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Le Zhang Qiujian Suanjing est l'un des dix textes mathematiques anciens qui forment la collection de Suanjing Shi Shu et qui constitue la base du developpement de l'arithmetique

L'A. commente l'histoire de l'arithmetique en Chine avec l'adoption d'un nouveau systeme numerique permettant le developpement des operations fondamentales a l'aide de fractions

The multi-faceted content of the SHCM and its collection of rules and tables made it an important mathematical work not only in China, but also in Korea and Japan. This book clearly demonstrated Chu's predominant interest in the field of algebra and his contribution to the solution of numerical equations of higher degree, which was a prelude to his famous treatise the Ssu-yuan yu-chien.

Bonaventura Cavalieri (1598–1647) was noted for his method of indivisibles which led to the principle which bears his name. In the third century, while attempting to derive the volume of a sphere, Liu Hui applied a similar principle to determine the ratio of the volumes of a sphere and a solid circumscribing the sphere. This solid is formed by the intersection of two perpendicular cylinders circumscribing the sphere and is called mou he fang gai . Liu Hui left unresolved the problem of finding the volume of the mou he fang gai . In the fifth century Zu Geng, also applying Cavalieri's principle, solved this problem and thus derived the volume of a sphere. The influence of Zu Geng's method on later mathematicians is discussed in the latter part of the article.

Notre systeme numerique (introduit en Europe au 12 siecle a travers la traduction latine du livre de al-Khwarizmi et actuellement utilise en Chine) aurait en fait ses origines dans l'ancien systeme numerique chinois)

The Hai dao suanjing [a] (Sea island mathematical manual) written by Liu Hui [b] in the year 263 is a unique work on mathematical surveying that is unmatched by any other similar work of the period. The book has nine problems, all on distant measurements, and is written in the traditional style of ancient Chinese mathematical treatises. First, the problem is stated, then the answer is given and this is followed by the procedure. Computation is by means of counting rods and the procedure consists of instructions for performing the computation. Since the traditional emphasis was on a packaged program to achieve the answer, there was naturally a scant emphasis on proofs and derivations of procedures. However, the absence of such proofs and methods of derivations should not be interpreted as a lack of such knowledge, for without them, the procedures for the counting board could not have originated. Liu wrote his methods of derivation in the commentary, which together with the accompanying diagrams, has been lost1.

In the history of Chinese mathematics the thirteenth century has been regarded as a period of vigorous growth and brilliant achievements. Among the mathematicians who featured prominently during this epoch was Li Ye [a]. Li Ye, whose literary name was Renjing [b], and whose appellation was Jingzhai [c], was born into a bureaucratic family in 1192. He was originally known as Li Zhi [d] but when he discovered that this name was the same as the Tang emperor whose dynastic title was Tang Kaozong [e], he changed it to Li Ye1. Li Ye's father, Li Yu [p], was a scholar-official serving as a state counsellor in the Jurchen court. To his chagrin, Li Yu found much oppression and corruption within and around the bureaucratic circle and being an upright man himself, he often spoke his mind and was critical of the imperial policy. This incurred the displeasure of his Jurchen superiors and consequently he was edged out of service and forced to lead a life of seclusion. Although the family misfortune left an indelible mark in Li Ye's mind, it did not prevent him from pursuing his education. In 1230 he passed the civil service examination and was offered a post in the government. However, his service was shortlived as the district to which he was assigned fell to the Mongols in 1232. He fled and took refuge in the north. When the Jurchen Kingdom finally collapsed in the face of the Mongol onslaught in 1234, he gave up all hope of

One of the significant contributions of Chinese mathematicians is the method of solving numerical equations of higher degree. A number of scholarly works have established similarities between ancient and medieval Chinese root-extraction procedures and Horner's method of solving a numerical equation of any degree. The conceptual development of the Chinese methods, which began with the procedures of extracting square and cube roots during the Han dynasty, culminated in the solution of higher numerical equations in the 13th and the beginning of the 14th centuries. This paper attempts to show the intimate role played by the triangular array of numbers (known in the West as the Pascal triangle) in the process of the development of the Chinese methods, especially when the original geometrical concept was being replaced by the more general algebraic method.

