Substituting, then, these four factors, in the place of the quantity which has been resolved into them, we have, S=1√(b+c+a)×(b+c—a)×(a+b—c)×(a−b+c, Here it will be observed, that all the three sides, a, b, and c, are in each of these factors. Let h=(a+b+c) half the sum of the sides. Then S=vh×(h—a) × (h—b) × (h—c) 210. For finding the area of a triangle, then, when the three sides are given, we have this general rule; From half the sum of the sides, subtract each side severally ; multiply together the half sum and the three remainders; and extract the square root of the product. APPLICATION OF TRIGONOMETRY TO THE MENSURATION OF HEIGHTS AND DISTANCES. ART. 1. The most direct and obvious method of determining the distance or height of any object, is to apply to it some known measure of length, as a foot, a yard, or a rod. In this manner, the height of a room is found, by a joiner's rule; or the side of a field by a surveyor's chain. But in many instances, the object, or a part, at least, of the line which is to be measured, is inaccessible. We may wish to determine the breadth of a river, the height of a cloud, or the distances of the heavenly bodies. In such cases it is necessary to measure some other line; from which the required line may be obtained, by geometrical construction, or more exactly, by trigonometrical calculation. The line first measured is frequently called a base line. 2. In measuring angles, some instrument is used which contains a portion of a graduated circle divided into degrees and minutes. For the proper measure of an angle is an arc of a circle, whose centre is the angular point. (Trig. 74.) The instruments used for this purpose are made in different forms, and with various appendages. The essential parts are a graduated circle, and an index with sight-holes, for taking the directions of the lines which include the angles. 3. Angles of elevation, and of depression are in a plane perpendicular to the horizon, which is called a vertical plane. An angle of elevation is contained between a parallel to the horizon, and an ascending line, as BAC. An angle of depression is contained between a parallel to the horizon, and a descending line, as DCA. The complement of this is the angle ACB. D B H P C 4. The instrument by which angles of elevation, and of depression, are commonly measured, is called a Quadrant. In its most simple form, it is a portion of a circular board ABC, on which is a graduated arc of 90 degrees, AB, a plumb line CP, suspended from the central point C, and two sight-holes D and E, for taking the direction of the object. H D A E B To measure an angle of elevation with this, hold the plane of the instrument perpendicular to the horizon, bring the centre C to the angular point, and direct the edge AC in such a manner, that the object G may be seen through the two sight-holes. Then the arc BO measures the angle BCO, which is equal to the angle of elevation FCG. For as the plumb line is perpendicular to the horizon, the is a right angle, and therefore equal to BCG. Taking from these the common angle BCF, there will remain the angle BCO=FCG. angle FCO In taking an angle of depression, as HCL, the eye is placed at C, so as to view the object at L, through the sight-holes D and E. 5. In treating of the mensuration of heights and dis tances, no new principles are to be brought into view. We have only to make an application of the rules for the solution of triangles, to the particular circumstances in which the observer may be placed, with respect to the line to be measured. These are so numerous, that the subject may be divided into a great number of distinct cases. But as they are all solved upon the same general principles, it will not be necessary to give examples under each. The following problems may serve as a specimen of those which most frequently occur in practice. PROBLEM I. TO FIND THE PERPENDICULAR HEIGHT OF AN ACCESSIBLE OBJECT STANDING ON A HORIZONTAL PLANE. 6. MEASURE FROM THE OBJECT TO A CONVENIENT STATION, AND THERE TAKE THE ANGLE OF ELEVATION SUBTENDED BY THE OBJEСТ. If the distance AB be measured, and the angle of elevation BAC; there will be given in the right angled triangle ABC, the base and the angles, to find the perpendicular. (Trig. 137.) As the instrument by which G H D B the angle at A is measured, is commonly raised a few feet above the ground; a point B must be taken in the object, so that AB shall be parallel to the horizon. The part BP, may afterwards be added to the height BC, found by trig. onometrical calculation. Ex. 1. What is the height of a tower BC, if the distance AB, on a horizontal plane, be 93 feet; and the angle BAC 35 degrees? Making the hypothenuse radius (Trig 121.) Cos. BAC : AB :: Sin. BAC ; BC=69.9 feet. For the geometrical construction of the problem, see Trig. 169. 2. What is the height of the perpendicular sheet of water at the falls of Niagara, if it subtends an angle of 40 degrees, at the distance of 163 feet from the bottom, measured on a horizontal plane? Ans. 136 feet. 7. If the height of the object be known, its distance may be found by the angle of elevation. In this case the angles, and the perpendicular of the triangle are given, to find the base. Ex. A person on shore, taking an observation of a ship's mast, which is known to be 99 feet high, finds the angle of elevation 3 degrees. What is the distance of the ship from the observer? Ans. 98 rods. 8. If the observer be stationed at the top of the perpendicular BC, whose height is known; he may find the length of the base line AB, by measuring the angle of depression ACD, which is equal to BAC. D B H P Ex. A seaman at the top of a mast 66 feet high, looking at another ship, finds the angle of depression 10 degrees. What is the distance of the two vessels from each other? Ans. 22 rods. We may find the distance between two objects which are in the same vertical plane with the perpendicular, by calcu lating the distance of each from the perpendicular. Thus AG is equal to the difference between AB and GB. |