Mayan

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The Maya

One of the most amazing cultures of the New World inhabited a region encompassing today's Guatemala, Belize, Honduras and El Salvador, and parts of southern Mexico (the states of Yucatan, Campeche, Quintana Roo, Tabasco and Chiapas). Today this area is occupied by the descendants of the ancient Maya, the vast majority of whom have to some extent preserved their cultural heritage and still speak the Mayan language.

By 5000 BC, the Maya had settled along the Caribbean and Pacific coasts, in fishing communities. By 2000 BC the Maya had also moved inland and adopted agriculture for their subsistence. Maize and beans formed the Maya diet then as today, although many other foodstuffs--manioc, squash, tomatoes, peppers, fruit, and game-were supplements.

To study the Maya their development has been divided into periods, the earlier Maya culture is called Formative or Pre Classic (2000 BC-AD 300), the Classic period goes from AD 300 to AD 900, and subsequent civilization is known as Post Classic (AD 900-conquest).

Now we know that the Maya began to develop intensive agriculture and sophisticated water management during the Middle Pre Classic (900-300 BC), surely to help support the population explosion of the Late Pre Classic (300 BC-AD 300). During this same period, writing was invented in Mesoamerica, and the Maya began to use it during the Late Pre Classic.

The Maya were the first people of the New World to keep historical records, and even if writing in the New World did not originate among the Maya, they developed and used it extensively. The Maya wrote a mixed script, with ideographic and phonetic elements.

Most of their writing survived on stelae, stone monuments very common in the Maya cities, they recount mostly civil events and record their calendric and astronomical knowledge.

Maya pottery gives testimony of their religion and elaborate mythology. Four Post Classic Maya screenfold manuscripts, called codices have survived, They reveal Maya calendric and astronomical calculations, as well as rituals, offerings, and auguries for the year.

The Maya used several calendars simultaneously. One of them called the "long count", is a continuous record of days from a zero date that correlates to Aug. 13, 3114 BC, and is more precise than the Julian calendar revised in Europe in 1582. The Maya were great astronomers and kept track of the solar and lunar years, eclipses and the cycles of visible planets. To carry out their calendric and astronomical calculations they developed a sophisticated mathematical system where units are written with dots and bars are used to represent five units. They discovered and used the zero as well as a vigesimal positioning system, similar to the decimal positioning system we use today.

During the Classic period monumental architecture and stelae with historical records were erected, on these monuments the Maya rulers reigned as divine kings. The Maya thrived during the Late Classic (AD 550-900), and art, architecture, writing, commerce and intensive agricultural practices flourished all through the Maya lands. More than 2 million people may have lived in the area, and it is estimated that Tikal, the largest center, had a population of 75,000-100,000.

However, the Classic Maya cities did not survive into the 10th century. It seems that the system of rule that had served them well for centuries failed. Probably faced with famine, foreign invasion, chronic warfare, adverse climatic conditions and perhaps disease, the Classic period ended in what is called the Classic Maya collapse. The Maya continued to live in both highlands and lowlands but the period of their greatest splendor was over. In the northern Yucatan peninsula, civilization continued at Uxmal and the surrounding area. The Post Classic saw the splendor of Chichen Itza. Chichen Itza was probably abandoned by the 12th century. Trading towns survived along the Caribbean coast. Tulum, a spectacular walled city and a major trading town, located above the coastline of what is now the state of Quintana Roo on Mexico's Caribbean seashore is a great example of these. This city when seen from a Spanish ship was compared to Seville.

The Maya of Yucatan finally broke up into small states and the Spanish took advantage of this division to take control in 1542. In that year, after having been fought back during 15 years, they were able to establish their own capital at Mérida, (in today's State of Yucatan, Mexico) on the site of a Maya city called Tiho. The last of the Maya kingdoms, Tayasal, in Lake Peten Itza (Guatemala), was conquered by the Spanish in 1697, 155 years after the conquest of Merida..

The Calendar

Calendar (L. calendarium)

1.- A catalog that registers all the days of a year, distributed in weeks and months, with astronomical data, such as time of sunrise and sunset, the moon phases, or with religious information such as patron saints and festivities.

2.- A time division system, all of the world's cultures have their calendars initially lunar and afterwards lunar-solar. The Chaldeans and Babylonians passed their calendric knowledge to the Egyptians, these in turn to the Greeks and these finally to the Romans who adopted it for their common use.

From the beginning of civilization there has been a very close link between astrology and the development of the calendar. The importance of this connection is evident considering the need to determine the times for the most basic functions of early societies such as agriculture and the celebration of religious events.

The most ancient calendars were probably based on lunar observation since the Moon's phases take place in an easily observed interval. It is most likely that the sighting of the crescent Moon marked a new time period. It was observed that recurrent Moon's phases were about 29 days apart. This gave birth to the first lunar calendars containing 29-30 days per time period (month), but since the sum of twelve or thirteen months differ from the length of a tropical year this calendar was not completely suitable for agricultural practices.

Due to this difference and in order to keep in step with the Sun, the lunar-solar calendars were born, adding a complementary time period to the total of days in the Moon's cycles so as to equal the solar year. Many of these calendars, with variations, existed through time in different areas of the world. In pre-Columbian America the Maya and Aztec calendars were very important. They are remarkably accurate and are made of 18 months of 20 days plus five supplemental days.

Later the solar calendars came to be, for example the Julian calendar which was instituted in Rome by Julius Caesar in 46 BC. This calendar set the year's length at 365 days and added one day to the year every four years. Pope Gregory XIII modified this calendar in 1582. And even though the Gregorian calendar is a solar calendar in the sense that it does not take into consideration the Moon in its calculations, it does contain rules for determining Easter and other religious holidays which are based on both the Sun and the Moon. The Gregorian calendar is used today in most of the world, it is divided into the twelve months we all know.

Historically people have sensed the need to have a fixed point to start their time calculations. In order to do this generally the starting point has been determined either by a historical event (the birth of Jesus) or by a hypothetical event (the date of the world's creation). Of all known cultures the Maya seem to have been the first to discover the need for such a date, using probably an astronomically significant or a hypothetical event they placed at 3114 BC.

Chapter 1: Maya and European Dates

Long count dates
Tzolkin dates
Haab dates
The Maya calendar round
Julian dates
Gregorian dates
Julian day numbers
Dresden Codex

1. Long count dates
The Mayas used three different calendrical systems (and some variations within the systems). The three systems are known as the tzolkin (the sacred calendar), the haab (the civil calendar) and the long count system. The tzolkin is a cycle of 260 days and the haab is a cycle of 365 days (these cycles are explained in Sections 2 and 3 of this chapter). The tzolkin cycle and the haab cycle were combined to produce a cycle of 18,980 days, known as the calendar round. 18,980 days is a little less than 52 solar years. Thus the Mayas could not simply use a tzolkin/haab date to identify a day within a period of several hundred years because there would be several days within this period with the same tzolkin/haab date.
The Mayas overcame this problem by using a third dating system which enabled them to identify a day uniquely within a period of 1,872,000 days (approximately 5,125.36 solar years). To do this they used a vigesimal (i.e. based on 20) place-value number system, analogous to our decimal place-value number system.
The Mayas used a pure vigesimal system for counting objects but modified this when counting days. In a pure vigesimal system each place in a number is occupied by a number from 0 to 19, and that number is understood as being multiplied by a power of 20. Thus in such a system:
    2.3.4    = 2*20*20 + 3*20 + 4*1 = 864
    11.12.13 = 11*20*20 + 12*20 + 13*1 = 4653 and 
    1.3.5.7  = 1*20*20*20 + 3*20*20 + 5*20 + 7*1 = 9307
When counting days, however, the Mayas used a system in which the first place (as usual) had a value of 1, the second place had a value of 20, but the third place had a value not of 400 (20*20) but of 360 (18*20). (This may have been due to the fact that 360 is close to the length of the year in days.) The value of higher places continued regularly with 7,200 (20*18*20), 144,000 (20*20*18*20), etc. In such a system:
     1.3.5.7       = 1*20*18*20 + 3*18*20 + 5*20 + 7*1 = 8,387
and 11.12.13.14.15 = 11*20*20*18*20 + 12*20*18*20 + 13*18*20 + 14*20 + 15*1
                   = 11*144,000 + 12*7,200 + 13*360 + 14*20 + 15
                   = 1,675,375.
A Maya long count date is a modified vigesimal number (as described above) composed of five places, e.g. 9.11.16.0.0, and interpreted as a count of days from some base date. There are many long count dates inscribed in the stellae and written in the codices. Calculation of the decimal equivalent of a long count yields a number of days. This is regarded as a number of days counted forward from a certain day in the past. It is the number of days since the day 0.0.0.0.0. The obvious question is: What day was used as the base date? This question has two aspects: (1) What day was used by the Mayas as the base date? (2) What day was that in terms of the Western calendar? We shall return to these questions below.
Just as we have names (such as week) for certain periods of time, the Mayas had names for periods consisting of 20 days, 360 days, 7,200 days, etc., in accord with their modified vigesimal system of counting days. A day is known as a kin. Twenty kins make a uinal, 18 uinals a tun, 20 tuns a katun and 20 katuns a baktun. Thus we have:
            1 kin    =  1 day
            1 uinal  = 20 kins   = 20 days
            1 tun    = 18 uinals = 360 days
            1 katun  = 20 tuns   = 7,200 days
            1 baktun = 20 katuns = 144,000 days
The numbers at the five places in the long count are thus counts of baktuns, etc., as follows:

baktuns . katuns . tuns . uninals . kin
Thus, for example, 9.15.9.0.1 denotes a count of 9 baktuns, 15 katuns, 9 tuns, no uinals and 1 kin, or in other words, 9*144,000 + 15*7,200 + 9*360 + 0*20 + 1*1 days, or 1,407,201 days. It is a count of days from the Maya base date of 0.0.0.0.0.
Most of the long count dates which occur in the stone inscriptions have a baktun count of 9. The period 9.0.0.0.0 through 10.0.0.0.0, the period of the Classic Maya, is now thought by scholars to coincide with the period (approximately) 436 A.D. through 829 A.D. There are, however, some strange anomalies. Morley [39] deciphers two long count dates (found at Palenque) as 1.18.5.4.0 and 1.18.5.3.6 (14 days apart) which are some 2,794 solar years prior to 9.0.0.0.0. Since there is no evidence that the Mayas existed before about 500 B.C., what could these early long count dates possibly be referring to?
We would expect that the next higher unit after the baktun would consist of 20 baktuns, and it appears there was such a unit, called a pictun. However, no long count date occurs with a baktun count of more than 12, except that 13.0.0.0.0 occurs (see Morley [39], pp.203-4, for an example). A widely-accepted school of thought holds that in the Maya long count system 13.0.0.0.0 marks the beginning of a new cycle, and so is equivalent to 0.0.0.0.0. In this view, 13 baktuns make up a great cycle or, Maya era, of 13*144,000 = 1,872,000 days (approximately 5125.37 solar years).
Here is the date sequence for the entire 12.19.7 tun (this was produced using the Mayan Calendrics software).
2. Tzolkin dates
The tzolkin, sometimes known as the sacred calendar, is a cycle of 260 days. Each tzolkin day is denoted by a combination of a number from 1 through 13 and a name from the following set of twenty (in the order: Imix, Ik, Akbal, Kan ....):
   Imix        Cimi    Chuen   Cib
    Ik         Manik   Eb      Caban
    Akbal      Lamat   Ben     Edznab
    Kan        Muluc   Ix      Cauac
     Chicchan   Oc      Men    Ahau
The days cycle through the numbers and through the names independently. The sequence of tzolkin days thus runs:
            1 Imix
            2 Ik
            3 Akbal
            4 Kan
              . . .
            13 Ben
            1 Ix        (here we repeat the cycle of numbers)
            2 Men
            3 Cib
            4 Caban
            5 Edznab
            6 Cauac
            7 Ahau
            8 Imix      (here we repeat the cycle of names)
            9 Ik
            10 Akbal
              . . .
There are 260 elements in this sequence. That is because 260 is the least common multiple of 13 and 20. Thus the cycle of (13) tzolkin day numbers combined with (20) tzolkin day names repeats each 260 days.
In order to explain this 260-day calendrical cycle some have speculated that the Mayas chose this number of days because their allegedly advanced astronomical knowledge revealed to them that a period of 260 days fits well with certain astronomical periods, such as the eclipse-year. A more prosaic explanation is that there were originally two branches of Maya society, one of which used a 13-day cycle of numbered days and the other a 20-day cycle of named days. (There is a set of thirteen Maya gods, which may be the origin of the 13 numbered days, similar to our week.) Then at some point in early Maya history the two groups merged, combining the two calendars so that neither group would lose their method of day-reckoning, resulting in the 260-day cycle as described above.
3. Haab dates
The Mayas also maintained a so-called "civil" calendar, called the "haab". This was similar to our calendar in that it consisted of months, and within months, of days numbered consecutively. However, unlike our calendar, the haab cycle is made up of eighteen months of twenty days each, plus five days at the end of the year. The eighteen names for the months (in the order: Pop, Uo, Zip ...) are:
     Pop    Xul     Zac   Pax
     Uo      Yaxkin  Ceh   Kayab
    Zip    Mol     Mac   Cumku
    Zodz   Chen     Kankin
    Zec     Yax     Muan
The five extra days formed the "month" of Uayeb, meaning "nameless". The five "nameless" days were considered unlucky. One did not get married in Uayeb. The haab cycle thus consisted of 18*20 + 5 = 365 days, the integral number of days closest to the mean solar year of 365.2422 mean solar days.
The sequence of days from the first day of the year to the last thus runs as follows:
    0   Pop
    1   Pop
    ...
    19  Pop
    0   Zip
    1   Zip
    ...
    19  Zip
    0   Zodz
    ...
    19  Cumku
    0   Uayeb
    ...
    4   Uayeb
For most of Maya history the first day of Pop was denoted by 0 Pop and the last by 19 Pop. However, on the eve of the Spanish conquest the first day of Pop began to be numbered 1, and the last day 20 (except for Uayeb), so that the year began with 1 Pop and ended with 5 Uayeb.
There is some uncertainty as to whether (what has usually been taken to be) the first day of each haab month (e.g., 0 Zip) is really the last (i.e., the 20th, or the 5th) day of the preceding month (Pop in this case), or in other words, whether the last day of each month was actually written as "the day before the beginning of (the next) month", where the glyph translated as "the seating of" was used with the meaning of "the day before the beginning of the next month, namely ...". 0 Zip can be interpreted either as the first day of Zip or as the last day of Pop, but unfortunately the classic Maya are no longer here to tell us how they understood this date.
4. The Maya calendar round
The tzolkin and the haab are each cycles of days; the former is a cycle of 260 days and the latter is a cycle of 365 days. When specifying a day the Maya usually used both the tzolkin date and the haab date, as in 4 Ahau 3 Kankin. For the Mayas these two cycles ran together and concurrently, as shown by the following sequence of days:
 Tzolkin date    Haab date
    10 Ben          11 Kayab
    11 Ix           12 Kayab
    12 Men          13 Kayab
    13 Cib          14 Kayab
    1 Caban         15 Kayab
    2 Edznab        16 Kayab
    3 Cauac         17 Kayab
    4 Ahau          18 Kayab
    5 Imix          19 Kayab
    6 Ik            0 Cumku
    7 Akbal         1 Cumku
    8 Kan           2 Cumku
                       . . .
    12 Imix         19 Cumku
    13 Ik           0 Uayeb
    1 Akbal         1 Uayeb
    2 Kan           2 Uayeb
    3 Chicchan      3 Uayeb
    4 Cimi          4 Uayeb
    5 Manik         0 Pop
    6 Lamat         1 Pop
    7 Muluc         2 Pop
                       ...
Since 260 = 4*5*13 and 365 = 5*73, the earliest that a tzolkin/haab date combination can repeat is after 4*5*13*73 = 18,980 days, or just short of 52 solar years. This cycle of 18,980 days is called the Maya calendar round.
Maya long count dates are often given in association with the corresponding tzolkin/haab date, as in:
         8.11.7.13.5    3 Chicchan 8 Kankin
        10.1.19.15.17   12 Caban 0 Yax
        10.3.8.14.4     6 Kan 0 Pop
        10.6.2.0.9      9 Muluc 7 Yax
        10.6.10.12.16   3 Cib 9 Uo
A particular tzolkin/haab date recurs every 18,980 days, whereas a long count date (assuming that the long count starts over at 0.0.0.0.0 on reaching 13.0.0.0.0) recurs every 1,872,000 days (once in 5,125.37 years). The combination of a long count date and a tzolkin/haab date occurs only once every 136,656,000 days (approximately 374,152 years or 73 Maya eras).
5. Julian dates
The Julian calendar, introduced by Julius Caesar in 46 B.C., is the basis of our modern calendar. It consists of a system of twelve months, January, February, etc. (although New Year's Day has not always been January 1st). If the number of the year is divisible by 4 then February has 29 days, otherwise it has 28. A date in the Julian calendar is termed a Julian date.
The Romans identified their years as a number of years supposed to have elapsed since the founding of Rome (which we now date as having occurred in 753 B.C.) Following the merger (under Constantine) of the Christian Church and the Roman Imperium years came to be numbered with reference to the year of the birth of Christ (now regarded as actually having occurred in 4 B.C.) In this system the year immediately before the year 1 A.D. is the year 1 B.C.
Astronomers use a system, which is also used in Mayan Calendrics, in which the year prior to the year 1 is the year 0. Thus 1 B.C. is the year 0, 2 B.C. is the year -1, 3 B.C. is the year -2, and so on. More generally the year n B.C. in common usage is said by astronomers to be the year -(n-1). (See more on this in section 7.)
According to Aveni [5], p.127, "the serial numbering of the years as we know them did not actually begin until the sixth century ..." Thus dates prior to 600 are always uncertain. The Emperor Augustus also tinkered with the lengths of the months during his reign, introducing a further element of uncertainty, and it is also possible that the Council of Nicea (325 A.D.) readjusted the calendar by a couple of days.
6. Gregorian dates
The average length of a year in the Julian calendar is 365.25 days, differing from the value of the mean solar year by about .0078 days. This resulted in a slow shift of the Julian calendrical year with respect to the solar year (i.e. to the solstices and equinoxes). By the 16th Century the Julian calendar was seriously out of synch with the seasons and Pope Gregory XIII introduced the Gregorian Calendar. This involved three changes:
(a) The day following October 4, 1582, was declared to be October 15, 1582, thereby excising ten days from the calendar.
(b) A year was declared to be a leap year if (i) it was divisible by 4 but not by 100 or (ii) it was divisible by 400.
(c) New rules for determining the date of Easter were introduced.
The Gregorian Calendar is now commonly used throughout the West and is the de facto international common calendar. There have been numerous suggestions for replacing it with a more "rational" calendar, but old habits die hard and any change would be expensive.
7. Julian day numbers
Astronomers use a system of dating days known as the Julian day number system, in which a day is identified as that day which is a certain number of days before or after the day -4712-01-01 (January 1st, 4713 B.C.) in the Julian calendar. Thus, for example, the day whose Julian day number is 584,283 is September 6, -3113 in the Julian calendar, 584,283 days after January 1st, -4712 J. This day is also August 11th, -3113 in the Gregorian calendar. By 2001-01-01 G we will have reached the day whose Julian day number is 2,451,991, by which time nearly two-and-a-half million days will have elapsed since -4712-01-01 J.

