Introduction
2012.06.12  New evidence implies Maya astronomers possessed precise astronomy as early as 800 C.E. Given the arrangement of older monuments it seemed such must be the case, albeit written evidence was lacking. I just noticed new evidence via a Science article. In May, William A. Saturno, David Stuart, Anthony F. Aveni, and Franco Rossi reported the astronomical tables found in an excavated room in Xutlun, Guatemala, "Many of these hieroglyphs are calendrical in nature and relate astronomical computations..." One numerical array presents several large numbers that imply precise astronomical constants.
The Xultun Murals
In March of 2010, Maxwell Chamberlain discovered mural paintings on a wall at Xultun in the Peten. Subsequent excavation revealed two numerical arrays written 1,200 years ago along with paintings of individuals and glyphs. Previously, Maya astronomical tables were only evidenced from Late Postclassic period codices
(post1300 C.E.), and best knowledge of the precision of Maya astronomy relied on only several Maya codices that survived Spanish book burnings. Last month's reporting of older astronomical tables was headline news in the archaeological world and in popular media, albeit I heard no interpretations regarding precise astronomy.
My interest was reignited by online fora discussing the Xultun numbers, but again no one commented that precision was achieved by the Maya. Finally, via a listserver on June 11, Dr. Barbara MacLeod noted, "Number B is 341,640 = 11569.021 x 29.530587 ... error in relation to the modern LSM (29.530587) is .000055 ...." This large Xultun number interpreted as days represents a whole number of full moons. To check accuracy during the Maya Classic, I converted 341,640 using the days per full moon value for 700 C.E. (Table 1). To know accuracy accurately given analysis of large number ratios, it is important to employ astronomical "constants" adjusted for the era in question.
Albeit the Maya Long Count has been shown conclusively to represent days, in 2009 I hypothesized Maya counts may represent intervals other than days. When I converted 341,640 to other astronomical motions, I noted a very precise ratio, 341,640.0 solar orbits to 4,225,659 moons, a difference of hours every four million plus full moons (1.0 : 1.000 000 009). Is Maya research hampered by an assumption—that Maya Long Count numbers refer to day counts, not other intervals? The 341,640 : 4,225,659 ratio precisely equaled the solar orbits to moons ratio around 600 C.E. In decimal using the same number of numerals, moons per solar orbit (mo) was 12.36874778762, more decimal numerals than modern astronomy constants employ.
Table 1. Cosmic Motion Ratio Accuracy

Given 700 C.E. astronomy constants (TT), where d = day, m = moon,
o = solar orbit.
(I introduce the AeGeo code for cosmic motions in a Fundamental Astronomy reading.) 
Period 
Ratio 
Accuracy 
Constant 
341,640 d 
11,569 m 
1.000 001 928 
29.53064223 
341,640 o 
4,225,659 m 
1.000 000 009 
29.53058525 
160 o 
1,979 m 
1.000 000 190 
29.53058001 
2,305 m 
68,068 d 
1.000 000 002 
29.53058568 
2,739 o 
33,878 m 
1.000 000 005 
29.53058546 
Maya Classic: 341,640 solar orbits =
Today: 341,640 solar orbits =

