## The Metonic Cycle

We have met with a variety of year and month lengths. From the perspective of calendar definition, we saw that the month of interest is the synodic month, the cycle time for the brightness phases of the Moon, currently lasting for an average of 29.53059 days (it is necessary to quote that to at least seven figures). Those readers with pocket calculators on hand may multiply that number by 235, for reasons that will soon be apparent, deriving a total of 6,939.69 days after rounding-off.

One could now argue about the proper length to use for a year, but the mean tropical year of 365.2422 days will do for these sums. If you multiply by 19 you get 6,939.60 days, rounded off. (In reality any particular set of 19 calendar years will contain either 6,939 or 6,940 days, depending upon whether five or only four leap years are counted among them.) It is immediately obvious that 235 synodic months last for almost exactly 19 years, the difference amounting to only 125 minutes. This 19-year period is called the Metonic cycle; we mentioned it earlier, in the Preface and in Chapter 2. There is more to be said about it, however.

The actual years we count in the Gregorian/Western calendar average to 365.2425 days, so that 19 of those will average to 6,939.6075 days. People often claim that the Gregorian calendar reform was necessary simply because the mean year in the Julian system (365.2500 days) was too long, and over the 16 centuries from Julius Caesar through to Pope Gregory XIII this resulted in the equinox arriving about 12 days too early. But that is only half the story.

From A.D. 532 the Metonic cycle had been employed in calculating the dates of Easter. For the cycle to be precise the average year length would need to be 365.2468 days (that is, 6,939.69 days divided by 19). Under the Julian calendar the average year lasted about 0.0032 days longer than this, and between 532 and 1582 these little differences had accumulated to exceed 3 days. In consequence the Moon in the sky was nowhere near the ecclesiastical moon followed by the Church tables for Easter, making Easter deviate substantially from full moon.

The Gregorian reform was therefore necessary to correct not only the Sun, but also the Moon, in terms of how closely the imaginary bodies encoded in the tables used to calculate Easter followed the movements of the real astronomical objects. The correction was designed to set those orbs right according to their parameters in A.D. 325, the time of the Council of Nicaea, when the fundamental tenets of the Christian faith were laid down.

Since 1582 the Catholic Church (joined later by many other Christian Churches) has continued to follow the Metonic cycle, but with two types of correction having been made. One is well known: the leap day corrections with three out of four century years being omitted and counted as common years instead, thus allowing the solar motion to be followed more accurately. But there is also a lunar correction, unrecognized by most people. This involves eight steps each of one day spread over 2,500 years. Using the figures cited above this correction appears to be near-

perfect: 2,500 divided by 8 gives an average of once every 312.5 years, which is the same as the reciprocal of 0.0032 days (although more decimal places are really required in the calculations to be precise). Nevertheless it is a pretty good approximation to the real behavior of the Moon. (Note that many of the Eastern Orthodox Churches continue to follow the Julian calendar, so that their Easter is often on a different date.)

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