Hence, making the necessary simplifications, we have surf. AD': surf. BECE' = vol. AD': vol. BECE' = 3:2. Q.E.D.

This interesting theorem is known as the Theorem of Archimedes, it having been discovered by that celebrated geometer.

720. SCHOLIUM. If we have a cone having the same base and altitude as the cylinder circumscribed about the sphere, then, since the volume of such a cone is π AB BC= π · AB3, we obtain the relation :

cylinder: sphere: cone 3:2:1.

EXERCISES.

NUMERICAL.

886. Find the area of the surface of a sphere whose radius is 3 in. 887. The surface of a sphere is to be 100 sq. in. What radius should be taken ?

888. Two spheres have radii of 9 in. and 5 in. respectively. What is the ratio of the surfaces of those spheres? Of their volumes ?

889. The areas of the surfaces of two spheres are as 125 to 27. What is the ratio of their diameters? Of their volumes ?

890. Two parallel planes intersect a sphere of 18 in. radius at distances of 9 in. and 13 in., respectively, from the center. Find the area of the intercepted zone.

891. What is the volume of the spherical sector that has for base the zone just mentioned ?

892. Find the altitude of a zone whose area is 100 sq. in. on the surface of a sphere of 12 in. radius.

893. In the same sphere, what is the altitude of a zone that contains one fourth of the surface of the sphere?

894. What is the volume of a sphere whose diameter is (1) 1 ft. ; (2) 18 in. ?

895. The surface of a sphere is 64 sq. in.

896. The volume of a sphere is 5 cu. ft. surface.

Find its volume.

Find its diameter and

897. Find the difference of the volumes of two spheres whose radii are 12 in. and 7 in. respectively.

898. The volumes of two spheres are as 27 to 8. Find the ratio (1) of their diameter; (2) of their surfaces.

899. The radii of two spheres are as 4 to 5. Find the ratio (1) of their volumes; (2) of their surfaces.

900. In a sphere whose radius is 6 in., find the altitude of a zone whose area shall be that of a great circle.

901. The area of a zone forming the base of a spherical sector is 50 sq. in.; the radius of the sphere is 12 in. Find the altitude of the zone and the volume of the sector.

902. The volume of a spherical sector is 25 cu. in.; the diameter of the sphere is 14 in. Find the area of the zone that forms the base of

the sector.

903. The altitude of a cylinder circumscribing a sphere is 5 in. Find the surface and volume of the sphere.

904. The volume of a sphere is one cubic foot. Find the surface of the circumscribing cylinder.

905. The surface and volume of a sphere are expressed by the same number. Find its diameter.

906. Find the volume of a sphere inscribed in a cube whose volume is 1331 cu. in.

907. Find the surface of a cube circumscribed about a sphere whose surface is 150 sq. in.

908. If a spherical shell have an exterior diameter of 12 in., what should be the thickness of its wall so that it may contain 696.9 cu. in.?

909. If an iron sphere, 6 in. in diameter, weigh n lbs., what will be the weight of an iron sphere whose diameter is 8 in. ?

910. In a sphere 10 in. in diameter, the radius of the lower base of a spherical segment is 8 in. Find the volume of the segment, its altitude being 2 in.

Geom.-23

THEOREMS.

911. The lateral area of a cylinder of revolution is equal to the area of a circle whose radius is a mean proportional between the altitude and diameter of the cylinder.

912. The lateral areas of the two cylinders generated by revolving a rectangle successively about each of its containing sides, are equal.

913. If the containing sides of the above rectangle are as m is to n, the total areas, and also the volumes, of the cylinders generated, will be as n is to m.

914. If the slant height of a cone of revolution is equal to the diameter of its base, its total area is to that of the inscribed sphere as 9 is to 4.

915. The arms of a right triangle are a and b. Find the area of the surface generated by revolving the triangle about its hypotenuse.

916. An equilateral triangle revolves about one of its altitudes. What is the ratio of the lateral surface of the generated cone to that of the sphere generated by the circle inscribed in the triangle ?

917. An equilateral triangle revolves about one of its altitudes. Compare the volumes generated by the triangle, the inscribed circle, and the circumscribed circle respectively.

918. A circle of cardboard being given, what is the angle of the sector that must be cut from it so that with the remainder, a cone with a vertical angle of 90° may be formed?

919. If the diameter of a sphere be divided by a perpendicular plane in the ratio m to n, the zones thus formed will also be as m to n.

920. The volume of a cylinder of revolution is equal to one half the product of its lateral area by the radius of its base.

921. If the altitude of a cylinder of revolution is equal to the diameter of its base, its volume is equal to one third the product of its total area by the radius of its base.

922. The base of a cone is equal to a great circle of a sphere, and the altitude is equal to a diameter of the sphere. What is the ratio of their volumes ?

923. The volume of a sphere is to that of the inscribed cube as π is to 2√3.

924. A sphere is to the circumscribed cube as π is to 6.

721. DEFINITION. Two points are said to be symmetrical with respect to a third point, if this point bisects the line joining the two points.

P

Thus the points P and P' are symmetrical with respect to 0, if the line PP' is bisected in 0. The point is then called the center of symmetry.

722. DEFINITION. Two figures are said to be symmetrical

with respect to a point, called their center of symmetry, if every point in the one has its symmetrical point in the other.

Thus the figures A B C, A'B'C', are symmetrical with respect to the center O, if every point in ABC has its symmetrical point in A'B'C'.

B

Β'

723. DEFINITION. In symmetrical figures, sides whose extremities are mutually symmetrical are said to be homologous. Thus AC is homologous to A'C', since 4 is symmetrical with A', and C with c'.

EXERCISE 925. The opposite vertices of a regular polygon of an even number of sides have a common center of symmetry.

926. The opposite vertices of a parallelopiped have a common center of symmetry.

PROPOSITION I. THEOREM.

724. If two polygons are symmetrical with respect to a center, any two homologous sides are equal and parallel, and drawn in opposite directions.

Given: Two polygons AB... E, A'B' E', symmetrical with respect to 0;

also AB, A'B', are drawn in opposite directions; (115) and similarly for BC and B'C', CD and C'D', etc. Q.E.D.

725. COR. 1. Any line MM', intercepted between two homologous sides, AE, A'E', and passing through O, is bisected in 0. For since AE is to A'E', the triangles AOM, A'OM', are equiangular; and 0404'; .. OM OM' (70).

0A =

726. COR. 2. If two polygons have their sides respectively equal and parallel, and drawn in opposite directions, they have a center of symmetry.

For if AB and A'B' are equal and parallel, and drawn on opposite sides of AA', BB', then A A', B B', are diagonals of