The properties associated with the sides of a right-angled triangle had always been a subject of intriguing interest among mathematicians of ancient civilisations. The first and most outstanding of these properties is what is generally known in the West as the theorem of Pythagoras. This single theorem is considered by many historians of mathematics as one of the most profound concepts and an important milestone in the history of mathematics. In ancient China the study of the relationships of the sides of a right-angled triangle formed an important and substantial foundation of knowledge which had considerable influence on the mathematical thought and reasoning of early mathematicians. The knowledge led to the pursuit of other subjects such as similar triangles, measurement of distances, geometry involving triangles and circles, methods of extracting square and cube roots and methods of solving quadratic equations. A discussion on the right-angled triangle is found in the Zhoubi suanjing [a] (The arithmetic classic of the gnomon and the circular paths of heaven). This is the oldest existing Chinese mathematics classic and its date and authorship are unknown. It is generally regarded as being essentially a Han book stretching as far back as the 1st century BC. The last of the nine chapters in the Jiu zhang suanshu [c] (Nine chapters on the mathematical art) is devoted entirely to the right-angled triangle. Again nothing is known of the authors nor the exact date of this book which is generally placed between 100 BC and 100 AD. The bulk of the early knowledge on right-angled triangles is confined to the above mentioned two books and this is considerably and substantially enlarged by the commentaries of Zhao Shuang [d] and Liu Hui [e] on the relevant sections of the Zhou bi suanjing and the Jiu zhang suanshu respectively. Both Zhao and Liu were well-known commentators and were contemporaries living in the 3rd century AD. Zhao was sometimes known by his literary name Zhao Junqing [f]. A thorough analysis of a subject such as the relationship of the sides of a right-angled triangle would in general divide into three stages of development. The initial stage pertains to special numerical results obtained from particular

One of the many commentators of the Chiu chang suan shu (Nine Chapters on the Mathematical Art), written approximately in the first century, is Yang Hui, who wrote Hsiang chieh chiu chang suan fa (A Detailed Analysis of the Mathematical Methods in the “Nine Chapters”) and the Chiu chang suan fa tsuan lei (Reclassification of the Mathematical Methods in the “Nine Chapters”) in 1261. The seventh chapter of the Chiu chang suan shu is called ying nu or ying pu tsu meaning “too much and not enough” or “surplus and deficiency.” It has been pointed out by historians of mathematics that the method known in Europe as the Rule of False Position is the same as the Chinese ying pu tsu method used in this chapter. Yang has made a detailed and thorough study of the twenty problems in this chapter. He has not only explained and expanded the method of ying pu tsu, but he has also suggested alternative and simpler solutions for some of the problems. This paper endeavours to translate the twenty problems, to analyse the ying pu tsu method, and above all to present a critical study of Yang's alternative methods.

A brief survey of the different methods of solving linear equations in traditional China through the rod numeral system is given. These equations range from the simple one in one unknown to a set of simultaneous equations in five unknowns. Problems from the various mathematical treatises are chosen to illustrate the methods. It is hoped that this study will provide an insight into the early Chinese attempts to express mathematical concepts by means of positional notations through the rod numeral system.

Before 1600 the most common method of division used in Europe was the galley, batello, or scratch method, and this was still popular up to the end of the eighteenth century. This method is commonly supposed to be of Hindu origin, being based on a method found in India about the fourth century. It will be shown that the Hindu method is identical to the Chinese method of division (ch'u) mentioned in very early Chinese texts such as the Chiu Chang Suan Shu (Nine Chapters on the Mathematical Art, c. first century a.d.). The earliest detailed description of this method is found in the Sun Tzŭ Suan Ching (The Mathematical Manual of Master Sun, third century a.d.).

Yang Hui1 was one of the four outstanding Chinese mathematicians during the thirteenth century. Apart from the fact that he was a native of Ch'ien-t'ang,2 practically nothing else is known of his life. His existing books indicate that he was a prolific writer, who concentrated on the practical applications of mathematics. His most important work, the Yang Hui Suan Faz {Yang Hui s Computational Methods), includes among other subjects a detailed study of the various methods of multiplication and division, certain types of series, the solutions of the quadratic equations in different forms, and magic squares and magic circles. The Hsiang Chieh Suan Fa* (A Detailed Analysis of the Methods of Computation), written in 1261, was one of Yang Hui's earliest works. This book, which is now lost, is said to deal with various methods of multiplication and division. A few fragments of the book may be found in the Yung-lo Ta Tien0 Chapters 16,343 and 16,344 and the Chu Chia Suan Fa.6 Li Yen7 has collected them together and printed them in his book Chung Suan Shih Lun Ts'ung.8* Altogether there are six problems with their methods of solutions and a section on what is now known as the " Pascal Triangle ".

This paper discusses the method of Liu Hui (3rd century) for evaluating the ratio of the circumference of a circle to its diameter, now known as π. A translation of Liu's method is given in the Appendix. Also examined are the values for π given by Zu Chongzhi (429–500) and unsurpassed for a millenium. Although the method used by Zu is not extant, it is almost certain that the applied Liu's method. With the help of an electronic computer, a table of computations adhering to Liu's method is given to show the derivation of Zu's results. The paper concludes with a survey of circle measurements in China.

This paper discusses the method of Liu Hui (3rd century) for evaluating the ratio of the circumference of a circle to its diameter, now known as π. A translation of Liu’s method is given in the Appendix. Also examined are the values for π given by Zu Chongzhi (429–500) and unsurpassed for a millenium. Although the method used by Zu is not extant, it is almost certain that he applied Liu’s method. With the help of an electronic computer, a table of computations adhering to Liu’s method is given to show the derivation of Zu’s results. The paper concludes with a survey of circle measurements in China.