Chinese Mathematical Texts

**By: Tao Steven Zheng (鄭濤)**

## **Suan Shu Shu 算數書**

### **Historical Background**

The *Suan Shu Shu* (算數書, *Book on Numbers and Computation*) is the oldest surviving mathematics problems book from ancient China. It was excavated in a tomb near Zhangjiashan, Jiangling County, in Hubei province between December 1983 and January 1984. The tomb belonged to a civil servant and was sealed in 186 BC, dating the text to the early Western Han period. Like many works of the time, the *Suan Shu Shu* was written on bamboo strips and strung together with string to produce a scroll.

### **Content**

The *Suan Shu Shu* consists of 69 mathematics problems. Each problem has a question, an answer, followed by a method. The problems cover a range of topics, including elementary arithmetic, fractions, proportion, factorization of numbers, geometric progressions, unit conversions, the double false position method, the extraction of square roots, volume calculations, and the calculation of the unknown side of a rectangle given its area and one side.

## **Zhoubi Suanjing 周髀算經**

### **Historical Background**

The *Zhoubi Suanjing* (周髀算經, *Mathematical Classic of the Zhou Gnomon*) is the oldest surviving astronomy text from ancient China. This influential text was compiled around the 1st century BC during the Han dynasty (汉朝, 206 BC – 220 AD); however, much of the content suggests that it dates back to the Zhou dynasty (周朝, 1046 BC – 256 BC). Zhao Shuang (趙爽, fl. 3rd century AD), a mathematician and astronomer who lived in the Kingdom of Wu during the Three Kingdoms period, authored a commentary of the *Zhoubi Suanjing*. He gave a detailed explanation on the *xiantu* and several formulas for solving right triangle problems.

### **Content**

The *Zhoubi Suanjing* consists of two chapters. Chapter one is a collection of mathematical and astronomical knowledge captured in two lengthy conversations, the first conversation is between the Duke of Zhou (周公, fl. 11th century BC) and his chief astronomer Shang Gao (商高); the second conversation is between Master Chen (陳子) and Rong Fang (榮方) around the 4th century BC. Chapter two explains in detail the *gaitian shuo* (蓋天說, celestial dome cosmological theory) and calculations related to the observation of shadows cast by a gnomon. A dissection proof of the Pythagorean theorem, called the *xiantu* (弦圖, hypotenuse diagram) is presented in Chpater One of the *Zhoubi Suanjing*.

## **Jiuzhang Suanshu 九章算術**

### **Historical Background**

The *Jiuzhang Suanshu* (九章算術), or *Nine Chapters on the Mathematical Art*, is a collection of 246 problems with answers and accompanying solution methods. The text dates back to the Han dynasty (206 BC – 220 AD), and served as a standardized compendium of mathematical knowledge of its time. According to the mathematician Liu Hui (劉徽, fl. 3rd century AD), the *Jiuzhang Suanshu* was initially compiled by the minister Zhang Cang (張蒼, 256 – 152 BC) from mathematical works that were lost from the book burnings of the First Emperor of China, Qin Shihuang. It is known that Zhang Cang was well-versed in the exact sciences of mathematics, astronomy, and harmonics; however, there is little evidence for the destruction of such knowledge during the Qin dynasty. In fact, mathematical texts written on bamboo slips dating to the Qin and early Han period have been excavated, with the most notable example being the *Suan Shu Shu* (算數書, *Book of Numbers and Computation*). What is certain is that some of the contents found in the *Jiuzhang Suanshu* must have originated from early antiquity. The Han dynasty scholar Zheng Xuan (鄭玄) stated that the title of each chapter *Jiuzhang Suanshu* was inspired from the nine mathematical topics taught to the aristocratic youth of the Zhou dynasty (1046 – 256 BC); albeit with some minor changes. Furthermore, the mathematical theory and practice would have been crucial in running state affairs by all the feudal kingdoms of the Warring States period (476 – 256 BC).

