Chinese mathematics

Summary: Chinese mathematicians have a long history of investigation and discovery, sometimes predating similar findings in other cultures.

Chinese mathematics has a very long history, and its development is quite independent of other civilizations before the thirteenth century. Roughly speaking, it has four periods of developments before the middle of the Qing dynasty, namely

• The early development period: from ancient times to the Qin dynasty (2700–200 b.c.e.)
• The foundation period: from the Han dynasty to the Tang dynasty (200 b.c.e–1000 c.e.)
• The golden period: from the Sung dynasty to the Yuan dynasty (1000–1367)
• The east meets west period: from the Ming dynasty to the middle of the Qing dynasty (1367–1840)

Early Chinese mathematics is problem based and is motivated by various practical problems, including astronomy, trade, land measurement, architecture, and taxation.

The Early Development Period

It is written in Yi Jing (I-ching or Book of Changes) that, “In early antiquity, knotted cords were used to govern with. Later, our saints replaced them with written characters and tallies.” In other words, the ancient Chinese used knotted cords to record numbers. Later, written symbols and tallies were used instead. In the Shang dynasty (1600–1050 b.c.e.), numerals were invented and inscribed on oracle-bones or tortoiseshells for recording numbers. It was a decimal system and was widely used at the time.

In the Zhou dynasty (1050–256 b.c.e.), mathematics was one of the Six Arts (Liu Yi), which were taught by teachers at schools. The other five arts were rites, music, archery, charioteering, and calligraphy. From this dynasty onward, the ideas of Taichi, Ying Yang, Trigrams, and Hexagrams largely influenced the developments of sciences, mathematics, philosophy, arts, architecture, and many other areas in Chinese culture. For example, Luoshu (3×3 magic square) is closely related to the eight Trigrams. It has both ceremonial and metaphysical importance, which plays a significant role in Chinese philosophy for several thousands of years.

From the Kingdoms of Spring and Autumn (720–480 b.c.e.) to the period of Warring States (480–221 b.c.e.), the Chinese used counting rods to do calculations. Numbers were expressed by nine symbols, and blanks were used to denote zeros. The numeration system was already a decimal place-valued system.

The first definitive work on geometry in ancient China was the Mo Jing, which was compiled after the death of Mozi (470–390 b.c.e.). Many basic concepts of geometry can be found in this book. For example, the Mo Jing defines a point to be the smallest unit that cannot be divided, and points on a circle to be equidistant from the center. The book also mentions the definitions of endpoints, straight lines, parallel lines, diameter, and radius.

In the Qin dynasty, the famous Great Wall and many huge statues, tombs, temples, and shrines were built, which required sophisticated skills and mathematical knowledge for calculating proportions, areas, and volumes. Unfortunately, not much is known about the actual mathematical development in the Qin dynasty now, because of the burning of books and burying of scholars ordered by Emperor Qin Shi Huang.

The Foundation Period

In 1984, a Chinese mathematics text called Suan Shu Shu, completed at about 200 b.c.e., was discovered in a tomb at Zhangjiashan of the Hubei Province. It is about 7000 characters in length, and is written on 190 bamboo strips. Its content is mainly concerned with basic arithmetic, proportions, and formulas of areas and volumes. The next complete surviving text is the Zhou Bi Suan Jing, written between 100 b.c.e. and 100 c.e. Although it is a book on astronomy, it contains a clear description of the Gougu Theorem (the Chinese version of the Pythagorean theorem), which is very useful in solving problems in surveying and astronomy. This work is perhaps the earliest recorded proof of the Pythagorean theorem.

After the book burning in 212 b.c.e., the Han dynasty (202 b.c.e.–220 c.e.) began to edit and compile the mathematical works lost in the Qin dynasty. The most important one is the Nine Chapters on the Mathematical Art (Jiuzhang Suan Shu), completed at around 179 c.e. Although the editor is unknown now, this book had a great impact on the mathematical developments in China and its neighboring countries, such as Japan and Korea. It contains a collection of 246 mathematical problems on agriculture, engineering, surveying, partnerships, ratio and proportion, excess and deficit (the method of double false positions), simultaneous linear equations, and right-angled triangles.

