Mathematical Treatises of the Ten Books

Author: Guo Shuchun
Publisher:
Publishing Date: 1998-12-01
Features: He was ordered to organize the scholars of the Imperial Academy, such as Liang Shu and Wang Zhenru, to annotate and compile the "Zhoubi Suanjing" and the "Jiuzhang Suanjing," among other ten classics, which were completed that year. He also wrote "Yisi Zhan," one of the most important astrological works in ancient China. Li Chunfeng and others criticized the theory that "the shadow of the sun differs by one inch over a thousand miles north and south" as unrealistic, pointing out that the ground being measured could not be flat. Therefore, he introduced four methods of inclined plane measurement in his annotations, another creation in the history of Chinese mathematics. He also pointed out that Zhao Shuang's use of the arithmetic interpolation method to calculate the shadow sizes of the twenty-four solar terms did not match actual measurements. The Ming Hu Zhenyu edition also includes Tang Yin's annotations, which are of little significance. Since the middle Qing Dynasty, people have attached importance to the study of the "Zhoubi Suanjing." The first among them was Dai Zhen, who conducted a comprehensive collation of the "Zhoubi Suanjing." Later, during the Daoguang era, Gu Guanwang and during the Guangxu era, Sun Zhirang also made collations. In recent years, the study of the "Zhoubi Suanjing" has been pioneered by Qian Baoqiong, who has published numerous research papers and conducted a comprehensive collation.
The "Jiuzhang Suanjing" was also known as the "Jiuzhang Suanjing" during the Tang and Song dynasties and is one of the most important mathematical classics in ancient China. It compiles the mathematical knowledge of China from the pre-Qin period to the Western Han Dynasty, consisting of nine chapters with nearly 100 general abstract formulas, methods, and 246 examples. The main part either lists one or more examples first, followed by abstract methods, where examples generally only include a question and an answer; or it presents abstract methods first, followed by several examples, where examples include a question, an answer, and a method, which is the application of abstract general methods; there are also about 50 examples (mainly in the chapters on "Shao Guang," "Jun Shu," and "Gou Gu") whose methods, despite their universal nature, lack abstraction and are specific problem-solving methods, which can be considered in the form of applied problem collections. The main body of the "Jiuzhang Suanjing" is clearly in the form where methods govern examples.
The compilation of the "Jiuzhang Suanjing" underwent the efforts of several generations and countless individuals. Wei Liuwei said, "Zhou Gong established rites and numbers, and the nine numbers are the continuation of this. In the past, the brutal Qin burned books, scattering the classics. Since then, the Northern Ping Marquis Zhang Cang and the Minister of Agriculture Zhongcheng Geng Shouchang were renowned for their mathematical skills. Cang and others used the remaining fragments of old texts to compile them separately. Therefore, while the titles may differ from the ancient ones, the discussions are mostly in modern language." Here, the "nine numbers" mentioned appear in the "Rites of Zhou·Department of Earthly Officials": "The guardian master advises the king of faults and nurtures the state officials with the six arts: the first is the five rites, the second is the six music, the third is the five archery, the fourth is the five charioteering, the fifth is the six writings, and the sixth is the nine numbers." The "Rites of Zhou" is a pre-Qin work that preserves various systems of the Western Zhou Dynasty (possibly including later periods). During the late Eastern Han Dynasty, the master of classics Zheng Xuan (12–200) said, "The nine numbers: Fang Tian, Suli, Chafen, Shao Guang, Shang Gong, Jing, Fang, Yingbu, and Pao Yao. Today, they include Chong Chao and Gou Gu." Chafen (pronounced "d") is the same as Shao Guang, and Yingbu is the same as Yingbu, showing that the pre-Qin "nine numbers" and the "Jiuzhang Suanjing" differ mainly in that the latter replaces Pao Yao with Gou Gu. According to Bei Liu Xian's "Huangdi Jiuzhang Suanjing Juchao," the method of fitting a rectangle into two right-angled triangles (dividing a rectangle diagonally into two right-angled triangles, with two equal areas of rectangles sharing a common point on the diagonal) is called the Pao Yao method. Pao Yao refers to methods like measuring the area of right-angled triangles, circles, and other shapes, which is the part of the "Gou Gu" chapter where methods govern examples. If this part of the "Jiuzhang Suanjing" is restored to its original title of "Pao Yao," excluding the content on solving right-angled triangles, and removing the non-Chafen and non-Jun Shu content from the two chapters, it will perfectly match the "nine numbers" described by the two Zhengs and adopt the form where methods govern examples. This shows that Liu Xie's statement that "the continuation of the nine numbers is the Jiuzhang Suanjing" and that "the titles may differ from the ancient ones" are entirely correct.
