I.Calendar is a system of reckoning time over extended periods. Observations of the motions of celestial bodies such as the Sun, the Moon and stars carried out from a remote time enabled people to reckon time, i.e. make a calendar. The moments of rising and setting of some stars allowed to predict the coming of year seasons and to plan economic activity. Cultivators planned the earth plough, sowing and harvesting works according to the rise and set of a certain star, for example Sirius, as in ancient Egypt. Cattle-breeders also were guided by movement of stars to determine the time of cattle birthing, shearing of sheep, changing the location. Such star "markers" for Kazakhs were Pleiades, Orion and Sirius.
II.Any calendar is based on the natural periods. Duration of lunar phase cycle is 29,53 days. This period is called the synodic month. The Moon makes a complete revolution round the Earth with respect to stars in 27,32 days. This period is called the sidereal, i.e. star month. The earth rotates round the Sun in 365 days 5 hours 48 minutes 46 seconds or in 365,2422 days. Such year is called tropical year and is defined as a period between two successive transits of the Sun center through the point of vernal equinox.
For many centuries different people invented various types of calendars. There are several types of calendars based on periodical motion of celestial bodies: lunar, star, solar and lunisolar.
III.The most ancient system of reckoning time is the lunar calendar which was created for several millennia B.C. It was based on synodic month and was important in calendars of the ancient people: Babylonians, Chinese, Jews and Indians. The lunar calendar was widely adopted in the Muslim countries. In this calendar each of 12 months in a year had to begin with the first sighting of the narrow lunar crescent. Odd months consisted of 30 days and even months consisted of 29 days. Therefore, the average duration of a month is 29,5 days. As this period is shorter than the synodic period, a certain discrepancy between the first day of a month and the day when a new Moon occurs gradually increases in the calendar. In 30 years the calendar lags behind for 11 days. For the solution of this problem 11 years in each 30-year cycle of this calendar are made leap. One day in a leap year is added to the last month. Hence, the duration of a year in a Muslim calendar is 354 or 355 days. Therefore the beginning of a year in a Muslim calendar (the first day of the month Muharram) every year drifts to the beginning of year in the Gregorian calendar on 10 - 11 days. The Muslim era (chronology) begins from hijri - resettlement of the prophet Mohammed and his associates from Mecca to Medina (in 622, June 16 by old style).
IV. The star calendar, based on the sidereal cycle of the Moon, in practice is used very seldom. It is also the most ancient calendar. Such calendar under the name togys (from kazakh togysu meaning «to intersect») were used by Kazakhs.
The beginning of months coincides with the moment when the Moon passes the Pleiades and the names of months correlate with lunar phases.
As the difference between the synodic and sidereal cycles of the Moon is about 2 days, each subsequent meeting of the Pleiades and the Moon occurs on a lunar phase that is 2 days earlier than the previous lunar phase. There is a certain relation between the names of togys¢s months and seasons. So, for example, in the spring such «meetings» occur on the 5th, 3rd, 1st days of a lunar month, in the winter - on the 11th, 9th and 7th days.
The beginning of a year in this calendar coincides with the month when the waxing (young) Moon meets the Pleiades. This usually occurs in a period between the end of April and the beginning of May. In an ancient Kazakh star calendar there are 13 such "meetings" in a year from which only 11 are visible. If every month contains 28 days then the duration of the year is 364 days.
V. The calendars based on motion (visible) of the Sun pursue two aims. The first aim is to have the year almost of the same duration as well as in the tropical year, and the second aim to have the vernal equinox always precisely on March 21.
The year in an Ancient Egyptian calendar consisted of 365 days. It begins on the day of the first rise of the Sirius at dawn, which at that time coincided with the summer solstice and the beginning of flood of Nile. In this calendar a year consisted of 12 months with 30 days in each and in the end of the year an extra 5 days were added.
