SOLUTION OF THE TWELVE CASES OF OBLIQUE ANGLED SPHERICAL TRIANGLES. GENERAL PROPOSITION. In an oblique angled spherical triangle, of the three sides and three angles, any three being given, the other three may be found. 1 Given. Sought. B, D,&C. a side op- 27. 2 B, C, and D. BC, two angles & the side between them. 3 BC, CD, BD. and B. 4 BC, DB, CD. and B. Co-S, BC R: Co-T, B: T, BCA. 19. Likewise by 24. Co-S, B: S, BCA:: Co-S, D: S, DCA; wherefore BCD is the sum or difference of the angles DCA, BCA according as the perpendicular CA falls within or without the triangle BCD; that is (16. of this,) according as the angles B, D are of the same or different affection. Co-S, BC: R:: Co-T, B: T, BCA. 19. and also by 24. S, BCA : S, DCA: CoS, B: Co-S, D; and according as the angle BCA is less or greater than BCD, the perpendicular CA falls within or without the triangle BCD; and therefore (16. of this,) the angles B, D will be of the same or different affection. R: Co-S, BT, BC: T, BA. 20. and Co-S, BC: Co-S, BA: Co-S, DC: CoS, DA. 25. and BD is the sum or differjence of BA, DA. R: Co-S, BT, BC: T, BA. 20. and Co-S, BA: Co-S, BC :: Co-S, DA: CoS, DC. 25. and according as DA, AC are of the same or different affection, DC will be less or greater than a quadrant. 14. 5 Given. Sought. B, D, and DB. 6 BC, BDD. and B. 7 BC, DCC. and B. 8 B, C and DC. BC. 9 BC, CD D. and B. 10B, D and DC. BC. 11 BC, BA, B. AC. Fig. 25. R: Co-S, BT, BC: T, BA. 20. and T, D: T, B:: S, BA: S, DA. 26. and BD is the sum or difference of BA, DA. R: Co-S, B: T, BC: T, BA. 20. and S, DA: S, BA : : T, B : T, D; and according as BD is greater or less than BA, the angles B, D are of the same or different affection. 16. Co-S, BC: R:: Co-T, B: T, BCA. 19. and T, DC: T, BC :: Co-S, BCA: Co-S, DCA. 27. the sum or difference of the angles BCA, DCA is equal to the angle BCD. Co-S, BC: R:: Co-T, B: T, BCA. 19. also by 27. Co-S, CDA : Co-S, BCA :: T, BC: T, DC. 27. if DCA and B be of the same affection; that is (13.) if AD and CA be similar, DC will be less than a quadrant. 14. and if AD, CA be not of the same affection, DC is greater than a quadrant. 14. S, CD: S, B:: S, BC: S, D. S, DS, BC: S, B: S, DC. S, AB XS, BC: Rq:: S, AC+AD × 2 S, AC-AD: Sq. ABC. See Fig. 25. AD being the difference of the sides BC, BA. Given. Sought. The 12 A, B, C. See Fig. 7. In the triangle DEF, DE, EF, FD are respectively the supplements of the mea sures of the given angles B, A, C, in the triangle BAC; the sides of the triangle DEF are therefore given, and by the preceding case the angles D, E, F may be found, and the sides BC, BA, AC are the supplements of the measures of these angles. The 3d, 5th, 7th, 9th, 10th cases, which are commonly called ambiguous, admit of two solutions, either of which will answer the conditions required; for, in these cases, the measure of the angle or side sought, may be either greater or less than a quadrant, and the two solutions will be supplements to each other (Cor. to def. 4. 6. Pl. Tr.) If from any of the angles of an oblique angled spherical triangle, a perpendicular arch be drawn upon the opposite side, most of the cases of oblique angled triangles may be resolved by means of Napier's rules. FINIS. |