Calendars

CALENDARS Roberto Bartali

Introduction All cultures in the world, from past to present time, observed the sky and they saw that the heaven is not static nor unchanging. After a few months it is easy to see changes, even in a day after day basis, the Moon moves respect to the background stars. After a few years, we can see that the temperature change over time, if there are not leafs on trees, the temperature is cold, if they are green, temperature is hot, if they are red and yellows, then follows a cold time. After a few cycles, people start to think that maybe there are some method to predict the season changes. Very soon, the calendar, would be a fundamental tool for the survival. Figure 1 This work is about the relationship between An artistic representation of the people and calendars. I will divide the context into two Aztec calendar. main parts: the first is about general concepts and the From: my archives second is about the description of the calendars used by different cultures in the world. In the first one, there is a description of the principle at the basis of a calendar, the rotation of the Earth and its orbit around the Sun; a discussion of how and why peoples make and needs a calendar. The second part contains a description of one of the most impressive calendars ever made, the Maya, and a general description of other culture solutions for their time keeping necessity. Then in the conclusion, I will discuss about the precision of our current calendar system, I will show that there is no possibility to make radical changes, even if they lead to a better system, so we only try to make the current system as precise as possible.

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What is a calendar? A calendar is a system that people developed to help tracking the seasons. Seasons are changes in the whether of the Earth (Figure 2) and depends on the inclination of its rotation axis respect to the plane of the orbit and on the relative position of the planet during the orbit around the Sun (Figure 3). From this picture we can see that seasons are not the same on all the planet, they are opposite on northern hemisphere respect to the southern, when is Winter in the north, it is Summer in the south. But climatological conditions are very different from a latitude to the other, near tropics there are very little changes in the seasons, but at latitudes near 45 degree (north or south) Figure 2 the difference between Seasonal changes in the northern hemisphere at latitude near 45 Winter and Spring are degree. From left to right and up to down: Winter, Spring, Summer and Autumn. From: my archives great (Figure 2). If an hypothetical planet orbit its star in a perfect circular orbit, at perfect constant speed, having its axis fixed at some inclination respect to the plane of the orbit, with a rotation rate also constant and without any kind of perturbations, each season duration would be the same, the climatological characteristics would be perfectly predictable and always the same year after year. But this ideal planet don´t exist, our Earth orbital and rotational parameters are very far from being ideal. The orbit is elliptical (Figure 3), so we have a perihelion and aphelion position (nearest and farthest distance respectively). The rotation is greatly dependant on the position of the Moon, the axis suffer a precession, many kinds of tidal effects, instability of the tectonic plates and inner core, perturbations from other planets, make either the revolution and the rotational movements very unstable and difficult to measure and then to modeling. So we can know with some precision, but not exactly, when seasons starts or ends. This is the main problem one encounter when try to make a calendar. Now let me talk about the various type of calendars and which are the basic concept they manage. A calendar is a system been able to track time during certain period, after which, it start again. The basic unit of a calendar is a day, the full period is a year. I have then to define the day and the year.

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Ours watches and calendars measure the day and the year with a constant duration of 24 hours, for the day, and 365 days for the year.

Figure 3 Diagram of the orbit of the Earth around the Sun during a year. From: Freedman and Kaufman, Universe, page 29

A normal day is not a full rotation of the Earth on its axis, a sidereal day of 23h 56m 04.091s, because after this time, the Sun is in the same position on the sky respect to the fixed stars, but the planet also has moved some distance on the orbit. At noon, the real position of the Sun is not the same day after day. The day we measure is called a mean solar day and it is the time lapsed from one meridian transit to another. By definition, this time is exactly 24 hours. We can say the same for the year, a full revolution of the Earth around the Sun, a sidereal year of 365d 6h 9m 10s or 365.2564 mean solar days, is not the time required for the returning of some season (Spring for example). Precession of the axis and other perturbations, make the time between two successive passages of the sun on Vernal Equinox a little less than the sidereal year. This time is called a tropical year and it is 365d 5h 48m 46s or 365.2422 mean solar days. Which is the reason for having a 365 days? Obviously we have to choose a whole number of days, many ancient astronomers from various cultures in Central America, Africa and Asia, measured a year of about 365 days. Julius Caesar, the roman emperor, take the length of the year of exactly 365.25 days. Almost all cultures reached the same value, this is because of the lower precision of their instruments and poor mathematical knowledge. Even this little error of 0.0078 days per year, respect to the average tropical year duration is important, and after 1000 or 2000 years the seasons are completely out of phase. Our calendar is based on that statement and some kind of adjustment is needed, we can see how it is accomplished in other section. After this introduction about the calendar division and definition, we have to say that not all people have used the relative movement of the Sun for their calendar. A calendar based on this is called a Solar calendar. Many past and, some present cultures, like Jewish and Islamic, are still using the movement of the Moon, this kind of calendar is called Moon Calendar. This calendars are much more problematic, because the Moon and the Sun are not synchronized, almost in a short period of time. The Moon orbit the Earth at the same time it rotate on its axis, so we can see always only one face; the relative position of the Earth, Moon, Sun, is the reason 3

