To find the Hypothenuse A B :

Here, because the three circular parts are joined together, the hypothenuse AB 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, co-tangent angle A x co-tangent

Radius

angle B.

co-sine hypothenuse A 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. A B 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 x co-sine angle B And because the angle A is stand first in the proportion. Hence,

As the angle A=

Is to radius=

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

To the leg AC =

55:40:38 Log, co-sine =

9.751168

So is the angle B = 64. 20. 25

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,

9.851598

So is the angle A = 50. 10. 20

To the leg BC= 44:43:11 Log. co-sine =

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

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 extremes 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 A B, 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

A

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 bc of the said dotted triangle, and the required angle B the leg a c.

Then, in the right-angled spherical triangle a b c, given the hypothenuse ab 59:40:30, and the leg be 47:30:20", to find the leg ac the angle B in the quadrantal triangle; the angle a the leg B C, and the angle b = the leg A C, of the said quadrantal triangle.

4°30′20′′

To find the Leg ac the Angle B in the Quadrantal Triangle :

Here the hypothenuse ab is the middle part, and the legs bc and a c are the extremes disjunct; therefore,

Radius x co-sine hyp. ab co-sine leg bc x co-sine leg a c.

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

Note. The leg ac is acute, because the hypothenuse and the given leg are of the same affection: hence the angle B (in the quadrantal triangle), represented by the leg a c, is also acute

41:37:54?

To find the Angle a the Leg B C in the Quadrantal Triangle :—

Here the leg bc is the middle part, and the hypothenuse a b and angle a are the extremes disjunct; therefore,

Radius sine of leg bc sine of hypothenuse ab x sine of the angle a.

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

Note. The angle a is acute, because the hypothenuse and the given leg are of the same affection: hence the leg BC (of the quadrantal triangle), represented by the angle a, is also acute = 58:40:26%

To find the angle b = the Leg A C in the Quadrantal Triangle :

In this case the angle b is the middle part, and the hypothenuse ab and the leg bc are the extremes conjunct; therefore,

Radius x co-sine of the angle b = co-tangent hypothenuse abx tangent of leg bc.

And radius, being connected with the required part, is, therefore, to stand first in the proportion. Hence,

Note.-The angle b is acute, because the hypothenuse and the given leg are of the same affection. Hence, the leg AC (of the quadrantal triangle), represented by the angle b, is also acute 50:19:19%.

PROBLEM II.

Given the Quadrantal Side and the other two Sides, to find the three

Example.

Angles.

Let AB, in the spherical triangle ABC, be the quadrantal side = 90:; the side AC, 115:19:45"; and the side BC, 117:39:35": required the angles A, B, and C ?

"0;0;06

B

b

A

117039.35

115.19:45

3.5

Solution. Let the angle c, in the dotted spherical triangle abc, be radius, and represent the side AB = 90% of the quadrantal triangle ABC. Let the angle a, of the dotted triangle, represent the side B C of the quadrantal triangle = 117:39:45, and let the angle b represent the side AC of the said quadrantal triangle = 115:19:45. Then, in the right-angled spherical triangle a b c, right-angled at c, given the angle a 117:39:35", and the angle = 115:19:45%, to find the hypothenuse a b, the leg ac, and the leg bc; the first of which represents the supplement of the angle

C opposite to the quadrantal side A B, in the triangle ABC; the second represents the angle B; and the third the angle A, in the said quadrantal triangle.

To find the Hypothenuse a b= the Supplement of the Angle C,
subtending the Quadrantal Side A B :-

Here the hypothenuse ab is the middle part, and the given angles a and b are the extremes conjunct; therefore,

Radius

co-sine hypothenuse ab

co-tangent of angle a x co-tangent of 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 are of the same affection:-but since it only represents the supplement of the angle C; therefore the angle C is obtuse, or 104:21:49".

To find the Leg ac = the Angle B in the Quadrantal Triangle.

The angle b, in this case, is the middle part, and the angle a and leg a c extremes disjunct.-Therefore, radius × co-sine of angle b = sine of angle ax co-sine of leg a c.

And the angle a being connected with the required part, is, therefore, to be the first term in the proportion.-Hence,

9.683962

To the side ac = 118:52:57" Log. co-sine =

Note. The side a c is obtuse, or of the same affection with its opposite angle band since a c represents the angle B; therefore the angle B, in the quadrantal triangle, is obtuse, or 118:52:57".

To find the Leg bc the Angle A in the Quadrantal Triangle.

In this case the angle a is the middle part, and the angle ↳ and leg b c extremes disjunct.-Therefore, radius x co-sine of the angle a = sine of the angle bx co-sine of the leg b c.