Plane and solid geometry . igned value, however and s approach a common limit. Keasons 6. § 54, 7 a. 7. § 543, I. 8. §686. 9. § 587. 10. §309. 11. § 594. 856. Note. The above proof is limited to regular prisms, but it canbe shown that the limit of the lateral area of any inscribed (or circum-scribed) prism is the same by whatever method the number of the sidesof its base is successively increased, provided that each side approacheszero as a limit. (See also § 649.) Compare the proof of § 865 with thatof § 646, I. 857. Def. The lateral area of a right circular cylinder is the com
Plane and solid geometry . igned value, however and s approach a common limit. Keasons 6. § 54, 7 a. 7. § 543, I. 8. §686. 9. § 587. 10. §309. 11. § 594. 856. Note. The above proof is limited to regular prisms, but it canbe shown that the limit of the lateral area of any inscribed (or circum-scribed) prism is the same by whatever method the number of the sidesof its base is successively increased, provided that each side approacheszero as a limit. (See also § 649.) Compare the proof of § 865 with thatof § 646, I. 857. Def. The lateral area of a right circular cylinder is the common limit which the successive lateral areas of circum-scribed and inscribed regular prisms (having bases containing3, 4, 5,- etc., sides) approach as the number of sides of thebases is successively increased and each side approaches zeroas a limit. 396 SOLID GEOMETRr Proposition VII. Theorem 858. The lateral area of a right circular cylinder isequal to the product of the circumference of its base andits Given a rt. circular cylinder with its lateral area denoted by5, the circumference of its base by C, and its altitude by prove S = C - H. Argument 1. Circumscribe about the rt. circular cylinder a regular prism. Denoteits lateral area by S, the perimeterof its base by P, and its altitudeby ^. 2. Then S^ = P - H, 3. As the number of sides of the base of the regular circumscribed prism isrepeatedly doubled, P approaches Cas a limit. 4. ., P • H approaches C • ^ as a limit. 5. Also S approaches .S as a limit. 6. But s is always equal to P • ^. 7. ,\ S= C H. Reasons1. § 852. §763.§650. 4. § 590. 5. § 857. 6. Arg. 2. 7. § 355. 859. Cor. If 8 denotes the lateral area, T the total area,H the altitude, and R the radius of the base, of a ri0htcircular cylinder, BOOK vm 39 S=2 7rliff] T=2 7tRH+ 2 7ri?^ = 2 7rR{H+ R). 860. Note. Since the lateral area of an oblique prism is equal tothe product of the perimeter of a right section and a later
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