for the phases (Figure 4). Normally all cultures uses the full Moon or the first visibility of the crescent moon, as soon as it reappears after the New Moon phase. The time needed for the Moon to return in the same position respect to the fixed stars, that is, a full revolution around the Earth, is called sidereal month and it is 27.322 days long. The time from one full phase to the next is the synodic month and it is 29.531 days long. It is longer because the Earth is moving in its orbit at the same time the Moon is rotating around the Figure 4 Diagram of the sequence of the phases of the Moon. Earth. The same phase is when the From: Freedman and Kaufmann, Universe, page 45. angle formed by the Sun, Moon, Earth is the same and this is true only about two days after the Moon complete one full orbit. In a tropical year we adjust for only 12 full synodic months (12.3681 synodic months per tropical year). The origin of the lunar calendar is because it is easiest to observe the position of the Moon respect to the fixed stars. There are no stars visible during the day, so for knowing the position of the Sun we have to do some calculations based on the observation of the stars visible immediately after the sunset or before the sunrise. The combination of a solar and a lunar calendar is the so called lunisolar calendar. This is based on the Moon synodic month but it try to remain in step with the seasons, so there is the need to include some leap days or months after a few years. Most important cultures in the world adopted this kind of calendar.

How people make a calendar There are no evidence on this topic, we just argue the way they may do that. In this section I try to explain the method I should used if I lived thousands years ago. There is just one method and it is the constant, persistent and carefully observation and recording of the sky. First we have to observe and learn that things happened in periodic fashion, not randomly. After that, we have to select a start point in time and other important thing is to Figure 5 select a fixed observing place from where take A 4.7 m high menhir in Lanvenael, France measurements. The selected place must have From:http://www.stonepages.com/france/fr some reference points like a mountain top, a ance.html rock, or any other natural elevation in order to compare the position of the stars, Sun and Moon every day. This process may last decades,

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this knowledge have to be passed from a generation to the other in oral form. The first step is to erect some heavy and tall rock, a menhir (Figure 5), aligned with some natural feature of the landscape, or erecting some more with a well defined direction, normally north-south or east-west. When we have this well defined observatory and the right methodology of observing and recording, we have to try to make predictions. Maybe the first attempt to record observation was the stone engraving, or cutting marks into a wood table, later, eventually, over a sheet of leather or paper, but all of this evidence are lost, the only ones left are the stones. There are a logically step sequence for knowing the season cycle. First, measuring the position of the sun day after day and record when it reach the highest position in the sky, then the lowest position. Another important measurement is the length of the day and the night. After a few cycles it is clear that if the length of the day is less than 12 hours and the elevation of the sun at noon is increasing each day, and, also is increasing the length of the day, it is winter, because the winter solstice mark the shortest day and its length increase until the Spring equinox when the day and the night duration is the same (12 hours). If the length of the day is more than 12 hours and the elevation of the sun at midday increase, it is Spring. On the Summer solstice the length of the day is the maximum. If the day is more than 12 h but the Sun elevation decrease each day, and also decrease the length of the day, it is Summer. On Autumn equinox the length of the day and the night are the same, as it is on Spring equinox, 12 hours, and, after that date it decrease again. If the day last less than 12 hours and the elevation of the Sun decrease each day, it is Autumn, so we have to check on following years if the cycle repeat, if yes, than we have a calendar and we can make prediction, after a few centuries, we are able to predict eclipses too if we learned the way the Moon is moving in the sky. If we can do that, our power over most people increase. To enhance our ability we have to build some kind of monument from where people could see the phenomenon we predict. The construction of this monuments are very important for two things. One is to demonstrate our knowledge and power over the environment, the other is for intimidation, common people must think very carefully what to do if they want to be our enemy. Making a big monument as our home is another form to say that we are on a superior level than common people. Knowing how the universe works, is like elevating ourselves at the same level of a God, because only a God can give order to the Sun or the Moon to move. I think Figure 6 Warrior or priest monument below what impressive must be to see a priest telling to the a “Door of the Sun” in Tihuanaco, people that the Sun or the Moon will be passing through Bolivia. It is though that the Sun a small window in the building or through an arc with a stay over the head of the warrior on solstice or the equinox. monument inside (Figure 6) at some precise time of the From::www.vic.com/bolivia/image year. What kind of respect and power he earn in that s/ lapaz/tihuanaco.html moment. For sure, all the people are thinking that if he has the power or the influence over the Sun to tell it how to move and where to go, there are nothing impossible for him, so the better choice is to obey him.