4,225,659.0 moons (29.53058525) 4,225,658.5 moons (29.53058886) 
Not only does the interval 341,640 days equal an integer number of full moons (11,519.022), interpreting the number as solar orbits also presents an integer number of full moons, and at a ratio with far greater accuracy. Using the date 700 C.E., my research applet returned the value of 4,225,658.96 moons instead of a perfect integer. Large numbers considered as solar orbits generate even larger numbers for moons or days, etc., such that a few centuries significantly alters integer accuracy. Table 1 illustrates and compares the degree of accuracy possible with three to five digit ratios, up to one part in 200,000,000. Note how much the accuracy of 160 orbits to 1,979 moons exceeds the 341,640 days to 11,569 moons ratio. While it takes large integers to accomplish what deep decimals do, about onefortieth of the 341,640 amount presents an accurate integer ratio of orbits to moons at 8,537 orbits to 105,592 moons.
With caution about such large numbers, I nonetheless was drawn to analyze more intervals because the orbits ratio presented an accuracy exceeding prior evidence. I emailed Dr. Saturno and requested a PDF of the article which arrived minutes later. The authors express this view:
"Many of these hieroglyphs are calendrical in nature and relate astronomical computations, ... The columns of five numbers appear at first glance to be Long Count dates, yet they are not—at least not in the conventional sense..." 
I converted the Maya number 341,640, along with the ratio I noted, to lunar synodic periods per solar orbit: 4,225,659.0 divided by 341,640.0 equals 12.3687478. Given 700 C.E., days (TT) per orbit (do) equaled 365.2563615, thus days per orbit divided by 12.3687478 equals 29.53058525 days per lunar synodic period (Table 1). While ancient intentions are not yet clearly shown, if the Maya number 341,640 does "relate astronomical computations" as the authors suggest, the number implies precise astronomy in the Maya world during the Classic period. More later after examining the other numbers and further considerations of the implications of using such large numbers to express astronomy ratios.
The Large Number Problem
2012.06.15 
One Xultun array, the lunar table of 27 columns of three numerals each, is a 177
or 178 additive sequence resembling the Dresden Codex lunar tables. The 27 vertical columns of baranddot numbers reach a total of 4,784, the number of days in 162.0 full moons. This together with glyphs associated with the moon atop some columns indicated the astronomical nature of the tables painted on the room walls. Likewise over a century ago, multiples of days per full moon led to recognition of the astronomical content in the Dresden Codex. Dr. MacLeod noticed the same relationship with regard to number B in the Long Count array (Table 1).
The four numbers in the Long Count array are fivenumeral columns, much larger numbers than the 4,784day lunar table of threenumeral columns. The four numbers are all multiples of 260, 365, 780, and 18,980 (260 * 365 = 18,980 * 5), and their least common divisor, 56,940 (18,980 * 3). The large quantities express the simplified ratio 21 : 6 : 43 : 31 (Table 2).
Table
2. Xultun Long Count Array


Number 
A 
B 
C 
D 
Maya Long Count 
8.6.1.9.0 
2.7.9.0.0 
17.0.1.3.0 
12.5.3.3.0 
Baktun = 144,000 
8 
2 
17 
12 
Katun = 7,200 
6 
7 
0 
5 
Tun = 360 
1 
9 
1 
3 
Uinal = 20 
9 
0 
3 
3 
Kin = 1 
0 
0 
0 
0 
Decimal Value 
1,195,740 
341,640 
2,448,420 
1,765,140 
Ratio 
21 
6 
43 
31 
Because astronomy "constants" actually change slightly with time, the Xultun Long Count numbers are near the limit of number size having utility to accurately express astronomy ratios. Utilizing such large numbers for integer ratios is difficult, as discussed following, even without the accuracy changing temporally. The easiest means of understanding this problem is to consider the difference between one million and one million and one, only one part in one million. Therefore changing one side of a large number ratio from one integer to the next integer results in a small change in inaccuracy compared to how changing to the next integer impacts small number ratios. When large numbers hypothetically represent astronomy constant values, a question certainly arises, "Why not use smaller numbers which yield the same accuracy?" In other words, "Why say 2,305,000 m = 68,068,000 d instead of 2,305 m = 68,068 d?"
Analysis of Number B reveals that 341,640 days is not the most accurate integer for 11,569 moons. Saturno, et.al., report "texts associated with the main figures of the mural and their actions seem to cluster around 814 C.E." Given 814 C.E., 11,569.0 lunar synodic periods equals 341,639.3 days (TT). The four large Xultun numbers share a series of divisors common to Maya counting, while the more accurate 341,639 does not. Moreover, 341,939 solar orbits does not equate to integer moons with the precision of 341,640. At the same time, the ratio 4,225,648 moons to 341,639 orbits is far more accurate than 11,560 moons to 341,640 days (29.530575 versus 29.530642). The 160 numbers in a series from 341,640 orbits to 341,800 orbits demonstrates the large number problem; the greatest deviation from a precise integer ratio is 1.0 : 1.000 000 118 and several of the ratios are more accurate than 341,640. Thus, any greater random number of orbits will produce a more accurate ratio than 11,569 : 341,640 by a very large factor. This clearly illustrates both the large integer ratio accuracy capability and the large number problem, and why to proceed with caution especially given the other four numbers are much larger than number B.
Great Ratios
2012.06.24  The large number problem complicates analysis. Can large Maya numbers be shown to have astronomical significance nonetheless? Far fewer numbers have an accurate integer ratio to two motions, even fewer given more than two other motions. Historical knowledge of integer ratios in astronomy provides useful context. In Ancient
Astronomy, Integers, Great Ratios, and Aristarchus I discussed the topic and defined a "great ratio" as an accurate integer ratio for three or
more astronomical periods.
One great ratio known to ancient Babylonian astronomers was a 345year eclipse period. The interval closely matches integer solar orbits, so days and rotations have near the same decimals as also do lunar orbits and moons. This great ratio has six
integer periods, seven when doubled. While integer accuracy expression is typically less precise with small numbers, in this case four periods equate with significant accuracy.
The 345YearEclipse
Great Ratio
345 y : 4,267 m : 4,573 a : 4612 l : 4,630.5 n :
126,007 d : 126,352 r

