40. If a side, or an angle, be 90°, the opposite angle, or side, and the hypothenuse, will be each 90°; and the other side and angle will be of the same number of degrees. 41. If a side be less than the hypothenuse, their sum will be less than 180°; and if it be greater than the hypothenuse, their sum will be greater than 180o. 42. If a side be less than its opposite angle, their sum will be less than 180°; and if it be greater than its opposite angle, their sum will be greater than 180°. 43. The difference of the two oblique angles is less than 90°, and their sum is greater than 90°, and less than 270°. 44. Each of the three sides is either equal to or less than 90°; or two of the sides are greater than 90°, and the third side is less. 45. A right-angled spherical triangle may have 1. One right angle, and two acute or two obtuse angles; 2. Or two right angles, and one acute or one obtuse angle; 3. Or all its angles right angles. L SPHERICAL TRIGONOMETRY. 46. SPHERICAL trigonometry is that science which treats of the analogies of the sides and angles of spherical triangles, and of the methods of computing the quantities of their sides and angles. Let ABC (Plate, fig. 4) be a right-angled spherical triangle, having a right angle at B; produce BC to a, and make Ba=90°; then a is the pole of AB (6). From A as a pole describe the arc abD, meeting AB, AC (produced if necessary) in D and b. The triangle abC thus formed is called the complemental triangle to ABC. For Ca is the complement of CB, Cb is the complement of AC, the angle Cab is measured by BD, which is the complement of AB, and ab is the complement of bD, which measures the angle A. Also, the angle aCb = ACB (22), and the angle abC is a right angle (21). == Because the sine and tangent of an arc or angle are the cosine and cotangent of its complement (23 Pl. Tr.), the sines and tangents of Ca, Cb, Cab, ba, are the cosines and cotangents of CB, AC, AB, and the angle A respectively; and the cosines and cotangents of the former are the sines and tangents of the latter. 47. Let ABC (plate, fig. 5) be a right-angled spherical triangle, having a right angle at B, and let O be the centre of the sphere. Draw OA, OB, OC. If we suppose the plane OAB to coincide with the plane of the paper, we must conceive the plane OBC to be perpendicular to OAB, because the angle at B is right. Draw CF perpendicular to OB, and also Fm perpendicular to OB in the plane OAB; then the angle CFm measures the inclination of the planes OBC, OBA (Def. 4. 2. Sup.). But these two planes are perpendicular to each other; therefore CFm is a right angle. Hence CF being perp. to the lines OFB and Fm, is perp. to the plane OBA (4. 2. Sup.). Draw FE and BD perp. to OA, and join CE; then CE will be perp. to AO. For CFE, CFO being right angles, CE2- EF2CF2 = CO2- OF2 (47. 1), therefore CO2CE2-OF-EF2OE2, therefore CEO is a right angle (48. 1). Now, since EC, EF are perp. to AO, the angle CEF is the inclination of the planes AOC, AOB (Def. 4. 2. Sup.), therefore the angle CEF CAB (23). = By construction CE is the sine, and OE is the cosine of the arc AC, CF is the sine, and OF is the cosine of CB, BD is the sine, and OD is the cosine of AB. Also, the sine, cosine, and tangent of the rectilinear angle CEF are the sine, cosine, and tangent of the spherical angle CAB. PROP. I. 48. In a right angled spherical triangle ABC radius is to the sine of the hypothenuse AC, as the sine of either of the oblique angles A is to the sine of the opposite side BC. Plate, fig. 5. In the rectilinear triangle CFE, having a right angle at F, R: CE :: s. CEF: CF (49 Pl. Tr.). But the angle