The Developing History Of Bernoulli Numbers And Euler Numbers

The treatise here is about the history of Bernoulli numbers and two kinds of Euler numbers in both East and West, on which compare between ancient Chinese mathematics and that of western Europe has been done, and some new results are presented. The whole treatise is divided into five chapters.The first chapter focuses on introducing the history of Bernoulli numbers that were formally presented in Bernoulli's post-life book Ars Conjectandi, two kinds of Euler numbers, and tangent numbers. All achievement presented in this part was obtained by mathematicians in west Europe, from which we can see the efficient methods used by them. Many important results were obtained especially after using generating functions and calculus.The second chapter presents the work done by Chinese mathematicians in late Qing Dynasty related to two kinds of Euler numbers. Using Chinese traditional methods, they got the same results as west mathematicians did. The work is hard but elegant.The third chapter briefly scans the achievement attained by Japanese mathematicians. Their work is surprising in both methods and dates when they did it.The fourth chapter is relatively difficult one. Comparing Chinese mathematics with that of Europe invariably involves philosophies held by different kinds of nations under different cultures. As long as mathematics is concerned, differences between researching approaches, between reasoning ways, between independence, and between philosophies can be found.The last chapter briefly reviews the important results in critical periods along with the development of mathematics. Some contents included in previous parts may be repeated there, but only occasionally. The new results are presented there, some of which are accomplished by the author.