 
POUVILLON RESTORATION PROJECT  November 2011
A discussion of the Easter calculator's functions,
the computus.
Front elevation view of the Easter calculator set within the movement.
These photos of the front and side elevations of the
Easter calculator show how diminuitive the mechanism is.
The calculator has 7 indications. Starting from the top
dial and working clockwise we have the following:
Dominical Letter

Indiction

Golden Number

Day of week the 1^{st}
of January falls on

Solar cycle

Epact

The center dial
indicates the date that Easter will fall in
the given year.

I will explain what each dial means and if it is used
in the calculation of the date of Easter; not all dials are used for this
purpose. It is an ecclesiastical calculator involved with the Catholic and
Protestant Church’s celebration of Easter. While Christmas is what many
consider to be the most celebrated religious holiday, Easter is the most
important religious event for the church. Easter is what validates all of
Christ’s teachings.
These religious functions are not restricted to just
the calculator, although the calculator is the most obvious and unusual
mechanism in connection with this. Another portion of the clock is used in
this calculation as well as other various movable and fixed feasts as they
relate to the church’s main celebration of Easter. The rolling moon on the
right side of the clock has an unusual two month duration dial with one
sector having 29 divisions and the other 30. This was difficult to
understand until viewed within the ecclesiastical context. About a quarter
of the clock Pouvillon designed is with the ecclesiastical functions in
mind. This movement is very much a religious device with many other fancy
complications added in. It appears that the calculator
was added quite late in the life of the clock before the passing of
Pouvillon in 1969 at the age of 91. We have a date stamped on the Easter
disc of 1946 and two newspaper articles from 1953 and 1955 indicating that
he was still perfecting the Easter calculator. It is also possible that he
added the movable feast indicators on the tellurian dial during this time.
In other words, most of the religious functions could have been added to the
clock when Pouvillon was in his late 60’s. The clock could have taken a more
religious tone in Pouvillon’s later life. I have left all of the links
originally provided in the Wikipedia references so one can further expand on
those items highlighted in blue.
In principle, the date of Easter is defined as the Sunday following the
full moon
following
vernal equinox (the socalled
Paschal Full Moon). In the current year it’s pretty easy to find the
vernal equinox, it’s commonly known as the first day of Spring, March
21st, when the
length of the day as well as the night are equal. Then look up the current
table of the moon’s phases, find when the first full moon occurs after the
first day of Spring and then pick the following Sunday. BUT, here the
simplicity ends. The phases of the moon change over the calendar and there
are irregularities in the calendar which require corrections from time to
time; the leap year every four years, and the 400 and 1000 year corrections.
So to be able to predict the day
Easter will fall in the future get’s quite complex.
The church turned to a series of complicated steps and tables to take
these variations into account using the
Dominical Letter, Golden Number,
Solar Cycle and
Epact all of which are meant to
take into account the variations of the calendar and celestial events to
reliably predict Easter. This they called the
computus or system of reckoning.
The computus is thus the procedure of determining the first Sunday
after the first ecclesiastical full moon that falls on or after March 21st.
The dial descriptions below are taken from references
in Wikipedia as well as the book Jens Olsen’s Clock, Otto Mortensen.
Please be aware that I do not personally have full command of the following
calculations and if you find inconsistencies, let me know.
Dominical Letter^{
}:
Dominical letters are letters A, B, C, D, E, F and G assigned to
days in a cycle of seven with the letter A always set against January 1st as
an aid for finding the day of the week of a given calendar date and in
calculating
Easter.
A common year is assigned a single dominical letter, indicating which
letter is Sunday (hence the name, from Latin dominica for Sunday).
Thus, 2011 is B, indicating that B days are Sunday. Leap years are given two
letters, the first indicating the dominical letter for January 1  February
28 (or February 24, see below), the second indicating the dominical letter
for the rest of the year.
In leap years, the leap day may or may not have a dominical letter. In the
original 1582 Catholic version, it did, but in the 1752 Anglican version it
did not. The Catholic version caused February to have 29 days by doubling
the sixth day before 1 March, inclusive, because 24 February in a common
year is marked "duplex", thus both halves of the doubled day had a dominical
letter of F.^{ }The Anglican
version added a day to February that did not exist in common years, 29
February, thus it did not have a dominical letter of its own.
In either case, all other dates have the same dominical letter every
year, but the days of the weeks of the dominical letters change within a
leap year before and after the intercalary day, 24 February or 29 February.
The dominical letter of a year determines the days of week in its calendar:
Dominical letters were a device adopted from the Romans by chronologers to
aid them in finding the day of the week corresponding to any given date, and
indirectly to facilitate the adjustment of the "Proprium de Tempore" to the
"Proprium Sanctorum" when constructing the ecclesiastical calendar for any
year.^{ }The Christian Church, due
to its
complicated system of movable and immovable feasts,
has long been concerned with the regulation and measurement of time. To
secure uniformity in the observance of feasts and fasts, it began, even in
the patristic age, to supply a system of reckoning (computus) by
which the relation of the solar and lunar years might be accommodated and
the celebration of Easter determined. It adopted the astronomical methods
that were available at the time, and these methods and their methodology
have become traditional and are perpetuated in a measure to this day, even
the reform of the calendar, in the prolegomena to the Breviary and Missal.
Calculation
The dominical letter of a year can be calculated based on any method for
calculating the day of the week,
with letters in reverse order compared to numbers indicating the day of the
week.
For example:
·
ignore
periods of 400 years

