Right-angled triangles in ancient China

Abstract

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