PROBLEMS.

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

Ans. 173.656 feet.

A

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,

CB 672 yards,

=

CA =588 yards,

ACB = 55° 40' ;

required the distance AB.

Ans. 592.967 yards.

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.

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'

viz: WAE= 57° 40'

ABE 42° 22'

EBW 71° 07';

=

Ans. 83.998.

A

B

5. Wanting to know the horizontal distance between two inacessible objects A and B, and not finding any station from which both of them could be seen, two points C and D, were chosen

F

D

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,

find the three distances PA, PC, and PB.

B

PA 710.193 yards.
PC1042.522

PB 934.291.

7. This problem is much used in maritime surveying, for the purpose of locating buoys and sounding boats. The trigonometrical solution is somewhat tedious, but it may be solved geometrically by the following easy con

as a centre, and OA or OC as a radius, describe the circumference of a circle: then, any angle inscribed in the segment APC, will be equal to 33° 45'.

Subtract, in like manner, twice CPB-45°, from 180°, and lay off half the remainder = 67° 30', at B and C, determining the centre of a second circle, upon the circumference of which the point P will be found. The required point P will be at the intersection of these two circumferences. If the point P fall on the circumference described through the three points A, B, and C, the two auxiliary circles will coincide, and the problem will be indeterminate.

BOOK I I.

PLANE SURVEYING.

SECTION I.

DEFINITIONS.-MEASUREMENT OF ANGLES AND LINES.

1. SURVEYING, in its most extensive signification, comprises all the operations necessary for finding,

1st. The area or contents of any portion of the surface of the earth;

2d. The lengths and directions of the bounding lines; and,

3d. For making accurate delineations of the surface and bounding lines on paper.

It is divided into two branches, Plane and Geodesic Surveying.

2. The radius of the earth being very large, the curvature may be neglected, when the survey is limited to small portions of the surface. This branch is called Plane Surveying.

When the curvature is taken into account, as it must be in all extensive surveys, the method of measurement and computation is called Geodesic Surveying.

3. If at any point of the surface of the earth, regarded as a sphere, a plane be passed perpendicular to the radius, it will be tangent to the surface. Such a plane, and all planes parallel to it, are called horizontal planes.

4. A plane perpendicular to a horizontal plane, at a

5. All lines of horizontal planes are called horizontal lines.

6. Lines which are perpendicular to a horizontal plane, are called vertical lines; and all lines which are inclined to it, are called oblique lines.

Thus, AB and DC are horizontal lines; BC and AD are vertical lines; and AC and BD are oblique lines.

D

A

B

7. The horizontal distance between two points, is the horizontal line intercepted between the two vertical lines passing through those points. Thus, DC or AB is the horizontal distance between the two points A and C, or the points B and D.

8. A horizontal angle is one whose sides are horizontal; its plane is also horizontal.

A horizontal angle may also be defined to be, the angle included between two vertical planes passing through the angular point, and the two objects which subtend the angle.

9. A vertical angle is one, the plane of whose sides is vertical.

10. An angle of elevation, is a vertical angle having one of its sides horizontal, and the inclined side above the horizontal side.

Thus, in the last figure, BAC is the angle of elevation from A to C.

11. An angle of depression, is a vertical angle having one of its sides horizontal, and the inclined side under the horizontal side. Thus, DCA is the angle of depression from C to A.

12. An oblique angle is one, the plane of whose sides is oblique to a horizontal plane.

13. All lines, which can be the object of measurement, must belong to one of the classes above named, viz.: 1st. Horizontal lines:

2d. Vertical lines:

3d. Oblique lines.

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