Epakt and contestants

Apparently, due to the fact that when writing numbers with Roman numerals it is very easy to make a mistake, Western European historians and calculators( "komputists") have developed a whole "set" of various calendar characteristics that were widely used when dating documents. Here are three typical examples.

The first is the act of September 15, 1011 "from the incarnation of our Lord Is. Chr. "Is dated as follows: anno ab incarnatione Dom.nostri I. Ch. MXI, indictione IX, littera VII, luna XIV, XVII Kal. Octobr.

Further, the diploma from the Lyons Bishopric of 11 March 1134: Die dominico. .. V idus Martii, luna decima secunda, anno ab incarn. Dom.millesimo cente-simo trigesimo quarto, indict. VII, epacta XXIII, concurrente VII.

And one more dating: a.d.inc.1223, epacta XXVIII, concurrente VI, indictione XII.

As in the East. Solar cycle, golden number and indict - these elements of dating are inherently identical with those used in the Christian East - in Byzantium and in Russia. There is some difference in the "reference points" and notations.

As in the East, the cycles mentioned above were first counted in the era from the "creation of the world."According to one of the variants of its era was attributed to 4713 BC.Therefore, the number of "year of the world" M - Annus Mundi is found by the formula M = 4713 + R, where R is the year number n.e.

The Western European circle of the Sun, more precisely the "solar cycle"( cyclus solaris - CS), is defined as the remainder of dividing the number of "year of the world" by 28:

However, the era of the year of the world in the late Middle Ages was practically not used, since Western Europe, VII century.quite quickly passed to the account of the years from the "Nativity of Christ."Therefore, the solar cycle was usually divided by the 28th day of the year.e. R, increased by 9:

In particular, for 1986 we have CS-7( VII).Therefore, with respect to the eastern circle of the Sun, Q the Western European solar cycle lagged 11: CS = Q - 11.

The golden number( numerus aureus - AM) - the year number in the 19-year lunar cycle is determined by dividing by 19year of the world M or increased by the 1st of the year and.e. R:

For the same year 1986 we find NA = 11( XI).Earlier it was noted that the golden number of the year is 3 more than the circle of the moon: NA = L + 3.

It is obvious that since the transition from the sun circles Q to the Western European solar cycles CS and from the lunar L circles to the gold numbers NA is elementary, The tables on which they are determined for any year and century. For this it is sufficient to take the numbers from Table.and make an appropriate amendment.

The era of the year of the world M is such that the numeric value of the indicator in Western and Eastern Europe was the same:

For 1986, we have I = 9( IX).

Putting all the characteristics of the year in brackets in Roman numerals, we reminded the reader that it was in this form that they were quoted on all documents. The table of indicators is already given earlier.

Sundays. The days of the year, from January 1 to December 31, medieval "komputists" denoted cyclically by seven Latin letters A, B, C, D, E, F, G, called calendar letters( litterae calendarum).The "binding" of letters to the dates of the months is carried out in the direct order: January 1 - A, 2-e - B, 3rd - C, 4th - D,. .., 8th - A, 9th - Band so on. As a result, for the first number of months, the following letters had the following letters:

Jan. 1,- A, 1 Apr.- G, 1 July -G, 1 Oct.-А,

1 february.- D, 1 May - B, 1 Aug.- S, November 1 - D,

March 1 - D, June 1 - E, September 1.- F, Dec. 1- F.

In the January calendar, the leap year has two Sunday letters. The first - "regular" - indicates the date of Sunday from January 1 to February 29, the second of a series of calendar letters( written in the reverse order: A, G, F, E, D, C, B, A) - from March 1 to December 31.

The distribution of Sunday letters( LD) in a 28-year solar period is given in Table. We note that the 1st, 5th, etc. years of the period are leap years, and the first year starts on Monday.

