Jiājiǎn chéngchú shì 加減乘除釋
Exposition of Addition, Subtraction, Multiplication, and Division by 焦循 (撰)
About the work
The opening monograph in 焦循 Jiāo Xún’s collected Lǐtáng xuésuàn jì (KR3fc042), in 9 juàn. Treats the four basic arithmetic operations on rational numbers, with detailed exposition of the procedural and theoretical foundations.
Abstract
The work is the systematic Qīng exposition of the four elementary operations, taking as its central concern the explication of why the standard procedures work — the theoretical-justificatory dimension that elementary practical-mathematical works of the Suànfǎ tǒngzōng lineage had taken for granted. Jiāo Xún’s treatment is characteristically integrative: each procedure is examined both in its indigenous Chinese form (counting-rod and abacus traditions) and in its European-derived form (the jiègēnfāng and Euclidean-arithmetic traditions), with explicit reconciliation of the two.
The nine juàn cover:
(1–2) Addition and subtraction of integers and fractions, including the systematic treatment of positive and negative numbers — going back to the KR3f0032 Jiǔzhāng Fāngchéng chapter’s introduction of negative numbers and the zhèngfù shù 正負術 rules.
(3–5) Multiplication: the elementary multiplication procedures, the jiǔjiǔ multiplication table and its theoretical foundations, the multiplication of fractions, and the systematic theory of proportional multiplication (the rule of three and its generalizations).
(6–7) Division: the elementary division procedures, the theory of fractions and reduction, and the systematic theory of proportional division.
(8) The interrelations among the four operations, with attention to the abstract algebraic structure (the zhèngfù shù rules essentially constitute the rules for arithmetic on a commutative ring).
(9) Applications: a closing miscellany of problem-types illustrating the four operations.
The work is methodologically the most theoretically-developed Qīng treatment of elementary arithmetic. Jiāo Xún’s systematic concern for the theoretical-justificatory dimension — present here as in the parallel Tiānyuányī shì (KR3fc044) and Kāifāng tōngshì (KR3fc048) — reflects the deeper kǎozhèng commitment to explicit demonstrative reasoning rather than mere procedural recipe-following.
For Jiāo Xún’s broader project and the collected Lǐtáng xuésuàn jì see KR3fc042.
Dating: bracketed by Jiāo Xún’s productive period 1797–1820.
Translations and research
For Jiāo Xún’s mathematical project see references at KR3fc042.
- Hé Bǐngyù 何丙郁 (John Hoe). 1977. Les systèmes d’équations polynomes dans le Siyuan yujian (1303). Paris: Collège de France.
- Bréard, Andrea. 1999. Re-Kreation eines mathematischen Konzepts im chinesischen Diskurs. Stuttgart: Steiner.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 7.