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For, the rectangle contained by the sines of any two sides is to the square of the radius, as the rectangle contained by the sines of half the perimeter and its excess above the base, to the square of the cosine of half the angle opposite to the base, by Prop. 30; and as the rectangle contained by the sines of the two excesses of half the perimeter above each of the sides to the square of the sine of half the angle opposite to the base, by Prop. 29: therefore, the rectangle contained by the sines of half the perimeter and its excess above the base, is to the rectangle contained by the sines of the two excesses of half the perimeter above each of the sides, (as the square of the cosine to the square of the sine of half the angle opposite to the base, that is,) as the square of the radius to the square of the tangent of half the angle opposite to the base. Q. E. D.

* Solution of the several Cases of Right Angled and Oblique Angled Spherical Triangles.

GENERAL PROPOSITION.

In a right angled spherical triangle, of the three sides and three angles, any two being given, besides the right angle, the other three may be found.

1. Rules for the sixteen Cases of Right Angled Spherical Trigonometry. See Fig. 16.

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The two quantities, in the same analogy, marked with asteriscs, are both of the same affection; that is, both at the same time less, or both at the same time greater, than 90°.

Cases 7, 8, 9, are doubtful, for two triangles may have the given things, but have the things sought in one of them the supplements of the things sought in the other. In the remaining cases, if the two given quantities which occur in the previous terms of the same analogy are both of the same affection, the last term will be less than 90°; but if they are of different affection, the last term will be greater than 90°. These limitations are founded on the 13th, 14th and 15th Propositions.

2. Rules for the sixteen Cases of Right Angled Spherical
Trigonometry, expressed in general terms.

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A side and its opposite angle.

The two sides.

4.

The two angles.

5.

6.

7.

8.

9.

13.

14.

Sought.

The side opposite
to given angle.

15.

10.

The other side.

The hypotenuse 11. Angle opposite and a side.

16.

The other side.

The hypotenuse.

The other angle.

tan given sidecot given angle_sine.

rad

rad sin given side sin given angle radXcos given angle cos given side radXcos hypotenuse cos given side radxsin given side to given side. sin hypotenuse 12. Angle adjacent 2 tan given sidecot hypotenuse to given side. rad cos one sideXcos other side rad radxsin adjacent side tan other side radXcos opposite angle sin other angle cot one angleXcot other angle_cosine.

=cosine.

rad

The other side.

The hypotenuse.

The other angle.

The hypotenuse.

An angle.

A side.

Rule.

The hypotenuse.

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Given.

3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles.

FIG. 26, 27.

Two sides AB, AC,
and the included
angle A.

Two angles A and ACB, and AC, the side between them.

Two sides AC and BC, and an angle A, opposite to one of them BC.

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Rule.

Draw the perpendicular CD from C the unknown angle, not required, on AB.

R: cos A

tan AC: tan AD, less or greater than 90°,

as A and AC are of the same or of different affection. BD AB AD.

sin BD sin AD: tan A: tan B, of the same or of different affection, as AD is less or greater than AB.

cos AD: cos BD:: cos AC: cos BC, less or greater than 90°, as A and BD are of the same or of different affection.

Draw the perpendicular CD, from C the end of the given side next to BC, the side sought.

R: cos AC tan A: cot ACD, less or greater than 90o, as AC and A are of the same or of different affection. BCD ACB ACD.

cos BCD cos ACD:: tan AC: tan BC, less or greater|. than 90o, as A and BCD are of the same or of different affection.

sin ACD: sin BCD cos A: cos B, of the same or of different affection with A, as ACD is less or greater than ACB.

sin BC: sin AC : : sin A: sin B. The affection of B is doubtful; unless it can be determined by this rule, that according as AC+BC is greater or less than 180°, A+B is also greater or less than 180o.

R: cos AC:: tan A: cot ACD, less or greater than 90°,
as AC and A are of the same or of different affection.
tan BC: tan AC:: cos ACD: cos BCD greater than
90°, if one or all of the 3 terms ACD, AC, BC, are
greater than 90°; otherwise less than 90°.
ACD+ BCD=ACB, doubtful. If ACD+BCD ex-
ceed 180°, take their difference; if BCD is greater
than ACD, take their sum, for the angle ACB.

::

R: cos A tan AC: tan AD, less or greater than 90o, as A and AC are of the same or of different affection. cos AC cos BC: cos AD: cos BD, greater than 90°, if one or all of the 3 terms AD, AC, BC, are greater than 90°; otherwise less than 90o. AD+BD AB, doubtful. If AD+BD exceed 180°, take their dif ference; if BD is greater than AD, take their sum for AB.

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Two sides and the included

angle.

9.

Sought.

8. The side BC
opposite to the
other given
angle A.

10.

11.

12.

Given. Case.

1.

2.

The side AB adjacent to the given angles A, B.

The third angle ACB.

An angle
A.

A side
AC.

Sought.

One of the other angles.

The third
side.

Rule.

sin B: sin A: sin AC: sin BC, doubtful; unless it can be determined by this rule, that according as A+B is greater or less than 180°, AC+BC is also greater or less than 180o.

R: cos A tan AC: tan AD, less or greater than 90°, as A and AC are of the same or of different affection. tan B: tan A: sin AD: sin BD, doubtful.

:

AB AD+BD. If AD+BD be greater than 180°, take their difference; if BD is greater than AD, take their sum for AB.

:

R: cos AC: tan A: cot ACD, less or greater than 90°, as AC and A are of the same or of different affection.

4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles,

expressed in general terins.

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Rule.

Find an arc x, so that
radXcos given angle
cot side opposite angle sought

tan x;

and let y=difference between x and the other given side,

tan given angleXsin x
sin y

then will

tan angle sought.

tan x ; and

Find an arc x, so that
radXcos given angle
cot one of the given sides
difference between r and the other given side,
cos first sideXcos y

let y
then will.

cos side sought.

COS X

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