Measure from the foot of the object, in the horizontal plane, any convenient distance, as A C, 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 A; therefore we can determine the height BC by Art. 121.

EXAMPLES.

B

A

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 C A B, or the angle of depression DBA, 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 A C by Art. 121.

EXAMPLES,

D

B

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 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. 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.

tree.

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.

C

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.

top of the object, and let it be required to find the height BC. Now, it is not convenient to meas

B

B

suppose ure a horizontal base line directly toward the object, and we measure it in any direction, A B', also measuring the angles CAB and CB'A. Then, in the horizontal triangle A B 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 A B 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

found the horizontal angles of its extremities with the object to be 33° 55' and 96°, and also the angle of elevation at the latter to be 45°. Required the height of the nest above the water.

Ans. 240 feet.

137. To find the distance between two objects separated by an impassable barrier.

Let A and B be two objects separated by an impassable barrier, and let it be required to find the distance, A B, between them.

C

B

Take any point, C, from which A and B are both visible and accessible. Measure CA and CB, and also note the angle A CB. Then, since in the triangle ABC the two sides CA and CB, with their included angle, are known, the distance AB can be found by Art. 128.

EXAMPLES.

1. Two bounds of a lot have between them an impassable morass, and, wishing to find their distance apart, I have taken their distances from a third point, which could be seen from each. These distances are 124.75 and 171.41 rods, and the angle at that point subtended by the bounds is 99° 25'. How far are the bounds apart? Ans. 227.91 rods.

2. The distance between two trees cannot be directly measured, in consequence of an intervening obstacle, but within sight of each is a third tree, and their distances from this are known to be 274.65 and 396.11 yards, and the angle at that point subtended by the two trees is 8° 56' 5". Required the distance between the two trees.

138. To find the distance between

two inaccessible objects.

Let C and D be the objects, and A and B two accessible points, from which both the objects are visible. Measure the base line AB, and observe the angles D A B, DBA,

CAB, and CBA. Then, in the

triangle DAB, since we have the side A B and all the angles, we can find the side BD by Art. 125. In the triangle ABC we have the side A B and all the angles, hence we can find BC. Then, BD and B C being found, we have in the triangle B C D the sides BD and BC, with their included angle; therefore we can find the distance CD by Art. 128.

EXAMPLE.

1. Wanting to ascertain the distance between a tree, D, and a flagstaff, C, on the opposite side of a river from me, I measured along the shore, on the horizontal plane with the objects, a base line, A B, of 110 yards. At A, the angle D A B equals 96°, and CAB equals 29° 56'; at B, the angle D B A equals 33° 55', and C B A equals 133° 50'. Required the distance between the tree and the flagstaff. Ans. 261.81 yards.

139. To find the distances from a given point, of three objects whose distances from each other are known.

Let it be required to find the distances from D, a given point, of three objects, A, B, and C, who e distances from each other are known.

Observe the angles ADC and BDC. Describe a circle about the triangle A DB, and draw A E and EB; then the angle A B E is equal to the angle A D E, since both are measured by half of the same arc A E (Geom., Prop. XVIII. Bk. III.); also the an

D

E

B

gle BAE is equal to the angle B D E, for a like reason. Now, in the triangle A E B, the side A B and all the angles are known, hence the side A E may be found by Art. 125. Again, the sides of the triangle A B C being given, we may find the angle B A C by Art. 129; then, in the triangle A E C, there will be known the two sides A C, A E, and the included angle CA E, so that the angle A CE may be found by Art. 128. Then, in the triangle AC D, we shall know the side A C and the angles ACD and AD C; therefore we can find the distance AD