Mesoamerican Math and Calendars (300-1600)

For those of us raised in the scientific traditions of Europe, mathematics may seem to be the epitome of rationality, the intellectual opposite to religion and spirituality. Yet for many Mesoamerican groups, computation, arithmetic and accurate astronomical measurement were practiced as part of the religion because time itself was the omnipresent manifestation of the divine. To solve the mathematical problems that made time comprehensible within the Mayan cosmology was not only a pursuit of knowledge, but a pursuit of wisdom and cosmic truth.

The mathematics of the Mayan world can be classified by modern analysts as possessing both practical and religious elements, but it is important to remember that the two were inextricably linked in the world view of Mayan sages. Mayan innovations in what we would readily call "science" are numerous, the most notable of which are the number zero and their calendar. The existence of the zero, which first appeared in Europe (via India and the Islamic world) hundreds of years after its invention in the Yucatan Peninsula, where the Maya lived, allowed sophisticated results from mathematical procedures like multiplication, division, and square roots, unable to be performed in Europe un the zero was introduced in the twelfth century .

The Mayan calendar remains one of the most remarkable innovations of the ancient world. They defined the "vague year" as a 365.2420 day period (which was only .0002 days off of modern measurements). Yet this solar calendar was only half of the full calendar, which included a 260 day lunar calendar, a ritual almanac that conveyed the relationship of specific gods and goddesses to certain days. In fact, according to J.E.S. Thompson, each day was itself a living god and time itself was carried by the gods. An accurate calendar of the 260 day cycle thus became an obsession among Mayan cosmologists as it was necessary to correlate astronomical observations with the ritual calendar. To accomplish this, elaborate tables were created for the cycles of the moon and the synodical revolutions of Venus, like those seen in the Dresden Codex (Thompson 1974). Included among these tables were appendices to account for necessary corrections, and the sum result was an almost totally accurate temporal calendar.

Mathematics also permitted the measurement of time over the largest scale, both into the future and the past and, as the Maya had no definite beginning or end for time, they calculated far in both directions. Calendrically correct dates have been found on stellae for days that would have occurred over 400 million years ago and their astronomical tables were pushed well into the future, allowing the prediction of eclipses and other cosmic phenomena. The largest temporal cycle is a period of 374,440 years in which no date is repeated. The focus on such immense periods reflects just how central time and its measurement were to the classical Maya.

Entire cities were even arranged to coincide with astronomical cycles. Commercial and ritual centers like Teotihuacan in Central Mexico and Chichen Itza in the Yucatan peninsula were laid out to reflect the divine order of celestial motions, and several cities constructed observatories for better measuring these phenomena.

Although the Aztecs are less renowned for their mathematics than their Maya predecessors, they too used a sophisticated calendar that had its roots in more ancient Mesoamerican civilizations, most notably the Olmecs. With the European conquest of Mexico, however, Christian missionaries found it necessary to annihilate indigenous time systems in order to convert the natives to Christianity because calendars were central to the religion and rituals of the Aztec religion. As Christianity and Spanish rule became more entrenched, Mesoamerican societies were stripped of the institutions and professionals that produced and promulgated their mathematics and, despite being far less accurate, Christian time and counting replaced those of Mesoamerica. 

Mayan mathematics is an example of "non-Western science" that contradicts Eurocentric ideas about what science is and who excels at its implementation. By wedding mathematics with religion and demonstrating excellence in its execution, the Maya blur lines between "center and periphery" in the history of science.

Questions for further exploration:

1. How do religion and astronomy merge in the Dresden Codex? You will need to consult secondary sources to posit a convincing answer.

2. Why would Christian missionaries spend so much effort interpreting Mesoamerican calendars, such as the interpretive wheel created by Diego Valades (in this topic's sources)? Was it to destroy culture, proselytize, preserve indigenous curiosities, or genuine scholarly interest? Make an argument based on contemporary context.

3. Ethnomathematics and archaeoastronomy are umbrella terms classifying science in traditions other than that of the modern West. By differentiating between "our" science and "their" science, these terms question the universality of science. Using the Maya as an example, argue for or against mathematics as universal.

4. Compare and contrast how the Maya used mathematics and astronomy with another Indigenous American civilization (Olmec, Aztec, Inca, Navajo, etc.). How did science contribute to the articulation and reinforcement of their worldviews?

Further reading:

Anderson, W. French. "Arithmetic in Maya Numerals." American Antiquity. 36: 1 (January 1971): 54-63.