·
considering
the second letter in the case of a leap
year:

o
for one
century within two multiples of 400, go
forward two letters from BA for 2000, hence
C, E, G.

o
for remaining
years, go back one letter every year, two
for leap years (this corresponds to writing
two letters, no letter is skipped).

o
to avoid up
to 99 steps within a century, there is a
choice of several shortcuts, e.g.:

§
go back one
letter for every 12 years

§
ignore
multiples of 28 years (note that when
jumping from e.g. 1900 to 1928 the last
letter of 1928 is the same as the letter of
1900)

§
apply steps
between multiples of 10, writing from right
to left:


2000 1990 1980 1970 1960 1950 1940 1930 1920 1910 1900 
BA G FE D CB A GF E DC B .G 

·
Note the
dummy step (we skip A between 1900 and 1910)
because 1900 is not a leap year. For
example, to find the Dominical Letter of the
year 1913:

·
1900 is G

·
1910 is B

·
count B A GF
E, 1913 is E Similarly, for 2007:

·
2000 is BA

·
count BA G F
E DC B A G, 2007 is G, For 2065:

·
2000 is BA

·
2012 is AG,
2024 is GF, 2036 is FE, 2048 is ED, 2060 is
DC, then B A G FE D, 2065 is D

·
or from 2000
to 2060 in steps of 10, written backward: DC
B AG F ED C BA, starting from 2000 is BA we
get 2060 is DC, then again B A G FE D, 2065
is (or,writing the last part backward
too: D FE G A B <DC> B AG F ED C BA)

