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'''Dominical letters''' or '''Sunday letters''' are a method used to determine the [[day of the week]] for particular dates. When using this method, each year is assigned a letter (or pair of letters for leap years) depending on which day of the week the year starts on.

Dominical letters are derived from the Roman practice of marking the repeating sequence of eight letters A–H (commencing with A on 1 January) on stone calendars to indicate each day's position in the eight-day market week (''[[nundinae]]''). The word is derived from the number nine due to their practice of [[Counting#Inclusive counting|inclusive counting]]. After the introduction of Christianity a similar sequence of seven letters A–G was added alongside, again commencing with 1 January. The '''dominical letter''' marks the Sundays. Nowadays they are used primarily as part of the [[computus]], which is the method of calculating the date of Easter.

A common year is assigned a single dominical letter, indicating which lettered days are Sundays in that particular year (hence the name, from Latin ''dominica'' for Sunday). Thus, 2017 is A, indicating that all A days are Sunday, and by inference, 1 January 2017 is a Sunday. Leap years are given two letters, the first valid for January 1 – February 28 (or February 24, see below), the second for the remainder of the year.

In [[leap year]]s, the leap day may or may not have a dominical letter. In the Catholic version it does, but in the 1662 and subsequent Anglican versions it does not. The Catholic version causes 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 have a dominical letter of F.<ref>Peter Archer, ''The Christian Calendar and the Gregorian Reform'' (New York: Fordham University Press, 1941) p.5.</ref><ref>Bonnie Blackburn, Leofranc Holford-Strevens, ''The Oxford Companion to the Year'' (Oxford: Oxford University Press, 1999), p.829.</ref><ref>[http://hermes.ulaval.ca/~sitrau/calgreg/calend.html Calendarium] {{webarchive |url=https://web.archive.org/web/20050215030750/http://hermes.ulaval.ca/~sitrau/calgreg/calend.html |date=February 15, 2005 }} (Calendar attached to the papal bull "[[Inter gravissimas]]").</ref> The Anglican version adds a day to February that did not exist in common years, 29 February, thus it does not have a dominical letter of its own.<ref>"Anno vicesimo quarto Georgii II. c.23" (1751), ''The Statutes at Large, from Magna Charta to the end of the Eleventh Parliament of Great Britain, Anno 1761'', ed. Danby Pickering, p.194.</ref><ref>J. K. Fotheringham, "Explanation: The Calendar", ''The Nautical Almanac and Astronomical Ephemeris for the year 1931'', pp.735-747, p.745, ''... 1938'', pp.790–806, p.803.</ref>

In either case, all other dates have the same dominical letter every year, but the days of the dominical letters change within a leap year before and after the intercalary day, 24 February or 29 February.

==History==
Per Thurston (1909), dominical letters were:{{quotation|1=

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."<ref name=CatholicFull>Thurston, H. (1909). Dominical Letter. In The Catholic Encyclopedia. New York: Robert Appleton Company. see New Advent at [http://www.newadvent.org/cathen/05109a.htm], accessed 27 January 2015.</ref>}}

Thurston continues that the Christian Church, with its [[Liturgical year|"complicated system of movable and immovable feasts]]" has long been concerned with the regulation and measurement of time; he states: "To secure uniformity in the observance of feasts and fasts, [the Church] began, even in the patristic age, to supply a ''computus'', or system of reckoning, by which the relation of the solar and lunar years might be accommodated and the celebration of [[Easter]] determined."<ref name=CatholicFull/> He continues, that naturally it "adopted the astronomical methods then available, and these methods and the methodology belonging to them having become traditional, are perpetuated in a measure to this day, even the reform of the calendar, in the prolegomena to the Breviary and Missal."<ref name=CatholicFull/>

