四元術
吴文俊
Jiri Hudecek
Department of History & Philosophy of Science
Needham Research Institute, Cambridge

• 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?

吴文俊
• See Li Wenlin, 李文林 (2001).
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

• “ 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 Baocong can be consulted for detailed introduction. ”
1
2
3

• “ 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 handheld programmable computer

1. Geometric statement  system of polynomial equations
2. Triangulation of premises polynomials
3. Division of conclusion by triangulated premises – final remainder must be zero
• Repeated use of polynomial division – polynomial multiplication and subtraction

Wu’s Method (2)
Triangulated polynomial set:
A
1
 I
1 x
1 m
1
 a
1
...
A r
 I r x m r r
 a r
.
– Example:
A
2
: (
A
1
: x 2 – u = 0 x -1) y 2 + xy + v = 0
I
1
2
= x I
1 r =2,
1
(1)
= x
1
… x, x
2
… y m
1
= m
2
= 2

Wu’s Method (2)
• Polynomial division:
I
G x M  m
F
G
F
 I
F x m  f


I
G x M  g reciprocal multiplication
I
G
I
F x M  I
G x M  m f
I
F
G  I
F
I
G x M  I subtraction
F g
R
1
 I
F
G  I
G x M  m F  I
R 1 x M  1  r
1

Wu’s Method (2)
• Polynomial division:
F  I
F x m  f
R
2
 I
F
R
1
R
1

…
 I
R 1 x M  1
M - k < m – remainder becomes divisor:
 r
1
I
R 1 x M reciprocal multiplication, subtraction
 m  1 F  I
R 2 x M  2  r
2
R k
 I
Rk x m  1
… F  I
F x m 
R
M
...
x
 r k f

Triangulated polynomial set:
Conclusion polynomial:
C  I
C x n r r
 c
A
1
 I
1 x
1 m
1
 a
1
...
Remainder polynomial
(pseudodivision):
R
1
 I r
C  I
C x n r r
 m r
(1)
A r
A r
 I r x m r r
 a r
.
R
1
 I r
I
G x n x r

Lower degree of
I r c  I
G
I r x n r
 I
G x m r x r r
 n r a r
Finally:
I s
1
1
I s
2
2  I s r r
C  Q
1
A
1
 Q
2
A
2
  Q r
A r
 R r

• 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

• 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
1
天乘地
0 太
0
天
以地
乘之
1 地 0 太
1 天
1
地之方
0
地
0
太
以天
乘之
0
地
0
太
2
天乘地
0 天
并之 1
天乘地
0 天
1
天之方
1
天之方

“Reciprocally hidden are equalised in parts”
互隠通分 y 2
  x  1

前式
 y
 x 2  x  1

 2 x 2 y 3
 x  2  0
 y 2
  2 x  2
  y
 x 2  4 x  2


後式
2 x 2  2 x  0
1 1 -2
1 -2 2 0
-1 1 -1
-2 4 -2
1 -2 1 -2
1 1 -2 0
-1 1 -1 0
1 -2 0
1 -2 2 0
-1 0 2 -2
2 -3 0
-1 2
0 0 0 0
-1 2 -2 0
2 -4 2
-1 2

“The two eliminate each other”
兩位相消 y 2
  x  1

前式
 y
 x 2  x  1

 2 x 2 y 3
 x  2  0
 y 2
  2 x  2
  y
 x 2  4 x  2

 2 x 2  2 x  0
1 1 -2
1 -2 2 0
-1 1 -1
-2 4 -2
1 -2 1 -2
R  I
F
G  I
G yF
1 1 -2 0
-1 1 -1 0
1 -2 0
-3 4 0
-1 3 -2
1 -1 0
-1 2
1 -2 2 0
-1 0 2 -2
2 -3 0
-1 2
0 0 0 0
-1 2 -2 0
2 -4 2
-1 2

[Reciprocally hidden are equalised in parts”
互隠通分 (2)]
1 1 -2
-1 1 -1
1 -2
-3 4 0
2 -1 -2
2 -4 2
-1 0 2
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

1 1 -2
-1 1 -1
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)]
-3 4 0
-1 3 -2
1 -1 0
-1 2
左式
7 -6
3 -7
-1 -3
1
-3 -3 6
2 -4 5
2 -3 5
-1 0 1
1 -2

[Reciprocally hidden are equalised in parts”
互隠通分 (3)]
7 -6
3 -7
-1 -3
1
1 1 -2
-1 1 -1
1 -2
7 7 0
-4 10 0
-4 9 0
1 2 0
-1 0
7 7 -14
-4 10 -13
-4 9 -15
1 2 -5
-1 2

7 -6
3 -7
-1 -3
1
[The two eliminate each other”
兩位相消 (3)]
1 1 -2
-1 1 -1
1 -2
7 7 0
-4 10 0
-4 9 0
1 2 0
-1 0
右式
13 -14
11 -13
5 -15
-2 -5
2
7 7 -14
-4 10 -13
-4 9 -15
1 2 -5
-1 2

左式
7 -6
3 -7
-1 -3
1
“The inner columns [multiply]”
内二行得
14
11
-2
-98
-133
-130
-67
右式
13 -14
11 -13
5 -15
-2 -5
2

左式
7 -6
3 -7
-1 -3
1
“The outer columns [multiply]”
外二行得
10
11
-2
-78
-157
-146
-43
14
11
-2
-98
-133
-130
-67
右式
13 -14
11 -13
5 -15
-2 -5
2

“The inner and outer eliminate each other and are simplified by four”
内外相消四約之
左式
7 -6
3 -7
-1 -3
1
10
11
-2
-78
-157
-146
-43
-5
6
4
-6
1
开方式
14
11
-2
-98
-133
-130
-67
右式
13 -14
11 -13
5 -15
-2 -5
2

• 《中国数学史》 , 1964, p. 184-185:

 Deterministic chain of necessary causes
 Driving force and trigger facilitated by a documented transmission channel