Quick Answer: Who Was Liu Hui?
Liu Hui (刘徽, Liú Huī) was a Chinese mathematician who worked in the state of Wei during the Three Kingdoms period. His commentary on the Nine Chapters on the Mathematical Art (九章算术, Jiǔzhāng Suànshù) is traditionally dated to 263 CE.[1]
- Commentary: Liu explained the reasoning behind procedures that an older text gave as instructions.[1]
- Pi: He used regular polygons inside a circle to improve the estimate of pi.[4]
- Surveying: The work later called the Sea Island Mathematical Manual (海岛算经, Hǎidǎo Suànjīng) uses geometry to measure objects beyond direct reach.[6]
Little is known about Liu’s personal life. The two works show his approach to problems in the history of Chinese mathematics.[1]
An old rule can tell you what to calculate. Liu Hui’s commentary asks why the rule works and where it gives only an approximation. He used that approach for familiar problems such as finding the area of a circle and for problems where the object being measured was an island away from shore.
What Liu Hui Added to the Nine Chapters
The older text organized problems by subject. It covered tasks such as measuring fields, calculating quantities for construction and distributing goods or obligations. A problem could state a result and then give a procedure for arriving at it.[2]
Liu added explanations that showed how a procedure made mathematical sense. Those explanations were passed down with the earlier text. In the seventh century, Li Chunfeng and other scholars prepared a further commentary. A reader of a later edition therefore encounters several layers of work.[3]
How Liu Hui Approximated Pi
Liu began with a regular hexagon whose vertices lay on a circle. Doubling the number of sides gave a polygon that filled more of the circle. Each repetition made the region left between the polygon and the circle smaller. Calculating the polygon’s area let Liu narrow down the area of the circle and improve the estimate of pi.[4]
Liu carried the calculation through a polygon with 192 sides. He used the remaining space around the polygon to constrain the possible value of pi. One approximation he gave was 3.1416.[4]
How Liu Hui Measured an Island from Shore
The manual’s first problem uses two poles of equal height set a known distance apart. An observer moves back from each pole until its top lines up with a distant island peak. The two distances the observer moves back differ. The two lines of sight form related triangles. The method uses the difference in the observer’s setbacks, the separation of the poles and their known height to infer the height of the peak and its distance from the nearer pole.[5]
The nine surveying problems continued work on right-triangle procedures in the earlier text. During the Tang dynasty they circulated as a separate work under the title used today. They give a mathematical example of measurement for early Chinese engineering and surveying.[6]
FAQs About Liu Hui
Did Liu Hui Write the Nine Chapters?
The main text was already in circulation when Liu wrote his commentary. Later editions preserve both the older text and Liu’s explanations.[3]
Is Liu Hui’s Approximation of Pi Exact?
The value 3.1416 is an approximation. The polygon procedure gave Liu a way to improve and assess it.[4]
When Did Liu Hui Live?
His work places him in third-century China. His commentary is traditionally dated to 263 CE, while his birth and death years are uncertain.[1]
出典
[1] MacTutor History of Mathematics, University of St Andrews. Liu Hui
[2] Shen Kangshen, John N. Crossley and Anthony W.-C. Lun. The Nine Chapters on the Mathematical Art, Companion and Commentary
[3] Karine Chemla. Reading Instructions of the Past, Classifying Them, and Reclassifying Them
[4] Eric W. Weisstein, Wolfram MathWorld. Liu Hui’s Exhaustion Method
[5] Liu Hui. Sea Island Mathematical Manual, Chinese Text
[6] Oxford University Press. Sea Island Mathematical Manual, Translation and Commentary