He then goes on to note that:
{{quotation|1=The Romans were accustomed to divide the year into [[Roman calendar#Nundinal cycle|''nundinæ'']], periods of eight days; and in their marble ''[[fasti]]'', or [[calendar]]s, of which numerous specimens remain, they used the first eight letters of the alphabet [A to H] to mark the days of which each period was composed. When the Oriental seven-day period, or week, was introduced in the time of [[Augustus]], the first seven letters of the alphabet were employed in the same way to indicate the days of the new division of time… [noting as well that] fragmentary calendars on marble still survive in which both a cycle of eight letters — A to H — indicating ''nundinae'', and a cycle of seven letters — A to G — indicating weeks, are used side by side (see "Corpus Inscriptionum Latinarum", 2nd ed., I, 220… [where the] same peculiarity occurs in the [[Philocalian Calendar]] of A.D. 356, ibid., p. 256)...}}
and that this device was imitated by the Christians.<ref name=CatholicFull/>

==Dominical letter cycle==
Thurston (1909) goes on to note that "the days of the year from 1 January to 31 December were marked with a continuous recurring cycle of seven letters: A, B, C, D, E, F, G... [and that the letter] A is always set against 1 January, B against 2 January, C against 3 January, and so on…" so that G falls to 7 January.<ref name=CatholicFull/>

He notes that A falls again on "8 January, and also, consequently on 15 January, 22 January and 29 January. Continuing in this way, 30 January is marked with a B, 31 January with a C, and 1 February with a D."<ref name=CatholicFull/>

When this is carried on through all the days of a [[common year]] (i.e. ordinary, or non-leap year) then "D corresponds to 1 March, G to 1 April, B to 1 May, E to 1 June, G to 1 July, C to 1 August, F to 1 September, A to 1 October, D to 1 November, and F to 1 December"; the resulting ADDGBEGCFADF sequence Thurston observes, is one  "which [[Guillaume Durand|Durandus]] recalled by the following [[distich]]:{{quotation|1=

Alta Domat Dominus, Gratis Beat Equa Gerentes 
Contemnit Fictos, Augebit Dona Fideli."<ref name=CatholicFull/>}}

Another one is "Add G, beg C, fad F," and yet another is "At Dover dwell George Brown, Esquire; Good Christopher Finch; and David Fryer." 
{| class="wikitable" 
|-	
!Months|| {{mvar|L}}	
|-	
| Jan Oct || A	
|-	
| May || B	
|-	
| Aug || C	
|-	
| Feb Mar Nov || D	
|-
| Jun || E
|-	
| Sept Dec || F	
|-	
| Apr July || G
|}
*If the letter ({{mvar|L}}) of the first day of a month is the dominical letter of the year, the month will have a Friday the 13th. That is to say, if the first day is Sunday, the 13th day will be Friday.

Clearly, Thurston continues, "if 1 January is a Sunday, all the other days marked by A will be Sundays; [i]f 1 January is a Saturday, Sunday will fall on 2 January which is a B, and all the other days marked B will be Sundays; [i]f 1 January is a Monday, then Sunday will not come until 7 January, a G, and all the days marked G will be Sundays."<ref name=CatholicFull/>

Thurston then notes that a complication arises with leap years, which have an extra day.<ref name=CatholicFull/> Traditionally, the Catholic ecclesiastical calendar treats 24 February (the "[[bissextus]]") as the day added, as this was the Roman leap day (''bis sextus ante Kalendas Martii''), with events normally occurring on 24–28 February moved to 25–29 February. The Anglican and civil calendars treat 29 February as the day added, and do not shift events in this way. But in either case, with leap years, Thurston explains, "1 March is then one day later in the week than 1 February, or, in other words, for the rest of the year [from leap day onward] the Sundays come a day earlier than they would in a common year."<ref name=CatholicFull/>

Thus a leap year is given two Dominical Letters, as Thurston explains, "the second being the letter which precedes that with which the year started."<ref name=CatholicFull/> For example, in 2020 (= ED), all E days preceding the leap day will be Sundays, and all D days for the rest of the year.

