A Research On Two Treatises On The Understanding And Amendment Of Equations

Solving algebraic equations is the direct purpose of algebra.Taking the fourth part,Two Treatises on the Understanding and Amendment of Equations(1615),of Vieta's The Analytic Art as the main content,this paper will explore his contribution to the theories of algebraic equations.In the previous part,On Understanding Equations,Vieta analyzed the structures of equations by the Zetetics,Plasmatic Modification and Syncrisis.In the latter part,On the Amendment of Equations,Vieta proposed the transformations corresponding to some difficult equations,in order to change them into easily solved ones.Vieta used several propositions to explain his research results,but he only proved part of them.According to the principle of restoration of ancient syndromes,this paper aims at analyzing the propositions and theorems in the two parts.The main work is as follows:First,during studying Vieta's basic principles of the formulating equations,it is obvious that he focused on the relation between equations and proportion.Consequently,while interpreting the previous part,the author restored some of the propositions and theorems,which were not explicated in Zetetics and Syncrisis,to analyze and prove the sources of his ideas,and draw a conclusion that the essence of the thought is identity deformation.Second,this paper will investigate the sources of the five kinds of transformations by analyzing the mathematical thoughts and methods of transformation implied in the latter part and combining with the three approaches in the previous part.In addition,the author will point out a mistake in one of Vieta's propositions.