The method by the tangent of half the angle is precise, and requires the use of but four logarithms.

NOTE. This case may also be solved by drawing a perpendicular from the vertex to the base of the triangle, thus dividing it into two right-angled triangles, of which the hypothenuses are known, and the sum of whose bases is the base of the original triangle. Let s and s represent CD and DA (Fig. Art. 113), then (Geom., Prop. XI. Bk. IV.),

a form to which logarithms can be readily applied.

Knowing s+s' and s- -s', s and s' can at once be found, and thence the angles A, C, and B, by Art. 122.

EXAMPLES.

1. Given of any triangle ABC, the side a equal to 216 yards, the side b equal to 217 yards, and the side c equal to 235 yards; to find the angles A, B, and C.

Solution. By (134), (135), and (136) we have

A=28° 27' 47"; B 28° 40' 4".4; C 32° 52′ 8′′.6. ≥ B= } =

Ans. A 56° 55' 34"; B 57° 20' 8".8; C 65° 44' 17".2.

2. Given the three sides of a triangle equal to 432, 543, and 654; to solve the triangle by means of the cosine.

3. Given the three sides of a triangle equal to 95.12, 162.34, and 98; to solve the triangle by means of the tangent.

Ans. The angles, 32° 14′ 53′′; 114° 24' 9"; 33° 20′ 58′′.

BOOK IV.

PRACTICAL

APPLICATIONS.

DETERMINATION OF HEIGHTS AND DISTANCES.

131. A HORIZONTAL PLANE is one which is parallel to the horizon.

A VERTICAL PLANE is one which is perpendicular to a horizontal plane.

A HORIZONTAL LINE is one which is parallel to the hori

zon.

A VERTICAL LINE is one which is perpendicular to a horizontal plane.

132. A HORIZONTAL ANGLE is one the plane of whose sides is horizontal.

A VERTICAL ANGLE is one the plane of whose sides is vertical.

An ANGLE OF ELEVATION is a verti- D cal angle having one side horizontal and the inclined side above it; as the angle CAB.

An ANGLE OF DEPRESSION is a vertical angle having one side horizontal and

the inclined side under it; as the angle A

D BA.

B

133. To determine the height of a vertical object standing on a horizontal plane.

Let B be the top of the object, and let it be required to find its height BC.

Measure from the foot of the object, in the horizontal plane, any convenient distance, as AC, as a base line, and at A observe the angle of elevation CA B. Then, in the right-angled triangle ABC, we have known the side A C and the acute angle 4; therefore we can determine the height BC by Art. 121.

EXAMPLES.

A

B

1. Standing on the edge of a moat 40 feet wide, I observe that the wall of a fort upon the opposite brink subtends an angle at the point of observation of 36° 52′ 12′′; required the height of the wall. Ans. 30 feet.

2. The angle of elevation of the top of a flag-staff, measured on a horizontal plane, at a distance of 89 feet from the foot of the staff, is 41° 29'; what is the height of the staff?

134. To find the distance of a vertical object, its height being given.

Let BC be the object whose height is given, and let it be required to find the distance A C.

Measure the angle of elevation C A B, or the angle of depression D B A, which is equal to CA B. Then, in the rightangled triangle ABC, we have known the side B C and the angles; therefore we can find the distance AC by Art. 121.

EXAMPLES.

D

B

A

C

1. A tree 91 feet in height stands on the same horizontal plane with a dial, at which the angle of elevation subtended by the tree is 32° 22'; required the distance of the dial from the foot of the Ans. 143.6 feet.

tree.

2. From the top of a house whose height is 30 feet, I observe that the angle of depression of an object standing on the same horizontal plane with the house is 36° 52′ 12′′; required the

distance of the object from the base of the house, and also the length of the line that will just connect the object with the top of the house.

135. To find the distance of an inaccessible point on a horizontal plane.

the triangle AB C, there will be known the side AB and the angles; therefore the sides AC and BC can be found by Art. 125.

EXAMPLES.

1. Wanting to know the distances of two objects from a tree, inaccessible by reason of an intervening river, I measured the distance in a straight line between the two objects, and found it to be 772.45 feet; I also found the horizontal angles formed by the extremities of the straight line with the tree to be 80° 58' 4" and 43° 33' 44". Required the distances of the objects from the tree. Ans. The one, 926.01 feet; the other, 646.16 feet. 2. Two ships are engaged in cannonading a fort by the seaside; the ships are 131.89 rods apart, and the two angles at the ends of the straight line connecting the ships, formed by that line and lines drawn to the fort, are 18° 52' 13" and 152° 11' 42". Required the distance of each ship from the fort.

136. To find the height of an inaccessible object above a horizontal plane.

First Method. Let B be the top of the object, and let it be required to find the height B C

Measure a horizontal base line, A C, of any convenient length, directly toward the object, and observe the angles of elevation at A and C. Then, in the triangle ABC, since

B

BCA is the supplement of CCB, we have known the side AC and all the angles; therefore we can find the side AB by Art. 125. Then, in the right-angled triangle ABC, we have known the hypothenuse AB and the angles; therefore we can find the height BC by Art. 120.

EXAMPLES.

1. Required the altitude of a hill whose angle of elevation, taken at the foot of it, was 55° 54′, and 300 feet back, on the same horizontal plane with the foot, the angle was 33° 20′.

Ans. 355.71 feet.

2. Two observers at sea, 800 yards apart, noticed at the same instant a meteor bearing due east from each; to the one its angle of elevation was 57°, and to the other the same angle was 31° 28'. Required the altitude of the meteor above the horizontal plane of the ships.

Second Method.

Let B be the

top of the object, and let it be re

B

quired to find the height BC. Now,

A

suppose it is not convenient to measure a horizontal base line directly B toward the object, and we measure it in any direction, A B', also measuring the angles CAB and CBA. Then, in the horizontal triangle AB C, we know the side A B' and all the angles; therefore the side A C can be found by Art. 125. Then, also, by observing the angle of elevation CA B, we shall, in the right-angled triangle AB C, know the side AC and all the angles; therefore the height B C can be found by Art. 121.

EXAMPLE.

1. A person on one side of a river observed an eagle's nest on an inaccessible mountain-crag on the opposite side, and being desirous of ascertaining its height above the level of the river, he measured along the shore a straight line 110 yards in length, and