Leap year
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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 synchronised with the astronomical or seasonal year. For example, February would have 29 days on a leap year instead of the usual 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. In fact, the Earth takes slightly under 365 1/4 days to revolve around the Sun.
[edit] Gregorian calendar
The Gregorian calendar, the current standard calendar in most of the world, adds a 29th day to February in all years evenly divisible by four, except for centennial years (those ending in -00), which receive the extra day only if they are evenly divisible by 400. Thus 1600, 2000 and 2400 are leap years but 1700, 1800, 1900 and 2100 are 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.[1]
- 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.
 This graph shows the variation between the seasonal year versus the calendar year due to unequally spaced 'leap days' rules. See Iranian calendar to contrast with a calendar based on 8 leap days every 33 years. |
[edit] 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 four years. This would be implemented by making a leap year every year divisible by 4 unless that year is divisible by 100. If it is divisible by 100 it would only be a leap year if that year was also divisible by 400.[2][3] So, in the last millennium, 1600 and 2000 were leap years, but 1700, 1800 and 1900 were not. In this millennium, 2100, 2200, 2300, 2500, 2600, 2700, 2900 and 3000 will not be leap years, but 2400 and 2800 will be. 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.
[edit] Leap year algorithms
Calculating leap years is simple, and is provided here by two different pseudocodes that determine whether a year is a leap year or not:
- Standard
if year modulo 400 is 0 then leap else if year modulo 100 is 0 then no_leap else if year modulo 4 is 0 then leap else no_leap
- Vectorized
mask400 ← year modulo 400 EQ 0 ; this is a leap year mask100 ← year modulo 100 EQ 0 ; these are not leap years mask4 ← year modulo 4 EQ 0 ; this is a leap year return ((mask4 and ~mask100) or mask400)
where ~ is the bitwise logical NOT operator. These algorithms are for a Proleptic Gregorian calendar, which include leap years before the official inception in 1582.
[edit] Leap day
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 Kalendas 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).[4]
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 Kalendas Martii (February 24) was doubled, producing ante diem bis sextum Kalendas Martii. Hence, the year containing the doubled day was a bissextile (bis sextum, "twice sixth") year. Originally, the first was regarded as bis sextum, 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 1970, 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, with the Vigil of St. Matthias shifting from February 23 to the leap day of February 24. Other feasts normally falling on February 25–28 in common years are also shifted to the following day in a leap year (although they would be on the same day according to the Roman notation). The practice is still observed by those who use the older calendars.
[edit] Julian, Coptic and Ethiopian calendars
The Julian calendar adds an extra day to February in years evenly divisible by four.
The Coptic calendar and Ethiopian calendar also add an extra day to the end of the year once every four years before a Julian 29-day February.
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.
[edit] Revised Julian calendar
The Revised Julian calendar adds an extra day to February in years divisible by four, 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 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.
[edit] Chinese calendar
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" (Traditional Chinese: 閏二月; Simplified Chinese: 闰二月; pinyin: rùn'èryuè).
[edit] Hebrew calendar
The Hebrew calendar is also lunisolar with an embolismic month. This 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 days 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, first day of the year] 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 (Passover) 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. However,Yom Kippur can be on Saturday.
[edit] Islamic Calendar
In the Islamic Calendar, leap months are forbidden. They are forbidden by Allah in the Qur'an saying:
| “ | The number of months with Allah has been twelve months by Allah's ordinance since the day He created the heavens and the earth. Of these four are known as sacred; That is the straight usage, so do not wrong yourselves therein, and fight those who go astray. But know that Allah is with those who restrain themselves. | ” |
| “ | Verily the transposing (of a prohibited month) is an addition to Unbelief: The Unbelievers are led to wrong thereby: for they make it lawful one year, and forbidden another year, of months forbidden by Allah and make such forbidden ones lawful. The evil of their course seems pleasing to them. But Allah guideth not those who reject Faith. (Qur'an 9:36-37) | ” |
[edit] 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 Gregorian date.
[edit] Hindu calendar
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.
[edit] Iranian calendar
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.
[edit] 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: perhaps by avoiding leap years in years divisible by 8,000.
(The most common such proposal is to avoid leap years in years divisible by 4,000.[5] 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.[6])
A system of 128-year-based leap years has 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.
[edit] Marriage proposal
There is a tradition, said to go back to Saint Patrick and Saint Bridget in 5th century Ireland, but apparently not attested before the 19th century, whereby women may make marriage proposals only in leap years.
Supposedly (but disputed), in a 1288 law by Queen Margaret of Scotland (then age five and living in Norway), fines were levied if the proposal was refused by the man; compensation ranged from a kiss to £1 to a silk gown, in order to soften the blow.[7] Because men felt that put them at too great a risk, the tradition was in some places tightened to restricting female proposals to 29 February.
Others regard these supposed folk traditions as unhistorical.[8]
[edit] Birthdays
A person born on February 29 may be called a "leapling". In common years they usually celebrate their birthdays on 28 February or 1 March.
For legal purposes, their legal birthdays depend on how different laws count time intervals. In Taiwan, for example, the legal birthday of a leapling is 28 February in common years, so a Taiwanese leapling born on February 29, 1980 would have legally reached 18 years old on February 28, 1998.
| “ | If a period fixed by weeks, months, and years does not commence from the beginning of a week, month, or year, it ends with the ending of the day which proceeds the day of the last week, month, or year which corresponds to that on which it began to commence. But if there is no corresponding day in the last month, the period ends with the ending of the last day of the last month.[9] | ” |
In some situations, March 1 is used as the birthday in a non-leap year since it then is the day just after February 28.
There are many instances in children's literature where a person's claim to be only a quarter of their actual age turns out to be based on counting their leap-year birthdays. A similar device is used in the plot of the Gilbert and Sullivan operetta The Pirates of Penzance.
[edit] References
- ^ E.G. Richards, Mapping time: The calendar and its history, p. 240, ISBN 0-19-282065-7
- ^ Royal Observatory, Greenwich: Which years are leap years?
- ^ USNO (U.S. Naval Observatory): Leap Years
- ^ Thomas Hewitt Key, Calendarium (1875)
- ^ Google search for 4000-year rule.
- ^ The Straight Dope with 4000-year rule
- ^ Virtually no laws of Margaret survive. Indeed, none concerning her subjects are recorded in the twelve volume Acts of the Parliaments of Scotland (1814–75) covering the period 1124–1707 (two laws concerning young Margaret herself are recorded on pages 424 & 441–2 of volume I).
- ^ The Privilege of Ladies by Barbara Mikkelson
- ^ Article 121 of the Civil Code Part I General Principles of the Republic of China in effect in Taiwan
