Cardano’s Rule of Proportional Position in Artis Magnae

1. Cardano’s Rule of Proportional Positionin Artis Magnae ZHAO Jiwei Department of Mathematics, Northwest University, Xi’an, China, 710127 Email: zzhaojiwei@tom.com

2. 1 Overview of Cardano’s Artis Magnae • Gerelamo Cardano (1501-1576) Italian Mathematician, physician, natural philosopher a prolific writer on medicine, mathematics, astronomy, astrology, music, philosophy and so on more than 200 books Cardano’s Opera (1663) more than 7000 pages

3. The main achievements of Artis Magnae(1545) the solutions of the cubic equations (reducible cases) the discriminant of the three-term cubic equations the solutions of some kinds of quartic equations access to an approximate value of higher degree equations the introduction and calculation of the complex number

4. The general solutions of quartic equations in Artis Magna Cardano and hispupil L. Ferrari (1522-1565) found the general method to solve quartic equations which do not contain the third power or the first power: the 4th power, the cubic term, the quadratic term, the constant; the 4th power, the first power, the quadratic term, the constant. 20 types quartic equations (Chapter 39)

5. The special solutions of quartic equations in Artis Magnae Aiming to solve the quartic equations contain both the third power and the first power. Chpater 26 Chpater 30 Chapter 34 Chapter 39 Chapter 40 It seems that Cardano and Ferrari had not the general method to solve such types of quartics.

6. 2 The proposed problem in chapter 33 of Artis Magnae • The problem To find two numbers such that (1) their difference or sum is known; (2) if taking the sum of the square of one part of one of the number and the square of another part of the other number, then this sum plus the square root of it is also known. • In modern expression

7. the substitution method in chapter 5

8. Cardano’s emotion Cardano did not express his emotion clearly, but from his method its believable that he wanted to solve the original equation by the traditional method. i.e., A quartic equation contains both the third power and the first power.

9. 3 Cardano’s method in chapter 33 • The method in chapter 33 Cardano wanted to find linear expressions of x for the two quantities and such that the sum of the squares has no first power of x. Therefore by eliminating and squaring on both side of the eqaution, he would have a bi-quadratic equation Find x, then a and b through the linear expressions.

10. The method for finding the linear expressions Cardano explains this method through 7 numerical examples. Example:

11. Modern expression of the calculation

12. The rule To summarize Cardano’s calculation in the 7 examples, the rule of proportional position can be expressed as (the case ): If then

13. 4 How cardano deduced the rule? • Cardano only said that it is based on the method in chapter 9 • Chapter 9: group of linear equations with 2 variables elimination method • Reconstruction of Cardano’s deduction two quantities to be squared new position of the unknowns

14. The coefficients of the first power ① • The position of the unknowns ②

15. By the method of elimination ①× ﹣② × ①× ﹣② × the method of undetermined coefficients

16. ConclusionFirst, the rule of proportional position indicates Cardano’s endeavor to the solution of some special quartic equations.Second, through reconstruction of the rule of proportional position the elimination method, Cardano had access to the the method of undetermined coefficients.

17. Thank you very much!