Tag Archives: Nine Chapters on the Mathematical Art

3010 Sea Island in the The Nine Chapters on the Mathematical Art

The Sea Island Mathematical Manual. 1726 Tu Shu Ji Cheng 窥望海岛之图.

I never read anything about The Nine Chapters on the Mathematical Art before taking this History of Math class. I heard about this book when I was a middle school student. It is interesting that I started paying attention 13 years after I graduated from a middle school. There is a chapter called “The Sea Island Mathematical Manual” in the book. I found this topic very interesting because I want to know how our ancient people solved real world problems without using any modern technology. In this blog, I will try to explore more about these sea island problems. (In this blog, I will use Nine Chapters to substitute the book title and “Sea Island” to substitute the chapter.).

Before talking about the sea island questions, I want to briefly talk about the book’s history. Nine Chapters was formed in the Han dynasty and it was a Chinese mathematics book that composed by several generations of scholars from the 10th – 2nd century BCE. This book has 246 real world questions, which relate to agriculture, business, engineering and solving equations, etc. It divides those questions into nine chapters. The Nine Chapters flourished between the Three Kingdoms period and earlier Tang dynasty in China. At that time this book was the primary math textbook in China and it also spread to Korea and Japan. The Nine Chapters was undoubtedly one of the cornerstones of Chinese modern math.

“The Sea Island” is one of the extension chapters of the book that was written by Liu Hui. This chapter has nine problems: surveys of Sea Island, height of a hill top pine tree, the size of a square city wall viewed afar, the depth of a ravine, the height of a building on a plain seen from a hill, the breadth of a river-mouth seen from a distance on land, the depth of a transparent pool, the width of a river as seen from a hill, and the size of a city seen from city. We can see all these questions are very similar to each other. Let’s take a look at the first question. Survey of sea island says “there is a sea island, and set up two three zhang (zhang is a distance and 1 zhang equals 3.3 meters) poles at one thousand steps apart and set the two poles and the island in a straight line. Step back from the front post 123 steps, with eye on the ground level, the tip of the pole is on a straight line with the peak of island. Step back 127 steps from the rear pole, eye on ground level also aligns with the tip of pole and tip of island. What is the height of the island, and what is the distance to the pole?” (Wikipedia)


According to Nine Chapters, Liu Hui could not measure the distance of the front pole to the island, so he sets up the two poles assuming they have the same height. Liu Hui gave two formulas: height of island AB = CD * DF / (FH – DG) + CD and distance of front pole to island BD = DG * DF / (FH – DG). How do we know these two formulas are correct? I thought about these two formulas but I could not convince myself until I read a proof about them. We have to take a look at Liu Hui’s theorem for the survey first. In the above figure he proved that FH * AI = IB * BF. He called it “‘In-out-complement’ principle which showed that the area of two inscribed rectangles in the two complementary right angle triangles have equal area” (Wikipedia). This proof is very straightforward if we know his “In-out-complement” principle. From the above figure, we know □EJ = □EB and □CK = □CB, then we use □EJ – □CK = □DE. Therefore, we know the height of island formula is correct. How do we get the distance of the front pole to island? That’s from □CB = □CK. An interesting proof huh?

The rest of eight problems used the “In-out-complement” principle to be solved too. I really like Liu Hui’s way to solve those real world problems, especially without using modern electronic technology. As you can see his ancient way to solve these problems was extremely important for geographical measurement and navigation industrial. If you want to learn more about rest of the questions, I highly recommend you to read the chapter 8 on the book Nine Chapters to dig more.


Shen K., John NC, Anthony W.-C, The Nine Chapters on the Mathematical Art. 1999.

JiMin L, 九章算术中的比率理论.

Zu Chongzhi and his mathematics


Image:Zu Chongzhi. Author: Gisling, via Wikimedia Commons.

Zu Chongzhi was a Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. He did a lot of famous mathematics during his life. His three most important contributions were studying The Nine Chapters on the Mathematical Art, calculating pi, and calculating the volume of sphere.

As we know,  The Nine Chapters on the Mathematical Art is the most famous book in the history of Chinese mathematics. In ancient China, most people could not understand “The Nine Chapters on the Mathematical Art”. Zu Chongzi read the book and then he used his comprehension to explain the formulas of the book. Zu Chongzhi, and his father wrote the “Zhui Shu”(缀术) together. The book made The Nine Chapters on the Mathematical Art easier to read. And the book also added some important formula by Zu. For example, the calculation of pi and the calculation of sphere volume. “Zhui Shu” also become math textbook at the Tang Dynasty Imperial Academy. Unfortunately, the book was lost in the Northern Song Dynasty.

Zu’s ratio, also called milü is named after Zu Chongzhi. Zu’s ratio was an early accurate approximation of pi. It was recorded in the “Book Of Sui” and “Zhui Shu”. (Book Of Sui is the official history of the Sui dynasty). According to the “Book Of Sui”, Zu Chongzhi discovered that pi is between 3.14159276 and 3.14159277. Today, we know the actual number is in accord with Zu’s ratio. But “Book Of Sui” did not record the method used to get the number. Most historians and mathematicians think Zu Chongzhi used Liu Hui’s π algorithm to get the number. Liu Hui’s algorithm means approximating circle with a 24,576 sided polygon. Japanese mathematician Yoshio Mikami pointed out, “22/7 was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however milü π = 355/113 could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe”. Hence Mikami strongly urged that the fraction 355/113 be named after Zu Chongzhi as Zu’s fraction.( Yoshio Mikami)


Image:Zu Chongzhi’s method (similar to Cavalieri’s principle) for calculating a sphere’s volume includes calculating the volume of a bicylinder. Author: Chen Bai, via WIkimedia Commons.

Zu Chongzhi’s other important contribution was calculation volume of the sphere. Together with his son Zu Geng, Zu Chongzhi used an ingenious method to determine the volume of the sphere.(Arthur Mazer). In The Nine Chapters on the Mathematical Art, the author used Steinmetz solid to get the volume of the sphere. The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid.


Image:Steinmetz solid. Author: Van helsing, via Wikimedia Commons.

But the book did not give the formula of how to get the volume of the sphere. Zu Chongzhi used “Zu Geng principle” (another name: Cavalieri’s principle) to show the volume of the sphere formula is (π*d³)/6. In order to commemorate the fact that Zu Chongzhi found the significant contribution of the principle with his son, people called the principle “Zu Geng principle”. “Zu Geng principle” is the same as “Cavalieri’s principle”, but “Zu Geng principle” is earlier than “Cavalieri’s principle”. “Cavalieri’s principle” means two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.(Kern and Bland 1948, p. 26).

Work cited:

Yoshio Mikami , (1947). Development of Mathematics in China and Japan. 2nd ed. : Chelsea Pub Co;.

Arthur Mazer , (2010). The Ellipse: A Historical and Mathematical Journey. 1st ed. : Wiley;

Kern, W. F. and Bland, J. R. “Cavalieri’s Theorem” and “Proof of Cavalieri’s Theorem.” §11 and 49 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 25-27 and 145-146, 1948.



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.