Liu Hui

Liu Hui (fl. 3rd century CE) was a Chinese mathematician and writer who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art, in which he was possibly the first mathematician to discover, understand and use negative numbers. He was a descendant of the Marquis of Zi District (菑鄉侯) of the Eastern Han dynasty, whose marquisate is in present-day Zichuan District, Zibo, Shandong. He completed his commentary to the Nine Chapters in the year 263. He probably visited Luoyang, where he measured the sun's shadow.

Along with Zu Chongzhi (429–500), Liu Hui was known as one of the greatest mathematicians of ancient China. Liu Hui expressed all of his mathematical results in the form of decimal fractions (using metrological units), yet the later Yang Hui (c. 1238-1298 AD) expressed his mathematical results in full decimal expressions.

Liu provided commentary on a mathematical proof of a theorem identical to the Pythagorean theorem. Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".

In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. In his commentaries on the Nine Chapters, he presented:

• An algorithm for calculation of pi (π) in the comments to chapter 1. He calculated pi to ${\displaystyle 3.141024<\pi <3.142074}$ with a 192 (= 64 × 3) sided polygon. Archimedes used a circumscribed 96-gon to obtain the inequality ${\displaystyle \pi <{\tfrac {22}{7}}}$, and then used an inscribed 96-gon to obtain the inequality ${\displaystyle {\tfrac {223}{71}}<\pi }$. Liu Hui used only one inscribed 96-gon to obtain his π inequality, and his results were a bit more accurate than Archimedes'. But he commented that 3.142074 was too large, and picked the first three digits of π=3.141024 ~3.14 and put it in fraction form ${\displaystyle \pi ={\tfrac {157}{50}}}$. He later invented a quick method and obtained ${\displaystyle \pi =3.1416}$, which he checked with a 3072-gon (= 512 × 6). Nine Chapters had used the value 3 for π, but Zhang Heng (78-139 AD) had previously estimated pi to the square root of 10.
• Gaussian elimination.
• Cavalieri's principle to find the volume of a cylinder and the intersection of two perpendicular cylinders although this work was only finished by Zu Chongzhi and Zu Gengzhi. Liu's commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng.
• Through his work in the Nine Chapters, he could have been the first mathematician to discover and compute with negative numbers; definitely before Ancient Indian mathematician Brahmagupta started using negative numbers.

Survey of sea island

Liu Hui also presented, in a separate appendix of 263 AD called Haidao Suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers. This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them". With this, the following cases are considered in his work:

• The measurement of the height of an island opposed to its sea level and viewed from the sea
• The height of a tree on a hill
• The size of a city wall viewed at a long distance
• The depth of a ravine (using hence-forward cross-bars)
• The height of a tower on a plain seen from a hill
• The breadth 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 a hill
• The size of a city seen from a mountain.

Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps. Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.

Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.