对一个重要极限证明中使用循环论证问题的质疑

有的著作认为在极限公式limx→0sinxx=1的证明中,要用到圆(扇形)的面积公式S=21Lr,而对后者的证明中必须要用到重要极限limx→0sinxx=1,从而犯了循环论证的错误。魏晋刘徽的“割圆术”是对无限问题的独特认识和致用的处理方式,是为证明圆面积公式而设计出来的一种方法。刘徽之前,希腊的阿基米德用穷竭法也证明了圆的面积公式。其中,刘徽的证法强调计算的程序性和构造性,而阿基米德则倾向于演绎的严谨性。这两种证明说明limx→0sinxx=1证明所谓的循环论证是可以避免的。

Oppugning the Issue of the Important Limit Arguing in a Circle

LIU Hui's Cyclotomic Method is one of the greatest achievements in the history of mathematics in China,even in the world,in which the special approach to understand the infinity and the method to solve it are presented.At the same time,Archimedes gives an exact proof on the formula of area of circle,with the method of exhaustion method depending on the Cyclotomic.Moreover it can be answered the question argued in nowadays concerning about the proof on that.