For an example, we determine on which day of the week it was on 11 March 1134 Adding to the year 9 and dividing by 28, we find in the remainder the solar cycle CS = 23. From Table.it follows that the Sunday letter of the 23rd year of G, which in March falls on the 4th, 11th, 18th and 25th. Consequently, March 11, 1134 is Sunday.

In medieval documents, often instead of the Sunday letter, its ordinal number was indicated in a series of calendar letters: A -littera I, B-litters II, C-littera III, D-littera IV, E-littera V, F-littera VI and G-litteraVII.

Sunday letter G and "encrypted" as "littera VII" in the act of September 15, 1011.

Competitors. Two of the above datings contain a little known element in our country - competition-rent. Meanwhile, the contestants, or solar epakt-you( concurrentes septimanae, epactae solis - ES) have been widely used since the VIII century.to identify the calendar date with the day of the week. The first but not the main purpose of the contestants was to indicate the day of the week, which occurred on March 24 in one or another year: with the competitor 1 this Sunday, 2 - Monday, 3 - Tuesday, 4 - Wednesday, 5 - Thursday, 6 - Friday and7 - Saturday.

Distribution of the competitor by years of the solar cycle is given in Table. As you can see, there is a one-to-one relationship between Sunday letters and contestants: F = 1, E = 2, D = 3, C = 4, B = 5, A = 6 and G = 7. This is understandable. If Sunday is, for example, the letter G, then March 24 falls on a Saturday, etc. In a leap year for January - February, you should take the competitor, corresponding to the first Sunday letter, ie, one less than the number indicated in the table.

Table. Location of Sunday letters and the competitor in the 28-year solar cycle

Number of the year in cycle	Sunday letter	Competition annuity	Number of year in cycle	Sunday letter	Competitor	Number of year in cycle	Sunday letter	Competitor
1 *	GF	1	11	A	6	21 *	CB	5
2	E	2	12	G	7	22	A	6
3	D	3	13 *	F E	2	23	G	7
4	C	4	14	D	3	24	F	1
5 *	VA	6	15	C	4	25 *	E D	3
6	Q	7	16	In	5	26	C	4
7	F	1	17 *	AQ	7	27		5
8	E	2	18	F	1	28	A	6
9 *	D WITH	4	19	E	2	•
10	IN	5	20	D	3
Note.		The asterisk	marked leap years.

Take for example 1340 g. Its solar cycle CS - 5. From the table.it can be seen that the competitor of this year( solar epaktoy) was the number 6( Sunday letter for March-December - A).Consequently, March 24, 1340 accounted for Friday.

However, the main role of solar epact( competitor) is as follows. Solar epact is the number indicating how many positions in a particular year of the solar cycle with the CS number( or Q for the Byzantine account) the day of the week calculated for a certain calendar date has advanced in comparison with the initial( "zero") year of the cycle. Obviously, when calculating solar effects, one must take into account the position of leap years in the 28-year solar cycle.

As already noted, in the Western European 28-year cycle, leap-days are the 1st, 5th, 9th, and so on. Therefore, from March 1 of the 1st year of the cycle, the days of the week shift by two positions in comparison with the last year of the cycle. This will happen again in the 5th, and so on. Thus, the solar effect of the year having the CS number in the 28-year cycle can be determined by such a simple formula:

In January-February, the ES value is one unit less than it follows from the formula.

In the Byzantine 28-year cycle, leap years are the 3rd, 7th, etc. March or the 4th, 8th, 12th,. .. years of the January style. Therefore, when calculating solar effects here it is necessary to use a slightly different formula:

It is noteworthy that the distribution of solar effects over the years depends on which cycle the calculator uses. In the case of the eastern cycle, their series looks like this: 1, 2, 3, 5, 6, 7, 1,3,. ..( as in Table 1), in the western cycle we have 1, 2, 3, 4, 6, 7, 1, 2, 4,. .. This difference is due to the fact that the beginning of the cycles are shifted relative to each other( Q = CS + 11!) Just enough that in fact both correspond to a single distribution of solar effects over the years.e.

