Xiangjie jiuzhang suanfa

work by Yang Hui

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July 16, 2026

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The Xiangjie jiuzhang suanfa (1261) is a significant work by the Chinese mathematician Yang Hui, who was active during the Southern Song dynasty. This book, meaning "A Detailed Analysis of the Nine Chapters on the Mathematical Procedures," is a reclassification and explanation of problems from the earlier Han dynasty classic, Jiuzhang suanshu.12

A notable inclusion in Yang Hui's work is the oldest known representation of what is recognized in the West as Blaise Pascal's triangle. Yang Hui stated he copied this from an earlier text by Jia Xian, who flourished around 1050. The Xiangjie jiuzhang suanfa also details a geometric method for solving quadratic equations and includes various magic squares. The work was influential, with fragments found in a Ming dynasty encyclopedia and later editions circulating in Korea and Japan.13

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Yang Hui
…first found fragments of Yang’s Xiangjie jiuzhang suanfa (1261; “A Detailed Analysis of the Nine Chapters on the Mathematical Procedures”) in a handwritten copy of an imperial Ming dynasty encyclopaedia, and he later discovered in Suzhou a Song dynasty edition of Yang Hui suanfa (1275; “Yang Hui’s Mathematical Methods”). The…
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…first found fragments of Yang’s Xiangjie jiuzhang suanfa (1261; “A Detailed Analysis of the Nine Chapters on the Mathematical Procedures”) in a handwritten copy of an imperial Ming dynasty encyclopaedia, and he later discovered in Suzhou a Song dynasty edition of Yang Hui suanfa (1275; “Yang Hui’s Mathematical Methods”). The…
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Jia Xian
…been incorporated in Yang Hui’s Xiangjie jiuzhang suanfa (“A Detailed Analysis of the Nine Chapters on the Mathematical Procedures”), compiled in 1261 and preserved in manuscript form in the Yongle dadian (1408; “Great Encyclopaedia of the Yongle Reign”) and in a printed edition from 1842.
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…been incorporated in Yang Hui’s Xiangjie jiuzhang suanfa (“A Detailed Analysis of the Nine Chapters on the Mathematical Procedures”), compiled in 1261 and preserved in manuscript form in the Yongle dadian (1408; “Great Encyclopaedia of the Yongle Reign”) and in a printed edition from 1842.
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