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Almost all cultures, from Neolithic age to the Middle age in Asia, Africa and America followed the same methodology, even when many of them have invented some form of writing like Egyptians, Babylonians, Chinese, Maya, among others. Suddenly many ancient cultures disappears or start to loose their culture and influence, some in a very few years, others in a much greater time and many due to the invasion of less evolved ones. As communities grows, there is the needs for more food and the commerce with other countries start, communication is the best way to learn more, but it is also the best way to loose their own identity. We start to see mixed cultural expressions, people with less scientific knowledge absorb the best of the other, but the more evolutioned one, take the worst of the other culture. For example, in Mexico and Central America, Maya was the more sophisticated culture, when they came in contact with Toltecs and Aztecs, we see the start of their ends, but Aztecs and Toltecs, with a much less culture and scientific knowledge, take the Maya calendar and they just changed the name of the days and months.

Why people needs a calendar From at least 10 thousands years, we can find evidences that people knows and follows some kind of calendar. They needs a system for knowing which season is, otherwise their survival may be in serious trouble. Before people established in communities and develop agriculture, they were hunters and collectors. They have to know when the vegetables and fruits they eat, was good, surely, their aliments was not the same all the year. They have to know when and where collect them at that time they was nomads, so each place have their own characteristics. The knowledge of the season was fundamental, otherwise they might go to the wrong place at the wrong time. Animals they hunted, also follows some kind of cycle, during Winter time they must have difficulties to hunt, so during Autumn, they have to collect and storage as much aliments as possible. When people established agricultural communities, the knowledge of the seasonal cycles was extremely more important. They have to know when put seeds into the ground, when plants grows and fruit or vegetables matures. An established community needs more organization. At that time, most people developed some kind of religion. This imply periodic ceremony too. As the community grows and the organization is best, there was the necessity to collect taxes or duties. They need to establishes the right time to do that, because it is impossible to pay taxes before collect aliments or sell them. Another very important activity is the storage, so there are very good reasons to store goods at very precise time of the year. Until a few centuries ago, the main reasons to follow and develop a calendar, was for religion ceremonies and tax collect. This is true now too, but we have much many other reasons. Our life is strictly dependent on a calendar. Our activities are much more and more different than they was a thousand years ago. We go to school, work, we have social and cultural activities, we do sporting, and many more. We have to coordinate all of them, so time is one of the most important things we have to take into account in our lives. When doing business, in a world that we see each day more and more boundary less, the calendar is fundamental, we have to send and receive merchandise on time, pay or receive money at a well defined time, so we need some kind of universal calendar, valid in every country.

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The Maya calendar In the American continent we can find many developed cultures with a high degree of scientific knowledge. Some of this cultures are completely lost, all we have are relics, but nothing more. Most important are Anasazi in North America, Olmec in Mexico, Nazca in South America, we believe that their best period was around 500 or 1000 BC, even earlier. Other more recent cultures was completely destroyed by the Spanish conquers: Maya, Zapotec, Toltec, Aztec in Mexico and Inca in South America. About this cultures we know a little more because they are more recent and many more archaeological sites are now discovered and restored, also we have a very little written documents saved from the massive Spanish destruction. Perhaps the most interesting of these cultures is the Maya. They occupy some Figure 7 parts of the Central and all the Segment of the Dresden Codex, It contains calendrical and astronomical information. South of Mexico and extended in From: www.archaeoastronomie.de/codex/ Central America until Honduras. Just for completeness, I want to introduce Mayans, this way the reader can identify when and where they lived. The Maya civilization begin about the year 2600 BC and ended when the Spaniard invasion in the XVI century. Most important periods of this civilization are: 2000 – 900 BC Old Preclassic 900 – 300 BC Middle Preclassic 300 BC – 250 AD Late Preclassic 250 – 600 AD Old Classic 600 – 900 AD Late Classic 900 –1500 AD Post classic 1500 – 1697 AD Colonial

The first known calendar was developed around 400 BC, but there are evidences that the Olmec civilization, before and, at their end, contemporary of the Maya in the Old Pre Classic period, invented it and Maya adopted and perfectioned. A hieroglyphic writing system

