《數》、《筭數書》和《九章筭術》中一類楔形體研究－兼論中國早期求積算法的某些特點

The Method for Finding the Volume of a Set of Wedges in ＂Shu, Suanshu Shu＂ and ＂Jiuzhang Suanshu＂: With a Discussion of Some Characteristics of the Methods for Calculating Solid Volumes in Early China

鄒大海(Da-Hai Zou)

《漢學研究》 32卷3期 (2014/09) Pp. 69-94

摘要

本文對出土的秦代竹簡算書《數》、西漢初年竹簡算書《筭數書》和傳世數學經典《九章筭術》中一類在古代較難處理的楔形體進行比較研究，挖掘它們之間的聯繫。利用《數》中的材料為認識秦及先秦處理體積問題時採用的基本立體提供了新的證據，在此基礎上為《數》中楔形體題目的術文提出了校補方案，推論這種立體的求積方法產生於推導，並提出了復原方案。借鑒生物學中進化和基因的觀念，論述三項文獻中的算法具有共同的特點和淵源，而這種淵源流傳到《九章》的先秦祖本比另兩項文獻很可能要早，同時也就上古時代體積算法的產生與流傳問題提出了新的認識。

關鍵字

秦簡《數》 ； 漢簡《筭數書》、《九章筭術》 ； 劉徽 ； 楔形體 ； 算法演化

並列摘要

This paper compares methods for calculating the volume of a particular type of wedge-shaped solid described in the Qin dynasty bamboo-slip manuscript Shu (數Numbers) , the Western Han dynasty bamboo-slip manuscript ＂Suanshu shu＂ (筭數書Book on Calculations), and the ＂Jiuzhang suanshu＂ (九章筭術Nine Chapters on Mathematical Procedures) , and identifies the relation between them. The text of Shu provides new evidence to support the opmion that four basic solids, the cuboid, ＂qiandu＂ (壍堵right triangular prism), ＂yangma＂ (陽馬right rectangular-based pyramid), and ＂bienao＂ (鱉腝tetrahedron with four right triangle faces) were used to determine solid volumes in the Qin and pre-Qin periods. Based on the above, this paper sets out a design that supplements the text in Shu to describe a mathematical method for finding the volume of a wedge. It reasons that the method for calculating the volume of the solid described on bamboo strip no. 0456 was found by mathematical derivation, and then attempts to reconstruct how the ancients obtained this method. Drawing on the concepts of evolution and genes from biology, this paper argues that the mathematical methods set out in the three sources have common characteristics and thus share the same origin. That common origin might have been transmitted and incorporated into an ancestor of the ＂Jiuzhang suanshu＂ that predates the other two documents. This paper also proposes a new understanding of how the methods for finding solid volumes were worked out and then transmitted throughout early China.

並列關鍵字

Shu on Qin dynasty bamboo slips ； Suanshu shu on Han dynasty bamboo slips ； Jiuzhang suanshu ； Liu Hui ； wedge solids ； evolution of mathematical methods

參考文獻

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