1 link, 1 chain, or any other fixed distance.

It has been already observed (Bk. II., Art. 18), that Gunter's chain of four rods or 66 feet in length, divided into 100 links,

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

49. The unit of measure for surfaces is a square described on the unit of linear measure.

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When, therefore, the linear measures are feet, yards, 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.

50. AN ACRE, which is the common unit of measure for land, 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.

A ROOD, is one quarter of an acre.

Since the chain is four rods in length, 1 square chain con

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