**Liu Hui** (fl. 3rd century) was one of the most renowned mathematicians in ancient China (primarily account his fame to his method for approximating pi).

(One version of portrait of Liu Hui)

He solved all the problems on 九章算術 (which is more prevalently known as ‘*The Nine Chapters on the Mathematical Art*‘ (In ancient Chinese, arithmetics has the same meaning as ‘mathematical art’)) and made his own version of compilation out of it named 九章算術註 (*‘Commentary on the Nine Chapters’*). *‘Nine Chapters’* is one of the earliest written ancient Chinese mathematics book where various math problems on measuring, architecture, engineering and surveying, and mathematicians of several generations ranging from 10th century to 2nd century B.C.E. together composed the whole book, thus it took great knowledge on mathematics to comment on the book and was considered to be a great honor for doing this job.

(image of ‘*Nine Chapters on the Mathematical Art*‘)

Furthermore, Liu Hui also wrote and published his own book 海島算經 (‘*Sea Island Mathematical Manual*‘) in year 263 C.E.. The book was originally known as the 10th chapter of *Commentary on Nine Chapters*, and was subsequently re-edited and solely published as a independent book during Tang Dynasty. An original copy of the book is kept in the library of Cambridge University; it contained 9 sets of problems. The name of the book ‘sea island’ came from the problems of the books which were basically about surveying the distance/size of an island and height of towers using the method of trigonometry.

(image of ‘*Sea Island Mathematical Manual*‘)

Liu Hui devoted his whole life working on exploring the subject of mathematics, applied mathematics and surveying methodology. He lived through the period of Three Kingdoms, where a lot of wars were going on. Despite the warring atmosphere he used to live, Liu kept his personality and his enthusiasm about math study. He used both deduction and intuition when dealing with practical and theoretical math problems. Even though he was suffering from poverty throughout his whole life, yet he never gave up discovering the field of mathematics.

Liu Hui’s achievements on mathematics could be roughly categorized into two aspects:

1. Number Theory (Algebraic):

a. He elaborated his way of doing elementary arithmetics (including the way of finding common numerator when adding/subtracting fractions, the way of simplifying fractions), and he was one of the earliest pioneers who discussed about the irrationality of some square roots, based on his study of an infinite process to approach some of the square roots. Also, he suggested to approach the precise value of these irrational numbers with the 10-based (decimal) numeral system.

b. He provided a comparatively clearer definition of ‘率’ (lv, which means ‘ratio’ in ancient Chinese), and used it as an expression of the ancient Chinese version of augmented matrices.

c. He presented his proof of the Pythagorean Theorem, and gave methods for determining some parameters concerning with right triangles. Also, his work on trigonometry, as well as the similarity of right triangles shall also be remembered.

(Liu Hui’s proof of Pythagorean Theorem)

2. Theorem on calculating areas/volumes (specially area of circle and volume of sphere)

a. HIS PI ALGORITHM is famous worldwide. He attempted to estimate the area of a circle using n-gons. He calculated the area of hexagon inscribed in a circle, and then 12-gon, 24-gon and so on till 3092-gon. Afterwards he used his result on the determination of pi, and got pi=3927/1250=3.1416, which is a pretty accurate approximation in his era. This ratio was named after him by ‘Hui ratio’.

b. In his comment on *Nine Chapters*, he pointed out that the formula of the area of a sphere indicated on the book (which is V=9d^3/16 with d being the diameter) is incorrect, and elaborated his proof with the introduction of a exclusively new 3-D model called 牟合方蓋 (‘Hegai’ literally means ‘double box lid’).

(simulated model of ‘double box lid’)

Bow to this great mathematician!