## The Nine Chapters on the Mathematical Art: Companion and CommentaryThe Nine Chapters on the Mathematical Art has been the most important mathematical source in China for the past 2000 years, comparable in significance to Euclid's Elements in the West. The Nine Chapters quickly acquired a distinguished reputation, and was the standard mathematics textbook in China and the surrounding regions until Western science was introduced in about 1600. This volume contains the first complete English translation of the Nine Chapters, together with the illuminating commentary of Liu Hui written in the 3rd century AD and other early century commentaries and further insights provided by the translators. The Nine Chapters contains 246 problems and their solutions, which fall into nine categories that are firmly based on practical needs. There are methods for solving problems in areas such as land measurement, construction, agriculture, commerce, and taxation, while the Chinese commentators provide the algorithms needed to solve the problems, and explanations of how the algorithms work. The translators' commentary provides clear and accessible background material for the Western reader, explanations of technicalities, and notes on the treatment of similar or identical mathematical problems in other countries. This first, full-English translation gives us an idea of the distinctive style and important contributions that have been made by traditional Chinese mathematics. |

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### Contents

Introduction | 1 |

Liu Huis Preface to his Running Commentary | 52 |

Field Measurement | 59 |

Millet and Rice | 134 |

Distribution by Proportion | 157 |

Short Width | 175 |

Construction Consultations | 251 |

Fair Levies | 307 |

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### Common terms and phrases

Algorithm altitude Answer arithmetic Aryabhata Aryabhatiya base bie'nao borrowed rod bundles of top calculated century China Chinese mathematics circle circumference column cone cost counting rods cube root cubic denominator diagram diameter difference digit distance divide dividend divisor Dynasty equations Euclid's Elements Extract the square formula fractions frustum given number given rate giving the number gnomon Gougu Han Dynasty height Hui's hulled millet hypotenuse inscribed integer length liang Liu Hui Liu's low grade paddy mathematicians means Method millet is required Nine Chapters number of coins obtained pole Problem Proposition pyramid Pythagorean triples qiandu Qilu Rule Qing Dynasty ratio rectangle remainder right triangular prism right-angled triangle Rule of Three Seki Takakazu sheng side silk solving sought rate soya beans sphere square root subtract Take Tell traditional Chinese mathematics triangular prism upper volume width xianchu Yang Hui yangma zhang Zu Chongzhi