A year does not divide into days

The tropical year is about 365.2422 days. The fraction is why the calendar needs leap days, and why no integer calendar stays honest forever.

A day is a spin. A year is an orbit, measured as the time to return to the same equinox. Divide and the remainder is about a quarter of a day, minus a thinner slice. Julius Caesar’s leap day every four years overcorrects. Gregory’s rule — skip three leap days in four centuries — leaves a residue of one day in a few thousand years.

There is no last calendar. The fractions are irrational for every practical purpose, and the day itself is lengthening, so the target moves. Leap seconds were a second patch, on the other clock. Any human date is a negotiated approximation to two motions that do not owe us an integer.