SOLUTION of the twelve CASES of oblique-angled SPHERICAL TRIANGLES. GENERAL PROPOSITIO N. IN any oblique angled spherical triangle, of the three fides and three angles, any three being given, the other three may be found. GIVEN. SOUGHT. 5 B, D and DB. R: CoS, B::T, BC: T, BA. 20. and T, D: T, B:: S, BA: S, DA. 26 and BD is the fum or difference of BA DA. 6 7 8 and B. R: Co S, B::T, BC: T, BA. 20. and S, DA: S, BA :: T, B: T, D; and BC, BD D. according as BD is greater or lefs than BA, the angles B, D are of the fame or different affection. 16. CoS, BC: R:: Co T, B: T, BCA. 19. and T, DC: T, BC :: Co S, BCA : BC, DC C. Co S, DCA. 27. the fum or difference of and B. the angles BCA, DCA is equal to the angle BCD. Co S, BC R:: Co T, B: T, BCA. 19. alfo by 27. CoS, DCA: CoS, BCA :: T, BC: T, DC. 27. if DCA and B be B, Cand DC. of the fame affection; that is, (13.) if AD and CA be fimilar, DC will be lefs than a quadrant, 14. and if AD, CA be not of the fame affection, DC is greater than a quadrant. 14. BC. II S, ABXS, BC: Rg :: S, AC+ AD 2 BC, BA, B. X S, AC-AD: Sq, ABC. See Fig. 25. The 3d, 5th, 7th, 9th, 10th, cafes, which are commonly called ambiguous, admit of two folutions, either of which will anfwer the conditions required; for, in thefe cafes, the meafure of the angle or fide fought, may be either greater or less than a quadrant, and the two folutions will be fupplements to each other. (cor. to def. 4. 6. Pl. Tr.) If from any of the angles of an oblique-angled fpherical triangle, a perpendicular arch be drawn upon the oppofite fide, moft of the cafes of oblique angled triangles may be refolved by means of Napier's rules. FINI 5. |