==Dominical letters of the years==
The dominical letter of a year provides the link between the date and the day of the week on which it falls. The following are the correspondences between dominical letters and the day of the week on which their corresponding common and leap years begin:
*'''A''': [[common year starting on Sunday]]
*'''B''': [[common year starting on Saturday]]
*'''C''': [[common year starting on Friday]]
*'''D''': [[common year starting on Thursday]]
*'''E''': [[common year starting on Wednesday]]
*'''F''': [[common year starting on Tuesday]]
*'''G''': [[common year starting on Monday]]
*'''AG''': [[leap year starting on Sunday]]
*'''BA''': [[leap year starting on Saturday]]
*'''CB''': [[leap year starting on Friday]]
*'''DC''': [[leap year starting on Thursday]]
*'''ED''': [[leap year starting on Wednesday]]
*'''FE''': [[leap year starting on Tuesday]]
*'''GF''': [[leap year starting on Monday]]

The [[Gregorian calendar]] repeats every 400 years (i. e., every four centuries). Of the 400 years in one Gregorian cycle, there are:
* 44 [[common year]]s for each single Dominical letter D and F;
* 43 common years for each single Dominical letter A, B, C, E, and G;
* 15 [[leap year]]s for each double Dominical letter AG and CB;
* 14 leap years for each double Dominical letter ED and FE;
* 13 leap years for each double Dominical letter BA, DC, and GF.

The [[Julian calendar]] repeats every 28 years. Of the 28 years in one Julian cycle, there are:
* 3 common years for each single Dominical letter A, B, C, D, E, F, and G;
* 1 leap year for each double Dominical letter BA, CB, DC, ED, FE, GF, and AG.

==Calculation==
{{Overly detailed|date=January 2015|section=yes}}
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:
**for one century within two multiples of 400, go forward two letters from BA for 2000, hence C, E, G.
**for remaining years, go back one letter every year, two for leap years (this corresponds to writing two letters, no letter is skipped).
**to avoid up to 99 steps within a century, the table below can be used.
{| class="wikitable" style="text-align:center"
|-
! Year mod 28 ||{{mvar|#}}
|-
| 00 06 <font color="red">12</font> 17 23 ||0
|-
| 01 07 12 18 <font color="red">24</font> ||6
|-
| 02 <font color="red">08</font> 13 19 24 ||5
|-
| 03 08 14 <font color="red">20</font> 25 ||4
|-
| <font color="red">04</font> 09 15 20 26 ||3
|-
| 04 10 <font color="red">16</font> 21 27 ||2
|-
| 05 11 16 22 <font color="red">00</font> ||1
|}
<font color="red">Red</font> for the first two months of leap years.

For example, to find the Dominical Letter of the year 1913:

*1900 is G and 13 corresponds to 5
*G + 5 = G − 2 = E, 1913 is E

Similarly, for 2007:
*2000 is BA and 7 corresponds to 6
*A + 6 = A − 1 = G, 2007 is G

For 2065:
*2000 is BA and 65 mod 28 = 9 corresponds to 3
*A + 3 = A − 4 = D, 2065 is D

===The odd plus 11 method===
A simpler method suitable for finding the year's dominical letter was discovered in 2010. It is called the "odd plus 11" method.<ref>Chamberlain Fong, Michael K. Walters: [https://arxiv.org/abs/1010.0765 "Methods for Accelerating Conway's Doomsday Algorithm (part 2)"], 7th International Congress of Industrial and Applied Mathematics (2011).</ref>

The procedure accumulates a running total ''T'' as follows:
# Let ''T'' be the year's last two digits.
#If ''T'' is odd, add 11. 
#Let ''T'' = {{sfrac|''T''|2}}.
#If ''T'' is odd, add 11.
#Let ''T'' = ''T'' mod 7.
#Count forward ''T'' letters from the century's dominical letter (A, C, E or G see above) to get the year's dominical letter.

The formula is
:<math> \left( \frac{y+11(y \bmod 2)}{2} + 11 \left(\frac{y+11(y \bmod 2)}{2}\bmod 2\right)\right) \bmod 7.</math>

