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Calendar Calculations

The tropical year is the period of time required by the sun to pass from vernal equinox to vernal equinox. It is equal to 365 days, 5 hours, 48 minutes, and 46 seconds, or 365.2422 days. The tropical year is used to keep track of seasons, planting, and harvesting. Let's try to develop a calendar with an integral number of days per calendar year that will keep track of the tropical year and not get out of step with the seasons over time.

We begin with a calendar of 365 days per year. Our calendar year is shorter than the tropical year by 0.2422 days. So to correct (approximately), we add 1 day every four years (leap year). Thus, three calendar years are 365 days long; the fourth calendar year is 366 days long. The average length of the calendar year in days now becomes: (3 x 365 + 366)/4 = 365.25 days.

This calendar system was actually instituted for use in the Roman Empire by Julius Caesar around 46 BC. But since the Julian calendar was 0.0078 days (11 minutes and 14 seconds) longer than the tropical year, errors in timekeeping gradually accumulated. Between 46 BC and 1582 AD, this accumulated error amounted to a total of: 0.0078 x (1582 + 46) = 12.7 days. In 1582, Pope Gregory XIII reformed the calendar by specifying that all years divisible by 4 are to be leap years except for century years, which must be divisible by 400 to be leap years. Now, in 1200 years:

1. A total of 300 years (including all century years {since any century year = N x 100, where N = an integer}) are divisible by 4, and are therefore candidate leap years.
2. A total of 900 years are not divisible by 4, and are therefore regular years.
3. Twelve century years are possible leap years.
4. But only 3 century years (out of the 12) are divisible by 400 (i.e., {400, 800, 1200}, {1600, 2000, 2400}, etc.), so only 3 century years are actually leap years⁹ .

Since 12 - 3 = 9, Gregory's rule eliminates 9 leap years out of 1,200. Thus: 300 - 9 = 291 years are actual leap years, and 900 + 9 = 909 years are regular years. The average length of the year becomes (291 x 366 + 909 x 365)/1,200 = 365.2425 days, with an error of 365.2425 - 365.2422 = 0.0003 days per year, or one day every 3,333.3 years.

The Gregorian calendar came into use in Roman Catholic countries in October 1582 when the seasons were brought back into step by eliminating 10 days from the calendar then in use. Thursday, October 4, was followed by Friday, October 15 (which caused some consternation among the populace, especially those with birthdays on the eliminated dates!). Britain and its colonies did not introduce the Gregorian calendar until September 1752 by which time an additional one day correction was required (actually, {1752 - 1582} x 0.0078 = 1.33 day). Some British documents from the period before the British reform actually contain two dates, an old and a new.

In each century, one out of every four years is divisible by 4. Of the century years, only 400, 800, and 1200 are divisible by 400, leaving 100, 200, 300, 500, 600, 700, 900, 1000, and 1100 that are not.

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