Brief history
The Maya lived and continue to live in Mesoamerica: in the Yucatan part of Mexico, as well as parts of current Belize, Honduras, and Guatemala. The period of the Classic Maya, when their culture was at its height, was from about 200 AD to 800-900 AD. During this period, Maya cities were ruled by strong kings who waged war on other cities. Aspects of the warfare, including the taking of captives, had a ritualized nature. Bloodletting ceremonies carried out by the nobels were another aspect of the Maya ritual life.
At the time of the Spanish conquest in the 1500's, most of the Maya cities had been abandoned. The Maya people and culture still survived, but in a much reduced form from the earlier Classic period. The decline appears to have occurred around 800-900 AD; it was spread over time and cannot be attributed to a single catastrophic event. It seems that the Classic Maya had a precarious existence, in which they had trouble providing for the growing population. The soil was subject to overcultivation and the Maya were highly susceptible to problems with droughts and storms (hurricanes). The collapse of the Classic Maya can be attributed to a combination of problems with supplying food for the population and the stress of frequent warfare. At the same time, the decline of the institution of kingship decreased the coherence of the culture.
During the Classic period, the ruling class employed scribes and experts in astronomy. They wrote books on a bark paper that were related to rituals and prognostications. Most of the books existing at the time of the Spanish conquest were destroyed because the Spanish believed that the books promulgated heretical beliefs. Diego deLanda, the bishop of the Yucatan, played an important role in the destruction. However, the destruction was not complete. In the 1800's, Maya codices turned up in libraries in Dresden, Paris, and Madrid. They were generally in poor condition, but the Dresden Codex survived fairly well and it has a significant astronomical content. In the twentieth century fragments of another book that had been pillaged from a Maya site turned up. It is known as the Grolier Codex.
The interpretation of the Maya documents and other written information started in the 1800's. Rapid progress was made on the Maya number system, their calendar system, and the astronomical information in codices in stelae. However, most of the the Maya glyphs that appear on stelae and other monuments were not initially deciphered. For many years, the assumption was that all the Maya writing was concerned with astronomy and mathematics. In the last part of the 20th century, progress was made on deciphering the glyphs, using a knowledge of current Maya languages. The result was that the writing is generally not concerned with astronomy and mathematics, but with the warfare undertaken by the Maya kings and with occasions for the transfer of power.
Number system
The Maya number system had some attributes that are similar to our own that make it especially practical for calculations. The properties are use of a place value and the concept of zero. Our own number system has these properties in that, e.g., 203=2x10^2 + 0x10^1 +3x10^0. The place of the digit determines the power of 10 that it multiplies, so that large numbers can be expressed in a compact form. The zero is necessary in the place value system to show that there is nothing times a particular power of 10.
Our number system is decimal in that it is 10 that is raised to various powers. The Maya system used a base 20 and was thus a vigesimal system. They used a place value notation and had a symbol for zero, sometines in the form of a shell. The basic structure is thus much like our system, although the numbers are written vertically and use bars and dots, so that the appearance is quite different from our numbers.
The place value number system is convenient for addition and subtraction, and there is ample evidence in the the Dresden Codex for addition up to large numbers. There is no evidence that they carried out multiplication and division, or that they developed the use of fractions. When they wanted to express a number with a fractional quantity, such as the number of days in a lunation (synodic month), they expressed it as some large number of lunations equals some large number of days.
When dealing with days, which applies to all the easily accessible Maya writing, there was a deviation from the straight base 20 system. Their word for day (or sun or time) was kin. There were 20 kin in one uinal. However, there were 18 uinals in one tun. This was evidently so that one tun would be 360 days and would thus be close to one year. This modification of the system adds a level of complexity in making calculations. However, it allows the simplification that the number of tuns is approximately the number of years.
Calendar and Dates
The Maya are known for the complex information that was included with a date on a monument. They appeared to have an obsessive interest in the passage of time and the variety of influences on a particular date.
The first part of a date was the position in the Long Count, measured as the number of days from a zero point 0.0.0.0.0. The corresponding Christian date depends on the correlation with the Maya calendar. For the standard GMT (Goodman-Martinez-Thompson) correlation, the date in the Gregorian calendar is August 11, 3114 BC. In the GMT correlation, the day 0.0.0.0.0 occurs on Julian Day 584,283. The five numbers in a Long Count date are baktuns, katuns, tuns, uinals, and kins. The system uses a base 20 except that 1 tun = 18 uinals. A baktun is thus 400 tuns (about 400 years) or 144,000 days. Most of the dates from the Maya Classic period are from baktun 9.
The Maya believed that a creation cycle takes 13 baktuns, where the number 13 probably comes in because of its use in the ritual 260 day cycle. At the end of each cycle, a catastrophic event would occur, probably a great flood. The last page of the Dresden Codex depicts such a great inundation. The current cycle of 13 baktuns ends December 21, 2012, in the Gregorian calendar. However, this cycle was apparently embedded in a larger cycle of 13 piktuns (1 piktun = 20 baktuns). A stela at Coba shows increasingly large cycles, going up to 13 cycles of 20^21 tuns. Rather than referring to some event in the distant past, the stela appears to refer to vast stretches of time which dwarf the lifetime of a human being. The time is much larger than the time scientists currently believe is the age of the universe since the Big Bang.
The Long Count date played a role in ceremonial occasions in that many of the dates of stelae are katun-ending dates (ending in 0.0.0) or half-katun-ending dates (ending in 10.0.0). These may have been auspicious days for special events to take place.
