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Leap year

A leap year is a calendar year with an additional day added to align the civil calendar more closely with the Earth's revolution around the Sun, which takes approximately 365.2422 days, known as the tropical year.^[1] In the predominant Gregorian calendar, a year is designated as a leap year if its number is evenly divisible by 4, except for century years (divisible by 100), which are leap years only if also divisible by 400; this rule results in an average year length of 365.2425 days over 400 years.^[2] The extra day, February 29, is inserted after February 28, extending the year to 366 days and preventing seasonal drift over time.^[2] Introduced in 1582 by Pope Gregory XIII through the papal bull Inter gravissimas to reform the earlier Julian calendar, the Gregorian system corrected accumulated errors that had shifted the vernal equinox by about 10 days and ensured more accurate timing for religious observances like Easter.^[2] Adoption varied by region—Catholic countries implemented it immediately, while Protestant and Orthodox nations followed centuries later, with Britain and its colonies switching in 1752 and Russia in 1918—but it is now the international standard for civil purposes.^[2] Leap years occur roughly every four years, with exceptions like 1900 and 2100 (not leap years) versus 2000 (a leap year), balancing the calendar's precision against simplicity.^[2]

General Concept

Definition and Purpose

A leap year is a calendar year that includes an additional day (or days) to better align the calendar with the tropical year, which measures approximately 365.2422 days.^[3] In solar calendars, this adjustment distinguishes a common year of 365 days from a leap year of 366 days.^[3] Lunisolar calendars achieve similar alignment by adding extra days or an entire intercalary month during leap years.^[2] The fundamental purpose of leap years is to counteract seasonal drift, ensuring that key astronomical events like equinoxes and solstices remain consistent with calendar dates over centuries.^[3] Without periodic additions of time, the calendar would gradually lose synchronization with Earth's orbit around the Sun, causing seasons to shift and misaligning cultural, agricultural, and religious observances with natural cycles.^[4] Intercalation—the practice of inserting extra days or months—has been essential in solar and lunisolar calendars since ancient civilizations, as the solar year does not consist of an integer number of days or lunar months.^[2] This mechanism maintains the calendar's utility for tracking seasonal changes and celestial phenomena across diverse historical systems.^[2]

Astronomical Justification

The tropical year, defined as the interval between two successive vernal equinoxes, measures the time required for Earth to complete one full cycle of seasons and has a mean length of 365 days, 5 hours, 48 minutes, and 46 seconds, or 365.2422 mean solar days.^[1] This duration arises from Earth's orbital period around the Sun, during which the planet's position relative to the equinox points—determined by the alignment of its rotational axis with the orbital plane—repeats.^[5] The slight excess over 365 whole days, specifically the fractional 0.2422 days, represents the core astronomical imbalance that calendars must address to maintain alignment between civil dates and natural seasonal cycles.^[6] Without correction, this fractional component accumulates annually, causing the calendar to drift relative to the equinoxes and solstices. For instance, the shortfall of approximately 5 hours, 48 minutes, and 46 seconds per year builds to nearly one full day after four years (4 × 0.2422 ≈ 0.9688 days), shifting seasonal events like the onset of spring by a full day and progressively misaligning agricultural and climatic patterns with fixed dates.^[7] Over centuries, unchecked accumulation would displace the vernal equinox by weeks or months, rendering a 365-day calendar inadequate for tracking the solar year's true progression.^[8] To approximate the tropical year's length, calendars employ a mean year duration that incorporates the fractional day through periodic additions. A foundational mathematical model posits the average as 365 integer days plus a quarter day to account for the excess:

$365 + \frac{1}{4} = 365.25

days, yielding an extra day every four years to offset the accumulation.^[7] This yields an average slightly exceeding the true 365.2422 days (by about 0.0078 days per year), necessitating precision adjustments in refined systems to minimize long-term drift, though the core 1/4-day rule stems directly from the orbital mechanics.^[1] Earth's axial precession, a slow wobble of the rotational axis caused by gravitational torques from the Sun and Moon on the planet's equatorial bulge, further differentiates the tropical year from the sidereal year—the time to orbit relative to fixed stars, which spans 365.256363 days. Precession advances the equinox points westward along the ecliptic by roughly 50.3 arcseconds annually, shortening the tropical year by about 20 minutes compared to the sidereal year and completing a full cycle every 25,772 years. For short-term calendar adjustments like leap years, however, precession exerts a negligible influence, as its annual effect is far smaller than the daily fractional discrepancy and primarily affects long-term alignments over millennia rather than seasonal synchronization.^[9]

