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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク On the Sphere and Cylinder ： ウィキペディア英語版 On the Sphere and Cylinder ''On the Sphere and Cylinder'' ((ギリシア語:Περὶ σφαίρας καὶ κυλίνδρου)) is a work that was published by Archimedes in two volumes c. 225 BC. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.〔 Retrieved on 2008-06-22〕 ==Contents== The principal formulae derived in ''On the Sphere and Cylinder'' are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. In his work, Archimedes showed that the surface area of a cylinder is equal to: :$A\_\; =\; 2\; \backslash pi\; r^2\; +\; 2\; \backslash pi\; r\; h\; =\; 2\; \backslash pi\; r\; (\; r\; +\; h\; ).\backslash ,$ and that the volume of the same is: :$V\_\; =\; \backslash pi\; r^2\; h.\; \backslash ,$ On the sphere, he showed that the surface area is four times the area of its great circle. In modern terms, this means that the surface area is equal to: :$A\_\; =\; 4\backslash pi\; r^2.\backslash ,$ The result for the volume of the contained ball stated that it is two-thirds the volume of a circumscribed cylinder, meaning that the volume is :$V\_\; =\; \backslash frac\backslash pi\; r^3.$ Archimedes was particularly proud of this latter result, and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave—since ''h''=2''r'', he had shown that both the volume and the surface area of the sphere were two-thirds that of the cylinder. Later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation. The argument Archimedes used to prove the formula for the volume of a ball was rather involved in its geometry, and many modern textbooks have a simplified version using the concept of a limit, which, of course, did not exist in Archimedes' time. Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he then determined the volume. It seems that this is not the original method Archimedes used to derive this result, but the best formal argument available to him in the Greek mathematical tradition. His original method likely involved a clever use of levers. A palimpsest rediscovered in the 19th century contained many of Archimedes works, including The Method of Mechanical Theorems, in which he describes a method to determine volumes which involves balances, centers of mass and infinitesimal slices.〔(【引用サイトリンク】 title=Archimedes' Secret (BBC Documentary) )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「On the Sphere and Cylinder」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.