===De Morgan's rule===
This rule was stated by [[Augustus de Morgan]]:
#Add 1 to the given year. 
#Take the quotient found by dividing the given year by 4 (neglecting the remainder). 
#Take 16 from the centurial figures of the given year if that can be done. 
#Take the quotient of III divided by 4 (neglecting the remainder). 
#From the sum of I, II and IV, subtract III. 
#Find the remainder of V divided by 7: this is the number of the Dominical Letter, supposing A, B, C, D, E, F, G to be equivalent respectively to 6, 5, 4, 3, 2, 1, 0.<ref name=CatholicFull/>
So the formulae (using the [[Floor and ceiling functions|floor function]]) for the Gregorian calendar is 
:<math>1. \left(1 + \text{year} + \Big\lfloor\frac{\text{year}}{4}\Big\rfloor + \Big\lfloor\frac{\text{year} - 1600}{400}\Big\rfloor - \Big\lfloor\frac{\text{year} - 1600}{100}\Big\rfloor\right) \bmod 7. </math>
It is equivalent to
:<math>2. \left(\text{year} + \Big\lfloor\frac{\text{year}}{4}\Big\rfloor + \Big\lfloor\frac{\text{year}}{400}\Big\rfloor - \Big\lfloor\frac{\text{year}}{100}\Big\rfloor - 1\right) \bmod 7 </math>
and
:<math>3. \left(y + \Big\lfloor\frac{y}{4}\Big\rfloor + 5(c\bmod4) -1\right) \bmod 7 </math> &nbsp; &nbsp; (where <math>\text{y}</math> = last two digits of the year, <math>\text{c}</math> = century part of the year).
For example, to find the Dominical Letter of the year 1913: 
*(1 + 1913 + 478 + 0 − 3) mod 7 = 2
*(1913 + 478 + 4 − 19 − 1) mod 7 = 2
*(13 + 3 + 15 -1) mod 7 = 2
Therefore, the Dominical Letter is E.

De Morgan's rules no. 1 and 2 for the Julian calendar:
:<math>1.</math> and <math>2. \left(\text{year} + \Big\lfloor\frac{\text{year}}{4}\Big\rfloor - 3\right) \bmod 7</math>
To find the Dominical Letter of the year 1913 in the Julian calendar:
*(1913 + 478 − 3) mod 7 = 1
Therefore, the Dominical Letter is F in the Julian calendar.

In leap years the formulae above give the Dominical Letter for the last ten months of the year. To find the Dominical Letter for the two first months of the year to the leap day (inclusive) subtract '''1''' from the calculated number representing the original Dominical Letter; if the new number is less than '''0''', it must be changed to '''6'''.

===Dominical letter in relation to the Doomsday Rule===

The "doomsday" concept in the [[doomsday algorithm]] is mathematically related to the Dominical letter. Because the letter of a date equals the dominical letter of a year (DL) plus the day of the week (DW), and the letter for the [[Doomsday rule|doomsday]] is C except for the portion of leap years before [[February 29]] in which it is D, we have:
:<math>\begin{align}
\text{C} &= (\text{DL} + \text{DW}) \bmod 7 \\
\text{DL} &= (\text{C} - \text{DW}) \bmod 7 \\
\text{DW} &= (\text{C} - \text{DL}) \bmod 7
\end{align}</math>
Note: G = 0 = Sunday, A = 1 = Monday, B = 2 = Tuesday, C = 3 = Wednesday, D = 4 = Thursday, E = 5 = Friday, and F = 6 = Saturday, i.e. in our context, C is mathematically identical to 3.

Hence, for instance, the doomsday of the year 2013 is Thursday, so DL = (3 − 4) mod 7 = 6 = F. The dominical letter of the year 1913 is E, so DW = (3 − 5) mod 7 = 5 = Friday.

{| class="wikitable" style="text-align:center"
! rowspan=2 valign="bottom" | Doomsday !! colspan=2 | Dominical letter
|-
!<small>Common year</small>!!<small>Leap year</small>
|-
| Sunday    || C || DC
|-
| Monday    || B || CB
|-
| Tuesday   || A || BA
|-
| Wednesday || G || AG
|-
| Thursday  || F || GF
|-
| Friday    || E || FE
|-
| Saturday  || D || ED
|}

===All in one table===
If the year of interest is not within the table, use a tabular year which gives the same remainder when divided by 400 ([[Gregorian calendar]]) or 700 ([[Julian calendar]]). In the case of the Revised Julian calendar, find the date of [[Easter]] (see the section "Calculating Easter Sunday", subsection "Revised Julian calendar" below) and enter it into the "Table for days of the year" below. If the year is a leap year, the dominical letter for January and February is found by inputting the date of [[Easter Monday]]. Note the different rules for leap years:
*'''Gregorian calendar''': every year which divides exactly by 4, but of century years only those which divide exactly by 400; therefore ignore the left-hand letter given for a century year which is not a leap year.
*'''Julian calendar''': every year which divides exactly by 4.
*'''Revised Julian calendar''': every year which divides exactly by 4, but of century years only those which give the remainder 200 or 600 when divided by 900.