Historical Development

Pre-Julian Roman Calendar

The original Roman calendar, attributed to the legendary founder Romulus around the 8th century BCE, consisted of ten months totaling 304 days, beginning with Martius (March) and ending with December. This structure aligned loosely with the agricultural year and lunar cycles, omitting winter days entirely as they were not considered part of the formal calendar. The months alternated between 30 and 31 days for even and odd positions, respectively, but the system left a gap of approximately 61 days unaccounted for each solar year.^[10] Numa Pompilius, the second king of Rome in the 7th century BCE, reformed the calendar to better approximate the lunar year by adding two months—Januarius (January) at the beginning and Februarius (February) at the end—resulting in a 12-month, 355-day lunar calendar. January received 29 days, February 28, and other months were adjusted to odd numbers of days to align with Roman cultural preferences for even numbers as unlucky. To reconcile this lunar framework with the longer solar year of about 365.25 days, an intercalary month called Mercedonius (or Intercalaris) was introduced, consisting of 27 or 28 days and inserted every second year after February 23 or 24. This addition aimed to keep the calendar in synchrony with the seasons, effectively creating a leap-like adjustment roughly every biennium.^[11]^[10] However, the administration of intercalation fell to the pontifices, the college of priests responsible for religious and calendrical matters, who applied it irregularly and often manipulated it for political purposes. Intercalations were frequently omitted during periods of war or instability, such as the Second Punic War, and the priests extended or shortened terms of office by adjusting the calendar to favor allies or delay unfavorable elections—for instance, in 59 BCE, consul Marcus Calpurnius Bibulus attempted to nullify Julius Caesar's legislation by declaring unfavorable omens and halting intercalation. This abuse led to significant seasonal drift over time, with the calendar year gradually decoupling from astronomical reality. By 46 BCE, during Julius Caesar's consulship, the Roman calendar had fallen three months behind the seasons, resulting in anomalies such as winter festivals occurring in summer and elections for cold-weather offices held in the height of warmth.^[11]^[10] This profound misalignment underscored the need for a more reliable system, setting the stage for Caesar's comprehensive reform.

Julian Calendar Reform

In 46 BCE, Julius Caesar undertook a comprehensive reform of the Roman calendar, advised by the Alexandrian astronomer Sosigenes, to establish a solar year of 365.25 days on average. This reform addressed the misalignment caused by the previous lunisolar system's irregular intercalations, which had shifted the calendar by several months relative to the seasons. To realign it, Caesar decreed that 46 BCE would be a transitional year of 445 days, incorporating two extra months (Intercalaris Prior and Intercalaris Posterior) alongside the standard 12, making it the longest recorded year in history.^[12]^[13] The core of the Julian leap year rule specified that every fourth year would have 366 days, achieved by adding an extra day initially inserted after February 23, effectively doubling that date (ante diem bis sextum Kalendas Martias). This insertion was later standardized to the end of February as February 29 in subsequent Western calendars. The reform took full effect in 45 BCE, with the year beginning on January 1, marking a shift to a fixed, sun-based system without reliance on lunar cycles or priestly adjustments.^[14]^[12] Although innovative, the Julian calendar overestimated the tropical year's length at 365.25 days, exceeding the actual approximately 365.2422 days by about 11 minutes and 14 seconds annually. This error resulted in a cumulative drift of roughly 3 days every 400 years, gradually causing seasonal misalignment over centuries.^[1] The Julian calendar spread throughout the Roman Empire following its adoption, becoming the standard dating system across Europe by the 4th century CE as Roman administration and culture expanded.^[15]^[16]

