从《算数书》盈不足问题看上古时代的盈不足方法

盈不足方法是古代数学中一项具有一般性的重要方法。通过将《算数书》和《九章算术》及其他文献结合起来，探讨它在中国上古时代的形成与流传。认为：先秦时期实际工作中经常出现某种东西过多或过少的情况，为了得到合适的数量，人们通过运用比、比例和分数的知识，找到了解决问题的盈不足方法。这种方法及其应用形成了先秦数学的一个科目——盈不足，并记载于《九章算术》在先秦的祖本中。受它直接或间接的影响，先秦到汉代的学者们根据需要设置了很多盈不足问题。《算数书》中的盈不足问题即由此而来。

The Method of Excess and Deficiency in Early China: An Investigation on the Basis of the Problems of Excess and Deficiency in the Suanshushu

The method of excess and deficiency that had general applications was important in ancient mathematics. Systematically considering the unearthed Suanshushu (Writings on reckoning), the Nine Chapters on Mathematical Procedures and other sources, this paper discusses the formation and spread of this method in early China. It argues that in Pre-Qin Period, there often appeared the situation that something was more in one case while was less in the other. In order to find the proper amount, ancient mathematicians created a method to tackle the problem on the basis of knowledge about rate, proportion and fraction. This method and its applications constituted a branch of mathematics in Pre-Qin China, which was recorded in the forerunner edition of the Nine Chapters. Influenced by it directly or indirectly, scholars in Pre-Qin to Han Period designed many problems of this method for social needs. And a few of the problems were compiled into the Suanshushu.