Jiuzhang Suanshu 九章算術

## **Description**

***Nine Chapters on the Mathematical Art*** (九章算術 jiǔ zhāng suàn shù) is a Chinese mathematics book composed by generations of scholars during the Han dynasty (206 BC - 220 AD). The book is one of the earliest surviving mathematics texts from China. It lays out an approach to mathematics that centers on finding the most general methods or algorithms for solving problems. The book consists of 246 mathematical problems organized into nine chapters. Each problem consists of a problem statement, followed by the answer, and an explanation of the method to obtain the answer. A majority of the problems involve the computation of fractions, rates, proportion, square roots, and cube roots. These computation methods are extended to solve applied problems concerning areas, volumes, simultaneous linear equations, quadratic equations, and right triangles.

## **Contents**

Chapter 1, entitled *Rectangular Fields* (方田 fāng tián), contains 38 problems on computing the areas of planar shapes with an emphasis on the arithemtic of fractions. Algorithms for reducing, adding, subtracting, multiplying, and dividing fractions are explained in detail, as well as the Euclidean algorithm for finding the greatest common divisor of two numbers. Formulas for calculating the area of rectangles, triangles, trapezoids, circles, annulus, and circle segments are given.

Chapter 2, entitled *Grains* (粟米 sù mǐ), contains 46 problems on solving proportion problems using the *rule of three* (今有術 jīn yǒu shù). It begins with a list of exchange rates among different types of grains, beans, and seeds. There are problems on solving for an unknown quantity of an item given the quantity of another item and the relevant exchange rates. In addition, there are problems on evenly dividing a set amount of some commodity among people.

Chapter 3, entitled *Proportional Distribution* (衰分 cuī fēn), contains 20 problems on the distribution of commodities and money at proportional rates. Certain problems illustrate arithmetic and geometric series.

Chapter 4, entitled *Small Widths* (少廣 shǎo guǎng), contains 24 problems that involve solving for an unknown side length. The first eleven problems ask for the length of a rectangular field given that the width is incrementally increased by a unit fraction but the area is kept constant. These problems are an exercise on the addition of fractions and division by mixed numbers. The remaining thirteen problems involve the extraction of square roots and cube roots. Certain areas of squares and circles as well as volumes of cubes and spheres are given such that the side length or diameter are to be determined using the square root algorithm (開方術 kāi fāng shù) and cube root algorithm (開立方術 kāi lì fāng shù).

Chapter 5, entitled *Estimating Workloads* (商功 shāng gōng), contains 28 problems on computing the volumes of solids. Solids such as prisms, cylinders, pyramids, wedges, and frustums are named after construction terms and architectural features such as excavated earth, walls, fortresses, pavillions, pools, and dykes.

Chapter 6, entitled *Equitable Distribution* (均輸 jūn shū), contains 28 problems concerning more complex problems on rate and proportion with applications such as taxation, work, and travel.

Chapter 7, entitled *Excess and Deficit* (盈不足 yíng bù zú), contains 20 problems on solving equations using the *rule of double false position* (盈不足術 yíng bù zú shù). Majority of the problems deal with linear equations in two unknowns and are solved simultaneously by making two guesses (one excess, one deficit) in which the answer is obtained by interpolating from the two guesses.

Chapter 8, entitled *Rectangular Arrays* (方程 fāng chéng), contains 18 problems that involve solving systems of linear equations by the method of elimination (方程術 fāng chéng shù) and the use of negative numbers. The method is essentially Gaussian elimination except the coefficients are placed in columns instead of rows . The problems involve up to six equations in six unknowns. Negative numbers are used in many of the solutions and rules on how to compute with them are explained under the method of positive and negative numbers (正負術 zhèng fù shù).

Chapter 9, entitled *Right Triangles* (勾股 gōu gǔ), contains 24 problems on right triangles. Majority of the problems are solved using the Pythagorean theorem (勾股術 gōu gǔ shù) and derivative formulas. There are also problems concerning Pythagorean triples, shapes inscribed within a right triangle, and similar triangles. Quadratic equations of the form

x
2

b
# x
c

{\displaystyle x^2 + bx = c}

 are solved by extending the square root algorithm in Chapter 4.

## **About the Authors**

**Zhang Cang** (張蒼 zhāng cāng, c. 253 - 152 BC) is credited for compiling and editing the earliest edition of the text. Archaeological evidence suggests that much of its contents is a collection of Chinese mathematical knowledge from the 11th – 2nd century BC. Later mathematicians wrote commentaries explaining the theory behind the solution methods. The most famous commentary was written by the mathematician **Liu Hui** (劉徽 liú huī, fl. 3rd century AD). Another important commentary was written by the astronomer-mathematician **Li Chunfeng** (李淳風 lǐ chún fēng, 602 - 670 AD) as part of the **Ten Computational Canons** (算經十書 suàn jīng shí shū).

Zhang CangLiu HuiLi Chunfeng

## **Trivia**

* In modern times, "算術" (pinyin: suàn shù) refers to arithmetic only; however, "算術" referred to mathematics in ancient China. Hence, the original title should be translated as *Nine Chapters on Mathematics*.