Gregorian Calendar Reform

In 1582, Pope Gregory XIII issued the papal bull Inter gravissimas on February 24, based on proposals by the Italian astronomer Aloysius Lilius and finalized by the German Jesuit mathematician Christopher Clavius, promulgating a reform to the Julian calendar to address the accumulated drift of approximately 10 days in the date of the vernal equinox since the Julian reform in 45 BCE.^[17]^[18]^[19] The bull mandated the omission of 10 days in October 1582, with Thursday, October 4, immediately followed by Friday, October 15, to realign the calendar with the seasons, and introduced refined rules for future leap years to prevent further discrepancies.^[20] The Gregorian leap year rule specifies that a year is a leap year if it is divisible by 4, except for century years, which are leap years only if divisible by 400; thus, years such as 1700, 1800, and 1900 are not leap years, while 2000 is.^[1] This algorithm can be expressed as: a year Y is a leap year if Y \mod 4 = 0 and either Y \mod 100 \neq 0 or Y \mod 400 = 0.^[1] Over a 400-year cycle, this results in 97 leap years, yielding an average year length of 365.2425 days.^[2] Implementation began immediately in Catholic countries including Italy, Spain, Portugal, and parts of France, where the 10-day skip occurred in October 1582.^[21] Protestant countries like Britain and its colonies adopted the reform later, in 1752, skipping 11 days in September (September 2 followed by September 14), while Orthodox nations delayed further; for instance, Russia transitioned in 1918 by a decree of the Soviet government, advancing dates by 13 days after January 31 (followed by February 14).^[21]^[22]^[23] The Gregorian calendar's average year length of 365.2425 days approximates the tropical year of 365.2422 days, resulting in a drift of only about one day every 3,300 years.^[1]^[2]

Leap Day Observance

February 29 in Western Calendars

In the Julian and Gregorian calendars, the leap day is inserted immediately after February 28, making February 29 the 60th day of the year in leap years.^[24] This placement ensures that the year consists of 366 days, with the additional day compensating for the fractional part of the solar year. In non-leap years, March 1 serves as the 60th day, but the insertion of February 29 shifts all subsequent dates forward by one day. Historically, the leap day originated in the Julian calendar reform of 45 BCE, where it was not initially designated as February 29 but as a duplication of February 24, known as the "bis sextus" or second sixth day before the Kalends of March (March 1). This insertion after February 23 reflected the Roman numbering system, which counted days backward from the month's end; the extra day effectively repeated the date ante diem sextum Kalendas Martias. Over time, as the calendar's month lengths stabilized and numbering conventions evolved during the early centuries CE, the leap day standardized as February 29 by the late Roman period and was retained in the Gregorian reform of 1582. The Gregorian calendar, which refined leap year rules to better align with the tropical year, adopted this same positioning without alteration. The addition of February 29 extends the month to 29 days in leap years, altering the sequential numbering of dates from March onward and affecting weekday assignments throughout the remainder of the year. For instance, dates after February 29 in a leap year fall on a weekday one day later than they would in a non-leap year (assuming the same starting weekday for January 1). This shift also affects comparisons to the previous year, depending on whether it was a leap year. This impacts date calculations, such as determining intervals or anniversaries, requiring algorithms to account for the extra day to avoid discrepancies in financial, legal, or computational contexts. In modern civil usage, February 29 is universally recognized under the ISO 8601 standard, which specifies the Gregorian calendar for international date representation and explicitly includes the leap day as YYYY-02-29 in applicable years.^[25] This standardization facilitates consistent global data exchange, ensuring that software and systems handle leap years uniformly without regional variations.

