BD and CD; and the figure is constructed. For, as was proved in the 6th example, and whence, AD: BD :: AE : EB :: 7 : 6; AD : CD :: AG : GC :: 7 : 5; Calculation. Join DF, DH and FH; then, in the triangle ABC, we have all the sides to find the angle BAC; then, in the triangle AFH, we have the sides AF, AH, and the included angle, to find the angle AFH and side FH; in the triangle FDH, the three sides are then known to find the angle DFH; whence the angle AFD becomes known: then, in the triangle AFD, we have the sides AF, FD, and the contained angle, to find the angle FAD and the side AD; from which BD and CD are found from the given ratios. Results: BAD = 25° 59′ 8′′; CAD = 25° 27' 15"; AD = 283.688; BD = 243.161; CD = 202.635. [The following ingenious construction of this problem, which admits of a simpler calculation than that already given, has been kindly furnished the author by SAMUEL ALSOP, Principal of Friends' Select School, Philadelphia.] the same or any other scale, lay down AB = 464, and complete the triangle ABC. Draw BD parallel to EF, cutting AF in D, which will be the point required. For, join CD; draw EH parallel to BC; and join FH. Then, since AEH is similar to FEG, being both similar to BAC; therefore (6.6), AEG and HEF are similar; and But AB: BC :: AE : EH :: AG : FH. AB BC: AG: 5; : therefore, FH = 5. Consequently, AD: BD: CD :: AF : FE : FH :: 7 : 6 : 5. = Calculation. In the triangle ABC, with the given sides, find the angle BAC FGE; also find AG, GF and GE. From the three sides of the triangle AGF, find the angle AGF; whence AGE becomes known. In the triangle AGE, find AE; then AE EF: AB: BD; from which AD and CD are found from the given ratios. * SECTION III. SPHERICAL TRIGONOMETRY. ARTICLE 45. The business of Spherical Trigonometry is, to investigate the properties of triangles formed on the surface of a sphere, by the arcs of circles whose planes pass through the centre. As the diameter of a circle is the greatest straight line in it (15.3), so the diameter of a sphere is necessarily the greatest straight line in it. Hence, when a plane passes through the centre of the sphere, the diameter of the circle which is formed by the section of this plane and the sperical surface, is greater than any other line in the sphere which is not a diameter. A plane cutting the sphere, but not passing through its centre, forms, by its section with the spherical surface, a circle whose diameter is less than the diameter of the sphere. That the section is a circle, is readily inferred from 14.3; and that the diameter of that circle is less than the diameter of the sphere, is plain from 15.3. Definition 1. Those circles whose planes pass through the centre of the sphere, are called great circles; but circles whose planes do not pass through the centre of the sphere, are called less circles. Corollary 1. The diameter of every great circle is also a diameter of the sphere. Cor. 2. The common section of the planes of two great circles, is a diameter to each of those circles. (78) Cor. 3. Every great circle in the sphere divides every other great circle into two equal parts. Def. 2. The axis of a circle is the right line which passes through its centre, and is at right angles to the plane of the circle; and the poles of a circle are the points where its axis meets the surface of the sphere. Def. 3. A spherical angle, or the angle formed by two great circles, is the inclination of their planes. Cor. When two great circles are at right angles to each other, each of them passes through the poles of the other; and if they pass through the poles of each other, they are at right angles. Also, when the plane of a great circle is at right angles to the plane of a less one, the former circle passes through the poles of the latter. circle passes through the centre of the angles to the plane of its own circle. For the axis of every sphere, and is at right Def. 4. A spherical triangle is formed by the arcs of three great circles, each of which cuts the other two, but in such manner that each of the arcs composing the triangle is less than a semicircle. B F E Def. 5. If AD and DF, two quadrants of great circles, are placed at right angles to each other; and through the points A, F, two other great circles, AE, FB, are described, cutting each other in C; the triangles ABC, FEC, are called complemental triangles. ART. 46. The arc of a great circle, intercepted between another great circle and its pole, is a quadrant. Let AEBF be a great circle, whose centre is C, and axis DCG, its poles being D and G; DAGB another great circle passing through the axis DCG; these great circles are at right angles to each other (17.2 sup.), and CA their common section at right angles to CD; hence the arcs AD, BD, AG and BG, are quadrants, ART. 47. The angle made by two great circles is measured by the arc intercepted between them, at the distance of 90° from the angular point. plane ACB, the other in the plane ADB; and let the plane |