Definition

Noun: * Euclid's Third Axiom: A fundamental postulate in Euclidean geometry stating that it is possible to draw a circle with any given radius centered at any given point.

Usage

This term is used exclusively in the context of geometry, mathematics, and the history of mathematics to refer to one of Euclid's foundational assumptions. * The construction of an equilateral triangle in Proposition 1 of Euclid's "Elements" relies on Euclid's third axiom. * A key difference between Euclidean and some non-Euclidean geometries is the modification or rejection of Euclid's third axiom.

Advanced Usage

• In Axiomatic Systems: The axiom is often phrased as "To describe a circle with any center and distance." It establishes the possibility of a specific geometric construction without proof, serving as a starting point for deriving further theorems.
• In Discussion of Geometric Foundations: The axiom is sometimes discussed in relation to the compass equivalence theorem or the concept of "collapsible" vs. "rigid" compasses in geometric constructions.

Variants and Related Words

• Euclid's Postulates: The collective term for the five fundamental assumptions, including the third axiom, upon which Euclidean geometry is built.
• Axiom: A statement or proposition that is regarded as being established, accepted, or self-evidently true, serving as a starting point for further reasoning.
• Compass and Straightedge Construction: The method of geometric construction that this axiom explicitly permits.

Synonyms

• Euclid's Third Postulate
• The Circle Postulate

Related Phrases

• "Based on Euclid's axioms": Describing a geometric proof or system that uses Euclid's postulates as its foundation.
  • The entire proof is based on Euclid's axioms.
• "Invoking the third postulate": The act of using this specific axiom within a geometric argument.
  • The next step in the proof involves invoking the third postulate to draw arc BC.

Noun

1. a circle with any radius can be drawn around any point