Aristarchus' interval, which I reconsidered as representing 2,438 solar orbits, equates to integeraccurate lunar orbit, moons, rotations and days (along with Mercury and Venus orbits and synodic periods).
Aristarchus' Great Ratio
4,876 o : 60,310 m : 65,186 l : 1,785,866 r : 1,780,990 d

I have so far discussed the four Xultun large numbers without consideration of their relationships. Are they intended to express relationships to each other? Are they a sequence like the threenumeral numbers? Should their differences be considered as temporal spans? Long Count expressions are typically understood as elapsed time. The Xultun authors report:
"... the Xultun intervals do not link the different tzolk’in day stations noted at the top of each column, suggesting that each column is selfcontained, expressing accumulations of days wedded, for reasons still undetermined, to the individual
tzolk’in days recorded at the top.
"... Why these four particular numbers
were used, which range in duration from
935 to 6703 years, is uncertain." Saturno, et.al. 2012

The authors consider the numbers as representing days. As days, no integer expressions to other periods are noted, not even when the six differences between the four numbers are also considered. The six differences also present the large number problem; all are far larger quantities than the smallest of the four numbers, 341,640.
One property of the four numbers offers another front of analysis. The least common denominator of 56,940 also presents accurate integer ratios (Table 3). A fairly precise rotations (UT) to lunar orbits ratio during the Xultun epoch occurs given four times the common divisor. An accurate great ratio (l : m :o) is seen with the sixteen times multiple, albeit once again the large number problem pertains to this quantity.
Table
3. The Common Divisor of the Xultun Long Count Array

Multiple 
Period 
Ratio 
Accuracy 
1 
56,940 l 
57,169.003 n 
1.000 000 057 
4 
227,760 l 
6,239,819.02 d 
1.000 000 003 
16 
911,040 l 
842,893.003 m 
1.000 000 004 
68,146.997 o 
1.000 000 045 
The Ring Number
Another smaller number offers further evidence. The incised Ring Number between the Lunar Table and the Long Count Array is depicted to the right. Below the tzolkin number 10 Kimi, the four numeral column reads 4.15.5.14.
The number 14 is encircled with a cartouche, a variant of undetermined meaning known also from the codices. Saturno, et.al. comment:
"The relationship between this Ring Number
and the other Xultun texts is at present unclear."


