given, the other three may be found. For practical application they are transformed and simplified in numerous ways.

98. Permutation of parts. We have deduced the equations (1) from (a) by merely permuting the letters. This process may be applied generally in accordance with the following theorem :

If we have found between the parts of a spherical triangle any equation which is true for all triangles, it will remain true when we permute the sides in any way; provided that we also permute the opposite angles in the same way.

For if we have proved our equation by calling the three sides a, b, and c, and the opposite angles A, B, and C, we could apply the same proof to the other parts of the triangle, substituting

and

Side a for side b, and vice versa,

Angle A for angle B, and vice versa,

in the demonstration. We should then have a result in which a and 6 changed places, and A and B changed places.

By interchanging a and c, and b and c, with their opposite angles in the same way, we should form all the six equations which could be written by permuting the symbols in the way described.

If, however, we made any supposition respecting any side or angle such that the reasoning applied to it would not apply to the others, then the symbol of this side or angle could not be permuted. For instance, we cannot permute all the parts in a formula true only for right triangles.

It follows from this that any true formula which expresses the value of one part in terms of the two remaining pairs of parts must be symmetrical with respect to the other pairs of parts. For example, equation a remains unchanged when we interchange b and c, else it would be wrong.

99. THEOREM OF SINES. In a spherical triangle the sines of the sides are proportional to the sines of the opposite angles.

Proof. Let O-ABC be the trihedral angle of the spherical tritriangle, and let A be any point on

the edge OA.

Through A pass a plane perpendicular to the edge OB, intersecting the faces AOB and BOC in the lines AB and BP.

Through A pass another plane perpendicular to OC, intersecting the

faces AOC and COB in the lines AC and CP.

B

AP will then be the line of intersection of these planes.

Because the planes ABP and ACP are perpendicular to the lines OB and OC respectively, they are each perpendicular to the plane BOC of these lines (Geom.) Therefore their line of intersection, AP, is also perpendicular to this plane, and the triangles APB and APC are right-angled at P. Hence

AP AB sin ABP = AB sin B.

Also, because AB is perpendicular to OB,

We may show in the same way, by permuting the parts,

The common value of these three ratios is called the modulus

of the spherical triangle.

100. The theorem of sines may also be obtained directly from the fundamental equations as follows:

From the first fundamental equation (1) we obtain

The second member of this equation is symmetrical with respect to a, b, and c, and so remains unchanged when these quantities are permuted among themselves. But if we derive the

values of

sin' B sin' b

and

sin' C sin' c

from the last two fundamental equa

tions, the results will be simple permutations of the last equation,

sin' b sin' c'

Extracting the square roots, the general results will have double (±) algebraic signs; but as the angles are all supposed to be less than 180°, the positive signs are to be taken.

Hence

the reciprocal of the relations (2).

101. Polar triangles. Def. When two triangles are so related that the vertices of the one are the poles of the sides of the other, the one is said to be the polar triangle of the other.

It is shown in geometry that the relation of a triangle to its. polar triangle is reciprocal; that is, if X and Y are two triangles, and Y is the polar triangle of X, then X is the polar triangle of Y. This reciprocity arises from the theorem:

If A, B, and C be the three poles of the sides QR, RP, and PQ of a triangle PQR, then P, Q, and R will be the poles of the sides BC, CA, and AB.

This theorem is readily proved by the geometry of the sphere.

Since every great circle has two poles, one at each end of a diameter, it follows that the three sides of a triangle have six poles in all. We may form a polar triangle to ABC by taking either of the poles of AB, either of the poles of BC, and either of

the poles of CA, and joining them by ares of great circles. Hence there are eight possible polar triangles to every given triangle. To avoid doubt which triangle is to be chosen, we take for each vertex of the polar triangle that pole of each side of the given triangle

which is on the side toward the triangle.

R

B

For example, if ABC is the given triangle, we take that pole of AB which is on the side toward C, and so with the other sides.

EXERCISES.

1. What must be the sides and angles of a triangle that it may coincide with its polar triangle?

2. Show that if each side of a triangle is greater than 90° the polar triangle will fall wholly inside of it, and if each side is less than 90° it will be wholly within its polar triangle.

3. If two sides exceed 90° and the third side is less than 90°, what will be the character of the polar triangle, and how will it be situated relatively to the given one?

102. Use of the polar triangle. It is shown in geometry that each side of the polar triangle is the supplement of the opposite angle of the other, and vice versa. This principle is applied to find new relations between the parts of a triangle in the following way:

1. We imagine ourselves to construct the polar of the given triangle.

2. We write any or all the equations between the parts of the polar triangle.

3. We substitute in these equations the supplementary parts

of the given triangle, and thus obtain equations between these parts.

the sides and opposite angles of the polar triangle. Since the general equations (1) are true for every triangle, they are true of this polar triangle. Hence

cos a' cos b' cos c' + sin b' sin c' cos A'.

But the polar triangle is so related to the original triangle that

Making these substitutions in the equations (1), we find

From every relation between the parts of a spherical triangle we may derive another relation by changing the cosine of each part into the negative of the cosine of the opposite part, and the sine of each part into the sine of the opposite part.

one.

But this relation will not always be different from the original If we apply the process to the equations (2), for instance, the same relations will be reproduced, each term being changed to its reciprocal.

It is also to be remarked that the use of the polar triangle is not absolutely necessary to deduce the new relations (3), because they can all be obtained from the fundamental equations (1) by eliminating first b and c, then e and a, then a and b. But the use