Ascher, Marcia. "Before the Conquest." Mathematics Magazine. 65: 4 (October 1992): 211-218.

Aveni, Anthony F. Skywatchers of Ancient Mexico. Austin: University of Texas Press, 1980.

Hassig, Ross. Time, History, and Belief in Aztec and Colonial Mexico. Austin: University of Texas Press, 2001.

Leon-Portilla, Miguel. Time and Reality in the Thought of the Maya. Second ed. Norman: University of Oklahoma Press, 1988.

Malmstrom, Vincent H. Cycles of the Sun, Mysteries of the Moon: The Calendar in Mesoamerican Civilization. Austin: University of Texas Press, 1997.

Milbrath, Susan. Star Gods of the Maya: Astronomy, Folklore, and Calendars. Austin: University of Texas Press, 1999.

Paxton, Meredith. The Cosmos of the Yucatec Maya: Cycles and Steps from the Madrid Codex. Albuquerque: University of New Mexico Press, 2001.

Read, Kay Almere. Time and Sacrifice in the Aztec Cosmos. Bloomington: Indiana University Press, 1998.

Rice, Prudence M. Maya Calendar Origins: Monuments, Mythistory, and the Materialization of Time. Austin: University of Texas Press, 2007.

Thompson, Sir John Eric Sidney. "Maya Astronomy." Philosophical Transactions of the Royal Society of London, Series A, Mathematical and Physical Sciences. 276: 1257, The Place of Astronomy in the Ancient World (May 1974): 83-98.

Aztec Calendar Stone

Date: 1510
Owner: Beinecke Rare Book and Manuscript Library, Yale
Source Type: Artifacts

 

This Aztec calendar stone, famous for its well-preserved condition and what it tells us about Mesoamerican time keeping, was made in the early sixteenth century and was probably placed flat in the Great Temple of Tenochtitlan and used as a platform for human sacrifice. The face in the center, considered to be that of Tlalteuctli, is framed by the calendrical glyphs of the four previous "suns," divisions of cosmological epochs. The band of glyphs directly outside of the face and sun symbols is a band containing the twenty day glyphs of the Aztec calendar, itself placed within another ring with images of flowing blood, jade, and sun rays. The outermost ring of this calendar stone depicts the bodies of two serpents, with their tails meeting at the top and heads at the bottom. Between the tails (at the top of the stone) is the glyph 13 Atlatl, which may represent the first year of the reign of King Itzcoatl (1427), founder of the Aztec empire. The heads of Xiuhteuctli, god of fire and time (left), and Tonatiuh, the sun (right), are shown coming out of the serpents' mouthes at the bottom of the stone (Hassig 2001).

The fact that the Aztec calendar, much like the calendar of the Maya, was based on repeating fifty-two year periods and usually took the shape of a circle or wheel has led many scholars to emphasize how the Aztecs perceived time as cyclical. Yet, anthropologist Ross Hassig has pointed out that scholars support this idea of cyclical time because of myths promulgated by the Aztecs themselves who, like all other advanced civilizations, actually experienced and noted time as both linear and cyclical. Cycles are manifest in yearly life (agriculture, festivals, and seasons), while historical events, like wars and politics, are usually represented as unique and thus non-cyclical. 

The genius of the Aztecs is that they manipulated time, using the calendar as a means of imposing control over their empire. Time became an instrument of control over their numerous tributaries. For example, they demanded tribute on calendrical days established as important to them and thereby undermined local time keepers and forced provinces into a rythm where yearly events became subject to Aztec time. By downplaying the linearity of political time and increasing the perception that time was heavenly and cyclical, the Aztecs were able to better legitimize their right to rule, not unlike their European and Asian counterparts at the time.

Reference: Hassig, Ross. Time, History, and Belief in Aztec and Colonial Mexico. Austin: University of Texas Press, 2001.
 


CITATION: Aztec Calendar Stone, Mexico City, no. 51150. Courtesy of the Beinecke Rare Book and Manuscript Library, Yale University. Call number: WA Photos 121.

DIGITAL ID: 13079

 

Diagram of Aztec Calendar

Date: 1580
Owner: John Carter Brown Library, Brown University
Source Type: Images

DIGITAL ID: 13037

Dresden Codex

Date: 1250
Owner: FAMSI
Source Type: Publications

 

The Dresden Codex was painted on amatl paper in the thirteenth century. As one of only three Mayan codices to survive the Spanish conquest and the ravages of time, the Dresden (named for where the original is housed) is an invaluable source on Mayan culture and civilization, but it is perhaps most well known as a work of astronomy and mathematics. The Dresden includes charts based on hundreds of years of observations and predictions both of eclipses (pages 51-58) and the cycles of Venus (pages 46-50).

