Elements of geometry and trigonometry . PROPOSITION XIV. THEOREM. The solidity of a sphere is equal to Us surface multiplied by «third of its radius. a. 182 GEOMETRY. Inscribe in the semicircle ABCDE aregular semi-polygon, having any numberof sides, and let 01 be the radius of thecircle inscribed in the polygon. If the semicircle and semi-polygon berevolved about EA, the semicircle willdescribe a sphere, and the semi-polygon asolid which has for its measure fTiOPxEA (Prop. XIII.) ; and this will be truewhatever be the number of sides of thepolygon. But if the number of sides ofthe polygon be i
Elements of geometry and trigonometry . PROPOSITION XIV. THEOREM. The solidity of a sphere is equal to Us surface multiplied by «third of its radius. a. 182 GEOMETRY. Inscribe in the semicircle ABCDE aregular semi-polygon, having any numberof sides, and let 01 be the radius of thecircle inscribed in the polygon. If the semicircle and semi-polygon berevolved about EA, the semicircle willdescribe a sphere, and the semi-polygon asolid which has for its measure fTiOPxEA (Prop. XIII.) ; and this will be truewhatever be the number of sides of thepolygon. But if the number of sides ofthe polygon be indefinitely increased, the E semi-polygon will become the semicircle, 01 will becomeequal to OA, and the solid described by the semi-polygon willbecome the sphere : hence the solidity of the sphere is equalto fTiOA^xEA, or by substituting 20A for EA, it becomes^^ X OA, which is also equal to 4nOA^ x J OA. But ^is equal to the surface of the sphere (Prop. X. Cor.) : hencethe solidity of a sphere is equal to its surface multiplied by athird of its radius. Scholium 1. The solidity of every spherical sector is equal tothe zone which forms
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