Masters: Why is Euclidean geometry called Euclidean space geometry, and what does this have to do with linear space?

The traditional description of Euclidean geometry postulate Euclidean geometry is an axiom and postulate system, and all "true propositions" are proved by finite axioms and postulates.

The five postulates of Euclidean geometry are:

1, any two points can be connected by a straight line.

2. Any line segment can be infinitely extended into a straight line.

3. Given any line segment, you can make a circle with an endpoint as the center and a line segment as the radius.

4. All right angles are congruent.

5. If two straight lines intersect with the third straight line, and the sum of internal angles on the same side is less than the sum of two right angles, then the two straight lines must intersect on this side.

The five axioms of Euclidean geometry are:

1, equal amounts are equal to each other.

2, the same amount plus the same amount, the sum is still equal.

3. If the same amount is subtracted from the same amount, the difference is still equal.

4. Objects that can overlap each other are congruent.

5, the whole is greater than the local.

The fifth axiom of the derived proposition is called the parallel axiom, and the following proposition can be derived:

Through points that are not on a straight line, only one straight line is parallel to this straight line. Parallel axioms are not so obvious as other axioms. Many geometricians tried to prove this axiom with other axioms, but all failed. 19th century, by constructing non-Euclidean geometry, it was proved that the parallel axiom could not be proved. If the parallel axiom is removed from the above axiom system, we can get a more general geometry, that is, absolute geometry. )

On the other hand, the five axioms of Euclidean geometry are incomplete. For example, there is a theorem in this geometry: any line segment is part of a triangle. He constructed it in the usual way: taking the line segment as the radius, taking the two endpoints of the line segment as the center respectively, and taking the intersection of two circles as the third vertex of the triangle. However, his axiom does not guarantee that these two circles must intersect. Therefore, many revised versions of axiomatic systems have been proposed, including Hilbert axiomatic system.