Dominical letter
From Academic Kids

The days of the year are sometimes designated letters A, B, C, D, E, F and G in a cycle of 7 as an aid for finding the day of week of a given calendar date and in calculating Easter. These letters are known as dominical letters.
A common year has a dominical letter, which is simply the dominical letter of its first Sunday. For example 2003 has 5 January as its first Sunday so has Dominical letter E.
In leap years, the leap day has no dominical letter. This ensures that each date has the same dominical letter every year, but causes the days of the weeks of the dominical letters to change within a leap year. Hence leap years have two dominical letters: the first for January and February and the second for March to December. The second dominical letter is the dominical letter of the first Sunday of October (which is the same as for January in a common year). The year 2004 has Dominical letters DC.
Examples include:
The dominical letter of a year determines the days of week in its calendar:
 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
Contents 
History
A device adopted from the Romans by the old 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 Church, on account of her complicated system of movable and immovable feasts (see Christian calendar), has from an early period taken upon herself as a special charge to regulate the measurement of time. To secure uniformity in the observance of feasts and fasts, she 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. Naturally she 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.
The Romans were accustomed to divide the year into nundinę, periods of eight days; and in their marble fasti, or calendars, of which numerous specimens remain, they used the first eight letters of the alphabet to mark the days of which each period was composed. When the Oriental sevenday period, or week, was introduced in the time of Cęsar Augustus, the first seven letters of the alphabet were employed in the same way to indicate the days of this new division of time. In fact, fragmentary calendars on marble still survive in which both a cycle of eight letters — A to H — indicating nundinę, and a cycle of seven letters — A to G — indicating weeks, are used side by side (see "Corpus Inscriptionum Latinarum", 2nd ed., I, 220; the same peculiarity occurs in the Philocalian Calendar of A.D. 356, ibid., p. 256). This device was imitated by the Christians, and in their calendars 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. A was always set against 1 January, B against 2 January, C against 3 January, and so on. Thus F fell to 6 January, G to 7 January; A again recurred on 8 January, and also, consequently, on 15 January, 22 January, and 29 January. Continuing in this way, 30 January was marked with a B, 31 January with a C, and 1 February with a D. Supposing this to be carried on through all the days of an ordinary year (i. e. not a leap year), it will be found that a 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 — a result which Durandus recalled by the following distich:
 Alta Domat Dominus, Gratis Beat Equa Gerentes
 Contemnit Fictos, Augebit Dona Fideli.
Now, as a moment's reflection shows, if 1 January is a Sunday, all the days marked by A will also be Sundays; If 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; if 1 January is a Monday, then Sunday will not come until 7 January, a G, and all the days marked G will be Sundays. This being explained, the Dominical Letter of any year is defined to be that letter of the cycle A, B, C, D, E, F, G, which corresponds to the day upon which the first Sunday (and every subsequent Sunday) falls.
It is plain, however, that when a leap year occurs, a complication is introduced. February has then twentynine days. Traditionally, the Anglican and civil calendars added this extra day to the end of the month, while the Catholic ecclesiastical calendar counted 24 February twice. But in either case, 1 March is then one day later in the week than 1 February, or, in other words, for the rest of the year the Sundays come a day earlier than they would in a common year. This is expressed by saying that a leap year has two Dominical Letters, the second being the letter which precedes that with which the year started. For example, 1 January 1907, was a Tuesday; the first Sunday fell on 6 January, or an F. F was, therefore, the Dominical Letter for 1907. The first of January, 1908, was a Wednesday, the first Sunday fell on 5 January, and E was the Dominical Letter, but as 1908 was a leap year, its Sundays after February came a day sooner than in a normal year and were Ds. The year 1908, therefore, had a double Dominical Letter, ED. In 1909, 1 January was a Friday and the Dominical Letter was C. In 1910 and 1911, 1 January fell respectively on Saturday and Sunday and the Dominical Letters are B and A.
Calculation
This, of course, is all very simple, but the advantage of tile device lies, like that of an algebraical expression, in its being a mere symbol adaptable to any year. By constructing a table of letters and days of the year, A always being set against 1 January, we can at once see the relation between the days of the week and the day of any month, if only we know the Dominical Letter. This may always be found by the following rule of De Morgan's, which gives the Dominical Letter for any year, or the second Dominical Letter if it be leap year:
 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.