{|class="wikitable" style="text-align:center"
|-
!Julian<br>calendar
!valign="top" scope="col"| <br/>700<br/>1400
!valign="top" scope="col"|100<br/>800<br/>1500
!valign="top" scope="col"|200<br/>900<br/>1600
!valign="top" scope="col"|300<br/>1000<br/>1700
!valign="top" scope="col"|400<br/>1100<br/>1800
!valign="top" scope="col"|500<br/>1200<br/>1900
!valign="top" scope="col"|600<br/>1300<br/>2000
!colspan=2|Paschal full moon date<br>{{font color|red|Year mod 19}}
|-
!Gregorian<br>calendar
!valign="top" scope="col"|1700<br/>2100
!
!valign="top" scope="col"|1800<br/>2200
!
!valign="top" scope="col"|1900<br/>2300
!valign="top" scope="col"|1600<br/>2000
!
!Gregorian<br><small>(1900–2199)</small>
!Julian<br>calendar
|-
!scope="row" style="text-align:left"|{{font color|red|00}} 28 56 84
| DC || ED || FE || GF || AG || BA || CB||14 Apr||5 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|01}} 29 57 85
| B || C || D || E || F || G || A ||3 Apr||25 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|02}} 30 58 86
| A || B || C || D || E || F || G ||23 Mar||13 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|03}} 31 59 87
| G || A || B || C || D || E || F ||11 Apr||2 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|04}} 32 60 88
| FE || GF || AG || BA || CB || DC || ED ||31 Mar||22 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|05}} 33 61 89
| D || E || F || G || A || B || C ||18 Apr||10 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|06}} 34 62 90
| C || D || E || F || G || A || B ||8 Apr||30 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|07}} 35 63 91
| B || C || D || E || F || G || A ||28 Mar||18 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|08}} 36 64 92
|AG||BA||CB || DC || ED || FE || GF ||16 Apr||7 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|09}} 37 65 93
| F || G || A || B || C || D || E||5 Apr||27 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|10}} 38 66 94
| E || F || G || A || B || C || D ||25 Mar||15 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|11}} 39 67 95
| D || E || F || G || A || B || C ||13 Apr||4 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|12}} 40 68 96
| CB || DC || ED || FE || GF || AG || BA ||2 Apr||24 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|13}} 41 69 97
| A || B || C || D || E || F || G ||22 Mar||12 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|14}} 42 70 98
| G || A || B || C || D || E || F ||10 Apr||1 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|15}} 43 71 99
| F || G || A || B || C || D || E ||30 Mar||21 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|16}} 44 72
| ED || FE || GF || AG || BA || CB || DC ||17 Apr||9 Apr
|-
!scope="row" style="text-align:left"|{{font color|red|17}} 45 73
| C || D || E || F || G || A || B ||7 Apr||29 Mar
|-
!scope="row" style="text-align:left"|{{font color|red|18}} 46 74
| B || C || D || E || F || G || A ||27 Mar||17 Apr
|-
!scope="row" style="text-align:left"|19 47 75
| A || B || C || D || E || F || G
! colspan=2, rowspan=3|Month
|-
!scope="row" style="text-align:left"|20 48 76
| GF || AG || BA || CB || DC || ED || FE
|-
!scope="row" style="text-align:left"|21 49 77
| E || F || G || A || B || C || D
|-
!scope="row" style="text-align:left"|22 50 78
| D || E || F || G || A || B || C
!colspan=2|Feb Mar Nov
|-
!scope="row" style="text-align:left"|23 51 79
| C || D || E || F || G || A || B
!colspan=2|Aug
|-
!scope="row" style="text-align:left"|24 52 80
| {{font color|red|B}}A || {{font color|red|C}}B || {{font color|red|D}}C || {{font color|red|E}}D || {{font color|red|F}}E || {{font color|red|G}}F || {{font color|red|A}}G
!colspan=2|Jan {{font color|red|May}} Oct
|-
!scope="row" style="text-align:left"|25 53 81
| G || A || B || C || D || E || F 
!colspan=2|Apr Jul
|-
!scope="row" style="text-align:left"|26 54 82
| F || G || A || B || C || D || E 
!colspan=2|Sep Dec
|-
!scope="row" style="text-align:left"|27 55 83
| E || F || G || A || B || C || D
!colspan=2|Jun
|-
!rowspan=5|Day||1||2||3||4||5||6||7
!rowspan=5, colspan=2|Table of letters for<br>the days of the year
|-
!8||9||10||11||12||13||14
|-
!15||16||17||18||19||20||21
|-
!22||23||24||25||26||27||28
|-
!29||30||31 || || || ||
|}
===Years with special dominical letters===
When a country switched to the Gregorian calendar, there could be some unusual combinations of dominical letters.
===== Some examples =====
* '''1582''': Many catholic countries switched to the Gregorian calendar Friday 15 October. The table above indicates that year 1582 had the dominical letter G in the Julian calendar and C in the Gregorian one. So the dominical letters for 1582 in these catholic countries became GC, a special combination not seen before.
* '''1752''': Great Britain and her colonies switched to the Gregorian calendar Thursday 14 September. The leap year 1752 had in the Julian calendar dominical letters ED and in the Gregorian one dominical letters BA. So year 1752 had in Great Britain and her colonies the dominical letters EDA, a very special combination.

