Draw the chord AB, and produce oo' to meet the circumferences.

oo' is to AB at its mid point C.

(75)

If we now revolve the upper part of the figure about 00', the two semicircles will generate the two spheres (583), while the point 4 will generate the line of intersection of the surfaces.

Also, since AC during the revolution remains

to oo',

AC will generate a circle whose center is C ;

i.e., the intersection of the surfaces of the spheres is the circumference of a circle to oo'.

Q.E.D.

606. DEFINITION. Two spheres are tangent if their surfaces have but one common point.

607. COR. If two spheres are tangent to each other, the point of tangency is in their line of centers.

For if we conceive the centers, 0, 0', to remove farther from each other till the circumferences of the great circles become tangent, the points A and B will coincide with, and the circumference of intersection be reduced to, a point C.

608. SCHOLIUM. Two spheres being given in any relative position, a plane passed through their centers will cut them in great circles; and according as these circles are within or without, are tangent or intersecting, the spheres will have corresponding positions in regard to each other.

609. DEFINITION. A plane is tangent to a sphere when it has but one point in common with the surface of the sphere.

EXERCISE 788. Denoting by a and r the altitude and radius of a cone rolling as in Ex. 784, find the ratio of the base of the cone to the entire circle generated.

789. What fraction of that circle is described by one revolution of the cone ?

PROPOSITION IX. THEOREM.

610. A plane perpendicular to a radius of a sphere at its extremity is tangent to the sphere.

M

P

N

Given: OP, a radius of a sphere, and a plane MN to OP at its extremity;

To Prove: MN is tangent to the sphere.

Take any point except P in MN, as 4, and join OA, PA.

Since OP is to AP,

OA > OP;

.. A lies without the sphere;

(Hyp., 427) (99")

.. MN is tangent to the sphere, Q.E.D. (609)

(since every point in MN, except P, lies without the sphere.)

611. COR. 1. A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact.

612. COR. 2. Any straight line drawn in the tangent plane through the point of contact is tangent to the sphere.

613. COR. 3. Any two straight lines tangent to the sphere at the point of contact determine the tangent plane at that point.

SPHERICAL ANGLES AND POLYGONS.

614. DEFINITION. The angle of two intersecting curves is the angle contained by the two tangents to the curves at the common point.

This definition applies, whatever be the surface upon which the curves are described.

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

PROPOSITION X. THEOREM.

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

Given: AB, an arc of a great circle described from the vertex of spherical angle APB as pole, and included between the sides of angle APB;

To Prove: Spherical angle APB is measured by arc AB.

Draw PT, PT', tangents to PAP', PBP' respectively, and radii OA, OB.

Since PT is to PP' in plane PAP', (Const., 191)

and 04 is to PP' in plane PAP',

(PA being a quadrant,)

PT is to OA;

(Hyp.)

(106)

(455)

But

AOB is meas. by arc AB;

(262)

.. / TPT' is meas. by arc AB,

i.e., spher.

APB is meas. by arc AB. Q.E.D. (614)

617. COR. 1. A spherical angle has the same measure as the dihedral angle formed by the planes of its sides.

618. COR. 2. All arcs of great circles drawn through the pole of a given great circle are perpendicular to its circumfer

ence.

For their planes are each perpendicular to its plane (469).

A

619. SCHOLIUM. The foregoing corollary enables us, through any given point P on the surface of a sphere, to describe an arc of a great circle perpendicular to a given arc ABC of a great circle. From P as pole describe (428) an arc of a great circle cutting ABC in C, and from C as pole describe an arc of a great circle passing through

B

P and cutting ABC in B; then arc PB is to arc ABC (618).

620. DEFINITION. A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The bounding arcs are the sides of the polygon; the angles formed by these sides are the angles, and the vertices of the angles are the vertices, of the polygon.

621. Since the planes of all great circles pass through the center of the sphere, the planes of the sides of a spherical polygon form at the center a polyhedral angle whose edges are radii drawn to the vertices of the polygon; the face angles of the polyhedral angle are angles at the center measured by the arcs that form the sides of the polygon; while the dihedral angles of the polyhedral angle have the same measure as the angles of the polygon (616).

Thus the planes of the sides of the polygon ABCD (see diagram for Prop. XII.) form at O, the center of the sphere, the polyhedral angle O-ABCD. The face angles, AOB, BOC, etc., are measured by the sides AB, BC, etc., of the polygon; and the dihedral angle whose edge is the radius OA, has the same measure as the spherical angle BAD, etc.

622. From the relations thus established between polyhedral angles and spherical polygons, it clearly follows that from any known property of polyhedral angles we may infer a corresponding property of spherical polygons; and conversely.

623. DEFINITION. A diagonal of a spherical polygon is an arc of a great circle passing through two nonadjacent vertices.

624. DEFINITION. A spherical triangle is a spherical polygon having three sides. Like plane triangles, spherical triangles may be right or oblique, scalene, isosceles, or equilateral.

EXERCISE 790. Out of a circle with radius R a sector of 60° is cut, and the edges of the remaining sector are joined so as to form the lateral surface of a cone of revolution. Find the radius r, and the altitude a, of the cone thus formed.

791. Find general formulas for the radius r and the altitude a of a cone of revolution whose lateral surface is formed from a sector