Geometry/Five Postulates of Euclidean Geometry

Postulates in geometry is very similar to axioms, self-evident truths, and beliefs in logic, political philosophy, and personal decision-making. The five postulates of Euclidean Geometry define the basic rules governing the creation and extension of geometric figures with ruler and compass. Together with the five axioms (or "common notions") and twenty-three definitions at the beginning of Euclid's Elements, they form the basis for the extensive proofs given in this masterful compilation of ancient Greek geometric knowledge. They are as follows:

1. A straight line segment may be drawn from any given point to any other.
2. A straight line may be extended to any finite length.
3. A circle may be described with any given point as its center and any distance as its radius.
4. All right angles are congruent.
5. If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles.

Postulate 5, the so-called Parallel Postulate was the source of much annoyance, probably even to Euclid, as it is not a simple, concise statement, as are the other four. Mathematicians, and really most of us, value simplicity arising from simplicity, with the long complicated proofs, equations, and calculations needed for rigorous certainty done behind the scenes, and to have such a long sentence amidst such other straightforward, intuitive statements seems awkward. As a result, many mathematicians over the centuries have tried to prove the results of the Elements without using the Parallel Postulate, but to no avail. However, in the past two centuries, assorted non-Euclidean geometries have been derived based on using the first four Euclidean postulates together with various negations of the fifth.