(N.B. In this and all other editions of the Dresden, pages 46-74 are actually arranged on pages 25-53, thus the Venus tables appear on the pages numbered 25-29. The introduction to the Venus tables is found on page 24.)

The Venus tables contain a complicated series of numerals, glyphs, and gods that are interrelated but often evade accurate interpretation. Some aspects of it, though, are fairly certain. For example, the drawings on the right hand side of each of the five Venus tables are based on common themes. On each page, a regent deity holding an overturned jar (top of the page) observes a manifestation of the Morning Star, Venus (a different manifestation on each page), who is using an atlatl (a weapon to throw darts) to spear a victim (the victims are the bottom image on each page). Each spear-throwing Venus lords over about 1/5th of a given year (depending on celestial cycles) and the deities he is impaling might represent constellations, planets, or stars in retrograde motion during a particular Venus ascendancy. Each of the five pages notes Venus' cycles for a period of 584 days, and the total days covered in the five pages (2,920 days) are equal to the eight solar years and five synodic cycles of Venus. The glyphs and symbols throughout the pages relate both the "long count" time and the periods of particular movements.

Looking at these astronomical charts (and the rest of the Codex) exemplifies just how connected religion, math, space, and time were for Mayan mathematicians.  Scholars may never fully understand the significance of much of these writings; due to the god-like power of Time, which was rightly held in awe, much of the historical record has been lost, obliterating the full meaning of much Mesoamerican cosmology.

(There is far too little space here to do justice to the mathematic and iconographic intricacies of the astronomical tables in the Dresden Codex, especially those relating to Venus. 

Refernce: for an excellent and thorough summary of the Dresden, see: Susan Milbrath, Star Gods of the Maya: Astronomy in Art, Folklore, and Calendars (Austin: University of Texas Press, 1999), pp. 113-115 and 163-177.)
 

CITATION: Dresden Codex. Published by Ernst Forstemann, 1880. Courtesy of the Foundation for the Advancement of Mesoamerican Studies, Inc.

DIGITAL ID: 13081

 

Mayan Stela

Date: 700
Owner: Library of Congress
Source Type: Artifacts

 

This eighth-century Mayan stele, photographed here by some early twentieth century explorers from the United States, is located in Quirigua, a once thriving city in present day Guatemala. The front of this stele has a carving of a Classical era Mayan king, but its most fascinating aspect is the sequence of glyphs on its side. Quirigua's steles are renowned for their "Long Count" inscriptions, which are a series of glyphs containing precise calendrical dates for the past as well as future events.

Because civil time was reckoned in cycles of fifty-two "vague" (365.2420 day) years, it was necessary for them to create a chronology that existed outside of this period in order to record time over the very long term. Thus Long Counts are often found on monuments like this that were meant to record significant dates for posterity. Other steles at Quirigua include dates from eras that could not have possibly been known to the Maya, including one from 400 million years ago. Yet the calendrical dates given for these mythic times were (and are) precise, another testament to the abilities of Mesoamerican cosmographers to observe and calculate the passage of time.

The long count dates are told in a series of glyphs like the ones seen on the left-hand side of this stele. To calculate the date, these glyphs are read from the top down and from left to right. At the top is an introductory glyph that relates the year and the god associated with the twenty-day month that that glyph records. Below this are listed the date's count within various multi-day cycles: the baktuns of 144,000 days, katuns of 7,200 days, tuns of 360 days, uinals of 20 days, and, finally, the days, or kins (there are 20 kins per uinal). The Long Count date is then achieved by taking the sum of these counts, a time equal to the total number of days passed since the beginning of the last Great Cycle. These were periods of 374,440 years (or, in Mayan reckoning, 13 baktuns), the length of time necessary for every single permutation of the long count system to be realized once and only once.

Reference: Leon-Portilla, Miguel. Time and Reality in the Thought of the Maya. Second Edition. Norman: University of Oklahoma Press: 1988.

CITATION: Guatemala, Quirigua. [Between 1908-1919]. Library of Congress Prints and Photographs Division: LC-F81-2705 [P&P].

DIGITAL ID: 13084

 

Source References

Web Sites

Maya Civilization- Cosmology and Religion  (Canadian Museum of Civilization Corporation): Descriptions of Maya beliefs and incorporation of cosmology and religion into their lives.


Maya Civilization- Astronomy (Canadian Museum of Civilization Corporation): Description of Maya contributions and spiritual connections to astronomy.

