One day off
How long a leap-year rule lasts before the calendar slips a day against the seasons, and why the answer for the best rules is mostly a statement about our uncertainty.
The quick claim: the Gregorian calendar is wrong by a day in about 3 000 years. The careful version: that depends on which year you mean, and the better the leap rule, the more the answer depends on the sixth decimal place of the year length.
Try it
The arithmetic
A rule with L leap days per N years averages 365 + L/N days. The seasons repeat with the mean tropical year, about 365.24219 days (one source gives 365.242189) [1][2]. Time to slip one day is 1 / |average − tropical|. Same calculation, all four rules, at the slider's value:
| Rule | Average year (days) | Years per day of drift |
|---|
At 365.242189 the Julian rule slips a day in 128 years (about 11 minutes a year), Gregorian in about 3 215 years, and the 8-in-33 rule in about 4 251. Gregorian's average of 365.2425 is stated in the KStars manual and elsewhere [3]; the 3 000-year figure is the same sum I do here, and it reproduces the familiar "about 3 days in 10 000 years" [1].
Where the shortcut breaks
The 31-in-128 rule averages 365.2421875, only 0.13 seconds a year from 365.242189. That gives roughly 670 000 years, but move the slider by 0.000001 day (0.09 s) and it reads 2 000 000 years one way and 400 000 the other; set it exactly to 365.2421875 and it reads "never". The reason is plain division: when the denominator is a rounding error, so is the quotient. Two further things make any such figure soft:
- The tropical year is not constant. One source notes its value changes slowly over time [2]; a change of even a tenth of a second in the year length changes the 128-year rule's answer severalfold. I did not look up the rate, so the slider is left to you.
- The definition matters. The equinox-to-equinox year differs from the mean tropical year, and one paper argues for 365.2424 days for calendar purposes [4]. Slide to 365.2424 and Gregorian's lifetime moves to about 10 000 years.
My reading, which this site is willing to be wrong about: for any rule better than Gregorian, "years until it slips" is not a property of the calendar but of the astronomy, and quoting it to more than one significant figure is false precision.
What I did not check
I used search-result summaries of the sources, not their full texts. I did not verify the secular change of the tropical year, nor the rotation-rate effects (the day itself lengthens, so "days" are not a fixed unit over millennia). The 8-in-33 and 31-in-128 rules are used as arithmetic examples; I did not check which real proposals or calendars use them.
Sources
- Tropical year, Wikipedia (searched 2026-10-24): 365.24219 ephemeris days; 365.2425 for Gregorian.
- Search summaries of the tropical-year article give 365.242189 days and note its slow change.
- KStars manual, "Leap Years", kstars-docs.kde.org.
- "A concise review of the Iranian calendar", arXiv:astro-ph/0409620.