·
or ignore 56
years, 2056 is BA, count G F E DC B A G FE
D, 2065 is D

Epact:
The epact (Latin epactae, from Greek: epaktai hèmerai =
added days) was originally defined as the age of the moon in days on January
1, and occurs primarily in connection with tabular methods for determining
the date of Easter. The Epact arises from the inequality of the solar and
lunar year. It varies (usually by 11 days) from year to year,
because of the difference between the solar year of 365 days and the lunar
year of 354 days.
Epacts
as they
relate to the Lunar calendar
Epacts are used to find the date
in the lunar calendar from the date in the common solar calendar. As the
epact calculation gives 30 different epacts, while the time between two
consecutive mean new moons is only a little over 29 ½
days, it has been established the
cyclic lunar months should change at a rate of 29 and 30 days, respectively.
This explains why the lunar dial has the unusual
arrangement of the moon cycling over a two month period with the ring dial
bisected by divisions 1 through 29 on one sector and 1 through 30 on the
other.
Solar and lunar
years
A (solar) calendar year has 365 days (366 days in
leap years). A lunar year has 12 lunar months which alternate between 30 and
29 days (in leap years, one of the lunar months has a day added).
If a solar and lunar year start on the same day, then after one year, the
start of the solar year is 11 days after the start of the lunar year; after
two years, it is 22 days after. These excess days are epacts, and are added
to the day of the solar year to determine the day of the lunar year.
Whenever the epact reaches or exceeds 30, an extra (embolismic or
intercalary) month is inserted into the lunar calendar, and the epact is
reduced by 30.
Leap days extend both the solar and lunar year, so they do not affect
epact calculations for any other dates.
The 19year cycle
The
tropical year is about 365¼ days, while the
synodic month is also slightly longer than 29½ days on average.
This gets corrected in the following way. Nineteen tropical years are as
long as 235 synodic months (Metonic
cycle). A cycle can last 6939 or 6940 full days, depending on
whether there are 4 or 5 leap days in this 19year period.
After 19 years the lunations should fall the same way in the solar years,
so the epact should repeat after 19 years. However, 19 × 11 = 209 , and this
is not an integer multiple of the full cycle of 30 epact numbers (209
modulo
30 = 29, not 0). So after 19 years the epact must be corrected by +1 in
order for the cycle to repeat over 19 years. This is the saltus lunae
(leap of the moon). The sequence number of the year in the 19year cycle is
called the
Golden Number. The extra 209 days fill 7 embolismic months, for a total
of 19×12 + 7 = 235 lunations. This correction is
discussed first in our October installment, photos 49002 through 49006 and
the correction is implemented in the November installment photos 56001
through 56016.
Golden Number:
The number of the year in the lunar cycle is the golden number and was
found by Meton, the Athenian who in the year 430 discovered that 19 tropic
years coincided very nearly with 235 synodic periods of the moon, in other
words the moon is in the same position in the sky with respect to the
surrounding stars every 19 years (the Metonic cycle), and the ancient Greeks
thought this so extraordinary that they are said to have cut these numbers
on buildings and highlighted them in gold. This term
was first used in the computistic poem Massa Compoti by
Alexander de Villa Dei in 1200. A later scribe added the Golden Number
to tables originally composed by
Abbo of Fleury in 988.
The lunar cycle is a period of 19 years, after which the phases of the
moon again fall on approximately the same dates in the solar year. The
number of the year in the period is found by adding 1 to the number of the
year and dividing by 19; the remainder is the number of the year. If there
is no remainder, the number of the yearor the golden number is 19. (In
mathematics this can be expressed as (year number
modulo 19) +1.)
For example, 2011 divided by 19 gives 105, remainder 16. Adding the
number before to the number that you are using gives a golden number of 17.
In computing, the modulo operation finds the
remainder
of
division of one number by another.
Solar Cycle:
The number of the year in the solar cycle is found
according to this rule:
The solar cycle is the period of 4 times 7 = 28 years,
after which period of days of the week in the Julian calendar again fall on
the same dates in the solar year. The number of the year in the period is
found by adding 9 to the golden number of the year and dividing by 28; the
remainder is then the number of the year. If there is no remainder, the
number is 28.
Cycle of Indiction:
The cycle of indiction is a period of 15 years but has
no connection with astronomical periods. Nothing is known for certain about
its origin, perhaps it was used by the Romans, possibly as an interest or
fiscal term, but it is continuous from that time throughout the reckoning.
The number of the year in the indiction is found by adding 3 to the number
of the year and dividing by 15. The remainder is the number of the year. If
there is no remainder, the number of the year is 15.
The three cycles, the solar cycle, golden number and
indiction together form a period of 28 x 19 x 15 = 7,980 years. This period
has often been used in ancient times for the dating of documents; in
addition to the rather uncertain date, the number of the year in the three
cycles were given. In this way the year can be determined with perfect
certainty. If a year is denoted by, say, solar cycle 12, indiction 3 and
golden number 9, it will be an easy matter to fix the year at 255 after our
reckoning. The three cycle numbers will only fit in for that year or for
years 7,980 years before or after it. The period 7,980 years is called the
Julian period. Year 1933, which has solar cycle 10, indiction 1 and golden
number 15, is the 6,645^{th} year in the Julian period. This dial is
not needed for the Easter calculation
Day of the week that January
1^{st} falls on
