The Beauty of the Gregorian Calendar Heiner Lichtenberg and Peter H. Richter November 1998

Introduction The Gregorian calendar was developed in the later part of the 16th century, mainly by Aloysius Lilius and Christophorus Clavius [2]. It was named after Pope Gregory XIII who decreed its implementation in 1582 [3]. By that time the Julian calendar had run out of step with the astronomical data in two ways. In its solar part, it had accumulated an error of ten days; the true average vernal equinox fell on March 11 rather than March 21 as the calendar assumed. This was corrected by omitting the ten calendar days October 5 through October 14, 1582. In its lunar part, the Julian calendar was wrong by three days; the true average age of the moon (the number of days elapsed since the last new moon) was three days larger than its calendar prediction. This was corrected by a sudden increase of the epact (see below) by three. After that one-time correction, a new algorithm was put in effect which is about twenty times more accurate in its solar part, and four times more accurate in its lunar part. We are using the Gregorian calendar to this very day [15]. However, it does not seem to be widely known that the moon plays a role in our calendar. Most people believe the Gregorian calendar to be a pure solar calendar, as opposed to the pure lunar calendar of the Arab world. But, in fact, Lilius and Clavius were careful to preserve a tradition that goes back at least to Babylonean times, not unlike Rabbi Hillel II. who, in the middle of the fourth century, reformed the Jewish calendar so as to pay respect to astronomical knowledge on the one hand, and to the dignity of 1

sun and moon as the two major celestial bodies on the other. In the Jewish calendar, the lunar part is more conspicuous than in the Christian, because it determines the average length of months and the beginning of a new year. In the Christian calendar, Julian or Gregorian, the moon has been associated with the date of Easter, i. e., with the holiest day of the Christian religion (related to the Jewish calendar through the role of Pessah in the evangelical record). For Lilius and Clavius this was a matter of such obvious importance that their main intellectual effort concerned the question of how, in the light of improved astronomical data, the synodic lunar period might be reconciled with the length of a tropical year – the fundamental problem of any lunisolar calendar. The problem, of course, is the incommensurability of the average periods of sun and moon with the length of a day and with each other: year, month, and day are time units with irrational ratios. Even worse: in the long run, their ratios are not even constant. Calendars are algorithms which try to overcome this incommensurability in terms of more or less satisfactory rational approximations. The following is an attempt to recall the basic principles underlying the Gregorian solution of this task. Using the fact that continued fractions provide optimal rational approximations to given irrational numbers, we assess the relative accuracy of various possible calendars. Astronomical accuracy was an important aspect of the Gregorian reform, but not the only one. It will be argued that beauty and wisdom are contained in two principles which have not received much attention in the unending discussions of possible new calendars. One is the principle of secularity which decrees Julian calendar rules for all years that are no secular years (divisible by 100); corrections may only be applied at turns of centuries. This guarantees a minimum of changes with respect to well established traditions. The other principle is the openness for adjustments as need arises. Contrary to a general misconception, the Gregorian system is not fixed once and for all, bound to eventually run out of phase with the astronomical data. Lilius and Clavius were conscious of the fact that the knowledge of their time might be limited. They designed their calendar as perpetual. The history of the adoption of the Gregorian calendar has been a complicated one and will not be discussed here [15]. To this very day, it has not been accepted by the Orthodox Churches. In an attempt to unify Eastern and Western calendars, the World Council of Churches has recently been discussing a compromise in which the lunar part of the Gregorian calendar would effectively be sacrificed; the date of Easter would be determined by the lunar ephemerides for the location of Jerusalem [16]. We feel that before any such decision is made, a fair evaluation of the merits of the Gregorian system ought to be undertaken. A cultural asset of its caliber should not 2

lightly be disposed of.

