【摘要】：早期代數(shù)學(xué)最直接的目的是求解代數(shù)方程。本文以韋達(dá)(Francois Vieta,1540-1603)的著作合集《分析術(shù)》(TheAnalytic Art)中第四部分《方程的理解與修正》(Two Treatises on the Understanding and Amendment of Equations,1615)為主要研究?jī)?nèi)容,探究其對(duì)代數(shù)方程理論所做的貢獻(xiàn)。在前篇《方程的理解》(Firstreatise:On Understanding Equations)中,韋達(dá)分別運(yùn)用符號(hào)分析法、二項(xiàng)式展開法和方程比較法分析了方程的結(jié)構(gòu);在后篇《方程的修正》(Seconnd Treatse:On the Amendment of Equations)中,韋達(dá)針對(duì)各類無(wú)法進(jìn)行數(shù)值求解或者數(shù)值求解十分困難的方程提出了相應(yīng)的方程變換法則,使其可以變換為能夠或者容易進(jìn)行數(shù)值求解的新方程。韋達(dá)在前后兩篇中都是通過(guò)具體的定理或命題展示自己的研究結(jié)果,但僅對(duì)其中一部分給出了解釋或說(shuō)明。本文目的在于遵循“古證復(fù)原”的原則分析這兩篇中的定理或命題,主要工作如下:第一,在探究韋達(dá)列方程的基本原則時(shí),發(fā)現(xiàn)他強(qiáng)調(diào)方程與比例之間的聯(lián)系,所以本文研讀前篇《方程的理解》時(shí),利用比例的思想復(fù)原了韋達(dá)在符號(hào)分析法與方程比較法中沒(méi)有解釋或說(shuō)明的定理與命題,給出其較為合理的來(lái)源分析與證明,從而明確地得出,韋達(dá)思想的實(shí)質(zhì)可歸結(jié)為恒等式變形。第二,分析后篇《方程的修正》中韋達(dá)提供的各類方程變換背后所蘊(yùn)含的數(shù)學(xué)思想和方法,結(jié)合前篇中的符號(hào)分析法、二項(xiàng)式展開法和方程比較法對(duì)五種常用的方程變換進(jìn)行探源,復(fù)原了韋達(dá)關(guān)于方程變換的部分定理,并指出其中的一條錯(cuò)誤命題。
[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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O151.1

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