Calculating the Julian Easter

In the next two days we will compute the date of Easter in the Julian calendar, in two different ways. First, we will proceed much as Dionysius did, by compiling and using tables. Then, we will replace the tabular considerations with modular arithmetic computations.

We must deal separately with the solar calendar, the lunar calendar, and the day of the week. We introduce labels, which deal with each of these.

Let us begin with solar year. We label each of the 365 days of the year (starting with January 1^st) by the seven letters A, B, C, …, G in turn. We then call the dominical letter for a particular year to be the letter label on the first Sunday of the year. For example, the ordinary year 2009 starts on Thursday, January 1^st. This means that its dominical letter is D. Then every day of the year that is labeled with D will also be a Sunday. We do something a little different with a leap year like 2008. We still label the 365 ordinary days in order, but we do not label the leap day. Now, since 2008 starts on a Tuesday, its dominical letter is F. But later in the year (after February 29^th) Sundays will now be labeled with E. Consequently, leap years have associated with them two dominical letters: in the case of 2008, those letters are F and E. I labeled the 14 Julian years on the perpetual calendar I handed out with the appropriate letter (for common years) and letters (for leap years).

Now let’s consider the 28 years of the solar cycle, one by one. Let’s start with 1 BC. This is a leap year (why???). It turns out that this year started on a Thursday. Consequently, it dominical letters are D, C. The next year (1 AD) begins on a Saturday, and so its dominical letter is B. We can continue this process as follows:

Date	Dominical Letters	Calendar Number	J	Date	Year Type
1BC	D, C	12	9	1400	Leap
1AD	B	7	10	1401	Common
2AD	A	1	11	1402	Common
3AD	G	2	12	1403	Common
4AD	F, E	10	13	1404	Leap
5AD	D	5	14	1405	Common
6AD	C	6	15	1406	Common
7AD	B	7	16	1407	Common
8AD	A, G	8	17	1408	Leap
9AD	F	3	18	1409	Common
10AD	E	4	19	1410	Common
11AD	D	5	20	1411	Common
12AD	C, B	13	21	1412	Leap
13AD	A	1	22	1413	Common
14AD	G	2	23	1414	Common
15AD	F	3	24	1415	Common
16AD	E, D	11	25	1416	Leap
17AD	C	6	26	1417	Common
18AD	B	7	27	1418	Common
19AD	A	1	28	1419	Common
20AD	G, F	9	1	1420	Leap
21AD	E	4	2	1421	Common
22AD	D	5	3	1422	Common
23AD	C	6	4	1423	Common
24AD	B, A	14	5	1424	Leap
25AD	G	2	6	1425	Common
26AD	F	3	7	1426	Common
27AD	E	4	8	1427	Common

The important thing to notice is that this process starts all over in the next year, and will cycle indefinitely (as long as we follow the Julian leap rule). To emphasize this I have inserted some years in the 15^th century that follow the same pattern. There is exactly a 28-year solar cycle! The numbers in the 4^th column give a count of where we are in the 28 year cycle; notice that J=Y+9 (modulo 28), for any Julian year Y. For historical reasons (which we will discuss later), the cycle count begins in 20 AD (and repeats in 48, 76, etc.) Dionysius and the computists did not do the modular arithmetic calculation to get J, the year in the solar cycle. Instead, they would have listed these values in extensive year-by-year tables.

Notice that in the 28-year cycle, each common year appears 3 times, while each of the seven leap years occurs exactly once. That gives us the 28-year cycle.

We are now ready to consider the Christian theoretical moon. As you recall, this follows a basic Metonic cycle, with 12 common and 7 embolismic years. The common years each have 6 30-day months and 6 29-day months. The first six embolismic years add a 30-day month; the last one adds a 29-day month. This makes 115 29-day months and 120 30-day months, or 6935 days altogether. But 19 Julian years makes 6939.75 days. We then must add a leap day every fourth year, and we add that day to whichever lunation happens to include February 24^th. This brings the theoretical Christian sun and moon into perfect synchronization.

The Golden Number is the Dionysius’s way of keeping track of the nineteen years in the Metonic cycle; traditionally, they are denoted by Roman numerals – later we will also use ordinary integers when we want to do modular arithmetic. We will let G stand for the Golden Number for a given year. Dionysius set up his labeling of the Golden Numbers, so that 1 BC is year I of the cycle. This means that if Y is the Julian year, then G=Y+1 (modulo 19). Once again, Dionysius would not have done the modular arithmetic, but would instead have found the Golden Number in extensive year-by-year charts.

