# Question about the beginning of the year

As a result of a careful analysis of the chronicles, it was concluded that the New Year began in Russia with the advent of a new moon in the first spring days close to the spring equinox, when the snow came down from the fields and all nature woke up from the long winter

hibernation. But the past year could have 12 months, that is, 354 days, or 13 months - 384 days. Therefore, the beginning of the New Year in Russia could not occur on the same date of the Julian calendar, but slipped by the March numbers, falling in some years to February or April. This account of the time the chroniclers adhered for a very long time, although they dated their records using the Julian calendar.

So, for example, in the 1-st Novgorod Novgorod Chronicle we read: "In the summer of 6645 I will enter on March 7. ..".This entry would be completely incomprehensible if the New Year began on March 1.

In the Laurentian Chronicle "in the summer of 6646" the story does not end with reports of events that took place in February. On the contrary, here there is a record of Prince Vsevolod's accession to Kiev( March 5) and of the reign of Chernigov in his place of Vladimir Davidovich. The calculation shows that the March New Moon in 1139 took place on March 2.7, therefore, a new moon could be seen on the evening of March 5.Only after this moment began the next "summer 6647" of the ancient chroniclers.

Such a time account for the "heavenly" months, unlike the "book" months of the Julian calendar, persisted for centuries. For example, in the Nikon Chronicle, the total eclipse of the moon on March 10, 1476, is reported as follows: "March 10, February 15, February. .. a month has begun."In the Chernigov chronicle, in 1703, we read the following: "July - the heavenly month has come on the heels, the numbers of the fifth month of June( book). ..".

In the XIV century.a significant part of the territory of Kievan Rus fell into dependence first from Lithuania, and later - from the Polish state, in which, beginning in 1364, it is customary to celebrate the beginning of the year from January 1, and the calendar to lead from the "Nativity of Christ".Such a chronology is found in Ukrainian annals. For example: "rock 1432 Fyodor prince Ostrozhskoy, husband of great diyalnosti. .. obtained Smotrych."This is confirmed by the documents of the Lithuanian Metric, books of the State Chancellery of the Grand Duchy of Lithuania, which contain documents from the life and life of the Ukrainian people during the 13th-18th centuries. Soon after the Gregorian reform of 1582 the documents of the Lithuanian metry

were dated according to a new style. The fact that a new style was sometimes used in Ukraine at the beginning of the 17th century is evidenced by the record of the lunar eclipse made in the Kiev chronicle: "In the same rock, 1620 days, the 9th chest on the setting of the Sun was a terrible eclipse of the Month, which was one and two years old."Calculations showed that the eclipse really happened on November 29, 1620, according to art. Art.or on December 9 th. Art.at 7:24 pm and lasted 1 hour and 40 minutes.

## And the superstitious. .. calmed

Already noted the desire of Christian and Jewish interpreters of the Bible to "foresee" the fate of the world, proceeding from both the story of the six days of "creation" and the texts of the Bible as a whole. One such "foresight" of the Jewish scribes looked like this: "In the first and last verses of the Bible, the first letter alef occurs six times, and since six alef = 6000,. .. the existence of the world will last 6,000 years."

In terms of Christians, the numbers 6000 and 7000 also acquired a special meaning as some "equivalents" of the number of days of creation and full week. This is why, for example, Gregory, bishop of Nyssa, wrote about the year 375: "When this rapidly and transitory time ceases. .., then, of course, this week will also end, measuring time, and its place will be occupied by the eighth day, that is, the next century, which is all one day. .. ".

And yet the end of the "6th thousand years from the creation of the world"( according to the Byzantine era in 6000-5508 = 492) was rather calm, apparently because at that time there was not yet formed and accepted by the overwhelming majority of ChristiansThe notion of "when" must end this last thousand years.

To the X century.n.e. Western Europe practically passed to the chronology from the "Nativity of Christ", and at the end of the 10th century, as the 1000th anniversary of this era was over, many chroniclers ceased their records, people from cities fled to the forests and caves by thousands, everyone expected the end of the world and the terrible judgment.

