Given a material sphere ABCE.

Required the diameter of the sphere.

Construction. From any point P as center, with any [opening of compasses as] radius, draw circumference ABC.

Measure [with compasses] the three chords AB, BC, and CA, and in any plane construct AA'B'C', having its sides respectively equal to AB, BC, and CA.

Construct D'A' the radius of the circumscribed circle A'B'C'. Draw right triangle P"A"D", having the hypotenuse P"A" = PA, and one arm D"A"

=

: D'A'.

At 4" draw "E" 1 P"A", meeting P"D" produced in E". Then P"E" is the required diameter.

HINT. Prove the equality of ▲ PAE and P"A"E".

728. DEF. The angle between two intersecting curves is the angle formed by the tangents at the point of contact.

729. DEF. A spherical angle is the angle between two intersecting great circles.

730. A spherical angle is measured by the arc of a great circle described from its vertex as a pole, and included between its sides, produced if necessary.

Given DPE, the spherical angle formed by the great circles PAC and PBC; AB, an arc of a great circle having P for its pole.

and 40 and DP are in the plane of O PAO.

731. COR. 1. An angle formed by two great circles is equal to the plane angle of the dihedral angle formed by their planes.

732. COR. 2. If the ends of a diameter (PC) are joined by two semicircles, the two spherical angles thus formed are equal. (I.e. C = Z P.)

733. DEF.

SPHERICAL POLYGONS

A spherical polygon is a portion of the surface of

a sphere bounded by three or more arcs of great circles.

The arcs are the sides, their points of intersection are the vertices, and the spherical angles formed by the sides are the angles of the polygon.

Thus, ABCD is a spherical polygon, AB, BC, etc., its sides, A, B, C, etc., its vertices, and ABC, BCD, etc., its angles.

734. DEF. A diagonal of a spherical polygon is an arc joining any two non-adjacent vertices.

735. DEF. A spherical triangle is a spherical polygon of three sides. It is called isosceles, equilateral, etc., in the same cases in which a plane triangle would be so called.

The planes of the sides of a spherical polygon form at the center a polyhedral angle (0-ABCD) which is said to correspond with the spherical polygon.

The sides of the spherical polygon are measured by the face angles of the corresponding polyhedral angle; its angles are equal to the dihedral angles of the corresponding polyhedral angle.

736. REMARK. By means of the relations between the parts of a spherical polygon and those of its corresponding polyhedral angle, we can deduce from any theorem of polyhedral angles an analogous theorem of spherical polygons.

737. A spherical polygon is convex if its corresponding polyhedral angle is convex. All spherical polygons are supposed to be convex polygons unless stated otherwise.

738. Spherical polygons are symmetric if their corresponding polyhedral angles are symmetric. Evidently, their parts must be respectively equal, but follow in reverse order.

In general, two symmetric spherical polygons cannot be made to coincide.

The sides of a spherical polygon are usually measured in degrees.

Ex. 1663. State the theorems relating to spherical polygons which follow directly from the following propositions :

(a) Two trihedral angles are congruent or symmetric if their face angles are respectively equal.

(b) If two face angles of a trihedral angle are equal, the opposite dihedral angles are equal.

(c) If the three face angles of a trihedral angle are right angles, the three dihedral angles are right angles.

(d) The sum of the face angles of a polyhedral angle is less than 360°. (e) The sum of two face angles of a trihedral angle is greater than the third face angle.

(f) If the opposite face angles of a tetrahedral angle are equal, the opposite dihedral angles are equal.

PROPOSITION X. THEOREM

739. The sum of two sides of a spherical triangle is greater than the third side.

But the central angle is measured by the intercepted arc.

.. AB+BC > AC.

Q. E. D.

PROPOSITION XI. THEOREM

740. The sum of the sides of any spherical polygon is less than four right angles.

B

E

Given ABCDE, a spherical polygon of n sides.

To prove AB+BC + CD +DE+EA, etc. <360°.

HINT. Construct the corresponding polyhedral angle and compare Remark (736).

PROPOSITION XII. THEOREM

741. Two triangles on the same sphere are congruent :

(1) If two angles and the included side of the one are respectively equal to two angles and the included side of the other,