Basic principles of calendar design Any lunisolar calendar has to be based on three incommensurate astronomical periods, - the mean solar day dsol , i. e., the average time between two successive lower transits of the sun across a meridian; - the synodic month msyn , i. e., the average time between two successive new moons; - the tropical year atrop , i. e., the average time between two successive vernal equinoxes. Taking the day as a convenient unit of time, there are only two ratios that matter, M :=

msyn = 29.530 589 . . . dsol

and

Y :=

atrop = 365.242 19 . . . dsol

(1)

When the year is measured in numbers of months, the relevant ratio is N :=

Y = 12.368 266 . . . M

(2)

These numbers are best fits to long term astronomical observations, but they are not constant over long times. Their secular variations are of the order of 1 second per century, corresponding to drifts in the last digit, mostly due to tidal friction between earth and moon. (The short time variations of M and Y are much larger: perturbations by other planets affect the length of the year in the range of minutes, while months vary on the time scale of hours due to the complexity of the lunar orbit.) All cyclic lunisolar calendar algorithms make use of rational approximations to these numbers. As it is well known that optimal rational approximations to real numbers X are obtained from their continued fraction representation [5] X = x0 +

1 1 x1 + x2 + · · ·

3

=: [x0 , x1 , x2 , . . . ] ,

(3)

we give here the corresponding expressions for Y and N , as they will be needed later on: Y = [y0 , y1 , y2 , . . . ] = [365, 4, 7, 1, 3, . . . ] , N = [n0 , n1 , n2 , . . . ] = [12, 2, 1, 2, 1, 1, 17, . . . ] .

(4)

Let us recall a few elementary facts on rational approximations by continued fractions. Truncating the representation (3) of X at the kth level, one obtains a rational number pk Xk = [x0 , x1 , x2 , . . . , xk ] =: , (5) qk with integers pk and qk which are easily generated by the recursion pk = pk−1 xk + pk−2 ,

qk = qk−1 xk + qk−2 ,

(6)

starting with (p−1 , q−1 ) = (1, 0) and (p0 , q0 ) = (x0 , 1). The number X has the exact representation X=

pk + pk−1 Rk , qk + qk−1 Rk

(7)

where the remainder Rk is defined as 1/Rk = [xk+1 , xk+2 , . . . ] and obeys the recurrence relation 1 . (8) Rk = xk+1 + Rk+1 As a consequence, it may be concluded (Liouville 1851) that the distance |X − Xk | decreases as the square of the denominators qk , |X − Xk | ≤

1 xk+1 qk2

,

(9)

and that no rational number p/q with q ≤ qk comes closer to X than does Xk . (In contrast, the approximation of X by decimal numbers improves only with the first power of the denominators 10k .) Note that the kth approximation is particularly good when xk+1 is a large number. Let us analyze the numbers Y and N from this point of view, and consider their best rational approximations at increasing levels of precision. We start with the tropical year Y and its first continued fraction approximations Yk = dk /ak . They are listed in the following Table: The last column is 104 times the difference Y − Yk ; it tells us how many days in 10 000 years a calendar runs ahead (+) or lags behind (−) the true 4

level k 0 1 2 3 4

ak

(Y − Yk ) · 104

365 365 1 4 1 461 4 7 10 592 29 1 12 053 33 3 46 751 128

+2 421.9 −78.1 +8.1 −2.3 +0.0

yk

dk

tropical year if it distributes dk days over ak calendar years. The 0th approximation reflects the solar calendar of ancient Egypt; it runs too fast by 2 422 days in 10 000 years, or by one year in 1 461 (the Sothis period, see [14]). The 1st approximation gives the Julian calendar with its well known leap year rule to let every fourth year have 366 days; this calendar stays behind the true sun by 78 days in 10 000 years. The 4th approximation suggests an excellent fit to the natural length of the year by distributing 46 751 days over 128 years: just omit every 32nd leap year (only 31 leaps in 128 years). However, Lilius and Clavius, the fathers of the Gregorian calendar, had good reasons not to choose this particular improvement of the Julian calendar, as will be explained below. The number N = [n0 , n1 , n2 , . . . ] of synodic months per year is approximated by Nk = µk /αk according to the following table: level k 0 1 2 3 4 5 6

