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CALENDAR(from Latin calendae or kalendae, “calends” - the name of the first day of the month among the ancient Romans), a way of dividing the year into convenient periodic intervals of time. The main tasks of the calendar are: a) fixing dates and b) measuring time intervals. For example, task (a) involves recording the dates of natural phenomena, both periodic - equinoxes, eclipses, tides - and non-periodic, such as earthquakes. The calendar allows you to record historical and social events in their chronological sequence. One of the important tasks of the calendar is to determine the moments of church events and “drifting” holidays (for example, Easter). Function (b) of the calendar is used in the public sphere and in everyday life, where interest payments, wages and other business relationships are based on specific time intervals. Many statistical and scientific studies also use time intervals.
There are three main types of calendars: 1) lunar, 2) solar and 3) lunisolar.
Moon calendar
based on the length of the synodic, or lunar month (29.53059 days), determined by the period of change of lunar phases; the length of the solar year is not taken into account. An example of a lunar calendar is the Muslim calendar. Most peoples who use the lunar calendar consider the months to alternately consist of 29 or 30 days, so the average length of a month is 29.5 days. The length of the lunar year in such a calendar is 12·29.5 = 354 days. The true lunar year, consisting of 12 synodic months, contains 354.3671 days. The calendar does not take this fractional part into account; Thus, over 30 years, a discrepancy of 11.012 days accumulates. Adding these 11 days every 30 years restores the calendar to the lunar phases. The main disadvantage of the lunar calendar is that its year is 11 days shorter than the solar year; therefore, the beginning of certain seasons according to the lunar calendar occurs year after year on increasingly later dates, which causes certain difficulties in public life.
Solar calendar
coordinated with the length of the solar year; in it, the beginning and duration of calendar months are not related to the change of lunar phases. The ancient Egyptians and Mayans had solar calendars; Nowadays, most countries also use the solar calendar. A true solar year contains 365.2422 days; but the civil calendar, to be convenient, must contain an integer number of days, therefore in the solar calendar an ordinary year contains 365 days, and the fractional part of the day (0.2422) is taken into account every few years by adding one day to the so-called leap year. The solar calendar is usually based on four main dates - two equinoxes and two solstices. The accuracy of a calendar is determined by how accurately the equinox falls on the same day each year.
Lunar-solar calendar
is an attempt to reconcile the length of the lunar month and the solar (tropical) year through periodic adjustments. To ensure that the average number of days per year according to the lunar calendar corresponds to the solar year, a thirteenth lunar month is added every 2 or 3 years. This trick is required to ensure that the growing seasons fall on the same dates each year. An example of a lunisolar calendar is given by the Jewish calendar, officially adopted in Israel.
TIME MEASUREMENT
Calendars use units of time based on the periodic movements of astronomical objects. The rotation of the Earth around its axis determines the length of the day, the revolution of the Moon around the Earth gives the length of the lunar month, and the revolution of the Earth around the Sun determines the solar year.
Sunny days.
The apparent movement of the Sun across the sky sets the true solar day as the interval between two successive passages of the Sun through the meridian at the lower culmination. If this movement reflected only the rotation of the Earth around its axis, then it would occur very uniformly. But it is also associated with the uneven movement of the Earth around the Sun and with the tilt of the Earth's axis; therefore, the true solar day is variable. To measure time in everyday life and in science, the mathematically calculated position of the “average sun” and, accordingly, the average solar day, which have a constant duration, are used. In most countries, the beginning of the day is at 0 o'clock, i.e. at midnight. But this was not always the case: in biblical times, in Ancient Greece and Judea, as well as in some other eras, the beginning of the day was in the evening. For the Romans, in different periods of their history, the day began at different times of the day.
Moon month.
Initially, the length of the month was determined by the period of the Moon’s revolution around the Earth, more precisely, by the synodic lunar period, equal to the time interval between two successive occurrences of identical phases of the Moon, for example, new moons or full moons. The average synodic lunar month (the so-called “lunar month”) lasts 29 days 12 hours 44 minutes 2.8 seconds. In biblical times, lunation was considered equal to 30 days, but the Romans, Greeks and some other peoples accepted the value measured by astronomers as 29.5 days as a standard. The lunar month is a convenient unit of time in social life, since it is longer than a day, but shorter than a year. In ancient times, the Moon attracted universal interest as an instrument for measuring time, since it is very easy to observe the expressive change of its phases. In addition, the lunar month was associated with various religious needs and therefore played an important role in the preparation of the calendar.
Year.
In everyday life, including when compiling a calendar, the word “year” means the tropical year (“year of the seasons”), equal to the time interval between two successive passages of the Sun through the vernal equinox. Now its duration is 365 days 5 hours 48 minutes 45.6 seconds, and every 100 years it decreases by 0.5 seconds. Even ancient civilizations used this seasonal year; According to the records of the Egyptians, Chinese and other ancient peoples, it is clear that the length of the year was initially taken to be 360 days. But quite a long time ago the length of the tropical year was specified to 365 days. Later, the Egyptians accepted its duration as 365.25 days, and the great ancient astronomer Hipparchus reduced this quarter of a day by several minutes. The civil year did not always begin on January 1. Many ancient peoples (as well as some modern ones) began the year from the moment of the spring equinox, and in Ancient Egypt the year began on the day of the autumn equinox.
HISTORY OF CALENDARS
Greek calendar.
In the ancient Greek calendar, a normal year consisted of 354 days. But since it lacked 11.25 days to coordinate with the solar year, then every 8 years 90 days (11.25ґ8), divided into three equal months, were added to the year; this 8-year cycle was called an octaesteride. After about 432 BC. the Greek calendar was based on the Metonic cycle and then the Callippus cycle (see section on cycles and eras below).
Roman calendar.
According to ancient historians, at the beginning (c. 8th century BC) the Latin calendar consisted of 10 months and contained 304 days: five months of 31 days each, four months of 30 and one month of 29 days. The year began on March 1; from here the names of some months have been preserved - September (“seventh”), October (“eighth”), November (“ninth”) and December (“tenth”). The new day began at midnight. Subsequently, the Roman calendar underwent considerable changes. Before 700 BC Emperor Numa Pompilius added two months - January and February. Numa's calendar contained 7 months of 29 days, 4 months of 31 days and February with 28 days, which amounted to 355 days. Around 451 BC a group of 10 senior Roman officials (decemvirs) brought the sequence of months to its current form, moving the beginning of the year from March 1 to January 1. Later, a college of pontiffs was established, which carried out a reform of the calendar.
Julian calendar.
By 46 BC, when Julius Caesar became Pontifex Maximus, calendar dates were clearly at odds with natural seasonal phenomena. There were so many complaints that radical reform became necessary. To restore the previous connection of the calendar with the seasons, Caesar, on the advice of the Alexandrian astronomer Sosigenes, extended the 46th year BC, adding a month of 23 days after February and two months of 34 and 33 days between November and December. Thus, that year had 445 days and was called the “year of confusion.” Then Caesar fixed the duration of the ordinary year at 365 days with the introduction of one additional day every four years after February 24. This made it possible to bring the average length of the year (365.25 days) closer to the length of the tropical year. Caesar deliberately abandoned the lunar year and chose the solar year, since this made all insertions, except for the leap year, unnecessary. Thus Caesar established the length of the year exactly equal to 365 days and 6 hours; Since then, this meaning has been widely used: after three ordinary years there follows one leap year. Caesar changed the length of the months (Table 1), making February 29 days in a normal year and 30 days in a leap year. This Julian calendar, now often called the “old style,” was introduced on January 1, 45 BC. At the same time, the month of Quintilis was renamed July in honor of Julius Caesar, and the vernal equinox was shifted to its original date of March 25.