Such solar calendar with 365 days in a year was used by Persians and Saks (Scythians) tribes. This calendar called the sogdian calendar originated from ancient religions, such as the mitraizm and zoroastrism and had a wide circulation among the people of Central Asia. The extra 5 days added at the end of the year the Kazakhs called «five guests», or «five ending numbers».
The calendar, made in the XI century by the great east poet and the scientist, the mathematician and the astronomer Omar Khayyam (1040 — 1123), is considered to be one of the perfect calendars. In this calendar each 33 year cycle contains 11 leap years. The average duration of the year is 365,24242 days that is only on 22 seconds longer than the tropical year.
VI. The lunisolar calendar is based on the tropical year and synodic month, i.e. in this calendar the Moon movement is coordinated with yearly movement of the Sun.
In such calendar the year consists of 12 lunar months with 29 and 30 days in each and for coordination with movement of the Sun the leap years containing additional 13th month are periodically inserted. They are inserted so that the beginning of each calendar year, whenever possible, coincides, for example, with an equinox.
In an ancient Roman calendar the first and third years consisted of 355 days, the second year of (355 + 22) = the 377, the fourth year of (355 + 23) = 378 days that led to exact coincidence of the first day of each month with a new moon. The cycle consisted of 1465 days, the average duration of a year was of 1465: 4 = 366,25 days that was longer than tropical year for about 1,01days.
The lunisolar calendar was known and applied in Babylon, Judea, China, Ancient Greece.
VII. The history of the modern calendar applied at the international level, begins in 46 B.C. when the emperor of Ancient Rome Julius Caesar introduced a new calendar in the countries dependent on him. In this calendar called Julian (old style), three years in each four consist of 365 days, and each fourth (leap) year of 366 days. Thus, the average duration of the year is 365,25 days, that is longer than tropical year for about 11 minutes 13,9 seconds. Though the year has 12 months, their duration isn't connected with lunar movement.
Insignificant difference accumulates into discrepancy equal to 1 day in 128 years. In the XVI century this discrepancy equaled 10 days that led to shifting of the vernal equinox on March 11.
To return the day of vernal equinox on March 21 and prevent its shifting the next reform of a calendar took place in 1582 according to which 10 days were dropped from the calendar and those years at the end of the centuries, which in a Julian calendar were considered to be leap, in a new Gregorian calendar became common if they were not divisible by 400 (for example, 1700, 1800, 1900). Thanks to this reform 3 days accumulated in old style in 400 years will be eliminated. The average duration of a year calculated in a Gregorian calendar (new style) is longer than tropical year only on 26 seconds.
Nevertheless the second aim of developing a calendar to keep the vernal equinox always precisely on March 21remains unachieved. This event from year to year most often falls on March 19, 20 and 21.
On the territory of the former Soviet Union this calendar was accepted in 1918. By that time the difference between calendars of the old and new styles equals 13 days. Chronology in the Gregorian calendar begins from the presumed date of birth of Jesus Christ and is called our era (A.D.).
Exercise. Accepting the synodic month has 29,53059 days, calculate the accuracy of the Arab Muslim calendar. How much time is needed to get 1 day discrepancy?
The total of days in a 30-year cycle with 11 leap years is
At the given duration of synodic month we have:
29,53059 • 12 • 30 = 10631,012 days.
Thus, the discrepancy of the Arab cycle is 0,012 days in 30 years. The 1 day discrepancy in this calendar occurs only once in 30: 0,012 = 2500 years.
1. What is the astronomical definition of synodic and sidereal months? What is their duration?
2. What is the definition of a tropical year? How many days in this year?
3. How many days had a year has in an ancient Egyptian calendar?
4. It is known that a year doesn't consist of an integer number of days. In which way the fractional days were taken into account in calendars? Give examples.
5. What discrepancy appeared in the Julian calendar during its long application?
1. Show via calculation that one day discrepancy in a Julian calendar accumulates approximately in 128 years.
2. In how many years does the discrepancy in one year accumulate in an Ancient Egyptian calendar?