Although most modern calendar years have 365 days, a complete revolution around the Sun (one solar year) takes approximately 365 days and 6 hours. An extra 24 hours thus accumulates every four years, requiring that an extra calendar day be added to align the calendar with the Sun's apparent position. Without the added day, in future years the seasons would occur later in the calendar, eventually leading to confusion about when to undertake activities dependent on weather, ecology, or hours of daylight.
A solar year is actually slightly shorter than 365 days and 6 hours (365.25 days), which had been known since the 2nd century BC when Hipparchus stated that it lasted 365 + 1/4 − 1/300 days, but this was ignored by Julius Caesar and his astronomical adviser Sosigenes. The Gregorian calendar corrected this by adopting the length of the tropical year stated in three medieval sources, the Alfonsine tables, De Revolutionibus, and the Prutenic Tables, truncated to two sexagesimal places, 365 14/60 33/3600 days or 365 + 1/4 − 3/400 days or 365.2425 days. The length of the tropical year in 2000 was 365.24217 mean solar days, Adding a calendar day every four years, therefore, results in an excess of around 44 minutes every four years, or about 3 days every 400 years. To compensate for this, three days are removed every 400 years. The Gregorian calendar reform implements this adjustment by making an exception to the general rule that there is a leap year every four years. Instead, a year divisible by 100 is not a leap year unless that year is also divisible by 400. This means that the years 1600, 2000, and 2400 are leap years, while the years 1700, 1800, 1900, 2100, 2200, 2300, and 2500 are common years.
Modern (Gregorian) calendar
The Gregorian calendar repeats itself every 400 years, which is exactly 20,871 weeks including 97 leap days (146,097 days). Over this period, February 29 falls on Sunday, Tuesday, and Thursday 13 times each; 14 times each on Friday and Saturday; and 15 times each on Monday and Wednesday. Excepting when a century mark that is not a multiple of 400 intervenes, consecutive leaps days fall in order Thursday, Tuesday, Sunday, Friday, Wednesday, Monday, and Saturday; then repeating with Thursday again.
Early Roman calendar
Adding a leap day (after 23 February) shifts the commemorations in the 1962 Roman Missal
The calendar of the Roman king Numa Pompilius had only 355 days (even though it was not a lunar calendar) which meant that it would quickly become unsynchronized with the solar year. An earlier Roman solution to this problem was to lengthen the calendar periodically by adding extra days to February, the last month of the year. February consisted of two parts, each with an odd number of days. The first part ended with the Terminalia on the 23rd, which was considered the end of the religious year, and the five remaining days formed the second part. To keep the calendar year roughly aligned with the solar year, a leap month, called Mensis Intercalaris ("intercalary month"), was added from time to time between these two parts of February. The (usual) second part of February was incorporated in the intercalary month as its last five days, with no change either in their dates or the festivals observed on them. This followed naturally, because the days after the Ides (13th) of February (in an ordinary year) or the Ides of Intercalaris (in an intercalary year) both counted down to the Kalends of March (i.e. they were known as "the nth day before the Kalends of March"). The Nones (5th) and Ides of Intercalaris occupied their normal positions.
The third-century writer Censorinus says:
When it was thought necessary to add (every two years) an intercalary month of 22 or 23 days, so that the civil year should correspond to the natural (solar) year, this intercalation was in preference made in February, between Terminalia [23rd] and Regifugium [24th].
The set leap day was introduced in Rome as a part of the Julian reform in the 1st century BC. As before, the intercalation was made after February 23. The day following the Terminalia (February 23) was doubled, forming the "bis sextum"—literally 'twice sixth', since February 24 was 'the sixth day before the Kalends of March' using Roman inclusive counting (March 1 was the Kalends of March and was also the first day of the calendar year). Inclusive counting initially caused the Roman priests to add the extra day every three years instead of four; Augustus was compelled to omit leap years for a few decades to return the calendar to its proper position. Although there were exceptions, the first day of the bis sextum (February 24) was usually regarded as the intercalated or "bissextile" day since the 3rd century AD. February 29 came to be regarded as the leap day when the Roman system of numbering days was replaced by sequential numbering in the late Middle Ages, although this has only been formally enacted in Sweden and Finland. In Britain, the extra day added to leap years remains notionally the 24th, although the 29th remains more visible on the calendar.