For Christians of the Eastern rite, particularly in Byzantium and Rus, the 7th thousand years from the "creation of the world" ended at 7000-5508 = 1492. The end of it was alarming: in 1453 the Turks conquered the capital of Eastern Christianity - Constantinople. The expectation of an "end of the world" was reflected in the lists of Easter eggs. In particular, although in 1408 the 13th Great Indiction( 877-1408) ended - the 532-year-old circle of the Easter tables, neither in Greece nor in Russia did anyone dare to write down the Easter dates for the new 532th anniversary, but limited themselves tojust short lists for. .. 1492 Because of superstition, no one dared "step over" in the years account for 7 thousand years!

On the contrary, on the last page of one of these Easter tables, a following note was made: "Here fear, grief, where trouble is great in the crucifixion of Christ, this circle is, and that summer is at an end, in the same tea, your worldwide coming about the lord.". ".Obviously, we are talking about the circle of the Sun or the Moon. Of course, even at that time there were sensible people: on the other preserved Easter leaf, there is no prediction of the end of the world. On the contrary, the scribe calmly noted the following: "The nations speak, then the second coming of the Lord will be;but the holy Evangelist Marco says: no one knows the days and the hour. .. "However, even here the table is still limited to the 7000th = 1492th year, although the 14th Indiction began in 1409 and it should continue until 1940.

The writer and historian N.M. Karamzin says: "The seventh thousand came to an endyears from the creation of the world in the Greek chronographs: superstition with the end of it was waiting for the end of the world. "

But then came September 1( 7000 - 5509 =) in 1491, then on March 1( 7000 - 5508 =) in 1492, again on September 1, already in 7001 and "The superstitious calmed down, saw that the earth was standing and the celestial vault did notfluctuates with the outcome of the seventh thousand. .. ".

And since the "fatal" in 1492 passed safely, the Moscow church council immediately in September 1492 approved the Paschalia for the following years. The Council at the same time decided to postpone the beginning of the year from March 1 to September 1.

## "I keep summer in summer"

For many centuries, calendar calculations in Russia were carried out "on the fingers" in the literal sense of the word. ..

Today, looking at the calendar, we can immediately tell what day of the week it was, for example, 1March or August 26, on which day of this or that month will the new moon. In our dynamic time such information is necessary for everyone. But it was needed 500, and 800 years ago. Calendars at that time did not print. And yet people came out of the situation, scoring the days and phases of the moon on their fingers, In particular, such calculations were used to determine the day of Easter. The method used was called "the hands of Damascene", possibly by the name of the Greek theologian John Damascene( 680 - 760), although in his surviving works no mention was made of this method. In Russia, the method was better known as the "hand of the theologian".The "hands" drawing for calendar calculations is placed in the Uzhgorod Peninsula, a manuscript of the 16th century, which describes in detail the calendar characteristics of the year and gives their tables. The description of the method and drawings are given in the "Psalter with the Heraldman", printed by Ivan Fedorov in Zabludov( 1569).

"Damascene's Hand" can be "implemented" in different versions, than this or that level of simplicity is achieved in the account of tens, hundreds and thousands of years on the fingers. In particular, in Petrov's book The Hand of the Theologian( M., 1787) we find about 70 separate drawings of the right and left hands with a corresponding arrangement of letters and numbers on them. The use of this method for calendar calculations, the development of its various options certainly indicates a high level of mathematical culture in Russia.

In one of the ancient manuscripts we read: "If the philosopher knocks the paschalians. .. and begins to praise," and you ricul to him in the middle: If you are a great philosopher and philosopher. .., find me mi. .. on cue day, the moon of heavenwill come and at the cue hour,. .. find me, a philosopher, with the hand of the easter easter and the Easter Christian. .. "

Well, let's try. ..

** "Summer" is in the hand of **.Let's see first how the day of the week is calculated on the fingers for any calendar date, that is, as a year( "summer") you can "hold in your hand".

Let's place the letters on the index finger of the left hand so that the letter A occupies the first position from the palm of the hand, above it we put the letter B, even higher-G, at the end of the finger - D and then on the back side respectively E, S, 3Fig.).

