Qín Jiǔsháo 秦九韶
Style name Dàogǔ 道古. Native of Lǔ jùn 魯郡 (the ancestral seat per his own preface; he himself was born and raised in Sìchuān, his father having moved there). Born around 1208; died 1261. The Sìkù 提要 of KR3f0041 Shùshū jiǔzhāng (the work whose title is later corrupted to Shùxué jiǔzhāng) notes the discrepancy that Qín claims to be of Lǔ jùn in a 1247 preface but Lǔ jùn had long fallen to the Yuán by then — concluding that this is an ancestral-genealogical claim rather than a current-residence claim.
The greatest mathematician of the Southern Sòng. Qín Jiǔsháo’s life was tumultuous: his career encompassed military service against the Mongols, official posts in Sìchuān and the lower-Yangzi region, repeated periods of disgrace-and-exile. He was twice impeached for corruption and once accused of murder; he died in disgrace at Méizhōu 梅州 (Guǎngdōng) in 1261. Despite these biographical complications, his mathematical work is universally recognized as the supreme achievement of SòngYuán mathematics — and one of the great mathematical works of any pre-modern tradition.
His sole surviving work is the Shùshū jiǔzhāng 數書九章 (KR3f0041 Shùxué jiǔzhāng in the Sìkù-recorded title — apparently corrupted), in 18 juàn containing 81 problems organized into 9 chapters by topical-administrative concern (parallel to the Jiǔzhāng’s 9-chapter structure but with completely different topical content):
(1) Dàyǎn 大衍 — the Chinese Remainder Theorem in fully-systematic form (Dàyǎn qiúyī shù 大衍求一術), generalizing the KR3f0033 Sūnzǐ suànjīng’s problem-26 procedure into a complete algorithm for any system of linear congruences with pairwise-coprime moduli — and the more general case where the moduli have common factors. This is one of the great algorithmic achievements of pre-modern mathematics.
(2) Tiānshí 天時 — astronomical-calendrical computation, including planetary motion.
(3) Tiányù 田域 — area calculations, including the Hǎidǎo-style remote-sensing problems.
(4) Cèwàng 測望 — depth-and-distance measurement.
(5) Fùyì 賦役 — taxation and labor service.
(6) Qiángǔ 錢榖 — currency and grain.
(7) Yíngjiàn 營建 — civil engineering.
(8) Jūnlǚ 軍旅 — military formations and supply.
(9) Shìyì 市易 — commerce.
The work is the foundational source for several major mathematical innovations:
(a) Lì tiānyuányī shù 立天元一術 — the systematic algebraic-equation methodology, expressing polynomial equations symbolically using counting-rod arrangements. This is the foundation of SòngYuán algebra and the precursor of all subsequent Chinese algebraic work.
(b) Héyīn shù — what amounts to the Horner method for evaluating polynomials at specific values; later named the Héyīn shù in the SòngYuán literature, fully systematized by Zhū Shìjié in the 14th century.
(c) Indeterminate-equation systematic methodology — generalizing the KR3f0039 Zhāng Qiūjiàn suànjīng’s Hundred Fowls problem into a full algorithm for determining the solvability of systems of indeterminate linear equations.
The 提要 of KR3f0041 judges Qín Jiǔsháo as the lone Southern-Sòng figure who successfully revived the technical mathematical tradition: “During the Sòng dynasty the various Confucians esteemed empty-discussion and despised practical-use; mathematics, although one of the Sage-Gate Six Arts, was likewise scorned and not spoken of… The work has both flaws-and-merits visible alongside, but the refined-and-correct passages are the majority”. Through his work, the SòngYuán mathematical tradition — which would also produce Lǐ Yě 李冶 (KR3f0042 / KR3f0043), Yáng Huī 楊輝 (KR3f0046), and Zhū Shìjié 朱世傑 — was foundational for the late-imperial Chinese mathematical inheritance.