两河流域六十进制位值记数法早期发展的新证据及其分析  

Early Development of the Sexagesimal Place Value Notation in Mesopotamia: Analysis of Some New Evidence

作　　者：欧阳晓莉[1] Christine Proust 

机构地区：[1]复旦大学历史系,上海200433 [2]法国科研中心 [3]巴黎狄德罗大学

出　　处：《自然科学史研究》2015年第2期201-221,共21页Studies in The History of Natural Sciences

基　　金：European Research Council项目(编号269804);上海市2013年度"东方学者"人才项目(编号2013-02)的阶段性成果

摘　　要：乌尔第三王朝（前约2112～2004）是六十进制位值记数法早期发展过程中的重要阶段。在这一时期的经济管理文书中出现了与这一记数法相关的若干新证据。称为“边缘数字”（书写在文书正文外空白处的数字）。作者通过分析两篇文献中的边缘数字，发现这些边缘数字既采用了随后在古巴比伦时期应用广泛的完全六十进制记数法，又采用了首次发现的部分六十进制记数法。使用后一记数法的数字整体不是一个六十进制数字，但其每个组成部分可视为一个六十进制数字。由此推断，这两篇文献中的边缘数字产生于以度量衡单位表示的数量与以六十进制数字表示的数值间的转换。边缘数字的形态取决于运算的具体性质：部分六十进制记数法数字促使加法和减法的计算更加简便，而完全六十进制记数法数字适用于乘法。The use of the sexagesimal striking features of cuneiform mathematics. place value notation （ hereafter SPVN） is one of the most The earliest attestations of a systematic use of the SPVN are found in a small set of Ur Ⅲ mathematical texts, but traces of positional notations also appear in about two dozen Ur III administrative texts. These notations are located outside the main text, and hence we refer to such notations as ＂marginal numbers. ＂This article studies the marginal numbers in two Ur III documents. The numbers testify to diversepractices with positional notations and sexagesimal factors in administrative contexts during this peri-od. A close analysis of the evidence reveals a diversity of graphical systems for what has been consid-ered until now as a uniform notion of the SPYN. Thus, in addition to the SPVN fully developed dur-ing the Old Babylonian period, we identify other systems that we label as ＂partially sexagesimal place value notation＂ （ hereafter partial-SPVN）. The partial-SPVNs do not appear to be floating, while the SPVN does. The partial-SPVNs convert fractions of a measurement unit into an integer number of a sexagesimal sub-unit. This transformation would facilitate subtractions of measurement values with fractions. This diversity of place value notations reflects computational methods that vary according to the operation （ multiplication, reciprocal, addition, subtraction） or the context （ administrative or mathe-matical）. The observations above may lead to a reconstruction of the process of calculation behind the Ur III administrative records. Some steps in the calculation process seem to have taken place external to the tablet, as no trace of their details occurs on the tablet. The marginal numbers thus provide evi-dence of an intermediate step, which converts quantities noted in the main text into the input of an external device where calculations are executed, or conversely, converts the output of the device into quantities noted in the main text.

关 键 词：六十进制位值记数法 乌尔第三王朝 边缘数字 两河流域数学 楔形文字 

分 类 号：O11[理学—数学]