アルキメデスの球と球状体の取り扱いを巡って

doi:10.34336/jhsj.26.164_193

183 181 0 0 OA アルキメデスの球と球状体の取り扱いを巡って

著者: 佐藤徹
出版者: 日本科学史学会
雑誌: 科学史研究 (ISSN:21887535)
巻号頁・発行日: vol.26, no.164, pp.193-206, 1987 (Released:2021-09-21)

Archimedes gave geometrical demonstrations to find the volume of a sphere and the ellipsoids of revolution in totally different ways, although both lead to the same integral,∫[2a,0]x (2a — x)dx. Nicolas Bourbaki -a group of French mathematicians-state their views on why Archimedes had no concept of integral calculus as follows: "Might it not be that Archimedes regarded such a standpoint as extreme 'abstraction,'and dared to concentrate on studying characteristic properties of each figure he was working on ?" Certainly there is something in what Nicolas Bourbaki say. However, their views do not answer fully the question To the solution of this difficult problem, in my opinion, an important clue can be found by considering Archimedes' scholastic career in chronological order. It was not until Archimedes wrote On Spirals in his late forties or early fifties that he could work out the summing of the series 1²+2²+…n². In his later work On Conoids and Spheroids Archimedes could obtain for the first time the sum of a series ∑Xk(2a—Xk), necessary to give geometrical proofs about the volume of the ellipsoids of revolution However,it was difficult for Archimedes,in writing On the Sphere and Cylinder I, to obtain the sum. Therefore, he proved the theorem about the volume of a sphere in a way not making use of such summation When the same integral appeared, Archimedes could not notice the internal connection unifying them. This may be because, for one thing, he excluded from geometry, due to their mechanical nature, the discussions using indivisibles found in The Method which could have been a clue toward noticing the internal connection. Secondly, obtaining the sum of a series was not a simple matter to Archimedes who lacked the necessary algebraic symbols.

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ギリシャ人が「微積分を発明できなかった理由」はかつて熱心に研究されたことがありました．こちらはその論文で，今ではフリーで読めるようです．古代ギリシャの数学者は二次式の積分に帰着できる問題を，なぜ統一的に解かなかったのか？という「謎」への一つの回答です． https://t.co/fEDjpvVOTW https://t.co/ZVk7NPqxfb

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