A spherical triangle is the figure bounded by three arcs (of great circles) on the surface of a sphere. Spherical trigonometry has for its principal problem that of determining the numerical values of the three remaining parts of a spherical triangle when three parts are given. This note is confined to triangles whose sides are each less than a semi-circumference and whose angles are each less than x, or 180°. Any spherical angle is measured by the corre sponding diedral angle, and the latter by the plane angle of intersecting lines (in the faces of the diedral angle) drawn perpendicular to its edge. For convenience the triangle will be supposed to be on a sphere of unit radius; the sides will then be measured by their correspond ing central angles.. In all mathematics, every problem of angular measurement is ultimately that of measuring the angle between intersect ing lines, as all distance problems are reducible to that of determining the distance between two points.
The Right Spherical Triangle. Let 0 be the centre of the unit-sphere containing the tri angle CAB right-angled at A. BP being drawn perpendicular to OA, and similarly PQ to OC, BQ is perpendicular to OC. By aid of the figure one may readily find in order the relationships: (1) cos a = cos b cos c, (2) sin c = sin a sin C, (3) cos C= cot a tan b (4) sin b = tan c cot C, (5) sin b = sin a sin B, (6) cos B = cot a tan c, (7) sin c= tan b cot B, (8) cos a = dot B cot C, (9) cos B = cos b sin C, (10) cos C = cos c sin B.
Sufficient for the solution of any right tri angle, these are less convenient than the equiva lent derived set presented under the title Napier's Circular Parts and These parts are : 90° a, B, 90° C, b and c. By substitution in the preceding formula, these parts are seen to be related as follows: (1') sin (90° a) = cos b cos c (2') sin c = cos (90" a) cos (§0° C), (5') sin b = cos (90° a) cos B), (9') sin (90° B) = cos b cos C), (10') sin (90° C) = cos c cos (90" B) ; (8') sin = tan tan (7') sin c = tan b tan (90° B), (4') sin b = tan c tan C), (6') sin (90° B) = tan (90° a) tan c, (3') sin (90° C) = tan a) tan b.
Arranging the parts in some such cyclical scheme as in Fig. 11, it will be seen that any part being taken as middle part, there are two adjacent parts, and two others that may be and are called opposite. By inspecting the first half of the preceding table it appears that the sine of any middle part is equal to the product of the cosines of its opposite parts, and, from the second half. that the sine of a middle part is the product of the tangents of its adjacent parts. Such are Napier's rules for circular parts, the more readily remembered by vir tue of the assonances appearing in their statement.
In the .solution of right spherical triangles it should be observed : (1) that a is less or greater than 90° according as 90° is not or is intermediate to b and c, these being supposed not equal to 90° ; (2) b or c and the opposite angle are both less or both greater than 90° ; (3) that corresponding to the data, b or c and the opposite angle, there are two solutions.
Quadrantal are so named that have a side equal to 90°. The supple mental polar triangle of a quadrantal is right angled. Hence to solve a quadrantal, solve its polar and subtract its parts each from Oblique Spherical The theory of the oblique spherical triangle is contained in the following numbered equations deducible by help of the figures. From Fig. 12 and analogy it is obvious that (11) sin a:sin b = sin A :sin B, (12) sin c = sin B : sin C, (13) sin c : sin a= sin C: sin A, three propositions constituting the Law of Sines for Spherical Trigonometry. The Law of Cosines, readily found from Fig. 13 (in which CP is perpendicular to the plane AOB, PD, and PE are perpendicular to OA and OB, and i PG and DF are parallel to OF and PE), the triplet : (14) cos a= cos b cos c + sin b sin c cos A, (15) cos b = cos c cos a -I- sin c sin a cos B, (16) cos c = cos a cos b -I- sin a sin b cos C. From these, by passing to the polar triangle (of equal generality with that of Fig. 13), one finds three relations of type (17) cos A = sin B sin C cos a cos B cos C.