There existed many commentaries of the *Jiuzhang Suanshu* written by mathematicians during the Han dynasty and later dynasties. The first commentary was written by Xu Yue (徐岳, c. 180 - 220 AD), but it had been lost by the Song dynasty. Other lost commentaries include the one written by Zu Chongzhi (祖沖之) in the 5th century AD, and another by Zhen Luan (甄鸞) in the 6th century AD. The earliest extant and most famous commentary of the *Jiuzhang Suanshu* was composed by Liu Hui in 263 AD. During the Tang dynasty, Li Chunfeng (李淳風, 602 – 670 AD) and his associates were tasked to annotate the *Jiuzhang Suanshu* alongside nine other mathematical texts for the Imperial College (國子監 *guo zi jian*). The texts formed a mathematical corpus called the *Suanjing Shi Shu* (算經十書, ‘*Ten Mathematical Classics*’), which heavily influenced the mathematical education and research in East Asia.

### **Content**

The contents of the *Jiuzhang Suanshu* follows a set style that formed the basis found in most Chinese mathematics texts it preceded. Each problem consists of a problem statement followed by the question. The answer is then supplied, and a solution method or principle formula is explained in words. Problems grouped under one solution method are usually presented before the method. The *Jiuzhang Suanshu* categorizes its problems under nine chapters:

(1) *Fang tian* (方田, “rectangular fields”) contains 38 problems. It introduces the rules for calculations involving fractions, and is primarily concerned with computing the areas of planar figures (referred to as ‘fields’). Formulas for finding the area of rectangles, triangles, trapezoids, circles, circle segments, circular sectors, and annuli (ring-shaped fields) are written procedurally in words.

(2) *Su mi* (粟米, ‘grains’) contains 46 problems. The chapter begins with a table of exchange rates for various types of grains. The values in the table are then used to calculate the quantity of one grain type for that of a different grain type. This method is identical to the ‘rule of three’, a common proportion calculation used in commerce. Aside from the problems on exchanging grains, there are problems that involve calculating the price of various commodities.

(3) *Cui fen* (衰分, ‘differential allocation’) contains 20 problems. The chapter consists of a class of problems that deals with the distribution of quantities according to a given list of proportions.

(4) *Shao guang* (少廣, “diminishing breadth”) contains 24 problems. The chapter involves solving the unknown length of one side of planar and solid figures. The chapter introduces the square root algorithm for determining the side of a square given its area, and the cube root algorithm for determining the side of a cube given its volume.

(5) *Shang gong* (商功, ‘consultations on works’) contains 28 problems on calculating the volume of various solid figures, including prisms, pyramids, frustums, and wedges. Like the area formulas in chapter 1, the formulas for calculating the volumes of shapes are written procedurally in words.

(6) *Jun shu* (均輸, ‘equal supply’) contains 28 problems with multiple rates. Applications include taxation, pursuit problems, and work allocation problems.

(7) *Ying bu zu* (盈不足, ‘excess and deficiency’) contains 20 problems solved using a form of linear interpolation called *ying bu zu*. This method is equivalent to ‘double false position’.

(8) *Fang cheng* (方程, ‘rectangular arrays’) contains 18 problems. The chapter describes a general method for solving systems of simultaneous equations. The *fang cheng shu*  (方程术, ‘method of rectangular arrays’) is equivalent to Gaussian elimination. This chapter offers the earliest description of the use of positive and negative numbers.

(9) *Gou gu* (勾股, ‘base and altitude’) contains 24 problems. This chapter deals with problems about right triangles, more specifically, the application of the *gou gu method* (Pythagorean theorem) and geometric proportion. Some problems involve the solution of quadratic equations.

## **Shushu Jiuzhang**

### **Historical Background**

The Shushu Jiuzhang (Mathematical Treatise in Nine Chapters)
The Shushu Jiuzhang (Mathematical Treatise in Nine Chapters) is the representative work of Qin Jiushao (1208 – 1268 AD).

### **Content**

The book contains 81 problems, which are divided into 9 categories. Qin Jiushao's main innovations include the dayan zongshushu, sanxie qiujishu and the Qin Jiushao suanfa. The dayan zongshushu is a generalized method of solving systems of linear modular congruences, which is generally recognized today as the Chinese Remainder Theorem. The sanxie qiujishu is a method of finding the area of a triangle given its three sides. The Qin Jiushao suanfa is a method of finding positive roots of higher-order polynomial equations. The Shushu Jiuzhang sums up much of the mathematical development during the Song dynasty and marks the peak of ancient Chinese mathematics.