Figure 8 Maya vigesimal numeric system. A seashell symbol is zero, a dot is the unit, a line is 5 units, no more than 4 dots are allowed and no more than 3 line are possible. For number greater than 19, they shift to the left and/or up one position each digit depending on the form they are written, either vertically or horizontally. From:www.mayacalendar.com/loscale ndariosmaya.html

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was in use from 700 BC. The best known and most important Maya period was the Classic. Unfortunately, all literature, containing the full scientific knowledge of the Maya, was destroyed by Spaniard between 1517 and 1524 by Fray Diego de Landa because he believed that manuscript was something satanic. Figure 9 One of the Maya specialized buildings for astronomical Only some sections of four observations, called “The Caracol” in Chichen Itza, documents survived called “Dresden México. Codex” (Figure 7), “Paris Codex”, From: www.atharley.com/ images/Cozumel_3-01/ “Grolier Codex” and “Madrid Codex” because people hide them in caverns. To understand the Maya calendar we have to know their vigesimal counting system (Figure 8). They invented the concept of zero and the positional arithmetic. They used many calendars, each one for some purpose, we know almost 17 different, but 2 of them are most important: the Haab and the Tzolk’in. Haab calendar is for civil purposes, it is based on the Sun and it is 365 days long. Tzolk’in calendar is a ceremonial calendar and it is 260 days long. The calendar and all astronomical activities, divination and ceremony organization was duties of the priest, the second most important people in the Maya hierarchy. It is interesting to see that almost all Central and South America cultures uses a 260 days long ritual calendar, many attempt to identify the reason was done, but there are not a good explanation for that. It is clear, seeing the incredible ability they have as astronomers, that it is based on some astronomical cycle, but nobody knows which Figure 10 is. The names and glyphs for the 18 periods (plus the 5 days Uayeb) of the Maya Haab calendar. The 365 days calendar, From: www.mayacalendar.com/loscalendariosmaya.html used for more than a thousand year during the old and middle preclassic period, was reformed in the year 249 BC. Maya astronomers, invented then the leap year concept. After three year of 365 days, there was a leap year of 366 days. This calendar works well, but the little error produce a shift of the start of the seasons, so another reform was done in the year 775 AD. This time, astronomers modify the leap year system, and not all multiple of four years will be leap years, they suppress 4 leap years during a period of 500 years. Globally this system reach a better accuracy than our Gregorian calendar. This is the most precise calendar in the world, past and present. Maya 8

astronomers obtain this kind of precision by using a scientific method, they carefully record their observation over very large periods of time, then using mathematics, they were been able to calculate and predict the position of the Sun, the Moon and the planets. Venus was an important celestial body for Mayans, but maybe they was interested also in Mars and Jupiter. Their precise calculations, let them to Figure11 Names and Glyphs for the 20 days of theTzolk’in Maya predict eclipses of the Sun and the ritual calendar. Moon. Astronomical observations From: www.mayacalendar.com/loscalendarios maya.html was done in very special building, erected for that purposes, like the Caracol (Figure 9) in Chichen Itza (Mexico). The Haab (Figure 10), or solar calendar, contains 19 periods, they are like our months. The first 18, contains 20 days each, the last, of 5 days, are not considered really a month, because they are bad days, so they are left to the end of the year and they have not a specific name. During this period of bad days, called Uayeb, people realize the coming new year ceremony. A date in the Haab calendar is specified by a number from 0 to 19 and the respective period name or glyph, for example 0 Pop, 1 Pop, …, 19 Pop, 0 Uo, 1 Uo, 2 Uo, etc. The name of the days are the same for the Haab and the Tzolk’in (Figure 11). The 260 days Tzolk’in, is divided into 13 periods of 20 days each, but they do not have a specific name, they are referred with a number from 1 to 13. To understand better how the Tzolk’in works, we can think as two encountered and rotating wheels (Figure 12), the secuence of days is as follows: 1 Imix, 2 Ik, 3 Akbal, …, 13 Ben, 1 Ix, 2 Men, etc. The Haab and the Tzolk’in calendars, was used alone or simultaneously. For Figure 12: civil duties the Haab was used and for agricultural Schematic representation of the labor and religious ceremony the Tzolk’in was used. Tzolk’in calendar of 260 days. But for important dates, like the coronation of the From: www.usu.edu/anthro/origins_of_writin king or a war end, they used both simultaneously. g/mayan_calendar/ The advantage of such system is that the repetition of the day name or number and the month occurs only after a period of 18980 days (52 years), so there are not the possibility to misunderstand a date.