The Dresden Codex — the Book of Mayan Astronomy
By Bohumil Böhm and Vladimir Böhm

A multi-page version of this article may be found at http://www.volny.cz/paib/dresden_codex.htm.

The advanced Mayan culture developed thanks to a complex synthesis of different culture streams arising from the home agricultural base, influenced by cultural values coming from regions lying out of the territory of Mayan settlement. Its forming falls to the so-called early phase of the initial period placed between 1500 – 800 BC. It was spread step-by-step to the regions of Guatemala, south-eastern Mexico, Belize, Salvador and north-western Honduras. The construction of beautiful and splendid cathedral cities, fine arts of sculpture and painting, use of their own hieroglyphic script, success in astronomy, existence of the literature and the development of handicraft and trade were the outer expression of this cultural-economic rise.
The results of Mayan observations and calculations of astronomical phenomena are concentrated in the Dresden Codex. It is a band of paper 3.5 meter long set up into 39 sheets making up 78 pages 8.5 x 20.5 cm. The paper was obtained from the bark of wild-growing species of fig tree. It is supposed that it originates from Yucatan as a latter transcription of an elder original. It contains calendrical data, written in the Mayan dating system, concerning astronomical data and the sky mechanics, and tables of multiple integers that are to be used for calculations of planetary movement ephemerids and tropical years, next to the hieroglyphic texts and numerous depicturings of the Mayan gods and ritual scenes.
The data contained in the Dresden Codex were studied by many researchers who suspected they contain astronomical data. M.Meinshausen (1913), C.E.Guthe (1921) and H.Spinden (1930) were the first who had been interested in the eclipses tables. E.Foerstemann has drawn our attention to Venus visibility ephemerides tables; he also issued the Dresden Codex with a commentary in 1892. The analysis of these ephemerides has been made by J.E.Teeple (1926). R.W.Wilson believed that some of the data could concern the observations of Mars, Jupiter and Saturn (1924). The above-mentioned researchers, and lots of others, worked with the calculation coefficients of 584,283 or 584,285 days accordingly to Goodman-Martinez-Thompson when converting the Mayan dates into the Christian dating system, or tried to calculate their own coefficient. For this reason their conclusions were very diverse.
One of the most important problems during the studies of various Mayan culture phenomena had been the problem of correlating the Mayan to our Christian dating system. In present times we are used to correlate the Mayan dates with the Christian ones using the Goodman-Martínez-Thompson correlation. Accordingly to it, a stable coefficient of 584,283 or 584,285 days is added to the Mayan dates expressing the counts of days which have passed from a particular day to the date of a certain event. The Mayan date is converted into a Julian day number and the latter finally to the corresponding day, month and year of the Julian calendar used in modern astronomy.
Working with the Mayan data of the Dresden Codex we found that the Goodman-Martínez-Thompson correlation is unusable, even for the dates evidently concerning certain astronomical phenomena, such as the observations of Venus visibility, or Sun and Moon eclipses. We have obtained a new coefficient of 622,261 days for the conversion of the Mayan dates to our dating system by a complete analysis of the mutual relations between the time intervals of all the Mayan dates in the Dresden Codex and 400 inscriptions from the cathedral cities. Using the so called Böhm correlation coefficient we were successful in proving that all data contained in the Dresden Codex are concerning astronomical phenomena.
The Mayan astronomical observations were carried out by simple measuring methods. It is therefore necessary to examine them statistically while respecting unavoidable accuracy scatter. It applies first of all to the sky phenomena calculated to the past and the future during several centuries recorded in the Dresden Codex. The dated astronomical observations are concerning following:

The observations of Venus visibility, when it had appeared for the first time after its conjunction with the Sun as a morning star in the sky shortly before the sunrise, or after its upper conjunction, when it had appeared in the sky as an evening star shortly after the sunset.

The observations of Mercury visibility. Its trajectory creates an eccentric ellipse. Thanks to this eccentricity the synodic circulations of the planet lasts from 104 to 132 days. The average length of the synodic circulation is 115.877484 days. The considerable proximity of the planet to the Sun makes its glow suppressed by dazzling sunshine. For that reason, the Mayan astronomers could have observed it only when the planet gets to the greatest angle distance during its circulation around the Sun, so called elongation. It is the western elongation, when Mercury rises over the horizon shortly before sunrise and the eastern elongation, when it is briefly visible over the western horizon right after sunset. The maximal angle distances are as a rule moving between 18 to 23 degrees. The maximal elongation of 27 degrees and 49 minutes happens when this elongation visible from the Earth runs during the epihelium, that means the greatest distance between Mercury and the Sun (Mercury gets there once for its sidereal circulation on its eccentric trajectory). The sidereal circulation is the real time of circulation of any planet around the Sun and it makes in this case 87.9693 days. During the perihelia (minimal distance from the Sun) is the maximal angle distance visible from the Earth 15 degrees and 55 minutes. It seems as if the planet stood in one place for 4-12 days in the time of maximal elongations. Its angle distance from the Sun changes between 1-2 degrees. This insignificant movement could not have been discovered by the Mayan observing methods. For that reason their determination of Mercury elongations moves on average inside the borders of that "mistake".

The solar eclipses and the fullmoons and newmoons.

The observation of the heliacal risings and settings of the planets. The heliacal rise sets in after the planets’ conjunction with the Sun, when they are visible in the morning sky before the sunrise. During their heliacal setting the planets are visible in the evening sky after sunset. In the period of conjunction the planets are invisible for a few days. By observing the heliacal risings and settings dates we are able to determine the length of the synodic circulations of the planets.

The observation of the planetary conjunctions (when two planets observable from the Earth get in line and are nearly covering each other). The Dresden Codex is mostly describing only close approaches of the planets because some of the dates are calculated to the past and to the future.

The determination of equinoxes and solstices.

All the astronomical phenomena in the Dresden Codex are counted for the visibility from the geographical area of the Mayan culture.

The Dresden Codex is divided into pages, that are numbered both after the older version set by E.W.Förstermann (F) and the newer one by Ju.V.Knorozov (D). The Mayan dates covering one, exceptionally more, pages are forming whole files. We keep their original topological placement during rewriting, in order to get a minimal distortion in comparison with the original. The Mayan dates in the Dresden Codex presented in brackets are not expressed by the number of passed days, but by a calendar circle made by combination of the dates from the 260 day tzolkin and 365 day haab. From them and from the relation to the other dates is calculated the real value of passed days.
Each Mayan date is at first rewritten in the Mayan calendar system, than transferred into the decadic counting system, by means of the Böhm coefficient (622,261 days) it is recalculated to the Julian day number (JD) and finally adjusted to the Julian calendar system.
All the computer programs were provided by Dr.Ing. Jan Vondrák from the Astronomical Department of the Science Academy of Czech Republic, for which we are very thankful.
The scheme of calculating the Mayan dates to the Christian system of dating using the Böhm coefficient of 622,261 days:
1/ The inscriptions of Mayan dates
2/ Transfers to the decimal arithmetical system
3/ Transfers to Julian day numbers by adding 622,261
4/ Days, months and years of the Julian calendar
5/ Analysis of the astronomical phenomena related to the single dates
PAGE F 24, 46-50, /D 24-29/
A/ 9.9.16.0.0 4 Ahau 8 Cumuku 1,366,560 days
- 6.2.0 - 2,200 days
B/ 9.9.9.16.0 1 Ahau 18 Kayab 1,364,360 days
C/ /9.14.2.6.0/ 1 Ahau 18 Uo 1,397,640 days
A/ J.D. 1,988,821 February 4, 733
B/ J.D. 1,986,621 January 27, 727
C/ J.D. 2,019,901 March 10,818
The rounded length of Venus synodic circulation of 584 days multiplied by five are added to the dates B and C 5-65 times and 130; 195 and 260 times. The real length of synodic circulation of Venus is 583.921 394 days. Time intervals of 185,120; 68,900; 33,280 and 9,100 days are added to the number B. And again the upper mentioned multiplies of 584 days. The total of B and interval of 33.280 days is the same as the date C, in the Dresden Codex mentioned only by the dates of calendar circle – 1 Ahau 18 Uo.
The total of B and 68,900 or B and 9,100 days is forming next dates D and E. Their last day always falls on 1 Ahau from 260 day tzolkin, as stressed in the Codex table.
B/ 9.9.9.16.0 1 Ahau 18 Kayab 1,364,360 days