CEF= spherical angle CAB (23.), and CE is the sine of the arc AC, and CF of BC. Hence R: s. hyp. AC :: s. CAB: s. op. side BC. In the same manner it may be proved that R : s. hyp. AC :: s. ACB: s. op. side AB. For if BC had been made the base, C would have been the angle at the base; consequently the angle ACB and the side AB would have entered the process, instead of the angle CAB and the side BC. Hence, in any proportion involving the angle A or C and the side BC or AB, the terms have similar relations, and may be substituted one for the other. PROP. II. 49. In a right-angled spherical triangle ABC radius is to the sine of one side AB, as the tangent of the angle A adjacent to that side is to the tangent of the other side BC. In the plane triangle CEF, having a right angle at F, R: tan. CEF or CAB:: EF: FC (50 Pl. Tr.), and tan. BC: R: FC OF (41 Pl. Tr.); therefore tan. BC: tan. CAB :: EF : OF (22. PROP. III. 50. In a right-angled spherical triangle ABC radius is to the tangent of either side AB, as the cotangent of the angle C opposite to that side is to the sine of the other side BC. R: s. BC:: tan. ACB : tan. AB (49), and cot. ACB: R:: R: tan. ACB (41 Pl. Tr.); therefore cot. ACB: s. BC :: R: tan. AB (22. 5), or R tan. AB :: cot. ACB: s. BC. PROP. IV. 51. In a right-angled spherical triangle ABC radius is to the cosine of the hypothenuse AC, as the tangent of either of the oblique angles ACB is to the cotangent of the other angle BAC. Plate, fig. 4. From the pole A describe the circle Db, meeting BC in a, and AC in b; then is the triangle Cba the complement of the triangle CBA (46). Therefore in the triangle Cba, right-angled at b, Cb is the complement of AC, the hyp. of the triangle ABC; ba is the comp. of the arc Dẻ, the measure of the angle A; aC, the hyp. of the triangle Cab, is the comp. of BC; the arc BD, the measure of the angle a, is the comp. of AB. Now in the triangle Cab, R: s. Cb :: tan. ¿Cà: tan. ba (49); that is, in the triangle ABC, R: cos. AC :: tan. ACB: cot. BAC. PROP. V. 52. In a right-angled spherical triangle ABC radius is to the cotangent of either of the oblique angles A, as the cotangent of the other angle ACB is to the cosine of the hypothenuse AC. For R: cos. AC :: tan. ACB: cot. A (51), and cot. ACB: R :: R : tan. ACB (41 Pl. Tr.); therefore cot. ACB: cos. AC:: R: cot. A (22. 5), or R: cot. A :: cot. ACB: cos. AC. PROP. VI. 53. In a right-angled spherical triangle ABC radius is to the tangent of the hypothenuse AC, as the cosine of either of the oblique angles A is to the tangent of the side AB adjacent to that angle. The construction of prop. IV. remaining, in the complemental triangle Cba, R: s. ba :: tan. a : tan. Cb (49). But s. ba cos. A, tan. a cot. AB, tan. Cbcot. AC. Hence R: cos. A:: cot. AB: cot. AC. But cot. AB: cot. AC :: tan. AC: tan. AB (42 Pl. Tr.). Hence R cos. A :: tan. AC: tan. AB (11. 5), or R: tan. AC :: cos. A : tan. AB (16. 5). In the same manner it may be demonstrated that R: tan. AC :: cos. ACB: tan. BC. See remark on prop. I. 54. Cor. The tangents of any two arcs AB, AC, are reciprocally proportional to their cotangents. For tan. AB: tan. AC:: cot. AC: cot. AB. PROP. VII. 55. In a right-angled spherical triangle ABC radius is to the cotangent of the hypothenuse AC, as the tangent of either of the sides AB is to the cosine of the angle A adjacent to that side. R: tan. AC:: cos. A : tan. AB (53), and R tan. AC :: cot. AC: R; therefore cos. A: tan. AB :: cot, AC: R (11.5), or R: cot. AC :: tan. AB: cos. A (A. 5). PROP. VIII. 56. In a right-angled spherical triangle ABC radius is to the cosine of either of the sides BC, as the cosine |