The amount of time that must pass before the phases of the moon fall again on the same days that they do in the present year.
Neither the tropical year nor the synodic month are a whole number of days. Not only that, but they have no common factor. If there were a whole number X such that dividing the number of days in X tropical years by the number of days in a synodic month left no remainder, then the phases of the moon would fall on the same days of the year as they had X years before. Actually, there is always a remainder, but historically several approximations have proven useful in making calendars that predict the phases of the Moon.
The Metonic cycle of 19 years, or 235 lunations, or 6,940 days. Although known earlier in Babylon and China, the cycle is named for the Greek astronomer Meton (5th century bce), who wrote a book titled Enneadecaterides on the cycle. (Meton is also remembered for feigning insanity to avoid ”the draft,“ having correctly foreseen that the Athenian attack on Syracuse would end in disaster.)
Taking the length of the tropical year and synodic month in Meton’s time,¹ 235 synodic months would have been 6939.687 084 days, and 19 tropical years 6939.604 386 days. 6940 days divided into 19 years makes the year 365.263 158 days, which is too big by about half an hour.
The Metonic cycle became the basis of the Greek calendar, with the first cycle said to have begun on 16 July 433 bce. In this calendar years 1, 2, 4, 5, 7, 9, 10, 12, 13, 15, 16 and 18 of the cycle had 12 lunar months of alternately 29 and 30 days. The other 7 years had 13 months. The extra month was 30 days long, except in the last year of the cycle, when it was 29 days long.
The ratio 235:19 is embodied in the gear train of the Antikythera mechanism, a mechanical astronomical computer probably built in Rhodes before 80 bce, and recovered in 1900 from an ancient shipwreck off the Greek island of Antikythera. For Java animations of the device by Bill Casselman, which incidentally are a fine visualization of the Metonic cycle, see www.ams.org/new-in-math/cover/kyth1.html Photographs of a working model built by John Gleave can be seen at www.grand-illusions.com/antikyth.htm
In the Middle Ages in Europe, each year was given a “golden number” that identified its place in the Metonic cycle. The number was crucial in establishing the date of Easter and was painted gold in manuscripts.
To find the golden number for any year A.D., add 1 to the year, then divide by 19. Any non-zero remainder is the golden number; a zero remainder indicates a golden number of 19.
This golden number is completely unrelated to the golden number which arises from division in extreme and mean ratio, about 1.618...
The Callipic cycle of 76 years, or 940 lunations, or 27,759 days, or in other words 4 Metonic cycles, was introduced by the Greek astronomer Callippus of Cyzicus (about 370 bce-300 bce), a friend of Aristotle’s, in about 334 bce. Callippus had determined experimentally a more precise value for the length of the tropical year. 76 tropical years would actually have been about 27,758.417 04 days, and 940 months about 27,758.748 59 days. 27,759 days divided into 76 years makes years of 365¼ days, which is closer to the tropical year than the Metonic cycle (it is only about 11 minutes too long). Despite the improved accuracy the Callipic cycle did not win wide use; it remained a consideration of scholars rather than calendar makers.
Four Callipic cycles less one day; 304 years or 3760 lunations, less a day, or 111,035 days. It was introduced by the Greek astronomer Hipparchus of Nicea (190 bce - 120 bce). 111,035 days divided by 304 years makes a year of 365.24671 days, which is only about 6 minutes longer than the actual length of the tropical year (about 365.2423 days).
1. The length of the tropical year and synodic month change slowly. The estimate of the lengths of these periods has been calculated using the Explanatory Supplement's statement of the model of
M. Chapront-Touzé and J. Chapront.
ELP 200-85: A Semi-Analytical Lunar Ephemeris Adequate for Historical Times.
Astron. Astrophys. vol. 190, 342-352.
The best and certainly by far the most comprehensive software for observing the moon is Gary Nugent's Lunar Phase Pro:
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Last revised: 26 September 2006.