αk

(Nk − N ) · 104

12 12 1 2 25 2 1 37 3 2 99 8 1 136 11 1 235 19 17 4 131 334

−3 682.66 +1 317.34 −349.33 +67.34 −46.30 +1.55 −0.03

nk

µk

The number (Nk − N ) · 104 tells us how many months a lunar calendar which distributes µk months over αk years, runs ahead (+) or lags behind (−) the true moon’s motion in 10 000 years. The 0th approximation corresponds to the Islamic calendar. Its traces are visible in our civil calendar, with the division of years into 12 months; the misfit amounts to −3 682.66 · 29.53/10 000 = −10.9 days per year, i. e., the lunar year is shorter than the solar year by almost 11 days.

5

The remarkably good 5th 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 vis`a-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 = 365.25. 4

(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 = 12.368 421... . 19

(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

The Julian calendar served its purpose well for about a millenium, but in the 13th century the Oxford chancellor and bishop of Lincoln Robert Grosseteste noticed that the true vernal equinox had drifted with respect to the calendar prediction, by more than a week towards earlier dates [12]. The moon’s retardation with respect to the Metonic cycle was also observed, and a series of proposals for a calendar reform was worked out in the subsequent centuries. The problem was solved at last, and the solution approved, by the apostolic letter Inter gravissimas curas of 24 February 1582 [17] in which Pope Gregory XIII decreed with papal authority that 1. Thursday, October 4th of 1582, was to be followed by Friday, October 15th , i. e., ten days were to be omitted from the solar part of the calendar; 2. at the same time, three days were to be added to the lunar age; 3. from then on, the Julian calendar was to be replaced by the scheme as formulated by Lilius and Clavius. The resets of sun and moon were decreed in order to correct for errors accumulated since the Council of Nicaea. After October 15, 1582, the Gregorian calendar was to establish a better accuracy by effectively replacing (10) with YG =

146 097 = 365.242 5, 400

(12)

and (11) with 70 499 183 = 12.368 277... . (13) 5 700 000 The specific rules to obtain these values as long term averages are the following. The Julian scheme of adding a day to strictly every fourth year, is broken at the end of three out of four centuries: a year X with X mod 4 = 0 is not a leap year if it is divisible by 100 but not by 400. This implies the value YG given in (12). The rule for the lunar correction is more complicated and consists of two parts. First, the epact E is decreased by one for every omitted leap year, i. e. 3 times in 400 years, NG =

∆1 E = −3;

(14)

Clavius calls this the aequatio solaris anni. Second, E increases 8 times in 2 500 years, ∆2 E = 8; (15)

8

Clavius calls this the aequatio lunae. Together, this amounts to a net decrease of 25 · ∆1 E + 4 · ∆2 E = −75 + 32 = −43 in 10 000 years, and to the number NG given in (13), see reference [7]. Neither YG nor NG are taken from the continued fraction expansions (4). Would it not have been more natural (and more accurate) to construct the calendar with Y4 = 46 751/128 and N6 = 4 131/334? Not if the principle of secularity is taken into account! Lilius and Clavius were careful not to disturb the widely accepted Julian calendar by modifications that would have appeared incomprehensible and impractical. This would have been the case with leap omissions every 128th year (on the basis that 1 461/4 − 46 751/128 = 1/128), or with epact increments every 215th year (on the basis that 235/19 − 4 131/334 = 1/(19 · 334), suggesting a reset of one month, or 29.53 days, in 6 346 years). The secularity principle forbids corrections to the Julian-Metonic calendar other than at turns of a century. There cannot be a doubt that this was a wise decision which greatly added to the acceptability of the Gregorian reform. Let us investigate its consequences. The principle implies that corrections to the Julian length of the year (in days) must be of the form σ 1461 − (16) 4 100 S1 where S1 is the number of secular years in which σ leap years are omitted. To obtain an optimal rational value for σ/S1 , we equate (16) to the astronomical length of the year Y , and find σ = 36 525 − 36 524.219 = 0.781 = [0, 1, 3, 1, 1, 3, ...]. S1

