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新編直指算法統宗

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Text ID
KR3fc027
Date
1592–1592
Dynasty
明

Xīnbiān zhízhǐ suànfǎ tǒngzōng 新編直指算法統宗

Newly-Edited Direct-Pointing Comprehensive Compendium of Mathematical Methods by 程大位 (撰)

About the work

程大位 Chéng Dàwèi’s monumental abacus-arithmetic compendium of 1592, in 17 juàn (the present KX recension carries 22 juàn including the appended materials), more commonly known by its short title Suànfǎ tǒngzōng 算法統宗. The most influential mathematical work of the late Míng and the foundational textbook of MíngQīng abacus arithmetic.

Abstract

The Suànfǎ tǒngzōng organizes the substantive material of 吳敬 Wú Jìng’s KR3fc023 Jiǔzhāng xiángzhù bǐlèi suànfǎ dàquán (1450) under a comprehensive abacus-oriented presentation. The 17 juàn of the original (with the additional 5 of the KX recension carrying preface, sub-prefaces, and supplementary apparatus): (1–2) basic arithmetic on the abacus (the standardized kǒujué mnemonics for the multiplication-and-division operations); (3–8) the standard Jiǔzhāng topical chapters in abacus form, with substantially expanded problem-sets; (9–12) advanced commercial applications — taxation, currency exchange, equitable transport, labor distribution; (13–15) surveying and area-and-volume calculations; (16–17) the right triangle and a small selection of more advanced root-extraction problems.

Chéng Dàwèi’s principal contributions:

(a) The standardized abacus: the Tǒngzōng establishes the 1-and-5 (one-bead-above-the-bar, five-beads-below) arrangement as the standard Chinese abacus form. Earlier abacus designs had varied; Chéng Dàwèi’s standardization, made authoritative by the Tǒngzōng’s influence, became the universal Chinese-Korean-Japanese abacus pattern through the late-imperial period.

(b) The standardized abacus-mnemonics (kǒujué 口訣): the multiplication-and-division shortcut formulas that Chéng Dàwèi codifies became the universal training material for commercial computation. The jiǔjiǔ multiplication-table and the divisor-mnemonics in particular were memorized by every late-imperial merchant apprentice.

(c) The transmissional preservation of SòngYuán problem-types: like 吳敬 Wú Jìng before him, Chéng Dàwèi preserves substantial SòngYuán problem-content within the late-Míng abacus framework. Although the tiānyuányī algebra is absent, the substantive problem-types from the SòngYuán tradition — the Bǎi jī wèn indeterminate-equation problems, the Hánxìn diǎnbīng Chinese-Remainder-Theorem problems, the Sūnzǐ indeterminate problems — all survive in the Tǒngzōng in their classical form.

The work’s transmission is unmatched among MíngQīng mathematical works: over thirty editions by the early Qīng, with continuous reprinting through the late-imperial period. It was transmitted to Korea immediately (forming a key part of the Chosŏn sàn-hak curriculum) and to Japan by the early seventeenth century (where it shaped the wasan 和算 tradition through Yoshida Mitsuyoshi’s 吉田光由 Jinkōki 塵劫記 of 1627, which is in substantial part a Japanese adaptation of the Tǒngzōng). The Tǒngzōng is the principal late-Míng / early-Qīng channel by which Chinese mathematical material entered the East-Asian common stock.

The work is also of considerable historical-literary interest. Chéng Dàwèi’s preface gives a circumstantial account of his career as a Huīzhōu merchant-mathematician traveling on commercial business while gathering mathematical books at every stop, finally settling at Bīnqú 賓渠 in his retirement to compile the Tǒngzōng. The preface is one of the principal documentary records of late-Míng commercial-mathematical culture and of the role of the Huīzhōu merchant network in supporting practical-mathematical scholarship.

Dating: 1592 per Chéng Dàwèi’s preface.

Translations and research

  • Lam Lay-Yong. 1986. “Linkages: Exploring the Similarities between the Chinese Rod Numeral System and Our Numeral System.” Archive for History of Exact Sciences 37: 365–392.
  • Horiuchi Annick. 1994. Les mathématiques japonaises à l’époque d’Edo (1600–1868). Paris: Vrin. — Treats the Tǒng-zōng / Jinkōki transmission.
  • Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 6. Beijing: Běi-jīng shī-fàn dà-xué chū-bǎn-shè.
  • Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer.
  • Jami, Catherine. 1992. Une histoire chinoise du ‘nombre π’. Archive for History of Exact Sciences 38: 39–50. — Treats the Tǒng-zōng values of π and the late-Míng abacus-arithmetic π-extraction.

Other points of interest

The standardized abacus that Chéng Dàwèi codifies in the Tǒngzōng — the 1-and-5 suànpán with the kǒujué mnemonics — was the dominant Chinese mathematical-computing instrument from the late Míng through the twentieth century, displaced only by the electronic calculator in the 1980s–90s. The technological-historical significance of the Tǒngzōng is therefore difficult to overstate: it codifies the standard form of an instrument that was the principal calculating device of one of the major civilizations for nearly four centuries.

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