The Perpendicular or Leg BC lying between the given Angle at C and the Bafe A B, it is an Extreme disjunct, and the Bafe A B, the Part to be found, is the middle Part; therefore, by the third Rule, and its Note,

As the Radius

Is to the Sine of one Extreme A C, 75°. 20 ́,

10.

9.985613

So is the Sine of the other Extreme BCA, 57.16, 9.924897

To the Sine of the middle Part A B, 54°. 28′,

9.910510

The Base A B, being Part of a right Circle, may be measured by Prob. 43, Cafe 2.

Prob. 66. In the right-angled Spherical Triangle ABC, right-angled at B, given the Hypothenufe AC 75°.20′, and the Angle at C 57°. 16', to find the Angle at A. See the Figure at Prob. 64, the Conftruction being the fame.

The two Parts given and the Part required lying together, with no Part between them, this is an Extreme conjunct, and BAC, one of the Extremes, is to be found; which may be done by the second Rule, and its Note, as follows:

As the Co-tangent of B C A, 57°. 16',

one of the Extremes,

Is to the Radius,

Co. Ar. 0.191917

IO.

So is the Co-fine of AC, 75°. 20', the middle Part, 9.403455

To the Co-tangent of BA C, 68°. 30', the other 9.595372 BAC,

Extreme,

The Angle BAC, made by the Interfection of the right Circle BOE and the oblique Circle CAD, may be measured by Prob. 55.

Prob. 67. In the right-angled Spherical Triangle ABC, right-angled at B, given the Side AB 54°. 28', and the Angle at A 68°. 30', to find the Leg BC.

Conftruction. With the Chord of 60°, draw the primitive Circle BCEF; draw the right Circle B E, which will be

perpen

perpendicular to the primitive
Circle, because it paffes thro' its
Pole O; lay 54°. 28' from B to
A, by Prob. 42, Cafe 2; then,

C

by Prob. 57, draw an oblique

great Circle, CAF, to make an B+

Angle of 68°. 30' with the right

Circle BE, in the Point A,

which completes the fpherical Triangle ABC, right-angled at B.

F

Here nothing lying between the Side BC, the Side B A, and the Angle at A, it is an Extreme conjunct, and one of the Extremes is to be found; wherefore, by the fecond Rule, and its Note,

As the Co-tangent of A, 68°.30, the Co. Ar. 0.404602 given Extreme,

Is to the Radius,

So is the Sine of A B, 54°.28', the middle Part,

10.

9.910506

To the Tangent of BC, 64°. 10', the other} 10.315108

Extreme,

The Perpendicular B C may be measured on the Line of Chords, by Prob. 41.

Prob. 68. In the right-angled Spherical Triangle ABC, right-angled at B, given the Bafe or Side A B 54°.28', and the Angle at A 68°. 30, to find the Angle at C. See the Figure in the laft Problem, the Conftruction being the fame.

The Hypothenufe A C lying between the Angles A and C, it is an Extreme disjunct, and the middle Part C is to be found; hence, by the third Rule, and its Note,

As the Radius

Is to the Co-fine of the Extreme A B, 54°. 28',
So is the Sine of the Angle at A, 68°. 30, the

other Extreme,

To the Co-fine of the Angle at C, 57°. 16',,
The Angle at C may be measured by Prob. 54.

10.

9.764308 9.968678

9.732986

Prob.

Prob. 69. In the right-angled spherical Triangle ABC, right-angled at B, given the Side AB 54°. 28', and the Angle at A 68°. 30', to find the Hypothenufe A C. See the Figure at Prob. 67, the Conftruction being the fame.

This is an Extreme conjunct, and one of the Extremes is to be found; therefore, by the fecond Rule, and its Note,

As the Tangent of the given Extreme AB, 54°. 28', 10.146198
Is to the Radius
So is the Co-fine of the middle Part BAC, 68°. 30, 9.564075

10.

19.564075 10.146198

To the Co-tang. of the Hypothenuse AC, 75°.20′, 9.417877

The Logarithm-Tangent in the first Number being greater than the Radius, the arithmetical Complement could not be had. The Truth of the Work may be found by measuring the Hypothenufe, by Prob. 46.

Prob. 70. In the right-angled fpherical Triangle ABC, right-angled at B, given the Hypothenufe AC 75°. 20 ́, and the Bafe AB 54°.28', to find the Perpendicular BC.

