The Maths of February 29

January 31, 2016

The year 2016 is longer than 2015: since 2016 is a leap year, February has one day more than usual. Why does the calendar work this way?

Well, basically it is all to do with the rotation of the Earth around the Sun (or, for pre-Copernicans, the rotation of the Sun around the Earth). We have 365 days in an ordinary year, but the period of the Earth’s orbit is close to 365¼ days.

This means that, if there were no leap years, every four years the seasons would be displaced by roughly one day, and so spring would come later and later in the calendar as the years passed.

To deal with this problem, the Roman leader Julius Caesar (advised by the mathematician Sosigenes) reformed the calendar in 46BC, so that, after every three years with 365 days, there was a fourth year with 366. At the time, calendars with 365-day years were in use in Persia and Egypt, and these calendars showed the seasonal slipping effects due to the discrepancy between the calendar and solar years.

The Romans had previously had a more complicated system which might have avoided the slippage: they alternated years of 355 days with years containing an extra, intercalary, month of either 22 or 23 days This system potentially could have kept the calendar in step with the seasons, but the interpolations were not always regular, because they were determined by priests who sometimes had political reasons for shortening or lengthening the year.

Caesar’s innovation resulted in a calendar — the Julian calendar — in which every fourth year was a leap year with an extra day. This was reasonably consistent with the apparent motion of the sun in the short term. But the solar year — the time between successive spring equinoxes — is actually slightly shorter than 365¼ days, so over the centuries the seasons measured by the calendar would start slightly earlier than in reality. By the sixteenth century, the discrepancy, which by now amounted to ten days, was clear.

This had practical implications: some parts of the church used the spring equinox (determined by the Sun) to calculate the date of Easter, while the Church of Rome used March 25 (given by the calendar) in its calculation, with the result that the calendar drift meant that not all Christians were celebrating Easter on the same day. To address this problem, Pope Gregory XIII in 1582 introduced a calendar reform: instead of every fourth year being a leap year, century years would not be leap years unless the year number was divisible by 400.

This reduced the mean calendar year from 365.25 days to 365.2425 days, a 0.002% correction which brought the calendar year closer to the solar year.

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