Cultural and Religious Practices

In Christianity, leap years play a significant role in the liturgical calendar, particularly through the computus, the method used to determine the date of Easter Sunday as the first Sunday after the first full moon on or after the vernal equinox. This calculation incorporates the solar year's 365.25-day average, with the extra day in February ensuring alignment between the ecclesiastical and astronomical calendars, thereby preventing gradual drift in Easter's timing over centuries.^[26] The observance of saints' feast days is also affected by leap years; those assigned to February 29, such as St. Oswald of York, are commemorated on February 28 in common years to maintain their position within the liturgical cycle. For instance, the feast of St. Matthias the Apostle, traditionally on February 24, shifts to February 25 in leap years due to ancient Roman calendar precedents integrated into Christian practice.^[27]^[28] Folk traditions surrounding February 29 often emphasize role reversals and romance, most notably in Irish and Scottish lore where women may propose marriage to men on this date, a custom attributed to St. Brigid in the fifth century who advocated for female initiative every four years. If refused, the man traditionally offers a gift like a glove, silk gown, or rooster to compensate, with historical records from 1288 codifying such penalties in Scottish law.^[29]^[30] European folklore frequently associates leap years with misfortune; in Greek and Ukrainian traditions, marriages during these years are believed to lead to divorce, while Germans view the entire period as unlucky, advising against major undertakings like travel or contracts. Italian proverbs echo this, with "Anno bisesto, anno funesto" warning of a "fatal leap year" marked by calamity.^[31]^[32] French cultural observance of leap day centers on the publication of La Bougie du Sapeur, a satirical newspaper issued exclusively on February 29 since 1980, blending humor with commentary on current events and donating proceeds to charity. In Russia, the pre-Lenten festival of Maslenitsa, a week of pancake feasts symbolizing winter's end, may incorporate February 29 when it falls within the period, extending communal rituals like effigy burnings and games across the extra day.^[33]

Legal and Personal Implications

Individuals born on February 29, known as leaplings, face unique challenges in non-leap years when determining their official birthday for celebratory and legal purposes. In most jurisdictions, the legal date of birth remains February 29, but the anniversary for aging or rights attainment shifts to either February 28 or March 1 in common years. For example, in the United States, legal age is typically calculated as advancing on March 1 in non-leap years, affecting milestones like voting eligibility or driver's license issuance. Similarly, under Hong Kong's Age of Majority (Related Provisions) Ordinance, the relevant anniversary for those born on February 29 is March 1 in non-leap years. In contrast, Taiwan recognizes February 28 as the legal birthday for leaplings in common years. These variations can lead to inconsistencies in age verification for passports, contracts, or age-restricted activities. Age calculations for leaplings emphasize chronological time over calendar dates, ensuring they attain legal adulthood or other milestones annually rather than quadrennially. A child born on February 29, 2020, for instance, legally turns 1 year old on March 1, 2021, and reaches age 5 on March 1, 2025, after approximately four calendar years but five full years of elapsed time. This approach prevents delays in rights, such as entering school or consenting to medical treatment, and aligns with statutes like Scotland's Age of Legal Capacity Act 1991, which sets the anniversary as March 1 in non-leap years. The rarity of such births—occurring with a probability of about 1 in 1,461—means only roughly 0.07% of the global population are leaplings, amplifying the need for clear legal frameworks. Leap years also influence administrative and contractual matters due to the extra day. Contracts, leases, or subscriptions expiring on February 29 must specify handling in non-leap years to avoid ambiguity; for instance, a lease ending on that date might roll over to February 28 or March 1, as recommended in property law guidance to prevent disputes. Elections scheduled near February can be affected if fixed dates fall on the 29th, potentially shifting polling or term lengths in jurisdictions with rigid calendars, though most systems adjust seamlessly. In financial contexts, leap years require 366-day calculations for interest accruals, payroll, and budgeting; hourly workers may receive pay for an additional day, while mortgage interest or investment returns are prorated over 366 days, impacting projections by about 0.27%.

