euclid's fifth axiom
Noun: * Euclid's fifth axiom: A fundamental postulate in Euclidean geometry, also historically known as the parallel postulate. It states that through a given point not on a given line, exactly one line can be drawn that is parallel to the given line. This axiom is distinct from Euclid's other postulates and is the foundation for the geometry of flat planes.
This term is used exclusively in mathematical and geometric contexts to refer to a specific historical and foundational principle. * The consistency of Euclidean geometry depends on Euclid's fifth axiom. * For centuries, mathematicians tried to prove Euclid's fifth axiom from the other four. * The discovery of non-Euclidean geometries arose from questioning the necessity of Euclid's fifth axiom.
- As a defining concept: The axiom is often discussed in contrast to its alternatives, which define hyperbolic or elliptic geometries.
- Rejecting Euclid's fifth axiom leads to the different geometric rules found in hyperbolic space.
- In historical context: It is frequently mentioned in the history of mathematics regarding attempts to prove it as a theorem.
- The long effort to prove Euclid's fifth axiom culminated in the revolutionary creation of non-Euclidean geometries.
- Parallel Postulate: The more common modern name for Euclid's fifth axiom.
- Playfair's Axiom: A logically equivalent but simpler statement: "In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point."
- Non-Euclidean Geometry: A type of geometry that is based on axioms that modify or reject Euclid's fifth axiom.
- Parallel postulate
- The fifth postulate (of Euclid)
- Euclidean Geometry: The system of geometry based on all five of Euclid's axioms, including the fifth.
- Absolute Geometry: The body of geometric theorems that can be proven without using the parallel postulate (i.e., from Euclid's first four axioms alone).
- only one line can be drawn through a point parallel to another line