Tóngwén suànzhǐ 同文算指
Common-Language Computational Pointers translated by 利瑪竇 (Matteo Ricci, S.J., Lì Mǎdòu, 1552–1610, 明, yì 譯); elaborated by 李之藻 (Lǐ Zhīzǎo, 1565–1630, 明, yǎn 演), printed Wànlì 42 (1614)
About the work
Lǐ Zhīzǎo’s 10-juan elaboration of European pen-and-paper arithmetic translated from Matteo Ricci’s oral exposition. The work is the first systematic Chinese-language exposition of European arithmetic — supplementing the Ricci-Xú Guāngqǐ Euclid translation KR3f0013 Jǐhé yuánběn (1607) with the practical-computational counterpart. Together with the Jǐhé yuánběn and the KR3f0014 Cèliáng fǎyì / Yìtóng / Gōugǔ yì trilogy, the Tóngwén suànzhǐ completes the foundational Wàn-lì-period Jesuit mathematical curriculum in Chinese.
Structure:
Front Edition (前編 qiánbiān) in 2 juàn: pen-arithmetic position-fixing (bǐsuàn dìngwèi 筆算定位); addition, subtraction, multiplication, division; fractional reduction (yuēfēn) and unification (tōngfēn).
Comprehensive Edition (通編 tōngbiān) in 8 juàn — using Western methods to discuss the Jiǔzhāng topics:
- Juàn 1: Three-Rate Standard-Measurement (sānlǜ zhǔncè — the Western Rule of Three), in three forms (variant-multiplication-and-same-division yìchéng tóngchú; same-multiplication-and-variant-division tóngchéng yìchú; same-multiplication-and-same-division tóngchéng tóngchú) — corresponding to the Chinese yìchéng tóngchú tradition.
- Juàn 2-3: Combined-category proportional differences (hélèi chāfēn); proportional sum-and-difference (héjiào sānlǜ); the Hóngcuī hùzhēng (Wide-Decreasing Mutual-Verification) — i.e., the Chinese chāfēn / cuīfēn method.
- Juàn 4: Iterated-borrowing mutual-verification (diéjiè hùzhēng — the Chinese yíngbùzú / Rule of False Position).
- Juàn 5: Mixed sum-and-difference multiplication (zá héjiào chéng — the Chinese fāngchéng / Method of Equations).
- Juàn 6: Measurement Three-Rate (cèliáng sānlǜ — the Chinese gōugǔ); square-root extraction (kāi píngfāng); irregular-number square-root extraction (qílíng kāi píngfāng — the Chinese shǎoguǎng).
- Juàn 7: Sum-and-difference square-root extraction (jījiào hé kāi píngfāng).
- Juàn 8: Companion-coefficient various-power root-extractions (dàizòng zhūbiàn kāi píngfāng); cube-root extraction (kāi lìfāng); broad various-powers (guǎng zhūchéngfāng); irregular-number various-powers (qílíng zhūchéngfāng) — all under the broader Chinese shǎoguǎng category.
The 提要 articulates a refined comparative judgment on the work’s substantive Chinese-vs-Western mathematical content:
“Examining: the Jiǔzhāng is the bequeathed method of the Zhōulǐ . Its uses each [are] distinct; later-period number-discussers cannot easily change [them]. The Western method only opens-the-square-root ( kāifāng — i.e., the Chinese shǎoguǎng ) and right-triangle ( gōugǔ ) each have specialist procedures. The remainder all use the Three-Rate ( sānlǜ ) to govern. As to the fāngtián*, sùbù*, chāfēn*, shānggōng*, jūnshū five chapters: [these] originally can be governed by the Three-Rate. Reaching the yíngnü to govern hidden-and-mixed mutually-appearing [matters] and the fāngchéng to govern entangled positive-and-negative — then the Three-Rate cannot govern [them]. Indeed the Chinese method and the Western method each have what they are good-at; none can [completely] cover the other.”
This is one of the most balanced and historically-mathematically accurate assessments of the strengths and limits of European-vs-Chinese mathematical method in the Sìkù corpus. The Chinese Yíngnü (proportion-by-deficit, the Rule of False Position) and Fāngchéng (system-of-equations) cannot be reduced to the European sānlǜ (Rule of Three) — these Chinese methods handle problem-types that the Western basic arithmetic does not address. The 提要’s recognition of this asymmetry is an important late-imperial Chinese articulation of the complementarity of the two mathematical traditions, against the more polemical Xīfǎ Zhōngyuán framings of other 提要 entries.
The 提要 cites Méi Wéndǐng’s 梅文鼎 Fāngchéng yúlùn 方程餘論: “the Jǐhé yuánběn speaks fully on right-triangles and triangles. The Tóngwén suànzhǐ*‘s* yíngnü and fāngchéng take the ancients’ methods to transmit them — not what Mr. Lì transmitted [from the West]”; further: “the various books’ errors all follow [the original Western source] but cannot examine; what they [Ricci-Lǐ Zhīzǎo] necessarily-do-not-know they [therefore] do not use; what they can speak of they [therefore] do-not-know-completely. [This] also can be seen”. Méi Wéndǐng’s verdict is sharper: the Tóngwén suànzhǐ’s treatment of the more advanced Chinese topics (yíngnü, fāngchéng) drops out of the European source-tradition entirely and falls back on Chinese sources, indicating the limits of the Ricci-Lǐ collaboration.
The 提要 nonetheless commends the work as a useful late-Míng compilation: “[Lǐ Zhīzǎo] alone, not afraid of the trouble, accumulated days-and-months, took various methods and combined-and-fixed this compilation — also can serve as a resource for the calculation-house’s examination-of-the-ancient”.
For the broader Wàn-lì-period Jesuit mathematical project, see 利瑪竇 (Ricci), 李之藻 (Lǐ Zhīzǎo), KR3f0009 Qiánkūn tǐyì, KR3f0014 Cèliáng fǎyì, KR3f0015 Húngài tōngxiàn túshuō. For the related Méi Wéndǐng synthesis, see KR3f0026 Lìsuàn quánshū.
Tiyao
[Full text summarized above. Dated Qiánlóng 46 (1781), tenth month.]
Translations and research
- Engelfriet, Peter M. Euclid in China, Sinica Leidensia 40, Leiden: Brill, 1998 (background on the Ricci-Lǐ Zhīzǎo collaboration).
- Lam Lay Yong and Ang Tian Se. Fleeting Footsteps, rev. ed., Singapore: World Scientific, 2004.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
- Standaert, Nicolas (ed.). Handbook of Christianity in China, vol. 1, Leiden: Brill, 2001.
- Hashimoto Keizō 橋本敬造. Hō Yū-ran, Kyoto: Kansai University Press, 1988.
Other points of interest
The 提要’s balanced verdict — recognizing that Chinese and Western methods each have characteristic strengths and that neither completely dominates the other — is one of the more sophisticated comparative-cultural-mathematical assessments in the Sìkù tíyào corpus. It contrasts notably with the more polemical Xīfǎ Zhōngyuán claims in adjacent 提要 entries.
The work’s title Tóngwén 同文 (“Common Language”) plays on the classical Confucian phrase shū tóng wén, chē tóng guǐ (writing has a common language, chariots have a common track) from the Zhōngyōng, applied to the unification of the realm. By extension, Tóngwén suànzhǐ claims to provide a common (Chinese-Western) computational language — anticipating the synthetic Méi Wéndǐng project.