(17)

The continued fraction approximations, 1/1, 3/4, 4/5, 7/9, ... are the numbers of choice. The Gregorian calendar takes the second approximation σ/S1 = 3/4 which implies the average year YG as given in (12). It would be entirely in the Gregorian spirit to adopt any of the better rational approximations to σ/S1 . In fact, the choice 7/9 was already suggested by Barnaba Oriani (1752-1832) citeOri. The Greek modification of the Julian calendar as proposed by Milutin Milankovi´c (1879-1956) and introduced in 1923 [14, 11], chooses the same value; it considers as leap years only those secular years which, if taken modulo 900, are 200 or 600. The corresponding year has 7 328 718 1461 − = = 365.242 2... days (18) YM = 4 900 900 which is an excellent approximation. 9

The argument for an improved value of the length of the month, or the number of months per year N = Y /M , is analogous to the above argument for the length of the year. The principle of secularity implies that corrections to the Metonic cycle must be of the form 235 ε/30 + 19 100 S2

(19)

where S2 is the number of secular years in which ε resets of the epact are made. The divisor 30 enters because a unit reset of the epact effects an advancement or a retardation of the calendar moon by 1/30 lunations. For a more detailed discussion see [7]. To obtain an optimal rational value for ε/S2 , we equate (19) to the astronomical value of N , and find ε/30 23 500 = 100 N − S2 19

(20)

or

 ε 23 500  = −0.465... = −[0, 2, 6, 1, ...]. (21) = 30 1236.8266 − S2 19 The rational continued fraction approximations are −1/2, −6/13, −7/15. This reasoning could not have been familiar to the makers of the Gregorian calendar because (19) was not explicitly known to them. Their estimate ε/S2 ≈ −(75 − 32)/100 = −43/100 is nevertheless a remarkably good guess, only 3 lunar resets off the best value in 10 000 years. The number NG given in (13) is obtained with this estimate: NG =

43 1 235 − · . 19 10 000 30

(22)

Considering the values YG and NG in (12) and (13), we see immediately that the Gregorian calendar has a period of PG = 5 700 000 years. This is because the solar period of 400 years divides the lunar period PG , and the number of days in a solar period happens to be a multiple of 7: 146 097 days are 20 871 weeks. An analogous argument for the Julian values YJ and NJ , see (10) and (11), shows that 76 YJ = 940 MJ = 27 759 days. (23) This is the so called Callippic cycle known in astronomy from antiquity, and named after Callippos of Kyzikos, about 330 BC. As 27 759 is not a multiple of 7, the period of the lunisolar Julian calendar is seven times the Callippic cycle, 532 YJ = 6580 MJ = 194 313 days. (24) 10

This is the Easter cycle of the Julian calendar as described by Beda Venerabilis (672/3-735) in his famous book about time reckoning [1]. An improved calendar might be based on the Oriani-Milankovi´c value (18) for the length of the year, and ε/S2 = −6/13, or NM =

6 4 582 443 235 − = = 12.368 267..., 19 13 · 3 000 1 235 · 300

(25)

for the number of months per year, which is in perfect agreement with the astronomical value. It would not make sense to look for better approximations given that N decreases by about 10−6 per century. The calendar defined by YM and NM would have a period of PM = 1 235 · 900 · 7 = 7 780 500 years, the factor 7 necessitated by the fact that 900 Oriani-Milankovi´c years are not an integral number of weeks. Summing up, the essence of the Gregorian reform of the Julian-Metonic calendar may be expressed in terms of the following two calendar equations,  1 σ  1461 · =: F1 · YJ , YC = 1 − 36 525 S1 4 (26)  19 ε  235 · =: F2 · NJ . NC = 1 + 705 000 S2 19 Both solar and lunar correction factors F1 and F2 deviate very little from 1, given that σ/S1 and ε/S2 are numbers of the order of 1. The equations show clearly how the Gregorian calendar mildly improves the Julian scheme. The particular choices σ/S1 = 3/4 and ε/S2 = (−75 + 32)/100 are comparatively arbitrary, and even more so the rules according to which the resets are effected – as long as the principle of secularity is not violated. In all probability, Lilius and Clavius would have accepted the choices σ/S1 = 7/9 and ε/S2 = −6/13 as better ones in view of astronomical accuracy. The specifics of their implementation would not raise any fundamental issues.

