For astronomy and calendar studies, the Callippic cycle (or Calippic) is a particular approximate common multiple of the year (specifically the tropical year) and the synodic month, that was proposed by Callippus during 330 BC. It is a period of 76 years, as an improvement of the 19-year Metonic cycle.
A century before Callippus, Meton had discovered the cycle in which 19 years equals 235 lunations. If we assume a year is about 365 1⁄4 days, 19 years total about 6940 days, which exceeds 235 lunations by almost a third of a day, and 19 tropical years by four tenths of a day. It implicitly gave the solar year a duration of 6940⁄19 = 365 + 5⁄19 = 365 + 1⁄4 + 1⁄76 days = 365 d 6 h 18 min 56 s. Callippus accepted the 19-year cycle, but held that the duration of the year was more closely 365 1⁄4 days (= 365 d 6 h), so he multiplied the 19-year cycle by 4 to obtain an integer number of days, and then omitted 1 day from the last 19-year cycle. Thus, he computed a cycle of 76 years that consists of 940 lunations and 27,759 days, which has been named the Callippic cycle after him.
Although the cycle's error has been computed as one full day in 553 years, or 4.95 parts per million, in actuality 27,759 days in 76 years has a mean year of exactly 365 1⁄4 days, which relative to the mean northward equinoctial year is about 11 minutes too long per year, in other words the cycle drifts another day late per 130 10⁄11 years, which is considerably worse than the drift of the unrounded Metonic cycle. If the Callippic cycle is considered as closer to its unrounded length of 27,758 3⁄4 days (based on 940 lunations) then its accuracy is essentially the same as the unrounded Metonic cycle (within a few seconds per year). If it is considered as 940 lunations less one day then the Callippic mean year will be shortened by 1⁄76 of a day (18 minutes 57 seconds), making it grossly too short, and it will also grossly drift ahead with respect to the mean lunar cycle at the rate of 1⁄940 of a day (1 minute 31 seconds) per lunar month. If the cycle length is truncated to 27,758 days then the mean year is 365 days 5 hours 41 minutes 3 seconds, or almost 8 minutes too brief per year, and it will drift ahead of the mean lunar cycle by about ( 3⁄4)⁄940 day (1 minute 9 seconds) per lunar month. Altogether, the purported accuracy of this cycle is not impressive, but it is of historical interest.
The first year of the first Callippic cycle began at the summer solstice of 330 BC (28 June in the proleptic Julian calendar), and was subsequently used by later astronomers. In Ptolemy's Almagest, for example, he cites (Almagest VII 3, H25) observations by Timocharis during the 47th year of the first Callippic cycle (283 BC), when on the eighth of Anthesterion, the Pleiades star cluster was occulted by the Moon.
The Callippic calendar originally used the names of months from the Attic calendar. Later astronomers, such as Hipparchus, preferred other calendars, including the ancient Egyptian calendar. Also Hipparchus invented his own Hipparchic calendar cycle as an improvement upon the Callippic cycle. Ptolemy's Almagest provided some conversions between the Callippic and Egyptian calendars, such as that Anthesterion 8, 47th year of the first Callippic period was equivalent to day 29 of the month of Athyr, during year 465 of Nabonassar. However, the original, complete form of the Callippic calendar is no longer known.