Aristotle's axiom

Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens that states:

If is an acute angle and AB is any segment, then there exists a point P on the ray and a point Q on the ray , such that PQ is perpendicular to OX and PQ > AB.

Aristotle's axiom is a consequence of the Archimedean property, and the conjunction of Aristotle's axiom and the Lotschnittaxiom, which states that "Perpendiculars raised on each side of a right angle intersect", is equivalent to the Parallel Postulate.

Without the parallel postulate, Aristotle's axiom is equivalent to each of the following two incidence-geometric statements:

• Given two intersecting lines m and n, and a point P, incident with neither m nor n, there exists a line g through P which intersects m but not n.
• Given a line a as well as two intersecting lines m and n, both parallel to a, there exists a line g which intersects a and m, but not n.

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