An introduction to “Nine Chapters on the Mathematical Art(九章算术)”


The “Nine Chapters on the Mathematical Art” is an ancient Chinese mathematics book. It is one of the ten most important arithmetic books of ancient China. Although it is hard to find the accurate publishing time of this book, by the historical record, it had been published in 263 AD (Han dynasty). In the following dynasties, Chinese mathematicians kept revising it and supplementing it. Thus “Nine Chapters on the Mathematical Art” can be also seen as the essence of the ancient Chinese arithmetic. By the Qing dynasty (1644-1912), most Chinese mathematicians started studying math from this book. In the Tang dynasty (618-907) and Song dynasty (960-1279) “nine chapters of arithmetic” was the professional textbook by the government provision. Also in 1084 AD, the Chinese government published the printed version “Nine Chapters on the Mathematical Art”, which made it become the earliest printed version mathematics textbook in the world. As a famous mathematics textbook, “Nine Chapters on the Mathematical Art” was introduced in Japan and Korea in the Sui dynasty (581-618) and right now, it has been translated into Russian, German, French and other languages.
The content of “Nine Chapters on the Mathematical Art” is plentiful. It was written in “problems and solutions” form including 246 problems related to production and practical life, and they were distributed into nine chapters. Although many problems have several solutions, this book does not contain any proof, which is a common disadvantage of most Chinese ancient mathematics textbooks. The first chapter is called “Fang tian (方田)”. It is about computing the area of various plane geometrical figures such as sector, annulus arch and so on. Also in this chapter, it refers to the arithmetic of fractions, which is the earliest record of textbook referring to fractions. The second and third chapter are called “Su mi(粟米)” and “Cui fen (衰分)” which are about proration problems. The fourth chapter is named as “Shao guang (少广) ”. It narrates the methods of computing the length of a edge when you get the area of the figure. This chapter also introduces the method of extraction of square and cubic roots. The fifth chapter “Shang gong (商功)”, gives the formulas to compute the volume of many objects. The sixth chapter “Jun shu(均输)” focuses on collecting taxes. But it also involves in the conceptions direct, inverse compound proportions and other proportion theory. In western countries, these conceptions appeared after the 16th century. The seventh chapter “Ying bu zu (盈不足)” discusses the problems of profit and loss. Some solutions from this chapter are very advanced in the world. The eighth chapter is called “equation(方程)”. It uses the method “separation coefficient” to represent systems of liner equations, which is similar to matrices. It also gives the earliest complete solution of systems of liner equations. In the solutions, it even introduces the concepts of negatives. This is the first time in human history to expand the number system from positive numbers systems. In the last chapter “Gou gu(勾股) ”, it uses “Gou gu theorem” (also known as Pythagorean theorem in the west) to solve some problems which are related to practical life. Some stuff in this chapter are very advanced, the last problem of this chapter gives a formula. In the western world, this formula was put forward by American mathematician L.E.Dickson at the end of 19th century.
“Nine Chapters on the Mathematical Art” determined the framework of ancient Chinese mathematics. It focuses on computations related to practical problem and has a very profound effect on the following mathematics.九章算术


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