This dial is self explanatory and not used for the
Easter calculation.
Theory
behind the calculations
To each day in a calendar year, the Easter cycle implicitly assigns a
lunar age, which is a whole number from 1 to 30. The moon's age starts at 1
and increases to 29 or 30, then starts over again at 1. Each period of 29
(or 30) days of the moon's age makes up a lunar month. With occasional
exceptions, 30day lunar months alternate with 29day months. So a lunar
year of 12 lunar months is reckoned to have 354 days. The solar year is 11
days longer than the lunar year. Supposing a solar and lunar year start on
the same day, with a crescent
new
moon indicating the beginning of a new lunar month on 1 January,
then the lunar year will finish first, and 11 days of the new lunar year
will have already passed by the time the new solar year starts. After two
years, the difference will have accumulated to 22: the start of lunar months
fall 11 days earlier in the solar calendar each year. These days in excess
of the solar year over the lunar year are called epacts (Greek: epakta
hèmerai). It is necessary to add them to the day of the solar year to
obtain the correct day in the lunar year. Whenever the epact reaches or
exceeds 30, an extra (socalled
embolismic or
intercalary) month of 30 days has to be inserted into the lunar
calendar; then 30 has to be subtracted from the epact.
Note that
leap
days are not counted in the schematic lunar calendar: The cycle
assigns to the first day of March after the leapday the same age of the
moon that the day would have had if there had been no leapday. The
nineteenyear cycle (Metonic
cycle) assumes that 19
tropical years are as long as 235
synodic
months. So after 19 years the lunations should fall the
same way in the solar years, and the epacts should repeat. However, 19 × 11
= 209 ≡ 29 (modulo 30), not 0 (mod 30); that is, 209 divided by 30 leaves a
remainder of 29 instead of being an even multiple of 30. So after 19 years,
the epact must be corrected by +1 day in order for the cycle to repeat. This
is the socalled saltus lunae or moon's leap. The extra 209 days fill
seven embolismic months, for a total of 19 × 12 + 7 = 235 lunations. The
sequence number of the year in the 19year cycle is called the "Golden
Number", and is given by the formula:
GN = Y modulo 19 + 1
That is, the remainder of the year number Y in the
Christian era when divided by 19, plus one.
Using the method just described, a period of 19 calendar years is also
divided into 19 lunar years of 12 or 13 lunar months each. In each calendar
year (beginning on 1 January) one of the lunar months must be the first one
within the calendar year to have its 14th day (its formal
full moon) on or after 21 March. This lunar month is the Paschal or
Eastermonth, and Easter is the Sunday after its 14th day (or, saying
the same thing, the Sunday within its third week.) The Paschal lunar
month always begins on a date in the 29day period from 8 March to 5 April
inclusive. Its 14th day, therefore, always falls on a date between 21 March
to 18 April inclusive, and the following Sunday then necessarily falls on a
date in the range 22 March to 25 April inclusive. In the solar calendar
Easter is called a moveable feast since its date varies within a 35day
range. But in the lunar calendar, Easter is always the third Sunday in the
Paschal lunar month, and is no more "moveable" than any holiday that is
fixed to a particular day of the week and week within a month.
Tabular
methods  background
The claim by the Roman Catholic Church in the 1582
papal
bull
Inter gravissimas, which promulgated the
Gregorian calendar, that it restored "the celebration of Easter
according to the rules fixed by ... the great ecumenical council of Nicæa"
was based on a false claim by
Dionysius Exiguus (525) that "we determine the date of Easter Day ... in
accordance with the proposal agreed upon by the 318 Fathers of the Church at
the Council in Nicaea." The
First Council of Nicaea (325) only stated that Easter was to be
celebrated by all Christians on the same Sunday—it did not fix any rules to
determine which Sunday. The medieval computus was based on the Alexandrian
computus, which was developed by the
Church of Alexandria during the first decade of the 4th century using
the
Alexandrian calendar. The
Eastern Roman Empire accepted it shortly after 380 after converting the
computus to the
Julian calendar. Rome accepted it sometime between the sixth and 9th
centuries. The British Isles accepted it during the 7th century except for a
few monasteries.
Francia
(all of Western Europe except Scandinavia (pagan), the British Isles, the
Iberian peninsula, and southern Italy) accepted it during the last
quarter of the 8th century. The last
Celtic monastery to accept it,
Iona,
did so in 716, whereas the last English monastery to accept it did so in
931. Before these dates other methods were used which resulted in dates for
Easter Sunday that sometimes differed by up to five weeks.
This method for the computation of the date of Easter was introduced with
the
Gregorian calendar reform in 1582.
The general method of working was given by
Clavius
in the Six Canons (1582), and a full explanation followed in his "Explicatio"
(1603).
Easter Sunday is the Sunday following the Paschal Full Moon date. The
Paschal Full Moon date is the Ecclesiastical Full Moon date following 20
March. The Gregorian method derives Paschal Full Moon dates by determining
the epact for
each year. The epact can have a
value from * (=0 or 30) to 29 days. The first day of a lunar month is
considered the day of the crescent
new moon.
The 14th day is considered the day of the
full moon.
Historically the Paschal Full Moon date for a year was found from its
sequence number in the Metonic cycle, called the golden number, which cycle repeats the lunar phase on a certain date
every 19 years. This method was abandoned in the Gregorian reform because
the tabular dates go out of sync with reality after about two centuries, but
from the epact method a simplified table can be constructed that has a
validity of one to three centuries.
The epacts for the (2009) Metonic cycle are:
Year