== Calculating Easter Sunday ==

{| class="wikitable" style="text-align:center"
|-
!colspan=2|Golden number
|1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16||17||18||19
|-
!rowspan=2|Paschal<br> full moon<br> date
!Gregorian<br>(1900–2199)
|14<br>Apr||3<br>Apr||23<br>Mar||11<br>Apr||31<br>Mar||18<br>Apr||8<br>Apr||28<br>Mar||16<br>Apr||5<br>Apr||25<br>Mar||13<br>Apr||2<br>Apr||22<br>Mar||10<br>Apr||30<br>Mar||17<br>Apr||7<br>Apr||27<br>Mar
|-
!Julian
|5<br>Apr||25<br>Mar||13<br>Apr||2<br>Apr||22<br>Mar||10<br>Apr||30<br>Mar||18<br>Apr||7<br>Apr||27<br>Mar||15<br>Apr||4<br>Apr||24<br>Mar||12<br>Apr||1<br>Apr||21<br>Mar||9<br>Apr||29<br>Mar||17<br>Apr
|}

To find the golden number, add 1 to the year and divide by 19. The remainder (if any) is the golden number, and if there is no remainder the golden number is 19. Obtain the date of the paschal full moon from the table, then use the "week table" below to find the day of the week on which it falls. Easter is the following Sunday.

===Week table: Julian and Gregorian calendars===

For Julian dates before 1300 and after 1999 the year in the table which differs by an exact multiple of 700 years should be used. For Gregorian dates after 2299, the year in the table which differs by an exact multiple of 400 years should be used. The values "<small>r0</small>" through "<small>r6</small>" indicate the remainder when the Hundreds value is divided by 7 and 4 respectively, indicating how the series extend in either direction. Both Julian and Gregorian values are shown 1500–1999 for convenience.

The corresponding numbers in the far right hand column on the same line as each component of the date (the hundreds, remaining digits and month) and the day of the month are added together. This total is then divided by 7 and the remainder from this division located in the far right hand column. The day of the week is beside it. Bold figures (e.g., '''04''') denote leap year. If a year ends in 00 and its hundreds are in bold it is a leap year. Thus 19 indicates that 1900 is not a Gregorian leap year, (but '''19''' in the Julian column indicates that it ''is'' a Julian leap year, as are all Julian ''x''00 years). '''20''' indicates that 2000 is a leap year. Use '''Jan''' and '''Feb''' only in leap years.

