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

If, in (3), we change b and C into c and B, we have (6.)

cos B cot a tan c.

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

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

sin b sin c = tan b tan c cot B cot C.

Dividing both members by tan b tan c, we have

cos b cos c = cot B cot C;

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

Formula (6) may be written under the form

Substituting for cos a, its value, cos b cos c, taken from (1), and reducing, we have

Again, substituting for sin c, its value, sin a sin C, taken from (2), and reducing, we have

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

These ten formulas are sufficient for the solution of any right-angled spherical triangle whatever. For the purpose of classifying them under two general rules, and for convenience in remembering them, these formulas are usually put under other forms by the use of

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.

90°-C

C

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 two others by an intervening part. In the first case, the one lying between the two other parts is called the middle part, and the two others, adjacent parts. In the second case, the one separated from both the other parts, is called the middle part, and the two others, opposite parts. Thus, if 90°is the middle part, 90° B and 90° - C are adjacent parts; and b and c are opposite parts; if c is the middle part, b and 90° B are adjacent parts (the right angle not being considered), and 90° C and 90° - a are opposite parts: and similarly, for each of the other parts, taken as a middle part.

-

a

74. Let us now consider, in succession, each of the five parts as a middle part, when the two other 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.

Let us now take the same middle parts, and the other parts adjacent. Formulas (8), (7), (4), (6), and (3), Art.

Comparing these formulas with the figure, we see that

The sine of the middle part is equal to the rectangle of the tangents of the adjacent parts.

These two rules are called Napier's rules for circular parts, and are sufficient to solve any right-angled spherical triangle.

75. In applying Napier's rules for circular parts, the part sought will be determined by its sine. Now, the same sine corresponds to two different arcs, or angles, supplements of each other; it is, therefore, necessary to discover such relations between the given and the required parts, as will serve to point out which of the two arcs, or angles, is to be taken.

Two parts of a spherical triangle are said to be of the same species, when they are each less than 90°, or each greater than 90°; and of different species, when one is less and the other greater than 90°.

From formulas (9) and (10), Art. 72, we have,

cos B sin C = cos b'

and sin B

cos C

=

COS C

since the angles B and C are each less than 180°, their sines must always be positive: hence, cos B must have the same sign as cos b, and the cos C must have the same sign as cos c. This can only be the case when B is of the same species as b, and C of the same species as c; that is, each side about the right angle is always of the same species as its opposite angle.

From formula (1), we see that when a is less than 90°, or when cos a is positive, the cosines of b and c will have the same sign; and hence, b and c will be of the same species: when a is greater than 90°, or when cos a is negative, the cosines of b and c will have contrary signs, and hence b and c will be of different species:

therefore, when the hypothenuse is less than 90°, the two sides about the right angle, and consequently the two oblique angles, will be of the same species; when the hypothenuse is greater than 90°, the two sides about the right angle, and consequently the two oblique angles, will be of different species.

These two principles enable us to determine the nature of the part sought, in every case, except when an oblique angle and the side opposite are given, to find the remaining parts. In this case, there may be two solutions, one solution, or no solution.

There may be two cases:

1o. Let there be given B and b, and B acute. Construct B and prolong its sides till they meet in B'. Then will BCB' and BAB' be semi-circumferences of great

A

circles, and the spherical angles B and B' will be equal to each other. As B is acute, its measure is the longest arc of a great circle that can be drawn perpendicular to the side BA and included between the sides of the angle B (B. IX., Gen. S. 2); hence, if the given side is greater than the measure of the given angle opposite, that is, if b > B, no triangle can be constructed, that is, there can be no solution: if b B, BC' and BA' will each be a quadrant (B. IX., P. IV.), and the triangle BA'C', or its equal B'A'C', will be birectangular (B. IX., P. XIV., C. 3), and there will be but one solution: if b < B, there will be two solutions, BAC and B'AC, the required parts of one being supplements of the required parts of the other.

=

Since B 90°, if b < B, b differs more from 90° than B does; and if b> B, b differs less from 90° than B.