By cor. 1. theo. 5. sect. 1. 180-the sum of A and B-C. A 31°. 08' B 45. 37 180-76. 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 have been already fully treated of. Plate V. AB 46) A 4. Given, AC 92 B required. BC 52 C Having thus gone through plane trigonometry, we shall now proceed to apply the same, in determining the measures of inaccessible heights and distances. And first, OF HEIGHTS. Plate V. THE HE instrument of least expence for taking heights, is a quadrant, divided into ninety equal parts or degrees; and those may be subdivided into halves, quarters, or eights, according to the radius, or size of the instrument: its construction will be evident by the scheme thereof. (Fig. 18.) 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 the edge AC, to the top of the object whose height is required; and holding it perpendicular, so that the plummet 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 the 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 greater certainty. An angle of altitude may be also taken by any of the instruments used in surveying as shall be particularly shewn, when we treat of their descriptions and uses. Most quadrants have a pair of sights fixed on the edge AC, with small circular holes in them; which are useful in taking the sun's altitude, requisite to be known in many astronomical cases; this is effected by letting the sun's ray, which passes through 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. Plate V. fig. 19. PROBLEM I. To find the height of a perpendicular object at one station, which is on an horizontal plane. A steeple. The angle of altitude, 53 degrees. Distance from the observer to the foot Given, 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 all respects, as case 2. of right-angled trigonometry; only there must be a line drawn parallel to, and Plate V. fig. 19. beneath AB of 5 feet for the observer's height, to represent the plane upon which the object stands ; to which the perpendicular must be continued, and that will be the height of the object. Thus, AB is the base, A the angle of altitude, BC the height of the steeple from the instrument, or from 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 rightangled plane trigonometry. Their sum is 117. 8 or 118 feet, the height of the steeple required. |