Bisect three diedral angles at the base of a triangular pyramid with planes.

How does any point in either of the bisecting planes relate to the sides of the diedral angle bisected? How will the three bisecting planes meet?

Given: Any tetraedron, v- A B D.

Can you inscribe a sphere within it?
Write the theorem, and call it Prop. IX.

636.

Cor. Can you prove how the six bisecting planes of the diedral angles of a tetraedron meet?

Imagine part of a hollow glass sphere to receive in part

another smaller sphere.

Can you picture the section made by the surfaces coming in contact?

What is the section of the intersection of the the surfaces of two spheres? How is the section. related to the line of centers?

Sug. 1. Let S and S' represent two intersecting spheres whose centers are C and C'. Pass a plane through C and C'. What are the circles formed?

Sug. 2. Letter points of intersection of the circles P and P' and draw the chord P P'.

Extend the line C C' to meet the circumferences.

Sug. 3. How is C C' related to P P? Why?

Sug. 4. Imagine the lower half of the figure composed of the intersecting circles to revolve about the axis C C' produced. What will the two semicircles generate? the line O P? the point P?

Give a complete demonstration, and write Prop. X.

EXERCISES.

512. Three equal lines have their extremities in the surface of a sphere. How are these lines related to the center of

the sphere?

513. If a cone of revolution roll upon a plane with its vertex fixed, what kind of a surface is generated by the surface of the cone?

Can you prove that one, and only one, surface of a sphere may be passed through any four points not in the same plane?

Sug. 1. Suppose A, B, C, D to be the four points not in

he same plane.

What is the locus of all points equally distant from A and B, B and C? What is the intersection of these two loci? Sug. 2. What is the locus of all points equally distant from B, D?

Note the intersection of the loci.

The pupil will work out the demonstration.

SPHERICAL ANGLES AND POLYGONS.

DEFINITIONS.

638.

The angle of two intersecting curves is the angle of the two tangents to the curves at their point of intersection. This definition applies to all curved surfaces.

639.

A spherical angle is the angle included between two arcs of great circles of a sphere. The arcs are the sides of the angle and their intersection the vertex.

640.

A portion of the surface of a sphere bounded by three or more arcs of great circles is called a spherical polygon.

The sides of the spherical polygon are the bounding arcs;

the angles of the polygon are the angles of the intersecting arcs; the vertices of the polygon are the points of intersection of the arcs.

Thus, A B D E is a spherical polygon. Its sides may be expressed in degrees.

641.

The diagonal of any spherical polygon is the arc of a great circle joining any two vertices not adjacent.

642.

The planes of the sides of a spherical polygon form a polyedral angle whose vertex is at the center of the sphere. Thus, CA B D E is a polyedral angle with its vertex at C.

643.

A convex spherical polygon is a spherical polygon whose corresponding polyedral angle is convex.

644.

A spherical triangle is a spherical polygon of three sides.

645.

A spherical triangle may be right, acute, equilateral, isosceles, etc., under the same restrictions as plane triangles.

646.

Any two points on the surface of a sphere may be joined by two arcs of a great circle; one will usually be greater than a semicircumference, the other less.

The smaller arc is always meant unless otherwise stated.

647.

Two spherical polygons are equa! when one may be applied to the other so that they will coincide in all their parts. Thus,

B'

the spherical triangles A B C and A' B' C' are equal if A B, B C, C A are equal respectively to A' B', B' C', C' A', and the angles A, B, C respectively equal the angles A', B', C'.

648.

Two spherical polygons are symmetrical when the sides and angles of one are equal respectively to the sides and angles of the other when taken in reverse order. Study the figures.

Can you make these figures coincide?

[The question of equality and symmetry of spherical triangles may be cleared by using the rind of an orange.