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,[1] 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.[2]
Without the parallel postulate, Aristotle's axiom is equivalent to each of the following three incidence-geometric statements:[3]