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張丘建算經

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Text ID
KR3fc004
Date
466–656
Dynasty
北魏

Zhāng Qiūjiàn suànjīng 張丘建算經

Mathematical Canon of Zhāng Qiūjiàn by 張邱建 (撰) — the catalog title preserves the original form 丘; the Persons note uses the Qīng-period taboo-avoidance form 邱 — and 李淳風 (注)

About the work

A late-fifth-century / sixth-century mathematical problem-collection in 3 juàn, by Zhāng Qiūjiàn 張丘建 (字 / dates unknown; conventionally placed in the late Northern Wèi or Northern Zhōu, c. 460–490 CE), with the canonical Táng commentary by 李淳風 Lǐ Chúnfēng (commissioned c. 656). One of the Suànjīng shíshū 算經十書 (Ten Mathematical Classics) canonized by Lǐ Chúnfēng for the Táng Imperial-Academy mathematics curriculum. The KX edition here parallels the WYG / SBCK editions catalogued at KR3f0039 (note: the catalog spells the author 邱 — Qīng-period taboo-avoidance form for 丘 to avoid Kǒngzǐ’s given name).

Abstract

The work contains 92 problems-with-solutions in three juàn, covering essentially the same applied-mathematical topics as the KR3f0032 Jiǔzhāng — proportional exchange, area-and-volume calculation, root extraction, taxation, and right-triangle problems — but with two distinctive mathematical innovations. The first is a more refined treatment of the manipulation of fractions, particularly the “first formula” stated in Zhāng’s preface that “to set up a problem and not know the procedure for combining and reducing fractions is the great difficulty of the suànjīng art” (立術不知合分通約者,算經之難也). The second, in problem 38 of the third juàn, is the famous Bǎi jī wèn 百雞問 (Hundred Fowls Problem): “A rooster costs 5; a hen costs 3; three chicks cost 1. With 100 cash buy 100 fowls — how many of each kind?“. The problem is a system of two linear equations in three unknowns with positive-integer constraints — the earliest known systematic indeterminate-equation problem in the Chinese tradition, and the foundation on which 秦九韶 Qín Jiǔsháo’s Dàyǎn qiúyī shù 大衍求一術 would later build.

The work’s transmission was difficult: Zhāng’s own xiàjuàn 下卷 (third juàn) was already incomplete in the Sòng, and the surviving Hundred-Fowls solution gives only three of the four positive-integer solutions (4 roosters/18 hens/78 chicks; 8/11/81; 12/4/84). The fourth (0 roosters / 25 hens / 75 chicks) is not given by Zhāng, and Lǐ Chúnfēng’s commentary likewise omits it. The Sòng mathematician 甄鸞 Zhēn Luán (active c. 570) had supplied an alternate solution; this also is partly preserved in the TángSòng transmission. The Sìkù editors recovered the text from the Yǒnglè dàdiǎn 永樂大典 and printed it in nearly-complete form, supplementing the lacuna in juàn 3 from Sòng manuscript fragments.

Lǐ Chúnfēng’s commentary, composed under Táng Gāozōng’s imperial commission, supplies the implicit working of each procedure and clarifies obscure points; it is essentially of the same type and date as his commentaries on the other Suànjīng texts. Dating: the work is most plausibly placed in the Northern Wèi (notBefore 466, the official year of the Yánxìng era; notAfter takes Lǐ Chúnfēng’s Táng commentary as 656). Some scholars argue for a slightly earlier dating based on the work’s mention of monetary units; others (Qián Bǎocóng 1963) place the author firmly in the late fifth century.

The work was the principal Chinese mathematical text to introduce the bǎi jī indeterminate-equation problem-type, which entered the standard problem-collection across all subsequent Chinese mathematical traditions and was transmitted through Korea and Japan to become a standard pedagogical problem-type of East-Asian mathematics generally. The problem also appears, possibly through Arab intermediaries, in Fibonacci’s Liber Abaci (1202) and in subsequent European medieval and early-modern problem collections.

Translations and research

  • Lam Lay-Yong. 1969. “On the Existing Fragments of Yang Hui’s Hsiang Chieh Suan Fa.” Archive for History of Exact Sciences 6: 82–88. — Discusses the Hundred Fowls problem in its later Sòng commentarial context.
  • Lam Lay-Yong and Ang Tian-Se. 2004. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China, revised edition. Singapore: World Scientific. — Includes a full English presentation of the Zhāng Qiūjiàn mathematics.
  • Qián Bǎocóng 錢寶琮, ed. 1963. Suàn-jīng shí-shū 算經十書. Beijing: Zhōnghuá shūjú. — The standard modern critical edition; Qián’s introduction discusses the dating and transmission in detail.
  • Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer. — Treats the Bǎi jī wèn in the context of Chinese indeterminate-analysis history.

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