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Liu Hui

Liu Hui (Chinese: 劉徽; pinyin: Liú Huī; 225-295 AD) was a Chinese mathematician who lived during the Wei Dynasty. His contributions, especially the narrative he wrote called “Nine Chapters on the Art of Mathematics” include a proof of the Pythagorean Theorem, theorems of solid geometry, an improvement on Archimedes’ π approach, and a systematic method for deciphering linear equations in several unknowns.

APA 7: Çarıkçıoğlu, A. E. (2023, April 22). Liu Hui. PerEXP Teamworks. [Article Link]

Who is Liu Hui?

Liu Hui (He lived in the third century AD, but his star shone in 263 AD.) is a Chinese mathematician. The only thing about Liu Hui’s life that is known is that he died in 3 AD. He lived in the Northern Wei Kingdom for a century.

His fame is based on the narrative he completed in 263: “Jiuzhang Suanshu (九章算術, pronounced: “Cicuang Sanshu”/Nine Chapters on the Art of Mathematics)”. This work is a series of mathematical laws of the first century BC or AD, which have the same importance in the East as the geometry book “Euclid’s Elements” in the Western world. [1]

His evidence and details

Liu’s narratives about the “Nine Chapters” are proof of his algorithms’ accuracy. These proofs are the oldest known Chinese proofs in the modern sense. However, unlike the ancient Greek mathematical copywriters, Liu’s theorems did not set out to prove the accuracy of algorithms. For example, he has meticulously proven algorithms for determining the area of circles. And he has done this by the volume of pyramids by their territory into an infinite number of parts. Although this is the case, his first purpose was to perform calculations of such operations. He also proved the algorithms used for arithmetic and algebraic operations, such as adding fractions and solving simultaneous systems of linear equations, without having a similar purpose. [2]

An analysis of Liu’s evidence reveals some repeated transactions. His regular use of what might be called “Algebraic proofs” in an algorithmic context has perhaps contributed to the emergence of this particular type of proof in world mathematics. In all these cases, a few basic operations underlie all the algorithms in the “Nine Sections”. Thus, it is seen that he aims to show that he reduces the variety and complexity of transactions. [3]

In the preface to the Nine Chapters, Liu drew attention to a gap in his transactions that does not allow for solving problems related to celestial distances. Thus, he probably appeared in the seventh century and added measurement problems and algorithms that formed a kind of “Trigonometry” to fill this gap in his work “Haidao Suanjing (Sea Island Mathematics Manual)” attributed to him.

The mentioned book contained many practical geometry problems, including the meters of the heights of the towers of the Chinese pagoda (Multi-story place of worship). This small study outlined the procedures of how to measure distances and heights “With poles of tall architecture and horizontal bars fixed at right angles to them”. However, the following situations were taken into account in their work:

  • Measuring the height of an island opposite to sea level and seen from the sea,
  • The height of a tree on the hill,
  • The size of a city wall viewed from a long distance,
  • The depth of a valley (Using crossbars located on the building skeleton),
  • The height of a tower on a plain seen from the top,
  • The width of a river mouth seen from a distance on land,
  • The width of a valley seen from a cliff,
  • The depth of a transparent pool,
  • The width of a river as seen from the top,
  • The size of a city seen from a mountain. [4]

The philosophical movements he was influenced by

A philosophical point of view influenced Liu’s mathematical work. Liu quotes from a comprehensive variety of ancient philosophical texts, such as the Confucian laws, prominently the “Yijing (易經, Book of Changes)”. In addition, it can be said that he was inspired by the leading commentaries that make up Taoism, the Ancient Chinese teachings (Such as Zhuangzi/莊子), and Mohist (It is Ancient Chinese teachings, the forerunner of which is Mozi.). Moreover, his narratives regularly reflect contemporary philosophical developments.

Finally, Liu winks at Parmenides and Heraclitus. Since he sees mathematics as an algorithm that embodies the transformations that take place during play with all parts of space, it can be argued that his philosophical reflections on mathematics related to the concept of “Change” are the main research topic in China. [5]


  1. DICTIONARY ENTRY Chemla, K. Carole (2006, June 9). Liu HuiEncyclopedia Britannica. [Britannica]
  2. WEBSITE Reck, K. (n.d.). Liu Hui. Math Greats and Great Math. [Math Greats and Great Math]
  3. JOURNAL Straffin, P. D. (1998). Liu Hui and the First Golden Age of Chinese Mathematics. Mathematics Magazine, 71(3), 163. [JSTOR]
  4. WEBSITE Mathigon. (n.d.). Liu Hui. Mathigon. [Mathigon]
  5. WEBSITE O’Connor, J. J., & Robertson, E. F. (2003, December). Liu Hui. Maths History. [Maths History]

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