The general method of solutions is provided, but no proof is given in the Greek sense. Most of the methods are of computational nature, and they can be applied to solve problems algorithmically. For instance, square roots, or cubic roots, can be found in a finite number of steps by using a procedure called Kai Fang Shu. For skillful users of this method, the answers can be computed efficiently by manipulating the counting rods. For circular measurements, the approximated value of π is taken as 3. Some problems are expressed in terms of a system of linear equations and then solved by algebraic techniques. For instance, a problem in Chapter Eight leads to the system

3x+2y+z=39

2x+3y+z=34

x+2y+3z=26

which can be solved by a method like the matrix approach described in modern textbooks.

In the third century, Liu Hui wrote his famous commentary on the Nine Chapters on the Mathematical Art. He also wrote a book called The Sea Island Mathematical Manual (Haidao Suan Jing), to demonstrate how to apply the Gougu Theorem. He was the first Chinese mathematician to deduce that the value of π lies between 3.1410 and 3.1427, by repeatedly doubling the number of sides of a regular polygon inscribed in a circle. It is called the method of dissection of a circle. Liu Hui also discovered the Cavalieri’s Principle and used it to find the volume of a cylinder. About two centuries later, Zu Chongzhi (430–501) and his son, Zu Geng, found that the value of π lies between 3.1415926 and 3.1415927, based on the pioneer works of Liu Hui. He also obtained the remarkable rational approximation 355/113 for π, which is correct to six decimal places. Working with Zu Geng, he successfully applied the Cavalieri’s Principle to deduce the correct formula for the volume of the sphere by computing the volume of a special solid called Mouhe Fanggai (the double vault) as proposed earlier by Liu Hui.

Unfortunately, his own work called Zhui Shu was discarded from the syllabus of mathematics in the Song dynasty and was finally lost in the literature. Many believed that Zhui Shu probably describes the method of interpolation and the major mathematical contributions by Zu Chongzhi and Zu Geng.

At the beginning of the Tang dynasty, Wang Xiaotong (580–640) wrote the Jigu Suanjing (Continuation of Ancient Mathematics), a text with only 20 problems that illustrate how to solve cubic equations. His method was a first step toward the Tian Yuan Shu (the method of coefficient array), which was then further developed by other mathematicians in the Sung and Yuan dynasties.

In the sixth century, mathematics was a subject being included in the civil service examinations. Li Chunfeng (602–670) was appointed by the Chinese emperor as the chief editor for a collection of mathematical treatises for both teachers and students. The collection is called the Ten Classics or the Ten Computational Canons, which include the Zhou Bi Suan Jing, the Jiuzhang Suan Shu, the Haidao Suan Jing, the Sunzi Suan Jing, the Wucao Suan Jing, the Wujing Suan Shu, the Shushu Jiyi, the Xiahou Yang Suan Jing, the Zhang Qiujian Suan Jing, and the Jigu Suan Jing. The book Zhui Shu by Zu Chongzhi had been included in the Ten Classics at the beginning, but it was later replaced by the Shushu Jiyi because of it being lost in the Sung dynasty.

The Golden Period

No significant advances in mathematics were made between the tenth century and the eleventh century. However, Jia Xian (1023–1050) improved the methods for finding square roots and cube roots, and also extended them to compute the numerical solutions of polynomial equations by means of the Jia Xian Triangle (the Chinese version of the Pascal Triangle).

The golden period of mathematical development in China occurs in the twelfth and the thirteenth centuries, which is called the “Renaissance of Chinese mathematics” by some authors. Four outstanding mathematicians appeared in the Sung dynasty and the Yuan dynasty, namely Yang Hui (1238–1298), Qin Jiushao (1202–1261), Li Zhi (also called Li Yeh, 1192–1279), and Zhu Shijie (1260–1320). Yang Hui, Qin Jiushao, and Zhu Shijie all used the Horner–Ruffini method to solve quadratic, cubic, and quartic equations. Li Zhi, on the other hand, revolutionized the method for solving problems on inscribing a circle inside a triangle, which could be formulated as algebraic equations, and solved by using the Pythagorean theorem. Another mathematician, Guo Shoujing (1231–1316), worked on spherical trigonometry for astronomical calculations. Therefore, much of the modern mathematics in the West had already been studied by Chinese mathematicians in this period.

