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A leap year (or intercalary year) is a year containing an extra day (or, in case of lunisolar calendars, an extra month) in order to keep the calendar year synchronized with the astronomical or seasonal year. For example, February would have 29 days instead of just 28. Seasons and astronomical events do not repeat at an exact number of days, so a calendar which had the same number of days in each year would over time drift with respect to the event it was supposed to track. By occasionally inserting (or intercalating) an additional day or month into the year, the drift can be corrected. A year which is not a leap year is called a common year.
Leap years (which keep the calendar in sync with the year) should not be confused with leap seconds (which keep clock time in sync with the day).
The Gregorian calendar, the current standard calendar in most of the world, adds a 29th day to February in all years evenly divisible by 4, except for centennial years (those ending in -00), which receive the extra day only if they are evenly divisible by 400. Thus 2000 was a leap year but 1700, 1800, and 1900 were not.
The reasoning behind this rule is as follows:
The Gregorian calendar is designed to keep the vernal equinox on or close to March 21, so that the date of Easter (celebrated on the Sunday after the 14th day of the Moon that falls on or after 21 March) remains correct with respect to the vernal equinox.
- The vernal equinox year is currently about 365.242375 days long.
- The Gregorian leap year rule gives an average year length of 365.2425 days.
This difference of a little over 0.0001 days means that in around 8,000 years, the calendar will be about one day behind where it should be. But in 8,000 years, the length of the vernal equinox year will have changed by an amount which can not be accurately predicted (see below). Therefore, the current Gregorian calendar suffices for practical purposes.
Leap year rules
In order to get a closer approximation, it was decided to have a leap day 97 years out of 400 rather than once every 4 years. To implement the model, it was provided that years divisible by 100 would be leap years only if they were divisible by 400 as well.   So, 1600, 2000, 2400 and 2800 are leap years, but 1700, 1800, 1900, 2100, 2200, 2300, 2500, 2700, 2900, 3000 are common years. The years that are divisible by 100 but not 400 are known as "exceptional common years". By this rule, the average number of days per year will be 365 + 1/4 – 1/100 + 1/400 = 365.2425.
Leap Year Algorithms
- if year mod 400 eq 0 then leap
- else if year mod 100 eq 0 then common
- else if year mod 4 eq 0 then leap
- else common
- mask400 = year mod 400 EQ 0 ; this is a leap year
- mask100 = year mod 100 EQ 0 ; these are non-leap years
- mask4 = year mod 4 EQ 0 ; this is a leap year
- return mask4 and (~mask100 or mask400)
where ~ is the NOT operator
The Gregorian calendar is a modification of the Julian calendar first used by the Romans. The Roman calendar originated as a lunisolar calendar and named many of its days after the syzygies of the moon: the new moon (Kalendae or calends, hence "calendar") and the full moon (Idus or ides). The Nonae or nones was not the first quarter moon but was exactly one nundinae or Roman market week of nine days before the ides, inclusively counting the ides as the first of those nine days. In 1825, Ideler believed that the lunisolar calendar was abandoned about 450 BC by the decemvirs, who implemented the Roman Republican calendar, used until 46 BC. The days of these calendars were counted down (inclusively) to the next named day, so 24 February was ante diem sextum Kalendae Martii ("the sixth day before the calends of March") often abbreviated a. d. VI Kal. Mar. The Romans counted days inclusively in their calendars, so this was actually the fifth day before March 1 when counted in the modern exclusive manner (not including the starting day).
The Republican calendar's intercalary month was inserted immediately after Terminalia (a. d. VII Kal. Mar., February 23) or immediately after Regifugium (a. d. VI Kal. Mar., February 24). This intercalary month, named Intercalaris or Mercedonius, contained 27 days, 22 additional days to which the last five days of February were added. Because only 22 or 23 days were effectively added, not a full lunation, the calends and ides of the Roman Republican calendar were no longer associated with the new moon and full moon.
When Julius Caesar developed the Julian calendar in 46 BC, becoming effective in 45 BC, in addition to distributing an extra ten days among the months of the Roman Republican calendar he replaced the intercalary month by a single intercalary day, located where the intercalary month used to be. To create the intercalary day, the existing ante diem sextum Kalendae Martii (February 24) was doubled, hence the year containing the doubled day was a bissextile (twice sixth) year. Which of the two days was the intercalary day and which was the ordinary day is moot. Apparently the second half was originally regarded as the intercalary day, but in 238 Censorinus stated that the intercalary day was followed by the last five days of February, a. d. VI, V, IV, III and pridie Kal. Mar. (which would be those days numbered 24, 25, 26, 27, and 28 from the beginning of February in a common year), hence he regarded the bissextum as the first half of the doubled day. All later writers, including Macrobius about 430, Bede in 725, and other medieval computists (calculators of Easter), continued to state that the bissextum (bissextile day) occurred before the last five days of February.
Until quite recently, the Roman Catholic Church always celebrated the feast of Saint Matthias on a. d. VI Kal. Mar., so if the days were numbered from the beginning of the month, it was named February 24 in common years, but the presence of the bissextum in a bissextile year immediately before a. d. VI Kal. Mar. shifted the latter day to February 25 in leap years.