Augustian calendar.
After the death of Caesar, the pontiffs, apparently misunderstanding the instructions about leap years, added a leap year not every four years, but every three years, for 36 years. Emperor Augustus corrected this error by skipping three leap years in the period from 8 BC. to 8 AD From this point on, only years with a number divisible by 4 were considered leap years. In honor of the emperor, the month of Sextilis was renamed August. In addition, the number of days in this month was increased from 30 to 31. These days were taken from February. September and November were reduced from 31 to 30 days, and October and December were increased from 30 to 31 days, which maintained the total number of days in the calendar (Table 1). Thus, the modern system of months developed. Some authors consider Julius Caesar, not Augustus, to be the founder of the modern calendar.
| Table 1. DURATION OF MONTHS THREE ROMAN CALENDARS (in days) |
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| Name of the month | Calendar of the Decemvirs (c. 414 BC) |
Calendar Julia (45 BC) |
August calendar (8 BC) |
| Januarius | 29 | 31 | 31 |
| Februarius | 28 | 29–30 | 28–29 |
| Martius | 31 | 31 | 31 |
| Aprilis | 29 | 30 | 30 |
| Mayus | 31 | 31 | 31 |
| Junius | 29 | 30 | 30 |
| Quintilis 1) | 31 | 31 | 31 |
| Sextilis 2) | 29 | 30 | 31 |
| September | 29 | 31 | 30 |
| October | 31 | 30 | 31 |
| November | 29 | 31 | 30 |
| December | 29 | 30 | 31 |
| 1) Julius in the Julius and Augustan calendars. 2) August in the Augustan calendar. |
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Kalends, Ides and Nones.
The Romans used these words only in the plural, calling special days of the month. Kalends, as mentioned above, were called the first day of each month. The Ides were the 15th day of March, May, July (quintilis), October and the 13th day of the remaining (short) months. In modern calculations, the nones are the 8th day before the Ides. But the Romans took into account the Ides themselves, so they had nones on the 9th day (hence their name “nonus”, nine). The Ides of March was March 15 or, less specifically, any of the seven days preceding it: from March 8 to March 15 inclusive. The nones of March, May, July and October fell on the 7th day of the month, and in other short months - on the 5th day. The days of the month were counted backwards: in the first half of the month they said that so many days remained until the nons or ids, and in the second half - until the calendars of the next month.
Gregorian calendar.
The Julian year, with a duration of 365 days 6 hours, is 11 minutes 14 seconds longer than the true solar year, therefore, over time, the onset of seasonal phenomena according to the Julian calendar occurred on earlier and earlier dates. Particularly strong discontent was caused by the constant shift in the date of Easter, associated with the spring equinox. In 325 AD The Council of Nicaea issued a decree on a single date for Easter for the entire Christian church. In subsequent centuries, many proposals were made to improve the calendar. Finally, the proposals of the Neapolitan astronomer and physician Aloysius Lilius (Luigi Lilio Giraldi) and the Bavarian Jesuit Christopher Clavius were approved by Pope Gregory XIII. On February 24, 1582, he issued a bull introducing two important additions to the Julian calendar: 10 days were removed from the 1582 calendar - after October 4, October 15 followed. This allowed March 21 to be retained as the date of the vernal equinox, which it probably was in 325 AD. In addition, three out of every four century years were to be considered ordinary years and only those divisible by 400 were to be considered leap years. Thus, 1582 became the first year of the Gregorian calendar, often called the "new style". France switched to the new style the same year. Some other Catholic countries adopted it in 1583. Other countries adopted the new style over the years: for example, Great Britain adopted the Gregorian calendar from 1752; By leap year 1700, according to the Julian calendar, the difference between it and the Gregorian calendar was already 11 days, so in Great Britain, after September 2, 1752, September 14 came. In the same year in England, the beginning of the year was moved to January 1 (before that, the new year began on the day of the Annunciation - March 25). Retrospective correction of dates caused much confusion for many years, as Pope Gregory XIII ordered corrections of all past dates back to the Council of Nicaea. The Gregorian calendar is used today in many countries, including the United States and Russia, which abandoned the Eastern (Julian) calendar only after the October (actually November) Bolshevik Revolution of 1917. The Gregorian calendar is not absolutely accurate: it is 26 seconds longer than the tropical year. The difference reaches one day in 3323 years. To compensate for them, instead of eliminating three leap years out of every 400 years, it would be necessary to eliminate one leap year out of every 128 years; this would correct the calendar so much that in only 100,000 years the difference between the calendar and tropical years would reach 1 day.
Jewish calendar.
This typical lunisolar calendar has very ancient origins. Its months contain alternately 29 and 30 days, and every 3 years the 13th month Veadar is added; it is inserted before the month of Nissan every 3rd, 6th, 8th, 11th, 14th, 17th and 19th year of the 19-year cycle. Nissan is the first month of the Jewish calendar, although years are counted from the seventh month of Tishri. The insertion of Veadar causes the vernal equinox to always fall on a lunation in the month of Nissan. In the Gregorian calendar there are two types of years - ordinary and leap years, and in the Jewish calendar - an ordinary (12-month) year and an embolismic (13-month) year. In the embolismic year, of the 30 days inserted before Nissan, 1 day belongs to the sixth month of Adar (which usually contains 29 days), and 29 days make up Veadar. In fact, the Jewish lunisolar calendar is even more complex than described here. Although it is suitable for calculating time, but due to the use of the lunar month it cannot be considered an effective modern instrument of this kind.
Muslim calendar.
Before Muhammad, who died in 632, the Arabs had a lunisolar calendar with intercalary months, similar to the Jewish one. It is believed that the errors of the old calendar forced Muhammad to abandon additional months and introduce a lunar calendar, the first year of which was 622. In it, the day and the synodic lunar month are taken as the unit of reference, and the seasons are not taken into account at all. A lunar month is considered equal to 29.5 days, and a year consists of 12 months containing alternately 29 or 30 days. In a 30-year cycle, the last month of the year contains 29 days for 19 years, and the remaining 11 years contain 30 days. The average length of the year in this calendar is 354.37 days. The Muslim calendar is widely used in the Near and Middle East, although Turkey abandoned it in 1925 in favor of the Gregorian calendar.
Egyptian calendar.
The early Egyptian calendar was lunar, as evidenced by the hieroglyph for “month” in the form of a lunar crescent. Later, the life of the Egyptians turned out to be closely connected with the annual floods of the Nile, which became the starting point for them, stimulating the creation of a solar calendar. According to J. Breasted, this calendar was introduced in 4236 BC, and this date is considered the oldest historical date. The solar year in Egypt contained 12 months of 30 days, and at the end of the last month there were five more additional days (epagomen), giving a total of 365 days. Since the calendar year was 1/4 day shorter than the solar year, over time it became more and more at odds with the seasons. Observing the heliacal risings of Sirius (the first appearance of the star in the rays of dawn after its invisibility during the period of conjunction with the Sun), the Egyptians determined that 1461 Egyptian years of 365 days are equal to 1460 solar years of 365.25 days. This interval is known as the Sothis period. For a long time, the priests prevented any change in the calendar. Finally in 238 BC. Ptolemy III issued a decree adding one day to every fourth year, i.e. introduced something like a leap year. This is how the modern solar calendar was born. The Egyptians' day began with sunrise, their week consisted of 10 days, and their month consisted of three weeks.
Chinese calendar.