Leap Years in Non-Western Calendars

Hebrew and Islamic Calendars

The Hebrew calendar is a lunisolar system that synchronizes lunar months with the solar year through intercalation, employing a 19-year Metonic cycle known as the machzor gadol. In this cycle, 235 lunar months approximate 19 solar years, with 7 of the 19 years designated as leap years (embolismic years) to prevent seasonal drift. Leap years occur when the Hebrew year number modulo 19 yields 0, 3, 6, 8, 11, 14, or 17, adding an extra month called Adar I (or Adar Aleph) before the regular Adar (Adar II or Adar Bet), resulting in 13 months and typically 383, 384, or 385 days.^[2]^[34] The determination of leap years and the start of months relies on the molad, a calculated mean conjunction of the sun and moon, beginning from the molad of Tishri in year 1 AM (Anno Mundi) at 2 days, 5 hours, and 204 ḥalaqim (one ḥalaq equals 1/1080 of an hour). The mean synodic month is fixed at 29 days, 12 hours, and 793 ḥalaqim. To refine alignment and avoid certain weekday conflicts, four postponement rules called deḥiyyot may delay Rosh Hashanah (1 Tishri) by one or two days: (a) if the molad falls on Sunday, Wednesday, or Friday; (b) if the molad occurs at or after noon (18 hours); (c) if the molad is on Tuesday at or after 9 hours and 204 ḥalaqim in a common year; or (d) if the molad is on Monday at or after 15 hours and 589 ḥalaqim following a leap year. These mechanisms ensure that Passover (Nisan 15) falls in spring, maintaining agricultural and religious harmony with the equinox.^[2] In contrast, the Islamic calendar (Hijri calendar) is a purely lunar system with 12 months totaling 354 or 355 days, deliberately untethered from the solar year to prioritize lunar phases for religious observances like Ramadan. Unlike lunisolar calendars, it includes no intercalary months or solar adjustments, leading to an annual drift of about 10–11 days relative to the seasons, completing a full cycle through the solar year every 32–33 years. This design fulfills Quranic injunctions against pre-Islamic intercalation (nasi), establishing a fixed lunar reckoning from the Hijra in 622 CE.^[2]^[35] Tabular variants of the Islamic calendar, such as the Umm al-Qura calendar officially used in Saudi Arabia, approximate moon sightings through arithmetic rules over a 30-year cycle, with 19 common years of 354 days and 11 leap years of 355 days achieved by extending Dhu al-Hijjah from 29 to 30 days. These variants maintain the lunar character without solar synchronization, though they may diverge by 1–2 days from actual sightings for administrative consistency.^[36]^[37] The key distinction lies in their astronomical priorities: the Hebrew calendar's intercalation and postponements actively align lunar and solar cycles for seasonal festivals, while the Islamic calendar's lack of such mechanisms intentionally decouples it from solar progression, emphasizing the moon's independent rhythm.^[2]

Chinese and East Asian Calendars

The traditional Chinese calendar operates as a lunisolar system, reconciling lunar months of approximately 29.5 days with the solar year of about 365.24 days through the insertion of leap months, termed runyue or intercalary months. These leap months are added when the 12 lunar months fail to align with the 24 solar terms—astronomical divisions marking the sun's position every 15 degrees along the ecliptic—to prevent seasonal drift. On average, 7 such leap months occur every 19 years, following the Metonic cycle where 235 lunar months closely approximate 19 solar years.^[38]^[39] The precise rule for intercalation ensures that no lunar month between the winter solstice and vernal equinox lacks a major solar term, or zhongqi (principal term, occurring every other solar term). This period, spanning roughly three months, must each contain a zhongqi to maintain agricultural and seasonal timing, with the winter solstice fixed in the 11th month. Leap months are determined using calculations based on the mean sun's position; the first lunar month without a zhongqi—typically after the winter solstice—is designated as the intercalary month, repeating the numbering of the preceding regular month (e.g., a leap 4th month follows the regular 4th).^[40]^[41] Like the Hebrew calendar, the Chinese system employs intercalation for lunisolar synchronization, but relies on solar term alignments rather than fixed arithmetic cycles. The traditional Japanese calendar, adopted from China via Korea and used until its replacement by the Gregorian calendar in 1873, mirrored these rules by inserting leap months approximately every two to three years to harmonize lunar phases with solar seasons. Similarly, the historical Korean calendar followed the same lunisolar framework, adding intercalary months 7 times per 19-year cycle for traditional rites and farming. In modern Japan and Korea, the Gregorian calendar governs civil life, yet lunisolar variants with leap months persist for festivals like Lunar New Year.^[42]^[43] A recent example is the 2023 Chinese lunar year (Year of the Rabbit), which featured a leap 2nd month starting on March 22, 2023, in Gregorian reckoning, effectively creating a duplicated lunar February to realign with solar terms.^[44]

