How Accurate Were Maya Astronomical Predictions?

Maya astronomical predictions were accurate by the standards of pre-telescopic naked-eye astronomy, and they were especially good for Venus and the Moon. The arithmetic was tabular — built on a place-value number system described in the page on Maya number system and the concept of zero — and the accuracy figures below come from comparing the Maya tables to modern ephemerides and to the modern values for the synodic periods of the planets, the lunar month, the tropical year, and the eclipse nodes. The page on Maya astronomy, stars, planets, and eclipses sets the broader context, and the Maya Calendar System overview describes the calendrical structures that the astronomical tables were designed to support.

Venus: the most accurate Maya table

The Venus table in the Dresden Codex uses a 584-day mean synodic period for Venus, with a 5-cycle count of 2,920 days (5 × 584) and an 8-year integration into the 365-day haab’ (8 × 365 = 2,920). The mean synodic period of Venus, by modern measurement, is 583.92 days. The 5-cycle average of 583.92 days is therefore 2,919.6 days, an amount that is 0.4 days short of the Maya’s 2,920 days. The 4-day correction page in the Dresden Codex, applied every 5 cycles, brings the running of the table down to the true 5-cycle average.

The actual heliacal-rising accuracy of the Maya table, calculated against the moments of the heliacal rising of Venus as seen from a lowland Maya site, is on the order of 1 to 2 days per cycle. Over a 40-year span, the accumulated error is approximately 10 days, which the 1-day-per-4-years correction in the Dresden Codex brings back to within 1 to 2 days. The accuracy is comparable to the Greek and Babylonian Venus systems and is more than adequate for the practical purpose of timing a war or an inauguration.

The heliacal rising of Venus is sensitive to atmospheric conditions and the precise horizon, and an observer at a lowland site can see Venus first appear anywhere from 5 to 10 days before or after the astronomical “exact” heliacal rising. The Maya table gives the date of the rising within about 1 day, and the actual observation sharpens that to within 1 to 2 days of the true rising. The system is the most accurate of all the Maya astronomical tables and the one most often cited as evidence of a high level of observational skill.

The Moon: the 11,960-day eclipse cycle

The lunar tables in the Dresden Codex are built on the 11,960-day, 405-lunation, 46-month eclipse cycle. The mean synodic month by modern measurement is 29.530588 days, and 405 × 29.530588 = 11,959.89 days, very close to the Maya 11,960. The Maya use the round number 11,960 = 6 × 1,993.33… of the half-month, and the 6-month, 1,993-day and 12-month, 3,986-day cycles appear as the operating basis of the lunar tables.

The accuracy of the lunar tables is, in detail, on the order of 1 to 2 days over the 11,960-day cycle. The half-month of 1,993.33 days (6 synodic months) is, by modern measurement, 177.18 days; the Maya use a round 178. The full 6-month, 1,993-day and 12-month, 3,986-day counts include corrections that bring the running down to the true value. The corrections are described in detail in Maya astronomy, stars, planets, and eclipses.

The eclipse warning system is built on the 11,960-day cycle. A solar or lunar eclipse is possible only when the Sun and Moon are within about 18 degrees of a lunar node, and the 11,960-day cycle returns the same node configuration after 32.75 years. The Maya warning pages, on the eclipse table of the Dresden Codex, list specific Long Count dates on which an eclipse was possible, plus the corresponding tzolk’in and haab’ positions. A warning issued at the start of a 78-year forward search would identify the correct eclipse season within an 18-day window, with the actual eclipse occurring on a date that the observer could narrow down by sighting the Moon’s position. The system is comparable in accuracy to the Greek Saros cycle (18 years, 11 days, 8 hours = 6,585.32 days) and to the Chinese prediction system of the same period.

The Sun: the 365-day year and its drift

The haab’ solar year is exactly 365 days, while the true tropical year is approximately 365.2422 days. The haab’ therefore drifts from the tropical year by about 0.2422 days per annum, or one full day in roughly 4 years. Over a 52-haab’ Calendar Round (18,980 days, about 51.97 tropical years), the haab’ drifts by about 13 days, and the calendar is no longer aligned with the agricultural seasons. The Calendar Round is therefore a useful 52-year cycle for naming days but is not a stable astronomical year.

The Maya knew this, and they built correction cycles into the Dresden Codex to bring the haab’ back into alignment with the true tropical year. The principal correction is the 1,040-haab’ cycle (1,040 × 365 = 379,600 days), which is approximately 1,040 × 365.2422 = 379,851.89 days, an error of 252 days over 1,040 years, or about 0.24 days per year. The 1,040-haab’ cycle is essentially an admission that the haab’ is a vague year, and the correction brings the 1,040-year average back to within 0.001 days of the true tropical year. A second correction, the 1,296-haab’ cycle (1,296 × 365 = 473,040 days), is used to align the 52-haab’ Calendar Round with the tropical year. The same corrections are noted in the page on Maya astronomical cycles and observations.

