# 2. The Gregorian Calendar

The Gregorian calendar today serves as an international standard for civil use. In addition, it regulates the ceremonial cycle of the Roman Catholic and Protestant churches. In fact, its original purpose was ecclesiastical. Although a variety of other calendars are in use today, they are restricted to particular religions or cultures.

### 2.1 Rules for Civil Use

Years are counted from the initial epoch defined by Dionysius Exiguus, and are divided into two classes: common years and leap years. A common year is 365 days in length; a leap year is 366 days, with an intercalary day, designated February 29, preceding March 1. Leap years are determined according to the following rule:

these centurial 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 leap years. These rules can be applied to times prior to the Gregorian reform to create a proleptic Gregorian calendar. In this case, year 0 (1 B.C.) is considered to be exactly divisible by 4, 100, and 400; hence it is a leap year.

The Gregorian calendar is thus based on a cycle of 400 years, which comprises 146097 days. Since 146097 is evenly divisible by 7, the Gregorian civil calendar exactly repeats after 400 years. Dividing 146097 by 400 yields an average length of 365.2425 days per calendar year, which is a close approximation to the length of the tropical year. Comparison with Equation 1.1-1 reveals that the Gregorian calendar accumulates an error of one day in about 2500 years. Although various adjustments to the leap-year system have been proposed, none has been instituted.

Within each year, dates are specified according to the count of days from the beginning of the month. The order of months and number of days per month were adopted from the Julian calendar.

1. January | 31 | 7. July | 31 |

2. February | 28* | 8. August | 31 |

3. March | 31 | 9. September | 30 |

4. April | 30 | 10. October | 31 |

5. May | 31 | 11. November | 30 |

6. June | 30 | 12. December | 31 |

* In a leap year, February has 29 days.

### 2.2 Ecclesiastical Rules

The ecclesiastical calendars of Christian churches are based on cycles of movable and immovable feasts. Christmas is the principal immovable feast, with its date set at December 25. Easter is the principal movable feast, and dates of most other movable feasts are determined with respect to Easter. However, the movable feasts of the Advent and Epiphany seasons are Sundays reckoned from Christmas and the Feast of the Epiphany, respectively.

In the Gregorian calendar, the date of Easter is defined to occur on the Sunday following the ecclesiastical Full Moon that falls on or next after March 21. This should not be confused with the popular notion that Easter is the first Sunday after the first Full Moon following the vernal equinox. In the first place, the vernal equinox does not necessarily occur on March 21. In addition, the ecclesiastical Full Moon is not the astronomical Full Moon -- it is based on tables that do not take into account the full complexity of lunar motion. As a result, the date of an ecclesiastical Full Moon may differ from that of the true Full Moon. However, the Gregorian system of leap years and lunar tables does prevent progressive departure of the tabulated data from the astronomical phenomena.

The ecclesiastical Full Moon is defined as the fourteenth day of a tabular lunation, where day 1 corresponds to the ecclesiastical New Moon. The tables are based on the Metonic cycle, in which 235 mean synodic months occur in 6939.688 days. Since nineteen Gregorian years is 6939.6075 days, the dates of Moon phases in a given year will recur on nearly the same dates nineteen years laters. To prevent the 0.08 day difference between the cycles from accumulating, the tables incorporate adjustments to synchronize the system over longer periods of time. Additional complications arise because the tabular lunations are of 29 or 30 integral days. The entire system comprises a period of 5700000 years of 2081882250 days, which is equated to 70499183 lunations. After this period, the dates of Easter repeat themselves.

The following algorithm for computing the date of Easter is based on the algorithm of Oudin (1940). It is valid for any Gregorian year, Y. All variables are integers and the remainders of all divisions are dropped. The final date is given by M, the month, and D, the day of the month.

N = Y - 19*(Y/19),

K = (C - 17)/25,

I = C - C/4 - (C - K)/3 + 19*N + 15,

I = I - 30*(I/30),

I = I - (I/28)*(1 - (I/28)*(29/(I + 1))*((21 - N)/11)),

J = Y + Y/4 + I + 2 - C + C/4,

J = J - 7*(J/7),

L = I - J,

M = 3 + (L + 40)/44,

D = L + 28 - 31*(M/4).

### 2.3 History of the Gregorian Calendar

The Gregorian calendar resulted from a perceived need to reform the method of calculating dates of Easter. Under the Julian calendar the dating of Easter had become standardized, using March 21 as the date of the equinox and the Metonic cycle as the basis for calculating lunar phases. By the thirteenth century it was realized that the true equinox had regressed from March 21 (its supposed date at the time of the Council of Nicea, +325) to a date earlier in the month. As a result, Easter was drifting away from its springtime position and was losing its relation with the Jewish Passover. Over the next four centuries, scholars debated the "correct" time for celebrating Easter and the means of regulating this time calendrically. The Church made intermittent attempts to solve the Easter question, without reaching a consensus.

By the sixteenth century the equinox had shifted by ten days, and astronomical New Moons were occurring four days before ecclesiastical New Moons. At the behest of the Council of Trent, Pope Pius V introduced a new Breviary in 1568 and Missal in 1570, both of which included adjustments to the lunar tables and the leap-year system. Pope Gregory XIII, who succeeded Pope Pius in 1572, soon convened a commission to consider reform of the calendar, since he considered his predecessor's measures inadequate.

The recommendations of Pope Gregory's calendar commission were instituted by the papal bull "Inter Gravissimus," signed on 1582 February 24. Ten days were deleted from the calendar, so that 1582 October 4 was followed by 1582 October 15, thereby causing the vernal equinox of 1583 and subsequent years to occur about March 21. And a new table of New Moons and Full Moons was introduced for determining the date of Easter.

Subject to the logistical problems of communication and governance in the sixteenth century, the new calendar was promulgated through the Roman-Catholic world. Protestant states initially rejected the calendar, but gradually accepted it over the coming centuries. The Eastern Orthodox churches rejected the new calendar and continued to use the Julian calendar with traditional lunar tables for calculating Easter. Because the purpose of the Gregorian calendar was to regulate the cycle of Christian holidays, its acceptance in the non-Christian world was initially not at issue. But as international communications developed, the civil rules of the Gregorian calendar were gradually adopted around the world.

Anyone seriously interested in the Gregorian calendar should study the collection of papers resulting from a conference sponsored by the Vatican to commemorate the four-hundredth anniversary of the Gregorian Reform (Coyne et al., 1983).