The ring number, 34,314 in decimal notation, when considered as representing lunar orbits presents an integer ratio to both lunar nodal periods and lunar orbit turns (Table 4). A multiple of the Ring Number also presents a great ratio. The twotimes multiple, considered as 68,628 moons, equals integer days, and the fourtimes multiple (137,256 moons) additionally equates to integer solar orbits and lunar orbits.
Table
4. The Ring Number

Multiple 
Period 
Ratio 
Accuracy 
1

34,314 l 
34,452.005 n 
1.000 000 152 
34,314 y

12,532,922.99 d (TT) 
1.000 000 001 
2

68,628 y 
917,434.993 l 
1.000 000 007 
4

137,256 m

148,353.0006 l 
1.000 000 004 
11,097.0006 o 
1.000 000 060 
The twotimes multiple when considered as years equates to integer lunar orbits. However, another problem arises when large numbers of years are considered. Not only is the length of the year variable over long spans of time due to the temporal variations of the motions producing the year, the elliptical orbit of the earth also results in the earth orbiting at a variable speed during each orbit, slowing and hastening in relation to apogee and perigee of orbit. Therefore, arbitrarily determining the point in the orbit where the year begins and ends changes the length of the year. Starting the year at different points in the elliptical orbit results in a slightly different number of days per year. In cultural astronomy studies, it is important to understand when the culture in question started their year before determining a value for duration of a year. The factor is significant in relation to the large numbers and the accuracies considered herein. The year is not a very useful interval to astronomers and is also a difficult period to determine by direct astronomical observation.
Filtering Large Number Sets
2012.06.30  Another property of the Long Count Array numbers allows analysis of a smaller number set. The four Long Count numbers (Table 5) are all multiples of 260, 365, and the Calendar Round of 18,980 (equals 365 * 52 or 73 * 260). Accuracy of astronomical motion ratios assessed comparatively in relation to numbers divisible by one or more of these factors reduces the number set immensely and serves to correspondingly reduce the large number problem.
Ancient American astronomers employed a continuous Long Count with a modified vigesimal tally, with base 18 in the third position resulting in the 360day tun. In parallel, they also combined the Maya Haab, a 365day calendar, with the 260day Tzolkin of 20 13day periods. Haabs and Tzolkin intercalate every 52 years, resulting in the 18,980day Calendar Round. Every eighteenth multiple of the Calendar Round is divisible by 360 (18 * 18,980 = 949 tun). Every sixtythird multiple of 18,980 is divisible by 819 (63 * 18,980 = 819 * 1460). This mathematics underlays the Xultun Long Count Array with the common divisor of three Calendar Rounds.
Table 5. Ancient American Calendar Cycles

Xultun Numbers 
Tzolkin 260 
Tun 360 
Count 364 
Haab 365 
Count 819 
Round 18,980 
Divisor = 56,940 
219 


