学数学，读原著，勤思考，效果好

  在我们国内，读数学原著确实不容易。

  有的数学原著，没有思想，没有阅读价值。

  希尔伯特“几何基础”，是一本经受时间考验的经典数学名著。

  学数学，读原著，勤思考，效果好。这是我们的经验。所以，在一年前，我们将希尔伯特“Foundations of geometrg”（几何基础“（英文版）收录在“无穷小微积分”专业网站之中。

  只要百度一下“无穷小””，进入“无穷小微积分”网站，点击“Foundations of geometrg”下载即可阅读英文原文。

  附件是该文的第一节内容。开启现代数学之大门。

袁萌  陈启清  10月31日

附件：

THE FIVE GROUPS OF AXIOMS.

§1. THE ELEMENTS OF GEOMETRY AND THE FIVE GROUPS OF AXIOMS.

Let us consider three distinct systems of things. The things composing the rst system, we will call points and designate them by the letters A, B, C,...; those of the second, we will call straight lines and designate them by the letters a, b, c,...; and those of the third system, we will call planes and designate them by the Greek letters α, β, γ,...The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space. We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry. These axioms may be arranged in ve groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows: I, 1–7. Axioms of connection. II, 1–5. Axioms of order. III. Axiom of parallels (Euclid’s axiom). IV, 1–6. Axioms of congruence. V. Axiom of continuity (Archimedes’s axiom).

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