To find the sum of ang. A&C. To find the angles A and C. 180° As CB+AB 380 2.57978 69 is to CB-AB 20 1.30103 Soistan.ang. A+C55° 30' 10.16287 2)111 fum The three sides of any triangle being given, to find the angles. Fig. 57. plate 4. AB 100 AC 60 AB; AB: AC+BC:: BC. AC:BD-AD 100 : 140 : : 20 : 28 50 14 In all cases of this kind, the | To the base The greater feg. 64 10.00000 10.00000 To find angle A. To find angle B. As AD 36 1.55630 | As BD 64 1.80618 is to rad 90° is to rad. 90 So is AC 60 1.77815 So is BC 80 1.90309 1 To sec. ang. A 53° 8' 10.22185 ) Tofec.ang. B 36° 52' 10.09691 Angle C may be found thus : From 180, subtract the sum of angles A and B, the remainder will give angle C. Or add the complements of the angles A and B together, and the fum is | angle C. The preceding problem is frequently wrought according to the fol lowing Rule. Add the three fides together, and, from half the sum, subtract the sides severally; then add the complements of the logarithms of the half sum, and of the difference between the halfsum, and the side opposite to the angle sought, to the logarithms of the differences of the two other sides and half sum ; and half their sum will be the tangent of half the angle re. quired. Thus, let angle A be required: So The angles BC may be found by problem 1. of oblique angled trigonometry. We come now to the application of trigonometry, to the mensuration of heights and distancese MENSU The instruments commonly made use of in measuring heights and distances, are the Geometrical Quadrant, the Theodolite and the Geometrical square. The Geometrical quadrant is used for investigating vertical angles ; whether they be angles of * altitude, or angles of depression. The Theodolite serves for measuring angles on a horizontal plane, or on an inclined plane. A vertical plane, is that which is at right angles with the horizon. A horizontal plane, is that which is parallel to the horizon. The Geometrical quadrant, is the fourth part of a circle, and is divided into 90°, to which two fights are adapted, and a plumb line fuspended from the centre ; it is commonly made of brass or wood. Fig. 1. plate 4. The N. B. When the object is higher than the measurer's eye, it is said to sube tend an angle of elevation, but when lower, an angle of depresion. . The Theodolite is a semi-circle divided into 180°, with an index which turns about on its centre, and retains any situation given it, on which are two fights, called the moveable sights; there are also two other fights fixed on the diameter of the theodolite, which are called the fixed fights. Fig. 2. plate 4. Sights are small pieces of wood or brass, having small holes or slits in them, to view the object through ;---They are fixed perpendicular to the plane of the theodolite, but parallel to the plane of the quadrant. The geometrical square may be made of brass, wood, or any solid body, having equal fides and angles; from one of the angles, a thread is suspended, with a small weight at the end, so as to point always to the centre. The two fides opposite to the centre of suspension, are divided each of them into 100 equal parts; there is also an index, which, (when occasion serves), may be fixed to the centre of suspension, and is made fo as to turn round, and retain any fituation; on this index, are two fights. See fig. 3. plate 4. Heights and distances are of two kinds, viz. accessible and inaccessible: accessible objects are houses, growing trees, &c. inaccessible ones are all mountains, celestial bodies, also houses and trees, in certain situations. PROBLEM I. See Plate 4. fig. 58. To measure accessible heights. EXAMPLE I. Let AB be a horizontal plane and BC a tower, whose height is required : From B, the foot of the tower, measure any convenient distance, 80 feet upon the horizontal plane AB. Suppose the tower to subtend an angle of 39° 49' from A. What is its height? Ag |