History of Mathematics in China: a Factor in World History and a Source for New Questions I. Recent Developments in the History of Mathematics in Early Imperial China

In the last decades much research has been devoted to the Jiuzhang suanshu or The nine chapters on mathematical procedures (hereafter abbreviated to The nine chapters), a book which played a crucial part in the mathematical traditions which expressed themselves in Chinese, quite comparable to the one played by Euclid's Elements of geometry in the West. Compiled around the beginning of the common era, after the uniication of the Chi-nese empire, the book was soon to become a \Classic" from which most subsequent Chinese mathematicians drew inspiration. As for all the other writings granted the status of Classics in China, commentaries were composed on The nine chapters , some of which were selected by the tradition to be handed down together with the book. This is how commentaries ascribed to Liu Hui (third century) and Li Chunfeng (seventh century) survived until today. The talk will present some recent results obtained on the book itself and its commentaries. It will then discuss various aspects of the transmission of mathematical knowledge between China and the West. Finally, the talk will c onsider some new questions for the study of mathematical activity raised by research on The nine chapters. In conclusion, it is argued that only an international approach to history of mathematics can provide an adequate framework to capture the historical processes which have constituted mathematical lores around the world. The nine chapters are constituted of problems and general algorithms with which to solve them. Inspired by Donald Knuth, who suggested reading Old-Babylonian clay-tablets from the point of view of algorithmic theory, Wu Wenjun initiated a new approach to ancient Chinese mathematical sources along similar lines. Algorithms found in The nine chapters such as root extractions, rules of false double position, or algorithms to solve systems of simultaneous linear equations, evidence the use of sophisticated resources: assignment of variables, conditionals, iterations. They demonstrate the development , wit hin an algorithmic framework, of an algebra that I shall ICM 1998 Berlin Plenary and Invited Lectures