{| class=wikitable
!colspan=2 style="line-height:10px; border-bottom:none"|<small>Century digits</small>||colspan=4 rowspan=2|Remaining year digits||colspan=5 rowspan=2|Month||colspan=1 rowspan=2|Day of<br>week||colspan=1 rowspan=2|Number
|-
! style="border-top:none; line-height:10px"|<small>Julian<br />(r ÷ 7)</small>
! style="border-top:none; line-height:10px"|<small>Gregorian<br />(r ÷ 4)</small>
|-
|<small>'''r5'''</small> '''19'''||'''16''' '''20''' <small>'''r0'''</small>||00 06 {{0|00}} 17 23||'''28''' 34 {{0|00}} 45 51||'''56''' 62 {{0|00}} 73 79||'''84''' 90 ||Jan||{{N/A|}} ||{{N/A|}} ||{{N/A|}} || Oct||Sat||0
|-
|<small>'''r4'''</small> '''18'''||15 19 <small>r3</small>||01 07 '''12''' 18 ||29 35 '''40''' 46||57 63 '''68''' 74 ||85 91 '''96'''||{{N/A|}} ||{{N/A|}} || May||{{N/A|}} ||{{N/A|}} ||Sun||1
|-
|<small>'''r3'''</small> '''17'''||{{N/A}}||02 {{0|00}} 13 19 '''24'''||30 {{0|00}} 41 47 '''52'''||58 {{0|00}} 69 75 '''80'''||86 {{0|00}} 97|| '''Feb'''||{{N/A|}} ||{{N/A|}} || Aug||{{N/A|}} ||Mon||2
|-
|<small>'''r2'''</small> '''16'''||18 22 <small>r2</small>||03 '''08''' 14 {{0|00}} 25||31 '''36''' 42 {{0|00}} 53||59 '''64''' 70 {{0|00}} 81||87 '''92''' 98||Feb|| Mar||{{N/A|}} ||{{N/A|}} || Nov||Tue||3
|-
|<small>'''r1'''</small> '''15'''||{{N/A}}|| {{0|00}} 09 15 '''20''' 26|| {{0|00}} 37 43 '''48''' 54|| {{0|00}} 65 71 '''76''' 82|| {{0|00}} 93 99 ||{{N/A|}} ||{{N/A|}} || Jun||{{N/A|}} ||{{N/A|}} ||Wed||4
|-
|<small>'''r0'''</small> '''14'''||17 21 <small>r1</small>||'''04''' 10 {{0|00}} 21 27||'''32''' 38 {{0|00}} 49 55||'''60''' 66 {{0|00}} 77 83||'''88''' 94 ||{{N/A|}} ||{{N/A|}} ||{{N/A|}} ||Sep ||Dec||Thu||5
|-
|<small>'''r6'''</small> '''13'''||{{N/A}}||05 11 '''16''' 22||33 39 '''44''' 50||61 67 '''72''' 78||89 95 ||'''Jan'''|| Apr||  Jul||{{N/A|}} ||{{N/A|}} ||Fri||6
|}

For determination of the day of the week (1 January 2000, Saturday)
*<span style="background-color:#FFFF88">the day of the month: 1</span>
*<span style="background-color:#FFFF88">the month: 6</span>
*<span style="background-color:#FF8888">the year: 0</span>
*<span style="background-color:#88FF88">the century mod 4 for the Gregorian calendar and mod 7 for the Julian calendar 0</span>
*<span style="background-color:#BBBBFF">adding {{nowrap|1=1 + 6 + 0 + 0 = 7}}. Dividing by 7 leaves a remainder of 0, so the day of the week is Saturday.</span>

=== Revised Julian calendar ===

* Use the Julian portion of the table of paschal full moons. Use the "week table" (remembering to use the "Julian" side) to find the day of the week on which the paschal full moon falls. Easter is the following Sunday and it is a Julian date. Call this date ''JD''. 
* Subtract 100 from the year.
* Divide the result by 100. Call the number obtained (omitting fractions) ''N''.
* Evaluate {{sfrac|7''N''|9}}. Call the result (omitting fractions) ''S''.
* The Revised Julian calendar date of Easter is {{nowrap|1=''JD'' + S − 1}}.

