“Method of Four Unknowns” 四元術 as inspiration for Wu Wenjun 吴文俊

1. “Method of Four Unknowns”四元術as inspiration forWu Wenjun 吴文俊 Jiri Hudecek Department of History & Philosophy of ScienceNeedham 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. 1 2 3 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. ”

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

7. Triangulated polynomial set: I2 = x -1 I1 = 1 r =2, x1…x, x2…y m1 = m2 = 2 Wu’s Method (2) • Example: A1: x2 – u = 0A2: (x-1)y2 + xy + v = 0

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

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

10. Remainder polynomial (pseudodivision): Lower degree ofxr Finally: Wu’s Method (3) • Triangulated polynomial set: Conclusion polynomial:

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. 以地乘之 以天乘之 并之 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:

13. “Reciprocally hidden are equalised in parts” 互隠通分 後式 前式

14. “The two eliminate each other” 兩位相消 後式 後式 前式

15. [Reciprocally hidden are equalised in parts”互隠通分(2)]

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

17. [Reciprocally hidden are equalised in parts”互隠通分(3)]

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

19. “The inner columns [multiply]” 内二行得 右式 左式

20. “The outer columns [multiply]” 外二行得 右式 左式

21. “The inner and outer eliminate each other and are simplified by four” 内外相消四約之 右式 左式 开方式

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

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