Qin Jiushao (1202–1261) invented the symbol for “zero” in Chinese mathematics. Before this invention, blank spaces were used to denote zeros. Qin Jiushao also studied indeterminate problems and generalized the method of Sunzi to become the now-called “Chinese Remainder Theorem.” He wrote the Shushu Jiuzhang (Mathematical Treatise in Nine Sections), which marks the highest point in indeterminate analysis in ancient China.

Yang Hui was an expert in designing magic squares. He discovered elegant methods for constructing magic squares with an order greater than three. Some of the orders are as high as 10. He was also the first in China to give the earliest clear presentation of the Jia Xian Triangle in his book Xiangjie Jiuzhang Suanfa.

The famous work of Li Zhi is the Sea Mirror of the Circle Measurements (Ce Yuan Hai Jing). It is a collection of some 170 problems. He used Tian Yuan Shu or the Method of Coefficient Array to solve polynomial equations of degree as high as six. He also wrote the book Yi Gu Yan Duan (New Steps in Computation) in 1259, which is an elementary book related to solution of geometric problems by using algebra.

The most important text in the thirteenth century is the Precious Mirror of the Four Elements (Si Yuan Yujian), written by Zhu Shijie in 1303. This book marks the peak of the development of algebra in China. The unknowns that appeared in equations are called the four elements, namely heaven, Earth, man, and matter. This book describes how to solve algebraic equations of degrees as high as 14. The method is the same as the Horner–Ruffini method. Zhu Shijie also used the matrix methods to solve systems of equations. He was also an expert in summation of series. Many formulas on summation of series can be found in the Precious Mirror of the Four Elements. He also wrote an elementary mathematics text called the Introduction to Computational Studies (Suanxue Qimeng) in 1299, which had a significant impact on the development of Japanese mathematics later.

The East Meets West Period

In the Ming dynasty, not much original mathematics work emerged in China. Even the famous work Suanfa Tongzong (General Source of Computational Methods) by Cheng Dawei (1533–1606) was an arithmetic book for the abacus only. Its style and content were still influenced very much by the Nine Chapters on the Mathematical Art. It was only when the Italian Jesuit Matteo Ricci (1552–1610) came to China in 1581 that the development of mathematics in China was influenced by the West from this time onwards. For instance, Xu Guangqi (1562–1633) and Matteo Ricci translated a number of Western books on sciences and mathematics into Chinese, including the famous Euclid’s Elements, the influence of the Western culture on China became more apparent.

However, the Chinese mathematicians also did an excellent job in editing and recording their traditional mathematics and science works in the early Qing dynasty, so that much of them can come down to us now. For example, Mei Juecheng (1681–1763) edited the famous mathematical encyclopedia Shuli Jingyun in 1723, and Ruan Yuan (1764–1849) edited the Chouren Zhuan (Biographies of Astronomers and Mathematicians) in 1799. Both of these works are very valuable and useful references for historians to study the mathematical developments in China before the middle of the Qing dynasty.

Achievements in Chinese Mathematics

After the decline of Greek mathematics in the sixth century, Western Europe was undergoing the period of Dark Ages. On the other hand, many of the achievements of Chinese mathematics predated the same achievements before and shortly after the Renaissance. For instance, before the fifteenth century, China was able to (1) adopt a decimal placed-value numeral system, (2) acknowledge and use negative numbers, (3) obtain precise approximations for π, (4) discover and use the Horner–Ruffini method to solve algebraic equations, (5) discover the Jia Xian Triangle, (6) adopt a matrix approach to solve systems of linear equations, (7) discover the Chinese Remainder Theorem, (8) discover the method of double false position, and (9) handle summation of series with higher order. It was only after the fourteenth century that the development of Chinese mathematics began to decline and lag behind the Western mathematics in the Ming and Qing eras. However, it is worthy to note that the traditional Chinese mathematics still can find its contribution in mechanized geometry theorem proving in the twentieth century, because of its algorithmic characteristics.

Bibliography

Cullen, Christopher. Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing. Cambridge, England: Cambridge University Press, 1996.

Martzloff, J. C. A History of Chinese Mathematics. Corrected ed. New York: Springer-Verlag, 2006.

Shen, Kangshen, John N. Crossley, and W. C. Lun Anthony. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford, England: Oxford University Press, 1999.

Swetz, Frank. Legacy of the Luoshu: The 4000 Year Search for the Meaning of the Magic Square of Order Three. Wellesly, MA: A K Peters, 2008.