This historical nicety is, however, in the process of being discarded: the European Union declared that, starting in 2000, 29 February rather than 24 February would be leap day, and the Roman Catholic Church also now uses 29 February as leap day. The only tangible difference is felt in countries that celebrate feast days, but in the general Catholic calendar, all the last days of February, starting on the 24 February, are feria, meaning that the honored saints depend on the national rites. Today, most Catholic countries observe feast days in February on fixed days of the Gregorian calendar, with the exception of some oriental Catholic churches, who still celebrate the same saints as those celebrated by the neighboring Orthodox churches.
Julian, Coptic and Ethiopian Calendars
The Julian calendar adds an extra day to February in years evenly divisible by 4.
This rule gives an average year length of 365.25 days. However, it was 11 minutes longer than a real year. This means that the vernal equinox moves a day earlier in the calendar every 131 years.
Revised Julian Calendar
The Revised Julian calendar adds an extra day to February in years divisible by 4, except for years divisible by 100 that do not leave a remainder of 200 or 600 when divided by 900. This rule agrees with the rule for the Gregorian calendar until 2799. The first year that dates in the Revised Julian calendar will not agree with the those in the Gregorian calendar will be 2800, because it will be a leap year in the Gregorian calendar but not in the Revised Julian calendar.
This rule gives an average year length of 365.242222… days. This is a very good approximation to the mean tropical year, but because the vernal equinox tropical year is slightly longer, the Revised Julian calendar does not do as good a job as the Gregorian calendar of keeping the vernal equinox on or close to 21 March.
The Chinese calendar is lunisolar, so a leap year has an extra month, often called an embolismic month after the Greek word for it. In the Chinese calendar the leap month is added according to a complicated rule, which ensures that month 11 is always the month that contains the northern winter solstice. The intercalary month takes the same number as the preceding month; for example, if it follows the second month (二月) then it is simply called "leap second month".
The Hebrew calendar is also lunisolar with an embolismic month. In the Hebrew calendar the extra month is called Adar Alef (first Adar) and is added before Adar, which then becomes Adar bet (second Adar). According to the Metonic cycle, this is done seven times every nineteen years, specifically, in years, 3, 6, 8, 11, 14, 17, and 19.
In addition, the Hebrew calendar has postponement rules that postpone the start of the year by one or two days. These postponement rules reduce the number of different combinations of year length and starting day of the week from 28 to 14, and regulate the location of certain religious holidays in relation to the Sabbath. In particular, the first day of the Hebrew year can never be Sunday, Wednesday or Friday. This rule is known in Hebrew as "lo adu rosh", i.e. "Rosh [ha-Shanah] is not Sunday, Wednesday or Friday" (as the Hebrew word adu is written by three Hebrew letters signifying Sunday, Wednesday and Friday). Accordingly, the first day of Pesah is never Monday, Wednesday or Friday. This rule is known in Hebrew as "lo badu Pesah", which has a double meaning - "Pesah is not a legend", but also "Pesah is not Monday, Wednesday or Friday" (as the Hebrew word badu is written by three Hebrew letters signifying Monday, Wednesday and Friday).
One reason for this rule is that Yom Kippur, the holiest day in the Hebrew calendar, must never be adjacent to the weekly Sabbath (which is Saturday), i.e. it must never fall on Friday or Sunday, in order not to have two adjacent Sabbath days (Yom Kippur can be on Saturday, however).
Calendars with Leap Years synchronized with Gregorian
The Indian National Calendar and the Revised Bangla Calendar of Bangladesh organise their leap years so that the leap day is always close to February 29 in the Gregorian calendar. This makes it easy to convert dates to or from Gregorian.
The Bahá'í calendar is structured such that the leap day always falls within Ayyám-i-Há, a period of four or five days corresponding to Gregorian February 26 - March 1. Because of this, Baha'i dates consistently line up with exactly one Gegorian date.
In the Hindu calendar, which is a lunisolar calendar, the embolismic month is called adhika maas (extra month). It is the month in which the sun is in the same sign of the stellar zodiac on two consecutive dark moons.
The Iranian calendar also has a single intercalated day once in every four years, but every 33 years or so the leap years will be five years apart instead of four years apart. The system used is more accurate and more complicated, and is based on the time of the March equinox as observed from Tehran. The 33-year period is not completely regular; every so often the 33-year cycle will be broken by a cycle of 29 or 37 years.
Long term leap year rules
The accumulated difference between the Gregorian calendar and the vernal equinoctial year amounts to 1 day in about 8,000 years. This suggests that the calendar needs to be improved by another refinement to the leap year rule, e.g. by cancelling leap years in years divisible by 8,000.
The most common such proposal is to cancel leap years in years divisible by 4,000 ; this is based on the difference between the Gregorian calendar and the mean tropical year. Others claim (erroneously) that the Gregorian calendar itself already contains a refinement of this kind .)
Hypothetical 128-year based leap cycles have been proposed, and it can be adopted directly without any modification to current leap year calculations until the year 2048.
However, there is little point in planning a calendar so far ahead because over a timescale of tens of thousands of years, the number of days in a year will change for a number of reasons, most notably:
- Precession of the equinoxes moves the position of the vernal equinox with respect to perihelion and so changes the length of the vernal equinoctial year.
- Tidal acceleration from the sun and moon slows the rotation of the earth, making the day longer.
In particular, the second component of change depends on such things as post-glacial rebound and sea level rise due to climate change. We can't predict these changes accurately enough to be able to make a calendar that will be accurate to a day in tens of thousands of years.