The prehistoric Chinese calendar was lunar. Around 2357 BC Emperor Yao, dissatisfied with the existing lunar calendar, ordered his astronomers to determine the dates of the equinoxes and, using intercalary months, create a seasonal calendar convenient for agriculture. To harmonize the 354-day lunar calendar with the 365-day astronomical year, 7 intercalary months were added every 19 years, following detailed instructions. Although solar and lunar years were generally consistent, lunisolar differences remained; they were corrected when they reached a noticeable size. However, the calendar was still imperfect: the years were of unequal length, and the equinoxes fell on different dates. In the Chinese calendar, the year consisted of 24 crescents. The Chinese calendar has a 60-year cycle, which begins in 2637 BC. (according to other sources - 2397 BC) with several internal periods, and each year has a rather funny name, for example, “year of the cow” in 1997, “year of the tiger” in 1998, “hare” in 1999, “dragon” in 2000, etc., which are repeated with a period of 12 years. After Western penetration into China in the 19th century. The Gregorian calendar began to be used in commerce, and in 1911 it was officially adopted in the new Republic of China. However, peasants still continued to use the ancient lunar calendar, but since 1930 it was banned.
Mayan and Aztec calendars.
The ancient Mayan civilization had a very high art of counting time. Their calendar contained 365 days and consisted of 18 months of 20 days (each month and each day had its own name) plus 5 additional days that did not belong to any month. The calendar consisted of 28 weeks of 13 numbered days each, amounting to a total of 364 days; one day remained extra. Almost the same calendar was used by the neighbors of the Mayans, the Aztecs. The Aztec calendar stone is of great interest. The face in the center represents the Sun. The four large rectangles adjacent to it depict heads symbolizing the dates of the four previous world eras. The heads and symbols in the rectangles of the next circle symbolize the 20 days of the month. Large triangular figures represent the rays of the sun, and at the base of the outer circle two fiery serpents represent the heat of the heavens. The Aztec calendar is similar to the Mayan calendar, but the names of the months are different.
CYCLES AND ERAS
Sunday letters
– is a diagram showing the relationship between the day of the month and the day of the week during any year. For example, it allows you to determine Sundays, and based on this, create a calendar for the whole year. The table of weekly letters can be written like this:
Each day of the year, except February 29 in leap years, is indicated by a letter. A specific day of the week is always indicated by the same letter throughout the year, with the exception of leap years; therefore, the letter that represents the first Sunday corresponds to all other Sundays of this year. Knowing the Sunday letters of any year (from A to G) you can completely restore the order of the days of the week for that year. The following table is useful:
To determine the order of the days of the week and create a calendar for any year, you need to have a table of Sunday letters for each year (Table 2) and a table of the structure of the calendar of any year with known Sunday letters (Table 3). For example, let's find the day of the week for August 10, 1908. In the table. 2, at the intersection of the centuries column with the line containing the last two digits of the year, Sunday letters are indicated. Leap years have two letters, and for full centuries such as 1900, the letters are listed in the top row. For Leap Year 1908, the Sunday letters will be ED. From the leap year part of the table. 3, using the letters ED we find the string of days of the week, and the intersection of the date “August 10” with it gives Monday. In the same way, we find that March 30, 1945 was a Friday, April 1, 1953 was a Wednesday, November 27, 1983 was a Sunday, etc.
| Table 2. SUNDAY LETTERS FOR ANY YEAR FROM 1700 TO 2800 (according to A. Philip) |
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| Last two digits of the year | Centennial years | ||||||
| 1700 2100 2500 |
1800 2200 2600 |
1900 2300 2700 |
2000 2400 2800 |
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| 00 | C | E | G | B.A. | |||
| 01 02 03 04 |
29 30 31 32 |
57 58 59 60 |
85 86 87 88 |
B A G F.E. |
D C B A.G. |
F E D C.B. |
G F E DC |
| 05 06 07 08 |
33 34 35 36 |
61 62 63 64 |
89 90 91 92 |
D C B A.G. |
F E D C.B. |
A G F ED |
B A G F.E. |
| 09 10 11 12 |
37 38 39 40 |
65 66 67 68 |
93 94 95 96 |
F E D C.B. |
A G F ED |
C B A GF |
D C B A.G. |
| 13 14 15 16 |
41 42 43 44 |
69 70 71 72 |
97 98 99 . . |
A G F ED |
C B A GF |
E D C B.A. |
F E D C.B. |
| 17 18 19 20 |
45 46 47 48 |
73 74 75 76 |
. . . . . . . . |
C B A GF |
E D C B.A. |
G F E DC |
A G F ED |
| 21 22 23 24 |
49 50 51 52 |
77 78 79 80 |
. . . . . . . . |
E D C B.A. |
G F E DC |
B A G F.E. |
C B A GF |
| 25 26 27 28 |
53 54 55 56 |
81 82 83 84 |
. . . . . . . . |
G F E DC |
B A G F.E. |
D C B A.G. |
E D C B.A. |
| Table 3. CALENDAR FOR ANY YEAR (according to A. Philip) | |||||||||||||||||
| Normal year | |||||||||||||||||
| Sunday letters and starting days of the week | A G F E D C B |
Sun Mon W Wed Thu Mon Sat |
Mon W Wed Thu Fri Sat Sun |
W Wed Thu Fri Sat Sun Mon |
Wed Thu Fri Sat Sun Mon W |
Thu Fri Sat Sun Mon W Wed |
Fri Sat Sun Mon W Wed Thu |
Sat Sun Mon W Wed Thu Fri |
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| Month | Days in a month | ||||||||||||||||
| January October |
31 31 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
|||||||||
| February March November |
28 31 30 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
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|
April |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
||||||||||
|
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
|||||||||||
|
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 |
|||||||||||
|
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
|||||||||||
|
September |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
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| Leap year | |||||||||||||||||
| Sunday letters and starting days of the week | A.G. GF F.E. ED DC C.B. B.A. |
Sun Mon W Wed Thu Mon Sat |
Mon W Wed Thu Fri Sat Sun |
W Wed Thu Fri Sat Sun Mon |
Wed Thu Fri Sat Sun Mon W |
Thu Fri Sat Sun Mon W Wed |
Fri Sat Sun Mon W Wed Thu |
Sat Sun Mon W Wed Thu Fri |
|||||||||
| Month | Days in a month | ||||||||||||||||
| January April July |
31 30 31 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
|||||||||
|
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
|||||||||||
| February August |
29 31 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
|||||||||
| March November |
31 30 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
|||||||||
|
3 10 17 24 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
|||||||||||
|
September |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
7 14 21 28 |
1 8 15 22 29 |
||||||||||
|
7 14 21 28 |
1 8 15 22 29 |
2 9 16 23 30 |
3 10 17 24 31 |
4 11 18 25 |
5 12 19 26 |
6 13 20 27 |
|||||||||||
Metonic cycle
shows the relationship between the lunar month and the solar year; therefore, it became the basis for the Greek, Hebrew and some other calendars. This cycle consists of 19 years of 12 months plus 7 additional months. It is named after the Greek astronomer Meton, who discovered it in 432 BC, not suspecting that in China they had known about it since 2260 BC. Meton determined that a period of 19 solar years contains 235 synodic months (lunars). He considered the length of the year to be 365.25 days, so 19 years were 6939 days 18 hours, and 235 lunations were equal to 6939 days 16 hours 31 minutes. He inserted 7 additional months into this cycle, since 19 years of 12 months add up to 228 months. It is believed that Meton inserted extra months in the 3rd, 6th, 8th, 11th, 14th and 19th years of the cycle. All years, in addition to those indicated, contain 12 months, consisting alternately of 29 or 30 days, 6 years among the seven mentioned above contain an additional month of 30 days, and the seventh - 29 days. Probably the first Metonic cycle began in July 432 BC. The phases of the Moon are repeated on the same days of the cycle with an accuracy of several hours. Thus, if the dates of new moons are determined during one cycle, then they are easily determined for subsequent cycles. The position of each year in the Metonic cycle is indicated by its number, which takes values from 1 to 19 and is called golden number(since in ancient times the phases of the moon were inscribed in gold on public monuments). The golden number of the year can be determined using special tables; it is used to calculate the date of Easter.