Fig. Arrangement of the letters on the forefinger to determine the day of the week for a given calendar date

Now let us recall that the letters "forever" are strictly connected with the numbers of the calendar months and that the account of the letters of commencement begins on March 1 so that March 1 = G, March 2 - B, March 3 - A, etc. Thereforeon the finger next to the letter G and put the corresponding calendar date on March 1.Further, the days are counted down:

on March 2, where there is the letter B, 3- A, etc. The letter G, as well as in the table, corresponds also to the dates 8, 15, 22 and 29 March. Therefore, on April 1, we set up on the back of the finger where there is a letter 3. We go four times up and down, the position with the letter 3 will correspond to April 29, above it is 30, and on May 1 it will take place where the letter E stands, and t

We recall that the account is maintained according to the Julian calendar and that in our, XX century, the first day of the month under Art. Art.corresponds to the 14th to n. Art. To carry out calculations for a new style, it is necessary, leaving the dates in place, cyclically move all the letters of letters forward one-by-one( A to B, B to G, etc.).

Earlier it was found that the year 1986 - A. This means that all the numbers of calendar months( art.), Falling on this joint, this year will be Sundays: March 3, March 10, September 1. .December 1,. ..

So ancient "philosophers" kept the "summer" in their hands, that is, they set the day of the week for any calendar date. Earlier, however, in the same way it was necessary to find a good year.

** The circle of the sun and the ship. ** The breakdown of the year, as it follows from the table, is established at the place of the year in the 28-year solar cycle. Therefore, the search for a hand-craft is accompanied by a simultaneous calculation of the circle of the Sun for a given year. To do this, first arrange( Fig.) The numbers from 1 to 28( these are the circles of the Sun) and the letters( corresponding to the circles of the Sun according to Table) on the joints and ends of the fingers of the left hand, starting from the end of the middle finger( hand A and circle of the Sun 1).

Fig. Arrangement of letters and numbers on the fingers of the left hand to determine the circle of the sun and the rifle

At the end of the ring finger will be handed to B and the circle of the Sun 2, on the little finger-D and 3. Then the score continues on the back of the hand( these numbers and letters in the figure are placed side by sidewith fingers) from left to right and from top to bottom, and then on the palms also from left to right, but already from the bottom up. The account ends at the end of the index finger-the round of the Sun 28( = 0) and handed 3. If you want to continue counting, you must go back to the middle finger. It is useful to remember that all leap years are located on the little finger( they correspond to the circles of the Sun Q - 3, 7, 11. ..).

In order to find the circle of the Sun and the vestige of the year, it is necessary, starting from the year 1 of the Constantinople era, to "walk" around the Sun until the desired year. These calculations can be simplified, since it is not difficult to see that hundreds and thousands, as well as "twenty" years, fit on one index finger. This follows from the fact that 100 = 3 * 28 + 16.Therefore, counting is done first on the index finger. At the same time, for every 1000 years there are five joints, one hundred years four joints and for twenty years five joints. In what direction to keep score is a secondary matter( there are variants, both of which), since the transition, say, at the count of thousands from the end of the finger( where 28 = 0 is located) to five positions on the back side( "from oneself"),completely equivalent to the transition to two positions from the palm( "to yourself").For the sake of monotony, we shall count in the direction of growth shown in Fig.35 numbers of units - from 0 "down from yourself", i.e. first on the back side, then from the palm side upwards in the direction of the numbers 0 - 4 - 8 - 12 - 16 - 20 - 24 - 0.

Account of the years remainingafter the postponement of thousands, hundreds and twentieth years, is already on all fingers along the horizontal rows in the direction of the growth of numbers( numbers of the circle of the Sun).

For an example, let's define the circle of the Sun and ship for 1986.e.= 7494 of the Constantinople era.

1. Seven thousand years, five joints per thousand - this is 7 * 5 = 35 joints, i.e., exactly 5 cycles on the index finger of the left hand. Therefore, the account of hundreds also starts from the same place - from the tip of the index finger, where the number 28( = 0) stands.

2. Four hundred years, four joints per hundred yield 4 * 4 = 16 = 2 * 7 + 2. We move to two positions to the number 8( where the letter D is the letter D).If the year number were expressed by an integer number of hundreds, then the calculation would end there. In the case of the year 7400, it would be worth the G and the circle of the Sun Q = 8.