The following chart specifies how the months are distributed in each year; the arrangement is a little weird, and was carefully designed so that the lunar almanac we obtain from this has certain important calendric properties we’ll encounter when we get to it. We’ll also explain why we are starting the table with Golden Number III. This table essentially represents the pattern followed by the Christian “theoretical moon”.

G	1	2	3	4	5	6	7	8	9	10	11	12	13	Total
III	30	29	30	29	30	29	30	29	30	29	30	29		354
IV	30	29	30	29	30	29	30	29	30	29	30	29		354
V	30	29	30	29	30	29	30	29	30	30	29	30	29	384
VI	30	29	30	29	30	29	30	29	30	29	30	29		354
VII	30	29	30	29	30	29	30	29	30	29	30	29		354
VIII	30	29	30	30	29	30	29	30	29	30	29	30	29	384
IX	30	29	30	29	30	29	30	29	30	29	30	29		354
X	30	29	30	29	30	29	30	29	30	29	30	29		354
XI	30	30	29	30	29	30	29	30	29	30	29	30	29	384
XII	30	29	30	29	30	29	30	29	30	29	30	29		354
XIII	30	29	30	29	30	29	30	29	30	30	29	30	29	384
XIV	30	29	30	29	30	29	30	29	30	29	30	29		354
XV	30	29	30	29	30	29	30	29	30	29	30	29		354
XVI	30	29	30	29	30	29	30	29	30	30	29	30	29	384
XVII	30	29	30	29	30	29	30	29	30	29	30	29		354
XVIII	30	29	30	29	30	29	30	29	30	29	30	29		354
XIX	30	29	30	30	29	30	29	29	29	30	29	30	29	383
I	30	29	30	29	30	29	30	29	30	29	30	29		354
II	30	29	30	29	30	29	30	29	30	30	29	30	29	384

Once again: this gives us the total of 6935 days. We will later insert the extra leap days, which will then bring the average year to exactly 365.25 days.

We will now begin constructing our lunar almanac! This will be a chart with an entry for every day of the year, arranged in 12 columns (for the twelve solar months), and 31 rows (for the day of the month).

We label the calendar letters easily, just repeating A-G over and over, starting with A for January 1st. We then label days with Golden Numbers, which represent the days on which the new moons occur for that year in the Metonic cycle. This is why the table above starts with III – for this year the first new moon occurs exactly on January 1^st. We then count through the year, month by month, marking the new moons as they come; we use the numbers in the chart above to tell us how long our theoretical lunation lasts. The last new moon in our year occurs on December 21^st. We count this moon as belonging to year III.

The next moon we count as belonging to year IV. Its new moon falls on January 20^th. We continue in this way through all nineteen years of the Metonic cycle. At the very end of this process is a small exception. The last lunation of year II actually ends on December 31^st, and so the first lunation for year III starts on January 1, as we began the process.

The slightly peculiar ordering of the 29 and 30 day moons in the chart above has been arranged exactly so that no day of the year has been assigned more than Golden Number. Since we are inserting 235 labels into 365 days, we actually have days with no Golden Number assigned at all: these days never have a theoretical new moon. With that freedom, and the ability to switch month lengths, it was possible for Dionysius and other computists to arrange such a table.

(Notice that because the real moon has a lunation period that never synchronizes perfectly the real sun, actual lunations can and do start on any day of the year.)