Maya Civilization- The Maya Calendar (Canadian Museum of Civilization Corporation): Description of the Maya-devised calendar and the spiritual significance of the numbers in the calendar.

The Maya Astronomy Page (Michielb): Explanation of the numbers and symbols used in Maya mathematics and astronomy.

The Maya Calendar (Maya World Studies Center): An in-depth description of the Maya Calendar, including historical background information and the day's date and significance on the Maya calendar.

The Maya Mathematical System (Maya World Studies Center): Detailed explanation of the Maya mathematical system, including explanation of symbols and Maya names for numbers.

Indigenous Mathematics of Central and South America- Mathematics and the Liberal Arts (Truman State University): Bibliography and accompanying descriptions of books and articles related to the history of mathematics in Latin America.

Mayan Mathematics (University of St. Andrews, Scotland)

Publications

Anderson, W. French. "Arithmetic in Maya Numerals." American Antiquity. 36: 1 (January 1971): 54-63.

Ascher, Marcia. "Before the Conquest." Mathematics Magazine. 65: 4 (October 1992): 211-218.

Aveni, Anthony F. Skywatchers of Ancient Mexico. Austin: University of Texas Press, 1980.

Barsh, Russel Lawrence. Counting, Computation, and the Calendar in Mesoamerica. Brooklyn, NY: Brooklyn Children's Museum, 1965. 

Burke, James. Circles. New York: Simon & Schuster, 2003.

Calinger, Ronald S. A Contextual History of Mathematics. Englewood Cliffs, NJ: Prentice-Hall, 1999. 

Closs, Michael P. Native American Mathematics. Austin: University of Texas Press, 1996.

Hassig, Ross. Time, History, and Belief in Aztec and Colonial Mexico. Austin: University of Texas Press, 2001.

Hoffman, Mary Ann. History of the Maya: Using Computational Skills in Problem Solving. New York: PowerKids Press, 2005.

Leon-Portilla, Miguel. Time and Reality in the Thought of the Maya. Second ed. Norman: University of Oklahoma Press, 1988.

Malmstrom, Vincent H. Cycles of the Sun, Mysteries of the Moon: The Calendar in Mesoamerican Civilization. Austin: University of Texas Press, 1997.

Milbrath, Susan. Star Gods of the Maya: Astronomy, Folklore, and Calendars. Austin: University of Texas Press, 1999.

Montgomery, Roger. Twenty Count: Secret Mathematical System of the Aztec-Maya. Santa Fe, NM: Bear & Co, 1995.

Paxton, Meredith. The Cosmos of the Yucatec Maya: Cycles and Steps from the Madrid Codex. Albuquerque: University of New Mexico Press, 2001.

Read, Kay Almere. Time and Sacrifice in the Aztec Cosmos. Bloomington: Indiana University Press, 1998.

Rice, Prudence M. Maya Calendar Origins: Monuments, Mythistory, and the Materialization of Time. Austin: University of Texas Press, 2007.

Thompson, Sir John Eric Sidney. "Maya Astronomy." In Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 276, no. 1257, The Place of Astronomy in the Ancient World (May 2, 1974), p. 83-98.

Willson, Robert Wheeler. Astronomical Notes on the Maya Codices. London: Corinthian Press, 1940.

Films and Videos

Maya: Lords of the Jungle (PBS Home Video, 1997)


Maya: The Blood of Kings  (Time-Life Video & Television, 1995)

Teotihuacan

Date: 150
Owner: Jack Hynes
Source Type: Buildings

This photograph shows the ruins of Teotihuacan, a city built near the valley of Mexico that thrived from around 200 BCE to 700 CE. Although it remains unclear which Mesoamerican civilization built this city, Teotihuacan was a massive center of ritual, commerce, and political control and was larger than any contemporary urban center in either the Americas or Europe. Even the original name of the city is unknown; although the city was already a ruin by the Aztecs' arrival, they named it Teotihuacan ("place of the gods") because of the amazing size of its pyramids, especially the Pyramid of the Sun (seen in the distance in the left of this picture).

What is known, however, is that like many Mesoamerican urban centers, Teotihuacan is laid out with geometric precision to correspond with calendrical and astronomical standards. The "Street of the Dead," the broad avenue running through the center of this photo, aligned the entire city grid on an axis that was offset from the true cardinal directions by 15.5 degrees. Even the suburbs, countryside, and chinampas (see the Agriculture and Science topic) around the metropolis followed this directional orientation.

