For astronomy and calendar studies, the
Contents 1 Mathematical basis 2 Application in traditional calendars 3 Further details 4 See also 5 External links 6 References Mathematical basis[edit] At the time of Meton, axial precession had not yet been discovered, and he could not distinguish between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days). Most calendars, like the commonly used Gregorian calendar, are based on the tropical year and maintain the seasons at the same calendar times each year. Nineteen tropical years are about two hours shorter than 235 synodic months. The Metonic cycle's error is, therefore, one full day every 219 years, or 12.4 parts per million. 19 tropical years = 6,939.602 days (12 × 354-day years + 7 × 384-day years + 3.6 days). 235 synodic months (lunar phases) = 6,939.688 days (Metonic period by definition). 254 sidereal months (lunar orbits) = 6,939.702 days (19 + 235 = 254). 255 draconic months (lunar nodes) = 6,939.1161 days. Note that the 19-year cycle is also close (to somewhat more than half
a day) to 255 draconic months, so it is also an eclipse cycle, which
lasts only for about 4 or 5 recurrences of eclipses. The Octon is
1⁄5 of a
8 years = 99 lunations (an Octaeteris) to within 1.5 days, i.e. an error of one day in 5 years; and 11 years = 136 lunations within 1.5 days, i.e. an error of one day in 7.3 years. By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate ever more accurate cycles. For example, simple arithmetic shows that: 687 tropical years = 250,921.39 days; 8,497 lunations = 250,921.41 days. This gives an error of only about half an hour in 687 years (2.5
seconds a year), although this is subject to secular variation in the
length of the tropical year and the lunation.
External links[edit] Eclipses, Cosmic Clockwork of the Ancients References[edit] ^ Rare Full Moon on Christmas Day, NASA ^ Ask Tom: How unusual is a full moon on Christmas Day? ^ Livy, Ab Urbe Condita, I, XIX, 6. Mathematical
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