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.

Given.

Sought.
B, D, and DB.

Cases.

5

R: COS, B::T, 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: CoS, 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.

CoS, BC: R:: CoT, B: T, BCA. 19. and T, DC: T, BC :: CoS, BCA: CoS, DCA. 27. the sum or difference of the angles BCA, DCA is equal to the angle BCD.

CoS, BC R:: CoT, B: T, BCA. 19. also by 27. CoS, DCA: CoS, 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 quad14.

rant.

Cor. Def. 4. & 6 Pl.

Trig.

The 3d, 5th, 7th, 9th, and 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 *.

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 Napier's rules.

BOD

BIAN

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