部分翻译
This does not mean that one should not make use of geometry in studying properties of real numbers. On the contrary, the geometry often suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof (one depending entirely on the axioms for the real numbers).
这并不意味着研究实数的性质时不会应用到几何。相反，几何经常会为证明一些定理提供思路，有时几何讨论比纯分析式的证明更清楚。
In this book, geometric arguments are used to a large extent to help motivate or clarity a particular discuss. Nevertheless, the proofs of all the important theorems are presented in analytic form.
在本书中，几何在很大程度上被用于激发或者阐明一些特殊的讨论。不过，所有重要定理的证明必须以分析的形式给出。
2-5理解笛卡儿几何学的基本概念
basic concepts of Cartesian geometry
5-A the coordinate system of Cartesian geometry
As mentioned earlier, one of the applications of the integral is thecalculation of area. Ordinarily , we do not talk about area by itself ,instead,we talk about the area of something . This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must firstfind an effective way to describe these objects.
就像前面提到的?积分的一个应用就是面积的计算,通常我们不讨论面积本身，相反，是讨论某事物的面积。这意味着我们有些想测量的面积的对象（多边形区域，圆域，抛物线 弓形等），如果我们希望获得面积的计算方法以便能够用它来处理各种不同类型的图形?我们就必须首先找出表述这些对象的有效方法。
The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’idea was to represent geometric points by numbers.The procedure for points in a plane is this：?
描述对象最基本的方法是画图，就像古希腊人做的那样。R 笛卡儿提出了一种比较好的方法，并建立了解析几何（也称为笛卡儿几何）这门学科。笛卡儿的思想就是用数来表示几何点，在平面上找点的过程如下?
Two perpendicular reference lines (called coordinate axes) are chosen, onehorizontal (called the“x-axis”),the other vertical (the“y-axis”). Their point ofintersection denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance ,although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned apair of numbers, called its coordinates. These numbers tell us how to locate the points.