Callippus cycle.
Another Greek astronomer - Callippus - in 330 BC. developed Meton's idea by introducing a 76-year cycle (= 19ґ4). The Callippus cycles contain a constant number of leap years, while the Metonian cycle has a variable number.
Solar cycle.
This cycle consists of 28 years and helps to establish the connection between the day of the week and the ordinal day of the month. If there were no leap years, then the correspondence between the days of the week and the numbers of the month would regularly repeat with a 7-year cycle, since there are 7 days in a week, and the year can begin with any of them; and also because a normal year is 1 day longer than 52 full weeks. But the introduction of leap years every 4 years makes the cycle of repeating all possible calendars in the same order 28 years. The interval between years with the same calendar varies from 6 to 28 years.
Cycle of Dionysius (Easter). This 532-year cycle has components of a lunar 19-year cycle and a solar 28-year cycle. It is believed that it was introduced by Dionysius the Lesser in 532. According to his calculations, just in that year the lunar cycle began, the first in the new Easter cycle, which indicated the date of Christ’s birth in 1 AD. (this date is often the subject of dispute; some authors give the date of Christ's birth as 4 BC). The Dionysian cycle contains the complete sequence of Easter dates.
Epact.
Epact is the age of the Moon from new moon in days on January 1 of any year. Epact was proposed by A. Lilius and introduced by C. Clavius during the preparation of new tables for determining the days of Easter and other holidays. Every year has its own impact. In general, to determine the date of Easter, a lunar calendar is required, but epact allows you to determine the date of the new moon and then calculate the date of the first full moon after the spring equinox. The Sunday following this date is Easter. Epact is more perfect than the golden number: it allows you to determine the dates of new moons and full moons by the age of the Moon on January 1, without calculating the lunar phases for the whole year. The complete table of epacts is calculated for 7000 years, after which the entire series is repeated. Epacts cycle through a series of 19 numbers. To determine the epact of the current year, you need to add 11 to the epact of the previous year. If the sum exceeds 30, then you need to subtract 30. This is not a very accurate rule: the number 30 is approximate, so the dates of astronomical phenomena calculated by this rule may differ from the true ones by a day. Before the introduction of the Gregorian calendar, epacts were not used. The epact cycle is believed to have begun in 1 BC. with epact 11. The instructions for calculating epacts seem very complicated until you look into the details.
Roman Indicts.
This is a cycle introduced by the last Roman Emperor Constantine; it was used to conduct commercial affairs and collect taxes. The continuous sequence of years was divided into 15-year intervals - indicts. The cycle began on January 1, 313. Therefore, 1 AD. was the fourth year of indictment. The rule for determining the year number in the current index is as follows: add 3 to the Gregorian year number and divide this number by 15, the remainder is the desired number. Thus, in the Roman indict system, the year 2000 is numbered 8.
Julian period.
It is a universal period used in astronomy and chronology; introduced by the French historian J. Scaliger in 1583. Scaliger named it “Julian” in honor of his father, the famous scientist Julius Caesar Scaliger. The Julian period contains 7980 years - the product of the solar cycle (28 years, after which the dates of the Julian calendar fall on the same days of the week), the Metonic cycle (19 years, after which all phases of the Moon fall on the same days of the year) and the cycle of the Roman indicts (15 years). Scaliger chose January 1, 4713 BC as the beginning of the Julian period. according to the Julian calendar extended into the past, since all three of the above cycles converge on this date (more precisely, 0.5 January, since the beginning of the Julian day is taken to mean Greenwich noon; therefore, by midnight, from which January 1 begins, 0.5 Julian day). The current Julian period will end at the end of 3267 AD. (January 23, 3268 Gregorian calendar). In order to determine the year number in the Julian period, you need to add the number 4713 to it; the amount will be the number you are looking for. For example, 1998 was numbered 6711 in the Julian period. Each day of this period has its own Julian number JD (Julian Day), equal to the number of days that have passed from the beginning of the period until noon of this day. So, on January 1, 1993, the number was JD 2,448,989, i.e. By Greenwich noon of this date, exactly that many full days have passed from the beginning of the period. The date January 1, 2000 has the number JD 2 451 545. The Julian number of each calendar date is given in astronomical yearbooks. The difference between the Julian numbers of two dates indicates the number of days that have passed between them, which is very important to know for astronomical calculations.
Roman era.
The years of this era were counted from the founding of Rome, which is considered to be 753 BC. The year number was preceded by the abbreviation A.U.C. (anno urbis conditae - the year the city was founded). For example, the year 2000 of the Gregorian calendar corresponds to the year 2753 of the Roman era.
Olympic era.
The Olympics are 4-year intervals between Greek sports competitions held in Olympia; they were used in the chronology of Ancient Greece. The Olympic Games were held on the days of the first full moon after the summer solstice, in the month of Hecatombaeion, which corresponds to modern July. Calculations show that the first Olympic Games were held on July 17, 776 BC. At that time, they used a lunar calendar with additional months of the Metonic cycle. In the 4th century. During the Christian era, Emperor Theodosius abolished the Olympic Games, and in 392 the Olympiads were replaced by the Roman Indictments. The term "Olympic Era" appears frequently in chronology.
Era of Nabonassar.
It was one of the first introduced and named after the Babylonian king Nabonassar. The era of Nabonassar is of particular interest to astronomers because it was used to indicate dates by Hipparchus and the Alexandrian astronomer Ptolemy in his Almagest. Apparently, detailed astronomical research began in Babylon during this era. The beginning of the era is considered to be February 26, 747 BC. (according to the Julian calendar), the first year of Nabonassar's reign. Ptolemy began counting the day from the average noon on the meridian of Alexandria, and his year was Egyptian, containing exactly 365 days. It is not known whether the era of Nabonassar was used in Babylon at the time of its formal beginning, but in later times it apparently was used. Keeping in mind the “Egyptian” length of the year, it is easy to calculate that the year 2000 according to the Gregorian calendar is the year 2749 of the era of Nabonassar.
Jewish era.
The beginning of the Jewish era is the mythical date of the creation of the world, 3761 BC. The Jewish civil year begins around the autumnal equinox. For example, September 11, 1999 on the Gregorian calendar was the first day of 5760 on the Hebrew calendar.
Muslim era,
or the Hijri era, begins on July 16, 622, i.e. from the date of Muhammad's migration from Mecca to Medina. For example, April 6, 2000 according to the Gregorian calendar begins the year 1421 of the Muslim calendar.
Christian era.
Began on January 1, 1 AD. It is believed that the Christian era was introduced by Dionysius the Lesser in 532; time flows in it in accordance with the Dionysian cycle described above. Dionysius took March 25 as the beginning of the 1st year of “our” (or “new”) era, so the day is December 25, 1 AD. (i.e. 9 months later) was named the birthday of Christ. Pope Gregory XIII moved the start of the year to January 1. But historians and chronologists have long considered the day of the Nativity of Christ to be December 25, 1 BC. There was a lot of controversy about this important date, and only modern research has shown that Christmas most likely falls on December 25, 4 BC. Confusion in establishing such dates is caused by the fact that astronomers often call the year of Christ’s birth year zero (0 AD), which was preceded by 1 BC. But other astronomers, as well as historians and chronologists, believe that there was no zero year and just after 1 BC. follows 1 AD There is also no agreement on whether to consider years such as 1800 and 1900 the end of the century or the beginning of the next. If we accept the existence of a zero year, then 1900 will be the beginning of the century, and 2000 will also be the beginning of the new millennium. But if there was no year zero, then the 20th century does not end until the end of 2000. Many astronomers consider century years ending in "00" to be the beginning of a new century.
