Mathematical work

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 in the comments to chapter 1. He calculated pi to 3.141024 < \pi < 3.142074 with a 192 (= 64 × 3) sided polygon. Archimedes used a circumscribed 96-gon to obtain the inequality \pi <\tfrac{22}{7}, and then used an inscribed 96-gon to obtain the inequality \tfrac{223}{71} < \pi . Liu Hui used only one inscribed 96-gon to obtain his 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 = 3.141024 ~3.14 and put it in fraction form \pi = \tfrac{157}{50}. He later invented a quick method and obtained \pi =3.1416, which he checked with a 3072-gon (= 512 × 6). Nine Chapters had used the value 3 for , 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.

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.

生平

劉徽為《九章算術》作注，于三國魏景元四年（263年）成書，其中他提出用割圓術計算圓周率的方法，以內接正六邊形開始，逐次倍加邊數的方法，逐步逼近圓周率。《九章算術》僅以π=3，劉徽則計算出正192邊形的面積，先得到圓周率的近似值為\pi=\frac{157}{50}=3.14，和晉武庫王莽銅律嘉量比較，覺得「此術微小」，于是再用圓周率捷法計算出正3072邊形的面積，求得\pi=\frac{3927}{1250}=3.1416。作此書注時，他還依據其「割補術」為證勾股定理，另闢蹊徑作青朱出入圖。圖雖失傳，但據其「出入相補、以盈補虛」原理，後人參照書中類似方法還原了此圖。

劉徽的注釋兼用圖形和模型作說明，以圖形相互拼湊方法解決各種面積計算問題，相當於一般平面幾何學中所用的平移與疊合的方法；並用直截面積的方法來計算立體體積。他指出《九章算術》計算球體體積方法錯誤，並引入了「牟合方蓋」（垂直相交的兩個圓柱體的共同部分的體積）這一著名的幾何模型，認為只有「牟合方蓋」與球體積之比才正好等於正方形與其內切圓的面積之比，也就是：

::球體積 :\, 牟合方蓋體積 = \pi:4 \,

但劉徽沒有給出牟合方蓋的體積公式，所以也就得不出球體的體積公式。

劉徽並在《九章算術注》提出重差術，應用中國傳統的出入相補原理，以多達4次的觀測，測量山高水深等數值。在唐代，有關重差術的注文被抽出單行，因其第一題是測量海島高度和距離的問題，故題為《海島算經》，成為《算經十書》之一。劉徽創造的四次重差觀測術，被吳文俊稱為「使中國測量學達到登峰造極的地步」，美國數學家弗蘭克·斯委特茲讚譽這使得「中國在數學測量學的成就，超越西方約一千年」。

後世紀念

劉徽的卓越成就受到後人的重視，宋徽宗時代為恢復數學教學制度，便追封了部分歷代的天算家，其中便有劉徽。