Other cycles of time played important roles in Maya timekeeping. The ritual, or tzolkin, cycle of 260 days was made of a number (1-13) and a day glyph (one out of 20). In advancing one day at a time, both the number and day name advanced. The position in the tzolkin was clearly important for ritual life because much of the Dresden Codex, a book of divination, relates to particular days in this cycle. Use of the 260 day cycle was widespread in Mesoamerica. It first appeared in the 6th century BC and its use persists to the present time.
Another time cycle that commonly appears on dates is the position in the haab or vague year. This cycle of 365 days is clearly intended to be an approximation to the solar year. The cycle is made up of 18 "months" of 20 days, plus five extra days (uayeb days). The uayeb days were thought to be unlucky. The days advanced in the same way as our days and months, unlike the tzolkin. The cycle was a more accurate approximation to the year than the tun, but it would become seriously in error over a period of many hundreds of years, which is the period over which it was used. Information from after the Spanish conquest shows that religious festivals were celebrated in certain months. Some of these events, such as the coming of rain and the time to harvest bee honey, were clearly seasonal events. It is not known what was done about the inaccuracy of the vague year, which must have been recognized.
The combination of the tzolkin and the vague year make up the Calendar Round. From the lowest common multiple of the number of days in the tzolkin and the vague year, it is possible to show that same position in the Calendar Round is reached every 52 vague years. When the starting day in the Calendar Round was reached, it was a time for special celebration. We know that the day 0.0.0.0.0 occurred on 4 Ahau 8 Cumku in the Calendar Round. All other dates that have been found are consistent with the tzolkin and haab year date keeping step with the Long Count from that initial point.
Another cycle that appeared on dates is associated with glyph G, the Lord of the Night. There were 9 Lords of the Night and the days continually cycled through them. There is some evidence that their influence on a day alternated between positive and negative. The glyph G9 is especially common because it always occurs on a katun-ending date.
Finally, a number of glyphs gave information about the lunar period. One glyph (D or E) give the lunar age in the synodic month from some reference point. Two glyphs are needed because the number can be 20 or larger and the Maya just put a single digit connected with each glyph. The reference point for the age is not known with certainty. Aveni suggests that it is the new moon, but this is a phase of the moon that is not directly observable. Another glyph (C) gave the number of the lunation in a set of 6 lunations. The appearance of 6 lunations, and its appearance in the Dresden codex, suggests a relation to eclipses. Another lunar series glyph indicates whether the particular lunation has 29 or 30 days.
Tropical year
The Maya treatment of the tropical year remains something of a mystery, perhaps because fragmentary information is all that is available on the Classic Maya. The haab or vague year of 365 days has a degree of inaccuracy that should have become evident during the Classic Maya period. Information from the time of Spanish conquest indicates that they were aware that 365 and 1/4 days is a much better approximation to the tropical year. There have been claims that the more exact tropical year estimate appears in the Maya writings, but the available evidence is not convincing.
Eclipses
The Maya view of eclipses is based on the eclipse tables of the Dresden codex. The table includes intervals of 177 and 148 days, which can be identified with 6 lunations and 5 lunations. These numbers come into eclipses because they are the number of lunations between when the Sun/Earth/Moon line up in the way that is necessary for eclipses. The times refer to either the times between lunar or between solar eclipses, so the table must refer to one or the other. The time between a lunar eclipse and solar eclipse must involve 1/2 lunation, the time that the Moon takes to go from full moon to new moon. This time, about 15 days, does not appear in the Dresden codex tables.
Early commentators, such as Thompson, argued that the tables are concerned with solar eclipses. However, the time between solar eclipses being observable at one spot is long, so it is unlikely that observers would be able to find periodicities associated with solar eclipses. Lunar eclipses are more frequent, although it is still not clear how the table was constructed. The times between the pictures (apparently eclipse symbols) cannot be identified with the times between eclipses that were observable during the Maya era. However, the connection between the tables and eclipses is compelling. The time between lunar eclipses depends on a combination of it being full moon and the Sun/Earth/Moon lining up. The latter event depends on the line of nodes lining up with the Sun, which occurs every eclipse half-year, or 173.3 days. The time is less than half of a tropical year because of the regression of the line of nodes. The average time between the events listed in the Dresden codex, which covers about 33 years, is 173.3 days. The correspondance to the eclipse half-year demonstrates a knowledge of the timescale involved in eclipses, even if we do not know how they obtained it.
Venus
Our primary evidence for Venus observations is also from the Dresden codex. There are 5 pages giving the lengths of times for the appearance and disappearance of Venus (because of being lost in the glare of the Sun). The times of appearance are somewhat different from each other, although, on average, we would expect them to be the same. They may have done this so that the days fit better with their ritual cycle. A main purpose of the Dresden codex was to describe times for ritual in the tzolkin, or ritual, cycle. There is evidence that the Maya followed Venus in order to decide on times to carry out warfare, but there is conflicting evidence as to whether Venus as an evening star or morning star was more auspicious for war. Among the modern Maya, there is still concern at the time of heliacal rising of Venus.
Zodiac A zodiac provides for the identification of stellar groups along the ecliptic. As such, it provides a means by which the positions of the Moon and planets can be recorded. This would be necessary for determining phenomena such as the retrograde motion of Mars.
Calendar correlation problem As described in Aveni, there were several steps in determining the calendar correlation.