The math of leap years: Why Feb 29 does not come once every four years

We know 2024 is a leap year because 24 is divisible by 4, the same reason that 2020, 2016, 2012, 2008, 2004 and 2000, too, were leap years.

In 1924, months before the Gateway of India was inaugurated in December, our counterparts a century older would have reasoned along similar lines, but with a key difference. They knew 1924 was a leap year, as were 1920, 1916, 1912, 1908 and 1904. The difference was 1900 which, unlike 2000, was a non-leap year.

The year 1800 wasn’t leap either, nor was 1700, but 1600 was leap. Going ahead, 2100, 2200 and 2300 will not be leap years, but 2400 will be.

So, what is going on here? Years ending in 00 are not always leap, even though they are all divisible by 4. They are leap years only if the two digits preceding the 00 form a multiple of 4; hence 1600, 2000 and 2400 are leap. To put it simply, a leap year does not come exactly once every four years; the actual frequency is 97 times every 400 years.

This is something that is perhaps less widely known than the idea of leap years and February 29 at its most basic level. To be sure, many people eventually get to know about this rule when they come across the non-leap 00 year during their lifetimes. On the other hand, many of the people living today will have missed the point when they found that 2000 was a leap year, like all other multiples of 4. And many of these same people will never get to see the next exception, in 2100, unless they are very young today and blessed with a long life.

The first time anyone learns about the rule, the obvious question that pops up is: Why? There is mathematics, astronomy and history at play here, and the explanation must begin with why we have leap years in the first place.

The early leaps

The calendar we follow today is solar, and is supposed to coincide with one full orbit of the Earth around the Sun. Meanwhile, we mark our days in terms of one rotation of our planet around its own axis. Unfortunately, a solar year is not an exact multiple that can be expressed in a whole number of days.

We take a calendar year as usually consisting of 365 days, but a full revolution of the Earth actually takes a little longer: 365 days, 5 hours, 48 minutes and 46 seconds. This is commonly approximated as being 365 years and 6 hours. The Earth’s orbital period was understood as far back as 46 BC, when Julius Caesar engaged astronomers to design an early version of the modern calendar.

The calendar worked on the approximation of 365 days and 6 hours. Their reasoning went like this: every year, we count 365 days in the calendar, without accounting for the extra 6 hours. In two years, we have 12 hours unaccounted for, and in four years that becomes 24 hours. So, after four years, our calendar ends up being a day short. To compensate for this, our calendar should add an extra day once every four years.

And so began the practice of including leap years with 366 days. In the Julian calendar, the extra day was not February 29, but February 23 appeared twice every leap year.

This worked fine for centuries. But the calendar had an obvious error: the approximation. Although the Romans took the calendar year as being six hours short of the solar year, the actual difference was 5:48:46 hours. In four years, the calendar added 6 x 4 = 24 hours. The real correction, however, should have been 5:48:46 x 4 = 23:15:04 hours. In other words, the calendar had overcompensated by almost 45 minutes.

How could anyone, however, account for a little under 45 minutes over four years? The average error was only 11 minutes and 14 seconds per year. What did a mere 11 minutes matter in the course of a whole year?

Not much, the Romans thought, as they happily kept following their calendar. They did not foresee the far future, when the error would keep piling up over the centuries.

Deleting leap years

By the 16th century, the world was sitting up and taking notice. At an average of 11 minutes and 14 seconds extra every year starting from Caesar’s time, the calendar had overshot the solar calendar by 10 days by the mid-1500s.

Drastic corrections followed. In 1582, Pope Gregory III engaged scientists to reform the calendar. The first thing they did was to get rid of the 10 extra days that had accumulated. October 4, 1582, was followed directly by October 15; the 10 days between October 5 and 14, 1582, were never allowed to exist.

Now, there was the matter of preventing the same error from accumulating again. So, some years that should have been leap must no longer be so. Which years should that be? Something that would be easy to remember.

If leap years were allowed to continue as before, the average year would continue to be 11:14 minutes too long, which would accumulate to approximately three days over 400 years. So, every four hundred years, three leap years must be designated non-leap.

The Gregorian calendar thus adopted the following rule. To take away three years in 400, remove all four 00 years. But that’s one extra year, so restore one of them. The year that is multiple of 400 (it begins with a multiple of 4, like 2000) would be the odd one out.

It is still not a perfect rule, and never can be. The Julian error was taken to be three years every four centuries, but it is a few minutes longer than that. Those few minutes, however, can be happily ignored since they come over a period of 400 years.

What the Gregorian calendar prescribed, therefore, is the one that we still follow. A leap year comes approximately once every four years, but not exactly so. And we live in an age when we had to miss the exception to the rule; rather, 2000 was the exception to the exception to the rule.