Meton's cycle depended on the discovery that 235 synodic months or lunations (new moon to new moon) are very nearly equal to 19 tropical years (vernal equinox to vernal equinox). This can be easily checked, since the mean synodic month is about 29.5306 days and the tropical year, as we have seen in Appendix 1, is about 365.2422 days. The Metonic ratio can be attained by calculating the fifth convergent of the simple continued fraction for the decimal part of the number of months in the year. This gives the ratio as
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After 19 years the mean phases of the moon tend to recur on the same days of the month (with perhaps a shift of one day according to the number of leap years in the cycle) and within about two hours of their previous times. The months originally involved were 110 of 29 days and 125 of 30 days. The total number of days in the cycle was therefore 6,940, and consequently the average number of days in the year was a little in excess of 365.26. The particular cycle introduced by Meton began on the thirteenth day of the twelfth month of the calendar then used in Athens, which was 27 June 432 BC according to our reckoning. It appears that this day was chosen because Meton had determined astronomically that it was the summer solstice.
A more accurate version of Meton's cycle based on the assumption that the year is equal to 365.25 days was introduced about 330 BC by the astronomer Callippus, who found that Meton's 19-year cycle was slightly too long. He therefore combined four 19-year periods into one cycle of 76 years and dropped one day from the period, so that his cycle contained 27,759 days. Although it never came into general use, it became the standard for later astronomers and chronologists, for example Ptolemy. The number of days assigned to the year by Callippus became the basis of the Julian calendar.