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Maya was the civilization most interested in time keeping and time recording, every action, realm activity, astronomical observation, natural event, etc. was recorded on paper or graved on a stone (Figure 13). It seems that they wished to preserve their history and scientific baggage for future generations, because they know the importance of the event recording, otherwise they never made a calendar or predict an eclipse without the exact data from ancient Figure 13: observations. Important events are The Haab and Tzolk’in combination recorded on stones like was not enough for a large period of time, this that is known as Quirigua stelae C. again, they invented a powerful system called Long Count calendar. We can compare the Maya Long Count to our Julian date number, this way each date has a unique number starting from some initial date when Mayans believed correspond to the creation cycle. Most mayanist think that the initial date is around some day of Agust 3114 BC. Why this? Nobody knows. There are a lot of speculation on the year and also on the exact date, but it depends only on the way we make calculations. There is no consensus yet on the correlation constant because we do not know exactly one date on both our calendar and the corresponding Maya calendar. This give to people the possibility to run their imagination. Figure 14 An example of a long count date: 8.14.3.1.12. This is a sketch of the Leiden stone in Peten, From top to bottom: Introductory glyph 8 baktun 14 katun 3 tun 1 uinal 12 kin then, a series of other details about the date. From:www.proel.com/alfabetos/maya.html

The most accepted correlation is the Goodman, Martinez, Thompson (GMT) but it works well only half the time. There are a hundred more, each one works for some specific events only. The aim of this work is not the discussion about how to correlate a date, but just the description of how the Maya calendar works and why it is so interesting. The long count is a unique combination of 7 numbers, so a date consist of a series of 5 numbers and the Tzolk’in date followed by the Haab date: - baktun number - katun number - tun number - uinal number - kin number Each period is represented by a number and a glyph as in Figure 14. 10

For example, on a lintel 16 found in Yaxchitlan, Mexico, there is the date: 9 .13 .17 .12 .10 .8 Ok . 13 Yax. To understand it, I will show first the meaning of each period of time: 1 baktun = 20 katun = 144000 days 1 katun = 20 tun = 7200 days 1 tun = 18 uinal = 360 days 1 uinal = 20 kin = 20 days 1 kin = 1 day so the date correspond to 9 baktun = 1296000 + 13 katun = 93600 + 17 tun = 6120 + 12 uinal = 240 + 10 kin = 10 = 1395970 days from the creation and that particular day is: 8 Ok, 13 Yax from the Haab and Tzolk’in system. A similar date written in our Julian date system will be: JD 2443711 July 21 (1978). Maya conception of time is very wide, they have a name and a glyph for each period of time, but time do not ends at baktun level, it extend to a very large periods, they are: 1 pictun = 20 baktun = 2,880,000 days 1 calabtun = 20 pictun = 57,600,000 days 1 kinichiltun = 20 calabtun = 1,152,000,000 days 1 alaultun = 20 kinichiltun = 23,040,000,000 days (roughly 63 million years) The logic in their calendar is very clever, because for each system they have a specific glyph, so it is impossible to make a mistake. If the date is only specified by the Tzolk’in and the Haab system, there is not an introductory glyph (Figure 15), but if the date is in the long count system, there is an introductory glyph. If the date represents an extremely important event, they add also a supplementary series which Figure 15 contains a number from 1 to 9 and the glyph Introductory glyph. The date is in Long Count format of one Lord of the night, a series of glyphs that From: represents deities which is though, they rule http://www.mexconnect.com/mex_/travel/l during the night. The same day, using the dumois/maya/ldmayanumbers.html Haab, Tzolk’in and Lord of the night cycle, together, repeat only after a period of 467 years. In addition there is the corresponding lunar phase for that day, and the age of the moon. There are also other symbols not decoded yet. The long count starting point is for mayans the day of the creation of their present cycle, the fourth, because Maya believed in recurring cycles of creation and destruction.