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

Golden
Number

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Epact

29

10

21

2

13

24

5

16

27

8

19

*

11

22

3

14

25

6

17

Paschal
Full Moon
date

14A

3A

23M

11A

31M

18A

8A

28M

16A

5A

25M

13A

2A

22M

10A

30M

17A

7A

27M

(M=March, A=April)
This table can be extended for previous and following 19year periods,
and is valid from 1900 to 2199.
The epacts are used to find
the dates of the New Moon in the following way: Write down a table of all
365 days of the year (the leap day is ignored). Then label all dates with a
Roman number counting downwards, from "*" (= 0 or 30), "xxix" (29), down to
"i" (1), starting from 1 January, and repeat this to the end of the year.
However, in every second such period count only 29 days and label the date
with xxv (25) also with xxiv (24). Treat the 13th period (last eleven days)
as long, though, and assign the labels "xxv" and "xxiv" to sequential dates
(26 and 27 December respectively). Finally, in addition, add the label "25"
to the dates that have "xxv" in the 30day periods; but in 29day periods
(which have "xxiv" together with "xxv") add the label "25" to the date with
"xxvi". The distribution of the lengths of the months and the length of the
epact cycles is such that each civil calendar month starts and ends with the
same epact label, except for February and for the epact labels xxv and 25 in
July and August. This table is called the calendarium. The
ecclesiastical new moons for any year are those dates at which the epact for
the year is entered. If the epact for the year is for instance 27, then
there is an
ecclesiastical new moon on every date in that year that has the epact
label xxvii (27).
Also label all the dates in the table with letters "A" to "G", starting
from 1 January, and repeat to the end of the year. If, for instance, the
first Sunday of the year is on 5 January, which has letter E, then
every date with the letter "E" will be a Sunday that year. Then "E" is
called the
Dominical Letter for that
year (from Latin: dies domini, day of the Lord). The Dominical Letter
cycles backward one position every year. However, in leap years after 24
February the Sundays will fall on the previous letter of the cycle, so
leap years have two Dominical Letters: the first for before, the second for
after the leap day.
In practice, for the purpose of calculating Easter, this need not be done
for all 365 days of the year. For the epacts, you will find that March comes
out exactly the same as January, so one need not calculate January or
February. To also avoid the need to calculate the Dominical Letters for
January and February, start with D for 1 March. You need the epacts only
from 8 March to 5 April. This gives rise to the following table:
Label

