3. Given the two sides of a triangle equal to 153 rods and 137 rods, and the included angle equal to 40° 33′ 12′′; to find the other parts.

Ans. Side, 101.615 feet; angles, 78° 13′ 1′′ and 61° 13′ 47′′.

CASE IV.

129. Given the three sides.

Let.there be given (Fig. Art. 128) the three sides a, b, and c; to solve the triangle.

To find A, B, and C. By (102), (103), and (104), we have

log sin

Blog (s—a) + log (s—c) — log a — log c

(135)

(136)

2

log sin † C′ — log (s — a) + log (s—b) — log a — log b

That is,

The logarithmic sine of half of any angle of a triangle is equal to the logarithm of the difference between half the sum of the sides and one of the adjacent sides, plus the logarithm of the difference between half the sum and the other adjacent side, minus the logarithms of those two sides, divided by 2.

130. A, B, and C can also be determined by formulæ (106), (107), and (108) for the cosine of half an angle, and by formulæ (109), (110), and (111) for the tangent of half an angle.

When the half angle is less than 45°, the table will determine it from its sine with greater precision than from the cosine, and vice versa when the half angle is greater than 45°.

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 ss 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

2. Given the three sides of a triangle equal to

654; to solve the triangle by means of the cosine.

9.734577

32° 52′ 8′′.6.

65° 44′ 17′′.2.

432, 543, and

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′′.

2) 19.361995

2) 19.469154

9.680998

B 28° 40' 4".4;

C

C

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

DBA.

B

C

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 AC and the acute angle A; therefore we can deter- A mine the height BC by Art. 121.

EXAMPLES.

B

C

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

EXAMPLES.

D

B

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

BOOK IV.

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.

Let C be the point inaccessible

from A and B, and let it be required to find its distance from each of those points.

Measure as a horizontal base line the distance between A and B, and observe the horizontal angles CAB and CBA. Then, in

the triangle A B 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 Ans. The one, 926.01 feet; the other, 646.16 feet.

tree.

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.

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