3. The number of tens and units of the year 94 = 4 * 20 + 14.Since 20 years correspond to 5 joints, then all in front should be counted 4 * 5 = 20 = 2 * 7 + 6 - six joints( or 7 - 6 = 1 back!).Therefore, from the position marked with the number 8, we move to the position where number 4 is located,

4. In the same place, i.e. in this case on the back of the hand, we pass to the account on the other fingers, moving on 14 positions according to the numbers of the circle of the Sun: the first position is 5, the second position is 6, etc. As a result, we find the circle of the Sun Q = 18( = 4 + 14) and вру вру вру. If we stopped at the count of twenty years, for example, at position 24,9 more years, then at the final count, we would come to the circle of the Sun 5( 24 + 9 - 28 = 5) and to the right of S.

** Circle of the Moon and correct. ** We now describe the method for determining the circle of the moon and the working letter( corrections) on the fingers of the right hand( Fig.).

Fig. Arrangement of letters and numbers on the right hand to determine the lunar circle and the working letter

. Recall that the latter indicates the date of "damage" - the day immediately following the spring full moon. In the proposed version, the reader uses all five fingers and only on one side of the hand - on the palm of your hand. Four fingers have four positions, and on the big one there are three, which in total gives 19 the full cycle of the years of the lunar circle. Count the years in the cycle from the index finger to the little finger( 1 - 4), then go to the thumb, where the positions 5, 10 and 15 are. The cycle( 19 = 0) ends at the end of the little finger. The next, the 20th from the beginning of the account, the year will take the position indicated by the number 1. After the second 19-year cycle, the 40th year will have to be replaced by the number 2, the 60th - in place of the number 3, the 80th - in placethe number 4, the 100th - on the base of the thumb, indicated by the number 5. As can be seen, moving along the horizontal to the left from any position to the next corresponds to an increase in the number of years by 20.

It is easy to establish that the 200th year falls on the 2nd jointthumb( position 10), 300th - at its end( 15), 400th - at the lower joint of the index finger( position 1).The same finger places the years 500( 6), 600( 11) and 700( 16).On the middle finger, the years 800, 900, 1000 and 1100, etc., will be located from the bottom upwards. Hence, the transition from the bottom up to one position corresponds to an increase in the number of the year by 100.

Finally, the "even" thousands of years are located one above the other,thumbs: position 5 corresponds to 2000, 10 to 4000, 15 to 6000, 1 to 8000, etc. The odd thousand number is counted in the same way starting at position 12: 12 corresponds to 1000, 17( middle finger end) to 3000 andetc. 5508 of the Constantinople era( ie, 1 BC) falls on the end of the middle finger.

So, when calculating the circle of the moon on the fingers of the right hand( in the version used here!), We count:

1) for every thousand years-12 positions in the direction of increasing numbers of the lunar circle( left),

2) for every hundred years - fivepositions to the left( or, what is the same, one position up),

3) for every 20 years - one position to the left. The rest of the year we consider in the usual way one position per year.

Example. Find the lunar circle and a working letter for 1986 n.e.= 7494 of the Constantinople era.

1) Since one thousand years correspond to 12 joints, then all you need to count is 7 * 12 = 84 = 4 * 19 +8.Four complete 19-year cycles are discarded and, having started the account from the number 1, we find the position 7000 at the position 8.

2) Four hundred years corresponds to moving horizontally to a position to the left: 4 * 5 = 20 = 19 + 1( the complete cycle is discarded).Therefore, 7400 will take the position 9.

3) Bear in mind that 94 = 80 + 14 = 4 * 20 + 14.Therefore, firstly, a) we pass along the horizontal line 4 positions to the left, i.e. from 9 to 13, then b) we count 14 more positions( 13 + 14 = 27 = 19 + 8) up to the number 8. So we find that for 1986The circle of the Moon is L = 8, and the correct letter is C. From the table it follows that the calculated( !) spring full moon will be one day earlier than indicated by this letter, ie on April 15 st. Art.

** The date of Easter. ** Everything that has been done so far in these calculations is an elementary need of a purely civil calendar: according to the found day, the days of the week are brought into line with the numbers of the months, the moon's circle and a working letter allow you to compare the moon phases with these numbers.