March

DL

April

DL

*

1

D



xxix

2

E

1

G

xxviii

3

F

2

A

xxvii

4

G

3

B

xxvi

5

A

4

C

25

6

B

4

C

xxv

6

B

5

D

xxiv

7

C

5

D

xxiii

8

D

6

E

xxii

9

E

7

F

xxi

10

F

8

G

xx

11

G

9

A

xix

12

A

10

B

xviii

13

B

11

C

xvii

14

C

12

D

xvi

15

D

13

E

xv

16

E

14

F

xiv

17

F

15

G

xiii

18

G

16

A

xii

19

A

17

B

xi

20

B

18

C

x

21

C

19

D

ix

22

D

20

E

viii

23

E

21

F

vii

24

F

22

G

vi

25

G

23

A

v

26

A

24

B

iv

27

B

25

C

iii

28

C



ii

29

D



i

30

E



*

31

F




Example: If the epact is, for
instance, 27 (Roman xxvii), then there will
be an ecclesiastical new moon on every date
that has the label "xxvii". The
ecclesiastical full moon falls 13 days
later. From the table above, this gives a
new moon on 4 March and 3 April, and so a
full moon on 17 March and 16 April.
Then Easter Day is the first Sunday after
the first ecclesiastical full moon on or
after 21 March. This definition uses "on or
after 21 March" to avoid ambiguity with
historic meaning of the word "after". In
modern language, this phrase simply means
"after 20 March". The definition of "on or
after 21 March" is frequently incorrectly
abbreviated to "after 21 March" in published
and webbased articles, resulting in
incorrect Easter dates.
In the example, this Paschal full moon is
on 16 April. If the dominical letter is E,
then Easter day is on 20 April.
The label 25 (as distinct
from "xxv") is used as follows: Within a
Metonic cycle, years that are 11 years apart
have epacts that differ by one day. Now
short lunar months have the labels xxiv and
xxv at the same date, so if the epacts 24
and 25 both occur within one Metonic cycle,
then in the short months the new (and full)
moons would fall on the same dates for these
two years. This is not actually possible for
the real Moon: the dates should repeat only
after 19 years. To avoid this, in years that
have epacts 25 and with a Golden Number
larger than 11, the reckoned new moon will
fall on the date with the label "25" rather
than "xxv". In long lunar months, these are
the same; in short ones, this is the date
which also has the label "xxvi". This does
not move the problem to the pair "25" and
"xxvi," because that would happen only in
year 22 of the cycle, which lasts only 19
years: there is a saltus lunae in
between that makes the new moons fall on
separate dates.
The Gregorian calendar has a correction
to the solar year by dropping three leap
days in 400 years (always in a century
year). This is a correction to the length of
the solar year, but should have no effect on
the Metonic relation between years and
lunations. Therefore the epact is
compensated for this (partially—see
epact) by subtracting one in these
century years. This is the socalled
solar correction or "solar equation"
("equation" being used in its medieval sense
of "correction").
However, 19 uncorrected
Julian years are a little longer than
235 lunations. The difference accumulates to
one day in about 310 years. Therefore, in
the Gregorian calendar, the epact gets
corrected by adding 1 eight times in 2500
(Gregorian) years, always in a century year:
this is the socalled lunar correction
(historically called "lunar equation"). The
first one was applied in 1800, and it will
be applied every 300 years except for an
interval of 400 years between 3900 and 4300,
which starts a new cycle.
The solar and lunar corrections work in
opposite directions, and in some century
years (for example, 1800 and 2100) they
cancel each other. The result is that the
Gregorian lunar calendar uses an epact table
that is valid for a period of from 100 to
300 years. The epact table listed above is
valid for the period 1900 to 2199.