68,900 days

D/ /9.19.1.5.0/ 1 Ahau 1,433,260 days

B/ 9.9.9.16.0 1 Ahau 18 Kayab 1,364,360 days
1.5.5.0 9,100 days
E/ /9.10.15.3.0/ 1 Ahau 1,373,460 days
D/ J.D. 2,055,521 September 17, 915
E/ J.D. 1,995,721 December 27, 751
The rounded length of synodic circulation of Venus multiplied by five is 2,920 days. They contain:
5 times the length of synodic circulation of Venus
13 times the length of sidereal circulation of Venus
8 times the length of tropical year
2,920 days is the typical period when the sidereal and sinodical circulation of Venus meet the tropical year. In this time the planet sets or rises in the same place in the sky after 8 years.
Venus visibility ephemerides are mentioned in the next Codex tables. Its synodic circulation length is written like five times /236 and 90 and 250 and 8 days/. These are comprehensible approximate values, because of the synodic circulation length of this planet is moving between 577 and 592 days.
236 days - Venus is in the eastern sky as a morning star
90 days - Venus is around the upper conjunction with the Sun and not visible
250 days - Venus is in the western sky as an evening star
8 days - Venus is around the lower conjunction with the Sun and not visible
This specification is added to dates B and D five times.
Dates B, C and D are concerning the moment of Venus first observation as a morning star after its lower conjunction with the Sun.
Date E is concerning the moment of Venus first observation as an evening star after tens of days of invisibility because of its upper conjunction with the Sun.

B/ January 27, 727 Venus is 8 days after the lower conjunction with the Sun. For first time observed as a morning star in the eastern sky.

Venus rose at 5.39 am

The Sun rose at 6.34 am

C/ March 10, 818 Venus is five days after its lower conjunction with the Sun. For the first time observed as a morning star in the eastern sky.

Venus rose at 5.36 am

The Sun rose at 6.11 am

D/ September 17, 915 Venus is six days after its lower conjunction with the Sun. For the first time observed as a morning star in the eastern sky.

Venus rose at 4.54 am

The Sun rose at 5.49 am

E/ December 27, 751 Venus is 63 days after upper conjunction with the Sun. For the first time observed as an evening star in the western sky.

The Sun set at 5.42 p.m.

Venus set at 6.46 p.m.

There is a time interval of 2,200 days marked between dates A and B. It contains:

19 times Mercury synodic circulation length /115.877 484 days/

25 times Mercury sidereal circulation length /87.968 581 days/
6 times the tropical year /365.242 199 days/

2,200 days is the typical period when the sidereal and sinodical circulation of Mercury meet the tropical year. In this time the planet sets or rises in the same place in the sky after 6 years. The similar cyclical regularity was observed by the Mayan astronomers about Venus, with a period of 2,920 days. The dates A and B are marking the moments when Mercury was to be found in a close approach of the maximal western elongations from the Sun. The D date concerns the maximal eastern elongation. Only in those positions around the maximal elongations is the planet visible not long before the sunrise or shortly after the sunset.

A/ February 4, 733 Mercury is to be found in western elongation with angle distance 26.47 degrees from the Sun.

Mercury rose at 4.53 am
The Sun rose at 6.32 am
The maximal western elongation of 26.73 degrees was on January 31, 733. The difference between determined and real elongation is 0.26 degrees.
B/ January 27, 727 Mercury was in western elongation with angle distance 25.5 degrees from the Sun.
Mercury rose at 4.55 am
The Sun rose at 6.34 am
The maximal western elongation of 26.07 degrees was on January 21, 727. The difference between determined and real elongation is 0.57 degrees.
D/ September 17, 915 Mercury was in eastern elongation with angle distance 25.05 degrees from the Sun.
Mercury set at 7.10 p.m.
The Sun set at 5.57 p.m.
The maximal eastern elongation 25.12 degrees occurred just few hours later.
The autumn equinox.

In all three determined elongations the Mayan astronomers have made an average mistake 0.3 degrees less than reality. It is so slight difference that it can’t have been discovered.

E/ December 27, 751 The conjunction of Venus, Mercury and Jupiter. The position of planets is expressed in degrees of geocentrical ecliptical coordinates:

Mercury 295.489
Venus 294.289
Jupiter 294.987

The time interval 2,200 days between A and B dates contains 74.5 times the length of synodic circulation of the Moon /29.530 588 days/.

A/ February 4, 733 The full moon
B/ January 27, 727 The new moon
PAGE F 51 – 58, /D 30 – 37/
A/ 8.16.4./8/.0 4 Ahau 1,268,800 days
8 8 days
B/ 8.16.4./8/.8 12 Lamat 1,268,808 days
C/ 10.19.6./1/.8 12 Lamat 1,578,988 days
D/ 9.19./5/.7.8 7 Lamat 1,434,748 days
E/ 9.16.4./11/.18 3 Etznab 1,412,878 days
F/ 9.16.4.11.3 1 Akbal 1,412,863 days
G/ 9.16.4.10.8 12 Lamat 1,412,848 days
A/ J.D. 1,891,061 June 11, 465
B/ J.D. 1,891,069 June 19, 465
C/ J.D. 2,201,249 September 10, 1314
D/ J.D. 2,057,009 October 14, 919
E/ J.D. 2,035,139 November 28, 859
F/ J.D. 2,035,124 November 13, 859
G/ J.D. 2,035,109 October 29, 859
The multiplies of 11,960 days (2,3,4,5,16,17,18 and 39 times) and of 13,780; 71,880 and 371,020 days are added to dates B, C, D, E, F and G.
B/ June 19, 465 The summer solstice
There is an interval of 144,240 days between C and D dates. It contains (with little differences of few days):
247 times Venus synodic circulation length /583.921 394 days/
642 times Venus sidereal circulation length /224.700 800 days/
185 times Mars synodic circulation length /779.936 160 days/
210 times Mars sidereal circulation length /686.979 800 days/
394.916 tropical years
The period of 394.916 tropical years contains the basic cycles of Mars and Venus conjunctions:
305.352 years the first conjunction cycle
14 times 6.3974 years the second conjunction cycle
C/ September 10, 1314 Mars and Venus conjunction.
Mars 197.76 degrees.
Venus 187.99 degrees.
D/ October 14, 919 Mars and Venus conjunction.
Mars 214 degrees.
Venus 215.48 degrees.

The position of both the planets is expressed in degrees of geocentrical ecliptical co-ordinates.

The close approach (1.5 degrees) of the planets was visible, so the D date could be considered as an actual one. The C date is calculated 395 years to future. Therefore the mistake of 9.77 degrees occurred. The Mayan astronomers have been calculating only with the approximate values of the planets synodic circulation length. The approximate values make difference of several days in the end.

E/ November 28, 859 The new moon.
F/ November 13, 859 The full moon.
G/ October 29, 859 Annular solar eclipse.
Beginning at 2.28 p.m. ephemeride time
Maximum at 5.19 p.m. ephemeride time
End 8.10 p.m. ephemeride time
The eclipse maximum was visible at 11.19 a.m. local time in the Mayan area centre (90 degrees western length; 16 degrees northern width).
The multiple of 11,960 days is added to date G. It is the interval of solar eclipses counted by the Mayan astronomers. It is divided into shorter intervals of 148 and 177 and 178 days following in the tables in certain system. They contain the gathering of the Moon synodic circulation length /29.530 588 days/ and draconic circulation length /27.212 219 days/, which is the time, when the Moon on its way around the Earth crosses the ecliptic twice, this means a half of the draconic circulation length. In newmoon the solar eclipse sets, in fullmoon the moon eclipse sets. In 11,960 days, there are 405 synodic and 439.5 draconic circulations contained.
Adding shorter intervals of 148 and 177 and 178 days to the starting date G, we get dates concerning the Sun again. But after a longer period the mistake grows to 1 or 2 days or more. The Mayan astronomers were aware of that problem, so each theoretically counted number (in the tables) could be additionally enlarged (1 or 2 days) in frame of 260-days tzolkin. Comprehensibly, not each counted solar eclipse was visible from the Mayan region. 73 solar eclipses visible from different places on Earth occurred in 11,960 days starting from day G.
Due to adding intervals of 148 and 177 and 178 days, in some cases, the Mayan astronomers made such a mistake that no eclipse occurred. In spite of that they were successful in determining 63 solar eclipses very well visible in the Mayan area during the basic cycle of 11,960 days. So their method was quite right – adding shorter intervals to the starting date G and creating tables of all seven expected eclipses. Their course, i.e. the beginning, maximum and the end, is expressed in ephemeride time. The maximum is then expressed in the local time of approximate center of the Mayan area – 90° western longitude and 16° northern latitude.
April 15,869 Annular solar eclipse.
The beginning at 4.47 p.m.
The maximum at 7.44 p.m.
The end at 10.40 p.m.

The maximum in the centre of the Mayan area at 1.44 p.m.

April 4, 870 Annular solar eclipse.

The beginning at 6.17 p.m.
The maximum at 9.22 p.m.
The end at 0.27 a.m.

The maximum in the centre of the Mayan area at 3.22 p.m.

May 16, 877 Total solar eclipse.

The beginning at 4.48 p.m.
The maximum at 7.34 p.m.
The end at 10.21 p.m.

