A postulate is an assumption, that is, a proposition, or statement, that is assumed to be true without any proof. Postulates are the fundamental propositions used to prove other statements known as theorems. Once a theorem has been proven it is may be used in the proof of other theorems. In this way, an entire branch of mathematics can be built up from a few postulates. Postulate is synonymous with axiom, though sometimes axiom is taken to mean an assumption that applies to all branches of mathematics, in which case a postulate is taken to be an assumption specific to a given theory or branch of mathematics. Euclidean geometry provides a classic example. Euclid based his geometry on five postulates and five "common notions," of which the postulates are assumptions specific to geometry, and the "common notions" are completely general axioms.

The five postulates of Euclid that pertain...