Aristotle's axiom

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

Aristotle's axiom asserts that a line PQ exists which is parallel to AB but greater in length. Note that: 1) the line AB does not need to intersect OY or OX; 2) P and Q do not need to lie on the lines OY and OX, but their rays (i.e. the infinite continuation of these lines).

If ${\displaystyle {\widehat {\rm {XOY}}}}$ is an acute angle and AB is any segment, then there exists a point P on the ray ${\displaystyle {\overrightarrow {OY}}}$ and a point Q on the ray ${\displaystyle {\overrightarrow {OX}}}$, 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]

• Given a line A and a point P on A, as well as two intersecting lines M and N, both parallel to A 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.
• Given a line A and two distinct intersecting lines M and N, each different from A, there exists a line G which intersects A and M, but not N.

References

1. Pambuccian, Victor (2019), "The elementary Archimedean axiom in absolute geometry (Paper No. 52)", Journal of Geometry, 110: 1–9, doi:10.1007/s00022-019-0507-x, S2CID 209943756
2. Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms", Journal of Geometry, 51 (1–2): 79–88, doi:10.1007/BF01226859, hdl:2027.42/43033, S2CID 28056805
3. Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967

Sources

• Greenberg, Marvin Jay (1988), "Aristotle's axiom in the foundations of geometry", Journal of Geometry, 33 (1–2): 53–57, doi:10.1007/BF01230603, S2CID 122416844
• Greenberg, Marvin Jay (2010), "Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries" (PDF), American Mathematical Monthly, 117 (3): 198–219, doi:10.4169/000298910x480063, S2CID 7792750
• Greenberg, Marvin Jay (2008), Euclidean and non-Euclidean geometries, 4th edition, W H Freeman
• Martin, George E. (1982), The foundations of geometry and the non-Euclidean plane, Springer
• Pambuccian, Victor (2019), "The elementary Archimedean axiom in absolute geometry (Paper No. 52)", Journal of Geometry, 110: 1–9, doi:10.1007/s00022-019-0507-x, S2CID 209943756
• Pambuccian, Victor (1994), "Zum Stufenaufbau des Parallelenaxioms", Journal of Geometry, 51 (1–2): 79–88, doi:10.1007/BF01226859, hdl:2027.42/43033, S2CID 28056805
• Pambuccian, Victor; Schacht, Celia (2021), "The ubiquitous axiom", Results in Mathematics, 76 (3): 1–39, doi:10.1007/s00025-021-01424-3, S2CID 236236967