Anthropologists and astronomers have offered several compelling explanations for why Teotihuacan would be offset at the seemingly arbitrary 15.5 degrees. Some older theories suggest that it was arranged to coincide with the setting of the Pleiades in the year 150 CE (when the city was thought be have been laid out) or that the Street of the Dead lines up with the setting sun on the day when the zenithal sun passes over (this theory, though, involved errors in reckoning the site's latitude). Geographer Vincent H. Malmstrom posited the compelling idea that the offset was based on a relationship between the setting of the sun across from the Pyramid of the Sun, Izapa (an ancient city in southern Mexico), and the Mesoamerican calendar. Based on the sun's azimuth at Teotihuacan (at 19.5 north latitude), he found that the sun would have passed directly over Izapa (at 14.8) on August 13, the day when the Maya believed the world began. Malmstrom thus argues that the specific angular layout of Teotihuacan intentionally referenced the Olmecs, the ur-civilization of Mesoamerica that spread its culture and calendar throughout the region. Although Teotihuacan was built 1000 miles from Izapa and 1000 years after the Olmecs, it reflects the cosmological continuity that mathematics and the calendar gave to millennia of Mesoamerican cultures. The difficulty of conducting (or even explaining) such calculations today bears witness to the genius of ancient Mesoamerican mathematicians.

Reference: Malmstrom, Vincent H. Cycles of the Sun, Mysteries of the Moon: The Calendar in Mesoamerican Civilization. Austin: University of Texas Press, 1997.
 

CITATION: Jack Hynes. View of Avenue of the Dead and Pyramid of the Sun from the Pyramid of the Moon. 26 May 2006. Public Domain.

DIGITAL ID: 13080

The Mayan Numbers

Date: 200
Owner: Immanuel Giel
Source Type: Images

 

This chart helps to simplify Mayan counting, the first step to deciphering the deep and manifold meanings found in Mayan writings. Although both head-variant numerals and full-figure glyphs were also used to represent numbers and days, this system was the basis for calculations and numerical charts, such as those found in the Dresden Codex, and probably predated more complicated counting schemes.

The Maya only had three symbols with which to express numeric value, the dot (=1), the bar (=5), and the zero glyph. This chart shows how these symbols could be combined to make the numbers 1-20, the basis of Mesoamerica's vigesimal system (just as the modern West uses a decimal system based on multiples of 10, indigenous Mesoamericans based counting on sets of twenties). Combinations of numbers 0-20 would be stacked vertically to create larger numbers. The bottom layer would have a number like those seen on this chart for which place value is already assigned. Each upper layer is then multiplied by place value factors of 20. Thus the second layer (consisting of a number 0-20) was multiplied by twenty, the first place factor in a vigesimal system. The third layer's number was then multiplied by 20 twice (or 400), the fourth layer by 20 to the third power (or 8000), etc. This system may seem overly complex, but it is no less natural or intuitive than modern counting systems and would have been easy to manipulate for those accustomed to it.

The number zero was most likely "invented" by the ancient Olmecs and is one of the most advanced mathematic concepts found anywhere in the pre-modern world. The graphic representation of the absence of numeric value is not intuitive, but inventing a way to hold place value was necessary for advanced mathematics or calculating large numbers (like the days of the Long Count). Thus Mayans could write the number "60" simply by placing 3 (three dots) in the second layer (3x20=60) and a zero in the bottom layer. The top and bottom layers are then added together to get the total sum: 60+0=60.

A description of how to read a more complex number might prove useful for better understanding Mayan counting. Let's say there is a glyph with 3 layers, the highest is 11 (2 bars and 1 dot), the second layer is 8 (1 bar and 3 dots), and the bottom layer is 7 (1 bar and 2 dots). The third layer, 11, must be multiplied by 20 twice (or, 400), which equals 4400. The second layer, 8, must be multiplied by 20 once, which equals 160. The bottom layer is not multiplied by anything, and thus remains 7. These 3 sums are then added together to calculate the total numerical value of the 3 layer symbol: 4400+160+7=4567. See if you can draw this and other numbers out in Mayan symbols.

Reference: Barsh, Russel Lawrence. Counting, Computation, and the Calendar in Mesoamerica. Brooklyn, NY: Brooklyn Children's Museum, 1965.

Leon-Portilla, Miguel. Time and Reality in the Thought of the Maya. Second ed. Norman: University of Oklahoma Press, 1988.


CITATION: Immanuel Giel 14:47, 21 December 2006 (UTC). Public Domain.

DIGITAL ID: 13083