Calendar Quirks

Calendars represent our efforts to create frameworks that allow us to reckon time over extended periods.

We normally count the day—the time it takes Earth to rotate once on its axis—as the smallest unit of calendrical time. The measurement of fractions of a day fits, by convention, into the category of timekeeping.

Of the dozens of calendars presently used in the world, the most common ones group days into weeks, months, and years. They follow two astronomical cycles in addition to the day: the year (based on the revolution of Earth around the sun) and the month (based on the revolution of the moon around Earth).

The trouble is that the cycles of revolution do not comprise an integral number of days (see "Fractions, Cycles, and Time"). Those quirks of the solar system add a maddening complexity to any calendar based on astronomical cycles—especially the 365 days, 5 hours, 48 minutes, and 46 seconds of the solar (tropical) year.

Moreover, because of the precession of Earth's axis, a solar year is actually defined not by the time Earth takes to make one revolution about the sun but as the average time between two vernal equinoxes. A vernal equinox represents the instant at which the sun lies exactly between the north and south celestial poles.

How best to handle that fraction of a day has long spurred calendar reforms. In 46 B.C., Julius Caesar instituted a calendar that made a year 365 days long, with every fourth year having an additional day.

However, because a year actually runs 365.2421896698 instead of 365.25 days, there was a slight discrepancy. Over ensuing centuries, the Julian calendar gradually went out of step with the seasons, muddling the timing of various festivals and religious observances.

In 1582, Pope Gregory XIII introduced a new calendar to replace the old Julian system. It was based on a 400-year cycle. Leap years followed the rule: Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100; and those years are leap years only if they are exactly divisible by 400. As a result, the year 2000 is a leap year, whereas 1900 and 2100 are not.

The Gregorian cycle of 400 years contains exactly 20,871 weeks. Hidden within the machinery is a bias toward certain days of the week landing on certain days of the month. For example, the 13th is more likely to be a Friday than any other day.

Indeed, Bernard D. Yallop of Her Majesty's Nautical Almanac Office at the Royal Greenwich Observatory a while ago determined that there are 688 Friday-the-thirteenths every 400 years, but only 684 Thursdays. That also means a month is most likely to begin on a Sunday.

You can find additional quirks by creating a giant table representing the Gregorian cycle, sorting the data in various ways, and compiling the results.

It's also relatively easy to write a computer program that tells you what day of the week a particular date is, how many days there are between two given days, and so on. Indeed, the calendar can be regarded as a positional number system, Ilan Vardi has remarked.

Here's a formula for computing the day of the week, W, for a given day, D, of the month, M, and the year 100C + Y:

where months are numbered beginning with March = 1. Dates in January and February are considered to be in the 11th and 12th months of the previous year. Days of the week are numbered W = 0 for Sunday, W = 1 for Monday, and so on. The symbolmeans take the integer part, x, of the decimal x.y; mod 7 means divide by 7 and retain only the remainder.