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. 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. 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.A circle may be described with any given point as its center and any distance as its radius.A straight line may be extended to any finite length.A straight line segment may be drawn from any given point to any other.
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. The five postulates of Euclidean Geometry define the basic rules governing the creation and extension of geometric figures with ruler and compass. Postulates in geometry is very similar to axioms, self-evident truths, and beliefs in logic, political philosophy, and personal decision-making.
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