Leap Year Calculator

The leap year rule is a rational approximation to an irrational quantity. The mean tropical year, the interval that governs the return of the seasons, is about 365.24219 days, so a fixed 365-day calendar loses roughly 0.2422 days annually, close to a full day every four years. Adding one day every fourth year overcorrects to 365.25, an excess of about 11 minutes 14 seconds per year, or one day in roughly 128 years. The Gregorian rule removes three leap days per 400 years by skipping century years not divisible by 400: 97 leap days in 400 years gives 146,097 days and a mean year of exactly 365.2425 days. The residual error is about 0.0003 days per year, one day in roughly 3,200 years. Working through the rule in order explains the cases that surprise people. 1900 is divisible by 4 and by 100 but not by 400, so it was a common year; 2000 is divisible by 400 and was a leap year; 2100, 2200 and 2300 will all be common years. Most centuries therefore contain 24 leap years rather than 25. Two limitations are worth stating. The rule is defined only from the Gregorian calendar's introduction in 1582 onward, so applying it to earlier dates yields a proleptic result that will not match historical records. And the underlying targets are not constant: the tropical year is shortening by roughly half a second per century, while tidal friction lengthens the mean solar day by about 1.8 milliseconds per century, which is why leap seconds are handled separately from leap days.

The Egyptian civil calendar used a fixed 365 days and drifted through the seasons over a 1,460-year cycle; the Canopus Decree of 238 BCE proposed adding a sixth epagomenal day every fourth year but was not sustained in practice. The reform that stuck came from Julius Caesar, advised by the Alexandrian astronomer Sosigenes. Effective from 45 BCE, the Julian calendar fixed the year at 365 days with an intercalary day every fourth year, preceded by the 445-day year of confusion in 46 BCE that realigned the calendar with the seasons. Roman priests initially intercalated every third year by miscounting inclusively, an error Augustus corrected by suspending leap days until roughly 8 CE. The Julian year's 11-minute excess accumulated to about ten days by the sixteenth century, pushing the equinox away from March 21 and destabilising the calculation of Easter. The physician and astronomer Aloysius Lilius proposed the century exception rule, and the Jesuit mathematician Christopher Clavius developed it into the scheme promulgated by Pope Gregory XIII in the papal bull Inter gravissimas of 24 February 1582. Catholic states dropped 5 to 14 October 1582 immediately; Protestant and Orthodox states resisted for generations. Britain and its colonies converted in September 1752 by removing eleven days and moving New Year from 25 March to 1 January, Russia adopted the civil Gregorian calendar in 1918, and Greece followed in 1923, the last European state to do so.