SPHERICAL TRIGONOMETRY.

69. SPHERICAL TRIGONOMETRY is that branch of Mathematics which treats of the solution of spherical triangles.

In every spherical triangle there are six parts: three sides and three angles. In general, any three of these parts being given, the remaining parts may be found.

GENERAL PRINCIPLES.

70. For the purpose of deducing the formulas required in the solution of spherical triangles, we shall suppose the triangles to be situated on spheres whose radii are equal to 1. The formulas thus deduced may be rendered applica ble to triangles lying on any sphere, by making them homogeneous in terms of the radius of that sphere, as explained in Art. 30. The only cases considered will be those in which each of the sides and angles is less than 180°.

Any angle of a spherical triangle is the same as the diedral angle included by the planes of its sides, and its mea sure is equal to that of the angle included between two right lines, one in each plane, and both perpendicular to their common intersection at the same point (B. VI., D. 4). The radius of the sphere being equal to 1, each side of the triangle will measure the angle, at the centre, subtended by it. Thus, in the triangle ABC, the angle at A is

the same as that included between the planes AOC and

AOB; and the side a is the measure of the plane angle BOC, O being the centre of the sphere, and OB the radius, equal to 1.

71. Spherical triangles, like plane triangles, are divided into two classes, right-angled spherical

B

triangles, and oblique-angled spherical triangles. Each class will be considered in turn.

We shall, as before, denote the angles by the capital letters A, B, and C, and the opposite sides by the small letters a, b, and c.

FORMULAS USED IN SOLVING RIGHT-ANGLED SPHERICAL

TRIANGLES.

72. Let CAB be a spherical triangle, right-angled at A,

and let O be the centre of the

sphere on which it is situated.

B

Denote the angles of the triangle

by the letters A, B, and C, and the opposite sides by the letters a, b, and e, recollecting that B and C may change places, provided that, b and с change places at the same time.

P

A

Draw OA, OB, and OC, each of which will be equal to 1. From B, draw BP perpendicular to 04, and from P draw PQ perpendicular to points and B, by the line QB.

OC; then join the

The line QB will be

perpendicular to OC (B. VI., P. VI.), and the angle PQB.

will be equal to the inclination of the planes OCB and OCA; that is, it will be equal to the angle C.

From the right-angled triangles OQP and QPB, we have,

Multiplying both terms of the fraction

remembering that cot a tan (90° 2P=02 lan b

QP 02 QP
X

=

QB QB OQ;

QP

or. cos Ctan (90° — a) tan b. (3.)

QB

by OQ, and

- a), we have,

Multiply both terms of the fraction

QP

by PR, and

OP

C), we have,

remembering that cot C = tan (90°C), we Q Pak

QP PB QP

=

OP ОР PB

213

or, sin b tan c tan (90° C). (4.)

If, in (2), we change c and C, into b and B, we

If, in (3), we change b and C, into

If, in (4), we change b, c, and C, into c, b, and

Multiplying (4) by (7), member by member, we have,

sin b sin c = tan b tan c tan (90° – B) tan (90°— C').

Dividing both members by tan b tan c, we have,

cos b cos c = tan (90°-B) tan (90°— C');

and substituting for cos b cos c, its value, from (1), we have,

cos a = tan (90° – B) tan (90° — C')

Formula (6) may be written under the form,

Changing B, b, and C, in (9), into C, c, and B, we

These ten formulas are sufficient for the solution of any

right-angled spherical triangle whatever.

NAPIER'S CIRCULAR PARTS.

73. The two sides about the right angle, the complements of their opposite angles, and the complement of the hypothenuse, are called Napier's Circular Parts.

b

D-06

90-a

90-R

B

If we take any three of the five parts, as shown in the figure, they will either be adjacent to each other, or one of them will be separated from each of the other two, by an intervening part. In the first case, the one lying between the other two parts, is called the middle part, and the other two, adjacent parts. In the second case, the one separated from both the other parts, is called the middle part, and the other two, opposite parts. Thus, if 90°-a, is the middle part, 90° — B, and 90° — C, are adjacent parts; and b and c, are opposite parts; and similarly, for each of the other parts, taken as a middle part.

74. Let us now consider, in succession, each of the five parts as a middle part, when the other two parts are opposite. Beginning with the hypothenuse, we have, from formulas (1), (2), (5), (9), and (10), Art. 72,

Comparing these formulas with the figure, we see that,

The sine of the middle part is equal to the rectangle of the cosines of the opposite parts.