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We know that the Earth completes a revolution around the Sun about every 365.25 days. But we consider each year 365 days and add one day to the year every four years in order to have the leap year and also to adjust the time: $4*0.25=1$.

But there are some occasions where there is an 8-year interval between consecutive leap years. For example, the leap year after 1896 was 1904. 1900 wasn't a leap year. So why didn't they multiply 0.25 by 8 to have 2 as the number of additive days to the next leap year in this example(s)?

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    $\begingroup$ What would be the point of skipping the leap day in 1900 only to have two leap days in 1904? $\endgroup$
    – PM 2Ring
    Aug 6, 2023 at 4:33
  • $\begingroup$ Well, with a skipped leap day the adjustment of calendar and the Earth movement will be ruined in the future. $\endgroup$ Aug 6, 2023 at 4:39
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    $\begingroup$ Genuinely curious, and not intending to be mean, OP: prior to asking the question, did you have a hypothesis for why the 1900 leap year was skipped? $\endgroup$
    – notovny
    Aug 6, 2023 at 12:13
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    $\begingroup$ @Notovny I live in Iran and here there is a 33 years cycle for leap years. 7 leap years occur every 4 years and then there is a 5 years interval before the next one. Then again 7 four year intervals and a 5 years interval, and the cycle continues. Of course there's not always 33 years cycles. Sometimes we have 29 years cycles and even 34 years cycles. It has a complex math behind it which I'm not familiar with. But in the case of Gregorian calendar there isn't a complex math. The reason for my assumption in my question is simply this. $\endgroup$ Aug 6, 2023 at 12:26
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    $\begingroup$ I understand now. Thanks for the info. $\endgroup$
    – notovny
    Aug 6, 2023 at 14:56

2 Answers 2

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We changed from the Julian calendar of 365.25 days to the Gregorian calendar of 365.2425 days because the old calendar gave us too many leap days.

The mean tropical year has ~365.24219 days, so the Gregorian is still a bit too long, but we have several centuries to decide what to do about it. The error due to the calendar year length combined with the increase in the length of the mean solar day will equal 1 day some time around the year 3200.

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    $\begingroup$ The Wikipedia article for Earth gives the number $365.256363004$ as the number of days in an orbital period. $\endgroup$ Aug 6, 2023 at 4:52
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    $\begingroup$ @SnackExchange That's a sidereal year. But we use the tropical year, so that our calendar stays in synch with the seasons. en.wikipedia.org/wiki/Year#Astronomical_years lists several different astronomical years. $\endgroup$
    – PM 2Ring
    Aug 6, 2023 at 5:03
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    $\begingroup$ @SnackExchange Also see en.wikipedia.org/wiki/Tropical_year The Solar Hijri calendar used in Iran depends on astronomical observation of the (northern) spring equinox. The tropical year based on that equinox had a length of 365.242374 ephemeris days (of 86400 SI seconds) in the J2000.0 epoch. Note that the mean solar day is slightly longer than the ephemeris day, and gradually (and irregularly) getting longer. $\endgroup$
    – PM 2Ring
    Aug 6, 2023 at 15:31
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Because the year is actually slightly less than exactly 365.25 days, the standard "one leap-day every four years" pattern makes us overshoot a little; if we didn't fix it, the days would slip slightly in the opposite direction. To even out the excess, we have to skip a leap year every century -- so to make it easy to find, we just say we skip the leap day if the year ends in 00. So 1896 was a leap year, 1900 was not, 1904 was. We skipped the 1900 leap day because that put us back on schedule. We don't need to then add another "make-up" leap-day later; that would defeat the purpose of skipping the day in 1900.

But skipping one leap day per century still isn't quite right, so to get even more accurate, we added another rule that if the year is divisible by 400, it is a leap year after all. Since 2000 is divisible by 400, we did NOT skip the leap day that time, so 1996, 2000, and 2004 were all leap years. The next time we skip a leap day will be 2100.

That system gets us to an accuracy of just about one day every 3,000 years. We could keep adding more rules to get even more accurate, but there's no point to that -- the axial procession of the earth, which is slightly variable, means that over a few thousand years, we're going to wind up with calendar shifts building up no matter what we do. We'll have to adjust for them as we go along and see how well it's keeping up.

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  • $\begingroup$ Calendar designers have accounted for axial precession (to some extent) since Hipparchus discovered the precession of the equinoxes. But knowledge of the gradual irregular increase in the length of the sidereal day (and hence of the mean solar day) is more recent. And it's a bit tricky to deal with because it's irregular, with some unpredictable components. $\endgroup$
    – PM 2Ring
    Aug 9, 2023 at 6:11
  • $\begingroup$ Right. If it were regular we could build it in (and IIRC it is built in to some extent) but the slight variability across centuries means we can't really predict what adjustments will be needed that far ahead. $\endgroup$ Aug 9, 2023 at 15:00
  • $\begingroup$ My point is that the precession is fairly regular, and we can model it quite well over rather long timespans, although (from en.wikipedia.org/wiki/Axial_precession ) "The term (C − A)/C is Earth's dynamical ellipticity or flattening, which is adjusted to the observed precession because Earth's internal structure is not known with sufficient detail". But over the long term, the decrease in the axial rotation rate will cause the number of solar days per year to increase. Also see en.wikipedia.org/wiki/%CE%94T_(timekeeping) & ucolick.org/~sla/leapsecs/dutc.html $\endgroup$
    – PM 2Ring
    Aug 9, 2023 at 20:41
  • $\begingroup$ The procession is regular in the short term, but the rate varies over time in an unpredictable way because of the effect of the liquid and semi-liquid interior of the earth. It works within a few hundred years and up to a few thousand years it's more or less accurate, but... It's right in the wiki you posted: "For eras farther out, discrepancies become too large – the exact rate and period of precession may not be computed using these polynomials even for a single whole precession period [of 41,000 years]." $\endgroup$ Aug 10, 2023 at 15:15
  • $\begingroup$ Anyway, even if we acquired the ability to predict the precession properly for large time periods, it was unpredictable in the 1920s when the julian calendar was being revised, and that was the reason why they didn't try to get it more accurate than the three rules we use to decide on leap years. $\endgroup$ Aug 10, 2023 at 15:21

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