775. COROLLARY. The volume of a spherical pyramid is to the volume of the tri-rectangular pyramid, as the base of the pyramid is to the tri-rectangular triangle. And, since the volume of the tri-rectangular pyramid is š the volume of the sphere, and the area of the tri-rectangular triangle is ļ of the surface of the sphere ; the volume of a spherical pyramid is to the volume of the sphere as its base is to the surface of the sphere. 776. Def. A Zone is the part of the surface of a sphere included between two parallel circles of the sphere; as the surface included between the circles A B C and EFG. 777. DEF. The Bases of a zone are the circumferences of the intercepting circles; as circumferences ABC and E FG. If the plane of one base become tangent to the sphere, that base becomes a point, and the zone will have but one base. 778. Def. The altitude of a zone is the perpendicular distance between the planes of its bases. 779. DEF. A Spherical Segment is a part of the sphere included between two parallel planes. 780. DEF. The Bases of a spherical segment are the bounding circles. One of the planes may become a tangent plane to the sphere. In this case the segment has but one base. 781. DEF. The Altitude of a spherical segment is the perpendicular distance between the planes of its bases. 782. DEF. A Spherical Sector is a part of a sphere generated by a circular sector of the semicircle which generates the sphere; as AOC K. 783. DEF. The Base of a spherical sector is the zone generated by the arc of the circular sector; as ACK. The other bounding surfaces of a spherical sector may be one conical surface, or two conical surfaces; or one conical and one plane surface. Thus, let A B be the diameter around which the semicircle AC B revolves to generate the sphere. The solid generated by the circular sector AOC will be a spherical sector having the zone A C K for its base, and for its other bounding surface the conical surface generated by C 0. The spherical sector generated by COD has for its base the zone generated by C D, and for its other surfaces the concave conical surface generated by D 0, and the convex conical surface generated by Co. The spherical sector generated by EO F has for its base the zone generated by E F, and for one surface the plane surface generated by E 0, and for the other surface the concave conical surface generated by F 0. PROPOSITION XXXV. THEOREM. 784. The area of a zone is equal to the product of its altitude by the circumference of a great circle. Let A B C D E be the circumference of a great circle, BC any arc of this circumference, and 0 A the If the semicircle A B C D revolve about the diameter A D as an axis, the semi-circunference A B C D will generate the surface of a sphere; the arc B C, a zone, and the chord B C, a surface whose area is PRX 27 01. $759 Now if we bisect the arc B C, and continue this process indefinitely, the surface generated by the chords of these arcs will approach the zone as its limit; the I o I will approach the radius of the sphere as its limit; while P R will remain constant. Q. E. D. 785. COROLLARY 1. Zones on the same sphere, or equal spheres, have the same ratio as their altitudes. 786. Cor. 2. A zone is to the surface of the sphere as the altitude of the zone is to the diameter of the sphere. 787. Cor. 3. Let arc A B generate a zone of a single base. Then, zone A B=AP X 2 7 0 A. Hence, zone A B = 1 A P X AD=7 Ā B”. (§ 307.) That is, a zone of one base is equivalent to a circle whose radius is the chord of the generating arc. ON THE VOLUME OF THE SPHERE. PROPOSITION XXXVI. THEOREM. 788. The volume of a sphere is equal to the area of its surface multiplied by one-third of its radius. Let R be the radius of a sphere whose centre is 0, S its surface, and V its volume. From 0, the centre of the sphere, conceive lines to be drawn to the vertices of each of the polyhedral & A, B, C, D, etc. These lines are the edges of six quadrangular pyramids, whose bases are the faces of the cube, and whose common altitude is the radius of the sphere. The volume of each pyramid is equal to the product of its base by } its altitude. $ 574 .. the volume of the six pyramids, that is, the volume of the circumscribed cube, is equal to the surface of the cube multiplied by } R. Now conceive planes drawn tangent to the sphere, cutting each of the polyhedral of the cube. We shall then have a circumscribed solid whose volume will be nearer that of the sphere than is the volume of the circumscribed cube. From O conceive lines to be drawn to each of the polyhedral of the solid thus formed, a, b, c, etc. These lines will form the edges of a series of pyramids, whose bases are the surface of the solid, and whose common altitude is the radius of the sphere; and the volume of each pyramid thus formed is equal to the product of its base by } its altitude. in the sum of the volumes of these pyramids, that is, the volume of this new solid, is equal to the surface of the solid multiplied by } R. Now, this process of cutting the polyhedral & by tangent planes may be considered as continued indefinitely, and, however far this process is carried, it will always be true that the volume of the solid is equal to its surface multiplied by } R. But the sphere is the limit of this circumscribed solid. ...V=Sx} R. $ 199 Q. E. D. 789. COROLLARY 1. Since S=40 R” ($ 762), V=45 R2X R=fa R$ If we denote the diameter of the sphere by 790. Cor. 2. Denote the radius of another sphere by R' and its volume by V'; we have Vi=fa R"8. ::: . V _ $7h_R8 V R18 R18 • That is, spheres are to each other as the cubes of their radii. 791. Cor. 3. The volume of a spherical sector is equal to the product of the area of the zone which forms its base by one-third the radius of the sphere. Let R denote the radius of a sphere, C the circumference of a great circle, H the altitude of the zone, 2 the surface of the zone, and V the volume of the corresponding sector. |