As you know, the date of Easter is constantly changing: it can fall on any day from March 22 to April 25 inclusive. According to the rule, Easter (Catholic) should be on the first Sunday after the full moon following the spring equinox (March 21). In addition, according to the English Breviary, "... if the full moon occurs on a Sunday, then Easter will be the following Sunday." This date, which has great historical significance, has been the subject of much debate and discussion. Pope Gregory XIII's amendments have been accepted by many churches, but since the calculation of the date of Easter is based on the lunar phases, it cannot have a specific date in the solar calendar.
CALENDAR REFORM
Although the Gregorian calendar is very accurate and quite consistent with natural phenomena, its modern structure does not fully correspond to the needs of social life. There has been talk for a long time about improving the calendar and even various associations have emerged to carry out such a reform.
Disadvantages of the Gregorian calendar.
This calendar has about a dozen defects. Chief among them is the variability of the number of days and weeks in months, quarters and half-years. For example, quarters contain 90, 91, or 92 days. There are four main problems:
1) Theoretically, the civil (calendar) year should have the same length as the astronomical (tropical) year. However, this is impossible, since the tropical year does not contain an integer number of days. Because of the need to add an extra day to the year from time to time, there are two types of years - ordinary and leap years. Since the year can start from any day of the week, this gives 7 types of ordinary years and 7 types of leap years, i.e. a total of 14 types of years. To fully reproduce them you need to wait 28 years.
2) The length of months varies: they can contain from 28 to 31 days, and this unevenness leads to certain difficulties in economic calculations and statistics.
3) Neither ordinary nor leap years contain an integer number of weeks. Semi-years, quarters and months also do not contain a whole and equal number of weeks.
4) From week to week, from month to month and even from year to year, the correspondence of dates and days of the week changes, so it is difficult to establish the moments of various events. For example, Thanksgiving always falls on Thursday, but the day of the month varies. Christmas always falls on December 25th, but on different days of the week.
Suggested improvements.
There are many proposals for calendar reform, of which the following are the most discussed:
International fixed calendar
(International Fixed Calendar). This is an improved version of the 13-month calendar proposed in 1849 by the French philosopher, founder of positivism, O. Comte (1798–1857). It was developed by the English statistician M. Cotsworth (1859–1943), who founded the Fixed Calendar League in 1942. This calendar contains 13 months of 28 days each; All months are the same and start on Sunday. Leaving the first six of the twelve months with their usual names, Cotsworth inserted the 7th month “Sol” between them. One extra day (365 – 13ґ28), called the Day of the Year, follows December 28th. If the year is a leap year, then another Leap Day is inserted after June 28th. These “balancing” days are not taken into account in counting the days of the week. Cotsworth proposed abolishing the names of the months and using Roman numerals to denote them. The 13-month calendar is very uniform and easy to use: the year is easily divided into months and weeks, and the month is divided into weeks. If economic statistics used a month instead of half-years and quarters, such a calendar would be a success; but 13 months are difficult to divide into half-years and quarters. The sharp difference between this calendar and the current one also causes problems. Its introduction will require great effort to obtain the consent of influential groups committed to tradition.
World calendar
(World Calendar). This 12-month calendar was developed by decision of the International Commercial Congress of 1914 and was vigorously promoted by many supporters. In 1930, E. Ahelis organized the World Calendar Association, which has been publishing the Journal of Calendar Reform since 1931. The basic unit of the World Calendar is the quarter of the year. Every week and year starts on Sunday. The first three months contain 31, 30 and 30 days, respectively. Each subsequent quarter is the same as the first. The names of the months are kept as they are. Leap Year Day (June W) is inserted after June 30, and Year End Day (Peace Day) is inserted after December 30. Opponents of the World Calendar consider its disadvantage to be that each month consists of a non-integer number of weeks and therefore begins with an arbitrary day of the week. Defenders of this calendar consider its advantage to be similar to the current calendar.
Perpetual calendar
(Perpetual Calendar). This 12-month calendar is offered by W. Edwards of Honolulu, Hawaii. Edwards' perpetual calendar is divided into four 3-month quarters. Every week and every quarter starts on Monday, which is very beneficial for business. The first two months of each quarter contain 30 days, and the last - 31. Between December 31 and January 1 there is a holiday - New Year's Day, and once every 4 years between June 31 and July 1, Leap Year Day appears. A nice feature of the Perpetual Calendar is that Friday never falls on the 13th. Several times, a bill was even introduced into the US House of Representatives to officially switch to this calendar.
Literature:
Bickerman E. Timeline of the ancient world. M., 1975
Butkevich A.V., Zelikson M.S. Perpetual calendars. M., 1984
Volodomonov N.V. Calendar: past, present, future. M., 1987
Klimishin I.A. Calendar and chronology. M., 1990
Kulikov S. Thread of Times: Small Encyclopedia of the Calendar. M., 1991
Lunar number(L) is used to calculate the approximate age of the Moon using the formula:
B =D + M + L
IN – Age of the Moon
D – Day of the month
M – Number of the month of the year
L – Lunar number
The lunar number is a variable value and increases by 11 annually. This is due to the fact that the lunar year is 11 days shorter tropical And calendar year and, therefore, in the remaining 11 days before the end of the tropical year, the Moon will change phase compared to what was observed in the previous year. The repetition of lunar phases on the same day occurs only after 19 years, through the so-called Metonic cycle.
The Metonic cycle serves to coordinate the length of the lunar month and the solar (tropical) year. According to the Metonic cycle, 19 tropical years are approximately equal to 235 lunar (synodic) months.
A lunar or synodic month is the period of complete revolution of the Moon relative to the Sun between two identical phases of the Moon - new moons. The duration of the lunar month is 29d 12h 44m 03s = 29.5 days.
Example: calculate the age of the Moon on November 29, 2017.
D – Day of the month – 29
M – Number of the month of the year – 11
L – We select the lunar number from the table – 1
Substitute the values into the formula:
B = D + M + L = 29 + 11 + 1 = 41
If the age of the Moon turns out to be more than 30, then you need to subtract 30 from the result obtained. In our case, subtract 30 and get the age of the Moon - 11 days.
Let's check the result obtained with the age of the Moon in the Marine Astronomical Yearbook. In the Marine Astronomical Yearbook for the date November 29, 2017, we select the age of the Moon - 11 days. We compare it with what we obtained using the formula and see that the results are similar.
Having the Marine Astronomical Yearbook, you can calculate the lunar date for the current year. To do this, we will use the above formula. As of today, November 29, 2017, we have:
B = D + M + L
11= 29 + 11 + L
since if a number is greater than 30, then it is necessary to subtract 30 from it, then after the subtraction we have:
In Astronomy, the approximate age of the Moon is used to approximate: the time of the Moon's culmination - Tk, sunrise – TV and approach - Tk, right ascension – a.
- Time of Moon's Climax:
Tk = 12h + 0.8h* IN,
Tk = 12h + 0.8h* 11 = 12h + 8.8h =20.8h =20h 48m
12h– approximate time of the upper culmination of the Sun;
0.8h= 49 m – daily delay of the apparent movement of the Moon relative to the Sun;
IN– age of the Moon.
