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. 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. Liu utilized the method of inscribing a polygon within a circle to approximate to equal \frac{157}{50} 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. •
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.
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 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. ==See also==