Algorithms Let us briefly comment on the algorithmic aspects of the Gregorian calendar, both its standard and possible improved versions. As stated earlier, the computational ease with which the Easter date could be predicted, was an important feature of the reform. Clavius took great care to demonstrate this point. The algorithm that he published in 1595 and later [2] still serve their purpose in the Western Christian world. In the meantime, however, simple computational algorithms have been invented that give the same Easter dates. The best known of them is due to Gauß [4]. An equivalent alternative, due to Spencer Jones, may be found in 11

the book of J. Meeus [10]. These almost magic algorithms can be shown [9] to agree with the tables of Clavius for all 5 700 000 years of a period. Nonetheless they are far from being transparent. In particular, it is hard if not impossible to identify which of their parts reflect the essence of the Gregorian scheme, namely, the structure of the calendar equations (26), and where improvements in the numbers σ/S1 , ε/S2 , or changes in the scheme of resets might be inserted. In a recent analysis of the Gauß algorithm, Lichtenberg et al. [8] proposed a reformulation which disentangles an invariant skeleton from parts that are open for change. The analysis has three parts. Part A refers to the lunar motion; it determines the date EM (Easter Moon) of the first full moon in spring. Part B deals with the sun and gives the date MS of the first Sunday in March. Part C evaluates EM and MS for the date of Easter, DE. Let us start with part C which belongs entirely to the invariant part of the algorithm. Assume EM and MS to be given. The date DE of Easter follows MS by an integral number of weeks, DE = MS + v · 7 (v ≥ 3; a date DE in April is considered as March DE + 31). On the other hand, by the Nicaean rule, DE is the first Sunday after EM. Defining ∆, the “Easter distance”, as the number of days between EM and Easter, we have DE = EM + ∆, and hence MS + v · 7 = EM + ∆ . (27) Taking this modulo 7 and remembering that ∆ must be between 1 and 7, we obtain DE = EM + 7 − (EM − MS) mod 7 . (28) Consider now part B of the algorithm. To compute the date MS(X) of the first Sunday in March for the year X, notice that in regular years MS recedes by 1, in leap years by 2. This shows that MS(X), a number between 1 and 7, can be expressed as MS(X) = 7 − (X + int (X/4) + GS(K)) mod 7 ,

(29)

where K = int (X/100) numbers the century to which X belongs, and GS(K) takes account of the number of leaps that were omitted in secular years, after a certain standard year. This is where the variable part of the algorithm comes in. For the Julian calendar, GS(K) vanishes identically. In the Gregorian calendar, the leaps are omitted when X = 0 mod 100, but not when X = 0 mod 400. This leads to 3K + 3 (30) GS(K) = 2 − int (X/100) + int (X/400) = 2 − int 4 where the constant 2 may be determined by look-up from any year after 1582 (e. g., MS(1999) = 7). If the rules determining leap years were to be 12

changed, it is obvious that GS(K) is what must be modified. For example, the Milankovi´c calendar requires 7K + 2 . (31) 9 Between 1600 and 2799 it gives the same values for GS(K). The first deviation occurs in the year 2800 where the Gregorian rule gives 21, the Milankovi´c rule 22. (The Oriani calendar would require GS(K)= 2 − int ((7K + 7)/9).) Finally, consider the lunar motion, or part A of the algorithm. It determines the date EM(X) of the first full moon on or after March 21 in the year X. The basic reference is the Metonic cycle of 19 years. A lunar parameter A(X) = X mod 19 is introduced to label the year X by its position in this cycle. (A(X) + 1 is the golden number of the year X.) Without corrections to be discussed, EM either increases by 19 days or decreases by 11 (once per cycle by 12: saltus lunae), depending on which comes first on or after March 21. The expression for EM(X) thus takes the form GS(K) = 2 − int

