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.

AC R: sin BC:: sin B*:sin AC* (18.)
AB R:cos B : :tan BC:tan AB (20.)
С R: :cos BC::tan B :cot C (19.)
AB R: sin AC::tan C* :tan AB. (17.)
BC

cos C:R::tan AC : tan BC (20.)
B R:cos AC*:: sin C:cos B * (22.)
AB tan B :tan AC:: R: sin AB (17.)
BC sin B: sin AC:: R: sin BC (18.)

С cos AC:cos B :: R: sin C (22.) AB

:cos BC :: R:cos A (21.) B sin BC: sin AC*:: R: sin B* (18.) С tan BC:tan AC :: R:cos C (20.) BC R:cos AB::cos AC:cos BC (21.) B sin AB:R::tan AC* :tan B* (17. с

sin AC:R::tan AB* :tan C.(17.)

cos AC:

10. AC and CB 11.

12.

13. AB and AC 14.

14.

15. B and C | 15.

16.

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.

Ecosine.

Given. Case. Sought.

Rule.
1. The side opposite sin hypot. X sin given angle sine.
to giveu angle.

rad The hypotenuse 2. side adjacent to?

radXcos given angle and one angle.

=tangent.
given angle.

cot hypotenuse
3.
The other angle.
rad X cos bypotenuse

=cotangent.

cot given angle 4.

rad X sin given side The other side.

=tangent.

cot given angle A side and its 5.

rad X cos given angle adjacent angle. The hypotenuse.

=cotangent.

tan given side 6. The other angle.

cos given side X sin given angle

rad

tan given side Xcot given anglesine. 7. The other side.

rad A side and its The hypotenuse.

radxsin given side 8. opposite angle.

Esine.

sin given angle 9. The other angle.

rad X cos given angle

=sine. cos given side

rad Xcos hypotenuse 10. The other side.

=cosine.

cos given side The hypotenuse 11. Angle opposite ? rad x sin given side Jand a side.

=sine. to given side. S sin hypotenuse 12. Angle adjacent tan given side Xcot hypotenuse

=cosine. to given side.

rad
13. The hypotenuse.
cos one side Xcos other side

=cosine. The two sides.

rad
14. An angle.
radxsin adjacent side

=cotangent. tan other side

15.

'The two angles.

rad Xcos opposite angle A side.

=cosine. sin other angle

cot one angleXcot other angle cosine. The hypotenuse.

rad

16.

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

Fig. 26, 27.

Giren,

Case.

Sought.

Rule.

1.

Two sides AB, AC, and the included

angle A.

One of the Draw the perpendicular CD from the unknown
other angles angle, not required, on AB.
B. 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 A B. The third cos AD: cos BD: : cos AC: cos BC, less or greater side BC. than 90', as A and B D are of the same or of different

affection.

2.

3.

The side

BC.

Two angles A and ACB,

and AC, the side between them.

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 greaier than 90°, 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 90', as A and BCD are of the same or of different affection.

4.

The third
angle B.

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.

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

side AC. A+B is also greater or less than 180°. The included R: cos AC:: tan A : cot ACD, less or greater than 90°, angle ACB. 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 900. 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. The third R: cos A : : tan AC : tan AD, less or greater than 90°, side AB. as A and AC are of the same or of different affection.

7.

Given.

Case.

Sought.

Rule.

The side BC sin B : sin A : : sin AC : sin BC, doubtful; unless it opposite to the can be determined by this rule, that according as other given A+B is greater or less than 180°, AC+BC is also

angle A. greater or less than 1800. The side AB R: cos A : : tan AC : tan AD, less or greater than 90°, adjacent to the as A and AC are of the same or of different affection. given angles | tan B : tan A : : sin AD : sin BD, doubtful. AB= A, B. AD+BD. If AD+BD be greater than 180°, take

their difference; if BD is greater than AD, take their sum for AB.

10.

The third angle ACB.

R: : 008 AC: : tan A : cot ACD, less or greater than

90o, as AC and A are of the same or of different affec

tion. cos A : cos B : : sin ACD : sin BCD, doubtful. ACB

=ACD+BCD. If ACD+BCD exceed 180°, take their difference ; if BCD be greater than ACD, take their sum for ACB.

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

expressed in general terins.

Rule,

1.

| Sought.
One of the Find an arc x, so that
other angles. rad Xcos given angle

cot side opposite angle sought
and let y=difference between x and the other given side,

tan given angle X sin
then will

=tan angle sought.

=tan x;

sin y