To find the Angle B:

Here the circular parts concerned are, the leg AC, the given angle A, and the required angle B; and since the angle B is disjoined from the other two parts by the hypothenuse A B, it is the middle part, and the other two are the extremes disjunct, by rule 2, page 183; therefore, by equation 2, page 183,

Radius × co-sine angle B = sine of angle A × co-sine leg A C.

Now, since radius is connected with the required term, it is to stand first in the proportion. Hence,

Note.-The angle B is acute, or of the same affection with its opposite given leg A C.

To find the Leg BC:

In this case, since the right angle never separates the legs, the three circular parts are joined together: hence the leg AC is the middle part, and the leg B C and the angle A are the extremes conjunct, according to rule 1, page 183; therefore, by equation 1, page 183,

Radius sine of leg AC co-tangent angle Ax tangent of leg B C. And since the angle A is connected with the required part, it is to be the first term in the proportion. Hence,

Note. The leg B C is acute, or of the same affection with its opposite given angle A.

To find the Hypothenuse AB:

In this case, since the three circular parts which enter the proportion are joined together, the given angle A is the middle part, and the leg AC and the hypothenuse A B are the extremes conjunct: therefore,

Radius co-sine of angle A tangent of leg AC x co-tangent.hypothenuse A B.

Now, the leg A C, being connected with the required part, is therefore to be the first term in the proportion. Hence,

Note. The hypothenuse is acute, because the given leg and angle are of the same affection.

PROBLEM V.

Given the two Legs, to find the Angles and the Hypothenuse.

Example.

Let the leg AC, of the spherical triangle ABC, be 70:10:20%, and the leg B C 76:38:40%; required the angles A and B, and the hypothénuse AB?

To find the Angle A:

Here, since the right angle never separates the legs, the leg AC is the middle part, and the leg BC and the required angle A are the extremes conjunct, agreeably to rule 1, page 183; therefore, by equation 1, page 183, Radius sine leg AC = tangent leg BC x co-tangent angle A. Now, since the leg B C is connected with the required part, it is to be the first term in the proportion. Hence,

Log. tangent ar. comp. = 9.375506

As the leg BC =
Is to radius = . 90. 0. 0.
So is the leg AC =70. 10. 20

76:38:40

To the angle A= 77:24:37"

Log. co-tangent=

Note. The angle A is acute, or of the same affection with its opposite given leg B C.

To find the Angle B :

In this case the leg BC is the middle part, and the leg AC and the

required angle B are the extremes conjunct, according to rule 1, page 183; therefore, by equation 1, page 183,

Radius x sine of the leg BC= tangent of leg AC x co-tangent angle B.

And since the leg A C is connected with the required part, it is to be the first term in the proportion. Hence,

70:40 517 Log. co-tangent =

9.545083

To the angle B = Note. The angle B is acute, or of the same affection with its opposite given leg A C.

To find the Hypothenuse AB:

Here the hypothenuse A B is the middle part, because it is disjoined from the legs by the angles A and B: hence AC and BC are extremes disjunct, agreeably to rule 2, page 183; Radius x co-sine hypothenuse A B

therefore, by equation 2, page 183, co-sine leg AC x co-sine leg BC.

And radius, being connected with the middle part, is therefore to be the first term in the proportion. Hence,

Note.-The hypothenuse AB is acute, because the given legs AC and

BC are of the same affection.

PROBLEM VI.

Given the two Angles, to find the Hypothenuse and the two Legs.

Example.

Let the angle A, of the spherical triangle ABC, be 50:10:20%, and the angle B 64:20:25"; required the legs A C and B C, and the hypothenuse AB?

To find the Hypothenuse A B :

Here, because the three circular parts are joined together, the hypothenuse A B is the middle part, and the angles A and B are the extremes conjunct, agreeably to rule 1, page 183; therefore, by equation 1, page 183, Radius co-sine hypothenuse A B = co-tangent angle A x co-tangent angle B.

Now, since radius is connected with the required part, it is to be the first term in the proportion. Hence,

Note. The hypothenuse A B is acute, because the given angles A and C are of the same affection.

To find the leg AC:

Here, since the angle B is disjoined by the hypothenuse AB from the other two circular parts concerned, it is the middle part, and the angle A and the required leg AC are the extremes disjunct, agreeably to rule 2, page 183; therefore, by equation 2, page 183, Radius co-sine angle B And because the angle A is stand first in the proportion. Hence,.

As the angle A = 50:10:20

Is to radius= . 90. 0. 0
So is the angle B = 64. 20. 25

sine of angle Ax co-sine of leg AC. connected with the required part, it is to

To the leg AC = 55:40:38"

Note. The leg AC is acute, or of the same affection with its opposite given angle B.

To find the Leg BC:

In this case the angle A is the middle part, because it is disjoined from the other two circular parts by the hypothenuse AB: hence the angle B and the required leg B C are extremes disjunct; therefore,

Radius co-sine of angle A = sine of angle B x co-sine of leg BC. And as the angle B is connected with the required part, it is to be the first term in the proportion. Hence,

Note. The leg BC is acute, or of the same affection with its opposite given angle A.

SOLUTION OF QUADRANTAL SPHERICAL TRIANGLES,
BY LOGARITHMS.

PROBLEM I.

Given a Quadrantal Side, its opposite Angle, and an adjacent Angle, to find the remaining Angle and the other two Sides.

Remark. Since the sides of a spherical triangle may be turned into angles, and, vice versa, the angles into sides, all the cases of quadrantal spherical triangles may be resolved agreeably to the principles of rightangled spherical triangles; as thus: let the quadrantal side be esteemed the radius; the supplement of the angle subtending that side, the hypothenuse; and the other angles legs, or the legs angles, as the case may be. Then the middle part, and the extrem es conjunct or disjunct, being established, the required parts are to be computed, and the affections of the angles and sides determined, in the same manner precisely as if it were a right-angled spherical triangle that was under consideration.

Example.

Let AB, in the spherical triangle ABC, be the quadrantal side = 90°, the angle C 120:19:30, and the angle A 47:30:20"; required the sides A C and BC, and the angle B?

B

47:30:20'

Solution.-Let the supplement of the angle C (59:40:30"), subtending the quadrantal side AB, represent the hypothenuse ab of the dotted spherical triangle abc. Let the given angle A 47:30:20% represent the leg b c of the said dotted triangle, and the required angle B the leg a c.