The maximum in the centre of the Mayan area at 1.34 p.m.

March 14, 880 Annular solar eclipse.

The beginning at 3.32 p.m.
The maximum at 6.28 p.m.
The end at 9.24 p.m.

The maximum in the centre of the Mayan area at 12.28.

August 28, 881 Annular solar eclipse.

The beginning at 11.49 a.m.
The maximum at 2.48 p.m.
The end at 5.48 p.m.

The maximum in the centre of the Mayan area at 8.48 a.m.

June 26, 884 Total solar eclipse.

The beginning at 3.19 p.m.
The maximum at 6.02 p.m.
The end at 8.46 p.m.

The maximum in the centre of the Mayan area at 12.02.

July 27, 892 Annular solar eclipse.

The beginning at 11.44 a.m.
The maximum at 2.35 p.m.
The end at 5.26 p.m.

The maximum in the centre of the Mayan area at 8.35 a.m.

Precisely the Mayan eclipse cycle of 11,960 days is contained between October 29, 859 (the starting date G) and July 27, 892 (the last date). It could be compared to the Babylonian Saros of 6585.3 according to which the eclipses were foretold (since 6.century B.C.).

PAGE F 58 – 59 /D 37 – 38/
A/ 9.18.2.2.0 4 Ahau 1,426,360 days
- 12.11 - 251 days
B/ /9.18.1.7.9/ 13 Muluc 1,426,109 days
C/ 9.12.11.11.0 4 Ahau 1,386,580 days
- 1.7.11 - 511 days
D/ /9.12.10.3.9/ 13 Muluc 1,386,069 days
A/ J.D. 2,048,621 October 26, 896
B/ J.D. 2,048,370 February 18, 896
C/ J.D. 2,008,841 November 28, 787
D/ J.D. 2,008,330 July 5, 786
The multiples of (1 to 9)x78, (1 to 18)x780 and (1 to 9)x14,820 days are added to dates B and D. Any multiple of 780 could be reached by mere addition through their mutual combination.
The interval of 39,780 days is contained between dates A and C. It contains with little differences of just a few days:
68 times Venus synodic circulation length rounded to 585 days.
177 times Venus sidereal circulation length (224.700 640 days).
51 times Mars synodic circulation length rounded to 780 days.
58 times Mars sidereal circulation length (686.979800 days)
17 times Venus and Mars basic conjunction cycle rounded to 2,340 days.
153 times the holy cycle tzolkin of 260 days.
Venus synodic circulation length is moving between 577 to 592 days. Its real average length is 583.921 394 days. Seemingly irregular sling movement of the planet 542 days forwards and 42 days backwards arises from adding the movements of the Earth to the movements of Venus. Nearly the same movement is performed by Mars. Mars synodic movement length is moving between 764 and 810 days. The direct movement is about 706 days and the reverse one about 74 days. Mars average synodic circulation length is 779.936 160 days. The basic theoretical cycle of Venus and Mars conjunction cycle is 2,340 days. And again, it moves between 2,315 and 2,360 days.
The Mayan astronomers were obviously aware of Venus and Mars mutual complicated movements, so they have been working with the rounded values, which enabled them determining the planets conjunctions for a long period with little mistake. The position of Venus and Mars is expressed in degrees of geocentric ecliptic co-ordinates.
A/ October 26, 896 Venus and Mars conjunction.
Venus 176.267°
Mars 175.374°
The distance of the planets was 0.893°
The real conjunction held 2 days ago – October 24,896
Venus 173.916°
Mars 174.117°
The distance of both the planets was 0.201°.
C/ November 28, 787 Venus and Mars conjunction.

Venus 207.547°

Mars 204.765°

The distance of the planets was 2.782°
The real conjunction held 1 day ago – November 27, 787
Venus 206.870°
Mars 204.115°
The distance of both the planets was 2.755°.
In those cases the Mayan astronomers have determined Venus and Mars conjunction a day or two later. The angle distance has changed only slightly during this period, so it was not observable. Their mutual shift of distance was only 0.692° and 0.027°.
If the conjunction is held during Venus being as a morning star close to the eastern elongation with the Sun, then the next conjunction is held around the western elongation, when Venus shines as a morning star in the sky. The interval between two following conjunctions is during those constellations very short and it moves between 206 and 295 days. The average value of this scatter is 251 days and it is to be found between dates A and B.
The A date introduces Venus and Mars conjunction. At the same time Venus was after western elongation with the Sun. Since this date, the time interval of 241 days (average time from the last conjunction – date B) is counted, till Venus was shortly before eastern elongation with the Sun.
B/ February 18, 896 Venus and Mars conjunction.
Venus 10.896°
Mars 9.966°
The distance of the planets was 0.93°
The real conjunction held 2 days ago – February 16,896
Venus 8.504°
Mars 8.507°
The distance of both the planets was 0.003°.
In this case, the Mayan astronomers have made a mistake of 0.897°, of which the planets receded since the real conjunction.
In all three determined conjunctions the Mayan astronomers have made a mistake moving from 0.027°to 0.897°, of which the planets receded since the real conjunction. It is so slight difference, that the planets movement on ecliptics in such small angle distance was undiscoverable for the Mayan astronomers.
Mars and Jupiter conjunctions are in average repeating after 2.21 years. They are together around their oppositions with the Sun once in 49.14 years. It is the time of the best conditions for their observation. Repetition of triple conjunctions or close mutual approaches occurs during a short time period (tens of days) while the planets in this position.
The D date, July 5, 786 expresses Mars and Jupiter short-period maximal distance between two following conjunctions. At the same time the planets were close to their oppositions with the Sun.
The first maximal approach – May 4, 786.
Mars 256.336°
Jupiter 267.078°
The distance of both the planets was 10.742°.
D/ July 5, 786 The maximal angle distance after the first approach.
Mars 243.621°
Jupiter 260.162°
The distance of both the planets was 16.541°
The second conjunction – August 22, 786.
Mars 258.101°
Jupiter 258.314°
The distance of both the planets was 0.213°.
The Mayan astronomers have been observing the mutual conjunctions of the outer planets (Mars, Jupiter, Saturn) during their oppositions with the Sun. In these positions were the planets excellently observable. We can say the same of dating Jupiter and Saturn conjunction on page F 45 (D 74).
The table of 780 days multiples follows after the opening Mayan dates. It is slightly rounded length of Mars synodic circulation (779.936 16 days). Through the mutual combination of those multiples we can get any multiple of 780 days. The table serves to determining the next conjunctions, or close approaches, of Mars and Venus, or Mars and Jupiter.
Mars and Venus conjunction is repeating after three synodic circulations of Mars, i.e. 2.340 days. By mere adding this cycle to the B date (February 18, 896), we get the time of the next conjunction.
The mistake has increased and the planets receded more and more after every cycle because of the Mayan astronomers counting with the rounded values. In spite of that the method was usable for 185.795 years with 29 basic cycles of conjunctions. The angle distance of both the planets was after this time only 7.685°. But for calculation of the next conjunctions is the triple multiple of 780 days no longer usable.
For calculating the next conjunctions of Mars and Venus, it is necessary to change for another cycle. 140 synodic circulations of Mars are again added to the starting Mayan date B. 140 synodic circulations of Mars are 109,200 days, i.e. nearly 299 years, with 187 synodic circulations of Venus. The planets get into the near approach after this time again. For a certain time, it is possible to add a shorter basic interval of 2,340 days (3 times 780 days) to this conjunction to get their conjunctions or close approaches, before the mistake grow shows again. Out of the table of 780 days multiplies we can easily find out the multiply of 140 and 780 days and the total of 103,740 and 5,460 days is 109,200 days.
The parallel positions of Mars and Jupiter nearby their oppositions with the Sun (when they also get into mutual conjunction) are repeating every 49.14 years with 23 synodic circulations of Mars, i.e. 17,940 days. Out of the table of 780 days multiplies we can find this value by adding 14,820 days to 3,120 days, which is exactly 17,940 days. By gradual adding the multiplies of this value to the initial Mayan date D (July 5, 786), Mars and Jupiter are always reaching the opposition with the Sun, when they are good observable, but they are also situated approximately between two following conjunctions. Because of working with average values of astronomical data, the mistake grows approximately after 147.4 years.
PAGE F 61 – 64 /D 40 – 43/

A/ /?/.15.9.1.3 The Mayan date value is undeterminable.

