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CASE IV.

(O) When two angles of an oblique-angled spherical triangle, and the side adjacent to both of them, are given to find the rest.

RULE.

1. To find the other two sides.

Cosine of half the sum of the two given angles,
Is to cosine of half their difference;

As tangent of half the adjacent side,

Is to tangent of half the sum of the other two sides. Half the sum of these sides, must be of the same species as half the sum of the given angles.

two sides.

Secondly.

Sine of half the sum of the two given angles,

Is to sine of half their difference;

As tangent of half the adjacent side,

Is to tangent of half the difference between the other

Half the difference between these sides is always acute. (N. 189)

Lastly.

Half the sum of the two sides increased by half their difference, gives the side opposite to the greater angle, and diminished by the same, leaves the side opposite to the less. (C. 35.)

2. To find the third angle.

Find the two required sides by the first part of the rule. Then, Sine of half the difference between these sides, Is to sine of half their sum ;

As tangent of half the difference between the given angles,

Is to cotangent of half the third angle. (M. 188.)

OR, without finding the other two sides.

To the sum of the logarithmical sines of the given angles, add double the logarithmical cosine of half the given side, and reject 30 from the index.

Look for the remainder in the table of logarithmical sines, and take the degrees and minutes answering to it. Then take the difference between twice the natural sine of those degrees, &c. and the natural cosine of the difference between the given

angles; the remainder will be the natural cosine of the angle required. This angle is acute or obtuse, according as the double natural sine is greater, or less, than the cosine of the difference between the given angles. (Y. 193.)

CASE V.

(P) When the three sides, of an oblique-angled spherical triangle, are given to find the angles.

RULE I.

From half the sum of the three sides subtract the side opposite to the required angle, and note the half sum and remainder. Then add together,

The logarithmical co-secants of each of the sides containing the required angle, rejecting the indices; and the sines of the above half sum and remainder: half the sum of these four logarithms is the logarithmical cosine of half the angle sought. (G. 185.)

OR, RULE II.

Add all the three sides together, from the half sum subtract each side containing the required angle, and note the remainders. Then add together,

The logarithmical co-secants of each of the sides containing the required angle, rejecting the indices; and the sines of the above-noted remainders: half the sum of these four logarithms, is the logarithmical sine of half the angle sought. (F. 184.)

OR, RULE III.

From half the sum of the three sides subtract each side separately. Then add together,

The logarithmical co-secants of half the sum of the sides, and of the difference between that half sum and the side opposite to the angle required, rejecting the indices; the logarithmical sines of the difference between the half sum and each side containing the required angle, half the sum of these four logarithms is the logarithmical tangent of half the angle sought. (H. 186.)

CASE VI.

(Q) When the three angles, of an oblique-angled spherical triangle, are given to find the sides.

RULE I.

Add all the three angles together, take the difference between the half sum and the angle opposite to the side sought, and note the half sum and remainder. Then add together,

The logarithmical co-secants of each of the angles adjacent to the required side, rejecting the indices, and the cosines of the above half sum and remainder; half the sum of these four logarithms is the logarithmical sine of half the side sought. (I. 186.)

OR, RULE II.

Take the supplements of each of the angles, and use the remainders as sides in a new triangle.

Find the angles of this triangle, by any of the rules in Case V. the supplements of which will be the sides sought. (U. 137.) (R) CASE I. Given two sides of an oblique spherical triangle, and an angle opposite to one of them, to find the rest.

In the oblique spherical triangle ABC.

The side AC-80°.19'

Given The side BC=63°.50'

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Required the rest..

BY CONSTRUCTION. (Plate V. Fig. 15.)

1. With the chord of 60 degrees describe the primitive circle; through the centre p draw CPe, and apr at right angles to it. 2. Set off the side AC 80°.19′ from c to a, by the scale of chords.

3. Through a draw the great circle abûn, making an angle of 51°.30' with the primitive. (P. 160.)

4. Set off the side BC=63°.50' by a scale of chords, from c to m, and draw the parallel circle mbâm. (Z. 162.) Through the points b, B, where it cuts the oblique circle abûn, and the point c, draw the great circles cbe, CBе.

5. Then, abc or ABC is the triangle required, each having the same data, which shews this example to be ambiguous.

To measure the required parts.

6. The side ab (C. 163.) 28°.33', and AB=120°.47'. Acb (G. 164.)=24°.37′, and ▲ ACB=131°.29′;

7. The

the Abc 120°.44′, and the ABC 59°.16'.

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In the oblique spherical triangle ABC.
The side AC 57°.30'

Given The side BC=115°.20Required the rest.
The LA 126°.37')

BY CONSTRUCTION. (Plate V. Fig. 16.)

1. With the chord of 60° describe the primitive circle, through the centre P draw CPe, and apr at right angles to it. 2. Set off the side ac=57°.30′ from c to A, by the scale of chords.

3. Through A draw the great circle AB, making an angle of 126°.37' with the primitive. (P. 160.)

4. Set off the side BC=115°.20', by a scale of chords, from c to m, and draw the parallel circle mвm. (Z. 162.) Through the point B, where it cuts the oblique circle ABn, and the point c, draw the great circle све.

5. Then ABC is the triangle required; and though it has exactly the same data as the former example, none of the parts are ambiguous.

To measure the required parts.

6. The side AB (C.163.)=82°.26'.

7. The B (G. 164.) 48°.30', and the c=61°.40′.

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2. In the oblique spherical triangle ABC.
The side AC=119°. 5′

Given The side BC= 79°.13' Answer.

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B=130°.56'

ZC 50°.11'.40"
AB= 62°.42′.

3. In the oblique spherical triangle ABC.
The side AC

30°

The side BC24°.4' Ans.
The LA=36°.8'

Required the other parts.

ZB 46°.18', or 133°.42′. 2c=104°, or 11°.23'. AB= 42°.9', or 7°.51'.

(S) CASE II. Given two angles of an oblique spherical triangle, and a side opposite to one of them, to find the rest.

Given

In the oblique spherical triangle abc.

The

The

LA 51°.30'

B=59°.16'

The side BC 63°.50'

Required the rest.

BY CONSTRUCTION. (Plate V. Fig. 17.)

1. With the chord of 60 degrees describe the primitive circle, through the centre P draw BPе, and DPE at right angles to it.

2. Set one foot of your compasses on 90 degrees, on the line of semi-tangents, extend the other towards the beginning of the scale, till the degrees between them be equal to the angle B=59°.16′, and apply this extent from E to n (P. 160.); and through the three points вne draw a great circle.

3. Set off the side BC=63°.50'. taken from a scale of chords, from в to m, and draw the parallel circle mcm, cutting the oblique circle, вne in c.

4. With the tangent of the angle A=51°.30′ and pas a centre, describe an arc; and with the secant of the same angle, and c as a centre, cross it in o.

5. With the centre o, and radius oc, draw the great circle Then ABC is the triangle required.

aca.

To measure the required parts.

6. AB measured by a scale of chords will be 120°.47′, or 151°.27' ambiguous.

7. AC will be 80°.19′, or 99°.41′ (C. 163.) ambiguous.
8. The c=131°.30′, or 155°.22′ (G. 164.) ambiguous.

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