Definition

1. Noun:
   • A fundamental geometric principle: A Euclidean axiom is one of a set of five foundational, self-evident statements or postulates upon which the entire logical structure of Euclidean geometry is built. These axioms describe the basic properties of points, lines, and planes in a flat, two-dimensional space.

Usage Examples

• Noun:
  • The first Euclidean axiom states that a straight line segment can be drawn joining any two points.
  • To prove the theorem, we must rely on the established Euclidean axioms.
  • Non-Euclidean geometries are developed by altering one or more of the traditional Euclidean axioms.

Advanced Usage

• "The parallel postulate": This is the fifth and most famous Euclidean axiom. It states that through a point not on a given line, exactly one line can be drawn parallel to the given line. This axiom was historically the one challenged to create non-Euclidean geometries.
  • For centuries, mathematicians tried to prove the parallel postulate from the other four Euclidean axioms.

Variants and Related Words

• Axiom (n): A statement or proposition that is regarded as being established, accepted, or self-evidently true, serving as a starting point for further reasoning.
  • In logic, an axiom requires no proof.
• Postulate (n): A synonym for axiom, especially in mathematics and logic.
  • Euclid's postulates form the basis of classical geometry.
• Non-Euclidean geometry (n): A type of geometry where the parallel postulate (the fifth Euclidean axiom) does not hold.
  • The surface of a sphere is an example of a non-Euclidean geometry.

Synonyms

• Postulate: A fundamental assumption used as a basis for reasoning.
• Fundamental principle: A primary or basic rule or truth.

Related Concepts (Not Phrasal Verbs)

• Euclidean geometry: The system of geometry based on the five Euclidean axioms, describing the properties of flat space.
• Foundational assumption: A basic premise that is accepted without proof within a given system.

Noun

1. (mathematics) any of five axioms that are generally recognized as the basis for Euclidean geometry