method of exhaustion

Japanese: method of exhaustion（英語表記）methodofexhaustion

…Archimedes seems to have been good at observation and experiments from an early age, but later he studied in Alexandria, where he learned mathematics from the successors of Euclid. He got to know mathematicians of the time who looked down on technology and experiments, and he re-proved many of the results he had previously obtained through experiments using the method of exhaustion, which is unique to Greek mathematics. These works include On the Sphere and Cylinder and On the Cone (Parabolic and Hyperbolic Solids) and the Spheroid (Spheroid), and we can now read them in the Greek text compiled by JL Heiberg (Collected Works of Archimedes, 3 volumes, 1910-15). …

From Eudoxus

…As a mathematician, he perfected a general theory of proportions that could be applied to incommensurable quantities, and the results were compiled in Book 5 of Euclid's Stoichia. He also devised a method of proof for area calculations that was named in the early modern period as the "method of exhaustion," and was the first to rigorously prove that the volume of a pyramid (cone) is one-third the volume of a rectangular prism (cylinder) with the same base and height. He also appears to have proved that the area of a circle is proportional to the square of its diameter, and that the volume of a sphere is proportional to the cube of its diameter. …

*Some of the terminology used to refer to "method of exhaustion" is listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…このようにアルキメデスは，早くから観測や実験を得意にしていたと思われるが，その後アレクサンドリアへ留学し，そこでユークリッド(エウクレイデス)の後継者たちから数学を学んだ。そして技術や実験を軽んじる当時の数学者たちを知り，それ以前に実験で得ていた多くの結果を，ギリシア数学独特の帰謬法による証明法〈取り尽しの方法method of exhaustion〉によって証明し直した。それが《球と円柱について》や《円錐状体(回転放物線体，回転双曲線体)と球状体(回転楕円体)について》など多くの著作であるが，それらを現在われわれは，ハイベルクJ.L.Heibergの編纂したギリシア語テキスト(《アルキメデス全集》3巻，1910‐15)で読むことができる。…

【エウドクソス】より

…数学者としては，通約できない量にも適用できる一般的な比例論を完成し，その成果はユークリッド(エウクレイデス)《ストイケイア》第5巻にまとめられている。また，求積問題では，〈取尽しの方法method of exhaustion〉と近世になって名付けられた証明法を考案して，角錐(円錐)の体積は同底同高の角柱(円柱)の体積の3分の1であることを初めて厳密に証明した。また，円の面積が直径の平方に比例し，球の体積が直径の立方に比例することも証明したらしい。…

※「method of exhaustion」について言及している用語解説の一部を掲載しています。

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