从“蒲莞并生”再看《九章算术》中的“盈不足术”

The Method of Surplus and Deficiency in Jiuzhang Suanshu Revisited:Based on the Analysis of the Problem of“Growing Cattail and Bulrush”

“盈不足术”是我国古代数学中一种近似“万能”的解题方法,它在世界数学史上同样具有重要地位。通过对《九章算术》中“盈不足”章的研究分析,将“盈不足术”解决问题的类型进行了划分。对“蒲莞并生”问题进行了重新解读和分析,指出“蒲莞并生”实际上是分段线性问题,得到的是精确解,并非近似解。将该问题的求解看作是“盈不足术”从“求解线性问题”到“求解非线性问题”的拓展,是编史学上的误读。由此说明《九章算术》中的“盈不足术”并未超出线性问题的范围。

“Surplus and deficiency”is a kind of almost“omnipotent”solution method in mathematics of ancient China.It has great influence on history of mathematics in the world.Through textual research on the chapter“surplus and deficiency technique”in the Nine Chapters of Arithmetic,we made a systematic organization for the types of problem solvable with the method surplus deficiency.We also focused on“the growing problem of cattail(蒲)and bulrush(莞)”and demonstrated it to be a piecewise linear problem rather than a nonlinear problem regarded by previous researcher,and its solution was exact but not approximate.We pointed out that it is a misreading in historiography to regard the solution of this problem as an expansion of the method of surplus and deficiency from"solving linear problems"to"solving nonlinear problems"and concluded that the application of the method of“surplus and deficiency”in Nine Chapters of Arithmetic does not went beyond linear problems.