Prob. 74. In the right-angled spherical Triangle A B C, right-angled at B, given the Perpendicular BC 64°.10', and the Angle at A 680.30', to find the Hypothenufe A C. See the Figure of the laft Problem, the Construction being the fame. The Bafe lying between the two given Parts, BC and the Angle B A C, this is an Extreme disjunct, and the Hypothenufe AC, the Part to be found, is one of the Extremes; therefore, by the fourth Rule, and its Note, As the Sine of the Extreme BAC, 68°. 30', Co, Ar, 0.031322 Is to the Radius, So is the Sine of the middle Part B C, 64°, 10', To the Sine of the Hypothenuse A C, 75°. 20 ́, 10. 9.954274 9.985596 The Hypothenuse A C may be measured by Prob. 46. Prob. 75. In the right-angled fpherical Triangle ABC, right-angled at B, given the Perpendicular BC 64°. 10, and the Angle at A 68°. 30', to find the Angle at C. The fame Parts being given as in the two laft Problems, the Conftruction is the fame. See the Figure at Prob. 73. The Hypothenufe A C lying between the two Angles BAC and BCA, this is an Extreme disjunct, and the required Angle at C is one of the Extremes; therefore, by the third Rule, and its Note, As the Co-fine of the Extreme BC, 64°. 10, Co. Ar. 0.360758 To the Sine of the Angle at C, 57°. 16′, 10. 9.924833 The Angle at C, made by the oblique Circle CAE interfecting the primitive Circle at the Point C, may be measured by Prob. 54. Prob. 76. In the right-angled spherical Triangle ABC, right-angled at B, given the Perpendicular BC 64°. 10', and the Bafe AB 54°.28', to find the Hypothenufe A C. Conftruction. With the Chord of 60°, draw the primitive Circle BCDE; thro' the Pole or Center draw the right N n Circle Circle BD; take 64°. 10' from The Angles lying between the two given Parts and the Part required, it is an Extreme disjunct, and the middle Part A C is to be found; therefore, by the third Rule, and its Note, As the Radius 10. 9.764308 Is to the Co-fine of the Extreme A B, 54°. 28', To the Co-fine of the middle Part A C, 75°. 20, 9.403550 The Hypothenufe may be measured by Prob. 46. Prob. 77. In the right-angled spherical_Triangle ABC, right-angled at B, given the Perpendicular BC 64°. 10′, and the Bafe A B 54°.28′, to find one of the Angles, as C. See the laft Figure, the Construction being the fame as in the laft Problem. No Part lying between the two given Sides and the Angle required, it is an Extreme conjunct, and the Angle at C, one of the Extremes, is to be found; whence, by the second Rule, and its Note, As the Tangent of the Extreme A B, 54°.28′, So is the Sine of the middle Part BC, 64°. 10', To the Co-tangent of BCA, 57°. 16′, 10.146198 10. 9.954274 9.808076 The Angle at C, made by the oblique Circle CA E interfecting the primitive Circle at the Point C, may be measured by Prob. 54. If the fame Things are given, and the Angle at A required, it is ftill an Extreme conjunct, and the Bafe A B is now the middle Part; to find which, by the firft Rule, and its Note the Proportion is as follows: As the Tangent of BC, 64°. to', Is to the Radius, So is the Sine of AB, 54°. 28', 10.315032 IO. 9.910506 19.910506 10.315032 9.595474 To the Co-tangent of B AC, 68°.30', The Angle at A, made by the right Circle BAD and oblique Circle CAE interfecting each other at the Point A, may be measured by Prob.-55. Prob. 78. In the right-angled spherical Triangle ABC, right-angled at B, given the Angle at A 68°. 30', and the Angle at C 57°. 16', to find the Hypothenufe A C. Conftruction. With the Chord of 60°, draw the Circle E CFD; q D F G laid from C to p, will cut the primitive Circle in s; on the primitive Circle, from s to q, fet 68°. 30', the given Angle at A; then draw the right Circle qr, perpendicular to which draw the right Circle BG, interfecting the oblique Circle CAD in A, making an Angle, CA B, of 68°. 30', and ABC is the Triangle required. The three Parts lying together, this is an Extreme conjun&, and the Hypothenufe AC is the middle Part; therefore, by the firft Rule, and its Note, As the Radius Is to the Co-tangent of B AC, 68°. 30 ́, 10. 9.595398 9.808083 To the Co-Sine of the middle Part AC, 75°. 20', 9.403481 The Hypothenufe A C may be measured by Prob. 46. Prob. 79. In the right-angled spherical Triangle ABC, right-angled at B, given the Angle at A 68°. 30, and the Angle at C 57°. 16', to find one of the Legs, as the Base A B. The Conftruction is the fame as in the laft Problem. See the Figure there. The Hypothenufe A C lying between the given Angles, it is an Extreme disjunct, and the Bafe A B is one of the Extremes; therefore, by the fourth Rule, and its Note, As the Sine of the Extreme BAC, 68°. 30', Co. Ar. 0.031322 To the Co-fine of the Bafe A B, 54°.28′, 10. 9-764302 The Base A B may be measured by Prob. 43, Cafe 2. If the Perpendicular B C had been required, it would still have been an Extreme disjunct; but the Angle at A would then have been the middle Part, and the Perpendicular might be found by the fourth Rule, and its Note; thus, As the Sine of the Extreme BCA, 57°. 16', Co. Ar. 0.075102 10. To the Co-fine of the Perpendicular BC, 64°. 10', 9.639177 Of Of Oblique Spherical Trigonometry. The two firft Cafes of oblique-angled fpherical Triangles are folved by this Axiom: The Sines of the Sides of all spherical Triangles are in Proportion to the Sines of their oppofite Angles. Prob. 80. In the oblique-angled Spherical Triangle ABD, given the Angle BDA 64°. 15, the Angle BAD 59° 47', and the Side B D 47°. 21', oppofite to one of the given Angles, to find the Side B A, oppofite the other given Angle. Conftruction. With the Chord of 60°, defcribe the primitive Circle AEFD; at any Point, as D, of the Periphery, by Prob. 48, draw the oblique Circle DE, fo as to make an Angle with the primitive Circle, at D, of 64°. 15, and lay 47°21′ on this oblique Circle, by Prob. F 45, from D to B; thro' B, by Prob. 48, draw an oblique Circle, ACF, to make an Angle, DAB, of 59°.47′ with the primitive Circle, and form the oblique Triangle A B D. D E B Here being given the Angle at A, oppofite the given Side BD, and the Angle at C, oppofite the required Side B A, this may be found by the following Proportion, founded on the above Axiom.: As the Sine of BAD, 59.47', Co. Ar. 0.063422 Is to the Sine of the oppofite Side B D, 47°. 21, So is the Sine of BDA, 64°. 15', To the Sine of the oppofite Side A B, 50°, 3', 9.866586 9.954579 9.884587 The Side A B, being Part of the oblique Circle A B F, may be measured by Prob. 46. Prob. 81. In the oblique-angled Spherical Triangle ABD, given the Side AD 74°. 20', the Side A B 65° 34', and the Angle at D 59°. 50′, to find the Angle at B. Conftruction. With the Chord of 60°, defcribe the primitive Circle HD AI; on which lay 74°. 20 from any Point affumed |