Then BA'C' = & BAC',
X

and

X BC'A' = 4 BCA';
×

(678. The angles contained by the sides of a lune, at their two points of intersection,

for they have the common supplement AC'. Hence, keeping A and C on the line AC, slide ABC until AC comes into coincidence with A'C'. Then, the angles at A, C, A', C', being all right, AB will lie along A′B, and CB along С'B, and hence the figures ABC and A'BC' coincide. .. each of the half-lines ABA' and CBC' is bisected at B. In like manner, any other line drawn at right angles to AC passes through B, the mid point of ABA'.

Hence every sect from AC to B is a quadrant.

699. COROLLARY I. A line is a circle whose spherical radius is a quadrant.

700. COROLLARY II. A point which is a quadrant from two points in a line, and not in the line, is its pole.

701. COROLLARY III. Any sect from the pole of a line to the line is perpendicular to it.

702. COROLLARY IV. Equal angles at the poles of lines intercept equal sects on those lines.

703. COROLLARY V. If K be the pole, and FG a sect of

any other line, the angles and semilunes ABC and FKG are to one another as AC to FG.

704. We see, from 698, that a spherical triangle may have two or even three right angles.

If a spherical triangle ABC has two right angles, B and C, it is called a bi-rectangular triangle. By 698, the vertex A is the pole of BC, and therefore AB and AC are quadrants.

705. The Polar of a given spherical triangle is a spherical triangle, the poles of whose sides are respectively the vertices of the given triangle, and its vertices each on the same side of a side of the given triangle as a given vertex.

D

THEOREM VIII.

706. If, of two spherical triangles, the first is the polar of the second, then the second is the polar of the first.

HYPOTHESIS. Let A'B'C' be the polar of ABC.
CONCLUSION. Then ABC is the polar of A'B'C'.

PROOF. Join A'B and A'C.

Since B is the pole of A'C', therefore BA' is a quadrant; and since C is the pole of A'B', therefore CA' is a quadrant ;

.. by 700, A' is the pole of BC.

In like manner, B' is the pole of AC, and C' of AB.

Moreover, A and A' are on the same side of B'C', B and B' on the same side of A'C', and C and C' on the same side of A'B'. :. ABC is the polar of A'B'C'.

THEOREM IX.

707. In a pair of polar triangles, any angle of either intercepts, on the side of the other which lies opposite to it, a sect which is the supplement of that side.

Let ABC and A'B'C' be two polar triangles.

Produce A'B' and B'C' to meet BC at D and E respectively. Since B is the pole of B'C', therefore BE is a quadrant; and since C is the pole of A'B', therefore CD is a quadrant; therefore BE + CD half-line, but BE + CD = BC + DE. Therefore DE, the sect of BC which A' intercepts, is the supplement of BC.

EXERCISES. 109. Any lune is to a tri-rectangular triangle as its angle is to half a right angle.

THEOREM X.

708. If two angles of a spherical triangle be equal, the sides which subtend them are equal.

PROOF. For draw A'B'C', the polar of ABC.

Now, on B'C' and A'B' the equal s A and C intercept equal sects. Therefore B'C' and A'B', being, by 707, the supplements of these equal sects, are equal, × A' = × C',

(696. The angles at the base of an isosceles spherical triangle are equal.)

.. the supplements of BC and AB are equal,

709. If one angle of a spherical triangle be greater than a second, the side opposite the first must be greater than the side opposite the second.

B D

D

In λ ABC, C > X A.

:. by 708,

A

Make ACD = 4 A ;
X
AD = CD.

710. From 708 and 709, by 33, Rule of Inversion,

If one side of a spherical triangle be greater than a second, the angle opposite the first must be greater than the angle opposite the second.

THEOREM XII.

711. Two spherical triangles having two angles and the included side of the one equal respectively to two angles and the included side of the other, are either congruent or symmetrical.

For the first triangle can be moved in the sphere into coincidence with the second, or with a triangle made symmetrical to the second.

THEOREM XIII.

712. Two spherical triangles having three angles of the one equal respectively to three angles of the other, are either congruent or symmetrical.

Since the given triangles are respectively equiangular, their polars are respectively equilateral.

(702. Equal angles at the poles of lines intercept equal sects on those lines; and, by 707, these equal sects are the supplements of corresponding sides.)