cardano’s rule of proportional position in artis magnae zhao jiwei department of mathematics,...
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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: [email protected]
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!