Mathematics, history, and the leap year

The ancient Egyptians aligned temples and pyramids with celestial events, solstices, and equinoxes. The Mayans developed a complex calendar aligned with the movements of the sun, moon, and planets. Stonehenge is also believed to be influenced by solar alignments and other astronomical events, as were the rituals of native Native American, ancient Chinese, Greek, and Aboriginal Australian cultures.

We view those practices as primitive, yet the calendar that governs the rhythms of our contemporary world is driven by many of the same observations. We still define a day as a single rotation of the Earth and a year as the time it takes Earth to orbit the sun. Our individual lives, structured within that framework, are defined by working weekdays, leisurely weekends, monthly bills, quarterly budgets, and seasonal holidays, yet we seldom have time to stop and think about the detailed mathematics behind our calendar or its rich history.

The architecture of our calendar begins with the many ways to measure a year. A sidereal year is the time it takes for the Earth to complete one orbit around the Sun relative to the fixed stars. It is approximately 365.25636 days. A tropical year is the time it takes for the Sun to return to the same position in the sky as observed from Earth, taking into account the tilt of the Earth’s axis. It is approximately 365.24219 days. An anomalistic year is the time it takes for the Earth to return to its closest approach to the Sun (perihelion), approximately 365.25964 days.

Those measurements are used for a variety of astronomical applications, but the roots of our calendar are based upon the Julian year, introduced in 46 BC by no less than Julius Caesar and calculated as 365.25 days. We don’t have much use for quarter days and prefer to have our days sync with the rising and setting of the sun (except when we tinker with it for daylight savings time), so the Julian Calendar tackled the .25-day problem with a“bissextile” year, more commonly known as “Leap Year.”

2024 will be a leap year. Since our last leap year, 2020, we will have accumulated four of those extra quarter days, so in February, we’ll add an extra day to the month. When we add the 29th day, that will make everything even again, right?

Well, no. It’s not that simple. 365.25 days was close enough for 46 BC, but in 1582, Pope Gregory XIII introduced what is now known as the Gregorian calendar based on the aforementioned tropical year, measured at 365.24219 days. His calendar further refined leap year calculations to reflect the cumulative rounding error, so now, If the year is divisible by 100 but not by 400, then it is not a leap year (we would have noticed that in 2000, but we were too busy with Y2K software updates).

For instance, the years 2000 and 2400 are leap years because they are divisible by 400, while 1800, 1900, 2100, 2200, 2300, and 2500 are not leap years, despite being divisible by four. All of that effort results in keeping our calendar accurate to within one day every 3236 years, which is close enough for government work.

It is not close enough for watchmakers, which led to the introduction of the “perpetual” calendar, and legally, March 1 is recognized as a birthday for people born on a leap day, allowing them to obtain a driver’s license, register to vote, buy liquor, etc. Otherwise, people born on that day face a lifetime of inescapable jokes about either being younger than they appear (by counting actual “birthdays”) or, if darker humor is preferred, that they are destined to die in their twenties.

Leap year marks other events, like the summer Olympic Games, Presidential Elections (here and in Nigeria), major conferences, and family reunions. The events are often scheduled every four years for other reasons (as directed by our Constitution, for instance), so occurrences on leap years are often more coincidental than systematic but always easier to remember.

Those details pale in comparison to the mathematic contortions and history associated with our ancient understanding of the cosmos. Leap year reminds us that our busy, hectic, self-important lives are subject to the laws of an unimaginably large universe, and an unfathomable span of time. Too often, we allow ourselves to believe that it works the other way around. Have a safe and healthy new year.