A Classic from China: The Nine Chapters - Geometric Design and Surveying

In the Far East, just as in the West, land surveying and construction were major stimuli for the development of geometry. The first major steps in building roads and canals in China were taken by the emperors of the Qin and Han dynasties. A system of roads built around 100 BCE made it possible to journey from Beijing to Canton (Guangzhou) on horseback in only 32 days, a distance of about 1200 miles. The Grand Canal, nearly 200 miles long, was built to connect the two largest river systems, the Yangtze and the Huang Ho. Such roads and canals played a crucial role in conveying troops, traders, crops, and other things, helping to unify the country. Crop irrigation also required extensive networks of canals. Each year, in the intervals between seasonal farming tasks, every able-bodied man was required to donate about one month digging canals or toiling on other public-works projects for his feudal lord, a practice called corvée labor. Chinese technology during the Han period was the most advanced in the world.

Building a canal was a significant feat of mathematics and engineering. Plane and solid geometry were needed in designing and surveying the canal and in many phases of its construction. For instance, dirt dug from the canal bed was routinely used to build embankment dykes of packed mud on either side; calculations were done to ensure that the dyke design required exactly the volume of earth that would be excavated. The following problem from Chapter 5 asks for the volume of a certain qiandu, a word that originally meant “embankment dyke” but came to mean “triangular prism”. Chinese mathematicians had discovered formulae for the volumes of a wide variety of solids, including the qiandu.

Problem 8. Now given a qiandu with a lower breadth of 2 zhang, a length of 18 zhang 6 chi and an altitude of 2 zhang 5 chi. Tell: what is the volume? (Shen et al., pp. 267-8)

Your answer to the above problem should be given in cubic chi.

Above: Illustration of a Chinese dissection proof of the Gōugŭ Rule (Pythagorean Theorem) from the Zhoubi suanjing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), a book on astronomy and mathematics dated to approximately 100 BCE. This image is from a copy printed in 1603. (Source: Mathematical Treasures - Zhoubi suanjing, MAA Convergence)

Chapter 9 of the Nine Chapters is focused mostly on right triangles and their use in various types of problems. The Chinese were masters at this. The shorter leg of a right triangle was called the gōu; the longer leg, gŭ; the hypotenuse, xián, literally “bowstring”; the right triangle itself, gōugŭ; and the Gōugŭ Rule or theorem was what we call the Pythagorean Theorem.

The author(s) of the Zhou Shadow-Gauge Manual, the second of the Ten Mathematical Classics (c. 100 BCE), gave a ringing endorsement of the importance of the Gōugŭ Rule (Pythagorean Theorem) by writing,

Emperor Yŭ quells floods, he deepens rivers and streams, observes the shape of mountains and valleys, surveys the high and low places, relieves the greatest calamities and saves the people from danger. He leads the floods east into the sea and ensures no flooding or drowning. This is made possible because of the Gōugŭ theorem. (Lĭ and Dù, pp. 29-30)

The Zhou Shadow-Gauge Manual was an astronomical treatise describing uses of the shadow-gauge or gnomon, a right-angled ruler. The treatise included a proof of the Gōugŭ theorem by the dissection method. In proof by dissection, which was especially common in traditional Asian geometry, a figure is imagined to be cut into shapes that are rearranged to make clear the relationships among the areas involved.

Here is a typical problem from the Nine Chapters that was solved using the Gōugŭ theorem:

Problem 9. Now given a circular [i.e., cylindrical] log of unknown size buried in a wall. When sawn 1 cun deep, it shows a breadth of 1 chi. Tell: what is the diameter of the log? (Shen et al., p. 473)

In his written commentary on the Nine Chapters, Liu Hui also provided an appendix consisting of surveying problems. The appendix was later separated off as a treatise in its own right. It became known as the Sea Island Mathematical Manual as a result of its first problem:

Problem 10. Now survey a sea island. Erect two poles of the same height, 3 zhang, so that the front and rear poles are 1000 bu apart. They are aligned with the summit of the island. Move backwards 123 bu from the front pole, sighting at ground level, and find that the summit of the island coincides with the tip of the pole. Move backwards 127 bu from the rear pole, sighting at ground level, and find that the summit of the island also coincides with the tip of the pole. Tell: what are the height of the island and its distance from the [front] pole? (Shen et al., p. 539; see the illustration on p. 542).

Download solutions to Problems here: Students Explore the Nine Chapters from China.