Plane and solid geometry . 446 SOLID GEOMETRY 969. Cor. m. If halves of regular polygons vnth thesame even number of sides are civeumscvihed about, andinseribed in, a semicircle, then hy repeatedly doubling thenumber of sides of these polygons, and making the poly-gons always regular, the surfaces generated by the semi-perimeters of the polygons as they revolve about thediameter of the semicircle as an axis approach a com-mon limit, OcTLixE OF Proof 1. If iS and s denote the areas ofthe surfaces generated by the semi-perimeters Ab^c- and ABC- asthey revolve about ae as an axis, then S = ae . 2
Plane and solid geometry . 446 SOLID GEOMETRY 969. Cor. m. If halves of regular polygons vnth thesame even number of sides are civeumscvihed about, andinseribed in, a semicircle, then hy repeatedly doubling thenumber of sides of these polygons, and making the poly-gons always regular, the surfaces generated by the semi-perimeters of the polygons as they revolve about thediameter of the semicircle as an axis approach a com-mon limit, OcTLixE OF Proof 1. If iS and s denote the areas ofthe surfaces generated by the semi-perimeters Ab^c- and ABC- asthey revolve about ae as an axis, then S = ae . 2 TTi? (§ 967) ; and s = AE*2 ttci (§ 968). o S Ae 2 ttR AE R S AE • 2 ira AE a 3. But polygon ^^c-- ^ polygon ABC AE. (§ 438). 4. 5. 6. AE S s S — s ^^ = ^(§§419, 435).AB a Ra Ra a R - a S R 7. .*. by steps similar to Args. 6-11 (§ 855), S and s ap-proach a common limit. 970. Def. The surface of a sphere is the common limitwhich the successive surfaces generated by halves of regularpolygons with the same even number of sides approach, if thesesemipolygons fulfill the following conditions: (1) They must be circumscribed about, and inscribed in, agreat semicircle of the sphere; (2) The number of sides must be successively increased,each side approaching zero as a limit. BOOK IX 447 Proposition XXIII. Theorem 971. The area of the surface of a sphei^e is equal to fourtimes the area of a great circle of the sphere.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
