Euclid's axiom
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Definition
Noun: A fundamental, self-evident principle or statement in Euclidean geometry, specifically one of the five basic postulates upon which Euclid built his system of plane geometry in his work "Elements." These axioms are accepted without proof and form the foundation for deriving geometric theorems.
Usage
The term is used in the context of mathematics, geometry, and the history of science to refer to one of the foundational assumptions of classical geometry. * The parallel postulate is the most famous of Euclid's axioms. * For centuries, mathematicians tried to prove that Euclid's fifth axiom could be derived from the other four. * The consistency of the entire system relies on the truth of Euclid's axioms.
Advanced Usage
- "To be based on Euclid's axioms": Describes a geometric system or proof that uses Euclid's postulates as its starting point.
- Classical plane geometry is entirely based on Euclid's axioms.
- The discussion of these axioms often leads to the concept of non-Euclidean geometry, which is built by modifying or rejecting one of Euclid's postulates (typically the parallel postulate).
Variants and Related Words
- Euclidean axiom: A direct synonym.
- Euclid's postulate: A fully interchangeable term; "axiom" and "postulate" are often used synonymously in this context.
- Euclidean geometry: The system of geometry derived from Euclid's axioms.
- Parallel postulate: Specifically refers to Euclid's fifth axiom.
Synonyms
- Postulate (in a geometric context)
- Fundamental assumption
- Self-evident truth (within the Euclidean system)
Related Idioms/Phrases
- "Taken as an axiom": Used more generally to mean something accepted as true without proof. While not containing the exact phrase "Euclid's axiom," it uses the core concept.
- In their economic model, rational actor behavior is taken as an axiom.
Noun
- (mathematics) any of five axioms that are generally recognized as the basis for Euclidean geometry