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.

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