和算中的圓理表及其應用－開方溟式出商表

傳統開方術可處理一元高次方程式開方求數值解的問題，和算家小出兼政則進一步提出求無窮多項方程式之根的方法，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0 = r + a_1x + a_2x^2 + a_3x^3 + ...+ a_nx^n + ...之中的x表示成r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》中不僅利用圓理表將弧長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，考察並探討和算家如何用表求弧背公式，以及如何用表將弦表示成弧長與直徑之幂級數展開式的過程。

關鍵字

和算；數學史；圓理算經；開方溟式

並列摘要

The kaifang shu 開方術 can solve the problem of extracting the roots of polynomial equations. Koide moved forward a single step, apparently by analogy, to deal with the problem of finding the root of the equation of the form 0 = r + a_1x + a_2x^2 + a_3x^3 + ...+ a_nx^n + ... which he called kaiho meishiki 開方溟式. He constructed three tables by which he could write the root x of 0 = r + a_1x + a_2x^2 + a_3x^3 + ...+ a_nx^n + ... as a power expansion of r. 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 tables.