Tricky Math, and History, Lead to Leap Year

There will never be another day like today — at least not until 2016. Today's date (Feb. 29) pops up on the calendar only on leap years, once almost every four years.

It has taken millennia for our calendar, called the Gregorian calendar after the pope who modified it in 1582, to evolve to include this tweak — 97 leap years every 400 years. There are other alternatives, according to Yury Grabovsky, an associate professor in the department of mathematics at Temple University, who has studied the history and mathematics of the Gregorian calendar.

Leap year addresses a discrepancy between our 365-day calendar and the time it takes the Earth to circle the sun. The discrepancy: It takes 365.24219878 days for the planet to complete one trip, so over time, our calendar and the astronomical one fall out of step.

The oldest Babylonian calendar consisted of 12 months, based on the cycles of the moon. At 354 days per year, it fell a fair number short of the astronomical year. The ancient Egyptians brought this calendar up to 360 days, and later, added five days at the end of each year, according to Grabovsky.   

More than 3,000 years later, in 238 B.C., the Egyptian king Ptolemy III introduced a sixth day at the end of every fourth year — the original leap year. Julius Caesar learned of this change and included it when he reformed the existing Roman calendar, added the leap day to February in his new Julian calendar.

But adding an extra day every four years didn't fully solve the problem. The remaining discrepancy was small, showing up only as the calendar and the seasons gradually fell out of sync.

"The calendar was making a one-day error in 100 years," Grabovsky said.

Over the centuries this became apparent. [Is It Time to Overhaul the Calendar?]

"I think the records show it became evident around the 10th century, about 1,000 years after Julius Caesar's time. The 10-days' error would have accumulated and astronomical observations even at the time were able to find the 10-day error," he said. "But it took another more than 500 years before an official political change could be made."

In 1582, Pope Gregory XIII decreed that October would lose 10 days, so Oct. 4 would be immediately followed by Oct. 15. However, Protestants rejected this change for nearly 200 years. In 1752, Sept. 2 was followed by Sept. 14 in Great Britain and its colonies.

The pope also tweaked the leap-year cycle to address the remaining one-day-per century drift: Years ending in "00" are not leap years unless they can be divided exactly by 400, so 2000 was a leap year, while 1900 was not. (Hence the 97 leap years over 400 years.)

Grabovsky has used continued fractions — fractions in which the denominator, or bottom number, is expressed as a whole number plus a fraction with its own denominator that is a whole number plus a fraction, and so on, iteratively — to examine the ways a calendar could be constructed to handle the discrepancy between the astronomical year and the calendar year. 

The 400-year cycle Pope Gregory XIII installed accumulates an extra 26 seconds each year, resulting in an error of a full day every 3,320 years, he calculated. He compares this to a 500-year cycle, which would be 17 seconds shorter than the solar year, resulting in an error of one day each 5,031 years — an alternative the pope appears to have missed either because he didn't do his math or because astronomical measurements at the time weren't precise enough to justify this cycle, Grabovsky writes.

Another alternative, based on a 900-year cycle, is too complicated and has an inconveniently long cycle.

A tiny amount of error remains in the Gregorian calendar, and to correct for most of it, he recommends canceling leap year every 3,200 years.

"The new system would accumulate a one day error in 100,000 years, that is never," he writes as part of a presentation he gave on Feb. 29, 2000.

To add to the complication, the planet's speed along its orbit isn't constant, so occasionally, leap seconds are added to our clocks to compensate. 

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