156 

3 
A = 1,195,740

4,599 

3,285 
3,276

1,460 
63 
B = 341,640

1,314 
949 

936


18 
C = 2,448,420

9,417 


6,708


129 
D = 1,765,140

6,789 


4,836


93 
C  A = 1,252,680 
4,818 


3,432


66 
C  D = 683,280 
2,628 
1,898 

1,872


36 
Of the four Long Count numbers, Number B is the only one divisible by 360. There is no lower number divisible by the three divisors, 260, 360 and 365. Of the six differences, only 683,280, Number B times two, is divisible by 360 (683,280 = 1,890 * 360). Four times Number B, 1,336,560, is known from the Dresden Codex. Even though onethird of Number B presents an accurate orbits to moons ratio, that amount is not a multiple of 360. Additionally, 260 times 360 times 365 equals 34,164,000, one hundred times number B (26 * 36 * 365 = 341,640). All ten, the four numbers and their six differences, and their common divisor are divisible by 60, 260, 365, and 18,980. Number A is the smallest number in the defined set divisible by 819 and 18,980 (MacLeod and Kinsman) and it is also divisible by 364 (Table 5).
All numbers divisible by both 260 and 365 are multiples of 18,980 (18,980 = 73 * 260 = 52 * 365). When solar orbits divisible by both 365 and 260 are considered, Number B presents the most accurate integer ratio to moons. This best accuracy ratio prevailed from around 500 C.E. to about 900 C.E. For any integer of the eclipse nodal interval, decimals are equal for nodal periods and moons. Thus, sorting for the most accurate eclipse nodal interval ratio to lunar nodal periods produces the same accuracy result as for moons.
At this point I must add a caveat; the limitations of "modern" astronomy are impeding and delimiting consideration of the large numbers and of the eclipse nodal interval ratios in particular. Lunar orbit turning is an even larger multiplier of days than is solar orbits. Astronomy temporal formulas today are geocentric, given definitions of constants expressed in days. Days are a geocentric experience produced by several motions and the base unit, the day, changes with time as the periods of the motions that determine the day change. Several questions have surfaced. What limitations are imposed by modern astronomy's temporal formulations? What is the confidence level of their accuracy? Which modern astronomy temporal formulas are most adequate for the analysis of the large numbers in Maya astronomy?
Modern Astronomy Formulas
2012.07.27  As a result of analyzing the Xultun large numbers, I detected a problem in a modern astronomy formula. After considerable insistence on my part, Dr. Anton R. Peters recognized the error and corrected several of his formulas. The disparity was only in the determination of the slight changes in astronomy constants over time, specifically in the temporal terms for lunar orbit and for lunar orbit turns (lunar orbit precession). In early 2011, these errors were introduced in my applets when I updated the formulation. The small differences impacted large number considerations and the amount of error increased with depth of time.
This, of course, meant that my applets needed formula corrections (now completed) as did some of the deep decimals of numbers I posted. In a sense, the ancient Maya had pointed out the problem; their large numbers made a slight inaccuracy apparent to me. Some of the material above had to be amended to reflect the changes, especially the deepest decimals. One nowcontradicted statement has been deleted, "While the number 341,640 equates both days and solar orbits to moons, it is not a great ratio." In fact, with the formula corrected, I find that the number is a great ratio and an eclipse interval at that. Thus, it now falls into the Maya pattern of noting intervals equating to eclipses.
Time and Astronomy
2012.08.02  Advances in science have altered how we understand and use time, and these changes have impacted and improved astronomy. Modern astronomy currently relies on several centuries of observations, the more recent the more refined, and scientists then extrapolate the past from the observation data. With the advent of accurate clocks, astronomers were able to observe slight variations in astronomical motions. After the advent of atomic time, astronomy standardized time independent of the fluctuations of temporal references such as solar zenith days. As a result, time and astronomy formulations today are at variance with ephemeris or "oldfashioned time" based on the unsteady positions of the sun, moon, and planets.
Lunar and solar gravity impose tidal friction and dissipation on the rotating earth, slowing the earth and increasing the lunar distance, lunar orbit period, and lenght of day. Lunar constants change with time as the unit of time commonly expressing these, the length of day, increases about 2.4 milliseconds per century. For precision and constancy, astronomers adopted a standard time reference independent of the fluctuating day length, Terrestrial Time (TT). Modern astronomy constants are formulated using Terrestrial Time with days defined as 86,400 SI seconds.
Universal Time replaced Mean Solar Time due to a recognition of the nonuniform rotation rate of the earth. Coordinated Universal Time (UTC), the current standard regulating clocks and time, is based on International Atomic Time but allows for adding leap seconds to calibrate to Earth's slowing rotation. Universal Time (UT1) is determined by the rotation of the Earth with respect to extragalactic, fixed space (the International Celestial Reference Frame, the ICRF). UT1 is time equivalent to observed astronomical motions in the past, what observers would have counted.
2012.08.12  The temporal changes of rotation rate and lunar orbit require temporal formulas to extrapolate values for astronomy constants in the past. Ancient astronomers did not employ atomic time. Hence, conversion to Universal Time (time observed astronomically) is also a requisite for precision, moreso the deeper the time considered and/or the larger the numbers considered. The large Xultun numbers imposed a new requirement on my research programming, determination of the length of days in prior epochs. The 341,640 orbits to 4,225,659 moons ratio remains the same, likewise for other motions, when converted from TT time to UT1 time, to 4,225,659 moons. However, the number of days changes significantly, by a factor of 1.0 : 1.000 000 24 (Table 6).
Table 6. Conversions of 341,640 Solar Orbits, Time Comparisons.