Conftruction. With the Chord of 60, draw the primitive Circle EFGD, thro' the Pole or Center A; draw a right Circle, E AD, and cross it at right Angles with the right Circle FAG; take the Half-Tangent of 75°. 20', and, with one Foot of the Compaffes in the Center of the primitive Circle, draw the leffer Circle mne, parallel to the primitive

E

H

F

I

Ꮐ

Ꭰ

Circle; lay 54°. 28′ on the right Circle E D, from A to B, by Prob. 42, Cafe 1; then thro' the three Points F, B, and G, by Prob. 17, draw the oblique. Circle F BG, interfecting the leffer Circle m ne in C, thro' which and the Pole or Center of the primitive Circle draw the right Circle HACI, and ABC is the Triangle required.

The

The Angle BAC falling between the given Sides A B and A C, it is an Extreme disjunct, and the Extreme BC is to be found; whence, by the fourth Rule, and its Note,

As the Co-fine of the given Extreme A B, Co. Ar. 0.235692

54°. 28',

Is to the Radius,

10.

So is the Co-fine of the middle Part AC, 75°. 20', 9.403455

To the Co-fine of the other Extreme BC, 64°. 10', 9.639147

The Perpendicular or Leg BC, being a Part of the oblique Circle FBG, may be measured by Prob. 46.

Prob. 71. In the right-angled Spherical Triangle ABC, right-angled at B, given the Bafe A B 54°.28', and the Hypothenufe AC 75°.20', to find the Angle at A.

The Conftruction is the fame as in the laft Problem.

The three Parts lying together, with no Part between, this is an Extreme conjunct, and the Angle B A C, required, is the middle Part; whence, by the first Rule, and its Note,

As the Radius

10.

Is to the Co-tangent of the Extreme AC, 75°.20', 9.417842 So is the Tang, of the other Extreme AB, 54°.28', 10.146198 To the Co-fine of the middle Part BAC, 68°. 30′, 9.564040

The Angle at A, being at the Center of the primitive Circle, is measured as a plain Angle, by Prob. 53.

Prob. 72. In the right-angled spherical Triangle A B C, right-angled at B, given the Bafe A B 54° 28', and the Hypothenufe AC 75°. 20', to find the Angle at C.

The Conftruction is the fame as in the two laft Problems. See the Figure at Prob. 70.

The Angle BA C lying between the Bafe and the Hypothenufe, this is an Extreme disjunct, and the Angle at C, which is to be found, is one of the Extremes; therefore, by the fourth Rule, and its Note,

As

As the Sine of the given Extreme AC, Co. Ar. 0.014387

75°. 20',

Is to the Radius,

So is the Sine of the middle Part A B, 54°. 28',

To the Sine of the required Extreme ACB, 57°.16′, 9.924893

The Angle A CB, being made by the Interfection of an oblique Circle and a right Circle, may be measured by Prob. 55.

and

Prob. 73. In the right-angled Spherical Triangle ABC, right-angled at B, given the Perpendicular BC 64°. 10′, the Angle at A 68°. 30', to find the Bafe A B.

Conftruction. With the Chord of 60°, defcribe the primitive Circle BFDG; thro' which draw, at Pleasure, the right Circle BD, and cross it at right Angles with the right Circle FG; from B fet off 64°. 10' to C, and draw C E, and at right Angles to it draw HI; from F, both Ways, fet off 68°. 30 to c and d ; a Ruler, being laid from B to c, or from D to d, will cut F G in e;

Ble

H

F

E

72

thro' the three Points d, e, and c, by Prob. 17, draw the parallel Circle ced, interfecting the right Circle H I in r; a Ruler, being laid from C to r, cuts the primitive Circle in s, from whence lay 90° to t; a Ruler, being laid from C to t, will cut the right Circle HI in u; then, by Prob. 17, draw a Circle thro' the three Points C, u, and E, which will interfect the right Circle BD in A, making an Angle, BAC, of 68°. 30, and complete the required Triangle ABC.

The three Parts lying together, with no Part between them, (except the right Angle, which is always excepted) this is an Extreme conjunt, and the middle Part A B is to be found s then, by the first Rule, and its Note,

As the Radius

10.

Is to the Tangent of the Extreme BC, 64°. 10, 10.315032 So is the Co-tang. of the other Extreme BAC, 68°.30, 9.595398 To the Sine of the middle Part A B, 54°. 28', 9.910430 The Bafe A B may be measured by Prob. 43, Cafe 2.

Prob.