CHAPTER VII.

HEIGHTS AND DISTANCES.

66. The plane of the sensible horizon at any place is the plane which is tangent to the earth's surface at that place.

Any line or plane which is parallel to the plane of the horizon is said to be horizontal; and any line or plane which is perpendicular to the plane of the horizon is said to be vertical.

The visible horizon for any observer is the circumference of a small circle of the earth which limits his view of the earth's surface.

The plane of the sensible horizon coincides with the surface of tranquil water, when this surface is so small that its curvature can be neglected; and it is perpendicular to the plumb line.

67. The angle of elevation of an object is the vertical angle which a line drawn to the object from the place of the observer makes with the horizontal plane at that place, when the object is above this horizontal plane; the angle of depression is the same angle, when the object is below the horizontal plane.

The bearing of an object from the place of the observer is the horizontal angle which the vertical plane passing through the place and the object makes with the plane of the meridian of the place.

Various instruments have been devised for estimating the direction of any visible object from the observer, with reference to the plane of the horizon or to that of the meridian or to the directions of other visible objects. The most important of these instruments in landsurveying is the theodolite, which consists of a telescope, capable of being rotated on its stand, about a vertical axis, into the same vertical plane with any visible object, and also of being rotated in that plane, about a horizontal axis perpendicular to it. By measuring these rotations, we can measure the horizontal angle made by the vertical plane of the object with the plane of the meridian (which is indicated

by the compass) or with any other vertical plane, and also the angle of elevation or depression of the object. Other instruments, such as the quadrant, the sextant, and the azimuth compass, are used on shipboard for measuring angles.

68. Problem. To determine the height of a vertical tower situated on a horizontal plane. [B., p. 94.]

Solution. Observation. Let AB (fig. 23) be the tower whose height is to be determined. Measure off the distance BC on the horizontal plane of any convenient length. At the point C observe the angle of elevation BCA.

Calculation. We have, then, given, in the right triangle ACB, the angle C and the base BC, as in § 34 of Pl. Trig., and the leg AB is found by (26).

EXAMPLE.

At the distance of 95 feet from a tower, the angle of elevation of the tower is found to be 48° 19′. Required the height of the tower. Ans. 106.69 feet.

69. Problem. To find the height of a vertical tower situated on an inclined plane.

Solution. Observation. on the inclined plane BC. makes with the plane. venient length. At the

Let AB (fig. 24) be the tower, situated Observe the angle B which the tower Measure off the distance BC of any conpoint C, observe the angle BCA by which the top of the tower is elevated above the inclined plane.

Calculation. In the oblique triangle ABC, there are given the side BC and the two adjacent angles B and C, and BA may be found as in § 73 of Plane Trigonometry.

EXAMPLE.

Given (fig. 24) BC= 89 feet, B=113° 12', C=23° 27'; to find BA.

Ans. BA 51.595 feet.,

70. Problem. To find the distance of an inaccessible object. [B., pp. 89 and 95.]

Solution. Observation. Let B (fig. 2) be the point the distance of which is to be determined, and A the place of the observer. Measure off the distance AC of any convenient length, and observe the angles A and C.

Calculation. AB and CB are found by § 73 of Pl. Trig.

71. Corollary. The perpendicular distance PB of the point B from the line AC and the distance AP and PC are found, in the triangles ABP and BPC, by § 32 of Pl. Trig.

72. Corollary. Instead of directly observing the angles A and C, the bearings of the lines AB, AC, and CB may be observed, when the plane ABC is horizontal; and the angles A and C are then easily determined, since the meridians may be considered as parallel.

73. EXAMPLES.

1. An observer sees a cape which bears N. by E.; after sailing 30 miles N. W., he sees the same cape bearing east; find the distance of the cape from the two points of observation.

2. Two observers, stationed on directly opposite sides of a cloud, observe the angles of elevation to be 44° 56' and 36° 4', their distance apart being 700 feet; find the distance of the cloud from each observer and its perpendicular altitude.

Ans.

Distances from observers

Height

417.2 feet, and — 500.6 ft. 294.7 feet.

3. The angle of elevation of the top of a tower at one station is observed to be 68° 19', and at another station, 546 feet farther from the tower, the angle of elevation is 32° 34'; find the height and distance of the tower, the two points of observation being supposed to be in the same horizontal plane with the foot of the tower.

The distance from the nearest point of observ. 185.86 ft.

74. Problem. To find the distance of an object from the foot of a tower of known height, the observer being at the top of the

tower.

Solution. Observation. Let the tower be AB (fig. 23) and the object C. Measure the angle of depression HAC.

Calculation. Since

ACB HAC,

we know in the triangle ACB the leg AB and the opposite angle C, so that we can find BC as in § 33 of Pl. Trig.

EXAMPLE.

Given the height of the tower

sion=

150 feet, and the angle of depres: 17° 25′; to find the distance from the foot of the tower. Ans. 478.16 feet.

75. Problem. To find the height of an inaccessible object above a horizontal plane, by means of observations taken at any two points in that plane. [B., p. 96.]

Solution. Observation. Let A (fig. 25) be the object, and let D be the foot of the perpendicular dropped from A on the horizontal plane. At two different stations in the horizontal plane, B and C, whose distance apart and bearing from each other are known, observe the bearings of the object, which are the same as the angles made by BD and CD with the meridians of B and C. Also observe the angle of elevation of A at one of the stations, as B.

Calculation. In the triangle BCD, the side BC and its adjacent angles are known, so that BD is found by § 73 of Pl. Trig. In the right triangle ABD, the height DA is, then, computed by § 34 of Pl. Trig.

EXAMPLE.

At one station, the bearing of a cloud is N. N. W., and its angle of elevation 50° 35'. At a second station, whose bearing from the first station is N. by E. and distance 5000 feet, the bearing of the cloud is W. by N. Find the height of the cloud.

Ans. 7316.5.

76. Problem. To find the distance of two objects whose relative position is known. [B., p. 90.]

Solution. Observation. Let B and C (fig. 1) be the two known objects, and A the position of the observer. Observe the bearings of B and C from A.

Calculation. In the triangle ABC, the side BC and the three angles are known. The sides AB and AC are found by § 73 of Pl. Trig.

EXAMPLE.

The bearings of the two objects are, of the first N. E. by E., and of the second E. by S.; the known distance of the first object from the second is 23.25 miles, and the bearing N. W.; find their distance from the observer.

Ans. The distance of the first object is = 18.27 miles.
That of the second object = 32.25 miles.

77. Problem. To find the distance apart of two objects separated by an impassable barrier, and their bearing from each other. [B., p. 91.]

Solution. Observation. Let A and B (fig. 1) be the objects the distance and bearing of which from each other is sought. Measure the distances and bearings from any point C to both A and B.

Calculation. In the triangle ABC, the two sides AC and BC and the included angle C are known. The side AB and the angles A and B may be found by § 82 of Pl. Trig.

EXAMPLE.

Two ships sail from the same port, the one N. 10° E. a distance of 200 miles, the second N. 70° E. a distance of 150 miles; find their bearing and distance from each other.

The bearing of the first ship from the second = N. 36° 6' W.

78. Problem. To find the distance apart of two inaccessible objects situated in the same plane with the observer, and their bearing from each other. [B., p. 92.]