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.

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.

Given. Case. Sought.

Rule.
| . |The side opposite , sin hypot. X sin given angle _cine
to given angle. S

rad The hypotenuse 2 side adjacent to 1

rad Xcos given angle =tangent. and one angle.

given angle. Si cot hypotenuse
3. The other angle.
radx cos bypotenuse

=cotangent.
cot given angle

rad X sin given side The other side.

=tangent.

cot given angle A side and its adjacent angle. / 5. The hypotenuse. rad X cos given angle

=cotangent. tan given side

cos given side X sin given angle_onsine. The other angle. cos gi

rad

tan given side x cot given angle_sine. The other side.

rad A side and its

radxsin giveu side The bypotenuse. opposite angle. 8.

sin given angle

=sine. 9. The other angle. rad Xcos given angle

=sine.

cos given side 10. The other side.

rad Xcos hypotenuse

=cosine.

cos given side The bypotenuse u. lAngle opposite rad X sin given side and a side.

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

rad
The hypotenuse.
cos one sideXcos other side

=cosine. The two sides.

rad
An angle.
radxsin adjacent side

=cotangent. tan other side

222

14.

A side.

15. The two angles.

| 16.

rad xcos opposite angle =cosine.

sin other angle
cot one angle Xcot other angle_novine

rad

The hypotenuse.

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

Fig. 26, 27.

Given.

Case.

Sought.

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 900, as A and AC are of the same or of different affection.

BD=AB, AD. sin BD : sin AD: :lan A : tan B, of the same or of

different affection, as AD is less or greater than A B. cos AD: cos BD : : cos AC : cos BC, less or greater

than 90°, as A and B D are of the same or of different affection.

The third
side BC.

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 900, 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 900; 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. The third R: cos A : : tan AC : tan AD, less or greater than 90°, side AB. L 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 900; otherwise less than 90°. AD+BD=AB, doubtful. If AD+BD exceed 180°, take their difference ; if BD is greater than AD, take their sum for AB.

7.

The

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

expressed in general terins.

Given.

Rule.

Case. 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,
then will tan given angle x sin a

=tan angle sought.

sin y

Two sides and the included

angle.