The discovery of the "Suan Shu Shu" in the 1980s from the Zhangjiashan Han tomb awaits further study. The published "Shao Guang" problem is identical in content to the "Shao Guang" chapter of the "Jiuzhang Suanjing," but the text is archaic, providing evidence for Liu Xie's statement that "the discussions in the Jiuzhang Suanjing are mostly in modern language." Japanese scholars have compared the prices reflected in the "Jiuzhang Suanjing" with those in the "Records of the Grand Historian," the "Book of Han," and the Qin and Han bamboo slips, concluding that "the prices in the Jiuzhang Suanjing are not the prices of the Han Dynasty but rather those of the Warring States and Qin periods." Most problems reflecting Han Dynasty prices are in the applied problem collection part, while those reflecting pre-Qin prices are in the part where methods govern examples, further proving Liu Xie's reliability. Since the Jiang Zhen era, the view that denies Zhang Cang's compilation of the "Jiuzhang Suanjing" is incorrect.
Zhang Cang (？-152 BC), a political figure and astronomer-calculator of the early Western Han Dynasty, was proficient in the books and records of the world before serving in Liu Bang's rebellion and was enfeoffed as the Marquis of Pingbei for his merits. He "excelled in mathematics and calendrical calculations" (as recorded in the "Biography of the Grand Minister Zhang" in the "Records of the Grand Historian") and served as the main accountant in the Ministry of Revenue for four years, later becoming the Grand Minister and Prime Minister during the reigns of Emperor Wu and Emperor Wen. He lived to be over a hundred years old. During Emperor Xuan's reign (73–49 BC), Geng Shouchang, the Minister of Agriculture, was also an astronomer-calculator who "excelled in mathematics and economic calculations" (as recorded in the "Treatise on Food and Taxes" in the "Book of Han") and established the Ever-Normal Granaries, benefiting the people. Zhang Cang and Geng Shouchang successively collected the fragments of the "nine numbers" scattered by wars and conflicts, transformed them into the language of their time, organized and compiled them, and collected the mathematical innovations of the time, eventually forming the "Jiuzhang Suanjing."
Excerpt:
Zhou Gong said, "Great is the power of numbers! Understanding the meaning of mathematical techniques, I am filled with awe and exclaim, 'Great is the power of numbers!' May I ask about the principles of the square?" It refers to the proper use of the square and the methods of observation. Shang Gao replied, "Align the square to straighten the rope, using water to level it. Correct the square to determine the level, so that even the smallest difference in the millimeter can prevent errors over a thousand miles. Align the square to observe heights, cover arrows to measure depths, and lay the square to know distances. The application is limitless, adapting to the task, and the methods are in the 'Jiuzhang Suanjing.'"
The square can be transformed into a circle, and the circle can be combined to form a square. After tracing and clearing, it can also be used to create square figures. To resist external forces, nothing can reach the heavens. The square belongs to the earth, and the circle belongs to the sky. The sky is round, and the earth is square. Things have shapes, and numbers have odd and even. The sky moves in circles, and its number is odd; the earth is still in squares, and its number is even. This reflects the principle of yin and yang, not the actual creation of heaven and earth. The sky is infinite and cannot be seen, and the earth is boundless and cannot be observed. How can we determine its roundness and squareness?
Also: "Below the North Pole, the height of the inhabited world is sixty thousand li, and the depth of the four seas is also sixty thousand li." This shows that the shapes are the same, but the paths are different, and the heights are equal, making it easy to describe. Therefore, it is said, "The sky is like a lid, and the earth is like a bowl." The square is the standard, and the circle is derived from the square. When constructing a square, measure the shadow to ensure correctness, and when dealing with a circle, it is difficult to verify. The square is constant, while the circle is variable. It is necessary to create methods and regulate it. The method of regulation is to multiply half the circumference by half the radius to get the area of the square. Alternatively, multiply the circumference by the diameter and divide by four. Or multiply the radius by itself and multiply by three, then divide by four. Or multiply the diameter by itself and divide by twelve. Thus, it is said, "The circle is derived from the square."
The square represents the sky. The lid is also round, with a perfect circular shape, symbolizing the sky. The lid resembles the sky. The poem says, "What raincoat, what hat?" This is the meaning. The sky is blue-black, and the earth is yellow-red. The number of the sky is like a lid, with blue-black outside and red-yellow inside, symbolizing the positions of heaven and earth. After resembling its shape, it also follows its position. The square resembles the round, is it not similar? Therefore, to understand the earth is wisdom, and to understand the sky is sanctity. Speaking of the height of the sky and the vastness of the earth, unless one is wise and sage, who can achieve this? Wisdom comes from the square, and the square is also the shadow. By observing the increase and decrease of shadows, one can know the heights and distances of things, hence, "wisdom comes from the square." The square comes from the rectangle, and the shadow is called the square. The square does not move, and it is also the shadow. To make the shadow correct, it is said, "The shadow comes from the rectangle."
In short, the square and the rectangle contain the essence of everything, turning and returning in all directions. Zhou Gong said, "Excellent cutting!" Truly, it is a profound understanding of the meaning.