D(X) = (19 · A(X) + GM(K)) mod 30, EM(X) = 21 + D(X),

(32)

where GM(K) accounts for the secular resets of the lunar date: 8K + 13 3K + 3 − int , (33) 4 25 where the constant 15 may again be determined by look-up from any year after 1582, for example, EM(1998)=31, i. e., March 31. It is easy to give the modified rule that corresponds to (25) and agrees with the Gregorian values between 1583 and 2299: 6K + 3 . (34) GM(K) = 15 + int 13 (For K = 22, 23, 24, 25, the formula (33) gives the sequence of numbers GM = 25, 26, 25, 26, whereas (34) gives the more satisfactory sequence GM = 25, 25, 26, 26. The sequence (34) cannot decrease.) One last correction must be applied to EM(X) in order to comply with the Alexandrian canon. If EM(X) as calculated from (32) comes out as 50, corresponding to April 19, it is decreased by 1 because this could not happen with the Julian-Metonic rules. If EM(X) is 49, a similar Alexandrian rule requires to reduce it by 1, provided the golden number A is 11 or greater. This tribute to tradition can be paid in terms of GM(K) = 15 + int

EM(X) → EM(X) − R(X),   D D D A R(X) = int 29 + int 28 − int 28 int 11 . 13

(35)

To wrap it up, we present the algorithm in the form in which it should be entered into a computer (as usual, this is the reverse order in which our understanding proceeds). Given a year X, determine A. the date EM of the first full moon in spring, by the steps 1. K = int(X/100) 2. A = X mod 19 3. GM(K) = accumulated secular resets of the epact 4. D = (19 · A + GM(K)) mod 30   D D A D + int 28 − int 28 int 11 5. R = int 29 6. EM = 21 + D − R B. the date MS of the first Sunday in March, from 1. GS(K) = accumulated number of secular non-leap years 2. MS = 7 − (X + int (X/4) + GS(K)) mod 7 C. the date DE of Easter, from 1. ∆ = 7 − (EM − MS) mod 7 2. DE = EM + ∆ The Gregorian routines GM(K) and GS(K) are given in (33) and (30), respectively. Their possible modifications, based on ε/S2 = −6/13 and σ/S1 = 7/9, are given in (34) and (31). Between 1600 and 2300, and again from 2500 through 2800, there is perfect agreement in their predictions of Easter. Before 1600, and after 2800, there is a difference because 1600 and 2800 are leap years in the Gregorian system but not according to Milankovi´c. Between 2300 and 2500, the different epacts in the two systems give 34 different Easter dates, the first time for 2302 when the Gregorian calendar puts Easter on April 20, the modified calendar on April 13.

Beauty and wisdom of the Gregorian scheme Lilius and Clavius took it for granted that calendars ought to be based on the average motion of sun and moon. They were well aware of the fact that prediction of the astronomically exact vernal equinox and first full moon in spring is a formidable computational task, impossible at their time, and, on time scales of thousands of years, a non-trivial challenge even for modern 14

