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In the "Elements" of Euclid (3th century B.C.) the primitive notions (point, line, surface, straight line, flat surface) are defined in a very problematic way.


Book I, Definitions

1. A point is that which has no part.

2. A line is breadthless length.

3. The ends of a line are points.

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

5. A surface is that which has length and breadth only.

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.


The struggle to define the primitive geometric notions in Euclid can be understood as coming up of a philosophical tradition aiming at clarifying geometrical notions and determining them terminologically. In the writings of Aristotle (384-322 B.C.) there is an effort of this kind. However, Aristotle himself  is not primarily interested in establishing a geometric terminology, but in solving paradoxes (like those of Zeno) emerging from terminological insufficiencies or misconceptions (according to his analysis).  

Ancient Geometry and Philosophy

From Lobatchevsky to Poincaré History From Lobatchevsky to Poincaré History