Now that we have gone through the very tortured
path of using the tabular method to determine Easter, the second table below
is a mathematical algorithm that will give us the result we are looking for
where the only information needed is
the year we are looking to determine Easter’s date. Note the
mathematical expressions of mod
and floor. Mod or modulo has
been defined. The floor function is also called the
greatest integer function,
and its value at x is called the
integral part or
integer part of x.
An example below will clarify:
Sample value x

Floor
[x]

Ceiling [x]

Fractional part
{x}

12/5
= 2.4

2

3

2/5 = 0.4

2.7

2

3

0.7

−2.7

−3

−2

0.3

−2

−2

−2

0

We are only concerned with the
floor function and positive expressions. And now for the algorithm to
determine Easter in the Gregorian calendar.
Gregorian algorithm
"A New York correspondent" submitted the algorithm below for determining the
Gregorian Easter to the journal
Nature in 1876. It has been reprinted many times, in 1877 by
Samuel Butcher in The Ecclesiastical Calendar, in 1922 by
H. Spencer Jones in General Astronomy, in 1977 by the Journal
of the British Astronomical Association,
in 1977 by
The Old Farmer's Almanac, in 1988 by Peter DuffettSmith in
Practical Astronomy with your Calculator, and in 1991 by
Jean
Meeus in Astronomical Algorithms.
Expression

Y = 1961

Y = 2009

a = Y mod
19

a = 4

a = 14

b = floor (Y
/ 100)

b = 19

b = 20

c = Y mod
100

c = 61

c = 9

d = floor (b
/ 4)

d = 4

d = 5

e = b mod
4

e = 3

e = 0

f = floor ((b
+ 8) / 25)

f = 1

f = 1

g = floor ((b
− f + 1) / 3)

g = 6

g = 6

h = (19a
+ b − d − g + 15) mod
30

h = 10

h = 20

i = floor (c
/ 4)

i = 15

i = 2

k = c mod
4

k = 1

k = 1

L = (32 + 2e
+ 2i − h − k) mod 7

L = 1

L = 1

m = floor ((a
+ 11h + 22L) / 451)

m = 0

m = 0

month = floor ((h
+ L − 7m + 114) / 31)

month = 4
(April)

month = 4
(April)

day = ((h
+ L − 7m + 114) mod 31) + 1

day = 2

day = 12

Gregorian Easter

2 April 1961

12 April 2009



Just for fun, I present the graphic above. It shows the percentage of time
easter will fall on a certain date in March or April throughout the
complete 5,700,000 year cycle which is calculated as follows: The solar and
lunar corrections repeat after 4 × 25 = 100 centuries. In that period, the
epact has changed by a total of −1 × (3/4) × 100 + 1 × (8/25) × 100 = −43 ≡
17 mod 30. This is prime to the 30 possible epacts, so it takes 100 × 30 =
3000 centuries before the epacts repeat; and 3000 × 19 = 57,000 centuries
before the epacts repeat at the same Golden Number. This period has
(5,700,000/19) × 235 + (−43/30) × (57,000/100) = 70,499,183 lunations. So
the Gregorian Easter dates repeat in exactly the same order only after
5,700,000 years = 70,499,183 lunations = 2,081,882,250 days. However, the
calendar will already have to have been adjusted after some millennia
because of changes in the length of the vernal equinox year, the synodic
month, and the day due to the slowing of the Earth's rotation.


