Hipparchic cycle

The Greek astronomer Hipparchus introduced two cycles that have been named after him in later literature.

The first is described in Ptolemy's "Almagest" IV.2. Hipparchus constructed a cycle by multiplying by 17 a cycle due to the Chaldean astronomer Kidinnu, so as to closely match an integer number of synodic months (4267), anomalistic months (4573), years (345), and days (126007 + about 1 hour); it is also close to a half-integer number of draconic months (4630.53...), so it is an eclipse cycle. By comparing his own eclipse observations with Babylonian records from 345 years earlier, he could verify the accuracy of the various periods that the Chaldean astronomers used.

The second is a calendar cycle: Hipparchus proposed a correction to the Calippic cycle (of 76 years), which itself was proposed as a correction to the Metonic cycle (of 19 years). He may have published it in the book "Peri eniausíou megéthous" ("On the Length of the Year"), which is lost. From solstice observations, Hipparchus found that the tropical year is about 1/300 of a day shorter than the 365 + 1/4 days that Calippus used (see "Almagest" III.1). So he proposed to make a 1-day correction after 4 Calippic cycles, such that 304 years = 3760 lunations = 111035 days. This is a very decent approximation for an integer number of lunations in an integer number of days (error only 0.014 days). But it is in fact 1.37 days longer than 304 tropical years: the mean tropical year is actually about 1/128 day (11 minutes 15 seconds) shorter than the Julian calendar year of 365.25 days. These differences cannot be corrected with any cycle that is a multiple of the 19-year cycle of 235 lunations: it is an accumulation of the mismatch between years and months in the basic Metonic cycle, and the lunar months need to be shifted systematically by a day with respect to the solar year ("i.e." the Metonic cycle itself needs to be corrected) after every 228 years.