阿基米德和刘徽求积的"余部分割法"

古代的西方和东方,在相距万里之遥的希腊和中国,两位数学科学的巨人--阿基米德和刘徽,在他们各自独立地解决几何学的重大课题--求积问题时,不约而同地采用了"余部分割法",而且他们论证问题的方式和步骤也有着惊人的相似之处.这不能不说是数学史上的一个奇迹.在某些特定的场合,这种"余部分割法"在讨论问题时确实有它的特别便捷之处.

Methods of Partitioning Remaining Parts in Archimedes' and Liu Hui's Ideas for Finding Areas

The goal in this paper is to make a comparison research between Archimedes's and Liu Hui's ideas in finding areas and recent theory of definite integrals, it is pointed out that the partitioning manner used by the latter is actually a whole partition, however, when Archimedes and Liu Hui, who were great mathematicians respectively born in ancient Greece and China very far from each other, solved the problems of finding areas, they used simultaneously a method of partitioning remaining parts, and the ways and steps were very similar in the aspects of solving problems. This is a miracle in the history of mathematics. In some certain situations, the method of partitioning remaining parts really has its advantages and conveniences in the discussions of problems. It is concluded that for the ancient orient and western mathematics, the essentials of them are the same though different starts may lead to different ends. Furthermore, sometimes we can find the initial ideas including in some concepts of modern mathematics from the methods of partitioning remaining parts.