The practical accuracy of a haab’-based seasonal prediction is therefore about 1 day in 4 years, or roughly 13 days in a 52-year Calendar Round. For agricultural purposes, the question is whether the 13-day drift over a Calendar Round is small enough to be neglected. The Maya evidently judged that the haab’ alone was adequate for many practical purposes — the agricultural year has a window of several weeks, not days — and the 1,040-haab’ and 1,296-haab’ corrections were used for ritual and astronomical purposes, not for the agricultural year itself.

Mars, Jupiter, Saturn: the planet tables

The Mars table in the Dresden Codex uses a 780-day mean synodic period for Mars. The true value is 779.94 days, an error of 0.06 days per cycle. Over a 5-cycle count (3,900 days), the accumulated error is about 0.3 days, comparable to the Venus table. The Mars table is shorter than the Venus table and has fewer correction rows; it is used for the timing of war and other ritual events.

Jupiter’s synodic period is 398.88 days. The Dresden Codex uses a 399-day cycle in some inscriptions and a 442-day cycle in others; the discrepancy is not fully resolved, and the cycles may correspond to different rituals. Saturn’s synodic period is 378.09 days, and the Maya use a 378-day cycle, an error of 0.09 days per cycle. The 819-day cycle, the 4-planet cycle used at Palenque, integrates the four visible planets into a 1 × 819, 2 × 819, 3 × 819, and so on sequence, and the integrated accuracy is comparable to the single-planet accuracy of each of the four component tables.

The 260-day tzolk’in and the 365-day haab’ integration

The 260-day tzolk’in and the 365-day haab’ lock together to form a Calendar Round of 18,980 days. The tzolk’in is, like the haab’, a calendar with a definite ritual cycle and a drifting relationship to astronomical events. The tzolk’in is, however, integrated into the Venus and eclipse tables as a 9-day and 13-day cycle, and the integration yields a 260-day cycle that is essentially a ritual cycle rather than an astronomical one. The accuracy of the 260-day cycle as an astronomical measure is not in question because the 260-day cycle is not used as an astronomical measure; it is used to name days.

The same integration appears in the Long Count, the cumulative day-count system that uses baktun, katun, tun, uinal, and kin to specify a date over millions of years. The Long Count is a calendar, not a measurement, and the accuracy of the Long Count is essentially infinite — any Long Count date corresponds to a unique day in the past or future, with no drift or error. The question of the Long Count’s calibration to the modern Gregorian calendar is a separate one, treated in The Long Count and Calendar Round.

A comparison with contemporary systems

The Maya Venus and lunar tables are comparable in accuracy to the contemporary Greek and Babylonian systems. The Babylonian System B lunar system (c. 300 BCE) used a 223-month, 6,585.32-day Saros cycle for eclipse prediction, accurate to within 1 day over the cycle. The Greek Hipparchan system (c. 140 BCE) used the Babylonian data with a 7,300-day, 20-year correction, accurate to within 0.4 days per year. The Indian Siddhantas of the fourth through sixth centuries CE used similar corrections and were, in some cases, more accurate than the contemporary Greek systems. The Maya system is roughly in the same range of accuracy as these, and the eclipse warning cycle is comparable to the Saros.

The question of whether the Maya were uniquely accurate is therefore best answered “no”: they were accurate by the standards of contemporary pre-telescopic astronomy, and they were notably accurate for Venus and the Moon, but they were not in a different category of accuracy from the Babylonian, Greek, or Indian systems. The question of whether the accuracy implies a different kind of practice is more interesting. The Maya tables are not theoretical predictions; they are almanacs built from a long observational record, and the corrections are added as needed to keep the almanac accurate. The system is empirically driven, not theoretically driven, and the empirical record is the source of the accuracy.

What the accuracy figures do and do not show

The accuracy figures show that the Maya were competent, careful observers of the sky who had built an arithmetic that could express and correct their observations. They do not show that the Maya were “first” or “best” at any specific prediction. They do not show that the Maya had a theory of gravitation, light, or the solar system. They do not show that the Maya had a concept of celestial mechanics beyond the practical arithmetic of cycles. The system is best understood as a sophisticated naked-eye astronomy, on a par with the contemporary civilizations of Eurasia, but it is not the same thing as modern astrophysics. The details of the calendrical structures that the astronomy supported are treated in The Maya Calendar System.

Sources

Aveni, A. F. (2001). Skywatchers of Ancient Mexico. University of Texas Press. — link
Saturno, W. A. et al. (2012). Ancient Maya astronomical tables from Xultun, Guatemala. Science 336 (6082), 714–717. — link
Berlin, E. A. & Berlin, B. (1996). Medical Ethnobiology of the Highland Maya of Chiapas, Mexico. Princeton University Press. — link
Sharer, R. J. & Traxler, L. P. (2006). The Ancient Maya (6th ed.). Stanford University Press. — link
Coe, M. D. (1993). The Maya (5th ed.). Thames & Hudson. — link