By: Tao Steven Zheng (鄭濤)
Liu Hui was a Chinese mathematician who lived during the Three Kingdoms period (220 - 280 AD) and Wei-Jin dynasty (265 - 420 AD). Little is known about Liu Hui's life aside that he was a descendant of the Marquis of Zixiang, and that he completed his commentary on the Jiuzhang Suanshu (九章算術; Nine Chapters on the Mathematical Art) in the year 263 AD. Despite his relative obscurity, Liu Hui was known for his mathematical prowess and inventiveness. His contribution to Chinese mathematics was immense because he rigorously proved the validity of many algorithms and formulas known before his time. He also developed novel ideas in the study of geometry that are theoretically sound.
Discoveries[]
Theory of Fractions and Rates[]
Approximations of π[]
Liu Hui devised a geometric algorithm for computing π called the ge yuan shu (割圓術; circle cutting method). Using a 96-sided polygon, Liu obtained the bound . He later devised a quick method involving the use of geometric series to arrive at a more accurate approximation of . Liu Hui's quick method achieved the accuracy of a 3072-sided polygon using the result he obtained from a 96-sided polygon.
Fangcheng method (equivalent to Gaussian elimination）
Arithmetic with Negative Numbers[]
Liu Hui established the rules of arithmetic with negative numbers in his commentary of the Jiuzhang Suanshu (九章算術). However, it is likely that negative numbers were known in China during the Han dynasty (206 BC - 220 AD) or earlier. He then explained in great detail the use of negative numbers for solving fangcheng problems (方程). These problems are systems of simultaneous linear equations solved using the elimination method.
See: Jiuzhang Suanshu (Fangcheng 8)
Method of indivisibles
The Cavalieri Principle[]
Liu Hui discovered the Cavalieri principle (the method of indivisibles) 1400 years before Bonaventura Cavalieri (1598 - 1647). He applied this principle to derive precise formulas for calculating the volumes of prisms, pyramids, cylinders, wedges, and frustums. Liu Hui also applied the Cavalieri principle to determine the volumetric ratio of the sphere to the mou he fang gai (牟合方盖), a solid formed by the intersection of two perpendicular cylinders.
See: Jiuzhang Suanshu Shanggong 10, Liu Hui's Puzzle
Out-In Complementary Principle[]
In his book Haidao Suanjing (海島算經), Liu Hui created and solved several surveying problems by exploiting the chong cha method (重差術), a nascent form of trigonometry.
