BOOK X

SPHERES

636. A solid bounded by a surface, every point of which is equally distant from a point within, is called a Sphere.*

The point within is called the center.

A sphere may be generated by the revolution of a semicircle about its diameter as an axis.

637. A straight line drawn from the center to any point of the surface of a sphere is called a radius.

A straight line which passes through the center of a sphere, and whose extremities are in the surface, is called a diameter.

638. A line or plane which has one, and only one, point in common with the surface of a sphere is tangent to the sphere. The sphere is then said to be tangent to the line or plane.

639. Two spheres whose surfaces have one, and only one, point in common are tangent to each other.

640. When all the faces of a polyhedron are tangent to a sphere, the sphere is said to be inscribed in the polyhedron.

641. When all the vertices of a polyhedron lie in the surface of a sphere, the sphere is said to be circumscribed about the polyhedron.

*In teaching Spherical Geometry, the class-room should be furnished with a spherical blackboard, on which the student should draw the diagrams required. It is also advised that each student be provided with some sort of a blackened or slated sphere for use in the preparation of lessons in cases where figures are to be drawn on its surface. A hemispherical cup to fit the sphere will enable him to draw great circles on the sphere,

SOLID GEOMETRY.-BOOK X.

Ax. 18. All radii of the same sphere, or of equal spheres,

ll diameters of the same sphere, or of equal spheres, are

wo spheres are equal, if their radii or diameters are equal.

Proposition I

.. Form a sphere and cut it by any plane (§ 519 N.). What re is the section thus formed?

aline joins the center of the sphere with the center of a circle here, what is its direction with reference to the plane of the

a sphere by planes which are equally distant from its center. he sections thus formed compare?

he cutting planes are unequally distant from the center, which he larger?

em. Any section of a sphere made by a plane is

A sphere; its center 0; and any section, as ABD.

Draw Oc perpendicular to the plane ABD; draw the and OD to any two points in the perimeter of the secd draw CA and CD.

O is a point in the perpendicular OC,

and D are any two points in the perimeter of section

644. Cor. I.

The line joining the center of a sphere to the center of a circle of the sphere is perpendicular to the plane of the circle.

645. Cor. II.

Circles of a sphere made by planes equally distant from the center are equal.

646. Cor. III. Of two circles of a sphere made by planes unequally distant from the center, the nearer is the larger.

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

648. A section of a sphere made by a plane which does not pass through the center is called a Small Circle of the sphere.

649. The diameter, which is perpendicular to the plane of a circle of a sphere, is called the Axis of the circle.

650. The ends of the axis of a circle of a sphere are called the Poles of the circle.

651. 1. Form a sphere and cut it by any plane, thus forming a circle of the sphere. Through what point of the circle does its axis pass?

2. Form a sphere and cut it by two parallel planes, thus forming two parallel circles. How are their axes situated with reference to each other? How, then, are the poles of one of these circles situated with reference to the poles of the other?

3. By passing planes form any two great circles of the same, or of equal spheres. How do they compare with each other? 4. Form a sphere and divide it into two parts by a great circle. How do these parts compare with each other?

5. By passing planes form any two great circles of a sphere. Since their intersection passes through the center and is a diameter of each circle, how do two great circles divide each other?

6. Form two great circles of a sphere by passing two planes through it perpendicular to each other. Where do these circles pass with reference to each other's poles? If two great circles pass through each other's poles, what is the direction of their planes with reference to each other?

7. Form a sphere and pass a plane through its center and any two points on its surface. What kind of a circle is the section thus formed? What kind of an arc, then, may be drawn through any two points on the surface of a sphere?

8. Form a sphere and pass a plane through any three points on its surface. What plane figure is the section thus formed? How many planes may be passed through the three points? How many circles, then, may be drawn through any three points on the surface of a sphere?

652. The axis of a circle passes through the center of that circle. 653. Parallel circles have the same axis and the same poles. 654. Great circles of the same sphere, or of equal spheres, are equal.

655. Any great circle of a sphere bisects the sphere.

656. Two great circles of the same sphere bisect each other.

657. Two great circles whose planes are perpendicular pass through each other's poles; and conversely.

658. Through two given points on the surface of a sphere an arc of a great circle may be drawn.

659. Through three given points on the surface of a sphere one circle may be drawn, and only one.

Proposition II

660. Form a sphere and select any two points on its surface; through these points and the center of the sphere pass a plane. What kind of a circle is this section? Then, what kind of an arc joins the given points? Join them by any other line on the surface. Which line represents the shortest distance between the given points?

Theorem. The shortest distance on the surface of a sphere between any two points on that surface is the arc, not greater than a semicircumference, of the great circle which joins them.

D

E

Data: Any two points on the surface of a sphere, as A and B, joined by the arc of a great circle, as AB, not greater than a semi

circumference; also any other line on the surface joining 4 and B,

as AECB.

To prove Proof.

AB less than AECB.

Take any point in AECB, as D, and pass arcs of great circles through A and D, and B and D. Draw OA, OB, and OD from O, the center of the sphere.

Then, AOB, AOD, and BOD are the face of the trihedral hose vertex is at 0;

.. § 498, ▲ AOD + BOD is greater than ▲ AOB.

But, § 224, arcs AD, BD, and AB are the measures of AOD, BOD, and 40B respectively;

arc ADarc BD > arc AB.

In like manner, joining any point in AED with 4 and D, and any point in DCB with D and B by arcs of great circles, the sum of these arcs will be greater than arc AD + arc BD, and therefore greater than arc AB.

If this process is indefinitely repeated, the distance from A to B on the great circle arcs will continually increase, and always be greater than AB.

Hence, AECB, the limit of the sum of these great circle arcs, is greater than AB.

Therefore,

AB is less than AECB.

Q.E.D.

661. By the distance between two points on the surface of a sphere is meant the shortest distance; that is, the arc of a great circle joining them.

662. The distance from the nearer pole of a circle to any point its circumference is called the Polar Distance of the circle.

Proposition III

663. 1. Form a sphere and cut it by any plane; pass planes through the axis of the circle thus formed and any points in its circumference. What kind of arcs, then, connect the pole of the circle and the points of its circumference? How do these arcs compare? Then, how do the distances from the pole of a circle of a sphere to all the points in its circumference compare?