Chinese mathematics

Several factors led to the development of mathematics in China being, for a long period, independent of developments in other civilisations. The geographical nature of the country meant that there were natural boundaries (mountains and seas) which isolated it. On the other hand, when the country was conquered by foreign invaders, they were assimilated into the Chinese culture rather than changing the culture to their own. As a consequence there was a continuous cultural development in China from around 1000 BC and it is fascinating to trace mathematical development within that culture. There are periods of rapid advance, periods when a certain level was maintained, and periods of decline.

The first thing to understand about ancient Chinese mathematics is the way in which it differs from Greek mathematics. Unlike Greek mathematics there is no axiomatic development of mathematics. The Chinese concept of mathematical proof is radically different from that of the Greeks, yet one must not in any sense think less of it because of this. Rather one must marvel at the Chinese approach to mathematics and the results to which it led.

Chinese mathematics was, like their language, very concise. It was very much problem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes. By the fourth century BC counting boards were used for calculating, which effectively meant that a decimal place valued number system was in use. It is worth noting that counting boards are uniquely Chinese, and do not appear to have been used by any other civilisation.

Our knowledge of Chinese mathematics before 100 BC is very sketchy although in 1984 the Suan shu shu (A Book on Arithmetic) dating from around 180 BC was discovered. It is a book written on bamboo strips and was found near Jiangling in Hubei province. The next important books of which we have records are a sixteen chapter work Suanshu (Computational prescriptions) written by Du Zhong and a twenty-six chapter work Xu Shang suanshu (Computational prescriptions of Xu Shang) written by Xu Shang. Neither of these texts has survived and little is known of their content. The oldest complete surviving text is the Zhoubi suanjing (Zhou Shadow Gauge Manual) which was compiled between 100 BC and 100 AD (see the article on The Ten Classics). It is an astronomy text, showing how to measure the positions of the heavenly bodies using shadow gauges which are also called gnomons, but it contains important sections on mathematics. It gives a clear statement on the nature of Chinese mathematics in this period (see for example [2]:-

The method of calculation is very simple to explain but has wide application. This is because a person gains knowledge by analogy, that is, after understanding a particular line of argument they can infer various kinds of similar reasoning ... Whoever can draw inferences about other cases from one instance can generalise ... really knows how to calculate... . To be able to deduce and then generalise.. is the mark of an intelligent person.

The period from the tenth to the twelfth centuries is one where few advances were made and no mathematical texts from this period survive. However Jia Xian (about 1010 - about 1070) made good contributions which are only known through the texts of Yang Hui since his own writings are lost. He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle. Although Shen Kua (1031 - 1095) made relatively few contributions to mathematics, he did produce remarkable work in many areas and is regarded by many as the first scientist. He wrote the Meng ch'i pi t'an (Brush talks from Dream Brook) which contains many accurate scientific observations.

The next major mathematical advance was by Qin Jiushao (1202 - 1261) who wrote his famous mathematical treatise Shushu Jiuzhang (Mathematical Treatise in Nine Sections) which appeared in 1247. He was the first of the great thirteenth century Chinese mathematicians. This was a period of major progress during which mathematics reached new heights. The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle. Equations up to degree ten are solved using the Ruffini-Horner method.

Li Zhi (also called Li Yeh) (1192-1279) was the next of the great thirteenth century Chinese mathematicians. His most famous work is the Ce yuan hai jing (Sea mirror of circle measurements). written in 1248. It contains the "tian yuan" or "coefficient array method" or "method of the celestial unknown" which was a method to work with polynomial equations. He also wrote Yi gu yan duan (New steps in computation) in 1259 which is a more elementary work containing geometric problems solved by algebra. The next major figure from this golden age of Chinese mathematics was Yang Hui (about 1238 - about 1298). He wrote the Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications) in 1261, and his other works were collected into the Yang Hui suanfa (Yang Hui's methods of computation) which appeared in 1275. He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures. He also gave a wonderful account of magic squares and magic circles.