'''Example'''. What is the date of Easter in 2017?

{{nowrap|1=2017 + 1 = 2018}}. {{nowrap|1=2018 ÷ 19 = 106 remainder 4}}. Golden number is 4. Date of paschal full moon is 2 April (Julian). From "week table" 2 April 2017 (Julian) is Saturday. {{nowrap|1=''JD'' = 3 April}}. {{nowrap|1=2017 − 100 = 1917}}. {{nowrap|1=1917 ÷ 100 = 19 remainder 17}}. {{nowrap|1=''N'' = 19}}. {{nowrap|1=19 × 7 = 133}}. {{nowrap|1=133 ÷ 9 = 14 remainder 7}}. {{nowrap|1=''S'' = 14}}. Easter Sunday in the Revised Julian calendar is {{nowrap|1=April 3 + 14 − 1 = '''April 16'''}}.

==Clerical utility==
{{unreferenced section|date=January 2015}}
The dominical letter had another practical utility in the period prior to the annual printing of the ''Ordo divini officii recitandi'', in which period, therefore, Christian [[clergy]] were often required to determine the ''Ordo'' independently. Easter Sunday may be as early as 22 March or as late as 25 April, and consequently there are 35 possible days on which it may occur; each dominical letter includes 5 potential dates of these 35, and thus there are 5 possible ecclesiastical calendars for each letter. The Pye or [[Directorium]] which preceded the present ''Ordo'' took advantage of this principle by delineating all 35 possible calendars and denoting them by the formula "primum A", "secundum A", "tertium A", et cetera. Hence, based on the dominical letter of the year and the [[epact]], the Pye identified the correct calendar to use. A similar table, adapted to the reformed calendar and in more convenient form, is included in the beginning of every [[breviary]] and [[missal]] under the heading "Tabula Paschalis nova reformata".

[[Saint Bede]] does not seem to have been familiar with dominical letters, given his "[[De temporum ratione]]"; in its place he adopted a similar device of Greek origin consisting of seven numbers, which he denominated "''concurrentes''" (''De Temp. Rat.'', Chapter LIII). The "concurrents" are numbers that denote the days of the week on which 24 March occurs in the successive years of the solar cycle, 1 denoting Sunday, 2 (''feria secunda'') for Monday, 3 for Tuesday, et cetera; these correspond to dominical letters F, E, D, C, B, A, and G, respectively.

==Use for computer calculation==
Computers are able to calculate the Dominical letter in this way (function in [[C (programming language)|C]]), where:

* <tt>m</tt> = month
* <tt>y</tt> = year
* <tt>s</tt> = "style"; 0 for Julian, otherwise Gregorian.
<syntaxhighlight lang="c">
 char dominical(int m,int y,int s){
  int leap;
  int a,b;
  leap=(s==0&&y%4==0)||(s!=0&&(y%4==0&&y%100!=0||y%400==0));
  a=(y%100)%28;
  b=(s==0)*(4+(y%700)/100+2*(a/4)+6*((!leap)*(1+(a%4))+(leap)*((9+m)/12)))%7+
    (s!=0)*(2*(1+(y%400)/100+(a/4))+6*((!leap)*(1+(a%4))+(leap)*((9+m)/12)))%7;
  b=(b==0)*(b+7)+(b!=0)*b;
  return (char)(64+b);
 }
</syntaxhighlight>

==See also==
* [[Lectionary#3 year cycle]]
* [[Runic calendar]]

==References==
{{Reflist}}

==Bibliography==
{{Refbegin}}
*{{catholic}}
{{Refend}}

==External links==
*{{Cite EB1911|wstitle=Calendar|display=Calendar s.v. Ecclesiastical Calendar|volume=4|page=992}}
*[http://www.newadvent.org/cathen/05109a.htm Catholic Encyclopedia article on Dominical letter]

{{Time measurement and standards}}

[[Category:Gregorian calendar]]
[[Category:Julian calendar]]
[[Category:Easter date]]
[[Category:Latin script]]
[[Category:Sunday|Letter]]