And only the last stage of calculations-the definition of the key of boundaries-relates to the establishment of the Easter date. It uses the distribution of key letters on the joints of the left hand, as shown in Fig.

Fig. Arrangement of letters on the fingers of the left hand to determine the key.

First of all we find in the figure a valid letter corresponding to the given year. Then from the letter located on the nearest lower joint on the right, including in the count and its( !), We count in the direction of the working letter as many positions as they are contained in the number of the vestige.

As a result, we get the key letter( which, however, you can also define the days of the week).

In the specific case of 1986, as has just been established, a valid letter Ç is delivered A = 1.The key letter of the year will be b, since we begin the countdown from it, and the number of the voucher 1. It follows that in 1986 Easter will be on April 21 st.item, that is, on May 4,Art. If the working letter is, for example, H, and the right-hand side of G( 3), then the count is made in the same way - from the letter P located on the same finger - and then the key letter is also H;but if, in the same case, the letters D( 4), "is"( 5), etc. appear, then the count must be started from the next finger - from the letter "from";through a valid letter "skip" can not. For the little finger "neighbor" is the thumb.

Let's consider one more example. The Laurentian Chronicle reports that the Bishop of Pereyaslav, Sylvester, died "in the summer of 6631. .. April, April 12, Thursday."To check whether the year of this event is correctly indicated( and all the others, about which the chronicler reports under the same number of the year!), It is necessary to make sure whether Easter was really celebrated in 6631 from the "creation of the world" on April 15.

First find the circle of the Sun and the ship.

1) 6 * 5 = 30 = 4 * 7 + 2: the transition to Fig.two positions down, from "28" to "8".

2) 6 * 4 = 24 = 3 * 7 + 3: go from position "8" to "20".

3) 31 = 20 + 11:

a) go to 5 positions forward - from "20" to "12",

b) from position "12" move to the right and successively upward by 11 positions. So we find the circle of the Sun Q-23 and hand-delivered 3.

Now we find the circle of the moon and the working letter:

1) 6 * 12 = 72 = 3 * 19 + 15: starting from the digit 1 on the index finger of the right hand( see the figure), we stop at position 15.

2) 6 * 5 = 30 = 19 + 11, 15 + 11 = 19 + 7 - After making a circle, go to position 7.

3) 31 = 20 + 11;we have( 7 + 1) +11 = 19. Thus, for 6631 the circle of the Moon is L = 19 and the correct letter is "from".

And, finally, in Fig.we find the key word of the year. The correct letter "from" is located on the right side of the middle finger. The nearest lower joint to the right is indicated by the letter b. From it( including also her), we count 7 positions corresponding to the "land" party. Thus we come to the key letter C. Consequently, in 6631, from the "creation of the world"( = 1123 AD), Easter really was on April 15.Thus, the event mentioned in the annals is dated correctly.

Of course, in the past such calculations were used to determine the date of Easter. But, we emphasize once again that by counting on the fingers the Moon phase was determined for this or that time of the year, the day of one or another calendar date was set.

It is worth mentioning again that "Easter calculations"( though not on fingers, but on tables) allow historians to check( and sometimes set!) The date of an event.

** Indication. ** Thus, and "on the cue day the Moon of Heaven will come," and "Passover by the Jew and Easter Christian" with the nominee "Damascene's hands" we have already determined( the date of the second day of the Jewish Passover is indicated by a working letter, although in our time fromfor the inaccuracy of the metonic cycle, the astronomical full moon, and consequently, 15 Nisan are several days earlier).But why does the manuscript refer to the "hand of the indicium"?

It turns out, in addition to lunar circles and serviceable letters, the indict of the year was also determined on the same right hand. We distinguish this figure separately( Fig.).

Fig. Arrangement of numbers on the fingers of the right hand to determine the indicator

The numbers of the indices - from I to XV - are placed here on three fingers and palms, starting from the end of the index finger, three times five positions each. It is easy to establish that all tens, hundreds and thousands of years are in the lower horizontal row. In particular, there are also 20, 50, 80, 200, 500, 800, 2000, 5000 on the joint where there is a number of 5( V), to position 10( X) - 10, 40, 70, 100, 400,700, 1000, 4000 and 7000, to the position 15( XV) -30, 60, 90, 300, 600, 900, 3000, 6000, that is, in all cases the same triples of the original numbers are repeated. Having noticed these patterns, the calculations of the indicator can be simplified. In this case, the starting point is not I, but XV( = 0).

