To find the fum of ang. A & C. To find the angles A and C. The three fides of any triangle being given, to find the angles. Fig. 57. plate 4. AB 100 Ex. 1. Given BC 80 Required all the angles. LAC 60 AB: AB: AC+BC:: BC-AC: BD-AD 100 140 : 20: 28 In all cafes of this kind, the | To the base verfa. 50 Add difference 14 The greater feg. །༄ From the base, 64 50 14 36 1.80618 1.55630 | As BD 64 is to rad. 90 1.77815 So is BC 80 1.90309 10.00000 10.00000 To fec. ang. A 53° 8' 10.22185 To fec.ang.B 36° 52' 10.09691 Angle C may be found thus: From 180, fubtract the fum 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 following Rule. Add the three fides together, and, from half the fum, fubtract the fides severally; then add the complements of the logarithms of the half fum, and of the difference between the halffum, and the side oppofite to the angle fought, to the logarithms of the differences of the two other fides and half fum; and half their fum will be the tangent of half the angle required. Thus, let angle A be required: The angles BC may be found by problem 1. of oblique angled trigonometry. We come now to the application of trigonometry, to the menfuration of heights and distances MENSURATION OF HEIGHTS AND DISTANCES. THE inftruments commonly made ufe of in measuring heights and distances, are the Geometrical Quadrant, the Theodolite and the Geometrical fquare. The Geometrical quadrant is used for investigating vertical angles; whether they be angles of * altitude, or angles of depreffion. The Theodolite ferves 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 fufpended from the centre; it is commonly made of brafs or wood. Fig. 1. plate 4. The N. B. When the object is higher than the measurer's eye, it is faid to fub tend an angle of elevation, but when lower, an angle of depreffion. |