Euclid's axioms are five postulates about the behavior of geometric objects; they constitute the foundation upon which Euclid built the entire edifice of geometry that is known today as Euclidean geometry. Before Euclid wrote his famous book The Elements around 300 BC, many geometric ideas were well understood, but in a disorganized way that obscured their logical structure. It was often unclear which facts depended on which others for their proofs, and this vagueness opened the door to circular reasoning and other logical errors. To systematize the study of geometry, Euclid formulated five axioms, statements so simple he considered them self-evident, and then attempted to prove all other geometric facts using only these five axioms and the principles of logical reasoning. Euclid's analysis was so definitive and far-reaching that it laid the foundation for the study of geometry for the next 2000 years. Still more, it took the...