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Let ABC be a spherical triangle, of which the two sides are AB, BC, and base" AC, and let the less fide BA be produced, so that BD shall be equal to BC: AD therefore is the excess of BC, BA; and it is to be shown, that the rectangle contained by the fines of BC, BA is to the square of the radius, as the rectangle contained by the fine of half the sum of AC, AD, and the line of half the difference of the fame AC, AD to the square of the Gne of half the angle ABC, opposite to the base AC.
Since by prop. 28. the rectangle contained by the fines of the fides BC, BA is to the square of the radius, as the excess of the verled Gnes of the base AC and AD, to the versed sine of the angle B; that is, (1.6.) as the rectangle contained by half the radius, and that excess, to the rectangle contained by balf the radius, and the versed line of B; therefore (29. 30. of this), the rectangle contained by the fines of the sides BC, BA is to the square of the radius, as the rectangle contained by the line of the arch, which is half the sum of AC, AD, and the fine of the arch which is half the difference of the same AC, AD is to the square of the fine of half the angle ABC. Q. E. D.
SOLUTION of the twelve Cases of oblique
angled SPHERICAL TRIANGLES.
IN an oblique angled spherical triangle, of the three
fides and three angles, any three being given, the other three may be found.
s B, D and DB. R: Co S, B :: T, BC : T, BA. 2. BC.
and T, D:T, B :: S, BA:S, DA 26. land BD is the sum or difference of BA DA.
6 BC, BD D. R: COS, B:: T, BC: T, BA. 20. and B.
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.
7 BC, DCC.
COS, BC : R:: Co T, B : T, BCA. and B.
19. and T, DC: T, BC :: Co S, BCA : COS, DCA. 27. the sum or difference of the angles BCA, DCA is equal to the angle BCD.
COS, BC:R:: Co T, B:T, BCA BC.
19. also by 27. Co S, DCA: COS, BCA ::T, BC: T, DC. 27. if DCA and B be of the fame affection; that is, (13.) iff AD and CA be similar, DC will be less than a quadrant, 14. and if AD, CA be not of the fame affection, DC is greater chan a quadrant. 14.
12 A, B, C. The In the triangle DEF, DE, EF, FD
, Fig. 7. Gides. are respectively the supplements of the
measures of the given angles B, A,C in the triangle BAC; the fides of the triangle DEF are therefore given, and by the preceding case the angles D, E, F may be found, and the fides BC, BA, AC are the supplements of the meafures of these angles.
The 3d, 5th, 7th, 9th, Toth, cafes, which are commonly called ambiguous, admit of two solutions, either of which will an. swer 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 tri. angle, 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.