the great circle CAD (297); and ABC is the required triangle. Measure AB, BC, AC, as before. AC 117° 31', AB = 126° 42', and BC= 28° 28'. 308. Case V.-Given the side BC 140° 53′, and angle C 105° 53', to construct the triangle. Make BGC140° 53′; draw the diameter BAD, and through C describe the circle CAE, making angle FCE 74° 7', the supplement of 105° 53′; and ABC is the required triangle. = D A Measure AB, AC, and angle A. = E H 309. Case VI.-Given AB 40° 25', and angle C 44° 56′, to construct the triangle. B Describe the circle CAD, making angle ACB = 44° 56′ ; and about G, as a pole, describe the small circle AA' at a distance from G=49° 35', the complement of AB; then through A and A' draw the diameters BH, B'H', and ABC, A'B'C, E are two triangles, constructed from the same data, that is, having their H sides AB, A'B', of the given magnitude, and the angle C common. D Measure AC, BC, and angle A; and also A'C and B'C, and angle A'. AC: = 66° 38', BC= 58° 36′, and A = 68° 25′; and A'C=113° 22′, B’C = 121° 24′, and A'=111° 35'; the three latter parts are the supplements of the three former. PROJECTION OF CASES OF OBLIQUE-ANGLED SPHERICAL B F TRIGONOMETRY. 310. Case I.-Given the side AB = 132° 11', BC = 143° 46', and AC 67° 24', to construct the triangle. = D Make ADB = 132° 11′; and about A, as a pole, describe the small circle DCE at a distance AD of 67° 24′; and about B', the small circle FCG at a distance B'F 36° 14', the supplement of BC; then through A, C, and B, C, de- A scribe the great circles ACA', BCB'; and ABC is the required triangle. By measurement, angle A = 143° 18′, B = 111° 4′, and C = 131° 30'. Make AC 44° 14'; make angle ACB = 36° 45′ (301); draw the small circle IBH about C, as a pole, at a distance = 84° 14'; and through the points A, B, describe the circle ABK (296); and ABC is the required triangle. B 114° 30', B 311. Case II.-Given the angle A 83° 12', and C = 123° 20′, to construct the triangle. Describe the great circle ACA', making angle BAC = 114° 30'; then about G, as a pole, describe a small circle PP'R at a distance from it = 83° 12′ (298); and about the remote pole of ACA', describe the small circle P'PS at a distance from it 56° 40' the supplement of 123° 20'; then about either of the points of intersection P, P', as P, describe the great circle B'CB; and ABC is the required triangle. It will be found by measurement that the side BC = 125° 24'. By measurement, AB = 51° 6′, angle A 130° 5', and B=30° 26'. H E B A K A E 312. Case III.-Given the side AC 44° 14′, BC = 84° 14', and angle C = 36° 45', to construct the triangle. F D = B 313. Case IV.-Given angle A = 130° 5', B=30° 26', and the side AB = 51° 6', to construct the triangle. Make BC= 40°, and angle ABC 30° 28' (301); about the pole of BAD, and at a distance = 31° 34', describe a small circle PP'G, cutting the diameter PP', which is perpendicular to CK in P and P'; about P, as a pole, describe the great circle CAK, and ABC is the required triangle. › E The great circle described D Make AC A 314. Case V.- Given the side AC 80° 19', BC = 63° 50′, and angle A = 51° 30′, to construct the triangle. 80° 19', and angle BAC = 51° 30′ (301); about C, as a pole, describe B'B at a distance = 63° 50′; and through B and C describe the circle EBC; or through B' and C describe EB'C; and either ABC or AB'C is the re- F quired triangle. E D D By measurement, in the triangle ABC, AB 120° 46', angle B = 59° 16', and angle C131° 32'. In the triangle AB'C, angle B' is the supplement of B = 180° 59° 16′ = 120° 44′; but AB' is not the supplement of AB, nor angle ACB' of ACB. It is found that AB′ = 28° 34', and ACB′ = 24° 36′. E H H K B 315. Case VI.-Given angle A = 31° 34′, B = 30° 28′, and the side BC= 40°, to construct the triangle. D F с G about P' as a pole would cut the circle BAD at the given angle; but it would be an exterior angle of the triangle, to which the side BC belongs. But if A were B, there would then be two triangles; in this case, the two poles, P and P', would lie on the same side of CK. By measurement, AC 130° 3'. 38° 30', AB = 70°, and C = SPHERICAL TRIGONOMETRY. 316. Spherical Trigonometry treats of the methods of computing the sides and angles of spherical triangles. DEFINITIONS. 317. A sphere is a solid, every point in whose surface is equidistant from a certain point within it. This point is called the centre. A sphere may be conceived to be formed by the revolution of a semicircle about its diameter as an axis. 318. A line drawn from the centre to the surface of a sphere is called its radius; and a line passing through the centre of the sphere, and terminated at both extremities by its surface, is called a diameter. 319. Circles whose planes pass through the centre of the sphere are called great circles; and all others small circles. 320. A line limited by the spherical surface, perpendicular to the plane of a circle of the sphere, and passing through the centre of the circle, is called the axis of that circle; and the extremities of the axis are the poles of the circle. 321. The distance of two points on the surface of the sphere means the arc of a great circle intercepted between them. 322. A spherical angle is an angle at a point on the surface of the sphere, formed by arcs of two great circles passing through the point, and is measured by the inclination of the planes of the circles. The angle formed by the planes is the same as that made by two tangents to the circles at the angular point. 323. A spherical triangle is a triangular figure formed on the spherical surface by arcs of three great circles, each of which is less than a semicircle. When one of the sides of a spherical triangle is a quadrant, it is called a quadrantal triangle. 324. The sides of a spherical triangle being arcs of great circles of the same sphere, their lengths are proportional to the number of degrees contained in them, and hence the sides of spherical triangles are usually estimated by the number of degrees they contain. The definitions of trigonometrical lines given in plane trigonometry (Geom. vol. I.) are employed in reference to the sides and angles of spherical triangles. 325. A spherical angle is measured by that arc of a great circle, whose pole is the angular point, which is intercepted by the sides of the angle. M B Thus, the spherical angle ABC, which is the same as the angle contained by the planes ABF, CBF, of the two arcs AB, BC, that contain the angle, is measured by the arc AC of a great circle ACD, whose pole is the angular point B. A N E The spherical angle ABC is also measured by the angle MBN contained by the tangents MB, NB, to the arcs BA, BC. For angle MBN = angle AEC, and AEC is measured by AC. F P D 326. Two arcs are said to be of the same species, affection, or kind, when both are less or both greater than a quadrant; and consequently the same term is applied to angles in reference to a right angle. The species of the sides and angles of spherical triangles can generally be easily determined by means of the algebraical signs of their sines, tangents, &c. (article 327.) The arc being estimated from A in the direction ABCD... |