“Method of Four Unknowns”

四元術as inspiration for

Wu Wenjun 吴文俊

Jiri Hudecek Department of History & Philosophy of Science

Needham Research Institute, Cambridge

Outline

• Wu Wenjun’s encounter with ancient Chinese mathematics

• Polynomial techniques in Wu Wenjun’s mechanisation method

• Polynomial techniques in the Method of Four Unknowns

• Transmission through Qian Baocong• What is inspiration?

Wu Wenjun 吴文俊• See Li Wenlin, 李文林 (200

1). 古为今用的典范 - 吴文俊教授的数学史研究

• Preferred spelling – Wu Wen-Tsün (*1919)– Studies in France (1947-1951)– CAS since 1952– 1974 – re-evaluation of

Chinese traditional mathematics (during pi Lin pi Kong campaign)

– Mechanisation of mathematics since 1977

Inspiration announced?• “ On the decision problem and mechanisation of theorem-

proving in elementary geometry” , Zhongguo Kexue 1977

• “The algorithm we use for mechanical proofs of theorems in elementary geometry involves mainly some applied techniques for polynomials, such as arithmetic operations and simple eliminations of unknowns. It should be pointed out that these were all created by Chinese mathematicians in the 12-14 century Song and Yuan period, and already reached considerable development then. The work of Qian Baocongcan be consulted for detailed introduction. ”

1

2

3

Inspiration announced

• “ Mechanisation of Mathematics” ( 数学机械化 ), Baike zhishi 1980: David Hilbert (Grundlagen der Geometrie, 1899) and

Alfred Tarski (Decision Method for Elementary Algebra and Geometry, 1951) – forerunners, not inspiration

“we set out the question and came up with a method of solution under inspiration from ancient Chinese algebra”

• “New view on traditional Chinese mathematics” ( 对中国的古代数学的再认识 ), Baike zhishi 1987:– Designed a version of Zhu Shijie’s technique for a hand-

held programmable computer

Wu’s Method

1. Geometric statement system of polynomial equations

2. Triangulation of premises polynomials3. Division of conclusion by triangulated

premises – final remainder must be zero• Repeated use of polynomial division –

polynomial multiplication and subtraction

Wu’s Method (2)

– Example:A1: x2 – u = 0

A2: (x-1)y2 + xy + v = 0

Triangulated polynomial set:

11111 axIA m

...

rmrrr axIA r .

r =2,

x1…x, x2…y

m1 = m2 = 2

I1 = 1I2 = x -1

Wu’s Method (2)

• Polynomial division:

fxIF mF

gxIG MG

fxIxIIFxI mMG

MFG

mMG

gIxIIGI FM

GFF

reciprocal multiplication

11

11 rxIFxIGIR MR

mMGF

subtraction

Wu’s Method (2)

• Polynomial division:

fxIF mF

11

11 rxIR MR

reciprocal multiplication, subtraction

22

21

112 rxIFxIRIR MR

mMRF

…

M - k < m – remainder becomes divisor:

… fxIF mF

km

Rkk rxIR 1

xRM no ...

Wu’s Method (3)

Conclusion polynomial:

cxIC rnrC

Triangulated polynomial set:

11111 axIA m

...

rmrrr axIA r .

rnm

rGn

rGrnxGr axIxIIcIxIIR rrrr 1

Remainder polynomial(pseudodivision):

rmn

rCr AxICIR rr 1

Lower degree of xr

Finally: rrr

sr

ss RAQAQAQCIII r ...22112121

Method of Four Unknowns• Recorded in the book Jade Mirror of Four Unknowns

《四元玉鉴》 (Si Yuan Yu Jian) by Zhu Shijie (1247)• Used to solve intricate trigonometric problems• Only the final elimination from two to one unknown

( 二元术 ) actually mechanical• Only basic labels for intermediate steps:

– 互隱通分– 两位相消

• The meaning of the first operation is clear from the last step:– 内外行相乘

Method of Four Unknowns• Recorded in the book Jade Mirror of Four Unknowns

《四元玉鉴》 (Si Yuan Yu Jian) by Zhu Shijie (1247)• Used to solve intricate trigonometric problems• Only the final elimination from two to one unknown

