Euclid's Postulates

Geometric representation of the postulates of Euclides.

The postulates of Euclid refer to the treatise called The Elements, written by Euclid around the year 300 BC. C., exposing the geometric knowledge of classical Greece deducing them from five postulates, considered the most obvious and simple.

The postulates of The Elements are

1. Two different points any determine a straight segment.
2. A straight segment can be extended indefinitely in a straight line.
3. A given circumference can be traced to any center and radius.
4. All straight angles are the same.
5. Postulate of the parallels. If a straight line cuts to another two, so that the sum of the two inner angles on the same side is less than two straight, the other two straights are cut, by prolonging them, on the side on which are the minor angles than two straight.

This last postulate has an equivalent, which is the most used in geometry books:

• From an outside point to a straight line, a single parallel can be traced.

At the beginning of the 19th century Gauss, Lobachevsky and János Bolyai considered the possibility of a geometry without the fifth postulate, discovering hyperbolic geometry.

In current terms, these postulates were stated by Hilbert in his axioms.