Let B C equal 150 feet, the altitude of the place of observation; draw C E perpendicular to B C, and B A parallel to C E, for the purpose of making the angles of depression. Set off the angle A B D equal to 45 degrees; and the angle A B E equal to thirty degrees. The intersections at D and E determine the positions of the vessels. Measure D to E, and it will give the distance between the vessels.

PROBLEM VI.

To find the height and distance of an inaccessible object, standing on a horizontal plane.

Take the angles of elevation of the object at two accessible points of observation in the same vertical plane; draw the angles, and the points of intersection will mark the position, and the measure of the lines will give the height and distance of the object.

1. The angle of elevation of the top of a steeple on the opposite side of a river is found to be 66 degrees, and 125 feet further distant in the same vertical plane, the angle of elevation is found to be 42 degrees. What is the height of the steeple, and the distance of its base from the first place of observation?

Draw A C indefinitely, and set off from A to B 125 feet. At A, make an angle of 42 degrees, and draw the line A D. At B, make an angle of 66 degrees, and the intersection of B D with A D, determines the height of the steeple. Draw the perpendicular, C D, and its intersection with A C marks the position of its base; measure from B to C, and it will give the distance from the first place of observation; to which, add the distance from B to A, and it gives the distance from the second place of observation.

2. Two persons, 175 feet apart, on the same level, and in the same vertical plane, find the angles of elevation of the top of a tower standing on the opposite bank of a river, to be 39 and 25 degrees. What is the height of the tower, and its distance from each place of observation?

3. The altitude of a steeple, by observations from the centre of a village, is 28 degrees; and advancing 30 yards nearer in the same vertical plane, the altitude is found to be 45 degrees. What is the height of the steeple, and distance of its base from the last place of observation?

PROBLEM VII.

To find the height of an object by knowing the greatest distance at which it is visible.

To the square of the distance of the object, add one fourth of the square of the diameter of the earth; from the square root of this sum, subtract the radius of the earth, the remainder is the height of the object.

1. What is the height of a mountain, the summit of which is visible 40 miles at sea? ANS. 1056 feet...

2. If the top of a lighthouse can be seen 20 miles at sea, what is its height from the surface of the sea? ANS. 266 feet,

PROBLEM VIII.

To find the greatest distance at which an object is visible, the height of which is known.

Multiply the diameter of the earth by the height of the object; to the product, add the square of the height; the square root of this sum will give the distance required.

1. If a mountain be one fifth of a mile high, at what distance is its summit visible? ANS. 39 miles. 2. How far can a mountain, 2640 feet high, be seen! ANS. 63 miles. 3. At what distance is a mountain, 2 miles high, visible? ANS. 126 miles.

PROLEM IX.

To find the diameter of the earth, by knowing the height of a mountain, and the distance at which it is visible.

Divide the square of the distance by the height of the object; and from the quotient subtract the height; the remainder will give the diameter of the earth.

I. If the height of a mountain be 2640 feet, and its summit can be seen 62.93 miles, what is the diameter of the earth? ANS. 7922 miles.

If a mountain, one mile high, be visible 89 miles, what is the diameter of the earth?

ANS. 7921 miles.

CHAPTER III.

MISCELLANEOUS EXAMPLES.

The following questions may be solved by the rules for finding the base, perpendicular and hypothenuse of a right triangle.

1. From the top of a tower, 180 feet high, standing on the opposite bank of a river, the distance is 225 feet, what is the width of the river?

ANS. 8 rods, 1 yard. 2. What is the height of a house which requires a ladder 52 feet long, standing 20 feet from the house, to reach the top?

3. A fort on the bank of a river surmounted by a wall 74.5 feet high. tance from the opposite bank to the top

ANS. 48 feet. 225 feet wide, is What is the dis

of the wall?

ANS. 237 feet. and B, from the same How far apart are they?

4. A. travels south 105 miles; point, travels west 140 miles.

ANS. 175 miles. feet, and the span 20

5. The height of a roof is 7 feet. What is the length of the rafters ? The span is the breadth of the building.

ANS. 12 feet.

6. How long must a ladder be to reach the roof of a house, 50 feet high, the ladder standing 20 feet from the house? ANS. 54 feet.

7. It is required to prop the wall of a building with a timber, 39 feet long; the upper end of the timber to be placed against the wall 36 feet from the ground. How far from the side of the building must the other end be placed? ANS. 15 feet.

8. A fortified town is surrounded by a wall 32 feet high, and outside of the wall, by a ditch, 24 feet broad. What must be the length of a scaling-ladder with which to mount the wall? ANS. 40 feet.

9. Two vessels depart from the same port at the same time; one sails 9 miles an hour in a direction due east; the other sails 12 miles an hour due south. How far apart will they be at the end of ten hours?

ANS. 150 miles. 10. Two ships leave port at the same time; one sails due east 5 miles in an hour; the other due north 7 miles an hour. At the end of 12 hours, they cast anchor, and despatch a boat from each vessel to the other. How long will it take for the boats to meet, rowing at the rate of 5 miles an hour? and in what direction must the boats go?

ANS. The boats will meet in 10 hours and 30 minutes. One must go north-west, the other south-east.