By Cor. 1. Theo. 5. Sect. 1. 180-the Sum of A and B, C A 31°. 081 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 fecond Term, fince they are all of the fame Kind: If you extend to the fecond, 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 fecond to the fourth. The Methods of extending the other Proportions have been already fully treated of. An Example in each Cafe of Oblique Angular Trigonometry. SAC 290) A 1. Given, C 69°. 30' B required. AB 350) BC (C 24. 20, AB 2. Given, B 128o. 301 LAC 3246 BC required. AC 6 A 3. Given, C 124°. 30B required. BC 4. 5 AB Plate V. 4. Given, AB 46, A BC 52 C Having thus gone through Plane Trigonometry, we shall now proceed to apply the fame, in determining the Measures of inacceffible Heights and Distances. And first, Of Of HEIGHTS. Plate V. T HE Inftrument of leaft Expence for taking Heights, is a Quadrant, divided into go equal 90 Parts or Degrees; and those may be fubdivided into Halves, Quarters, or Eighths, according to the Radius, or Size of the Inftrument: its Conftruction will be evident by the Scheme thereof, (Fig. 18.) From the Center of the Quadrant let a Plummet be fufpended by a Horfe Hair, or a fine Silk Thread; of fuch 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 whofe Height is required; and holding it perpendicular, fo that the Plummet may neither fwing 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, fo as to bring the Inftrument into a perpendicular Pofition; the Angle of Altitude by this Means, can be acquired with much greater Certainty, Plate V. An Angle of Altitude may be alfo taken by any of the Inftruments used in Surveying; as shall be particularly fhewn, when we treat of their Defcriptions and Ufes. Moft Quadrants have a Pair of Sights fixed on the Edge AC, with fmall circular Holes in them; which are useful in taking the Sun's Altitude; requifite to be known in many aftronomical Cafes; this is effected by letting the Sun's Ray, which paffes 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 prefent Purpose, for looking along the Quadrant's Edge to the Top of the Object will be fufficient, as before. PRO B. I. Plate V. Fig. 19. 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. Given, the Steeple, or the Bafe, 85 Feet. Height of the Inftruments, or of the Obferver, 5 Feet. 1 Required the Height of the Steeple. The Figure is conftructed and wrought, in all Refpects as Cafe 2 of right angled Trigonometry; only there must be a Line drawn parallel to, and beneath Plate V. Fig. 19. beneath AB of 5 Feet for the Obferver's Height, to represent the Plane upon which the Object ftands; to which the Perpendicular must be continued, and that will be the Height of the Object. Thus, AB is the Bafe, A the Angle of Altitude, BC the Height of the Steeple from the Inftrument, or from the Obferver'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 Cafe 2 of rightangled Plane Trigonometry, Their Sum is 117.8 or 118 Feet, the Height of the Steeple required. PROB. |