---
title: "Ptolemy's equant"
type: "entity"
entity_kind: "concept"
status: "hub"
canonical_name: "Equant (punctum aequans)"
aliases: ["punctum aequans","equant","Ptolemy's equant"]
first_seen: "2026-10-10T00:00:00.000Z"
writer_model: "claude-sonnet-5"
connects_to: ["[[entity-claudius-ptolemy|Claudius Ptolemy]]","[[entity-johannes-kepler|Johannes Kepler]]","Kepler's second law","elliptical-orbit empty focus","[[entity-tycho-brahe|Tycho Brahe]]"]
seek_code_commit: "d40ee39"
---


In Ptolemaic astronomy, the equant (*punctum aequans*) is a geometric point — offset from both the orbit's center and the observer — about which a planet is defined to sweep out equal *angles* in equal time, even though its distance from that point varies. It is Ptolemy's device for reconciling observed non-uniform planetary motion with the ancient conviction that celestial motion must be uniform: not uniform speed along the path, but uniform angular speed as seen from one specific point.

For an ordinary Keplerian ellipse, the empty focus is only an *approximate* equant (uniform angular speed to within O(ε²) in eccentricity, never exact) — which is why the equant and [[entity-johannes-kepler|Kepler]]'s second law were long treated as mutually exclusive: an orbit satisfying one exactly could not satisfy the other exactly, given an ellipse. Matters to this vault via [[claim-astronomia-ova-orbit-family-satisfies-keplers-second-law-and-exact-equant]], a 2026 paper constructing non-elliptical orbits that satisfy both exactly at once, and [[claim-ptolemy-equant-mars-error-8-arcminutes-tycho-precision-forced-kepler-to-abandon-it]], which records the Mars data that eventually forced the historical equant's abandonment.

## References

- [[claim-astronomia-ova-orbit-family-satisfies-keplers-second-law-and-exact-equant]]
- [[claim-ptolemy-equant-mars-error-8-arcminutes-tycho-precision-forced-kepler-to-abandon-it]]