Let the following be postulated:
To draw a straight line from any point to any point.
从任一点到任一点，可以作一条直线。
To produce a finite straight line continuously in a straight line.
一条有限直线可以沿直线方向不断延长。
To describe a circle with any centre and distance.
以任一中心和任一距离，可以作一个圆。
That all right angles are equal to one another.
所有直角都彼此相等。
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.
一条直线同两条直线相交，如果同侧内角之和小于两个直角，那么这两条直线无限延长后，会在该侧相交。
Things which are equal to the same thing are also equal to one another.
同等于一物的量，彼此也相等。
If equals be added to equals, the wholes are equal.
等量加等量，所得整体相等。
If equals be subtracted from equals, the remainders are equal.
等量减等量，所得余量相等。
Things which coincide with one another are equal to one another.
能彼此重合的东西彼此相等。
The whole is greater than the part.
整体大于部分。
欧几里得给出作直线、延长直线、作圆和第五公设的基础许可；阿波罗尼奥斯的圆锥曲线会在这个综合几何传统中继续发展截面、直径与切线语言。可接着读 apollonius-conics/overview，再回到笛卡尔看这些曲线如何被方程重写。
第五公设和隐含公理的后续历史，见 math-meta/topic-noneuclidean-history 与 math-meta/topic-axiomatic-movement-deepening。
徐光启与利玛窦合译《几何原本》时，定义、公设、命题和证明的层级进入中文数学语境。相关长卷见 math-meta/topic-east-west-encounter。
第五公设的两千年困惑，在十九世纪被罗巴切夫斯基、鲍耶和黎曼改写成新的几何语言。原作选读见 non-euclidean-originals/lobachevsky-overview。
1607 年徐光启、利玛窦合译《几何原本》前六卷，把定义、公设、命题和证明的欧氏文体带入中文。十九世纪李善兰与伟烈亚力续译后九卷，这条线从明末证明文体延伸到晚清术语工程；可接着读 late-qing-translation/li-overview。