There are many different calendars on earth in use today: Hebrew, Chinese, Hindu, Ethiopian etc. These various calendarical systems may seem completely artificial, a pure product of human arbitrary choice, like a language. Yet, the basis of all calendars are the observed periodic motion of the Sun and the Moon across the skies. Here’s the numbers. One year is 365.24219878 days and one lunar month is 29.530589 days long. The ratio of these two numbers is 12.368267 and it it is equal to the number of lunations in a year. So, you see that the lunar month is between 29 and 30 days long, while there is about 12 lunar months a year. The oldest Babylonian calendar was a lunar one of 12 months consisting alternately of 29 and 30 days in accordance with these numbers. Observe that the calendarical year in such lunar calendar contains just 354 days, a gross underestimate. But by at least the 5th millennium BC this calendar was replaced by an Egyptian calendar of 12 months, each consisting of 30 days.
The “New” calendar had only 360 days in a year and the discrepancy was soon noticed. To adjust the calendar, five days, the epagomenes, were added at the end of the 360-day year in Pharaonic times. This 365 day calendar was in effect for more than 3000 years of Pharaohs until 238 B.C.
In a remarkable Decree of Canopus by Ptolemy III, a sixth epagomenal day was introduced every fourth year. This is so called Alexandrian calendar. It survives nowadays in the calendars of Coptic and Ethiopian churches.
Our calendar is a direct descendant of the ancient Roman calendar
Up until 46 B.C. Romans used a 365 day year. During his Egyptian campaign Julius Caesar learned about the Alexandrian calendar with its 4-year leap year cycle, that was much more precise than the current Roman calendar of 365 days. Along with him Caesar brought the Alexandrian astronomer Sosigenes, upon whose advice he based his calendar reform, creating the Julian calendar. The mean year length for Julian calendar is 365.25 days, which is very close to the more precise number 365.24219878.
The Julian calendar was so good that it accumulated only one day error in about a hundred years. Yet, over the next millennium, the discrepancy was noticed and suggestions were made to correct it. Finally, in 1582 Pope Gregory XIII assembled a commission to design a new more precise calendar system. The main author of the new system was the Naples astronomer Aloysius Lilius. Following the recommendation of his commission, Pope Gregory XIII decreed that the day following Oct. 4, 1582 would be Oct. 15; that the years ending in “00” would be common years rather than leap years – except those divisible by 400 and that New Year will start on January 1. The non-Catholic world perceived the calendar decree as a Catholic ploy. It took nearly 200 years for the change to come about. Great Britain and her colonies made the change in
1752 when September 2nd was followed by September 14 and New Year’s Day was changed from March 25 to January 1.
If your computer has a calendar program that can display calendars for 1582 on, you can check what your computer thinks the calendar should look like. For example, on Unix/Linux systems the results are
| October 1582 | September 1752 |
| Su Mo Tu We Th Fr Sa | Su Mo Tu We Th Fr Sa |
| 1 2 3 4 5 6 | 1 2 14 15 16 |
| 7 8 9 10 11 12 13 | 17 18 19 20 21 22 23 |
| 14 15 16 17 18 19 20 | 24 25 26 27 28 29 30 |
| 21 22 23 24 25 26 27 | |
| 28 29 30 31 |
The year 2000 is one of the rare leap years that end in “00”. The next time this happens will be 400 years from now.
The Gregorian calendar is both precise (1 day error in about 3,300 years) and convenient. Is it an art to come up with such a design or is there a science behind it? Continued fractions provide just such a science.
Next: Continued fractions
Yury Grabovsky 2000-03-03