| Ancient Greek mathematics affects modern maths in many aspects, such as maths system, methods and theory fruits. In the silver age of Greek mathematics, Pappus of Alexandria (about A.D.290—A.D.350) stood out as an accomplished researcher, his only survived work Mathematical Collection has much value for the study of the succession. Its contents were important to the development of the western and the whole world maths. This paper belongs to the comparison and communication study of the history between the Chinese and the Western maths. For the study it uses some methods, such as the analysis method of original documents, the quotation and the analysis method of the predecessors'results and the method of comparison research. The main works in this paper as follows:1. Part one faithfully based on the modern English translations and other former researchers'results, has explained and discussed nearly all around of the maths contents in the books, clarified certain details and the courses of some problems of changes and developments, compared the different attitudes and methods between Pappus and his predecessors for some problems, discussed his acute insight and foresee into maths development directions, Moreover, made it clearly about his position and effects in the history of maths. As a result, it concludes that Pappus was an important mathematician in the history of maths for his carried forwards the cause of the older generation and exploits some new fields. The other clue of the preceding part of the paper is that it is tried to introduce the study results of the western researchers about the books in more details and discuss their main ideas, methods and fruits.2. Part two discusses the diffusion of the relative problems in the books in the period from the late Ming Dynasty to the late Qing Dynasty and analyses the researches and the developments of the Chinese mathematicians to them in the Qing Dynasty. Especially, it helps to understanding the instance of the acceptance, comprehension and application better. Meanwhile, in this course, it could make it more clearly to understand the differences in the each of traditions, developments and outlets between the Chinese and the Western maths.3. Part three studies and discusses some original results, which are relative with Mathematical Collection, but were discovered independently by Chinese mathematicians in the Qing Dynasty. The key point is analyzed that the contributions made by Zhou Da about"shoemaker's knife". And it shows how to develop of the Chinese mathematicians in the Qing Dynasty from a case study of Zhou Da. It set out Zhou Da's independent result based on the affiliation between the traditional mathematics and the western one. And it was the high level reflection of his harmonization of the western maths into the Chinese mathematics.
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