Euclid's axioms
Around 300 BCE, Euclid of Alexandria wrote the Elements, organizing the geometry of his time into a deductive system. He started with five postulates and a handful of "common notions," and derived hundreds of theorems in strict logical order.
Euclid's postulates, in modern language:
- Any two points can be joined by a straight line.
- Any line segment can be extended indefinitely.
- Given a point and a radius, a circle can be drawn.
- All right angles are equal.
- The parallel postulate: through a point not on a given line, there is exactly one line parallel to the given line.
Key theorems
- The Pythagorean theorem — Proposition 47 of Book I.
- The sum of angles in any triangle equals 180°.
- Vertical angles are equal.
- In similar triangles, corresponding sides are proportional.
- The inscribed angle in a semicircle is a right angle.
The parallel postulate problem
For two thousand years, mathematicians suspected the parallel postulate was redundant — provable from the other four. They tried, and failed. In the 19th century, Lobachevsky, Bolyai, and Riemann independently showed non-Euclidean geometries exist in which the parallel postulate is false. Euclid's fifth postulate was a choice, not a necessity.
Why it still matters
Despite being 2300 years old, Euclidean geometry remains the foundation of high-school mathematics and everyday spatial reasoning. Every time you compute a distance with Pythagoras, draw a triangle, or use similar triangles, you're working in Euclidean space.
Modern analytic geometry gave us the tools to handle Euclidean shapes using algebra and coordinates. And topology studies the properties that survive even when Euclidean structure is removed.