来源 / Source: Project Gutenberg

Source / 来源: Project Gutenberg

几何原本：第一卷 - 基础

The Elements: Book I - Foundations

欧几里得 (Euclid)托马斯·L·希思 (Thomas L. Heath) 译 (1908)

EuclidTranslated by Thomas L. Heath (1908)

定义 (Definitions)

Definitions

1. 点 (point) 是没有部分的。

1. A point is that which has no part.

2. 线 (line) 是无宽度的长度。

2. A line is breadthless length.

3. 线的端点是点。

3. The extremities of a line are points.

4. 直线 (straight line) 是均匀地平放于其上诸点之线。

4. A straight line is a line which lies evenly with the points on itself.

8. 平面角 (plane angle) 是平面内两条相交且不共一直线的线之间的倾斜度。

8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

$$\mathrm{\angle }\alpha$$

$$\mathrm{\angle }\alpha$$

10. 当一条直线矗立在另一条直线上，使得邻角彼此相等时，这些相等的角每一个都是直角 (right)，而矗立在另一条上的直线叫做那条直线的垂线 (perpendicular)。

10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

$$\text{若 }\mathrm{\angle }\alpha =\mathrm{\angle }\beta ⟹\mathrm{\angle }\alpha ={90}^{\circ }$$

$$\text{If }\mathrm{\angle }\alpha =\mathrm{\angle }\beta ⟹\mathrm{\angle }\alpha ={90}^{\circ }$$

15. 圆 (circle) 是由一条线包围的平面图形，从图形内某一点向该线所作的所有直线彼此相等。

15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.

23. 平行 (parallel) 直线是在同一平面内，向两个方向无限延长而在任何方向都不相交的直线。

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

$$L_1\parallel L_2⟺L_1\cap L_2=\mathrm{\varnothing }$$

$$L_1\parallel L_2⟺L_1\cap L_2=\mathrm{\varnothing }$$

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公设 (Postulates)

Postulates

提出以下公设：

Let the following be postulated:

1. 从任一点到任一点作一条直线。

1. To draw a straight line from any point to any point.

2. 将有限直线连续地沿直线延长。

2. To produce a finite straight line continuously in a straight line.

3. 以任意中心和距离（半径）画圆。

3. To describe a circle with any centre and distance.

4. 凡直角都彼此相等。

4. That all right angles are equal to one another.

5. 若一条直线与另外两条直线相交，使同侧的内角和小于两个直角，则这两条直线若无限延长，必在内角和小于两直角的一侧相交。

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

注： 这就是著名的平行公设 (Parallel Postulate)。用现代符号表示： 若 $\alpha +\beta <{180}^{\circ }$，则两直线相交。

Note: This is the famous Parallel Postulate. In modern notation: If $\alpha +\beta <{180}^{\circ }$, then the lines intersect.

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公理 / 一般概念 (Common Notions)

Common Notions

1. 等于同量的量彼此相等。

1. Things which are equal to the same thing are also equal to one another.

$$A=C\wedge B=C⟹A=B$$

$$A=C\wedge B=C⟹A=B$$

2. 等量加等量，其和相等。

2. If equals be added to equals, the wholes are equal.

$$A=B⟹A+C=B+C$$

$$A=B⟹A+C=B+C$$

3. 等量减等量，其差相等。

3. If equals be subtracted from equals, the remainders are equal.

4. 彼此重合的东西彼此相等。

4. Things which coincide with one another are equal to one another.

5. 整体大于部分。

5. The whole is greater than the part.

$$\mathrm{\forall }x\in S,S>x$$

$$\mathrm{\forall }x\in S,S>x$$