Abstract
The earliest extant commentaries on the Han Canon The nine chapters on mathematical procedures (first century BCE or CE), the one completed by Liu Hui in 263 and the one presented by Li Chunfeng in 656, systematically seek to establish the correctness of the algorithms described in the Canon. In previous publications, I have attempted to describe the practice of proof to which they bear witness. The intention of this paper is to show that the commentaries testify to a kind of algebraic proof in an algorithmic context that is attested to nowhere so far in ancient traditions except in ancient China. Moreover, I suggest that this ancient use of algebraic proof could provide inspiration for new methods of introducing algebraic proof in the classroom.
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Notes
- 1.
Chemla and Guo Shuchun (2004) present a critical edition and a French translation of The Nine Chapters and the two ancient commentaries with which it was handed down. The book also contains a complete bibliography and discussion of all statements made here about The Nine Chapters and the commentaries, except those incorporating results published after the book’s publication.
- 2.
For more extensive argumentation and a discussion of other cases, see Chemla (in press).
- 3.
I represent the algorithm as follows: On the left of an arrow, appear the data entered in an operation and, under the arrow, the operation applied. Two successive arrows indicate that the results yielded by the left operation are entered as terms in the right operation.
- 4.
It is only when one uses this value for π that one can explain why the factor appearing in the algorithm is 12. The key point is that the area of a circle, whose circumference is C, is equal to C2/4π. The Nine Chapters uses this value of π throughout the book.
References
Chemla, K. (In press). ‘Reading proofs in Chinese commentaries: Algebraic proofs in an algorithmic context’. In K. Chemla (Ed.), The history of mathematical proof in ancient traditions.Cambridge: Cambridge University Press.
Chemla, K., & Guo S. (2004). Les neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
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My sincere thanks to Gila Hanna, John Holt and Sarah-Jane Patterson for the generosity with which they helped me improve the text. Naturally, I am responsible for all remaining shortcomings.
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Chemla, K. (2012). Using Documents from Ancient China to Teach Mathematical Proof. In: Hanna, G., de Villiers, M. (eds) Proof and Proving in Mathematics Education. New ICMI Study Series, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2129-6_18
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DOI: https://doi.org/10.1007/978-94-007-2129-6_18
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