Ivars Peterson's MathTrek

December 21, 1998

# 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 40 or so 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, Oct. 15, 1997). 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 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 has recently 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.

Thanks to the power of today's personal computers, 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, says Ilan Vardi of the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France.

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 symbol means take the integer part, x, of the decimal x.y; mod 7 means divide by 7 and retain only the remainder.

For example, for Dec. 25, 1998, D = 25, M = 10, C = 19, and Y = 98.

The answer is Friday. You would get the same result for Jan. 1, 1999.

***** Happy Holidays and Best Wishes for the New Year! **

References:

Battersby, S. 1998. Friday XIII. Nature 396(Nov. 21):113.

Dershowitz, N., and E.M. Reingold. 1997. Calendrical Calculations. Cambridge, England: Cambridge University Press.

Duncan, D.E. 1998. Calendar: Humanity's Epic Struggle to Determine a True and Accurate Year. New York: Avon.

| O'Neil, W.M. 1975. Time and the Calendars. Sydney, Australia: Sydney University Press.

Peterson, I. 1996. Fatal Defect: Chasing Killer Computer Bugs. New York: Vintage.

______. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.

Stewart, I. 1996. A guide to computer dating. Scientific American (November):116.

Vardi, I. 1991. Computational Recreations in Mathematica. Reading, Mass.: Addison-Wesley.

A brief, authoritative history of the calendar can be found at http://astro.nmsu.edu/~lhuber/leaphist.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.