In the Marine Astronomical Yearbook we find that 11/29/2017 is the time of the Moon’s culmination in 20h 29m. The formula found approximately 20h 48m.
- Moonrise time:
TV = Tk – 6h = 20h 48m – 6h =14h 48m
- Moonset time:
Tk = Tk + 6h = 20h 48m + 6h =02h 48m(next days)
- Moon's right ascension:
a = ac +12° c *B = 247° +12 ° c *1 = 247° +12 ° = 259 °
ac– direct ascension of the Sun;
12c– daily advance of the apparent movement of the Sun relative to the Moon – 12° per day;
B– age of the Moon.
Since on the day of the winter solstice, December 22, the direct ascension of the Sun will be equal to 270 ° , then it’s easy to find its approximate value on November 29: 270 ° – 23 (number of days until 22/12) = 247 ° .
Asia Minor) the celebration of Easter takes place on the first Sunday after the spring full moon, which occurs after or on the day of the vernal equinox, if this Sunday falls after the day of the celebration of the Jewish Passover; otherwise, the celebration of Christian Easter is transferred to the first Sunday after the day of Jewish Passover. Thus, the day of Easter celebration turns out to be from March 22 to April 25 of the old style or from April 4 to May 8 of the new style.
Calculating the time of Easter celebration
Calculation of the day of the Jewish Passover
Based on the prescriptions set out in the book of Exodus, as well as the lunisolar calendar, finally adopted by the Jews in the era of the second temple, the Jewish Passover is celebrated on the 15th of the month of Nisan (see Biblical Time Calculation). Thus, among the Jews, the holiday of Passover is immovable.
In the modern Jewish calendar, the months are no longer established, as was the case in ancient times, by direct observation of the lunar phases, but are determined by the cycle. Since the beginning of each month coincides with some essentially fictitious new moon (moled), the fifteenth day coincides with the full moon. The month of Nisan is closest to our March, so the ruling on the Jewish Passover can be formulated in such a way that it is celebrated on the first spring full moon, calculated according to well-known regulations.
The so-called starting point of the Jewish chronology is taken as moled of creation or moled of the month of Tishri of the first year, which took place, according to Jewish calculations, in the pre-Christian era, on October 7 at 5 o'clock 204 hlakim (khlak - 1/1080th of an hour) after six o'clock in the evening under the meridian of Jerusalem, or, according to our division day, October 6 at 11:11 pm.
According to some rabbis, this moled came in the year before creation, when, as the book of Genesis puts it (1:2), thohu webohu prevailed. Therefore, Jewish chronologists call this moled moled thohu. The time interval between two new moons is taken to be 29 days 12 hours 793 hlakim, which represents Hipparchus’ definition of the synodic month of the moon.
Since all changes occur in the first half of the year, from Tishri to Nisan, the number of days passing from Easter to the New Year is always 163 and therefore it makes no difference whether to calculate the day of Passover or 1 Tishri of the next year. Detailed calculation rules are set out in the book of Moses Maimonides “Kiddusch hachodesch” (“Kiddush ha-chodesh”).
The following rules, remarkable in their simplicity, for calculating the day of the Jewish Passover in the year of the Julian calendar were given by the famous mathematician Gauss without proof in the “Monatliche Correspondeoz” for the year. These rules were proven by Cysa de Cresy in the “Proceedings of the Turin Academy of Sciences” ().
Let B be the number of the Christian year, i.e. B = L – 3760, where A is the number of the year of the Jewish calendar. Let's call the remainder of 12B +12 divided by 19 by a; the remainder of B divided by 4 through b. Let’s compose the value: M + m – 20.0955877 + 1.5542418a + 0.25b – 0.003177794B, where M is an integer and m is a proper fraction. Finally, we find the remainder c from dividing the value M + 3B + 5b +1 by 7.
Then: 1) if c = 2 or 4, or 6, then the Jewish Passover is celebrated on M + March 1 (or, what is the same, M – April 30) old style; 2) if c = 1, moreover a > 6 and, in addition, m > 0.63287037, then Easter will take place M + March 2; 3) if immediately c = 0, a > 11 and m 0.89772376, then Easter day will be M + March 1; 4) in all other cases, Easter is celebrated on March 1st.
As a result of the above, the 1st Tishri of the next year will fall on P + August 10 or P – September 21, where P is the day of Passover in March. Generally speaking, it is enough to calculate to the second decimal place. A more accurate calculation is only necessary in extremely rare doubtful cases.
Example: if B = 1897, then a = 14, b = 1, M + m = 36.04, i.e. M = 36, m = 0.04, s = 0. Easter Day: March 36, or April 5 old style. The New Year began on September 15th.
Calculation of the day of Christian Easter
Due to the accepted rules, it is necessary to know for each year the Sundays in March and the day of the Easter full moon. Sunday days are determined from the fact that in the year preceding the Christian era (leap year), which is sometimes incorrectly called the zero year of our chronology, Sundays fell on March 7, 14, 21, 28; further, in every simple year, consisting of 52 weeks and 1 day, Sundays recede in numbers by one, in a leap year, consisting of 52 weeks and 2 days, by two units.
The Metonian lunar cycle comprises 19 Julian years in 365.25 days and almost 235 synodic months of the moon in 29.53059 days. The difference between these two periods is 0.0613 days. The lunar months in this cycle consist alternately of 30 and 29 days, and when the Julian year contains 13 new moons, an additional month of 30 days is inserted at the end of it, and at the end of the last, nineteenth year of the cycle - a month of 29 days. With this distribution, February is always counted as 28 days (permanent calendar), so that the lunar month that falls on February 25, the intercalary day of a leap year, is actually extended by one day.
Since January and February are 59 days long, it follows that the same cyclic phases of the moon will fall on the same dates in January and March. The ancients did not actually observe the new moon, but the first appearance of the new moon; the time interval between this appearance and the full moon is approximately 13 days, and therefore in Paschal the full moon is determined from the new moon by adding 13 days.
The Easter full moon is called the Easter limit. During the first year of the cycle, the Alexandrian Church adopted the so-called. the era of Diocletian (according to R. Chr.), when the Easter new moon fell on March 23, and the first new moon of the year on January 23; On the same day, according to the Metonic cycle, there is a sunrise in the year preceding the Christian era. This year was accepted as the original by Dionysius the Small.
The number showing the place of a year in the cycle is called the golden number. The origin of this name is controversial. The Jews, who also used the Metonic cycle, accepted its beginning three years later than the Alexandrian Church and Dionysius, and in this shifted cycle the new moon in the initial year falls on January 1.
This cycle, called the Easter circle of the moon, is used in the Paschal of the Orthodox Church. To distinguish it, Dionysius calls one of these cycles (Jewish) riclus lunaris, the other - ciclus decemnovennalis. The indicated excess of 19 Julian years over 235 synodic months causes the new moons calculated using the Metonic cycle to lag behind the actual astronomical ones. Every 310 years, one day is accumulated. By the end of the 19th century. this difference was more than five days, for example. the Easter new moon, calculated according to the cycle, was on March 27, while the astronomical one was on March 21 in the evening.
Of all the practical formulas proposed for calculating the day of Easter based on the above rules, by far the simplest and most convenient belong to Gauss.
They are as follows. Let us call through a the remainder of dividing the number of the year by 19, through b the remainder from dividing it by 4, and through c from division by 7. Next, we will call the remainder from dividing the value 19a + 15 by 30 d and the remainder from dividing it by 2b + 4c + 6d + 6 by 7 let it be e. Easter Day will be March 22 + d + e or, which is the same, d + e – April 9. These seven lines contain the complete Paschal of the Julian calendar, adopted by the Orthodox Church.
