the triangle is equal to the rectangle under the perpendicular and the diameter of the circumscribed circle. 22. The hypothenuse of a right-angled triangle is less than the sum of the two sides by the diameter of the inscribed circle. 23. If two triangles have an angle of the one equal to one angle of the other, and also another angle of the one equal to the supplement of another angle of the other, the sides about the two remaining angles shall be proportionals. 24. Given two sides of a triangle and the difference of the angles opposite to them; solve the triangle. 25. Given the area, the perimeter, and an angle; solve the triangle. 26. Given two sides of a triangle and the included angle, find either of the angles into which the given angle is divided by a line drawn from it to the middle point of the opposite side; and adapt the expression to logarithmic computation. 27. The tangents of the angles of a triangle are in harmonical progression; given one of the sides, and the difference of the first and third angle; solve the triangle. 28. The perimeter of a triangle: the diameter of the A B C inscribed circle :: 1: tan tan tan 29. 2 2 2' Given the base, the vertical angle, and the difference of the sides; find the remaining angles. 30. The sides of a triangle are in arithmetical progression, and its area is to that of an equilateral triangle of the same perimeter as 3 : 5. Find the ratio of the sides and the value of the largest angle. 31. The area of any triangle is to the area of the triangle, the sides of which are respectively equal to the lines joining its angular points with the middle points of the opposite sides, as 4 : 3. ; express the 32. The angles of a plane triangle form a geometrical progression of which the common ratio is greatest side in terms of the perimeter. 1. A river AC, the breadth of which is 200 feet, runs at the foot of a tower CB, which subtends an angle BAC of 25° 10′ at the edge of the bank. Required the height of the tower. 2. The angles of elevation of a balloon were taken at the same time by three observers, placed respectively at the two extremities and at the middle point of a base measured on the earth's surface. Find an expression for the height of the balloon. 3. In order to ascertain the height of a mountain, a base was measured of 2761 feet, and at either extremity of this base were taken the angles formed by the summit and the other extremity; these were 58° 29′ and 111° 52′; also at the extremity from which the latter angle was taken, the angular height of the mountain was 11° 18'. Required the mountain's height. 4. A person standing at the edge of a river observes that the top of a tower on the edge of the opposite side subtends an angle of 55° with a line drawn from his eye parallel to the horizon; receding 30 feet, he finds it to subtend an angle of 48°. Determine the breadth of the river. 5. A person on a tower can see the top of a pillar of known altitude, from which he wishes to know the distance, and the height of the tower. He can see also an object on the horizontal plane from which he has formerly observed the angular distance of the top of the pillar from that of the tower. Shew how he may find the required distances, having with him an instrument for measuring angles. 6. A person on the top of a tower, the height of which is 50 feet, observes the angles of depression of two objects on the horizontal plane, which are in the same straight line with the tower, to be 30° and 45° respectively. Find their distances from each other and from the observer. 7. Three objects A, B, C, form an isosceles triangle whose vertex is B, and whose angles are as the numbers 4, 1, 1; an observer walking from A toward C, measures a base AD of a feet, and observes the angle BDC; he then advances to E, b feet further, and observes that the angle BEC the supplement of BDC. From these observations find the sides of the triangle. = 8. A person walking from C to D on the horizontal road can plainly see the summit of a hill A from every point except E, where he can just see it over a hill B. He measures EC, and at C observes the angles of elevation of B and A, as well as the angles ACB, ACE. At E he observes the angle AEC. Shew how to find the heights of the hills. 9*. In ascending a tower of known height, a person observes from a window the angle of depression of a point in the horizontal plane upon which the tower stands; when he arrives at the top of the tower he observes the angle of depression of the same point; shew how to find the height of the window above the ground. 10. A person wishing to ascertain the height of a tower standing on a declivity, ascends 80 feet from its base, and it then subtends an angle of 30o. The inclination of the side of the hill to the horizon being 15o, find the height of the tower. 11. The elevation of a steeple standing on a horizontal plane is observed, and at a station a feet nearer to it its elevation is found to be the complement of the former. On advancing b feet nearer still, the elevation is found to be double the first; shew that the height of the steeple is 12. From the summit of a tower, the height of which is 108 feet, the angles of depression of the top and bottom of a column, standing on the same horizontal plane with the tower, are observed to be 30o and 60° respectively. Required the height of the column. 13. The top of a tower is visible from three stations A, B, C, in the same horizontal plane; at each of the stations the angular distance of the top of the tower from each of the other two stations is observed; given the distance between A and B, and the height of the tower, it is required to find the distance of C from each of the other stations and from the tower. 14. A person travelling along a straight road observes the elevation of a tower, the nearest distance of which from the road is known. At the same time he also observes the angular distance of the top of the tower from an object in the road. Required the height of the tower. 15. Describe the observations and calculations, necessary for determining the breadth of a river, from stations upon one of its banks. 16. From a station B, at the base of a mountain, its summit A is seen at an elevation of 60°; after walking 1 mile towards the summit, up a plane making an angle of 30° with the horizon, to another station C, the angle BCA is observed to be 135°. Find the height of the mountain in yards. 17. An object 6 feet high, placed on the top of a tower, subtends an angle the tangent of which is '015, at a place the horizontal distance of which from the foot of the tower is 100 feet; determine the height of the tower. ILLUSTRATIONS OF THE CIRCULAR MEASURE. 1. Find the number of degrees, minutes and seconds, in the angles, the circular measures of which are 1.5, 2, + 1, and 3.14 respectively. 2. Reduce to the circular measure the following angles; 14°, 15° 30′, 120°, 17° 8', and 92° 3'. 3. If two-thirds of a right angle be assumed as the angular unit, what will be the numerical value of an angle of 45° ? Determine the angular unit by the assumption of which the following equation would be numerically true, subtended arc 5. angle = 2 radius Find the circular measure of 1'. 03 6. It may be shewn, that if a very small angle ◊ is expressed according to the circular measure sin 0 = 0 nearly; what change must be made in the formula if 0 is expressed in seconds? 6 MISCELLANEOUS PROBLEMS. 1. Given three lines drawn from any point within a square to three of its angular points; determine a side of the square. 2. Express the diagonals of a quadrilateral inscribed in a circle in terms of the sides. . 3. The excess of the sine above the versed sine, in angles less than 90°, is greatest when the angle = 45°. Prove this and find the value of the maximum excess. 4. In any quadrilateral figure, the square of one side is less than the sum of the squares of the other sides, by |