Húshǐ suànshù 弧矢算術
Calculation Methods for Arc-and-Arrow (circle-segment computations) by 顧應祥 (Gù Yìngxiáng, 1483–1565, 明, zhuàn 撰), composed Jiājìng rénzǐ 3rd month (1552)
About the work
Gù Yìngxiáng’s 1-juan compendium of circle-segment (húshǐ 弧矢) computational methods. The húshǐ tradition treats the geometry of a circle’s segment defined by a chord (húxián 弧弦), the arc above the chord (húbèi 弧背), and the perpendicular from the chord’s midpoint to the arc (shǐ 矢, “arrow”). The work compiles and supplements earlier Chinese treatments of this geometry: Wú Xìnmín 吳信民’s late-Sòng / early-Yuán Jiǔzhāng fǎ (which treats the topic in only one entry), Zhū Shìjié’s Sìyuán yùjiàn (several entries but without methodological-explanation), Shěn Kuò’s Mèngxī bǐtán (with the gēyuán method but limited to chord-from-diameter-and-arrow). Gù Yìngxiáng’s preface laments the inadequacy of these earlier treatments and presents his own elaborated procedures.
The work is a Míng-period compilation that, like Gù Yìngxiáng’s KR3f0043 Cèyuán hǎijìng fēnlèi shìshù, suffers from Gù’s failure to understand the lì tiānyuányī 立天元一 algebraic methodology. Gù’s procedures are accordingly elementary-computational rather than algebraic — providing the suànshì (computational forms) without the underlying algebraic apparatus.
The 提要 is brief but identifies the work’s connection to the earlier Yuán-period Shòushí lìcǎo 授時厯草 of Guō Shǒujìng — which had originally formulated the húshǐ problem using lì tiānyuányī methodology to express the shǐ (arrow) as a polynomial root, but had stopped at displaying the polynomial coefficients without showing the actual root-extraction procedure. Táng Shùnzhī 唐順之 had emphasized to Gù Yìngxiáng that húshǐ is the foundational geometry for bù rìchán (sun-position computation) and yuèlí (moon-path computation) — i.e., for serious calendrical-astronomical work — and had presented his own Húshǐ lùn 弧矢論 to Gù Yìngxiáng to motivate this compilation. Gù Yìngxiáng’s response, the present work, supplies the elementary-computational expositions of the kāi dàizòng sānchéng (extraction of cubic-with-companion-coefficient) procedures that Guō Shǒujìng had used but not displayed.
The Sìkù 提要 is parallel to its 提要 of Gù’s other work KR3f0043: Gù’s elementary-computational expositions are useful as auxiliary computational displays but do not address the underlying methodological-algebraic content — and so the work, like KR3f0043, is preserved as auxiliary rather than as a primary mathematical contribution.
For Gù Yìngxiáng’s biography, see 顧應祥. For the parallel work, see KR3f0043 Cèyuán hǎijìng fēnlèi shìshù.
Tiyao
[Full 提要 in source file. Dated Qiánlóng 46 (1781), second month.]
Translations and research
- Limited substantial secondary literature. Treated in:
- Mei Rongzhao 梅榮照, Míng-Qīng shù-xué-shǐ lùn-wén jí 明清數學史論文集, Nánjīng: Jiāngsū Jiào-yù Chūbǎnshè, 1990.
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
Other points of interest
The work documents the late-Míng difficulty of accessing the SòngYuán algebraic-methodological tradition: even Táng Shùnzhī (one of the most learned figures of the late-Míng intellectual scene, also Wáng Yǎngmíng’s interlocutor and a major figure in the gǔwén prose-revival movement) was unable to direct Gù Yìngxiáng to the recovery of lì tiānyuányī. The methodological forgetting was profound and would not be remedied until the Kāngxī-period European-mediated recovery (cf. KR3f0042).