B/ 9 Kan 12 Kayab A cyclic date repeating every 18,980 days.
The Mayan date final value in the chronology system is undeterminable.
C/ 8.16.15.16.1 1,272,921 days
- 1.4.16 - 456 days
D/ /8.16.14.11.5/ 3 Chicchan 1,272,465 days
E/ 8.16.14.15.4 1,272,544 days
- 6.1 - 121 days
F/ /8.16.14.9.3/ 13 Akbal 1,272,423 days
G/ 8.11.8.7.0 1,234,220 days
- 11.15 - 235 days
H/ /8.11.7.13.5/ 3 Chicchan 1,233,985 days
I/ 8.16.3.13.0 1,268,540 days
- 0.17 - 17 days
J/ /8.16.3.12.3/ 13 Akbal 1,268,523 days
K/ 10.8.3.16.4 1,499,004 days
L/ 10.13.13.3.2 1,538,342 days
- 7.2.14.19 - 51,419 days
M/ /10.6.10.6.3/ 13 Akbal 1,486,923 days

N/ /8/.19.0.4.4 1,288,884 days

C/ J.D. 1,895,182 September 22, 476
D/ J.D. 1,894,726 June 24, 475
E/ J.D. 1,894,805 September 11, 475
F/ J.D. 1,894,684 May 13, 475
G/ J.D. 1,856,481 October 8, 370
H/ J.D. 1,856,246 February 15, 370
I/ J.D. 1,890,801 September 24, 464
J/ J.D. 1,890,784 September 7, 464
K/ J.D. 2,121,265 September 16, 1095
L/ J.D. 2,160,603 May 30, 1203
M/ J.D. 2,109,184 August 19, 1062
N/ J.D. 1,911,145 June 6, 520
Behind the set of the Mayan dates follow the tables containing multiplies of (1 to 20)x91 days, 15x364, 20x364, 40x364, 40x364, 60x364 and 80x364 days. The record of higher dates is damaged.
The Mayan dates C, E, G, I and N are concerning Jupiter heliacal sets, i.e. the moments of the last short visibility over the western horizon shortly after the sunset. The planet was invisible for the next few tens of days because of its conjunction with the Sun. Jupiter synodic circulation length multiplies are contained between the dates E, G, I and N. It moves between 395 and 404 days. The average is 398.884 070 days.
E/ September 11, 475
38,324 days = 96 Jupiter synodic circulation length
G/ October 8, 370
34,320 days = 86 Jupiter synodic circulation length
I/ September 24, 464
20,344 days = 51 Jupiter synodic circulation length
N/ June 6, 520

C/ September 22, 476 Jupiter is 41 days before conjunction with the Sun.

The heliacal set of the planet.
The Sun set at 5.56 p.m.
Jupiter set at 7.47 p.m.
The autumn equinox.
E/ September 11, 475 Jupiter is 23 days before conjunction with the Sun.
The heliacal set of the planet.
The Sun set at 6.05 p.m.
Jupiter set at 7.06 p.m.
11 days before the autumn equinox.
G/ October 8, 370 Jupiter is 49 days before conjunction with the Sun.
The heliacal set of the planet.
The Sun set at 5.45 p.m.
Jupiter set at 8.03 p.m.
16 days after the autumn equinox.
I/ September 24, 464 Jupiter is 35 days before conjunction with the Sun.
The heliacal set of the planet.
The Sun set at 5.54 p.m.
Jupiter set at 7.29 p.m.
2 days after the autumn equinox.
N/ June 6, 520 Jupiter is 44 days before conjunction with the Sun.
The heliacal set of the planet.
The Sun set at 6.27 p.m.
Jupiter set at 8.50 p.m.
14 days before the summer solstice.

The results of observed, or into past calculated heliacal sets of Jupiter can be summarized into following conclusion:

1/ The heliacal sets set in on average 39 days before conjunction with the Sun. This corresponds to the time, when the planet is for the last time shortly visible after the sunset over the western horizon, before it gets into conjunction with the Sun.
2/ Jupiter heliacal sets always happened approximately in the prominent parts of the tropical year. Four times around the autumn equinox and once around the summer solstice.
3/ The Sun set in a very short time interval of 20 minutes, since 5.45 till 6.05 p.m. in cases C, E, G and I. Jupiter set some 57 minutes, since 7.06 till 8.03 p.m.
The Mayan dates C, E, H, L and M are concerning Saturn heliacal rises, i.e. the moments when the planet was visible for the first time in the morning sky before the sunrise. It was unobservable for few tens of days before, because of its conjunction with the Sun. Saturn synodic circulation length multiples are contained in all these data. With a slight scatter the circulation makes 378.091 900 days.
C/ September 22, 476
377 days = 1x Saturn synodic circulation length
E/ September 11, 475
38,559 days = 102x Saturn synodic circulation length
H/ February 15, 370
304,357 days = 805x Saturn synodic circulation length
L/ May 30, 1203
51,419 days = 136x Saturn synodic circulation length
M/ August 19, 1062

C/ September 22, 476 Saturn is 43 days after conjunction with the Sun.

The heliacal rise of the planet.
Saturn rose at 3.23 a.m.
The Sun rose at 5.49 a.m.
E/ September 11, 475 Saturn is 45 days after conjunction with the Sun.
The heliacal rise of the planet.
Saturn rose at 3.13 a.m.
The Sun rose at 5.47 a.m.
H/ February 15, 370 Saturn is 63 days after conjunction with the Sun.
The heliacal rise of the planet.
Saturn rose at 2.53 a.m.
The Sun rose at 6.29 a.m.
L/ May 30, 1203 Saturn is 61 days after conjunction with the Sun.
The heliacal rise of the planet.
Saturn rose at 2.30 a.m.
The Sun rose at 5.26 a.m.
M/ August 19, 1062 Saturn is 51 days after conjunction with the Sun.
The heliacal rise of the planet.
Saturn rose at 2.44 a.m.
The Sun rose at 5.45 a.m.

Two files of following Mayan dates are concerning Mercury positions close to its maximal western or eastern elongations. The G, J, C, K, H and D dates are concerning the western elongations, when the planet was rising in the eastern sky before the sunrise. The F, N and E dates are concerning the eastern elongations. Mercury set in the evening sky after the sunset. The planet was practically observable only in those positions, when it got into the maximal angle distances from the Sun. Mercury synodic circulation length multiples are contained between the dates, incidentally the synodic and sidereal circulations and the approximate tropical year conformities.

G/ October 8, 370
34,303 days = 296x Mercury synodic circulation length
390x Mercury sidereal circulation length
94x the tropical year length
J/ September 7, 464
4, 398 days = 38x Mercury synodic circulation length
50x Mercury sidereal circulation length
12x the tropical year length
C/ September 22, 476
226,083 days = 1,951x Mercury synodic circulation length
2,570x Mercury sidereal circulation length
619x the tropical year length
K/ September 16, 1095
265,019 days = 2,287x Mercury synodic circulation length
H/ February 15, 370
38,480 days = 332x Mercury synodic circulation length
D/ June 24, 475
F/ May 13, 475
16,461 days = 142x Mercury synodic circulation length
187x Mercury sidereal circulation length
45x the tropical year length
N/ June 6, 520
16,340 days = 141x Mercury synodic circulation length
E/ September 11, 475
If Mercury synodic and sidereal circulation length meets the tropical year approximate length, the maximal elongations of the planet are repeating in positions concerning (with little differences) the conformity of:
1/ the planet rises and sets time,
2/ the Sun rises and sets time,
3/ the approximate position of the planet on its ecliptics,
4/ the date of the year with difference smaller than 30 days.

G/ October 8, 370 Mercury is close to the western elongation with angle distance 17.61° from the Sun.

Mercury rose at 4.44 a.m.
The Sun rose at 5.52 a.m.
The real maximal elongation of 18.83° was on October 12, 370.
The difference between the estimated and real elongation is 1.22°.
J/ September 7, 464 Mercury is close to the western elongation with angle distance 17.27° from the Sun.
Mercury rose at 4.37 a.m.
The Sun rose at 5.46 a.m.
The real maximal elongation of 17.92° was on September 10, 464.
The difference between the estimated and real elongation is 0.65°.
C/ September 22, 476 Mercury is close to the western elongation with angle distance 14.51° from the Sun.
Mercury rose at 4.55 a.m.
The Sun rose at 5.49 a.m.
The real maximal elongation of 18.33° was on September 29, 476.
The difference between the estimated and real elongation is 3.82°.
K/ September 16, 1095 Mercury is close to the western elongation with angle distance 17.61° from the Sun.
Mercury rose at 4.39 a.m.
The Sun rose at 5.49 a.m.
The real maximal elongation of 17.94° was on September 18, 1095.
The difference between the estimated and real elongation is 0.33°.
H/ February 15, 370 Mercury is close to the western elongation with angle distance 26.08° from the Sun.
Mercury rose at 4.58 a.m.
The Sun rose at 6.30 a.m.
The real maximal elongation of 27.69° was on February 22, 370.
The difference between the estimated and real elongation is 1.61°.
D/ June 24, 475 Mercury is close to the western elongation with angle distance 19.25° from the Sun.
Mercury rose at 4.11 a.m.
The Sun rose at 5.23 a.m.
The real maximal elongation of 19.96° was on June 27, 475.
The difference between the estimated and real elongation is 0.71°.

The Mayan dates are determining Mercury western elongations on average four days sooner, before the real moment of the maximal elongation. The Mayan astronomers have made an average mistake 1.39°, which was not measurable at all because of the difficulties connected with observing this planet. The biggest departure of –3.82° can be found on September 22, 476 – the date C. Mercury synodic and sidereal circulation length and the approximate tropical year meets between the dates G, J, C and K. The determined elongations are probably results of theoretical calculations. The date K into the future, the next dates into the deep past. Therefore a bigger mistake occurred by the C date, if only average values of the planet circulation length were used. Mercury synodic circulation length vary from 104 to 132 days.

F/ May 13, 475 Mercury is close to the eastern elongation with angle distance 23.41° from the Sun.

