Spb. Trig. PRO P. XXIII. by the fines of two sides, is to the rectangle contained by the fines of half the sum and half the difference of the base and the excess of the fides, fo is the square of the radius to the square of the fine of half the angle oppolite to the base. Let ABC be a spherical triangle, of which the fide AC is greater than AB; a d make AD equal to AC, and AE to AB; the rectangle contained by the fines of BA, AC is to the rectangle contained by the fine of half the sum of CB, BD, and the fine of half the difference of the same CB, BD, as the square of the radius to the square of the fine of half the angle BAC. Describe through C, D and B, E the great circles CFD, BGE, and bileet the angle BAC by the great circle AGF, and join BC, BD, DC, BE ; and let O be the centre of the sphere, and join OF, OG, meeting CD, BE in H, K, and join HK, and a 11. 1: draw a BL perpendicular to CD. And because in the spherical triangles CAF, DAF, the two fides CA, AF, are equal to the two ĎA, AF, and the angle CAF is equal to DAF; therefore the b 6. S. T. bale CF is equal to DF 6, and G the angle CFA to AFD; there fore the plane AGF is perpenc 3. Def. dicular c to the plane CFD: and CD is perpendicular to their which bisects the arch CFD, dCor.39.3. bisects CD at right angles d; e4. Def... therefore CD is perpendicu iar to the plane AGF, and DHK is same reason, BE is perpendicular to the plane AGF; therefore f 6.11. BK is parallel to DH': and BL, HK are parallel, because BLH, LHK are right angles; therefore BL HK is a parallelogram, ḥ 34. I. and BK is equal to LH 6: and because the sides CB, BD of the triangle CBD are placed in equal circles CB, ABD, the rect angle CH, HL, or CH, BK, is equal to the rectangle contained k LEM. by half the sum and half the difference of the arches CB, BDk; but in the right angled spherical triangle AGB, the fine of AB I 20. S. T. is to BK the line of BG, as the radius to the fine of BAG ! half of BAC; and in the right angled triangle ACF, the fine of KI S. T. OUI S. T. of AC is to CH the fine of CF, as the radius to the fine of FAC Spb. Trig. m 22. 6. PROP. XXIV. by the fines of two fides, is to the rectangle con- Let ABC be a spherical triangle; the rectangle contained by Produce BA, CA to D, E, and make AD equal to AC, and be demon- K В in A А G Sph, Trig. in H, the rectangle CH, HL is equal to the rectangle contained by the fines of half the sum, and half the difference of the arches BD, BC, and the sum of BD, BC is the sum of the three fides, and the excess of the half of BD above half of BC is the excess of half the sum of the three fides above BC; therefore the rectangle CH, HL is equal to the rectangle contained by the fines of half the sum of the three sides, and of its excess above BC: also, because the angles CAB, CAD are equal to two right angles, the half of CAB and the half of CAD are together equal to a right angle; therefore the fine of half CAD is the cofine of half CAB: But the rectangle contained by the fines of BA, AC, is to the rectangle CH, HL as the square of the radius to the square of the fine of half the angle CAD; therefore the rectangle contained by the fines of BA, AC, is to the rectangle contained by the fines of half the sum of the three sides, and of its excess above the base BC, as the square of the radius to the square of the cofine of half the angle BAC Cor. Because the rectangle contained by the fines of half the sum of the three fides, and of its excels above BC, is to the rectangle contained by the fines of BA, AC as the square of the cotine of half BAG to that of the radius ; and that the rectangle contained by the fines of BA, AC, is to the rectangle contained by the fines of half the sum, and half the difference of the base and the excess of the sides, as the square of the radius to the $ 22. 5. square of the fine of half BAC; therefore, by equality, the rectangle contained by the fines of half the sum of the three sides, and of its excess above the base, is to the rectangle contained by the lines of half the sum, and half the difference of the base, and the excess of the fides, as the square of the cofine of half the angle BAC to the square of its fine; that is, as the square of the radius to the square of the targent of half the b 2. Cor. angle BACO. Def. 10. P. T. SOLUTION of the Cases of OBLIQUE-ANGLED SPHERICAL TRIANGLES. GENERAL PROPOSITION. three angles, any three being given, the other three may be found. The The several cases of this proposition may be resolved by the spb. Trig. help of the three general proportions, together with the 22d, and any one of the subsequent propositions ; as in the following Table, in which ABC is any spherical triangle; and the perpen- The cases referred to, are those of the preceding Table. I Cases. Given. 1 Songht. Solution. B oppofite opposite to C. If the sum of BA, AC be less than 180°, and AB less ACB is ambiguous. B opposite side. (case 2.) and cos. AB : cos. AC: : cos. BD : cos. DC. trary affection. their fum is CB; if CD be Cases, Sph. Trig. Cafes. Given. | Souq: B oppofite contained (case 3.), and tan. AC: tan. If В be acute, DAC and AC are of the same affection, otherwise they are of different affection. If DAC be not less than BAD, their sum is BAC: if DAC be less than BAD, but their fum not less than 180°, their difference is BAC. In other cases BAC is ambiguous. B, C and AB, AC the fide Sin. C : fin. B :: fin. AB : fin. two angles opposite to AC. If the sum of B and C 4 and the side B. be less than 180°, and B less opposite to than C, AC is acute : or if the one of them fum of B and C be greater C. than 180°, and B greater than C, AC is obtufe. In other cafes, AC is ambiguous. B, C and AB, A the third R: cos. AB::tan. B: cot. BAD, two angles angle. (case 3 ), and cos. B : cos. C:: 5 and the side lìn. BAD: fin. DAC, (p. 3.), oppofite to which is less than BAD, if one of them B, C be of diff. affection, or C. less than the supplement of BAD, if B and C be of the same affection : In other cases it is ambiguous. When B and C are of the same affec. tion, BAСis the sum of BAD, DAC, otherwise it is their difference. Cases. |