Euclid's Five Postulates

A postulate is a statement that is assumed without proof, considered to be self-evident, and is often used as the basis for an argument.

The following five postulates are stated at the beginning of Euclid's Elements. All of classical geometry is built from these five assumptions.

  1. A straight line segment can be drawn joining any two points.

  2. Any straight line segment can be extended indefinitely in a straight line.

  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

  4. All right angles are congruent.

  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

    This last postulate is equivalent to what is known as the parallel postulate:
    Given any line and a point not on the line, there exists exactly one line through the point that is parallel to the given line.

As you can see, the 5th postulate is considerably less obvious than the first four. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements (perhaps indicating some doubt as to its self-evident nature), but was forced to invoke the parallel postulate on the 29th.
Examples of elliptic and hyperbolic space
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