The Sun set at 6.22 p.m.
Mercury set at 8.07 p.m.
The real maximal elongation of 23.51° was on May 10, 475.
The difference between the estimated and real elongation is 0.1°.
N/ June 6, 520 Mercury is close to the eastern elongation with angle distance 23.49° from the Sun.
The Sun set at 6.30 p.m.
Mercury set at 8.10 p.m.
The real maximal elongation of 25.15° was on May 25, 520.
The difference between the estimated and real elongation is 1.66°.
E/ September 11, 475 Mercury is close to the eastern elongation with angle distance 25.21° from the Sun.
The Sun set at 6.05 p.m.
Mercury set at 7.16 p.m.
The real maximal elongation of 25.70° was on September 6, 475.
The difference between the estimated and real elongation is 0.49°.

The Mayan astronomers determined the eastern elongation on average 5 days later, after the planets maximal distance from the Sun. They have made an average mistake 0.75°, which was due to their methods undiscoverable. The angle distance of Mercury and the Sun changes only slightly for a few days around the maximal elongations. The planet seemingly stands at one place. Therefore the Mayan astronomers could not have reached bigger accuracy.

The following time intervals a to h are added to the cyclic date B /9 Kan 12 Kayab/. We can not determine the real value of this date in the Mayan chronology system and therefore recalculate it into the Christian dating system.
a/ 4.6./1./11.3.1. 3 Chicchan 12,395,221 days
b/ 4.6./13./13.15.1 3 Chicchan 12,482,581 days
c/ 4.6.1.9.15.0 3 Kan 12,394,740 days
d/ 4.6.9.16.10.1 3 Chicchan 12,454,761 days
e/ 4.6.7.12.4.10 3 Ix 12,438,810 days
f/ 4.6.11.10.7.2 3 Cimi 12,466,942 days
g/ 4.6.9.15.12.19 13 Akbal 12,454,459 days
h/ 4.6.1.9.15.0 3 Kan 12,394,740 days
Basing on the analysis of single time sections and mutual intervals between them we found out, that they are concerning the observations of tropical year for very long time period, but it is impossible to match them with Christian calendar dates.
a/ 12,395,221 days 33,937 tropical years with +3 days departure
d/ 12,454,761 days 34,100 tropical years with –3 days departure
The interval between dates e – f contains 77 tropical years with –8 days departure.
The interval between dates g – b contains 77 tropical years with +2 days departure.
The interval between dates c – b contains 240.5 tropical years.
The interval between dates c – g contains 163.5 tropical years with -2 days departure.

The intervals between some time sections also contain some planets synodic circulations, what theoretically correspondences with their conjunctions.

Time interval of 87,360 days between dates a-b = Mars, Jupiter and Saturn conjunction.
Time interval of 72,202 days between dates c-f = Jupiter and Saturn conjunction.
Time interval of 12,483 days between dates f-g = Mars and Saturn conjunction.
The Mayan dates E, F, I, J, L and M are repeated on page F 31-32 /D 60-61/. Than follows the table of 91 and 364 multiplies, which are added to the dates.

A/ /?/ 15. 9. 4. 4 The value of the date is undeterminable.

B/ 9 Kan 12 Kayab The cyclical date repeating every 18,980 days. The final value of this date in Mayan chronological system is undeterminable.

PAGE F 69 – 73, / D 48 – 52/

A/ /?/.15.9.4.4 The Mayan date value is undeterminable.
B/ 9 Kan 12 Kayab The cyclic date repeating every 18,980 days.
The cyclic date is mentioned twice, its’ final value is undeterminable.

C/ 9.13.12.10.0 1,394,120 days

- 1.12.6 - 606 days
D/ /9.13.10.15.14/ 9 Ix 1,393,514 days
E/ 9.19.11.13.0 1,437,020 days
- 4.10.6 - 1,646 days
F/ /9.19.7.2.14/ 9 Ix 1,435,374 days
G/ 10.17.13.12.12 4 /Eb/ 1,567,332 days
H/ 10.11.3.18.14 9 Ix 1,520,654 days
I/ 8.6.16.12.0 1,201,200 days
- 4.6 - 86 days
J/ /8.6.16.7.14/ 9 Ix 1,201,114 days
K/ 8.16.19.10.0 1,274,240 days
- /9/.8 - 188 days
L/ /8.16.19.0.12/ 4 Eb 1,274,052 days
M/ /10/.14.2.16.12 4 Eb 1,541,852 days
N/ /9/.15.9.15.14 9 Ix 1,407,554 days
O/ /9/.11.11.15.14 9 Ix 1,379,474 days
P/ /9/.4.16.8.12 /4/ Eb 1,330,732 days

C/ J.D. 2,016,381 July 20, 808

D/ J.D. 2,015,775 November 22, 806
E/ J.D. 2,059,281 January 2, 926
F/ J.D. 2,057,635 July 1, 921
G/ J.D. 2,189,593 October 12, 1282
H/ J.D. 2,142,915 December 25, 1154
I/ J.D. 1,823,461 May 13, 280
J/ J.D. 1,823,375 February 17, 280
K/ J.D. 1,896,501 May 3, 480
L/ J.D. 1,896,313 October 28, 479
M/ J.D. 2,164,113 January 7, 1213
N/ J.D. 2,029,815 May 1, 845
O/ J.D. 2,001,735 June 14, 768
P/ J.D. 1,952,993 January 2, 635
The multiplies of number 54 are added to the Mayan dates ended by 9 Ix day: 1-13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 520, 780, 1,040, 1,300, 1,560, 1,820, 2,080, 2,340 and 2,600 times.
The multiplies of number 65 are added to the Mayan dates ended by 4 Eb day: 1-28, 56, 84, 224, 336, 448, 560, 672, 784, 1,008, /1,232 ?/, 1,456, 1,680, /1,904 ?/ and /2,128 ?/ times.
O/ June14, 768 4 days before the summer solstice.

All the visible planets were rising or setting close to the maximal northern declination of the Sun, i.e. in the place, where the Sun rises and sets during the summer solstice.

Declinations: Sun 23.40°
Mercury 24.41°
Venus 18.54°
Mars 23.56°
Jupiter 23.47°
Saturn 23.32°

The Mayan dates C, N and M are concerning the dates of Venus last visibility in the western sky before its lower conjunction with the Sun. Venus synodic circulation lengths (583.92139 days) are contained between those dates.

C/ July 20, 808
13,434 days = 23 Venus synodic circulation lengths.

N/ May 1, 845

134,298 days = 230 Venus synodic circulation lengths.
M/ January 7, 1213
C/ July 20, 808 Venus 8 days before the lower conjunction with the Sun.
Seen for the last time as an evening star in the western sky.
The Sun set at 6.32 p.m.
Venus set at 7.06 p.m.
N/ May 1, 845 Venus 6 days before the lower conjunction with the Sun.
Seen for the last time as an evening star in the western sky.
The Sun set at 6.18 p.m.
Venus set at 7.09 p.m.
M/ January 7, 1213 Venus 12 days before the lower conjunction with the Sun.
Seen for the last time as an evening star in the western sky.
The Sun set at 5.49 p.m.
Venus set at 7.15 p.m.

All the Mayan dates, except the date O /June 14, 768/, are concerning Mercury circulations. The basic positions of Mercury – Sun and Mercury – Earth during one synodic circulation /115.877 484 days/ are following:

The upper conjunction with the Sun
36 days
The maximal eastern elongation
22 days
The lower conjunction with the Sun
22 days
The maximal western elongation
36 days
The upper conjunction with the Sun
The intervals between Mercury positions are only approximate and can differ a few days after every finished synodic circulation with length varying between 104 to 132 days. This is caused by a great eccentricity of Mercury trajectory around the Sun.
The Mayan dates are recording Mercury positions close to the four basic positions that occur during its circulation around the Sun. They can be divided into four files:
File 1 – the maximal eastern elongation – dates L, D and G.
File 2 – the maximal western elongation – dates I and E.
File 3 – the lower conjunction with the Sun – dates C, F, P, H, M and N.
File 4 – the upper conjunction with the Sun – dates J and K.
The maximal elongations are observable when the planet sets in the longest time interval after the sunset during the eastern elongation, or rises before the sunrise during the western elongation. The lower and upper conjunction (the planet is unobservable) was probably counted by the Mayan astronomers as the middle of time intervals between two following maximal elongations. In the tables concerning Venus visibility they also counted 8 and 90 days, when the planet was unobservable, because it was around the lower or upper conjunction with the Sun – pages F 24, 46-50, /D 24-29/.
Mercury synodic circulation lengths are contained between all the dates in each file. Between some multiplies of synodic and sidereal circulation meetings (87.9693 days) with approximate tropical year length (365.242 199 days). That means approximately same mutual positions of Mercury, Sun and Earth were held after some time.
File 1 – the maximal eastern elongation - dates L,D and G
L/ October 28, 479
119,462 days = 1,031 Mercury synodic circulation lengths.
1,358 Mercury sidereal circulation lengths.
327 tropical year lengths.
D/ November 22, 806
173,818 days = 1,500 Mercury synodic circulation lengths.
1,976 Mercury sidereal circulation lengths.
476 tropical year lengths.
G/ October 12, 1282

L/ October28, 479 Mercury is close to the eastern elongation with angle distance 21.34° from the Sun.