For example, to find the Dominical Letter of the year 1913:
 (Steps 1, 2, & 4) 1914 + 478 + 0 = 2392
 (3) 19  16 = 3
 (4) 2392  3 = 2389
 (5) 2389 / 7 = 341, remainder 2.
Therefore, the Dominical Letter is E.
Practical use for the clergy
But the Dominical Letter had another very practical use in the days before the Ordo divini officii recitandi was printed annually, and when, consequently, a priest had often to determine the Ordo for himself. As can be seen in the article Epact, Easter Sunday may be as early as 22 March or as late as 25 April, and there are consequently thirtyfive possible days on which it may fall. It is also evident that each Dominical Letter allows five possible dates for Easter Sunday. Thus, in a year whose Dominical Letter is A (i. e. when 1 January is a Sunday), Easter must be either on 26 March, 2 April, 9 April, 16 April, or 23 April, for these are all the Sundays within the defined limits. But according as Easter falls on one or another of these Sundays we shall get a different calendar, and hence there are five, and only five, possible calendars for years whose Dominical Letter is A. Similarly, there are five possible calendars for years whose Dominical Letter is B, five for C, and so on, thirtyfive possible combinations in all. Now, advantage was taken of this principle in the arrangement of the old Pye or directorium which preceded the present "Ordo". The thirtyfive possible calendars were all included therein and numbered, respectively, primum A, secundum A, tertium A, etc.; primum B, secundum B, etc. Hence for anyone wishing to use the Pye the first thing to determine was the Dominical Letter of the year, and then by means of the Golden Number or the Epact, and by the aid of a simple table, to find which of the five possible calendars assigned to that Dominical Letter belonged to the year in question. Such a table as that just referred to, but adapted to the reformed calendar and in more convenient shape, will be found at the beginning of every Breviary and Missal under the heading, "Tabula Paschalis nova reformata".
The Dominical Letter does not seem to have been familiar to Bede in his "De Temporum Ratione," but in its place he adopts a similar device of seven numbers which he calls concurrentes (De Temp. Rat., cap. liii), of Greek origin. The Concurrents are numbers denoting the days of the week on which 24 March falls in the successive years of the solar cycle, 1 standing for Sunday, 2 (feria secunda) for Monday, 3 for Tuesday, and so on. It is sufficient here to state that the relation between the Concurrents and the Dominical Letter is the following:
 Concurrents 1 2 3 4 5 6 7
 Concurrent 1 = F (Dominical Letter)
 Concurrent 2 = E
 Concurrent 3 = D
 Concurrent 4 = C
 Concurrent 5 = B
 Concurrent 6 = A
 Concurrent 7 = G
Use for mental calculation
There exist patterns in the dominical letters, which are very useful for mental calculation.
Patterns for years:
To use these patterns, you need to remember a year to use as a starting point. I suggest using 2000=BA as a starting point.
Note that because of the complicated Gregorian leapyear rules, these patterns break near some century changes. Note the reverse alphabetical order.
1992 3 4 5 96 7 8 9 2000 1 2 3 04 5 6 7 2008 ED C B A GF E D C BA G F E DC B A G FE
and
(note the reversed order of the years as well as of the letters) 2040 2030 2020 2010 2000 1990 1980 1970 1960 1950 AG F ED C BA G FE D CB A           G FE D CB A GF E DC B AG 2046 2036 2026 2016 2006 1996 1986 1976 1966 1956
Patterns for days of the month:
The dominical letters for the first day of each month form the nonsense mnemonic phrase "Add G, beg C, fad F".
The following dates, given in month/day form, all have dominical letter C: 4/4, 6/6, 8/8, 10/10, 12/12, 5/9, 9/5, 7/11, 11/7. This was stolen from the Doomsday algorithm.
References
 This article incorporates text from the public domain Catholic Encyclopedia.
External links
 Catholic Encyclopedia article on Dominical letter (http://www.newadvent.org/cathen/05109a.htm)