A leap year (or intercalary year) is a year containing one or more extra days (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 in a leap year instead of the usual 28. Seasons and astronomical events do not repeat at an exact number of full 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.
Gregorian calendar
In the Gregorian calendar, the current standard calendar in most of the world, most years whose division by 4 equals an integer are leap years. In a leap year, the month of February has 29 days instead of 28. Adding an extra day to the calendar every four years compensates for the fact that a solar year is almost 6 hours longer than 365 days.
However, some exceptions to this rule are required since the duration of a solar year is slightly less than 365.25 days. Years which are evenly divisible by 100 are not leap years, unless they are also evenly divisible by 400, in which case they are leap years. For example, 1600 and 2000 were leap years, but 1700, 1800 and 1900 were not. Going forward, 2100, 2200, 2300, 2500, 2600, 2700, 2900, and 3000 will not be leap years, but 2400 and 2800 will be. By this rule, the average number of days per year will be 365 + 1/4 − 1/100 + 1/400 = 365.2425, which is 365 days, 5 hours, 49 minutes, and 12 seconds.
The Gregorian calendar was 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 about 365.242374 days long (and increasing), whereas the average year length of the Gregorian calendar is 365.2425.
The marginal difference of 0.000125 days means that in around 8,000 years, the calendar will be about one day behind where it is now. But in 8,000 years, the length of the vernal equinox year will have changed by an amount which cannot be accurately predicted (see below). Therefore, the current Gregorian calendar suffices for practical purposes, and Herschel's correction (making 4000 AD not a leap year) will probably not be necessary.
Algorithm
This algorithm determines leap years on the proleptic Gregorian calendar, which includes leap years before the official inception in 1582.
Pseudocode to determine whether a year is a leap year or not:
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
A more direct algorithm:
if (year modulo 4 is 0) and ((year modulo 100 is not 0) or (year modulo 400 is 0))
then leap
else no_leap
Leap day
February 29 is a date that occurs only every four years, and is called leap day. This day is added to the calendar in leap years as a corrective measure, because the earth does not orbit around the sun in precisely 365.000 days.
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).
The Republican calendar's intercalary month was inserted on the first or second day after the Terminalia (a. d. VII Kal. Mar., February 23). The remaining days of Februarius were dropped. This intercalary month, named Intercalaris or Mercedonius, contained 27 days. The religious festivals that were normally celebrated in the last five days of February were moved to the last five days of Intercalaris. 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.
The Julian calendar, which was developed in 46 BC by Julius Caesar, and became effective in 45 BC, distributed an extra ten days among the months of the Roman Republican calendar. Caesar also 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. For legal purposes, the two days of the bis sextum were considered to be a single day, with the second half being intercalated, but common practice by 238, when Censorinus wrote, was 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), i.e. the intercalated day was 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.
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 is 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 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 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.
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 then it is simply called "leap second month" (traditional Chinese; simplified Chinese; pinyin: rùn'èryuè).
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.
Years consisting of 12 months have between 353 and 355 days. In a k'sidra ("in order") 354-day year, months have alternating 30 and 29 day lengths. In a chaser ("lacking") year, the month of Kislev is reduced to 29 days. In a malei ("filled") year, the month of Cheshvan is increased to 30 days. 13-month years follow the same pattern, with the addition of the 30-day Adar Alef, giving them between 383 and 385 days.
Islamic calendar
In the Islamic calendar, leap months are not used. The Qur'an says:
“ 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) ”
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.
The Thai solar calendar uses the Buddhist Era (BE), but has been synchronized with the Gregorian since AD 1941.
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. Adhika Maas typically occurs once every 3 years, or 4 times over 11 years. So the yearly lag of a lunar year (which tends to have ten fewer days (355-356 days) per year than solar calendar) is adjusted every 3 years. Thus, Hindu festivals tend to occur within a given span. For example: the No Moon during Diwali festival tends to occur between October 22 and November 15.
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.
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 evenly divisible by 8,000.
(The most common such proposal is to avoid leap years in years evenly 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.)
A system of 128-year-based leap years has been proposed[citation needed], and it can be adopted directly without any modification to current leap year calculations until the year 2048 because no year between now and 2048 is divisable by 128. This rule gives a mean year of 365 + 1/4 − 1/128 = 365.2578125 days, which is 365 days, 6 hours, 11 minutes, and 15 seconds.
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.
Folk traditions
In the English speaking a world, it is a tradition that women may propose marriage only on leap years. While it has been argued that the tradition was initiated by Saint Patrick or Brigid of Kildare in 5th century Ireland, it is dubious as the tradition has not been attested before the 19th century. Supposedly, a 1288 law by Queen Margaret of Scotland (then age five and living in Norway), required that fines be levied if a marriage proposal was refused by the man; compensation ranged from a kiss to £1 to a silk gown, in order to soften the blow. Because men felt that put them at too great a risk, the tradition was in some places tightened to restricting female proposals to the modern leap day, 29 February, or to the medieval leap day, 24 February. According to Felten: "A play from the turn of the 17th century, 'The Maydes Metamorphosis,' has it that 'this is leape year/women wear breeches.' A few hundred years later, breeches wouldn't do at all: Women looking to take advantage of their opportunity to pitch woo were expected to wear a scarlet petticoat -- fair warning, if you will.".
In Greece, it is believed that getting married in a leap year is bad luck for the couple. Thus, mainly in the middle of the past century, couples avoided setting a marriage date in a leap year.
A person with a birthday on the leap day 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. ”
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 only their leap-year birthdays. A similar device is used in the plot of the Gilbert and Sullivan operetta The Pirates of Penzance.
year synchronised with the astronomical or seasonal year. For example, February would have 29 days in a leap year instead of the usual 28. Seasons and astronomical events do not repeat at an exact number of full 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.
equinox. The vernal equinox year is about 365.242374 days long (and increasing), whereas the average year length of the Gregorian calendar is 365.2425.
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).
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. ” 
(Toronto, Ontario) The ownership of three breakaway Ontario Anglican churches will be the subject of a courtroom battle Friday after the collapse of negotiations aimed at staving off legal proceedings. 
