THEOREM XIX.

604. If the vertices of a convex polygon be joined to a point not in its plane, the sum of the vertical angles of the triangles so made is less than a perigon.

PROOF. The sum of the angles of the triangles which have the common vertex S is equal to the sum of the angles of the same number of triangles having their vertices at O in the plane of the polygon. But

(603. If three lines not in a plane meet at a point, any two of the angles formed are together greater than the third.)

Hence, summing all these inequalities, the sum of the angles at the bases of the triangles whose vertex is S, is greater than the sum of the angles at the bases of the triangles whose vertex is 0; therefore the sum of the angles at S is less than the sum of the angles at O, that is, less than a perigon.

BOOK VIII.

TRI-DIMENSIONAL SPHERICS.

605. If one end point of a sect is fixed, the locus of the other end point is a Sphere.

606. The fixed end point is called the Center of the sphere.

607. The moving sect in any position is called the Radius of the sphere.

608. As the motion of a sect does not change it, all radii are equal.

609. The sphere is a closed surface; for it has two points on every line passing through the center, and the center is midway between them.

610. Two such points are called Opposite Points of the sphere, and the sect between them is called a Diameter.

611. The sect from a point to the center is less than, equal to, or greater than, the radius, according as the point is within, on, or without, the sphere.

For, if a point is on the sphere, the sect drawn to it from the center is a radius; if the point is within the sphere, it lies on some radius; if without, it lies on the extension of some radius.

612. By 33, Rule of Inversion, a point is within, on, or without, the sphere, according as the sect to it from the center is less than, equal to, or greater than, the radius.

THEOREM I.

613. The common section of a sphere and a plane is a circle.

D

A

CY

Take any sphere with center O.

Let A, B, C, etc., be points common to the sphere and a and OD the perpendicular from 0 to the plane. Then A

OA = OB = OC = etc.,

being radii of the sphere,

:. ABC, etc., is a circle with center D.

plane,

and cand to the fool d

(598. Equal obliques from a point to a plane meet the plane in a circle whose center is the foot of the perpendicular from the point to the plane.) Shed

op

614. COROLLARY.

The line through the center of any circle of a sphere, perpendicular to its plane, passes through the center of the sphere.

615. A Great Circle of a sphere is any section of the sphere made by a plane which passes through the center.

All other circles on the sphere are called Small Circles. 616. COROLLARY. All great circles of the sphere are equal, since each has for its radius the radius of the sphere.

617. The two points in which a perpendicular to its plane, through the center of a great or small circle of the sphere, intersects the sphere, are called the Poles of that circle.

618. COROLLARY. Since the perpendicular passes through the center of the sphere, the two poles of any circle are opposite points, and the diameter between them is called the Axis of that circle.

THEOREM II.

619. Every great circle divides the sphere into two congruent hemispheres.

For if one hemisphere be turned about the fixed center of the sphere so that its plane returns to its former position, but inverted, the great circle will coincide with its own trace, and the two hemispheres will coincide.

620. Any two great circles of a sphere bisect each other.

Since the planes of these circles both pass through the center of the sphere, their line of intersection is a diameter of the sphere, and therefore of each circle.

621. If any number of great circles pass through a point, they will also pass through the opposite point.

622. Through any two points in a sphere, not the extremi

ties of a diameter, one, and only one, great circle can be passed; for the two given points and the center of the sphere determine