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本刊1957年5月号“定义圆周长的一种方法”一文中我曾提到:可以通过各种途径来作出圆周长的定义,但是我们在考虑应该采取那一途径的时候,应当注意今后定义圆的面积、圆弧长、圆扇形面积的和谐性,也就是说要把定义圆周长的这个途径的精神贯徹到以后这些定义中去。在那篇文章中我介绍了定义圆周长的一种方法——从作圆的内接和外切同边数的正多边形开始,使边数无限倍增,得到一系列的内接和外切正多边形,这些正多边形的周长一个是无限递增有界数列,一个是无限递减有界数列,因此各有极限存在,证明这两数列有着共同的极限,就把这个极限定义为圆周长。本文将继续介绍根据这一精神来定义圆面积、圆弧长和圆扇形面积的方法。
In the May 1957 issue of the article “A Method for Defining the Circumference,” I mentioned that the definition of the circumference can be defined in various ways. However, when we consider that we should adopt that approach, we should pay attention to the future. Define the harmony of the area of the circle, the length of the arc, and the sector of the circle, that is to say, the spirit of the way of defining the circumference of the circle should be applied to these later definitions. In that article, I introduced a way to define the circumference of a circle—starting from the inscribed and out-of-round regular polygons of a circle, the number of edges is infinitely multiplied, resulting in a series of in- and out-cuts. Polygons, the perimeter of these regular polygons is an infinitely increasing bounded number sequence, and one is an infinitely decreasing bounded number sequence. Therefore, each limit has its existence. It is proved that the two numbers have a common limit, and this limit is defined as a circle length. This article will continue to introduce the method of defining the area of a circle, the length of an arc, and the area of a circle sector in accordance with this spirit.