spherical triangle. Arcs a, b, and c are the measures of a, B, and y respectively. BCA and D-OC-E are measured by the same plane angle, as also are ZABC and E-OD-C, and ZCAB and C-OE-D.

110. The sum of the sides of a spherical triangle is less than 360°. The sum of the angles of a spherical triangle is greater than 180° and less than 540°.

It is evident that the sides and angles of a spherical triangle can be greater than 180°; however, to simplify the subject, it is agreed to consider only those spherical triangles in which the sides and angles are each less than 180°.

111. Polar triangles. If the vertices of a spherical triangle are used as poles and great circles drawn, another triangle is formed called the polar triangle of the first.

Thus in Fig. 116, A is the pole of a', B the pole of b', C the pole of c', and A'B'C' is the polar triangle of ABC.

It is evident that, in general, the great circles drawn as stated will intersect so as to form eight triangles. The one of these is the polar triangle in which A and A', B and B', C and C' lie on the same side of a', b', and c' respectively.

If one triangle is the polar of another, then the latter is the polar triangle of the former.

The sides and angles of a spherical triangle are the supplements, respectively, of the angles and sides opposite in the polar triangle, and conversely.

Thus in Fig. 116, A' = π — a, B' =π

- C,

a'

c'

=π- - C.

These relations are of great importance, for, if any general theorem be proved with respect to the sides and angles of any

spherical triangle, it can at once be applied to the polar triangle. Thus any theorem of a spherical triangle may be at once transformed into another by replacing each side, or angle, by the supplement of its opposite angle, or side, in the polar triangle.

112. Since the side of a spherical triangle and the corresponding face angle of the trihedral angle have the same numerical measure, the plane trigonometric functions may be taken of the arcs as well as of the plane angles. Hence the identities of plane trigonometry are true for the sides of a spherical triangle.

113. A right spherical triangle is one which has an angle equal to 90°. A birectangular triangle is one which has two right angles. A trirectangular triangle is one which has three right angles.

RIGHT SPHERICAL TRIANGLES

114. In a spherical triangle there are six parts, three sides and three angles, besides the radius of the sphere which is supposed known. In general, if three of these parts are given the other parts can be found. If the triangle is a right spherical triangle, two given parts in addition to the right angle are sufficient to solve the triangle.

Since there are three unknowns to be found in solving a right triangle, it is necessary to have any two given parts combined. with the remaining three in three independent relations or formulas. Now, since the five parts taken three at a time form ten combinations, ten formulas are necessary and sufficient to solve all right spherical triangles.

If a, b, c, a, and ẞ are the five parts of the triangle, omitting the right angle, then the ten combinations of these taken three at a time are abc, aba, abß, aca, acß, aaß, bca, bcß, baß, and caß. It is necessary to derive a formula connecting the parts in each of these combinations.

115. Derivation of formulas for the solution of right spherical triangles. -Let O be the center of a sphere of unit radius, and ABC a right spherical triangle, with y the right angle,

(abc)

116. Napier's rules of circular parts. The preceding ten formulas for the solution of right spherical triangles are included in a theorem first stated and proved by Napier. The theorem is usually stated as two rules known as "Napier's rules of circular parts."

In the right spherical triangle ABC, omit the right angle at C and consider the sides a and b, and the complements of a, B, and c. Call these the circular parts of the triangle and designate them as a, b, co-a, co-ß, and co-c.

In the triangle or the circular scheme shown in the figure, any one of these five parts may be selected and called the middle part; then the two parts next to it are called adjacent parts, and the other two parts the opposite parts. For example, if

b is chosen as the middle part, then co-a and a are the adjacent parts and co-c and co-ẞ are the opposite parts. Napier's rules are then stated as follows:

(1) The sine of a middle part equals the product of the tangents of the adjacent parts.

(2) The sine of a middle part equals the product of the cosines of the opposite parts.

It may assist in remembering the rules to notice the repetition of a in (1) and of o in (2).

Napier's rules may be verified by showing that they give the ten formulas of Art. 115. A demonstration of the theorem as given by Napier may be found in Todhunter and Leathem's Spherical Trigonometry.

Example. Use co-a as the middle part and apply rule (1).

sin (co-a) = tan b tan (co-c).

.. COS α = tan b cot c, which is formula (8).

Exercise. Verify all the ten formulas by Napier's rules.

Napier's rules thus furnish a very convenient way for the determination of the formulas for the solution of right spherical triangles.

117. Species. Two angular quantities are said to be of the same species when they are both in the same quadrant, and of different species when they are in different quadrants.

Since any or all the parts of a right spherical triangle may be less than or greater than 90°, it is necessary to have a method for the determination of the species of the parts. The following rules will be found to cover all cases:

(1) An oblique angle and its opposite side are always of the same species.

(2) If the hypotenuse is less than 90°, the two oblique angles and therefore the two sides of the triangle are of the same species; if the hypotenuse is greater than 90°, the two oblique angles and therefore the two sides are of opposite species.

These rules are here verified in two cases. As an exercise, the student is asked to verify them in other cases.

If

Example 1. By formula (5), Art. 115, cos c = cot a cot B. c< 90°, cos c is +. Then the product cot a cot ẞ must be +, and this will be true if cot a and cot ẞ are both + or both that is, if a and B are both in the same quadrant.

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