The Julian Lunar Almanac

Day

Jan

Feb

Mar

Apr

May

June

July

Aug

Sept

Oct

Nov

Dec

1

A III

D

D III

G

B XI

E

G XIX

C VIII

F XVI

A XVI

D

F XIII

2

B

E XI

E

A XI*

C

F XIX

A VIII

D XVI

G V

B V

E XIII

G II

3

C XI

F XIX

F XI

B

D XIX

G VIII

B

E V

A

C XIII

F II

A

4

D

G VIII

G

C XIX*

E VIII

A XVI

C XVI

F

B XIII

D II

G

IB X

5

E XIX

A

A XIX

D VIII*

F

B V

D V

G XIII

C II

E

A X

C

6

F VIII

B XVI

B VIII

E XVI

G XVI

C

E

A II

D

F X

B

D XVIII

7

G

C V

C

F V

A V

D XIII

F XII

B

E X

G

C XVIII

E VII

8

A XVI

D

D XVI*

G

B

E II

G II

C X

F

A XVIII

D VII

F

9

B V

E XIII

E V*

A XIII

C XIII

F

A

D

G XVIII

B VII

E

G XV

10

C

F II

F

B II

D II

G X

B X

E XVIII

A VII

C

F XV

A IV

11

D XIII

G

G XIII*

C

E

A

C

F VII

B

D XV

G IV

B

12

E II

A X

A II*

D X

F X

B XVIII

D XVIII

G

C XV

E IV

A

C XII

13

F

B

B

E

G

C VII

E VII

A XV

D IV

F

B XII

D I

14

G X

C XVIII

C X*

F XVIII

A XVIII

D

F

B IV

E

G XII

C I

E

15

A

D VII

D

G VII

B VII

E XV

G XV

C

F XII

A I

D

F IX

16

B XVIII

E

E XVIII*

A

C

F IV

A IV

D XII

G I

B

E IX

G

17

C VII

F XV

F VII*

B XV

D XV

G

B

E I

A

C IX

F

A XVII

18

D

G IV

G

C IV

E IV

A XII

C XII

F

B IX

D

G XVII

B VI

19

E XV

A

A XV*

D

F

B

D I

G IX

C

E XVII

A VI

C

20

F IV

B XII

B IV*

E XII

G XII

C

E

A

D XVII

F VI

B

D XIV

21

G

C I

C

F I

A I

D IX

F IX

B XVII

E VI

G

C XIV

E III

22

A XII

D

D XII*

G

B

E

G

C VI

F

A XIV

D III

F

23

B I

E IX

E I*

A IX

C IX

F XVII

A XVII

D

G XIV

B III

E

G XI

24

C

F

F

B

D

G VI

B VI

E XIV

A III

C

F XI

A XIX

25

D IX

G XVII

G IX*

C XVII

E XVII

A

C

F III

B

D XI

G XIX

B

26

E

A VI

A

D VI

F VI

B XIV

D XIV

G

C XI

E XIX

A

C VIII

27

F XVII

B

B XVII*

E

G

C III

E III

A XI

D XIX

F

B VIII

D

28

G VI

C XIV

C VI*

F XIV

A XIV

D

F

B XIX

E

G VIII

C

E XVI

29

A

D

G III

B III

E XI

G XI

C

F VIII

A

D XVI

F V

30

B XIV

E XIV*

A

C

F

A XIX

D VII

G

B XVI

E V

G

31

C III

F III*

D XI

B

E

C V

A XIII

We now label the paschal new moons (with asterisks in the table above)! To do this, we need only look for first full moon after the spring equinox, which Dionysius has fixed at March 21^st. For the theoretical moon, we will decree that the full moon always occurs on day 14 of the lunation.

For year III, we have a new moon on March 1^st. But then the full moon occurs on March 14^th, which is still before March 21^st. The paschal new moon is then the one on March 31^st.

We will eventually be able to use the completed table to compute the date of Easter, assuming that we know the Golden Number and the Dominical letter. But to get this information, we need one more table, called a Paschal table. What this table tells us is the age of the moon on January 1^st. This is called the epact for the year (recall from your reading that Dionysius used the epact). Clearly the epact for year III is zero. Now III is a common lunar year with 354 days, and because the solar year is 11 days longer, this means that the epact for year IV is 11. Similarly, the epact for year V is 11 + 11 = 22. Now year V is an embolismic year with length 384. This is 19 days longer than the solar year. But note that the first moon is old enough (22 days) so that all thirteen of the moons of that lunar year will begin in the solar year! (In our particular case, this works because 384-22=362.) Sure enough, the thirteenth moon begins on Dec. 28^th, and so the epact for the year VI is 22+11=33=3 modulo 30.

Paschal Table, Preliminary Version

Golden Number	Epact	Type of Year	Date of Paschal New Moon
III	0	Common	March 31
IV	11	Common	March 20
V	22	Embolismic	March 9
VI	3	Common	March 28
VII	14	Common	March 17
VIII	25	Embolismic	March 6
IX	6	Common	March 25
X	17	Common	March 14
XI	28	Embolismic	April 2
XII	9	Common	March 22
XIII	20	Embolismic	March 11
XIV	1	Common	March 30
XV	12	Common	March 19
XVI	23	Embolismic	March 8
XVII	4	Common	March 27
XVIII	15	Common	March 16
XIX	26	Embolismic	April 4
I	8	Common	March 23
II	19	Embolismic	March 12

Now we can use the tables to compute the date of Easter for any given Julian year. Let’s try the year 1066. G=1067 (mod 19) = 3. In the 28-year solar cycle, we have J=1075 = 11 (modulo 28). From the chart above, we see that the Dominical letter is then A. We now look at the Julian lunar almanac. The Paschal new moon is on March 31^st, and so the Paschal full moon is on April 13^th. We then find the first available Sunday, which is April 16^th.

Let’s try the year 1500. Then G = 19. The year J in the Julian cycle is 25. It is a leap year, and so we use the second Dominical letter D. The Paschal new moon is on April 4^th, and so the Paschal full moon is on April 17^th. The first available Sunday is April 19^th.