Euclid's axioms ruled geometry for two millennia
Euclidean geometry is the system Euclid laid out in his textbook, the Elements: accept a few intuitive postulates, then prove everything else from them. Many results were known earlier, but he was the first to chain them into a logical structure where each theorem rests on axioms and earlier proofs.
The Elements mostly organised existing Greek knowledge, and it was judged so superior that older treatises stopped being copied and are now nearly all lost. Books I to IV and VI cover plane geometry, including the Pythagorean theorem as proposition 47 of Book I. Books V and VII to X handle number theory, treating numbers as lengths or areas, introducing primes and irrationals and proving there are infinitely many primes. Books XI to XIII turn to solids, showing a cone holds a third of the volume of a cylinder with the same base and height and constructing the Platonic solids.
Early in Book I come five postulates, phrased as constructions: join any two points with a straight line, extend a line, draw a circle of any centre and radius, treat all right angles as equal, and the parallel postulate. Five common notions add rules such as things equal to the same thing being equal to each other. Because the figures can be built with compass and unmarked straightedge alone, the system is constructive, more concrete than theories like set theory that assert objects exist without saying how to make them. Modern scholars agree the postulates fall short of a complete foundation, and today's versions use fuller axiom lists.
The fifth postulate always looked less obvious, and for centuries people tried to prove it from the others. Euclid himself seems to have set it apart, proving his first 28 propositions without it. Such a proof is now known to be impossible, since consistent geometries exist in which it fails. Playfair's axiom, allowing at most one parallel through a point off a line, is an equivalent version.
Hyperbolic and elliptic geometry are both self-consistent, and Einstein's general relativity implies physical space is Euclidean only approximately, over short distances and in weak gravity. Descartes's coordinate-based analytic geometry arrived almost 2,000 years after Euclid's synthetic approach.
Source: Euclidean geometry
