What is a leap year?

What is a leap year?

A leap year is a calendar year that contains one more day than the usual count, inserted to keep the calendar in step with the seasons. In the Gregorian calendar — the civil calendar used internationally for secular dating — a common year has 365 days and a leap year has 366, with the extra day designated 29 February and placed between 28 February and 1 March.¹ The Romans, whose calendar Caesar reformed in 46 b.c. into the direct ancestor of today's, called such years bissextile, a word still used in some technical writing for the same idea.¹⁴

The reason a leap day is needed at all is that the year — the time the Earth takes to return to the same point in its orbit relative to the Sun — is not a whole number of days. The relevant astronomical quantity is the tropical year, defined today as the time for the Sun's mean ecliptic longitude to increase by 360°.⁵⁶ In the year 2000 the mean tropical year was about 365.24219 days long; equivalently, 365 days, 5 hours, 48 minutes, and roughly 45 seconds.¹ A calendar with a fixed 365-day year would lag the seasons by almost a full day every four years; one with 366 days would race ahead by more than three days. A practical solar calendar has to interpolate between the two.

What is the Gregorian leap-year rule?

The Gregorian rule, in its operational form, is a single sentence with two exceptions to the basic four-year cycle:

Every year that is exactly divisible by 4 is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400.¹

So 2024, 2028, and 2032 are leap years, like every fourth year. 1700, 1800, and 1900 — divisible by 100 but not 400 — were not. 1600 and 2000 — divisible by 400 — were. The next centurial leap year is 2400; 2100, 2200, and 2300 will be common years of 365 days.²³ In source code the same rule fits in one line, as the C example given in the appendix to the internet timestamp standard demonstrates: year % 4 == 0 && (year % 100 != 0 || year % 400 == 0).⁷

The rule is occasionally misremembered. Two common errors are worth naming. First, it is not "every four years": three out of every four centurial years break the four-year pattern. Second, the centurial exception is not optional or country-specific. It is the entire point of the Gregorian reform; without it the calendar would be the older Julian calendar, which the Gregorian replaced precisely because the four-year rule on its own had drifted ten days off the seasons by 1582.¹

The numerical character of the rule is straightforward to verify by counting. In any 400-year span the calendar contains 100 multiples of 4. Four of those are also multiples of 100, and three of those four are not multiples of 400 — those three are the centurial years skipped by the rule (1700, 1800, and 1900 in the most recent cycle; 2100, 2200, and 2300 in the next). The 400-year cycle therefore contains 100 − 3 = 97 leap years and 303 common years: 303 × 365 + 97 × 366 = 146,097 days, which works out to a mean calendar year of exactly 365.2425 days.¹²

Why does the calendar need leap years at all?

Because the Earth does not rotate on its axis a whole number of times in one orbit around the Sun. Twelve months of 30 or 31 days that hold their place against the seasons require the year to average something close to the tropical year of 365.2422 mean solar days, and no calendar that uses only whole-number-day years can do that. A 365-day year, the version used by the ancient Egyptian civil calendar, drifts a full day against the seasons every four years and a full year — through every season, top to bottom — about every 1,500 years.¹

The fix in every solar calendar that has paid serious attention to the problem is to alternate years of different lengths in a regular cycle, so that the average matches the astronomical year as closely as possible. The Julian calendar of Caesar's reform did this with one rule — every fourth year a leap year, with no exceptions — for an average year of 365.25 days.¹ That rule is simple but slightly too long: the calendar's average year exceeds the tropical year by about 11 minutes 14 seconds, which works out to about one day every 128 years. The Gregorian rule reduces that error by an order of magnitude.

Why was the Gregorian reform necessary?

By the late sixteenth century the Julian calendar had been running for sixteen hundred years and the Julian year's small surplus over the tropical year had cumulated into a visible problem. The vernal equinox — the moment in spring when day and night are equal, used by the Christian church since the Council of Nicaea in 325 to anchor the date of Easter — had drifted from 21 March, where Nicaea had assumed it would stay, to 11 March. The astronomical new moon was occurring four days before the dates the church's tables assumed. Easter was being celebrated on the wrong Sunday.¹

Pope Gregory XIII convened a calendar commission in 1572.⁸ It adopted a proposal originated by the Italian physician and astronomer Aloysius Lilius and worked out in detail by the Jesuit mathematician Christopher Clavius. On 24 February 1582 the Pope issued the bull Inter gravissimas, which did three things at once. It deleted ten days from the calendar — 4 October 1582 was followed directly by 15 October 1582 — to bring the equinox back to about 21 March. It changed the rule for intercalating leap days to the divisible-by-4-except-centuries-unless-divisible-by-400 rule still in use today. And it introduced new tables for computing the date of Easter that took the corrected calendar into account.³¹

The bull is unusually specific about what the new rule meant in practice. It instructs that "the bissextile day every fourth year shall continue, as the custom is now, except in centurial years … only every fourth centurial year shall be bissextile, thus the years 1700, 1800 and 1900 shall not be bissextile."³ The reform was first promulgated in the Roman Catholic states. Protestant Europe took longer to accept it: Britain and her American colonies held out until 1752, by which point eleven days had to be skipped to catch up.¹ Eastern Orthodox churches still calculate Easter against the Julian calendar today.¹

How accurate is the Gregorian rule?