By the time the Gregorian calendar was introduced, the phases of the moon, calculated according to the cycle, were already three days behind the actual ones, so the papal commission led by Aloysius Lilius decided to move the lunar cycle by three days and, in addition, to avoid the accumulation of errors for the future instead of golden numbers enter the epact circle.
Epakta (ὲπάγειν - add) is the growth of the moon on January 1, i.e. the time elapsed since the last new moon of the previous year as a result of the excess of the solar year over the lunar year, consisting of 354 days. In the Julian calendar, the Roman epact is the waxing of the moon on January 1, calculated under the assumption that in the initial year of the lunar cycle, or at the golden number zero, the new moon falls on January 1, as occurs in the Jewish lunar cycle.
During the reform of the calendar, due to the rearrangement of the lunar cycle and the skipping of ten days, the new moon of the first year in the lunar cycle moved from January 23 to 30, and the previous one fell to December 31; therefore, the epact of the first year in cycle 1. The epacts of subsequent years are obtained by adding 11 each time and lowering numbers that are multiples of 30. To return to epact 1, when moving to a new cycle, you need to add 12; this was called saltus epactae or saltus lunae.
In order to avoid new errors, Lilius introduced epact amendments. One of them is called the solar equation and comes from throwing out three leap days for 400 years and therefore each time reduces the epact (reduces the number of days that have passed since the new moon). The second is called the lunar equation and aims to correct the discrepancy between 19 Julian years and 235 synodic months of the moon; it is added 8 times every 2500 years and each time increases the epact, since according to the Metonic cycle the phases of the moon are delayed. Both of these amendments are applied to epacts in the years that end centuries.
Nevertheless, Gauss presented them in the following elegant form. Let the remainders from dividing the number of the year by 19, 4 and 7 be a, b and c, respectively; the remainder of dividing the value 19a + M by 30 will be d and the remainder of dividing the value 2b + 4c + 6d + N by 7 will be e. Then Easter will come on March 22 + d + e or d + e - April 9 of the new style. The values of M and N are calculated as follows. Let k be the number of centuries in a given year, p the quotient of 13 + 8k divided by 25 and q the quotient of k divided by 4. Then M will be defined as the remainder of 15 + k – p – q divided by 30 and N as the remainder of dividing 4 + k – q by 7. Here, however, two exceptions must be borne in mind, namely: when, with d = 29, the calculation gives for Easter day April 26, one must take April 19 instead, and when, with d = 28, we get April 25 for Easter, and a > 10, then we need to take April 18. Calling by h the quotient of dividing a by 11 and by f the quotient of dividing d + h by 29, in addition, denoting d – f by d and considering e as the remainder of dividing 2b + 4c + 6d + N by 7, we obtain the formula for the day Easter: March 22 + d + e, which no longer requires any exceptions. Example: for 1897 a = 16, b = 1, c = 0, k =18, p = 6, q = 4, M = 23, N = 4, d = 27, e = 0. Easter Day April 18 (new style). Each of the values of M and N is constant, at least for a whole century, and therefore it is more convenient to calculate them in advance.
Their values will be:
- 1800-1899 M=23 N=4
- 1900-1999 M=24 N=5
- 2000-2099 M=24 N=5
- 2100-2199 M=24 N=6
- 2200-2299 M=25 N=0
- 2300-2399 M=26 N=1
- 2400-2499 M=25 N=1
The formulas given by Gauss for the Julian calendar can be obtained as a special case from the formulas for the Gregorian calendar, constantly assuming M = 15, N = 6. Using Gauss’s formulas, it is possible to solve the inverse Paschal problem for the Julian calendar: find those years in which Easter falls on given number. A general solution to a similar question for the Gregorian calendar, given the current state of numerical analysis, is impossible.
In the Paschal of the Orthodox Church, some terms have been preserved that require clarification. In church calendars, or monthly calendars, one of seven Slavic letters is assigned to each day of the year; Z, S, E, D, G, V, A, called adult letters. The year in church Easter begins on March 1; to this day, on the basis of some considerations concerning the biblical days of creation, the letter G is assigned; the following days have the letters B, A, Z, O, E, D, G, V, A, Z, etc. The letter to which Sundays correspond in a given year is called vrutselet.
Thus, knowing vrutseleto and having all the days of the year written in vrutseleto letters, you can easily find out the day of the week for any day of the year. T.N. The Easter circle of the moon coincides with the Jewish circle, i.e. deviates by three years from that adopted by Dionysius. The new moon in the initial year of this cycle falls on January 1st. The base is the number indicating the age of the moon by March 1, found under the assumption of the Easter circle of the moon. The Great Andiction is a period of 532 years; since the phases of the moon return to the same numbers of months after 19 years, and the days of the week (taking into account leap years) after 28 years, then after 28 x 195 = 32 years all these elements will return to their previous order, and the days of Easter according to the Julian calendar will repeat absolutely right. The boundary key is the number of days between March 21 and Easter. Since the latest Easter is April 25, the boundary key can reach a value of 35.
In the so-called of the sighted Easter, the key of boundaries is indicated instead of numbers by letters of the Slavic alphabet. For each year of the great indiction, a key letter is given, and from it, from another table, the day of Easter is found, as well as the days of other moving holidays associated with it. From the Gauss formulas it follows that the key of the boundaries K = d + e + 1. Then we have: the beginning of Maslenitsa (meat empty
The next conclusion is that the methods of calculating Christian Easter have changed several times. This, of course, is not the discovery of the author of this study. There is hardly any serious specialist who would deny this. This is common knowledge.
Here, among other things, additional attention will be drawn to the last revision of the Easter tables around the 15th century.
One of the most striking evidence of editing of the Easter tables is the placement of the “moon jump” after the 16th year of the nineteen-year cycle.
“Moon jump” is an amendment to the “lunar flow” schedule, which once every 19 years shifts the date of the full moon next year not by 11 days, but by 12. Thus, it compensates for the error that has occurred. Anyone who understands in detail the structure of the 19-year lunar cycle will understand that the “moon jump” can only be located after a year with the “circle of the Moon 19”. And nowhere else! Moreover, if it is placed where it should be, no one will even know about it, since from the year with “circle of the Moon 1” a new cycle will begin, repeating the same dates as in the previous cycle.
The “moon jump” shift most likely occurred in ancient times (although, of course, later times cannot be ruled out). It was probably associated with a change in views on the age of the Savior in the year of the Resurrection. This led to the construction of a new Biblical chronology. Most likely, such chronologies changed several times (it is very possible that different chronologies existed in different places at the same time), and it is not possible to accurately restore the sequence of changes. In any literature devoted to calendars and chronologies, various “eras” are mentioned (Alexandrian, Constantinople, etc.).
Around 1409, when a new Great Indiction began, the Easter tables were clearly corrected, since the dates of the March full moons of the 15th century correspond to the "foundations" and "epacts" of the Easter tables. If there were no correction, then real full moons would have serious deviations from the tabular ones. During the previous Great Indiction, a significant error would have accumulated.
“1409” in this case is a very arbitrary date. Editing of the Easter tables could well have happened later (during the conclusion of the Ferraro-Florentine Union, for example). It could have happened earlier.
The edit could have happened around 1492. Then they were waiting for the end of the world (since the summer of 7000 was approaching), and historical sources indicate that the dates of Easter were not calculated beyond the year 1492.
Easter tables could have been corrected several times during the 15th century.
