Euclid's Geometry

The word 'geometry' comes from the Greek 'geo' (earth) and 'metron' (to measure). It grew from very practical needs: measuring land after the Nile floods in Egypt, building pyramids, and laying out cities. Around 300 BCE a Greek teacher named Euclid collected all the scattered geometric knowledge of his time and arranged it into a single logical system in a set of books called The Elements. Instead of treating each result as a separate fact, he showed how every result could be proved from a tiny number of starting assumptions.

Euclid began with definitions of the basic objects: a point is that which has no part, a line is breadthless length, and so on. He then listed two kinds of assumptions accepted without proof. Axioms (also called common notions) are general truths used throughout mathematics, such as 'things which are equal to the same thing are equal to one another'. Postulates are assumptions specific to geometry, such as 'a straight line may be drawn from any point to any other point'. From these few starting points he proved theorems one after another.

Euclid stated five postulates. (1) A straight line can be drawn joining any two points. (2) A finite straight line can be extended indefinitely. (3) A circle can be drawn with any centre and any radius. (4) All right angles are equal to one another. (5) If a straight line crossing two straight lines makes the interior angles on one side add up to less than two right angles, then the two lines, if extended, meet on that side. The first four feel obvious; the fifth is long and complicated, which made mathematicians suspicious of it for two thousand years.

The fifth postulate did not look as self-evident as the others, so for centuries mathematicians tried to prove it from the first four. They all failed. Instead, their efforts led to a useful equivalent statement called Playfair's axiom: 'For every line l and every point P not on l, there is exactly one line through P parallel to l.' Later, mathematicians who assumed the fifth postulate was false discovered entirely new and consistent geometries, called non-Euclidean geometries, which describe curved surfaces and even the shape of space itself.