The Beauty of the Gregorian Calendar

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The remarkably good 5 th approximation is the basis of the Metonic cycle which was implemented into the calendar of Athens by the Greek mathematician Meton in the fifth century BC. Soon thereafter it was officially adopted in Babylon as well. Inspite of this historic succession, it is quite possible that the cycle was first found by Babylonian astronomers (this question has not been settled yet). It consists of periods comprising 19 years or 235 months, starting on July 16, 432 BC. The number of a year in that cycle was later called its golden number G, G = 1, . . . , 19. There were 12 regular years with 12 months each, and 7 leap years with 13 months. The leap years carried the golden numbers 3, 6, 8, 11, 14, 17, 19. This scheme is still at the heart of the Hebrew calendar. It underlies the determination of Easter within the framework of the Julian calendar, and up to corrections of epacts (see below) is also constitutive for the Gregorian calendar. The misfit of the Metonic cycle with respect to the correct relative length of year and month is 1.55 months, or 45.8 days, per 10 000 years, or 1 day in 218 years. This high level of precision is mathematically related to the high number n6 = 17 in the continued fraction expansion, cf. the inequality (9). The Gregorian calendar The Gregorian calendar as decreed in 1582 was designed by the physician and mathematician Luigi Giglio, or Aloisius Lilius, born around 1510 in Naples and deceased 1576 in Verona. It was worked out in detail and published in 1595 and later [2] by the mathematician and astronomer Christophorus Clavius S. J., born 1537 in Bamberg and deceased 1612 in Rome. Clavius called the new calendar a calendarium perpetuum, implying that it can be adjusted to astronomical data if need arises. Two guiding principles are characteristic of the Gregorian calendar: its faithfulness to traditions, and its openness for corrections. The respect visà-vis tradition is expressed in a threefold way: - validity of the Julian calendar rules within each century (the principle of secularity), - continuity of the Nicaean definition of Easter, - continuity of the lunisolar Metonic cycle for the calculation of Easter. Corrections are applied at the ends of centuries, in order to keep track with the average motion of sun and moon: - to get back in step with the sun, a certain portion of leap years is omitted; 6
- to get back in step with the moon, the epact is occasionally reset; - all this is done in an easily computable manner; the specific details may be adapted to improved astronomical data. Let us comment on these principles in some depth. The fundamental decision was to adhere to the ancient tradition of a lunisolar calendar. Given that the Council of Nicaea (325 AD) had determined to celebrate Easter on the first Sunday after the first full moon in spring, this decision was of course never in doubt. But the custom to number days according to their position in both the solar and the lunar cycle, antecedes Christian traditions and goes back at least to the Babylonians. It may not be widely known, but our calendar counts days not only from 1 through 365 (or 366) as members of the solar cycle, but also from 1 through 29 (or 30) in the lunar cycle. The epact E is related to this lunar counting; it is defined as the age of the moon on New Year’s day. The possible values of E are 0, 1, . . . , 29. E = 0 means new moon on New Year’s day; E > 0 means the last new moon was on December 32 − E of the old year. The original Julian calendar, as decreed by Julius Cesar in 45 BC, dealt only with the solar cycle, and fixed the length of the year to YJ = [365, 4] = 1461 4 = 365.25. (10) In the aftermath of the Council of Nicaea, it was combined with the Metonic 19-year cycle in order to account for the motion of the moon. This means the average number N of synodic months per year was fixed as NJ = [12, 2, 1, 2, 1, 1] = 235 19 = 12.368 421... . (11) In this scheme, the epact E turned out to increase by 11 days, from year to year, or to recede by 19 days. (At the end of the 19-year cycle, the recess was only 18; this special reset was called saltus lunae). There were 19 possible values of E = 11(G − 1) mod 30, depending on the golden number G. To compute Easter, one had to know, in addition, the weekday of New Year, and whether the given year was a leap year. It is not difficult to check that Easter could fall on any day between March 22 and April 25, in a 19 · 7 · 4 = 532 year cycle. The first day in spring was assumed to be March 21, hence the earliest possible full moon could be on that day, and the first next Sunday on March 22. The latest possible full moon in spring, according to this scheme, could occur on April 18; hence Easter could not be later than April 25. This Alexandrian canon was to be preserved in the Gregorian calendar. 7
Page 1 and 2: The Beauty of the Gregorian Calenda
Page 3 and 4: lightly be disposed of. Basic princ
Page 5: level k yk dk ak (Y − Yk) · 10 4
Page 9 and 10: Clavius calls this the aequatio lun
Page 11 and 12: This is the Easter cycle of the Jul
Page 13 and 14: changed, it is obvious that GS(K) i
Page 15 and 16: computing. It is part of the tradit
Page 17: [14] S. I. Seleschnikow, Wieviel Mo

The Beauty of the Gregorian Calendar

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