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iCHSTM 2013 Programme • Version 5.3.6, 27 July 2013 • ONLINE (includes late changes)
Index | Paper sessions timetable | Lunch and evening timetable | Main site |
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iCHSTM 2013 Programme • Version 5.3.6, 27 July 2013 • ONLINE (includes late changes)
Index | Paper sessions timetable | Lunch and evening timetable | Main site |
摘要
黃俊瑋
和田寧 (Wata Yasushi,1787~1840) 是和算末期最重要的數學家，他創造許多圓理表，解決許多圓理難題，然而，其著作因1836年的一場火災而亡佚。其弟子小出兼政 (Koide Kanemasa 1797~1865) 整理過去和田寧所授書籍，於1842年寫成《圓理算經》一書。本書的第三部份羅列許多表，這些表不僅是解決數學問題的工作，同時也包含了許多數學性質與公式。在沒有微積分方法的情況下，小出利用這些表解決各類複雜幾何圖形求積與求弧長，以及它們的交周與交積等問題。
傳統開方術可處理a=a1x+a2x2+a3x3+…+anxn之一元高次方程式開方求數值解的問題，而小出兼政則進一步發現如何求「a=a1x+a2x2+a3x3+…+anxn+…」此方程式的根，他並將這類式子稱為「開方溟式」。他在《圓理算經》中造出三種「開方溟式出商表」，利用這些表可將開方式a=a1x+a2x2+a3x3+…+anxn+…之中的x表示成a的幂級數展開式，並藉以求得此方程式的一個近似數值解。小出更進一步利用此開方表，處理圓理相關問題。
已知弦長與直徑求弧長之「弧背術」是傳統和算家們所感興趣的問題，而《圓理算經》不止用表求將弧背與弧積表示成直徑與弦長的幂級數展開式，並反過來
利用「開方溟式出商表」，在已知直徑與弧長的條件下，求得弦長之幂級數展開式。本研究主要透過文本分析，考察並探討其如何用「表」以求弧背公式，以及如何利用將弦表成弧長與直徑之幂級數展開式，並從和算發展的脈絡中，適當地還原小出造表之過程。
Wata Yasushi和田寧 (1787~1840) is the most important mathematician in the early period of the 19th century in Japan. He developed many yenri tables 圓理表 (Tables of circle principles ) and solved many difficult problems of geometry in wasan. Because of a fire disaster in 1836, his work does not survive to this day. Koide Kanemasa 小出兼政 (1797~1865), one of Wata’s disciples, sought to collect the mathematical knowledge from his teacher, and wrote the Yenri sankyo 圓理算經 (The mathematical classic of circle principles) in 1842. The third part of the treatise lists many numerical tables which not only are intellectual tools for solving problems but also manifest significant mathematical properties and formulas. Without knowledge of calculus, Koide used these tables to deal with the problem of finding the formulas of area, volume and arc length of various complicated geometry figures and their intersections in the first and second part of the treatise.
The kaifang shu開方術 can solve the problem of extracting the roots of polynomial equations such as a=a1x+a2x2+a3x3+…+anxn. Koide moved forward a single step, apparently by analogy, to deal with the problem of finding the root of the equation of the form a=a1x+a2x2+a3x3+…+anxn+… which he called kaiho meishiki開方溟式. He constructed three tables by which he could write the root x of a=a1x+a2x2+a3x3+…+anxn+… as a power expansion of a. Eventually, he found out an approximate solution of the equation. Moreover, he also used these tables to solve problems of the Yenri sankyo.
Traditionally, wasan practitioners were deeply interested in the problem of expressing the arc length in terms of the diameter and the chord in a circle. Koide was no exceptional at this point. However, he also turned around the problem by trying to express the chord in terms of the arc length and the diameter in a form of a power series with the previously mentioned numerical tables. This presentation will be devoted to analyzing the content of the Yenri sankyo and how Koide’s construction of these tables meant in the context of wasan on the eve of the Meiji Restoration.