Lunar letters, numbers and effects. Moon letters( litterae lunares) were used to determine the age of the moon on a specific date in the early Middle Ages. This is 20 letters of the Latin alphabet from A to U, which were recorded in three different versions and combined into two groups - only 30 + 29 = 59 - according to the number of days respectively in full and empty lunar months. The days from January 1 to January 20 were designated by the letters A, B, C,, null,null,null, U( they were called litterae nudae), then the same letters were put, but with an asterisk: January 21 - A *, 22-B *,. ..(this is litterae postpunctatae) and, finally, with the icon in front of the letter: February 10 - * A, 11 - * B,. ..( litterae praepuncta-tae), here the last letter * U was not shown. Then the cycle was repeated in the same order, so that, in particular, the letter A was distributed over the months as follows:

A = 1.01;1.03;29.04;27.06;25.08;23.10;21.12;

A * = 21.01;21.03;19.05;17.07, 14.09;12.11;

* A = 10.02;10.04;8.06;6.08;4.10;2.12.

The year ended with the letter L = December 31.

Therefore, if somehow it was established that in a certain year the new moon falls on, say, on January 3, indicated by the letter C, then the new moon of all other full months also falls on the calendar numbers indicated by the same letter. New moon of the empty months in 29 days will be in the days indicated by the letter N. . This establishes the schedule of the phases of the moon for a whole year. Especially these letters were useful for the days from March 22( B *) to April 25( * Q), that is, when calculating the date of Easter.

Used for dating and the numbers indicating the age of the moon reckoned from the nearest past new moon on Easter day( luna paschae).For example, in the table of the dates of Easter for the year 532, compiled by Dionysius the Small in 525, the luna paschae = XX is indicated, since in the year mentioned, Easter was April 11, and nepoteny on March 23.On the inclusive account on March 23 - the first day, and April 11 - the 20th.

For many centuries both for the calculation of the Easter dates and the dating of documents, lunar epakttes( epactae lunares, EL) were used - the age of the Moon on March 22.It has already been mentioned how this age varies during the 19-year cycle. Ibid in Table.and lunar effects are given for each year of the cycle. Moon epaticotes were usually indicated in datings instead of the golden date of the year, as is evident from the second and third of the examples above. The change of the epakt was made on September 1, so for the dates from September 1 to December 31, the epakta of the following year was indicated.

In particular, for 1134 we find: the golden number NA-14 and the lunar epakta EL = XXIII, as it is written in the document. For 1223 NA = 8 and corresponds to the epakta EL = XVII.The document, however, shows the epitaph XXVIII.Consequently, it was compiled after the September 1 , which is confirmed elsewhere.

It remains to clarify the meaning of the designation "luna decima secunda", ie "Luna-12" in the document dated March 11, 1134. This is the age of the moon, indicating its phase at the time of writing the document. From the "schedule" of the new moon for the 19-year cycle, we find that in 1134, with the golden number of the year NA = 14, the nearest last new moon was February 28, so by 11 March the age of the moon really was 12. In 1011( with NA =5) the calculated neobeniya was on September 2, and by September 15( inclusive account), the age of the moon was 14. This could give rise to the erroneous idea that in the latter case, "luna XIV" means lunar epact, which this year is also equal to XIV.

Regulators. Here it is appropriate to mention one more little known calendar elements - regulapax. They undoubtedly contributed to the development of a variety of "perpetual calendars" with auxiliary coefficients for each month of the year.

Solar regulars( regulares solares mensium, RS) are numbers, one for each of the months of the year, which must be added to the contest-there to get the day of the week on the 1st day of the month. Invented them, apparently, back in the VIII century. Church historian Bede Venerable. Here are the values ​​of these numbers:

January-2, May -3, September-7,

February-5, June-6, October-2,

March -5, July -1, November -5,

April -1, August - 4, December -7.

