its plane. Through opposite points, an indefinite number of great circles can be passed.

623. Through any three points in a sphere, a plane can be passed, and but one; therefore three points in a sphere determine a circle of the sphere.

624. A small circle is the less the greater the sect from its center to the center of the sphere. For, with the same hypothenuse, one side of a right-angled triangle decreases as the other increases.

625. A Zone is a portion of a sphere included between two parallel planes. The circles made by the parallel planes are the Bases of the zone.

626. A line or plane is tangent to a sphere when it has one point, and only one, in common with the sphere.

627. Two spheres are tangent to each other when they have one point, and only one, in common.

EXERCISES. 105. If through a fixed point, within or without a sphere, three lines are drawn perpendicular to each other, intersecting the sphere, the sum of the squares of the three intercepted chords is constant. Also the sum of the squares of the six segments of these chords is constant.

106. If three radii of a sphere, perpendicular to each other, are projected upon any plane, the sum of the squares of the three projections is equal to twice the square of the radius of the sphere.

THEOREM III.

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

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For, by 597, this radius, being perpendicular to the plane, is the smallest sect from the center to the plane; therefore every point of the plane is without the sphere except the foot of this radius.

629. COROLLARY. Every line perpendicular to a radius at its extremity is tangent to the sphere.

630. INVERSE OF 628. Every plane or line tangent to the sphere is perpendicular to the radius drawn to the point of contact. For since every point of the plane or line, except the point of contact, is without the sphere, the radius drawn to the point of contact is the smallest sect from the center of the sphere to the plane or line; therefore, by 597, it is perpendicular.

EXERCISES. 107. Cut a given sphere by a plane passing through a given line, so that the section shall have a given radius.

108. Find the locus of points whose sect from point A is a, and from point B is b.

THEOREM IV.

631. If two spheres cut one another, their intersection is a circle whose plane is perpendicular to the line joining the centers of the spheres, and whose center is in that line.

HYPOTHESIS. Let C and O be the centers of the spheres, A and B any two points in their intersection.

CONCLUSION. A and B are on a circle having its center on the line OC, and its plane perpendicular to that line.

PROOF. Join CA, CB, OA, OB. Then

A CAO = A CBO,

because they have OC common, CA = CB, and OA = OB, radii of the same sphere.

Since these As are, therefore perpendiculars from A and B upon OC are equal, and meet OC at the same point, D.

Then AD and DB are in a plane 1 OC; and, being equal sects, their extremities A and B are in a circle having its center at D.

632. COROLLARY. By moving the centers of the two intersecting spheres toward or away from each other, we can make their circle of intersection decrease indefinitely toward its center; therefore, if two spheres are tangent, either internally or externally, their centers and point of contact lie in the same line.

THEOREM V.

633. Through any four points not in the same plane, one sphere, and only one, can be passed.

Let A, B, C, D, be the four points. Join them by any three sects, as AB, BC, CD. Bisect each at right angles by a plane.

The plane bisecting BC has the line EH in common with the plane bisecting AB, and has the line FO in common with the plane bisecting CD. Moreover, EH 1 plane ABC, and FO 1 plane BCD,

(575. If one plane be perpendicular to one of two intersecting lines, and a second plane perpendicular to the second, their intersection is perpendicular to the plane of the two lines.)

For O is in three planes bisecting at right angles the sects AB, BC, CD.

(601. The locus of all points from which the two sects drawn to two fixed points are equal, is the plane bisecting at right angles the sect joining the two given points.)

634. A Tetrahedron is a solid bounded by four triangular plane surfaces called its faces. The sides of the triangles are called its edges.

635. COROLLARY I. A sphere may be circumscribed about any tetrahedron.

636. COROLLARY II. The lines perpendicular to the faces of a tetrahedron through their circumcenters, intersect at a common point.

637. COROLLARY III. The six planes which bisect at right angles the six edges of a tetrahedron all pass through a common point.