Plane and solid geometry . (§ 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
Plane and solid geometry . (§ 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 Given sphere 0 with its radius denoted by R, and the areaof its surface denoted by S, To prove 5 = 4 ttJ^, Argument 1. In the semicircle ACE inscribe ABODE, half of a regular polygon with aneven number of sides. Denote itsapothem by a, and the area of thesurface generated by the semiperime-ter as it revolves about AE as an axisby-s. 2. Then 5 = ^^ • 2 ira; Le. 5 = 2 i?. 2 7ra = 4 irRa, 3. As the number of sides of the regular polygon, of which ABODE is half, isrepeatedly doubled, s approaches Sas a limit. 4. Also a approaches i? as a limit. 5. .-. 4 TT-K • a approaches 4 7ri? • i?, Le, 4 irlf, as a limit. 6. But S^ is always equal to 4 7ri? • a. 7. 5 = 4 ttRK Reasons1. § 517, a. 2. § 968. 3. § 970. 4. § 543, I 5. § 590. 6. Arg. 2. 7. § 355. 448 SOLID GEOMETRY 972. Cor. I. The areas of the surfaces of two splieres areto each otiier as the squares of their radii and as thesquares of their diameters. OcTLixE OF Proof1. ;5 = 4 TTi?^ and S= 4 7ri?2. ...^ = iz^ = ^. i
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
