和算的積分法－以求橢圓周術為例

The Integration Method in Wasan : Take the Jutsu About the Formula of the Circumference of an Ellipse as an Example

黃俊瑋(Jyun-Wei Huang)

《中華科技史學會學刊》 26期 (2021/12) Pp. 42-51

摘要

求橢圓周長問題，是19世紀之前，中西數學家難以解決的問題。本文分析和田寧傳、小出兼政編的《圓理算經》，探討書中求橢圓周長的方法，藉以了解關流和算家用以求解圓理難題的積分方法與特色，並分析此方法的結構與程序。和田寧發明的積分法可分成「分割檢矩線表得微元、檢表展開、檢表疊之求和、依逐忖術得術」四個步驟。上述的結構與程序，為和田寧求解各類圓理難題的普遍性方法，廣泛地用於求解《圓理算經》其它問題的過程中。除了演示求得此橢圓周術的過程，並且「核證」了此「術」的正確性。亦即整個「解曰」可視為一種「建構」並「證明」橢圓周長公式的方法。

關鍵字

和算 ； 和田寧 ； 圓理算經 ； 橢圓周長

並列摘要

It was a difficult mathematical problem to find the formula of the circumference of an ellipse for mathematicians before 19th century. In this article, I will analyze and introduce the method about finding the formula of the circumference of an ellipse in Koide Kanemasa' Yenri Sankyo in order to understand the integration method used by Wasan mathematicians in Seki school in 19th century. And then, I can Analyze the structure and the procedures of this method. The method invented by Wada Yasushi contains for procedures: dividing the figure to get the differential elements, using the Yenri tables to get the binomial expansions of the differential elements, using the Yenri tables to integrate, and finding the recurrence relation to transform the formula into the form of a Jutsu. This integral method was generally used by Wada Yasushi to solve all kinds of difficult Yenri problems in Yenri Sankyo. Finally, This method was not only used to construct the formula of the circumference of an ellipse, but also considered a proof that verifies the certainty of the formula.

並列關鍵字

Wasan ； Wada Yasushi ； Yenri Sankyo ； the circumference of an ellipse

參考文獻

小出兼政，《圓理算經》，1842 年。收入徐澤林《和算選粹》。北京：科學出版社，頁504 – 650 頁，2008。

林典蔚，《關孝和《三部抄》之內容分析》，國立臺灣師範大學數學系碩士論文，2012。

徐澤林，《和算選粹》。北京：科學出版社，2008。

黃俊瑋，《關流算學研究及其歷史脈絡:1722-1852》。國立臺灣師範大學數學系博士論文，2014。

黃俊瑋，〈從圓周率發展探討和算家的數學知識需求〉。《科學史通訊》，第 40 期，頁17-30，2016．