Definition

Noun: A fundamental geometric postulate stating that for any two distinct points, exactly one straight line segment can be drawn connecting them. This is a foundational principle in Euclidean geometry.

Usage

This term is used exclusively in the context of geometry, mathematics, and logic to refer to a specific, self-evident truth upon which a system of reasoning is built. It is often cited when discussing the axioms of Euclidean geometry.

Examples

• The proof begins with Euclid's first axiom, establishing that a line exists between the vertices.
• In his , Euclid's first axiom is stated as "To draw a straight line from any point to any point."
• The entire structure of plane geometry relies on postulates like Euclid's first axiom.

Advanced Usage

• In discussions of non-Euclidean geometry: The term is often used to contrast with geometric systems where this axiom does not hold true (e.g., on the surface of a sphere, where multiple "straight" lines (great circles) can connect two antipodal points).
• In logic and philosophy: It can be cited as a classic example of an or self-evident truth in a deductive system.

Variants and Related Words

• Euclid's first postulate: A direct synonym.
• Axiom: A statement accepted as true without proof, serving as a starting point for further reasoning.
• Postulate: Another term for an axiom, especially in geometry.
• Euclidean geometry: The system of geometry based on Euclid's axioms, including this first one.

Synonyms

• The first postulate of Euclidean geometry.
• The line postulate.

Related Phrases

• "By Euclid's first axiom...": A phrase used to introduce a logical step in a geometric argument or proof, invoking this foundational rule.
  • By Euclid's first axiom, we can construct a line between point A and point B.

Noun

1. a straight line can be drawn between any two points