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 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
= 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 Milanković (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-Milanković 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-Milanković 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 Milanković 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 Milanković
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 Milanković.
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
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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 Milanković
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
Beförderung der Erd- und Himmelskunde, Aug. 1800. Werke VI, 73-79,
Göttingen 1874. Erratum: Z. f. Astron. u. verw. Wissensch. 1 (1816)
158. Werke XI/1, 201, Göttingen 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 entschlüsselt, 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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