Happy Leap Year 2024! A leap day is an extra day (February 29) that is added to the Gregorian calendar (the calendar most of us use day to day) in leap years. While leap years most commonly come in four-year intervals, they sometimes come every eight years. This is because a traditional leap day every four years is actually a slight overcompensation in the calendar. Thus, a leap year is skipped every one hundred years when those years are not divisible by 400 (this is actually the entire difference between the Julian and the Gregorian calendars).

## Leap Years in a Backward-Orbit Earth

Phileas Fogg (*Around the World in Eighty Days*) traveled around the world in fewer than 80 full days from his start in London, but he counted 81 sunrises because he was traveling opposite to the motion of the Sun in the sky. If he had traveled in the same direction, he would have counted 79 sunrises in the same period of time. If the Earth rotated backward, these numbers would be swapped, and Fogg would have needed to travel toward the west to win his bet.

The same phenomenon happens for all of us every year. The Earth travels a full orbit around the Sun in a year and, in the same time, it rotates approximately 366.25 times (this is the equivalent of 80 days for Fogg) with respect to the stars—well, it’s actually with respect to the vernal equinox point, which itself moves too due to precession, but that gets too complicated.

### Would We Remove a Leap Day if the Earth Rotated Backward?

Because the Earth rotates in the same direction, we count one day fewer, and so we get a year that has, on average, 365.25 solar days. If the Earth rotated backward, we would perceive that a year has 367.25 days!

Let us stop for a moment to measure what a day would be in each case. A full orbit around the Sun takes this amount of time:

It corresponds to this “Foggian” number of our solar days:

If we measure rotations with respect to the stars, we count one more, so the day is shorter:

This is the so-called “sidereal day”:

If the Earth rotated backward, the solar day would have this length:

That is, we would have 367.25 days in a year, but each day would be about eight minutes shorter. Or, perhaps, we should say that days would still have 24 hours, but each one would be about 20 seconds shorter:

It’s so easy to get all these precise numbers with Wolfram Language!

*Disclaimer:* Due to how the solar system was created, it is unlikely that the Earth would rotate backward, and if it did, tidal friction with the Moon would have given a very different duration of the day. But let us ignore all that here and assume that rotation with respect to the stars would have the same angular speed.

Now we can address our question: would we remove leap days if the Earth rotated backward? No. We see that the number of days in a year would be 367.25, so the natural thing would be to have normal years of 367 days and then add a leap day every four years (with a Gregorian correction!) The main consequence for our standard calendar is that we would have two more days. Presumably, February would also have 30 days in normal years, and 31 in leap years. Wouldn’t that be nicely symmetric with all other months?

## Math, Calendars, Leap Days and the Importance of Computation

So, how many total leap days have there been (including Julian and Gregorian calendars)? Has that math been done, and if so, was it right?

### Explanation

The physical year (i.e. an orbit of the Earth around the Sun) is called a “tropical year,” known to very good precision:

Put another way:

The difference with 365 days and 6 hours is only a bit more than 11 minutes.

This is the number of days (i.e. turns of the Earth with regards to the Sun) between January 1 of year 1 (in the Julian calendar) and January 1 of year 2025 (in the current Gregorian calendar), including one but not the other, so this is 2,024 full calendar years:

The difference with 2,024 tropical years is only 2.8 days:

This is a very good approximation in more than 700,000 days. But where did those 2.8 days come from?

Imagine all years had 365 days. Then 2,024 years would be:

And there would be a difference of more than a full year with respect to the physical counting of years!

The Julian calendar was introduced in 45 BCE to add one day every four years (extending by six hours the average length of a year). Then 2,024 Julian years would be this number of days:

That’s now too much by 15.8 days:

By the end of year 1581, exactly 395 leap days had been added since year 1, which was about 12 days too many:

The Gregorian reform of the calendar removed 10 days in 1582 (the day following October 4 was October 15). The new calendar also changed the rule of how leap days are added, to avoid accumulating 11 minutes of error every year (or, equivalently, one day every 128 years). Years that are a multiple of 100 but not of 400 are not leap years. This has happened so far for years 1700, 1800 and 1900. Therefore, the Gregorian calendar has corrected 13 days of the 15.8 days of error. The difference is the 2.8 days we saw before, most of it from the removal of 10 instead of 12 days. The other 0.8 is essentially because we are close to correcting another leap day in year 2100.

The important comparison is this: In 400 years of the Gregorian calendar, there are 97 leap days added. Therefore, the average year is:

So there is a difference of only 27 seconds per year, to be compared with the more than 11 minutes of error in the Julian calendar:

It will take more than 3,200 years to accumulate a day of error in the Gregorian calendar, while it takes only 128 years to have a day of error in the Julian calendar:

In short, yes, the math has been done… and it wasn’t *exactly* right—but with more precise computation, we’re getting closer by the second!

(The Newtonian calendar is slightly more precise, but that’s a rabbit hole for another day.)

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