Spherical Trigonometry

SPHERICAL TRIGONOMETRY Figures composed of lines and points in a plane are accessible to computation through triangles, and these in turn through right triangles. In the totality of lines and planes radiating from a point in ordinary space of three dimensions the useful instrument for computation is that consisting of three lines or rays—half lines starting from a common point—and the three plane angles which are bounded each by two of these three lines. It is usually agreed to restrict these angles to be not over i8o° each, and to fix the same limit for the dihedral angles whose edges are the three lines first mentioned. The plane angles are called face angles, angles formed by two planes at any edge are dihedral angles, and the complete figure of six parts is a trihedral or solid angle. If the point (or vertex) lies at the centre of a sphere, the edges and faces cut the spherical surface in points and arcs of great circles that constitute a spherical triangle. The sides and angles of this spherical triangle have the same measures (in de grees or radians) as the plane and dihedral angles of the tri hedral angle. The capitals A, B, C may be used for the positions and for the measures of the angles at the vertices, and a, b, c for the measures of the three plane angles (face angles) of the arcs of the triangle.

Right Trihedral Angles or Spherical Triangles.

If two planes of the trihedral angle are perpendicular, e.g., if the angle C of the intercepted spherical triangle is 9o°, the latter is a right spherical triangle and subtends a right trihedral angle. If a side (a face angle) is 9o°, the triangle and its trihedral angle are called a quadrantal triangle. In contrast to the plane triangle, two or even three angles of the triangle may be right angles ; in the one case two and in the other case three of the sides will then be quadrants or quarter circles of the sphere, and the tri angles will be biquadrantal, or else equilateral (trirectangular or triquandrantal).

Let a trihedral angle at 0, the centre, intercept on a sphere of radius R a right spherical tri angle ABC, right angled at C. Then the planes AOC and BOC will be perpendicular along the edge OC. (Fig. 3.) To measure the angle whose edge is OA, let a plane perpendicular to OA cut OB in B', and OC in C'. There

is then formed a figure in which all four triangles are right tri angles, viz., OAB', OAC', OB'C' and AB'C'. In three of these, the ratios of edges are trigonometric functions of c, b, a; in the other, they are functions of the angle A or of the dihedral angle OA. We may write from inspection these three relations: in cycle. Each Napier's part is adjacent to two in the cycle, the one next preceding and the one immediately following it in the cycle. The rules read: I) The sine of any middle part is the product of the tangents of the adjacent parts, 2) The sine of any middle part is the product of the cosines of the opposite parts.

Here opposite means non-adjacent. When any three Napier's parts are specified, of the five excluding the right angle C, either they are consecutive in the cycle, and then the first and third are called adjacent to the second; or only two are consecutive, in which case they are both called opposite to the third. These formulae solve most of the problems arising in geodetic navi gation, and the problems of celestial astronomy concerning an altitude and an azimuth.

Polar Triangles : Quadrantal Triangles.

There is a unique relation between two trihedral angles, each polar to the other. The interior and exterior spaces of a trihedral angle must be defined. Given three half-lines OA, OB, OC, and the plane angles, each less than 18o°, which they bound, produce indefinitely AO, BO, and CO, to and C1. Through 0 draw any other line not in any face plane (fig. 4). If from but not from D, a point can move continuously to without crossing the interior of any face angle or any edge, then D is an interior point and is an exterior point. If both and D can be so moved, then both are exterior to the trihedral angle 0—ABC. Take D an interior point, and from it draw three half-lines in tersecting perpendicularly the three faces of the first trihedral angle. On the half-line perpendicular to AOB mark any point C2, and similarly mark two other points A2, B2. The edges DA2, DB2, define a second trihedral angle called polar to the first.