Define the year's symbol 1986 = 7494 from the "creation of the world."

1) The number of thousands-7 = 2 * 3 + 1.Therefore, starting the score with XV( = 0), go to the right one position, where X.

2) The number of hundreds - 4 = 3 + 1: move to the right one more position, in place of V.

3) 94 = 90+ 4 = 30 * 3 + 4

: a) after completing 30 cycles on the bottom horizontal( V-XV -X-V), return to the same place V;

b) the remaining years are considered in the usual order of increasing the number of digits: 5 + 4 = 9. Therefore, the 1986 indict I = 9( IX).

** Algorithms "The Hands of Damascene".** Let's take a closer look at the basic principles of the "Damascene hand" method. This will, if desired, use a larger or smaller number of fingers, change the arrangement of the letters on them, arrange them on both sides of the hand, etc. Let's start by calculating the circle of the Sun.

A thousand and a hundred years can be represented as follows: 1000 = 35 * 28 + 20, 100 = 3 * 28 + 16.

In calculations, only the remainders of dividing thousands and hundreds by 28 are considered, and the complete cycles are discarded. Since four fingers are used in the calculation, and 20 = 4 * 5, one thousand years corresponds to the transition on the finger from which the account begins( index), to 5 positions( "five joints").In the case of hundreds, the remainder( 16) is 4 * 4, which corresponds to moving forward by 4 positions. This is the whole "secret" of the method.

We will write an arbitrary number of the year of the Constantinople era in the form K = TXZY, where T is the number of thousands, X is hundreds, Z is of tens and Y-units. Throwing out entire cycles upon counting thousands and hundreds of years, we write the words in the form of the formula

Q = |( 20T + 16X + ZY) / 28 |

For 1986 = 7494 in the Constantinople era

Q = |( 20 * 7 + 16 * 4 + 94) / 28 | = 18

When calculating the circle of the moon, we have 1000 = 52 * 19 + 12, 100 = 5 * 19+ 5 and 20 = 19 + 1 - transition to 12, 5 and 1 positions respectively, the whole cycles are discarded. Writing the year number in the same form K = TXZY, we have a formula for determining the lunar circle, which is the result of a "game on the leftovers":

L = |( 12T + 5X + ZY) / 19 |

Since the count is conducted in a closed circle, the sum of the remainders is divided by the definition of the circle of the sun by 28, the circle of the moon by 19, and the number of complete cycles is discarded. We note that here the symbols [] and ||, as before, mean, respectively, the whole part of the quotient and the remainder after division. For simplicity, the two-digit number of units and tens of years is written in the form ZY( strictly speaking, if Z is the number of tens, then the record should be 10Z + Y).

For a specific 7494 g we have [ZY / 20] = 4 and | ZY / 20 |= 14.

So, L = 8

And, finally, when determining the indices 1000 = 66 * 15 + 10, 100 = 6 * 15 + 10. But we have already seen that the transition to 10 units forward is equivalent to a horizontal shift by oneposition, if the numbers are located, as in the figure, and the formula for determining the indicator has the form

I = |( 10T + 10X + ZY) / 15 |

For the year 7494 we have

I = |( 10 * 7 + 10 * 4 + 94) / 15 |

Recall that in the period from March 1 to September 1 the numbers of years of both styles of the era from the "creation of the world" coincide. But the "indicator year" begins on September 1, hence the count of the years goes in the Byzantine era, whereas the reference point of the circles of the Sun and the Moon is shifted for six months in advance, and the years are counted here in the Constantinople era.

These formulas we have resulted not so much for calculations of circles of the Sun and the Moon and indices, how many for finding-out of bases of a method "Damascene's hands".It can hardly be objected to the assertion that the outlined approach to the solution of tasks in the form of "play on the leftovers" is evidence of a high mathematical culture. ..