大家知道，在欧几里得《几何原本》（电子版）中，有一条“公设”：通过线外一点，可作一条该直线的一条平行线。回来，人们咸其为欧几里得公理，也叫平行线公理。
在希尔伯特《几何基础》
电子版中，继承了这个事实，称为平行线公理，编号“III”。
给定平面P，在其上建立直角坐标系xoy。此时，平面P上任一几何点与其“坐标”（x,y）之间的一一对应关系就是由平行线公理保证的。
十分明显的是，没有点与其坐标的对应关系，函数，乃至导数、微分，……就无从谈起了。
平行线公理是整个现代数学的基础之一，十分重要。
袁萌 陈启清 3月19日
附件：现代几何学的平行线公理（III）英电子版:
§5. GROUP III: AXIOM OF PARALLELS. (EUCLID’S AXIOM.)
The introduction of thi saxioms im plifie sgreatly thef undamental principles of geometry and facilitates in no small degree its development. This axiom may be expressed as follows:
III. In a plane α there can be drawn through any point A, lying outside of a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A.
8
This statement of the axiom of parallels contains two assertions. The first of these is that, in the plane α, there is always a straight line passing through A which does not intersect the given line a. The second states that only one such line is possible. The latter of these statements is the essential one, and it may also be expressed as follows: Theorem 8. If two straight lines a, b of a plane do not meet a third straight line c of the same plane, then they do not meet each other. For, if a, b had a point A in common, there would then exist in the same plane with c two straight lines a and b each passing through the point A and not meeting the straight line c. This condition of affairs is, however, contradictory to the second assertion contained in the axiom of parallels as originally stated. Conversely, the second part of the axiom of parallels, in its original form, follows as a consequence of theorem 8. The axiom of parallels is a plane axiom.