( 二元术 ) actually mechanical• Notation – e.g. ( 天 + 地 )2:

1地之方

0 地 0 太

2天乘地

0 天

1天之方

1地 0 太1天

0 地 0 太1

天乘地0 天

1天之方

以天乘之

1地之方

0 地 0 太

1天乘地

0 天以地乘之

并之

“Reciprocally hidden are equalised in parts” 互隠通分

1 1 -2

-1 1 -1

1 -2

1 -2 2 0

-2 4 -2

1 -2

1 1 -2 0

-1 1 -1 0

1 -2 0

0 0 0 0

-1 2 -2 0

2 -4 2

-1 2

1 -2 2 0

-1 0 2 -2

2 -3 0

-1 2

前式 02 211 222 xxxxyxy

0 2224 2 2 2223 xxxxyxyy

後式

“The two eliminate each other” 兩位相消

前式後式

1 1 -2

-1 1 -1

1 -2

1 -2 2 0

-2 4 -2

1 -2

1 1 -2 0

-1 1 -1 0

1 -2 0

0 0 0 0

-1 2 -2 0

2 -4 2

-1 2

1 -2 2 0

-1 0 2 -2

2 -3 0

-1 2

-3 4 0

-1 3 -2

1 -1 0

-1 2

02 211 222 xxxxyxy

0 2224 2 2 2223 xxxxyxyy

yFIGIR GF

後式

[Reciprocally hidden are equalised in parts”

互隠通分 (2)]

1 1 -2

-1 1 -1

1 -2

-3 -3 6

2 -4 5

2 -3 5

-1 0 1

1 -2

-3 4 0

-1 3 -2

1 -1 0

-1 2

-3 4 0

2 -1 -2

2 -4 2

-1 0 2

1 -2

[The two eliminate each other” 兩位相消 (2)]

1 1 -2

-1 1 -1

1 -2

-3 -3 6

2 -4 5

2 -3 5

-1 0 1

1 -2

-3 4 0

-1 3 -2

1 -1 0

-1 2

-3 4 0

2 -1 -2

2 -4 2

-1 0 2

1 -2

7 -6

3 -7

-1 -3

1

左式

[Reciprocally hidden are equalised in parts”

互隠通分 (3)]1 1 -2

-1 1 -1

1 -2

7 7 -14

-4 10 -13

-4 9 -15

1 2 -5

-1 2

7 7 0

-4 10 0

-4 9 0

1 2 0

-1 0

7 -6

3 -7

-1 -3

1

[The two eliminate each other” 兩位相消 (3)]

13 -14

11 -13

5 -15

-2 -5

2

右式

1 1 -2

-1 1 -1

1 -2

7 7 -14

-4 10 -13

-4 9 -15

1 2 -5

-1 2

7 7 0

-4 10 0

-4 9 0

1 2 0

-1 0

7 -6

3 -7

-1 -3

1

“The inner columns [multiply]” 内二行得

13 -14

11 -13

5 -15

-2 -5

2

右式

7 -6

3 -7

-1 -3

1

左式-98

-133

-130

-67

14

11

-2

“The outer columns [multiply]” 外二行得

13 -14

11 -13

5 -15

-2 -5

2

右式

7 -6

3 -7

-1 -3

1

左式-98

-133

-130

-67

14

11

-2

-78

-157

-146

-43

10

11

-2

“The inner and outer eliminate each other and are simplified by four”

内外相消四約之

13 -14

11 -13

5 -15

-2 -5

2

右式

7 -6

3 -7

-1 -3

1

左式-98

-133

-130

-67

14

11

-2

-78

-157

-146

-43

10

11

-2

-5

6

4

-6

1

开方式

• 《中国数学史》 , 1964, p. 184-185:

Qian Baocong’s mediation

What is inspiration?

Deterministic chain of necessary causes

Driving force and trigger facilitated by a documented transmission channel