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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.

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Given Sought 5 B, D, and DB. R : Co-S, B:-T, B€:T, BA. 20. and BC.

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

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Given Sought

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See Fig. 7. 12 A, B, C. The In the triangles DEF, DE, EF, FD Fig. 7.

sides. are respectively the supplements of the

measures of the given angles B, A, 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 mea.

sures 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.

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