Cèyuán hǎijìng fēnlèi shìshù 測圓海鏡分類釋術
Categorically-Arranged Procedural-Explanation of the Sea-Mirror of Circle Measurements based on 李冶 Lǐ Yě’s KR3f0042 Cèyuán hǎijìng (Yuán); explanatory exposition by 顧應祥 (Gù Yìngxiáng, 1483–1565, 明, shìshù 釋術)
About the work
Gù Yìngxiáng’s 10-juan attempt to expound and make accessible Lǐ Yě’s KR3f0042 Cèyuán hǎijìng. Composed in the mid-Jiā-jìng period (c. 1540–1565). The work re-organizes Lǐ Yě’s 170 problems by fēnlèi (categorical-arrangement), removes the cǎo 草 (procedural-elaboration that uses lì tiānyuányī), and substitutes elementary computational procedures (shìshù 釋術 — explanatory-procedure) using simple addition-subtraction-and-square-root-extraction methods.
The Sìkù 提要 of this work is exceptionally sharp in its critique. The fundamental problem: Gù Yìngxiáng received Lǐ Yě’s Cèyuán hǎijìng from Táng Shùnzhī 唐順之 but did not understand the lì tiānyuányī method that is the algebraic foundation of Lǐ Yě’s procedural-elaborations. Lacking this understanding, Gù simply removed the tiānyuányī sections and substituted what he could understand — elementary computational steps — claiming that this was a pedagogical service to beginners. The 提要 demolishes this rationale:
“[Gù] Yìngxiáng obtained Lǐ Yě’s book from Táng Shùnzhī. On the words ‘establishing the heavenly-element-one’ [ lì tiānyuányī ], [the two] mutually deliberated-and-sought-but-could-not-obtain its understanding. Accordingly [Gù] removed the detailed-procedures and exclusively elaborated the calculation-methods, transforming [the work] into this book — claiming-himself it was convenient for the lower-students. [He] entirely did not recognize that the marvel of lì tiānyuányī is its ability to enable [the solution of] what other methods cannot obtain. If [the cǎo-procedure] is absent, then even [Lǐ] Yě [himself] would have had something he could not obtain by method — and yet [Gù Yìngxiáng] sticks-and-fixates on the addition-subtraction-and-extraction-of-roots numbers. [This] can be called: ‘following the branches-and-leaves and losing the root’“.
The 提要 then preserves Táng Shùnzhī’s letter to Gù Yìngxiáng about the exposition — one of the most important documents in the late-Míng history of mathematics:
“This book [Lǐ Yě’s Cèyuán hǎijìng]‘s lower-form numbers are too detailed; the upper-form meaning is somewhat brief. [The result] makes the viewer unable to escape the doubt ‘the numbers can be set out, but the meaning is hard to know — like giving someone the mandarin-duck pillow but not giving them the golden needle [for embroidering it]‘. My intent: hoping that [you, Gù Yìngxiáng,] would at the essential point select-and-bring-out one or two as the source-head of the law-method to bring out, allowing later-period mathematics-doers, [those] who recognize the great [purport, will] obtain its meaning, [those] who recognize the small [matters, will] obtain its numbers — then this book will become even more crystal-clear”.
The 提要 sympathetically observes: “[Táng Shùnzhī’s] dissatisfaction with [Gù] Yìngxiáng’s work is correct. Only the source-head of the law-method is just the one phrase ‘establishing the heavenly-element-one’. Once [Gù] Yìngxiáng has removed [this], how then is he to make the essential-point selection*?*”
The 提要 then notes a residual value: even though Gù Yìngxiáng’s exposition fails on its own terms, his elementary-computational presentations of the zhūchéngfāngliángyú (additions-subtractions-extractions of polynomial-square-root expressions up to the third, fourth, and fifth orders) provides a useful supplement to Lǐ Yě’s cǎo-procedures, by displaying the actual computational forms (suànshì 算式) that Lǐ Yě’s tiānyuányī setups would produce on the counting-board.
The work is therefore preserved in the Sìkù not as an adequate exposition of Lǐ Yě (it is not), but as a useful auxiliary to Lǐ Yě’s Cèyuán hǎijìng for the student trying to understand the computational forms.
For the original Lǐ Yě work, see KR3f0042 Cèyuán hǎijìng. For the principal exponent, see 顧應祥. For Lǐ Yě, see 李冶. For the late-Míng forgetting and Kāngxī-period recovery of tiānyuányī, see the 提要 of KR3f0042.
Tiyao
[Full text summarized above. Dated Qiánlóng 46 (1781), tenth month.]
Translations and research
- 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 (treats the late-Míng Lǐ Yě exposition tradition).
- Martzloff, Jean-Claude. A History of Chinese Mathematics, Berlin: Springer, 1997.
Other points of interest
The Táng Shùnzhī letter preserved in the 提要 is one of the most explicit late-Míng statements of the forgetting of SòngYuán algebra. The famous metaphor — yuānyāng zhěn (mandarin-duck pillow, the embroidered finished product) without jīnzhēn (golden needle, the tool for making it) — captures the late-Míng predicament: the SòngYuán mathematical works survived as physical books, but the mathematical understanding necessary to use them had been lost. The Sìkù 提要’s preservation of this letter is a deliberate documentation of the late-Míng-mathematical-impoverishment thesis.
The 提要’s residual judgment that Gù Yìngxiáng’s work has some useful auxiliary value — for displaying computational forms — is a generous editorial concession that allows the work to be preserved despite its principal inadequacy. This is one of the more nuanced editorial assessments in the Tiānwén suànfǎ 提要 corpus.