Ancient Chinese Mathematicians

**By: Tao Steven Zheng (郑涛)**

## **Liu Hui 劉徽**

Liu Hui (c. 225 - c. 295 AD) was a Chinese mathematician who lived in the northern Wei kingdom during the 3rd century AD. He is best known for his commentary of the *Jiuzhang Suanshu* (The Nine Chapters on the Mathematical Art), completed in 263 AD.

Liu’s commentary of the *Jiuzhang Suanshu*

```
proved the correctness of its algorithms. These proofs are the earliest-known Chinese proofs in the contemporary sense. However, in contrast to authors of ancient Greek mathematical texts, Liu did not set out to prove theorems so much as to establish the correctness of algorithms. For example, he rigorously proved algorithms for determining the area of circles and the volume of pyramids by dissecting the regions into infinitely many pieces. He also proved algorithms for arithmetic and algebraic operations, such as adding fractions and solving systems of simultaneous linear equations. An analysis of Liu’s proofs reveals some recurring procedures. For instance, he regularly used what can be called algebraic proofs within an algorithmic context, perhaps a contribution to the emergence of this specific kind of proof in world mathematics. In all these cases, it appears that he aimed to show that a small number of fundamental operations underlie all the algorithms in The Nine Chapters, thereby reducing their diversity. 3/27/22, 12:16 PM Liu Hui -- Britannica Online Encyclopedia https://www.britannica.com/print/article/344457 2/2 In his preface to the The Nine Chapters, Liu noted a gap in its procedures that did not allow one to tackle problems involving celestial distances. He thus appended surveying problems and algorithms that amounted to a kind of trigonometry to fill this gap. These problems were gathered, probably in the 7th century, in an independent book, Haidao suanjing (“Sea Island Mathematical Manual”), ascribed to him. A certain philosophical perspective permeates the mathematical work of Liu. He quotes a great variety of ancient philosophical texts, such as the Confucian canons, prominently the Yijing (I Ching; Book of Changes); Daoist key texts, such as the Zhuangzi; and Mohist texts. Moreover, his commentary regularly echoes contemporary philosophical developments. It can be argued that he considered an algorithm to be that which, in mathematics, embodies the transformations that are at play everywhere in the cosmos—thus his philosophical reflections on mathematics related to the concept of “change” as a main topic of inquiry in China.

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## **Wang Xiaotong 王孝通**

**Wang Xiaotong** 王孝通 (580 - 640 AD), was a Chinese mathematician and astronomer who made important advances in the solution of problems involving cubic equations and quartic equations. According to his autobiography, Wang Xiaotong became interested in mathematics at a young age. After studying Liu Hui's commentary on the *Jiuzhang Suanshu* 九章算術 (*Nine Chapters on the Mathematical Art*) and Zu Geng's *Zhui Shu* 綴術 (*Method of Interpolation*), he became a mathematics teacher. Wang was a *suanli boshi* 算歷博士 (Erudite Scholar of Mathematics and Calendrics) during the reign of **Emperor Gaozu of Tang** (618 – 626 AD), and was later appointed as the *taishi cheng* 太史丞 (Deputy Director of the Astronomical Bureau ). He took part in the revision of the Wuyin calendar 戊寅元歷 and wrote the *Jigu Suanjing* 緝古算經 (*Continuation of Ancient Mathematics*) in 625.

In 623, together with **Zu Xiaosun** 祖孝孫, Wang was assigned to report on problems with the calendar. Although only recently adopted, it was already out of step with the eclipses. In fact Wang did not approach this in a sophisticated way; he proposed to ignore the irregularity of the sun's motion and also the precession of the equinoxes—both had already been incorporated in calendar calculations by Zu Chongzhi in the fifth century.

The *Jigu Suanjing* contains 20 problems. Except for the first problem dealing with astronomy, the other problems involve quadratic, cubic, or quartic equations. Wang solved 25 cubic equations of the form

x
3

P

x
2

Q
# x
N

{\displaystyle x^3 + Px^2 + Qx=N}

where

P
,
Q
,
N

{\displaystyle P, Q, N}

. In 13 of the problems the volume of a polyhedron is given, along with some other facts involving its dimensions, and one is asked to find its dimensions or else, given the volumes of some polyhedral parts of the polyhedron, to find their dimensions. In solving some of these problems, Wang first used the formula for the volume of a dike of varying cross section. There are also problems that involve right-angled triangles.

In 656, the *Jigu Suanjing* became a text for the Imperial examinations. Wang's work would influence later Chinese mathematicians, such as **Jia Xian** 賈憲 (c. 1010 - 1070 AD) and **Qin Jiushao** 秦九韶 (1202 - 1261 AD) of the Song dynasty. His ideas on cubic equations would later be transmitted westward, influencing Islamic mathematics and the mathematics of late Medieval and early Renaissance Europe.

**Sources**

* J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).
* O'Connor, John J.; Robertson, Edmund F., "Wang Xiaotong", MacTutor History of Mathematics archive, University of St Andrews
* 中国大百科全书数学. 北京、上海：中国大百科全书出版社. 1988.
* 中国古代科学技术史纲-数学卷. 沈阳： 辽宁教育出版社, 2000.