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 1st century CE. This book is one of the earliest surviving ...

).'' He was a descendant of the Marquis of Zixiang of the

Eastern Han dynasty The Han dynasty was an Dynasties of China, imperial dynasty of China (202 BC9 AD, 25–220 AD) established by Liu Bang and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–206 BC ...

and lived in the state of

Cao Wei Wei () was one of the major Dynasties in Chinese history, dynastic states in China during the Three Kingdoms period. The state was established in 220 by Cao Pi based upon the foundations laid by his father Cao Cao during the end of the Han dy ...

during the

Three Kingdoms The Three Kingdoms of Cao Wei, Shu Han, and Eastern Wu dominated China from AD 220 to 280 following the end of the Han dynasty. This period was preceded by the Eastern Han dynasty and followed by the Jin dynasty (266–420), Western Jin dyna ...

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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, theorems in solid

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an improvement on Archimedes's

approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...

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 ( zh, s=洛阳, t=洛陽, p=Luòyáng) is a city located in the confluence area of the Luo River and the Yellow River in the west of Henan province, China. Governed as a prefecture-level city, it borders the provincial capital of Zheng ...

, where he measured the sun's shadow.

Mathematical work

Liu Hui expressed mathematical results in the form of

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

fractions that utilized

metrological Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fr ...

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 Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial the ...

(c. 1238–1298 CE) is considered to have introduced a unified decimal system. Liu provided a proof of a theorem identical to the

. 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 Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how t ...

solid geometry. For example, he found that a

wedge A wedge is a triangle, triangular shaped tool, 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 conver ...

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

tetrahedral In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

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

trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

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 to approximate to equal 3.14159, which is a more accurate approximation than the one calculated by Archimedes or Ptolemy. *

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...

. *

Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

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), courtesy name Wenyuan (), was a Chinese astronomer, inventor, mathematician, politician, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3.1415 ...

and

Zu Gengzhi Zu Geng or Zu Gengzhi (; ca. 480 – ca. 525) was a Chinese mathematician, politician, and writer. His courtesy name was Jingshuo (). He was the son of the famous mathematician Zu Chongzhi. He is known principally for deriving and proving the fo ...

. 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 Li Chunfeng (; 602–670) was a Chinese astronomer, historian, mathematician, and politician who was born in today's Baoji, Baoji, Shaanxi, during the Sui dynasty, Sui and Tang dynasty, Tang dynasties. He was first appointed to the Imperial Astr ...

. * 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 Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

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 Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Plane (mathematics), two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of Point (geom ...

. This book contained many practical problems of geometry, including the measurement of the heights of

Chinese pagoda A pagoda is a tiered tower with multiple eaves common to Thailand, Cambodia, Nepal, India, China, Japan, Korea, Myanmar, Vietnam, and other parts of Asia. Most pagodas were built to have a religious function, most often Buddhism, Buddhist, bu ...

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 mean, average surface level of one or more among Earth's coastal Body of water, bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical ...

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. Ravines are typically classified as larger in scale than gullies, although smaller than valleys. Ravines may also be called a cleuch, dell, ...

(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 Cartography (; from , 'papyrus, sheet of paper, map'; and , 'write') is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can ...

and state minister

Pei Xiu Pei Xiu (224–3 April 271), courtesy name Jiyan, was a Chinese cartographer, geographer, politician, and writer of the state of Cao Wei during the late Three Kingdoms period and Jin dynasty (265–420), Jin dynasty of China. He was very m ...

(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 ...

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 French may refer to: * Something of, from, or related to France ** French language, which originated in France ** French people, a nation and ethnic group ** French cuisine, cooking traditions and practices Arts and media * The French (band), ...

by Guo Shuchun, a professor from the

Chinese Academy of Sciences The Chinese Academy of Sciences (CAS; ) is the national academy for natural sciences and the highest consultancy for science and technology of the People's Republic of China. It is the world's largest research organization, with 106 research i ...

, 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 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