The Sun set at 5.34 p.m.
Mercury set at 6.46 p.m.
The real maximal elongation of 22.40° was on October 22, 479.
The difference between the estimated and real elongation is 1.06°.
D/ November 22, 806 Mercury is close to the eastern elongation with angle distance 20.21° from the Sun.
The Sun set at 5.28 p.m.
Mercury set at 6.48 p.m.
The real maximal elongation of 20.30° was on November 23, 806.
The difference between the estimated and real elongation is 0.09°.
G/ October 12, 1282 Mercury is in the eastern elongation with angle distance 23.61° from the Sun.
The Sun set at 5.37 p.m.
Mercury set at 6.52 p.m.

The Mayan astronomers have made an average mistake of 0.38° in those three elongations – for them an undiscoverable mistake.

File 2 – the maximal western elongation - dates I and E
I/ May 13, 280
235,820 days = 2,035 Mercury synodic circulation lengths.
E/ January 2, 926

I/ May 13, 280 Mercury is close to the western elongation with angle distance 23.06° from the Sun.

Mercury rose at 4.19 a.m.
The Sun rose at 5.29 a.m.
The real maximal elongation of 23.33° was on May 15, 280.
The difference between the estimated and real elongation is 0.27°.
E/ January 2, 926 Mercury is close to the western elongation with angle distance 21.54° from the Sun.
Mercury rose at 5 a.m.
The Sun rose at 6.33 a.m.
The real maximal elongation of 23.55° was on
December 22, 925.
The difference between the estimated and real elongation is 2.01°.

The Mayan astronomers have made an average mistake of 1.14° in those two elongations – for them an undiscoverable mistake.

File 3 – the lower conjunction with the Sun – dates C, F, P, H, M and N.
C/ July 20, 808
41,254 days = 356 Mercury synodic circulation lengths.
469 Mercury sidereal circulation lengths.
113 tropical year lengths.
F/ July 1, 921
104,642 days = 903 Mercury synodic circulation lengths.
P/ January 2, 635
189,922 days = 1,639 Mercury synodic circulation lengths.
2,159 Mercury sidereal circulation lengths.
520 tropical year lengths.
H/ December 25, 1154
21,198 days = 183 Mercury synodic circulation lengths.
241 Mercury sidereal circulation lengths.
58 tropical year lengths.
M/ January 7, 1213
134,298 days = 1,159 Mercury synodic circulation lengths.
N/ May 1, 845
The dates are concerning Mercury positions around the lower conjunction with the Sun. The Mayan astronomers have probably determined this position with help of the visible eastern and western elongations, when Mercury is best visible. They set an approximate middle of the time Mercury was unobservable (between two following maximal elongations and around the lower conjunction with the Sun).
C/ July 20, 808 Mercury 2 days before the lower conjunction.
26 days after the eastern elongation.
19 days before the western elongation.
F/ July 1, 921 Mercury 1 day after the lower conjunction.
29 days after the eastern elongation.
18 days before the western elongation.
P/ January 2, 635 Mercury 7 days after the lower conjunction.
23 days after the eastern elongation.
17 days before the western elongation.
H/ December 25, 1154 Mercury 5 days after the lower conjunction.
21 days after the eastern elongation.
19 days before the western elongation.
M/ January 7, 1213 Mercury 1 day before the lower conjunction.
14 days after the eastern elongation.
26 days before the western elongation.
N/ May 1, 845 Mercury 1 day before the lower conjunction.
22 days after the eastern elongation.
26 days before the western elongation.
The results of the Mayan astronomers calculations could be using a statistical average method simplified into following conclusion, that is concerning Mercury position close to the lower conjunction with the Sun:
1/ 1.3 days before the lower conjunction.
2/ 4.3 days after the lower conjunction.
3/ 22.5 days after the eastern elongation.
4/ 20.8 days before the western elongation.
The average length of time to the dated position of Mercury after eastern elongation and before western elongation is 21.7 days. This precisely corresponds with the real average time length from position - eastern elongation to lower conjunction and from lower conjunction to western elongation, which lasts for 22 days on average, as shown before.
File 4 – the upper conjunction with the Sun – dates J and K.
J/ February 17, 280
73.126 days = 631 Mercury synodic circulation lengths.
K/ May 3, 480
The dates are concerning Mercury positions around the upper conjunction with the Sun. In a similar way as with the lower conjunction, the Mayan astronomers have set an approximate middle of the time Mercury was unobservable between both the maximal elongations and around the upper conjunction with the Sun.
J/ February 17, 280 Mercury 14 day before the upper conjunction.
31 days after the western elongation.
40 days before the eastern elongation.
K/ May 3, 480 Mercury 1 day before the upper conjunction.
38 days after the western elongation.
36 days before the eastern elongation.
We can gather the results of dated positions of Mercury around lower conjunction with the Sun in a statistical average:
1/ 7.5 days before the upper conjunction.
2/ 34.5 days after the western elongation.
3/ 38 days before the eastern elongation.
The time lengths average, when Mercury was accordingly to the Mayan dating in position after the western elongation and before eastern elongation, is 36.25 days. This precisely correspondences with the real time length average from western elongation to upper conjunction and from upper conjunction to eastern elongation, which is on average 36 days, as shown before.
Following time interval a is added to the cyclical date B and time interval b is probably added to date A. Because we can not reliably determine the value of dates A-B in the Mayan calendar system, it is impossible to match their total with time intervals to particular dates of the Christian calendar.
a/ 4.5.19.13.12.8 4 Eb 12,381,728 days
b/ 4.6.19.0./12./ 10 9 Ix 12,521,050 days
The time intervals are probably concerning the tropical year observations for very long time intervals.
a/ 12,381,728 days = 33,900 tropical years with –18 days mistake.
b/ 12,521,050 days = 34,281.5 tropical years.
PAGE F 43 – 44, / D 72 – 73/
A/ 9.19.8.15.0 4 Ahau 1,435,980 days
- 17.12 - 352 days
B/ /9.19.7.15.8/ 3 Lamat 1,435,628 days
A/ J.D. 2,058,241 February 27, 923
B/ J.D. 2,057,889 March 12, 922
The multiplies of (1-9)x78, (1-5)x780, 20x780, 140x780, 168x780, 194x780,
3,380, 13,000, 30,940, 69,600 and 72,800 days are added to date B.
Dates A and B are concerning Mars visibility observations.
A/ February 27, 923 Mars is 19 days after opposition with the Sun.
The planet was in position, when it is best observable around the opposition with the Sun. The declination was 19.33°. This means, it was going through the centre of Mayan area at 10.42 p.m. nearly in zenith. It set at 5.12 a.m. The Sun rose at 6.18 a.m.
B/ March 12, 922 Mars is 59 days after conjunction with the Sun.
The heliacal rise of the planet.
Mars rose at 5.32 a.m.
The Sun rose at 6.08 a.m.
During the heliacal rise Mars was for the first time shortly visible in the morning sky shortly before the sunrise. It was unobservable for few tens of days, because it was going through the area of conjunction with the Sun.
PAGE F 45 / D 74/
A/ 8.17.11.3.0 4 Ahau 1,278,420 days
- 1.10 - 30 days
B/ /8.17.11.1.10/ 13 Oc 1,278,390 days
A/ J.D. 1,900,681 October 13, 491
B/ J.D. 1,900,651 September 13, 491
(2-5)x364, 10x364, 15x364, 40x364?, 60x364? and 80x364 days are added to date B.
A/ October 13, 491 Mars is 26 days before opposition with the Sun.
The planet was in position, when it is best observable around the opposition with the Sun. The declination was 19.05°, so quite same as by the date A (February 27, 923) on page F 43-44 /D 72-73/. This means, it was going through the approximate centre of Mayan area at 2.11 a.m. nearly in zenith.
B/ September 13, 491 Jupiter and Saturn conjunction.
Jupiter 307.827°.
Saturn 306.157°.
The distance between the planets 1.67°.
Jupiter was 41 days and Saturn 44 days after the opposition with the Sun. So they were in positions, when the best visible.
The real conjunction was held on October 2, 491.
Jupiter 307.221°.
Saturn 305.645°.
The distance between the planets 1.576°.
The Mayan astronomers have made a mistake of only 0.094°, which was undiscoverable.
In the 364 days multiplies table, added to date B (September 13, 491), are as a final value marked 80x364 days, which are 29.120 days. They contain:
73 Jupiter synodic circulation lengths.
77 Saturn synodic circulation lengths.
It is the period with 19.87 +- 0.8 years four times contained, which is the basic interval of Jupiter and Saturn conjunctions repeating. By adding 29,120 days to the starting date B (September 13, 491) marking a conjunction of both the planets, we get a new date – June 5, 571, when the planets got to a close approach again, with distance 4.081°near to their oppositions with the Sun. Jupiter 46 and Saturn 43 days after the opposition.
Last published:
1) Kdy zacal Mayský kalendár ? Vesmír. Prírodovedecký casopis Ceskoslovenské a
Slovenské akademie ved, c.2/1991.
2) Calculation of the Correlation of the Mayan and Christian System of dating.
Actes du XII^e Congrés International des Sciences Préhistoriques et
Protohistoriques. Bratislava, Slovakia, September 1991.
3) Results of Mayan Astronomy in the Dresden Codex and Correlation of the Mayan
System of Dating with the Christian one. The Sections of the XIII International
Congress of Prehistoric and Protohistoric Sciences. Forlí, Italy, September 1996.
4) Mayské datování. Vesmír. Prírodovedecký casopis, císlo 10/1999.

Last modified: Zoesday, 1 Amethes 98, 17 Abrasax 95 (a.k.a. 2000-02-05 CE)