大家知道，现代几何学的连接公理组共计有7条几何公理，规定了点、线、面之间的关系。
在数学发展史上，希尔伯特第一次完整地提出了现代几何学的连接公理组
本文附件全面列出了这7条几何公理，I.1-7.请读者查阅。
我们学习微积分，总是不自觉地用到这7条公理。由此可见，学习微积分离不开希尔伯特的《几何基础》。
只有理解了的东西才能更好地感知它的存在。
袁萌 陈启清 3月18日
附件：希尔伯特《几何基础》电子版连接公理（I. 1-7）
I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a.
Insteadof“determine,” wemayalsoemployotherformsofexpression; forexample, we may say A “lies upon” a, A “is a point of” a, a “goes through” A “and through” B, a “joins” A “and” or “with” B, etc. If A lies upon a and at the same time upon another straight line b, we make use also of the expression: “The straight lines” a “and” b “have the point A in common,” etc
I, 2. Any two distinct points of a straight line completely determine that line; that is, if AB = a and AC = a, where B6= C, then is also BC = a.
3
I, 3. Three points A, B, C not situated in the same straight line always completely determine a plane α. We write ABC = a.
We employ also the expressions: A, B, C, “lie in” α; A, B, C “are points of” α, etc.
I, 4. Any three points A, B, C of a plane α,which
h doieinthesamestraightline,completely determine that plane.
I, 5. If two points A, B of a straight line a lie in a plane α, then every point of a lies in α.
In this case we say: “The straight line a lies in the plane α,” etc.
I, 6. If two planes α, β have a point A in common, then they have at least a second point B in common.
I, 7. Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.
Axioms I, 1–2 contain statements concerning points and straight lines only; that is,
concerningtheelementsofplanegeometry.
We will callthem,there fore,theplane axioms of group I, in order to distinguish them from the axioms I, 3–7, which we will designate briefly as the space axioms of this group. Of the theorems which follow from the axioms I, 3–7, we shall mention only the following:
Theorem 1. Two straight lines of a plane have either one point or no point in common; two planes have no point in common or a straight line in common; a plane and a straight line not lying in it have no point or one point in common. Theorem 2. Through a straight line and a point not lying in it, or through two distinct straight lines having a common point, one and only one plane may be made to pass.