cardano’s rule of proportional position in artis magnae zhao jiwei department of mathematics,...

Cardano’s Rule of Proportional PositioCardano’s Rule of Proportional Positio

nn in Artis Magnaein Artis Magnae

Cardano’s Rule of Proportional PositioCardano’s Rule of Proportional Positio

nn in Artis Magnaein Artis Magnae

ZHAO Jiwei

Department of Mathematics, Northwest University,

Xi’an, China, 710127

Email: zzhaojiwei@tom.com

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

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

The general solutions of quartic equations in Artis Magna

Cardano and his pupil 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)

0234 cbxdxx024 caxbxx

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.

36121234 324 xxxx

xxxx 96814424 324 xxxx 82

1346 324

xxxx 12102006 324

xxx 221 34 xxxx 234 31

12 34 xxx

xxxx 1596 234

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

mba

ba

a

bb

a

ba

a

b 2

2

22

1

12

2

22

1

1 )()()()(nab

the substitution method in chapter 5

2

2

22

1

1 )()( ba

ba

a

bx

mxx

nab

mba

ba

a

bb

a

ba

a

b 2

2

22

1

12

2

22

1

1 )()()()(

Cardano’s emotion

Cardano did not express his emotion clearly, but from his method it

s believable that he wanted to solve the original equation by the tra

ditional method. i.e.,

A quartic equation contains both the third power and the first power.

mba

ba

a

bb

a

ba

a

b 2

2

22

1

12

2

22

1

1 )()()()(

22

2

22

1

12

2

22

1

1 ))()(()()( ba

ba

a

bmb

a

ba

a

b

anb

22

2

22

1

12

2

22

1

1 ))(()(())(()( ana

ba

a

bman

a

ba

a

b

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.

aa

b

1

1 ba

b

2

2

2

2

22

1

1 )()( ba

ba

a

b

222 )( srxmsrx

The method for finding the linear expressions

Cardano explains this method

through 7 numerical examples.

Example:

14 ab

110)4

()3

()4

()3

( 2222 baba

168)11(14)43(

25)14()13( 22

Modern expression of the calculation

'33a

xa '44b

xb

25

171

4

1

)13()14(

)11(14)43('

22

a

25

62

3

1

)13()14(

)11(14)43('

22

b

25

217

144

25)

25

171

4()

25

171

3()

4()

3( 22222 x

xxba

The rule

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

If

then

nab

nab '1

1

1

1 akxa

ba

a

b '2

2

2

2 bkxa

bb

a

b srxba

ba

a

b 22

2

22

1

1 )()(

2

22

122

21

1122

)()(

)()('

a

b

baba

banbaa

1

12

122

21

1122

)()(

)()('

a

b

baba

banbab

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

aa

b

1

1 )(2

2

2

2 ana

bb

a

b

'1

1

1

1 akxa

ba

a

b '2

2

2

2 bkxa

bb

a

b

The coefficients of the first power

①

The position of the unknowns

②

'2'22

2

1

1 kxba

bkxa

a

b ''

2

2

1

1 ba

ba

a

b

)'(1

1

1

1 akxa

b

b

aa )'(

2

2

2

2 bkxa

b

b

ab anb

nab

ab

b

a ''

1

1

2

2

By the method of elimination

①× ﹣② ×

①× ﹣② ×

the method of undetermined coefficients

2

22

122

21

1122

)()(

)()('

a

b

baba

banbaa

1

12

122

21

1122

)()(

)()('

a

b

baba

banbab

2

2

b

a

2

2

a

b

1

1

b

a

1

1

a

b

5 Conclusion First, 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.

Thank you very much!