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.

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THE ELEMENTS OF GEOMETRY.

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

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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

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.

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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

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.