In the right angled triangle ADC, there is AC 47, and AD 40. 23, given, to find the angle A. This is resolved by case 4. of right angled plane trigonometry, thus. AD: R: AC: Sec. A, 31°. 8'. Or it may be had by finding the angle ACD, the complement of the angle A; without a secant, thus, AC: RAD: S. ACD,=58° 52'. then 90-58° 52′ 31° 08′, the angle A.' Then, by theo. 1. of this sect. BC: S. A AC: S. B,-45° 37'. By cor. 1. theo. 5. sect. 1. 180-the sum of A and B=C: And 180°-76° 45′ 103° 15′, the angle C. By Gunter's Scale. The first proportion is extended on the line of numbers; and it is no matter whether you extend from the first to the third, or to the second term, since they are all of the same kind: If you extend to the second, that distance applied to the third, will give the fourth; but if you extend from the first to the third, that extent will reach from the second to the fourth. The methods of extending the other proportions, An example in each case of oblique angular trigono: Having thus gone through plane trigonometry, we shall now proceed to apply the same, in determining the measures of inaccessible heights and distances. THE OF HEIGHT. HE instrument of least expense for taking heights, is a quadrant, divided into 90 equal parts or degrees ; and those may be subdivided into halves, quaters, or eights, according to the radius, or size of the instru ment; its construction will be evident by the scheme thereof. (Fig. 78.) B fig. 78. A 70,80 90 From the centre of the quadrant let a plummet be suspended by a horse hair, or a fine silk thread, of such a length that it may vibrate freely, near the edge of its arc by looking along its edge AC, to the top of the object whose height is required; and holding it perpendicular, so that the plumet may neither swing from it, nor lie on it; the degree then cut by the hair, or thread, will be the angle of altitude required. If the quadrant be fixed upon a ball and socket on a three legged staff, and if the stem from the ball be turned into the notch of the socket, so as to bring the instrument into a perpendicular position, the angle of altitude by this means, can be acquired with much A angle of altitude may be also taken by any of the instuments used in surveying; as shall be particularly sewn, when we treat of their descriptions and uses. Most quadrants have a pair of sights fixed on the ege AC, with small circular holes in them; which are seful in taking the sun's altitude, requisite to be known in many astronomical cases; this is effected by etting the sun's ray, which passes thro' the upper sight, fall upon the hole in the lower one; and the degree then cut by the thread, will be the angle of the sun's altitude; but those sights are useless for our present purpose, for looking along the quadrant's edge to the top of the object will be sufficient, as before. PROB. I. fig. 79. B D To find the height of a perpendicular object at one station, which is on an horizontal plane.. Given, A STEEPLE. The angle of altitude, 53 degrees. Distance from the observer to the foot of the steeple, or the base, 85 feet. Height of the instrument, or of the observer, 5 feet. Required, the height of the steeple. The figure is constructed and wrought, in allespects, as case 2. of right angled trigonometry; aly there must be a line drawn parallel to, and beneth AB of 5 feet for the observer's height, to represat the plane upon which the object stands; to which th perpendicular must be continued, and that will be th height of the object. Thus, AB is the base, A the angle of altitude, BC the height of the steeple from the instrument, or fron the observer's eye if he were at the foot of it, DC the height of the steeple above the horizontal surface. Various statings for BC, as in case 2. of right-angled plane trigonometry. 90°-53° 37°=C. 4. S.C: AB:: S.A: BC=112. 8 2. R.: AB : : T.A: BC,=112. 8 3. T.C: AB:: R.: BC,=112. 8 To BC 5. the height of the observer. Their sum is 117. 8 or 113 feet, the height of the |