阿拉伯代数方程求解几何方法的比较研究

A COMPARATIVE STUDY OF THE ARABIC GEOMETRICAL METHODS OF THE SOLUTION OF ALGEBRAIC EQUATIONS

代数方程的求解是阿拉伯数学最突出的成就之一。该文主要从比较的角度讨论了阿拉伯代数方程求解的几何方法。几何方法在代数方程求解中的运用在阿拉伯学者那里得到了进一步的发展。考查阿拉伯学者这方面的工作,我们发现其思想发展的两条不同路线:一是以花拉子米为代表明确给出解的代数表述或算法,同时为之提供以"出入相补原理"为基础的几何证明;另一则是以奥马为代表,以二次曲线相交的几何方法为基础寻求代数方程的解。这两条路线有着不同的思想来源,并产生不同的历史影响。以花拉子米为代表的路线,本质上属于中国与印度传统,体现了东方数学的特色,这条路线对文艺复兴时期的数学家的代数方程研究有着不容忽视的影响;以奥马为代表的另一条路线,则明显地是希腊几何代数的延伸。但由于种种原因,这项本来可以推动代数与几何密切结合的重要成就,却随着阿拉伯文化的衰落被忽视,与花拉子米代数的影响形成鲜明对照。

The solution of algebraic equations is one of the most significant contributions and most influential work of Arabic mathematics. This paper discusses the geometrical methods used by Arabic scholars in solving algebraic equations from the viewpoint of comparative study. By investigating the Arabic works in this field, we expound here two different approaches for Arabian to apply the geometrical methods in solving algebraic equations. One was represented by al-Khowarizmi, which gave clearly the algebraic expressions or algorithms of solutions of the quadratic equations with the geometrical proofs based on the out-in complementary-like principle. The other was represented by Omar Khayyam, which looked for the solution of the cubic equations by means of the geometric method of the quadric curve intersection. Two approaches had different source and played different roles in the development of solution of algebraic equations, about which some arguments were presented in this paper. Khowarizmi's approach, in line virtually with Indian and Chinese tradition, embodied the spirit of Eastern mathematics and had produced certain impact upon the solution of algebraic equations of the Renaissance mathematics. In contrast, Omar's approach, apparently an extension of Greek geometric algebra, had long been ignored with the general decline of Arabic sciences, though it is a precious achievement in the history of mathematics.