Baháʼí, Coptic, and Regional Calendars

The Baháʼí calendar is a fixed solar calendar consisting of 19 months, each with 19 days, totaling 361 days, supplemented by four intercalary days known as Ayyám-i-Há inserted between the 18th and 19th months.^[45] In leap years, a fifth intercalary day is added to maintain alignment with the solar year, occurring every four years to ensure the New Year, Naw-Rúz, coincides precisely with the vernal equinox on approximately March 21.^[45] This structure, introduced by the Báb and elaborated by Bahá’u’lláh, emphasizes unity and harmony through its 19-based numerical symbolism, reflecting the calendar's role in fostering communal gatherings and spiritual reflection during the Ayyám-i-Há period.^[45] The Coptic calendar, used by the Coptic Orthodox Church, divides the year into 13 months: 12 months of 30 days each followed by an intercalary month, Nasie, of five epagomenal days in common years or six in leap years.^[46] Leap years occur every four years, following the Julian calendar's rule where the year number is divisible by four, adding the extra day to Nasie without century-year exceptions to synchronize with the solar cycle.^[46] The Ethiopian calendar, closely related and derived from the ancient Egyptian system, employs an identical structure with its 13th month, Pagume, gaining a sixth day in leap years every four years, ensuring 366 days total.^[47] These calendars' epagomenal adjustments historically tied to the Nile River's annual flooding cycles, which informed ancient Egyptian agriculture and ritual timing.^[46] In South and Southeast Asian solar calendars, leap year mechanisms vary but prioritize agricultural harmony. The Bengali calendar, officially revised in Bangladesh in 1987, features seven months of 31 days and five of 30 days, with the month of Falgun extended to 31 days in leap years—those divisible by four, mirroring Gregorian rules—to align with solar progression and support harvest festivals like Pohela Boishakh.^[48] The Indian national calendar, based on the Saka era adopted in 1957, is a solar system where leap years are determined by adding 78 to the Saka year and checking if the result qualifies as a Gregorian leap year, adding an extra day to Chaitra to maintain 365 or 366 days and coordinate with monsoon-dependent farming.^[49] Similarly, the Thai solar calendar, formalized in 1941 as an adaptation of the Gregorian with Buddhist Era numbering (adding 543 years), follows identical leap year rules every four years, inserting an extra day in February (Athikasurathin) to ensure seasonal accuracy for rice cultivation and Buddhist observances.^[50]

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Chinese Calendar - Chinese Customs - Nations Online
- The Winter Solstice, Principal Term 11, always falls in the 11th month. - An ordinary year has twelve lunar months; an intercalary year has thirteen lunar ...
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Why Do the Rules for the Chinese Calendar Work? - GitHub Pages
In the past, the Chinese used this 19-year cycle to insert leap months in the calendar. ... 7 leap months being inserted in the calendar every 19 years.
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Japanese date conversion - CWI
The lunisolar calendar. Before 1873-01-01 the Japanese calendar had a year with twelve or thirteen months, each of 29 or 30 days, where in principle the ...
[44]
Leap Month in a Leap Year - Gwangju News
Jun 9, 2020 · The lunar leap month, or intercalary month, occurs 3 times in 8 years and 7 times in 19 years .
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Chinese Calendar 2026: Convert Gregorian Date to Lunar Date
Leap Months - When Chinese Add a 13th Month The last Chinese calendar leap month was the second lunar month 2 in 2023. The next leap month will be in 2025 (the ...
[46]
The Bahá'í Calendar - Bahai.org
The Bahá’í calendar, introduced by the Báb, has 19 months of 19 days each, plus intercalary days. The New Year is fixed to the vernal equinox. Days are from ...
[47]
The Coptic Calendar of Martyrs & Easter Calculations
We have put up a page that calculates the date of all major Coptic feasts including the Orthodox Easter for any year after 1902 A.D. and also calculates a few ...
[48]
The Ethiopian Leap Year - Time and Date
A leap year in the Ethiopian calendar occurs every 4 years when an extra day is added to the last month of the year.Missing: rules | Show results with:rules
[49]
Bangla calendar and Pahela Boishakh - The Financial Express
Apr 13, 2025 · As for the leap year, the committee recommended that the years divisible by four would be leap years. ... Year's Day, according to Bangla calendar ...
[50]
National Identity Elements - National Calendar - Know India
The national calendar is based on the Saka Era, with Chaitra as the first month, 365 days, and 1 Chaitra falling on 22 March normally.
[51]
Thai Calendar - Nova Masters Consulting
Sep 20, 2025 · A leap day in the 7th lunar month; A leap month (13th month). The Royal Astronomical Institute sets the rules to align lunar months with solar ...