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# Developments in Chinese Mathematics

## Overview

Chinese mathematical progress during the medieval period reached its pinnacle between 1000-1300, during which time there was both innovation and discovery, and an increasingly systematic organization of the great traditions of Chinese math. Advances were made in solving problems for remainders, solving numerical equations, negative numbers, "magic squares" (vertical and horizontal arrangements of numbers that resulted in interesting properties among the columns), extractions of roots and calculations of congruences, and polynomials. Much of the mathematical effort took the form of proofs for existing problems and commentaries on mathematical histories and the problems presented in those histories. Despite the diversity of mathematical topics and challenges undertaken by scholars during this time, genuine mathematical progress was relatively limited, and theoretical mathematics all but unknown, other than in the form of diversions such as magic squares. The emphasis throughout was upon the pragmatic uses of math, particularly in surveying, astronomy, finance, and the keeping of accurate calendars.

## Background

The great classic of Chinese mathematical literature, Nine Chapters on Mathematical Procedures, dating from the first century a.d., gathered within its contents the bulk of Chinese mathematical knowledge of the time. Its centrality to Chinese mathematics served not only as a foundation for further mathematical development, but also as a tool for preserving fundamental mathematical knowledge, a base upon which mathematical education was built. While the book addressed the writing of numbers, its primary emphasis was on the use of physical aids to computation, especially counting rods, sticks of bamboo that symbolically represented numbers and, more importantly, the place value each number held in a computation. Counting rods enabled the Chinese to undertake and solve many complex problems, including those involving fractions, division, solutions for area and volume, extraction of roots, and calculations involving π.

For ten centuries much of the focus of Chinese math was directed at commenting upon and annotating the Nine Chapters. New explanations and proofs of solutions further enhanced the importance of the Nine Chapters, with the production of commentaries being among the major activities of Chinese mathematicians. Although many of the commentaries and proofs served to extend and improve the operations described in the Nine Chapters, it was not until some time after a.d. 1000 that a systematic attempt was made to completely revise the text.

## Impact

By 1050, commentators, particularly Jia Xuan, were working to simplify the extraction of both square and cube roots. Evidently Jia Xuan achieved some success in this quest, though this is surmised only from subsequent commentaries—the mathematician's own work does not survive.

During these centuries, the ability to perform rapid calculation using counting rods or similar devices, such as a "checkerboard" whose grid represented numerical positions, was highly prized by China's rulers. Indeed, much of the emphasis of Chinese mathematics during the period was focused on calculation and practical applications of the formulae in the Nine Chapters. Measurement, accurate calendars and date calculations, and financial math were the primary applications of computational ability.

So highly prized were mathematical abilities that by the eleventh century an "Office of Mathematics" was established by the government. While the purpose of the office was to employ mathematics for governmental uses, a side effect was the further codification of Chinese mathematical knowledge in a new book, called the Ten Classics of Mathematics. This instructional text accepted the essence of the Nine Chapters and extended its effectiveness through new solutions, proofs, and methods.

By the twelfth century Chinese mathematicians were beginning to range further into new territory. Advances were made in geometry, particularly the development of methods of incorporating negative numbers into mathematical coefficients, an advance vital to the development and evolution of equations.

Equations themselves were increasingly the focus of Chinese mathematical effort. The publication in 1247 of Mathematical Treatise in Nine Sections (a title obviously intended to honor the original text) saw mathematician Ch'in Chiushao (1202-1261) address the nature of algebraic equations, their properties, as well as geometric and astronomical properties, including the accurate handling of remainders in complex calculations. Much of this work anticipated mathematical advances that would not be made in Europe for several centuries.

At almost exactly the same time, in China's northern section (Ch'in Chiu-shao lived in southern China), another mathematician, Li Yeh (1192-1279) published Sea-Mirror of Circle Measurements (1248), a geometry text important for its contribution to the development of polynomials (constant numbers multiplied by variables in algebraic equations.) Working from the constants and their coefficients, Li Yeh showed how to solve for the indeterminate power, which he referred to as the "celestial unknown."

The more unknowns that could be solved, the more complex the problems that could be tackled. Chu Shih-chieh (1280?-1303) wrote Precious Mirror of the Four Elements (also known as Jade Mirror of the Four Unknowns) in which he showed how to solve polynomials, including multiple unknowns.

By the beginning of the fourteenth century increasing attention was being paid to the role of negative numbers, which when written were either depicted in a different color from the positive numbers, or were denoted by having a diagonal line drawn through them (or through one numeral if the number was larger than 9.) The attention paid to negatives, though, was not further pursued; some scholars argue that the Chinese sense of order and placement made the idea of negatives unappealing. Nor did the Chinese see large practical applications for negative values. The concept was identified, but allowed to languish.

Exerting a large appeal, on the other hand, was a mathematical construction that had little practical value. This was the "magic square," which, according to Chinese legend, had been discovered by the emperor Yu the Great around 2000 b.c.

The "magical" quality of the squares was found in the fact that with the proper arrangement of numbers in vertical columns, the sums of those columns would be reflected in the square's consequent horizontal and diagonal columns.

During the thirteenth century Yang Hui (1238?-1298?) devoted much effort to the construction of highly complex magic squares and magic circles, all of which demonstrate remarkable properties of constants.

More importantly, though, Yang Hui collected much important mathematical knowledge in a book that included a major section on the importance of systematic mathematical education, rather than the traditional path of simple memorization—a step toward understanding the importance of mathematics theory as well as the practical contributions of mathematics.

In addition, he presented the first clear explication of decimal notation, and anticipated Pascal's Triangle, a geometrical exercise that would be perfected (and named for) French mathematician and philosopher Blaise Pascal (1623-1662).

Yang Hui's achievements were in many ways the final flowering of the great age of Chinese mathematics. Following his work, most Chinese mathematical texts were recapitulations of earlier works. Once contact with the West was established in the centuries to come, Chinese mathematics texts were among the first books to be translated, and affected the far livelier development of mathematics in Europe. By the late 1600s the current was reversed, and Western mathematics would begin to flow into China, absorbing, and ultimately subsuming Chinese traditions and techniques.

The great and ongoing impact of Chinese mathematics is found in the sense of importance that the Chinese themselves ascribed to mathematics. While their culture neither encouraged—nor needed—the exploration of mathematics as a pure or theoretical endeavor, the culture did elevate mathematicians to a higher social rank and position within the bureaucracy, reflecting Chinese understanding of the practical value of mathematics in government and finance. The passage of Chinese mathematical ideas into the West would serve to accelerate Western mathematical development even as Chinese mathematics remained relatively stagnant.

KEITH FERRELL

## Further Reading

Dunham, William. Journey through Genius: The Great Theorems Of Mathematics. New York: John Wiley & Sons, 1990.

Gernet, Jacques. A History of Chinese Civilization. Cambridge: Cambridge University Press, 1996.

Katz, Victor, A History of Mathematics: An Introduction. 2nd ed. New York: Addison-Wesley, 1998.

Needham, Joseph. Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, Vol. 3. Cambridge: Cambridge University Press, 1959.

Swtz, Frank J., ed. From Five Fingers to Infinity: A Journey through the History of Mathematics. Chicago and Lasalle: Open Court, 1994.