computing. It is part of the tradition going back to Babylon and Egypt, preserved in both the Julian and the Hebrew calendars, to focus on average periods and ignore fluctuations. The only period that was not averaged until the middle of the 19th century, was the length of a solar day. But that was only because clocks weren’t precise enough to show the average time. It was generally considered to be a progress when the universal system of mean times was introduced in the 1890s. The average periods of sun and moon were well known around 1600, from long term measurements. The Gregorian calendar incorporates the best of that knowledge, and we may marvel at the high degree of that precision. The serious astronomers of the time, notably Johannes Kepler, were ardent supporters of the scheme, irrespective of their religious affiliation. If it took England until September 1752 before they adopted the reform, or Russia until February 1918, the reasons were purely politically. On the other hand, to the extent that the Gregorian values for the length of a year Y and the number of months per year N are off the true values, Lilius and Clavius would have been the first to call for amendments. For example, the value YG = 365.2425 has turned out to be three units too large in the last digit. As a result, the astronomical vernal equinox is gradually shifting to earlier dates (March 20 being the most frequent date at present). This could be corrected without compromising the Gregorian spirit; the Greek Milankovi´c proposal seems to be the best solution. Corrections of the average lunar period are less urgent because the Gregorian choice NG is amazingly good. Nevertheless, a slight improvement might be effected with the choice ε/S2 = −6/13 because six backsteps of the epact in 1300 years seem to be a simpler and more natural scheme than the somewhat artificial rule of going 75 steps one way, and 32 the other, every 10 000 years. Again, we do not doubt that Lilius and Clavius would have advocated this slight modification had they known the astronomical facts with better precision. Beauty and wisdom of the Gregorian calendar reside in its combination of astronomical accuracy (in the average values YG and NG ) and respect for tradition. It preserves three strands of human culture: the Babylonian-Greek Metonic cycle; the Julian calendar with its origin in Egypt and Rome; the Nicaean definition of Easter which is no less Jewish than Christian. None of these features could be removed without serious damage to the whole.

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References [1] Beda Venerabilis De temporum ratione, CChr, Ser. Lat. vol. 123b, Turnhout 1977. [2] Ch. Clavius, Romani Calendarii a Gregorio XIII. P. M. restituti Explicatio, Rome 1595 and 1603. Opera Mathematica, Tom. V, Mainz 1612. [3] G. V. Coyne, M. A. Hoskin, O. Pedersen, Gregorian Reform of the Calendar, Proc. Vatican Conf. to commemorate its 400th anniversary 15821982, Specola Vaticana 1983. [4] C. F. Gauß, Berechnung des Osterfestes, Monatl. Correspondenz zur Bef¨orderung der Erd- und Himmelskunde, Aug. 1800. Werke VI, 73-79, G¨ottingen 1874. Erratum: Z. f. Astron. u. verw. Wissensch. 1 (1816) 158. Werke XI/1, 201, G¨ottingen 1927. [5] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford 1979. [6] L. Ideler, Lehrbuch der Chronologie, Berlin 1831. [7] H. Lichtenberg, Die Struktur des Gregorianischen Kalenders, anhand der Schwankungen des Osterdatums entschl¨ usselt, Sterne und Weltraum 3/1994, 194-201. [8] H. Lichtenberg, L. Gerhards, A. Graßl, Z. Zemanek, Die Struktur des Gregorianischen Kalenders, Sterne und Weltraum 4/1998,326-332. [9] H. Lichtenberg, Zur Interpretation der Gaußschen Osterformel und ihrer Ausnahmeregeln, Historia Mathematica 24(1997), 441-444. [10] J. Meeus, Astronomical Algorithms, Willmann-Bell, Richmond VA 1991. [11] M. Milankovitch, Das Ende des Julianischen Kalenders und der neue Kalender der orientalischen Kirche, in: Astron. Nachr. 220(1923/24)/5279, 379-384. [12] J. D. North, The Western Calendar – “Intolerabilis, Horribilis, et Derisibilis”; Four Centuries of Discontent, in: [3], pp. 75-113. [13] B. Oriani, De usu fractionum continuarum ad inveniendos Ciclos Calendarii novi & veteris, Ephem. Astron. Anni 1786, Mediolani 1785, pp. 132-154.

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[14] S. I. Seleschnikow, Wieviel Monde hat ein Jahr?, Verlag MIR, Moskau, and Urania-Verlag, Leipzig, 1981. [15] C. Tondering, The Calendar FAQ, http://www.pip.dknet.dk/ ct/calendar.html [16] World Council of Churches, Towards a Common Date of Easter, Aleppo (Syria) 1997. [17] A. Ziggelaar The Papal Bull of 1582 Promulgating a Reform of the Calendar, in: [3], pp. 201-239.

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