Calendars

Summary: Various calendars use different methods of resolving the need for “leap” days, months, or years.

Even the earliest human beings must have noticed the astronomical cycles: the alternation of day and night, the pattern of the changes in the moon’s shape and position, and the cycle of the seasons through the solar year. It must have been frightening every autumn as the days became shorter, causing concern that the night might become permanent. This led to celebrations of light in many areas as the days began to lengthen again. Once the repetitions of the patterns were recognized, people could count them to keep track of time. Longer cycles helped avoid difficulties in keeping track of large numbers—once approximately 30 days had been counted, people could, instead, start counting “moons.” This same technique of grouping also occurred in the development of counting systems in general—leading to place-value structures in numeration systems.

The problem was that the shorter cycles did not fit evenly into the longer cycles. Trying to fit the awkward-length cycles together actually led to some mathematical developments: two different cycles would come together at the least common multiple of the lengths of their cycles; modular arithmetic and linear congruences were methods of handling leftover periods beyond the regular cycle periods.

The Julian and Gregorian Calendars

The Romans developed the Julian calendar (named for Julius Caesar), recognizing that the exact number of 365 days in one year was slightly too short and would soon throw the calendar off the actual cycle of the solar year. They found a remedy by assuming the solar year to be 365.25 days. To handle the one-quarter day, they added one full day every four years—the day that we call “leap-year day” on February 29 of years whose number is a multiple of four. This gives 3(365) + 366 = 1461 days in four years, or an average of 365.25 days per year as desired. However, the actual solar year is 365.2422 days long (to four decimal places), about 11 minutes less than the Romans’ value. Even in a human lifetime, this is negligible. Over centuries, however, the extra time builds up so that by the 1500s, the calendar was 10 days off from the solar cycle (for example, the vernal equinox seemed to be coming too late).

In 1582, Pope Gregory XIII assembled a group of scholars who devised a new system to fit better. It kept the Roman pattern except that century years (1600, 1700), which should have been leap years in the Roman calendar, would not have a February 29 unless they were multiples of 400. For example, 1900 was not a leap, year but 2000 was. In the full 400-year cycle, there are (400×365) regular days + 97 leap-year days = 146,097 days, making an average of 365.2425 days per year. This cycle is only .0003 days (about 26 seconds) too much; in 10,000 years, we would gain three extra days. This system was called the Gregorian calendar. Since the longer Julian calendar had fallen behind the solar year by about 10 days, the changeover to the Gregorian required jumping 10 days.

Various countries in Europe changed at different times, with each switch causing local controversy as people felt they were being “cheated” out of the skipped days. The effects of the change are noticed in history. When Isaac Newton was born, the calendar said it was December 25, 1642; but later England changed the calendar, so some historians today give Newton’s birthday as January 4, 1643. The Russians did not change their calendar until after the 1917 October Revolution, which happened in November by the Gregorian calendar.

The Lunar Calendar

The other incongruity of calendar systems is that the moon cycle of 29.53 days does not fit neatly in the 365.2422 days of the year. Twelve moon periods is 11 days shorter than a year, and 13 “moons” is 18 days too long. It is interesting to note that of the three major religious groups of the Middle East—the Christians, the Muslims, and the Jews—each chose a different way to handle “moons/months.” The Christians (actually, originally, the Romans) ignored the moon cycle and simply created months of 30 and 31 (and 28 or 29) days. The Muslims considered their year to be 12 moon cycles and ignored the solar year. This means that dates in the Muslim calendar are shifted back approximately 11 days each year from the solar calendar, and Muslim festivals move backward through the seasons.

People in the Jewish faith chose to keep both the solar and lunar cycles. After 12 lunar months, a new year begins—as in the Muslim calendar—11 days “too early.” However, after the calendar slips for two or three years—falling behind the solar calendar by 22 or 33 days—an extra month is inserted to compensate for the loss. There is a 19-year pattern of the insertion of extra months, which keeps the year aligned with the solar year. Interestingly, the traditional east Asian calendar follows a pattern very similar to the Jewish calendar.

The Mayan Calendar

The Mayans of Central America had a very complex pattern of cycles leading to a 260-day year for religious purposes, and a regular solar year that was used for farming and other climate-related activities. Their base-20 numeration system, which should have had place-value columns of 1-20-400-8000, was adjusted to 1-20-360 to fit into the 365+ days of the year. They were also notable for developing massive cycles of years lasting several millennia, including one ending in late 2012 of the Gregorian calendar.

Bibliography

Aslaksen, Helmer. “The Mathematics of the Chinese Calendar.” http://www.math.nus.edu.sg/aslaksen/calendar/chinese.shtml.

Crescent Moon Visibility and the Islamic Calendar. http://aa.usno.navy.mil/faq/docs/islamic.php.

Duncan, David Ewing. Calendar: Humanity’s Epic Struggle to Determine a True and Accurate Year. New York: Harper Perennial, 2001.

Rich, Tracey R. “Judaism 101: Jewish Calendar.” http://www.jewfaq.org/calendar.htm.

Richards, E. G. Mapping Time: The Calendar and Its History. New York: Oxford University Press, 2000.

Stray, Geoff. The Mayan and Other Ancient Calendars. New York: Walker & Company, 2007.