《方程的理解与修正》研究

发布时间：2018-12-12 00:37

【摘要】：早期代数学最直接的目的是求解代数方程。本文以韦达(Francois Vieta,1540-1603)的著作合集《分析术》(TheAnalytic Art)中第四部分《方程的理解与修正》(Two Treatises on the Understanding and Amendment of Equations,1615)为主要研究内容,探究其对代数方程理论所做的贡献。在前篇《方程的理解》(Firstreatise:On Understanding Equations)中,韦达分别运用符号分析法、二项式展开法和方程比较法分析了方程的结构;在后篇《方程的修正》(Seconnd Treatse:On the Amendment of Equations)中,韦达针对各类无法进行数值求解或者数值求解十分困难的方程提出了相应的方程变换法则,使其可以变换为能够或者容易进行数值求解的新方程。韦达在前后两篇中都是通过具体的定理或命题展示自己的研究结果,但仅对其中一部分给出了解释或说明。本文目的在于遵循“古证复原”的原则分析这两篇中的定理或命题,主要工作如下:第一,在探究韦达列方程的基本原则时,发现他强调方程与比例之间的联系,所以本文研读前篇《方程的理解》时,利用比例的思想复原了韦达在符号分析法与方程比较法中没有解释或说明的定理与命题,给出其较为合理的来源分析与证明,从而明确地得出,韦达思想的实质可归结为恒等式变形。第二,分析后篇《方程的修正》中韦达提供的各类方程变换背后所蕴含的数学思想和方法,结合前篇中的符号分析法、二项式展开法和方程比较法对五种常用的方程变换进行探源,复原了韦达关于方程变换的部分定理,并指出其中的一条错误命题。
[Abstract]:The most direct purpose of early algebra is to solve algebraic equations. In this paper, the main content of this paper is "understanding and revising the equation" in the fourth part of "Analytical technique" (TheAnalytic Art) by Francois Vieta,1540-1603 (Two Treatises on the Understanding and Amendment of Equations,1615). To explore its contribution to the theory of algebraic equations. In the previous "understanding of equations" (Firstreatise:On Understanding Equations), Veda uses symbolic analysis method, binomial expansion method and equation comparison method to analyze the structure of the equation. In the latter part of "Correction of equations" (Seconnd Treatse:On the Amendment of Equations), Veda proposes the corresponding equation transformation rules for all kinds of equations which can not be solved numerically or which are very difficult to solve numerically. It can be transformed into a new equation that can be solved numerically or easily. In both the preceding and the following chapters, Veda presents his research results through specific theorems or propositions, but only gives explanations or explanations for some of them. The purpose of this paper is to analyze the theorems or propositions in these two chapters in accordance with the principle of "restoration of ancient evidence". The main work is as follows: first, when exploring the basic principles of the Vedalier equation, it is found that he emphasizes the relation between the equation and the proportion. So in this paper, when we read the previous book "understanding of equation", we use the idea of proportion to restore the theorems and propositions that Veda did not explain or explain in symbolic analysis and equation comparison, and give its more reasonable source analysis and proof. It is clear that the essence of Veda's thought can be summed up as identity deformation. Secondly, it analyzes the mathematical ideas and methods behind the transformation of all kinds of equations provided by Veda in the latter part of "Correction of the equation", and combines the symbolic analysis method in the previous chapter. In this paper, the binomial expansion method and the equation comparison method are used to explore the source of five kinds of commonly used equation transformations, and the partial theorems of Vedar's equation transformation are restored, and one of the wrong propositions is pointed out.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.1

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