"technique" To Study

Cardano's Ars Magna is a landmark in the history of mathematics. It started the theoretical research on algebraic equations, solved the polynomial equations of third-degree and forth-degree in a general and systematical way, and studied the imaginary number as well as its operations for the first time.This paper reviews the historical development of the algebraic solutions of polynomial equations, introduces the life and scientific achievements of Cardano together with the historical background of Ars Magna. Based on this, the paper makes an elaborate study on Ars Magna.Cardano did not use the mathematical signals in Ars Magna, and the mathematical tradition then was synthetic instead of analytic, therefore, most rules in Ars Magna were arithmetical but not algebraical. In order to understand and orient this work more accurately, this paper analyzes the algebraic origin of the rules according to the principles in reconstructing the ancient proofs initiated by Mr. Wu Wen-jun.Firstly, the paper analyzes the mathematical bases and the source of Cardano's "golden rule". Grounded on this and by using the skills of estimating solutions, the conversion of equations and the reduction of power which Cardano had known, the paper consequently discusses the algebraic origins of Cardano's conclusions of the number of the positive and negative solutions. It also studies the algebraic source of the relationship given by Cardano between the three real solutions of the third-degree equation and the coefficient of its second power. Besides, the paper points out that these conclusions are based on algebraic reasoning without being influenced by the geometrical tradition of al-Tusi.Secondly, the paper analyzes systematically the algebraic origins of the general and special rules in the identical conversions between equations. Although these rules are about the conversions between the equations, Cardano did not point out on what conversions that the rules based. This paper finds out the specific conversions in eachrule, then classifies and analyzes them.Thirdly, this paper explains the algebraic origin of Cardano's general rules of the third-power equations by means of identical transformations of equations. And it also researches the sources of the special rules of the third-degree and the forth-degree equations in Ars Magna by way of the conversions and identical transformations of equations.Last but not least, this paper points out various mistakes in the English version of Ars Magna. Further more, it analyzes these mistakes systematically and gives referential amendment to each of them.