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19. The principal part of a treatise on Spherical Trigonometry consists of theorems relating to spherical triangles; it is therefore necessary to obtain an accurate conception of a spherical triangle and its parts.
It will be seen that what are called sides of a spherical triangle are really arcs of great circles, and these arcs are proportional to the three plane angles which form the solid angle corresponding to the spherical triangle. Thus, in the figure of the preceding Article, the arc AB forms one side of the spherical triangle ABC, and the plane angle AOB is measured by the fraction
radius OA; and thus the arc AB is proportional to the angle AOB so long as we keep to the same sphere.
The angles of a spherical triangle are the inclinations of the plane faces which form the solid angle; for since Ab and Ac are both perpendicular to 0 A, the angle bAc is the angle of inclination of the planes OAB and OAC.
20. The letters A, B, C are generally used to denote the angles of a spherical triangle, and the letters a, b, c are used to denote the sides. As in the case of plane triangles, A, B, and C may be used to denote the numerical values of the angles expressed in terms of any unit, provided we understand distinctly what the unit is. Thus, if the angle C be a right angle, we may say that C = 90', or that C according as we adopt for the unit a de
2' gree or the angle subtended at the centre by an arc equal to the radius. So also, as the sides of a spherical triangle are proportional to the angles subtended at the centre of the sphere, we may use a, b, c to denote the numerical values of those angles in terms of any unit. We shall usually suppose both the angles and sides of a spherical triangle expressed in circular measure. (Plane Trigonometry, Art. 20.)
21. In future, unless the contrary be distinctly stated, any arc drawn on the surface of a sphere will be supposed to be an arc of a great circle.
22. In spherical triangles each side is restricted to be less than a semicircle; this is of course a convention, and it is adopted because it is found convenient.
Thus, in the figure, the arc ADEB is greater than a semicircumference, and we might, if we pleased, consider ADEB, AC, and BC as forming a triangle, having its angular points at A, B, and C. But we agree to exclude such triangles from our consideration; and the triangle having its angular points at A, B, and C, will be understood to be that formed by AFB, BC, and CA.
23. From the restriction of the preceding Article it will follow that any angle of a spherical triangle is less than two right angles.
For suppose a triangle formed by BC, CA, and BEDA, having the angle BCA greater than two right angles. Then suppose D to denote the point at which the arc BC, if produced, will meet AE; then BED is a semicircle by Art. 10, and therefore BEA is greater than a semicircle; thus the proposed triangle is not one of those which we consider,
III. SPHERICAL GEOMETRY.
24. The relations between the sides and angles of a Spherical Triangle, which are investigated in treatises on Spherical Trigonometry, are chiefly such as involve the Trigonometrical Functions of the sides and angles. Before proceeding to these, however, we shall collect, under the head of Spherical Geometry, some theorems which involve the sides and angles themselves, and not their trigonometrical ratios.
25. Polar triangle. Let ABC be any spherical triangle, and let the points A', B', C' be those poles of the arcs BC, CA, AB
respectively which lie on the same sides of them as the opposite angles A, B, C; then the triangle A'B'C' is said to be the polar triangle of the triangle ABC.
Since there are two poles for each side of a spherical triangle, eight triangles can be formed having for their angular points poles of the sides of the given triangle ; but there is only one triangle in which these poles A', B', C' lie towards the sa e parts with the corresponding angles A, B, C; and this is the triangle which is known under the name of the polar triangle.
The triangle ABC is called the primitive triangle with respect to the triangle A'B'C'.
26. If one triangle be the polar triangle of another, the latter will be the polar triangle of the former.
Let ABC be any triangle, A'B'C' the polar triangle : then ABC will be the polar triangle of A'B'C'.
For since B' is a pole of AC, the arc AB' is a quadrant, and since C" is a pole of BA, the arc AC' is a quadrant (Art. 7); therefore A is a pole of B'C' (Art. 11). Also A and A' are on the same side of B'C'; for A and A' are by hypothesis on the same side of BC, therefore A'A is less than a quadrant; and since A is a pole of B'C', and AA' is less than a quadrant, A and A' are on the same side of B'C'.
Similarly it may be shewn that B is a pole of 'A', and that B and B' are on the same side of C'A'; also that C is a pole of A'B', and that C and Care on the same side of A'B'. Thus ABC is the polar triangle of A'B'C'.
27. The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.
For let the arc B'C', produced if necessary, meet the arcs AB, AC, produced if necessary, at the points D and E respectively; then since A is a pole of B'C', the spherical angle A is measured by the arc DE (Art. 12). But B'E and C'D are each quadrants; therefore DE and B'C' are together equal to a semicircle; that is, the angle subtended by B'C' at the centre of the sphere is the B'EacB
B's 1,20), HEC
supplement of the angle A. This we may express for shortness thus; B'C' is the supplement of A. Similarly it may be shewn that C'A' is the supplement of B, and A'B' the supplement of C.
And since ABC is the polar triangle of A'B'C', it follows that BC, CA, AB are respectively the supplements of A', B', C'; that is, A', B, C are respectively the supplements of BC, CA, AB.
From these properties a primitive triangle and its polar triangle are sometimes called supplemental triangles.
Thus, if A, B, C, a, b, c denote respectively the angles and the sides of a spherical triangle, all expressed in circular measure, and A', B', C', a', b', d' those of the polar triangle, we have
A'=T-a, B'= 7-6, C'=T-C,
28. The preceding result is of great importance; for if any general theorem be demonstrated with respect to the sides and the angles of any spherical triangle it holds of course for the polar triangle also. Thus
such theorem will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles. We shall see several examples of this principle in the next Chapter.
29. Any two sides of a spherical triangle are together greater than the third side. (See the figure of Art. 18.)
For any two of the three plane angles which form the solid angle at 0 are together greater than the third (Euclid, xi. 20). Therefore any two of the arcs AB, BC, CA, are together greater than the third.
From this proposition it is obvious that any side of a spherical triangle is greater than the difference of the other two.
30. The sum of the three sides of a spherical triangle is less than the circumference of a great circle. (See the figure of Art. 18.)