His major contributions as recorded in his commentary onThe Nine Chapters on the Mathematical Art include a proof of thePythagorean theorem, theorems in solidgeometry, an improvement onArchimedes'sapproximation ofπ, 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 visitedLuoyang, where he measured the sun's shadow.[3]
Liu Hui expressed mathematical results in the form ofdecimal fractions that utilizedmetrological units (i.e., related units of length with base 10 such as 1chǐ = 10cùn, 1cùn = 10fēn, 1fēn = 10lí, etc.); this led Liu Hui to express a diameter of 1.355 feet as 1chǐ, 3cùn, 5fēn, 5lí.[4] 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 andYang Hui (c. 1238–1298 CE) is considered to have introduced a unified decimal system.[5]
Liu provided a proof of a theorem identical to thePythagorean theorem.[3] 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."[6]
In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors toempirical solid geometry. For example, he found that awedge with rectangular base and both sides sloping could be broken down into a pyramid and atetrahedral wedge.[7] He also found that a wedge withtrapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid.[7] He computed the volume of solid figures such as cone, cylinder, frustum of a cone, prism, pyramid, tetrahedron, and a wedge.[2] However, he failed to compute the volume of a sphere and noted that he left it to a future mathematician to compute.[2]
In his commentaries onThe Nine Chapters on the Mathematical Art, he presented:
Analgorithm for the approximation ofpi (π). While at the time, it was common practice to assumeπ to equal 3,[8] Liu utilized the method of inscribing a polygon within a circle to approximateπ to equal on the basis of a 192-sided polygon.[9] 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 approximateπ to equal 3.14159, which is a more accurate approximation than the one calculated by Archimedes or Ptolemy.[10]
Cavalieri's principle to find the volume of a cylinder and the intersection of two perpendicular cylinders[11][12] although this work was only finished byZu Chongzhi andZu 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 theTang mathematician and Taoist believerLi Chunfeng.
Through his work in theNine Chapters, he could have been the first mathematician to discover and compute with negative numbers; definitely before Ancient Indian mathematicianBrahmagupta started using negative numbers.
Liu Hui also presented, in a separate appendix of 263 AD calledHaidao Suanjing orThe Sea Island Mathematical Manual, several problems related tosurveying. This book contained many practical problems of geometry, including the measurement of the heights ofChinese pagoda towers.[13] 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".[14] With this, the following cases are considered in his work:
The measurement of the height of an island opposed to itssea 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 aravine (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
Liu Hui's information about surveying was known to his contemporaries as well. Thecartographer and state ministerPei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of arectangular grid and graduated scale for accurate measurement of distances on representative terrain maps.[15] Liu Hui provided commentary on the Nine Chapter's problems involving buildingcanal and riverdykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.[16]
Although translated into English long beforehand, Liu's work was translated intoFrench by Guo Shuchun, a professor from theChinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.
^abcStewart, Ian (2017).Significant Figures: The Lives and Work of Great Mathematicians (First US ed.). New York: Basic Books. p. 40.ISBN978-0-465-09613-8.
^Needham, Joseph (1959).Science and Civilization in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. With the collaboration of Wang Ling. Cambridge University Press. pp. 84–85.ISBN978-0521058018.{{cite book}}:ISBN / Date incompatibility (help)
^Needham, Joseph (1959).Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. With the Collaboration of Wang Ling. Cambridge University Press. p. 86.ISBN978-0521058018.{{cite book}}:ISBN / Date incompatibility (help)
^Needham, Joseph (1959).Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. With the Collaboration of Wang Ling. Cambridge University Press. p. 99.ISBN978-0521058018.{{cite book}}:ISBN / Date incompatibility (help)
^Needham, Joseph (1959).Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. With the Collaboration of Wang Ling. Cambridge University Press. p. 100.ISBN978-0521058018.{{cite book}}:ISBN / Date incompatibility (help)
^Needham, Joseph (1959).Science and Civilisation in China, Volume 3, Mathematics and the Sciences of the Heavens and the Earth. With the Collaboration of Wang Ling. Cambridge University Press. p. 101.ISBN978-0521058018.{{cite book}}:ISBN / Date incompatibility (help)