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Calendars in history around the world As we saw, any organized culture or community, have their proper calendar. In this section I will show only some examples, because there are a lot of different calendars used in the past and some of them that people are still using today. Chinese Calendar One of the most precise calendars in the world is the Chinese. There are evidences of the uses of a calendar back in time until the 14 century BC. The initial date is the year 2953 BC. For Chinese people, the Heaven and the Emperor are connected, Gods observe how he govern and if they do not like it, they send signs that astronomers must interpret as a bad presage. Astronomy was the most important job and the most dangerous. Astronomers was employed by the emperor and they are under his order. They have to made observations of the sky and the planets, they have to make very precise calculations to predict eclipses and other phenomena. Their life depends on their ability, if they fail, the emperor may order their death. The Chinese calendar is lunisolar, based on the movement of the Sun and the Moon, very early, they calculated the year of 365.25 days and the synodic month to 29.5 days (Figure 16). It is the most ancient and the only non interrupted calendrical system already in use in the world. Today for business purposes, China adopted the Gregorian system, but they are still using the traditional calendar for agriculture and Figure 16 Chinese calendar, inscription on bones ceremony. The day is measured as we do showing the length of the year of 365.25 days now, from the midnight to the successive and synodic month of 29.5 days. midnight. From:webexhibits.org/calendars/calendarchinese.html The official Chinese calendar calculations must be done for the longitude of 120°, but many times they are in disaccord with the true calendar because of the parameters used. The precision of the calendar is due to the fact that it is based on the tropical year and the real movement of the Sun and the Moon in the sky, it is the only calendar in the world that track simultaneously the phases of the moon and the movement of the Sun on the Ecliptic. The solar year is divided into 24 parts of 15° each this is called a Solar Term. The seasons do not begin on fixed days (equinoxes or solstices) but at some day in between. The first day of the year (when the Chinese spring begins) is when the Sun ecliptic longitude is 315°, this occurs near the second New Moon phase after the Winter Solstice, this date is always from 2 weeks before to 2 weeks after the February 4 of our calendar. The lunar year is divided into 12 month of 29 and 30 days (due to the 29.5 days of the synodic month), at some regular intervals they add a leap month, this way the solar and the lunar year are in synchronization. The solar years are arranged in cycles of 60 years. Each year take the name of an animal, there are 12 possible names and they repeat after the cycle ends:

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1 – Zi (Rat) 2 – Chou (Ox) 3 – Yin (Tiger) 4 – Mao (Rabbit) 5 – Chen (Dragon) 6 – Si (Snake) 7 – Wu (Horse) 8 – Wei (Sheep) 9 – Shen (Monkey) 10 – You (Rooster) 11 – Xu (Dog) 12 – Hai (Pig) But each year name is composed of two names: the combination of the 12 animal names with a series of other 10 words called the Celestial Stems. These are: 1 – Jia 2 – Yi 3 – Bing 4 – Ding 5 – Wu 6 – Ji 7 – Geng 8 – Xin 9 – Ren 10 – Gui so the first year of the 60 year cycle is Jia-Zi, followed by Yi-Chou until Gui-You, then Jia-Xu, Yi-Hai, Bing-Zi, etc. The duration of the year is not constant, because depends on the sequence of the lunar month of 29 or 30 days, so they have years of 353, 354 and 355 days, but when the leap month is added, the year length is 383, 384 and 385 days. There are a complicated series of rules for adding the leap month that depends on the occurrence of the New Moon respect to one of the 15° segments of the Ecliptic. It is a very complicated calendar for us, but it worked well for more than 3000 years. Jewish calendar This is a strange calendar, because it is based on tables and not to the real occurrence of the astronomical phenomenon. The day begins at Sunset. There are no names for the days, only the Sabbath day has one. This is the most important day of the week and correspond to the time from the Sunset of Friday until the Sunset of Saturday. This calendar origin is very far in time, October 7 3760 BC, so years are counted from that date. Due to the secretness of the calendar in early history of the Jewish people, we do not known exactly the origin. The Patriarc Hillel in the year 359 AD, introduced new rules for the leap month and eliminate the secrets about the calendar making it available to all the people, but for another 6 centuries there are not enough information on the calendar. It seems that Jewish calendar is an heredity of the babilonian’s because there are similitude on the names of the months and on the rules for the leap month.

