Ptolemy's equant

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 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