Mathematics in Ancient and Medieval China: Abstract
We known the mathematics of Greece as well as the mathematics of two civilizations known to have influenced Greek mathematics - Mesopotamia and Egypt. But mathematics was done in other parts of the world, even in ancient times. In this article, we will look at some mathematical ideas from ancient and medieval China, some of which may have, through paths so far undiscovered, reached Europe.
Mathematics in Ancient and Medieval China: Introduction
Although there are legends that date Chinese civilization back 5000 or more years, the earliest solid evidence of such a civilization is provided by the excavation of ruins at Anyang, near the Huang River, which are dated to about 1600 BCE. It is to the society centered there, the Shang Dynasty, that the "oracle bones" belong, curious pieces of bone inscribed with very ancient writing, which were used for divination by the priests of the period and are the source of current knowledge of early Chinese number systems. Around the beganning of the first millennium BCE, the Shang dynasty was replaced by the Zhou dynasty, which in turn dissolved into numerous warring feudal states. In the sixth century BCE, there was a great period of intellectual flowering, in which the most famous philosopher was Confucius. Academies of scholars were founded in several of feudal states. Other feudal lords hired scholars to advise them in a time of technological growth caused by the development of iron.
The feudal period ended as weaker states were gradually absorbed by the stronger, until ultimately China was unified under the Emperor Qin Shi Huangdi in 221 BCE. Under his leadership, China was transformed into a highly centralized bureaucratic state. He enforced a severe legal code, levied taxes evenly, demanded the standardization of weights, measures, money, and especially the written script. Legend holds that this emperor ordered the burning of all books from earlier periods to suppress dissent, but there is some reason to doubt that the book burning was actually carried out. The emperor died in 210 BCE, and his dynasty was soon overthrown and replaced by that of the Han, which was to last about 400 years. The Han completed the establishment of a trained civil service, for which a system of education was necessary. Among the texts that began to be used for this purpose were two mathematical works, probably complied early in the Han dynasty but containing material from several hundred years earlier. These are the Zhoubi Suanjing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) and the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art). The latter work, in particular, became central to Chinese mathematical practice over the centuries. As originally written, it was a compilation of problems with answers and rules for determining the answers. Over the centuries, however, commentaries were written to explain or derive the rules.
The Han dynasty in China disintegrated early in the third century CE, and China broke up into several warring kingdoms. The period of disunity lasted until 581, when the Sui dynasty was established, followed 37 years later by the Tang dynasty, which was to the last nearly 300 years. Although another brief period of disunity followed, much of China was again united under the Song dynasty (960-1279), a dynasty itself overthrown by the Mongols under Ghengis Khan. Despite the numerous wars and dynasty conflicts, a true Chinese culture was developing throughout most of east Asia, with a common language and common values. The systems of imperial examinations for entrance into the civil service, instituted during the Han dynasty, lasted - with various short periods of disruption - into the twentieth century. Although the examination was chiefly based on Chinese literary classics, the demands of the empire for administrative services, including surveying, taxation, and calendar making, required that many civil servants be competent in various area of mathematics. The Chinese imperial government therefore encouraged the study of applicable mathematics, as indicated in this chapter's opening quotation. In fact, at various times, there was an imperial Institute of Mathematics, where officials were trained in practical mathematics. At other times, mathematical texts studies at the Institute of Astronomy or the Institute of Records. In general, the mathematical texts studies by candidates for such institute were collections of problems with methods of solution, including the Nine Chapters on the Mathematical Art. New methods were rarely introduced. The examination system often required recitation of relevant passages from the mathematics texts, as well as the solving of problems in the manner described in those texts. Thus, there was no particular incentive for mathematical creativity.
Nevertheless, creative mathematicians did appear in China, and they applied their talents not only to improving old methods of solving practical problems, but also to extending those methods far beyond the requirements of practical necessity. We will look at developments in four major areas: numerical calculations, geometry, equation solving, and the solution of linear congruences. The first two area are, in general, the product of the period prior to about 500 CE, while new discoveries in the latter two areas were being made into the thirteenth century.
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