Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (

The Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...

).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state of

Cao Wei Wei ( Hanzi: 魏; pinyin: ''Wèi'' < Middle Chinese: *''ŋjwei^C'' <

Three Kingdoms The Three Kingdoms () from 220 to 280 AD was the tripartite division of China among the dynastic states of Cao Wei, Shu Han, and Eastern Wu. The Three Kingdoms period was preceded by the Eastern Han dynasty and was followed by the West ...

period (220-280 CE) of China. His major contributions as recorded in his commentary on ''The Nine Chapters on the Mathematical Art'' include a proof of the Pythagorean theorem, theorems in solid

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, an improvement on Archimedes's approximation of , and a systematic method of solving linear equations in several unknowns. In his other work, '' Haidao Suanjing (The Sea Island Mathematical Manual)'', he wrote about geometrical problems and their application to surveying. He probably visited

Luoyang Luoyang is a city located in the confluence area of Luo River and Yellow River in the west of Henan province. Governed as a prefecture-level city, it borders the provincial capital of Zhengzhou to the east, Pingdingshan to the southeast, Nanyang ...

, where he measured the sun's shadow.

Mathematical work

Liu Hui expressed mathematical results in the form of decimal fractions that utilized metrological units (i.e., related units of length with base 10 such as 1 '' chǐ'' = 10 '' cùn'', 1 ''cùn'' = 10 ''fēn'', 1 ''fēn'' = 10 ''lí'', etc.); this led Liu Hui to express a diameter of 1.355 feet as 1 ''chǐ'', 3 ''cùn'', 5 ''fēn'', 5 ''lí''. Han Yen (fl. 780-804 CE) is thought to be the first mathematician that dropped the terms referring to the units of length and used a notation system akin to the modern decimal system and Yang Hui (c. 1238-1298 CE) is considered to have introduced a unified decimal system. Liu provided a 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 A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...

with rectangular base and both sides sloping could be broken down into a pyramid and a

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

wedge.Needham, Volume 3, 98-99. He also found that a wedge with

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...

base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. He computed the volume of solid figures such as cone, cylinder, frustum of a cone, prism, pyramid, tetrahedron, and a wedge. However, he failed to compute the volume of a sphere and noted that he left it to a future mathematician to compute. In his commentaries on ''The Nine Chapters on the Mathematical Art'', he presented: * An algorithm for the approximation of pi (). While at the time, it was common practice to assume to equal 3, Liu utilized the method of inscribing a polygon within a circle to approximate to equal

\frac

on the basis of a 192-sided polygon. This method was similar to the one employed by Archimedes whereby one calculates the length of the perimeter of the inscribed polygon utilizing the properties of right-angled triangles formed by each half-segment. Liu subsequently utilized a 3072-sided polygon the approximate to equal 3.14159, which is a more accurate approximation than the one calculated by Archimedes or Ptolemy. * 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 Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...

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.

Surveying

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 A pagoda is an Asian tiered tower with multiple eaves common to Nepal, India, China, Japan, Korea, Myanmar, Vietnam, and other parts of Asia. Most pagodas were built to have a religious function, most often Buddhist but sometimes Taoist, ...

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 Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardise ...

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 A ravine is a landform that is narrower than a canyon and is often the product of streambank erosion.transparent Transparency, transparence or transparent most often refer to: * Transparency (optics), the physical property of allowing the transmission of light through a material They may also refer to: Literal uses * Transparency (photography), a still, ...

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 Canals or artificial waterways are waterways or engineered channels built for drainage management (e.g. flood control and irrigation) or for conveyancing water transport vehicles (e.g. water taxi). They carry free, calm surface flo ...

and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.Needham, Volume 4, Part 3, 331. Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the

Chinese Academy of Sciences The Chinese Academy of Sciences (CAS); ), known by Academia Sinica in English until the 1980s, is the national academy of the People's Republic of China for natural sciences. It has historical origins in the Academia Sinica during the Republi ...

, who began in 1985 and took twenty years to complete his translation.

References

External links

*
Liu Hui and the first Golden Age of Chinese Mathematics, by Philip D. Straffin Jr
* * {{DEFAULTSORT:Liu, Hui Ancient Chinese mathematicians Cao Wei science writers Mathematicians from Shandong People from Zibo People of Cao Wei Writers from Zibo 3rd-century Chinese mathematicians

Mathematical work

Surveying

See also

Further reading

References

External links