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Like Chinese calendar, it is based on the Lunar synodic month. There are 12 months of 29 or 30 days long, after a few years they add a leap month. The calendar is based on the Metonic cycle that establishes a 235 synodic month to occurs in 19 solar years. The Leap month is added on the year 3, 6, 8, 11, 14, 17, 19 in a 19 years basis. The years are classified as Deficient, Regular and Complete, the number of the days are 353,354,355 respectively. If the year contains a leap month, then the Deficient year has 383 days, the regular year has 384 and the complete year has 385 days. Even when the days of the weeks has no name, they are called using a number (1 to 6), but the month do have names: 1 – Tishri 2 – Heshvan 3 – Kislev 4 – Tevet 5 – Shevat 6 – Adar 7 – Nisan 8 – Iyar 9 – Sivan 10 – Tammuz 11 – Av 12 – Elul To define if the month has 29 or 30 days there are some rules: 1 – if the year is complete Heshvan take 30 days 2 – if the year is deficient Kislev take 29 days 3 – when the leap month is added there are two Adar month, the first (Adar I) of 30 days and the second (Adar II) of 29 days. Due to the non uniform length of the day, that varies with seasons, the beginning of the day is by definition at 6 PM., this time is their hour 0. All calculations are done for the longitude of Gerusalem. The synodyc month is considered constant and its mean value is 29 days 12 hours 793 halakim, when an halakim correspond to 1/1080 of the day. This is another complicated system, and works well only on a short term, because of the variations of the movements of the Sun and Moon. As other peoples in the world do, they uses this calendar only for religious purposes and for business they occupy the standard Gregorian calendar. Islamic calendar This is a calendar based only on the lunar phases. It is based on the observation not to calculation like the Jewish, but of course there are tables to define a standard time. The month begins not on the astronomical New Moon phase, but on the visibility for the first time of the crescent Moon. If the weather is poor and the Moon is not visible, the beginning of the month may be declared after a period of 13 days. This is not, however a rule, it depends on the religious leader, scientist in charge of calculation, poor calculation methods, etc. The week contains seven days, each day begin on Sunset. The days are numbered and the first day of the week is the one that begins on the sunset of Saturday. The only day of the week with a name is the fifth, it is called Jum. It is the day when prayer congregate.

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The origin of this calendar is more recent than the Chinese and the Jewish, it is 15 July of 622 AD. This date represent the time when the Prophet migrate to Medina from Mecca. This first year is known as Muharram 1. Now in the Islamic world, they use a lunisolar calendar, slightly different from the original, because they use the leap month. This change was made by the Prophet and stated in the Sacred religious text of the Islam: Coran. Due to the many pray time during each day that Muslims has to observe, the determination of this needs a lot of complicated rules and calculations because the length of the day is not constant in the year. Before the Prophet reform, the season cycle was a completely “do not care concept”, because in a very short term ( roughly 33 year) the season are completely out of phase respect to the equinox. Even now there is no correlation between seasons and calendar.

Conclusion It is very interesting to see how different cultures uses their calendars, but for thousand of years the main use was for agricultural purposes. Then religion, share the importance with survival activities and in most cases it was more important. Some cultures adopted one calendar for civil activities and a different one for religious activities. There are cases where the religious calendar modified the civil one. When people get more organized and communities grows into cities and states, the calendar begin to have more importance because of the many more activities people have to do everyday, but religion grows up in importance too, so there was the necessity to unify all systems in only one. This task is not easy and really do not exist a calendar created in accordance with both civilian and religious duties. The Gregorian calendar, the one we use, that is almost a standard for all industrialized countries, is one of this categories. It was born as for civilian uses, but during the Dark Age and after, Catholic religion was the axis of the Western world, so most important dates was Easter and Christmas. During last 500 years, really just after the Gregorian reform in 1582, many people tried to modify the calendar, one of the most heavy reasons for that is precisely the religious background of it. Another very good reason is to try to have a more precise and easiest way to keep seasons synchronized with the calendar. We can find more than one hundred proposals, but many are totally unpractical and extremistic, because they are a radical change, really they want to change everything, from the number of the days in a week to the name of the days, months, etc. If we implement such type of calendar, the world will be falling in a chaos, many times worst than the famous Y2K. Other proposals are less radicals, but they do not take into account the real movement of the Earth, so maybe for some years it works well, but this imply to forget about the seasons, this is not good at all. Honestly, I can not find, (but I do not analyze all I have encountered and maybe there are a lot more that I ignore they exist) one that could be really implemented without a radical change, not only in the form to take into account time and dates, but how people think and live. Absolutely all the electronics equipment in the world (from watches to airplanes and life support systems) are designed to work with a non perfect, non liked, not easy, with all the possible defects we can find in it, Gregorian calendar system, so, just a little change in it

