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PROPOSITION XXXVIII. THEOREM.

667. A frustum of any cone is equivalent to the sum of three cones whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum.

d

k

K

E

Let V denote the volume of the frustum, B its lower

base, b its upper base, and H its altitude.

We are to prove V = { H (B + b + √ B × b).

Let V' denote the volume of an inscribed frustum of a pyramid, B' its lower base, b' its upper base.

Its altitude will also be H.

Then,

V' = } H (B' + v' + √ B' × b'),

$ 578 (a frustum of any pyramid is to the sum of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and a mean proportional between the bases of the frustum).

Now, let the number of lateral faces of the inscribed frustum be indefinitely increased,

the new edges continually bisecting the arcs in the bases of the frustum of the cone.

But however great the number of lateral faces of the frustum of the pyramid,

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668. COROLLARY. If the frustum be that of a cone of revolution, and R and r be the radii of its bases, we have B

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= π R2,

BOOK VIII.

THE SPHERE.

ON SECTIONS AND TANGENTS.

669. DEF. A Sphere is a solid bounded by a surface all points of which are equally distant from a point within called the centre. A sphere may be generated by the revolution of a semicircle about its diameter as an axis.

670. DEF. A Radius of a sphere is the distance from its centre to any point in the surface. All the radii of a sphere are equal.

671. DEF. A Diameter of a sphere is any straight line passing through the centre and having its extremities in the surface of the sphere. All the diameters

of a sphere are equal, since each is equal to twice the radius.

672. DEF. A Section of a sphere is a plane figure whose boundary is the intersection of its plane with the surface of the sphere.

673. DEF. A line or plane is Tangent to a sphere when it has one, and only one, point in common with the surface of the sphere. 674. DEF. Two spheres are tangent to each other when their surfaces have one, and only one, point in common.

675. DEF. A polyhedron is circumscribed about a sphere when all of its faces are tangent to the sphere. In this case the sphere is inscribed in the polyhedron.

676. DEF. A polyhedron is inscribed in a sphere when all of its vertices are in the surface of the sphere. In this case the sphere is circumscribed about the polyhedron.

677. DEF. A Cylinder or cone is circumscribed about a sphere when its bases and cylindrical surface, or its base and conical surface, are tangent to the sphere. In this case the sphere is inscribed in the cylinder or cone.

PROPOSITION I. THEOREM.

678. Every section of a sphere made by a plane is a circle.

B

Let the section ABC be a plane section of a sphere whose centre is 0.

We are to prove section ABC a circle.

From the centre O draw OD to the section, and draw the radii O A, O B, O C, to different points in the boundary of the section.

In the rt. AO DA, O D B and O DC,

OD is common, and O A, O B and O C are equal,
(being radii of the sphere).

.. the rt. AO DA, O D B and ODC are equal,

(two rt.

$ 109

are equal when they have a side and hypotenuse of the one equal respectively to a side and hypotenuse of the other).

.. DA, D B and DC are equal,

(being homologous sides of equal §).

.. the section ABC is a circle whose centre is D.

Q. E. D.

679. COROLLARY 1. The line joining the centres of a sphere and a circle of a sphere is perpendicular to the circle.

680. COR. II. If R, r and p, respectively, denote the radius of a sphere, the radius of a circle of a sphere, and the perpendicular from the centre of the sphere to the circle, then √ R2 p2. Therefore all circles of a sphere equally distant from the centre are equal, and of two circles unequally distant from the centre of the sphere the more remote is the smaller.

r=

Again, if p=0, then r = R, and the centre of the sphere and the centre of the circle coincide; such a section is the greatest possible circle of the sphere.

681. DEF. A Great circle of a sphere is a section of the sphere made by a plane passing through the centre.

682. DEF. A Small circle of a sphere is a section of the sphere made by a plane not passing through the centre.

683. DEF. An Axis of a circle of a sphere is the diameter of the sphere perpendicular to the circle; and the extremities of the axis are the Poles of the circle.

684. Every great circle bisects the sphere. For, if the parts be separated and placed with their plane sections in coincidence and their convexities turned the same way, their convex surfaces will coincide; otherwise there would be points in the spherical surface unequally distant from the centre.

685. Any two great circles, ABCD

and AECF, bisect each other.

For the

intersection AC of their planes passes through the centre of the sphere, and is a diameter of each circle.

686. An arc of a great circle may be drawn through any two given points A and E in the surface of a sphere. For

D

F

the two points A and E, and the centre O, determine the plane of a great circle whose circumference passes through A and E.

$443

If, however, the two given points are the extremities A and C of the diameter of the sphere, the position of the circle is not determined. For, the points A, O and C, being in the same straight line, an infinite number of planes can pass through them.

§ 441

687. An arc of a circle may be drawn through any three given points on the surface of a sphere. For, the three points determine the plane which cuts the sphere in a circle.

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

M

N

Let O be the centre of a sphere, and MN a plane perpendicular to the radius O P, at its extremity P.

We are to prove MN tangent to the sphere.

From O draw any other straight line O A to the plane MN.

OP<0A,

(a is the shortest distance from a point to a plane).

.. point A is without the sphere.

But OA is any other line than O P,

§ 448

.. every point in the plane MN is without the sphere,

except P.

.. M N is tangent to the sphere at P.

$673

Q. E. D.

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

690. COR. 2. A straight line tangent to a circle of a sphere lies in a plane tangent to the sphere at the point of contact.

691. COR. 3. Any straight line in a tangent plane through the point of contact is tangent to the sphere at that point.

692. COR. 4. The plane of any two straight lines tangent to the sphere at the same point is tangent to the sphere at that

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