The parallel postulate seems natural enough, but it's the kind of statement that seems like it should be a theorem-something we prove using other axioms and postulates-rather than a postulate. The parallel postulate says that if the angle measures of α and β add up to less than 180 degrees, then the dotted lines eventually intersect. The first four are short and sweet, but the fifth is a mouthful. In Thomas Heath's translation of Euclid's Elements (also known as the translation I have), the five postulates are stated as: 'Let the following be postulated:ġ) To draw a straight line from any point to any point.Ģ) To produce a finite straight line continuously in a straight line.ģ) To describe a circle with any centre and distance.Ĥ) That all right angles are equal to one another.ĥ) That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.' Euclidean geometry, codified around 300 BCE by Euclid of Alexandria in one of the most influential textbooks in history, is based on 23 definitions, 5 postulates, and 5 axioms, or 'common notions.' But as I mentioned in my recent post on hyperbolic geometry, one of the postulates, the parallel postulate, is not like the others.

Postulate geometry