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SOLUTION OF THE TWELVE CASES OF OBLIQUE ANGLED SPHE

RICAL 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.

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3 BC, CD, BD R: Co-S, B:: T, BC • T, BA. 20. und B.

and Co-S, BC : Co-S, BA :: Co-S, DC 1: Co-S, DA. 25. and BD is the sum or difference of BA, DA.

BC, DB, CD R: Co-S, B:: T, BC : T, BA. 20. and B.

and Co-S, BA : Co-S, BC :: Co-S, DA : Co-S, DC. 25. and according as DA, AC are of the same or different affecion DC will be less or greater than a juadrant. 14.

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Given.

So't

5

B, D and DB R: : Co-S, B::T, BC : T, BA. 20. BC.

and T, D: T, B::S, BA : S, DA. 26. and BD is the sum or difference of BA DA.

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B, C and DC. Co-S, BC : R :: Co-T, B : T, BCA. BC.

19. alsa by 27. Co-S, DCA : 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 siinilar, 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.

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AD being the difference of the sides BC,

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Given

So't

12 A, B, C.

Fig. 7.

See Fig. 7. The In the triangles DEF, DE, EF, FD sides. are respectively the supplements of the

measures of the given angles B, A, C in the triangle BAC; the sides of the riangle DEF are therefore given, and by the preceding case the angles D, E, IF 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.

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