Zégǔxī zhāi suànxué shísān zhǒng 則古昔齋算學十三種
Mathematical Works from the Zé-gǔ-xī Studio, in Thirteen Kinds by 李善蘭 (撰)
About the work
李善蘭 Lǐ Shànlán’s (1810–1882) collected mathematical writings, originally published 1867 from his Hǎi-níng-period studio Zégǔxī zhāi 則古昔齋 (“Studio for Following the Ancients”). The collection comprises thirteen separate treatises (shísān zhǒng 十三種) in 25 juàn total, representing the entirety of Lǐ Shànlán’s indigenous-tradition mathematical work — composed before his 1852 move to Shànghǎi and the start of the Mòhǎi Shūguǎn translation programme that would consume the rest of his career.
Abstract
The studio name Zégǔxī 則古昔 (“following / taking-as-pattern the ancients”) makes explicit the orientation of the collection: Lǐ Shànlán’s mathematical work of the 1840s and early 1850s is squarely in the indigenous SòngYuán algebraic and MíngQīng cyclotomic-series traditions, and explicitly continues those traditions rather than introducing new methods from European mathematics. The collection was assembled retrospectively in the mid-1860s for publication in 1867; the title and the framing of the works as a “studio collection” are Lǐ Shànlán’s later imposition on the earlier productive period, marking the boundary between his pre-translation indigenous work and his post-translation modernised mathematical writing.
The thirteen treatises (conventional list, with brief summary):
-
方圓闡幽 Fāngyuán chǎn yōu (1 juàn) — early work on cyclotomic series, extending the Míng Āntú tradition.
-
弧矢啓秘 Húshǐ qǐ mì (2 juàn) — extension of the arc-sagitta apparatus.
-
對數探源 Duìshù tàn yuán (2 juàn) — original investigation of logarithms, including independent derivations of the principal logarithmic series.
-
垛積比類 Duī duǒ bǐ lèi (4 juàn) — the famous “pile-summation” treatise. Contains the Lǐ Shànlán identity — a binomial-coefficient summation formula independently rediscovered in the indigenous Chinese tradition, equivalent to a result in the European combinatorial literature.
-
四元解 Sìyuán jiě (2 juàn) — solutions of four-variable algebraic systems in the 朱世傑 Zhū Shìjié Sìyuán yùjiàn tradition.
-
麟德術解 Líndé shù jiě (3 juàn) — reconstruction of the Tang Líndé 麟德 calendar of Lǐ Chúnfēng (cf. parallel work by 李銳 KR3fc051–KR3fc055).
-
椭圜正術解 Tuǒyuán zhèngshù jiě (2 juàn) — the ellipse and its principal procedures.
-
椭圜新術 Tuǒyuán xīnshù (1 juàn) — new methods for the ellipse.
-
椭圜拾遺 Tuǒyuán shíyí (3 juàn) — supplementary results on the ellipse.
-
火器真訣 Huǒqì zhēn jué (1 juàn) — applied ballistics (the projectile-motion calculations needed for artillery work).
-
尖錐變法解 Jiānzhuī biànfǎ jiě (1 juàn) — methods involving the “pointed-cone” — Lǐ Shànlán’s original geometric construction analogous to the European calculus of infinitesimals, used by him to derive results that European mathematics would derive via the calculus.
-
級數迴求 Jíshù huí qiú (1 juàn) — series-reversion (inverse-series) computations.
-
天算或問 Tiānsuàn huò wèn (1 juàn) — miscellaneous questions on astronomy and mathematics.
The collection is the principal monument of indigenous Chinese mathematical work at the moment immediately preceding the wholesale absorption of European mathematics. The Jiānzhuī biànfǎ jiě in particular has been the subject of substantial historical-philosophical attention: Lǐ Shànlán’s “pointed-cone” construction is in effect an independent indigenous-Chinese discovery of the integral calculus, derived from indigenous-tradition geometric apparatus rather than from European infinitesimal methods. The discovery is sometimes adduced as the principal counterexample to a strong “Western-priority” thesis about modern mathematics.
Dating: the constituent works are composed across the 1840s and early 1850s; the collection itself is assembled and published 1867. notBefore 1845 (early productive period for the earliest constituent works); notAfter 1867 (date of collected publication).
Translations and research
- Horng, Wann-Sheng [洪萬生]. 1991. “Li Shanlan, the Impact of Western Mathematics in China during the Late 19th Century.” Ph.D. diss., City University of New York. — Principal English-language biography and mathematical study.
- Wáng Yúsōng 王渝生. 2000. Lǐ Shàn-lán: zhōng-guó jìn-dài kē-xué de xiān-qū-zhě 李善蘭：中國近代科學的先驅者. Bě-jīng: Kē-xué chū-bǎn-shè.
- Tián Miǎo 田淼. 2003. Zhōng-guó shù-xué de xī-huà lì-chéng 中國數學的西化歷程.
- Martzloff, Jean-Claude. 1997 [2006]. A History of Chinese Mathematics. Berlin: Springer.
- Wú Wénjùn 吳文俊, ed. 1985. Zhōng-guó shù-xué shǐ dà-xì 中國數學史大系, vol. 8.