For those who doubt that the Easter tables were corrected around 1409, we present correspondence between the full moons calculated from the “epacts” and “foundations” of the currently existing Easter tables (according to their modern interpretation) and the real full moons of the beginning of the 15th centuries (that is: since “epakta” is the 20th day of the Moon, it means that the tabular full moon will occur 6 days earlier):
Table No. 12
“Circle of the Moon” “Epakta” Tabular Real
full moon full moon
1 7 1st March 2nd March 1409 2 26 20th March 21st March 1410
3 15 9th March 10th March 14114 4 March 28th March 28th, 14125 23 17th March 18th March 14136 12 6th March 7th March 14147 1 25th March 26th March 14158 20 March 14th March 14th, 14169 9 3rd March 4th March 141710 28 March 22nd March 23rd, 141811 17 11th March 12th March 1419
12 6 March 30th March 30th, 142013 25 March 19th March 19th, 142114 14 March 8th March 9th, 142215 3 27th March 27th March 142316 22 March 16th March 16th, 142417 10 4th March 5th March 142518 29 23rd March 24th March 142619 18 12th March 13th March 1427
The calculation of real full moons was carried out using the tables of N.I. Idelson, which give a fairly accurate result (with an error of up to 0.5 days).It can be seen that the Easter tables reflect the real “lunar flow” of the 15th century. Moreover, real full moons often occur later than tabular ones. This would never have happened if the "foundations" and "epacts" had been inherited from the previous Great Indiction.
The fact that “foundations” is the “age” of the Moon on March 1st, and “epakta” is the number of March on which the 20th day of the Moon falls, is confirmed by the schedule of the “Lunar Current” from the “Eye of the Church” (sheet 1174 on the back).
For example, for the “circle of the Moon 1” (“base 14”, “epact 7”) in the “Church Eye” the full moon is indicated on March 1st. Since the full moon is the 14th day of the moon, the “age” of the moon on March 1st will be 14 days, and this is the “base 14”. 6 days after the full moon, the 20th day of the moon will come. Since the full moon is on March 1st (day 14), then the 20th day will be March 7th, and this is “epakta 7”.
And for the “circle of the Moon 2” (“base 25”, “epact 26”) in the “Church Eye” the full moon is indicated on March 20th. Accordingly, 1st dayThe Moon will be on March 7th, the 30th day of the Moon will be on March 6th, and March 1st will be the 25th day of the Moon. That is, the “age” of the Moon on March 1st will be 25 days, and this is the “base 25”. 6 days after the full moon, the 20th day of the moon will come. Since the full moon is on March 20th (day 14), then the 20th day will be March 26th, and this is “epact 26”».
Correspondence of “grounds” and“Epact” to the Lunar Current schedule will be present in 15 out of 19 years. In 4 years, due to the inaccuracy of the Metonic cycle, there will be a discrepancy of one day.
Another evidence of the correction of Easter tables are tables preserved from ancient times, called the “hand of Damascus” (or “hand of the Theologian”).
Here is an example of such a table from the 17th century “Eye of the Church”:
And here is from the 14th century “Scaligerian Canon” (Leiden University Library, Netherlands):
These illustrations show how to calculate the date of Christian Easter using the “circles of the Sun” and “circles of the Moon”. Once upon a time, such tables were actually used for counting, using human hands and placing numbers on the folds, phalanges and ends of the fingers.
The right “hand” contains the so-called “Jew chamfers”. In a purely technical sense, “fasque yid” is the date, the first resurrection after which is Christian Easter. The “chamfer” duplicates the “good letter”. The “good letter” indicates the date one day after the “chamfer”.
The dates of the “chamfer” (in Slavic numerals) on the “hand” are located as follows.
Table No. 13
13 25 5
17 29 9 21
1 12 24 4
15 27 7 18
30 10 22 2
Dates refer to March and April. Dates from 21 to 30 are the dates of March. Dates from 1 to 18 are April dates. The order of arrangement is as follows: the rows start from the bottom, and the columns start from the thumb (from right to left).
That is, the dates of the “chamfers” are in the following order: 2, 22, 10, 30, 18, 7, 27, 15, 4, 24, 12, 1, 21, 9, 29, 17, 5, 25, 13.
There are no additional notes on the handwritten table from the canon. The table from the “Eye of the Church” contains explanatory notes. Small letters “m” and “a” indicate March and April. Red numbers from 1 to 19 indicate the “circles of the Moon” corresponding to the “chamfers” (they look gray in the black and white illustration).
The left “hand” contains “vrucelet” from 1 to 7, corresponding to “circles of the Sun” from 1 to 28.
The “vrucelet” are located on the “hand” as follows.
Table No. 14
3 4 5 6
5 6 7 1
7 1 2 3
2 3 4 5
4 5 6 7
6 7 1 2
1 2 3 4
The counting also goes “from the thumb” (in this case, from left to right). But there is already a strange complication here. Instead of starting the count from the bottom from the first position from the left (which would be completely consistent with both common sense and the right table), the count starts from the second position of the third row from the top! Then it goes to the second line from the top, then to the top one, then goes to the bottom one, from the bottom to the second, etc.
In order not to be mistaken, on the “hand” from the “Eye of the Church” next to the “vrucelet” there are marked (in red) the corresponding “circles to the Sun”.
There can only be one explanation for this strangeness. In the original version, the counting began (as expected) from the bottom line.
“Vrutselets” were in full compliance with leap years. That is, the table of correspondence between “circles of the Sun” and “vruceles” looked like this.
6) 5 11 16 22 -
7) 6 - 17 23 28
According to it, it turns out that it was not the fourth year “from the Creation of the World” that was a leap year, but the third! From a theological point of view, this is complete nonsense.
Of course, the explanation for this discrepancy is known. It consists in the fact that the year, they say, begins according to the Julian calendar in January. Therefore, starting the year from March, you still need to count leap years from January. This explanation is very dubious.
One can also doubt that the year after the Julian reform began in January. The consuls actually took office in January. But modern presidents, for example, take office at different times of the year. And no one can bear the New Year because of this. Additional days (and months) in calendars are usually inserted at the end of the year. In the Julian calendar this is done in February. We must also not forget that the words “September”, “October”, “November” and “December” in Latin are not names, but serial numbers (seventh, eighth, ninth and tenth). Why should the twelfth month be called the tenth? And the Old Russian (and Byzantine) year, which began in March, cannot be ignored either.
The shift of the “circles of the Sun” relative to the change cycle of the “vrucelet” was necessary so that the “circles of the Moon” could also be shifted. And the “circles of the Moon” were clearly shifting (as shown above). And for three years (this can be seen from the “moon jump”). And for an unknown number of years “around 1409” (to bring the real lunar phases into line with the “foundations” and “epacts”).
But it is impossible to “move” only the “circles of the Moon” and not touch the “circles of the Sun”. Due to the complex cyclical interaction of these quantities, if only one of them changes, the entire chronology will immediately collapse.
For example, summer 7519 (year 2011) has “circle to the Sun 15”, “circle to the Moon 14” and “indict 4”. If we increase the “circle of the Moon” by just 1 and get the “circle of the Moon 15”, then we will find ourselves in a different era. “Circle to the Sun 15”, “circle to the Moon 15” and “indict 4” correspond to the 3739th year from the Creation of the World. That is, 1770 BC!
Therefore, by “correcting” and “clarifying” the “circle of the Moon” of the current year, the correctors were inevitably forced to correct the “circle of the Sun” in order to obtain a new “clarified” meaning of the summer from the Creation of the World that is close (it is impossible to obtain exactly the same) to the current one. Most likely, it is the Easter reforms that explain the discrepancies in the dates of the same events in different chronicles.
