Calendar lab

The calendar on your wall is about 27 seconds too long every year. Can you write a better leap rule?

The Gregorian rule has three clauses. You get up to four. First guess how long it takes to slip by a full day, then try to beat it over 10,000 years.

Step 1. Guess

The Gregorian calendar averages 365.2425 days per year. The real mean tropical year (equinox to equinox) is 365.24219. After how many years has the Gregorian calendar drifted a full day away from the seasons?

1,000years

Step 2. Write your own rule

A year is a leap year if it passes the clauses below, checked top to bottom, and the last matching clause wins. Set a clause to 0 to switch it off. The Julian rule is clause 1 alone with 4.

1. Leap if divisible by

2. but not if divisible by

3. but leap again if divisible by

4. but not if divisible by

Drift against the seasons, years 1 to 10,000 Horizon

Positive: the calendar has added more days than the sun needed, so the equinox lands earlier on the calendar page. Negative: the opposite. The curve is a sawtooth because leap days arrive in whole days. The gray band is within half a day.

■ Julian■ Gregorian■ Yours

Step 3. Why 1582 needed a fix

The Julian year is 365.25 days. Per the sources below it gains one day every 128 years, and by 1582 the equinox had slipped about 10 days since the Council of Nicaea in 325, so Thursday 4 October 1582 was followed by Friday 15 October. Drag the year to see how far the Julian calendar has drifted from the seasons by that year, using the same simulator.

1582

What you just learned

Adding a clause trades complexity for accuracy. The Gregorian rule already needs two exceptions to get from 365.25 to 365.2425, and that remaining gap of 0.00031 days is the 27 seconds you saw. Any rule is just a fraction of leap years per year. Your rule works when its fraction lands near 0.24219. Spotting those fractions is what the reformers did, and a century clause is the cheapest way to shave 0.0075 off the Julian mean.

One caveat: the tropical year itself changes slowly, so all drift numbers use the fixed value 365.24219 days (mean in the year 2000). Over 10,000 years the true figure is therefore a bit uncertain; the Wikipedia article on the tropical year cites an estimate that Gregorian would end up between three hours and four days behind.

Sources

Wikipedia (secondary; official sites not consulted): Gregorian calendar. 97 leap years per 400, mean year 365.2425, 10-day jump in October 1582, 400-year cycle of 146,097 days.
Wikipedia (secondary): Tropical year. Mean tropical year 365.24219 days (value for 2000), estimate of 3 hours to 4 days behind after 10,000 years.
Wikipedia (secondary): Julian calendar. 365.25 day year, one day gained every 128 years.

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