Very accurate, but not perfect. The mean Gregorian year is 365.2425 days, against a current mean tropical year of about 365.24219 days; the calendar runs long by roughly 27 seconds per year.¹ On its own that error accumulates to a full day in about 3,300 years, or — phrased differently and a little more conservatively, since the tropical year itself is slowly shortening — to a one-day discrepancy with the vernal equinox after about another 8,000 years from the year 2000.¹² In practical terms the Gregorian calendar is good for the timescale on which civilisations plan, and the question of what a future society should do about its residual error has been left, sensibly, to that society.

Various proposals to refine the rule have been made over the centuries — typically by dropping the leap day in years divisible by some larger number, such as 4,000 — but none has been adopted. The 2,000-year track record of the Julian rule, and the post-1582 track record of its Gregorian successor, both suggest that the world is more comfortable with a calendar that drifts slowly and predictably than with one that is repeatedly tweaked.¹

How do computers represent leap years and dates before 1582?

Most modern date-handling software does not care about the historical date the Gregorian rule was adopted. It applies the rule to every year it is asked about, including years before 1582 and indeed before the year 1, treating the Gregorian calendar as if it had always been in use. This is called the proleptic Gregorian calendar, and it is the convention of the standard for date-and-time strings on the internet, of the IANA Time Zone Database used by every major operating system to map UTC to local time, and of the date arithmetic in nearly every programming language.⁷⁹ The same standard pins the proleptic rule down with the same wording the Gregorian reform used: a leap year is a year divisible by four, except centuries, unless they are divisible by 400.⁷

The convention works for two reasons. First, computer software that does date arithmetic almost never needs the difference of "what calendar was a 14th-century European document actually written in"; it needs a single consistent year-month-day-to-day-count function, and the proleptic Gregorian rule is the simplest such function whose answers match modern dates. Second, the alternative — switching from Julian to Gregorian on different dates depending on jurisdiction (1582 in Italy, 1752 in Britain, 1918 in Russia, and so on) — would make every program that handles historical dates a regional history exam.

Negative and zero years follow the astronomical convention introduced by Jacques Cassini in 1740: the year a.d. 1 is preceded by year 0, which is preceded by year −1, with no gap.¹ In this scheme the year 0 is divisible by 400, so it is a (proleptic) leap year. Historical writing instead labels years 1 b.c. and skips zero, which is convenient for chronologists and inconvenient for everyone doing arithmetic; the internet timestamp standard explicitly uses the astronomical convention.⁷

What does this mean in practice for date arithmetic?

Three concrete consequences are worth holding onto. First, "one year from 29 February 2024" is ambiguous: there is no 29 February in 2025, and software libraries handle the case differently. Most clamp to 28 February 2025; some advance to 1 March 2025. People born on 29 February — sometimes called leaplings — observe their birthday on whichever of those two days local custom assigns, with no formal international rule.

Second, day-of-year numbering is not stable: 1 March is day 60 in a common year and day 61 in a leap year. Anything that pins a recurring event to "the Nth day of the year" rather than to a calendar date will land on different dates in February depending on the year. The ordinal date form of the internet timestamp standard — YYYY-DDD — encodes this directly, with the day-of-year running 001 through 365, or 366 in a leap year.⁷

Third, the average month length over a 400-year cycle is exactly 30.436875 days — which is 365.2425 / 12 — but no individual month is that length. Date arithmetic that adds "one month" to a fixed date and expects a uniform result will be wrong; adding "30.436875 days" to mean the same thing will, on average, be right, but on any specific date will land on the wrong day. The right primitive for almost every application is to add calendar months and accept the resulting variation in day count.

Fourth, an extra calendar day on 29 February nudges every count of business days that crosses the leap day: the count rises by one in years where 29 February falls on a Monday through Friday, and is unchanged in years where it falls on a Saturday or Sunday. Deadline calculators and settlement-date tools built against a fixed-date start point therefore land on different calendar dates in leap years than in the surrounding common years.

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