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imply that all those apparatus go to the trash can, it is impossible today to change a system adopted worldwide. Today things are different from when Pope Gregory XIII ordered in 1582 the reform. We can now send spacecrafts out of the Solar System, we have a technology being able to do almost anything we want, but such thing like a simple and precise calendar escape to our knowledge, imagination and capacity. With our most precise and impressive atomic clocks, which accuracy is better than 1 part in 10-15 per year, in a few years the are out of synchronization with the Earth, so we have to adjust them introducing a leap second. As I said in the “What is a calendar” section, it seems to be easy to follows the seasons, but, all depends on the precision we want to do that, because every planet, star, etc, in the Universe is moving in a way that the whole of them, in more or less quantity, perturb or influence each other. Most important causal of perturbations of the Earth movement are: - elliptical orbit - the Moon and all effect produced by tidal forces like precession - synchronization of the Moon rotation - other planets - fused internal core There is no way to make a calendar that works exactly forever, there are too many variables, some of that not yet discovered, there are some changes in the rotation that seems to happened in a randomly way. We just try to reduce errors in our actual system, thinking on medium term, because at best we can make a calendar that works with an error of a few minutes for a period less than a thousand years. Even this kind of error is not absolute, it is the average error in relation of the entire period. Each year our calendar is going out of phase by more than 6 hours, just after 4 years we get in time again adding a day (in the leap year), but not exactly at all, because the error can reach more than 18 hours! The addition of a day, or a month, was used from thousand of years ago by the most important civilizations, our actual Gregorian calendar is the most precise in use, because other calendars, that are most precise on average, are the least if we take a single year, in fact, for example, the Chinese calendar has variable length years. The best uses of the leap years was the Maya, they add a day each four, just like the Gregorian do, but instead of make a leap year if the year number is divisible by 400, they do that if the year number is divisible by 500. This improve the accuracy and, on average, the mayan tropical years is 365.2420 days long. This figure is 0.0002 days closer to the real actual mean value of 365.2422 days. For the Gregorian calendar, the average tropical year is 365.2425 days long. Maybe if the Spanish bishop Diego de Landa should not destroyed the Maya documents, Pope Gregory XIII scientific advisors followed their system. An advantage of the Gregorian calendar over all others, is the uniformity of the length of the year, the difference from a leap year to a normal year is just one day instead of a full month of 30 days or a slightly shorter one of 29 days like the Chinese or the Jewish. As the space exploration technology get better results, it seems that in the near future (less than 100 years I think) we can be able to have colonies on other planets ( Mars at first) and on the Moon. What happened with time and calendars? Do we have a different calendar for any colony? My guess is not, it will be a very big problem, so “Terrestrials” (people from Earth) must institute a really Universal Time and Universal Calendar. But based on what? A day on the Moon is 27 terrestrial days, a day on Mars is half an hour

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more than on the Earth. Maybe we have to change radically our conception of time and use a Solar System Time Clock and Calendar based on the time and date of the Sun. As a final point, I will suggest a very easy way to have the year always in step with seasons and without the necessity of adding a day during the leap year. It is based on clocks that are running at a rate equal to the duration of the Tropical second. A tropical second is calculated as follows: 1 Tropical year = 365.24219 mean days of 86400 seconds = 31,556,925.216 seconds/year 1 Gregorian year = 365 days of 86400 seconds = 31,536,000 seconds/year 1 tropical second = 1 Gregorian second * (number of tropical seconds per year / number of Gregorian seconds per year) = 1*31,556,925.216/31,536,000= 1.0006635342. So we only have to accelerate our clocks 0.07 %. The international definition of a second must be changed to the follow: “…the duration of 9,198,731,396 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom in zero magnetic field” instead of 9,192,631,770 periods as stated now. We always still have days of 86400 divisions and years of 365 days. This system will works well and much better than Gregorian’s, but, on a long time basis, it will have errors because the length of the year is lengthening; this is because the Earth is slowing down. So we always have an error, but extremely much less than actual.

References Time and Calendar: - Murin P, Enciclopedia of Astronomy and Astrophysics, Institute of Physics, 2001 - Dogget L et al, Almanac for computers, USNO, 1978 Maya calendar: - Gonzalez T.Y, diccionario de mitologia y religion de Mesoamerica, Larousse, 1991 - Carter M., The Maya religion, East Tenn, State University, 1996 - Wolfgang R. D., History of Astronomy Topics Calendars, Time and Chronology, 2002 www.astro.unibonn.de/ - Rodriguez F.G, Astronomia en el mundo Maya, Sociedad Astronomica de La Laguna www.astro-digital.com - Dumois L., La civilización Maya, www.mexconnect.com - Blanco M., El calendario Maya, 2002 - El mundo cosmologico Maya, www.cervantesvirtual.com - El calendario Maya, Centro de estudios del mundo Maya, 2003 - Tippets N., The Mayan calendar, 2002, www.usu.edu/anthro - Vollemaere A.L., Electronic open University, 2003, http://users.skynet.be/fa039055/ Calendars in the World: - Dogget L.E, Calendars and their history, Explanatory supplement to the Astronomical Almanac, http://astro.nmsu.edu/lhuber/

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