1. “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

2. 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?

3. Wu Wenjun 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

4. 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 Baocong can be consulted for detailed introduction. ” 12 3

5. 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

6. Wu’s Method 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

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

8. Wu’s Method (2) Polynomial division: reciprocal multiplication subtraction

9. Wu’s Method (2) Polynomial division: reciprocal multiplication, subtraction … M - k < m – remainder becomes divisor: …

10. Wu’s Method (3) Conclusion polynomial: Triangulated polynomial set: Remainder polynomial (pseudodivision): Lower degree of x r Finally:

11. 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

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

13. “Reciprocally hidden are equalised in parts” 互隠通分 11-2 1 1-2 1 20 4 1 11 0 1 0 1-20 0000 -12-2-20 2-4-42 -12 1 20 -102 2-30 -12 前式 後式

14. “The two eliminate each other” 兩位相消 前式 後式 11-2 1 1-2 1 20 4 1 11 0 1 0 1-20 0000 -12-2-20 2-4-42 -12 1 20 -102 2-30 -12 -3-340 3-2 1-10 -12 後式

15. [Reciprocally hidden are equalised in parts” 互隠通分 (2)] 11-2 1 1-2 -3 6 2-45 2-35 01 1-2 -3-340 3-2 1-10 -12 -340 2-2 2-42 02 1-2

16. [The two eliminate each other” 兩位相消 (2)] 11-2 1 1-2 -3 6 2-45 2-35 01 1-2 -3-340 3-2 1-10 -12 -340 2-2 2-42 02 1-2 7-6 3-7 -3 1 左式左式

17. [Reciprocally hidden are equalised in parts” 互隠通分 (3)] 11-2 1 1-2 77-14 -410-13 -49-15 12-5 2 770 -4100 -490 120 0 7-6 3-7 -3 1

18. [The two eliminate each other” 兩位相消 (3)] 13-14 11-13 5-15 -2-5 2 右式右式 11-2 1 1-2 77-14 -410-13 -49-15 12-5 2 770 -4100 -490 120 0 7-6 3-7 -3 1

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

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

21. “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 右式右式 7-6 3-7 -3 1 左式左式 -98 -133 -130 -67 14 11 -2 -78 -157 -146 -43 10 11 -2 -5 6 4 -6 1 开方式

22. 《中国数学史》, 1964, p. 184-185: Qian Baocong’s mediation

23. What is inspiration?  Deterministic chain of necessary causes Driving force and trigger facilitated by a documented transmission channel