Let's check the day of the week that occurred on May 3, 1340. The solar cycle of the year is CS = 5, the competitor( solar effect) ES = 6, the solar regulator for May RSV = 3. Consequently, on May 1, 1340 was 6 + 3 =( 9 - 7 =) 2 - Monday, and 3 May - on Wednesday.

Summarizing the above, you can write down such a simple formula for determining the day of the week q:

Here D is the number of the month. It decreases by 1, since, paying tribute to the ancient Roman tradition, medieval calculators calculated the regulars by the 1st of each month of the "inclusive score", i.e.( having already included it in the value of the regulator.) Obviously, from today's point of view, it is more expedientthe numerator of formula( 1.3), rewrite that: ES +( RS-1) + D, ie, reduce the numerical values ​​of regulars by 1. In addition, it is customary to start the account of days in the week from Monday, so the regulars of the Honor should be reducedanother 1, As a result, the formula is rewritten as follows:

NumbersThe regular values ​​of the regulars are determined from the following considerations: On January 1, the initial simple year was on Wednesday( q - 3) If all the months of the year had 28 days, then their first numbers would be on the same day as the beginning of the year.in January there are 4 full weeks and 3 more days, in February - 4 weeks, in March - 4 weeks and 3 days, in April - 4 weeks and 2 days, etc. When writing down the days of the week by the number of months, we note thatOn February 2, a three-day shift occurred in their series, and it falls on 3 + 3 = 6th day, that is, on Saturday, March 1 - this is also Saturday. Further, since in March 28 + 3 days, on April 1 we will have to 6 + 3 = 9( -7 =) = 2 - on Tuesday, 1 May - at 2 + 2 = 4 -. Fourth, etc. Thereforethe regulator for January will be the number 2, for February 2 + 3 = 5, for March 5 + 0 = 5, for April 5 + 3( -7) = 1, etc. So the above table is composed. Obviously, the difference between the values ​​of the regulars remains the same regardless of whether it happened on Monday, Wednesday or Saturday on January 1: they are determined by the remainders from the division by the 7th day of the days in calendar months.

Note that the insertion of the 366th day at the end of February of a leap year can be taken into account by decreasing the January and February regulators by 1.Then the formula will be suitable for a whole calendar year.

From the formula it is clear that:

a) in each specific calendar year, the value of the competitor ES has a specific value and in the transition from month to month only the value of the RS regularizer changes;

b) in the transition from year to year of the 28-year solar cycle, the magnitude of the solar epatic effect ES varies in a known manner. Therefore, it is possible to compile the monthly coefficients

K = ES +( RS-2) for each of the 12 months of a certain year and their nameplate for all years of the 28-year solar cycle. Then the day of the week is defined as

q = |( K + D) / 7 |

It is obvious that in the Julian calendar the values ​​of the monthly K coefficients are completely repeated every 28 years and through 28 * 25 = 700 years. Comparing the years of the 28-year cycle with specific years n.e., we get a kind of "perpetual calendar" with monthly coefficients.

Lunar regulars( regulares lunares, RL) make it possible to calculate the age( phase) of the Moon for the 1st day of the calendar month in any year of the 19-year cycle by known phases on the first day of the month in the first year of the cycle. These last for the year with the gold number NA-1 are written as follows:

January 1 -9, 1may -11, September 1 -16,

February 1 -10, June 1 -12, October 1 -16,

March 1 -9, 1 July -13, 1 November -18,

1 April - 10, 1 August - 14, 1 December - 18.

To establish the age of the moon on the 1st day of any year of the 19-year cycle, it is enough to the lunar regular of the monthadd the lunar effect of the given year and subtract, if necessary, 30.

For an example, we will establish the age of the Moon on August 1, 1370. The golden number of the year is NA = 3. From Table.9 we find the lunar effect EL = XXII.Consequently, the desired age of the moon is 22 + 14 =( 36 - 30 =) 6 days( inclusive account!), So that the new moon( more precisely, the earthquake) in 1370 was July 27.