Universal Time (UT1) 
Terrestrial Time (TT) 

600 C.E. 
814 C.E.  600 C.E. 
814 C.E. 
m 
4,225,658.998 
4,225,658.921 
4,225,658.998 
4,225,658.921 
l 
4,567,298.998 
4,567,298.921

4,567,298.998 
4,567,298.921 
n 
4,585,667.986 
4,585,667.840 
4,585,667.986 
4,585,667.840 
e 
360,008.987 
360,008.919 
360,008.987 
360,008.919 
t 
18,368.987 
18,368.919 
18,368.987 
18,368.919 
d 
124,786,213 
124,786,208 
124,786,183 
124,786,183 
Table 6 illustrates the problem posed by using the day as a unit of measure, its inconstancy over time. Modern astronomy only recently figured this out and adjusted to their geocentric bias by defining a fixed "day" of 86,400 seconds (TT). Did Maya astronomers understand time to a sufficient degree to place greater import on equations not involving the day? Did they use solar orbits? Did eclipses serve as a more steadfast timekeeper instead? These questions and more are certainly brought to the foreground by the accurate integral equation of 341,640 solar orbits to the two lunar motions which produces the lunar nodal period, lunar orbit and turning of the direction of lunar orbit (Table 7). Did Maya astronomers understand that the ratios of solar orbits to the lunar motions were relatively constant while the day length was changing?
Table
7. Conversions of 341,640 Solar Orbits, Accuracy at 600 C.E.

Period 
Ratio 
Accuracy 
341,640 o 
4,225,658.998 m 
1.000 000 000 
4,567,298.998 l 
1.000 000 000 
4,585,667.986 n 
1.000 000 003 
Given the ratio of 341,640 solar orbits equates to 4,567,299.0 lunar orbits at 595 C.E., a Maya astronomer recording the number around 814 C.E. likely relied on observation over centuries centered on the 595 C.E. date, observations from about 400 to 800 C.E. This time span having a close correlation with the Maya Classic period adds further weight to the implication that the number is not a random multiple of Maya time cycles, but rather reflects an understanding of cosmic actuality. The implications of this discovery may reach further back in time also, to the beginning of Long Count mathematics.
Even when the temporal shoe fits, I am disinclined to build inference on interpretation. However, the questions posed by Xultun Number B are difficult to ignore. Did the Maya realize the day was not the best reference in astronomy, and if so, when? Did the Maya astronomers extrapolate the future and understand the change in length of days adequately when aligning the Long Count of days to place the Baktun 13 end on winter solstice in 2012? Does the later question answer the first? Or, is the number of coincidences irrelevant?
Afterwords
2012.08.12  The ancient Maya certainly knew enough to not only teach me a lot about astronomy and time, but also to reveal errors in modern astronomy formulas and bugs in my programming (I hope those are now both in good repair, but I remain vigilant). They certainly forced me to once again correct errors and to refine my research methods with greater precision, this time with an ephemeris perspective sufficient to deal with their day numbers in their time frame and sufficient for the long spans of time represented in their mathematics. For that, I thank them.
Given one current correlation of the Maya calendar with the Roman calendar prevailing today, the Maya Baktun 13 cycle ends on winter solstice, 2012. Knowing what we know about the day and how length of day and year changes with time, this is a predictive feat requiring accurate astronomical knowledge, knowledge which at a minimum must have been held when the calendar was created and the count of days began. While the Maya did not predict the end of the world as cataclysmic booksellers like to proclaim, they apparently did know enough to predict when their own calendar would complete a baktun period in relation to years completing. If intended, that is prediction of the future and certainly it is astronomy as science as well, not astrology and superstitions. Given such, it truly remains an impressive cultural milestone, one the modern world could only replicate and understand in very recent times.
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Related Readings
Jacobs, James Q., 2009, The Dresden Codex Lunar Series and Sidereal Astronomy http://jqjacobs.net/archaeology/maya_astronomy.html.
MacLeod, Barbara and Hutch Kinsman, 2012, Xultun Number A and the 819Day Count, "Xultun Number A is also the smallest unit which commensurates the 819Day Count with the Calendar Round."
Matson, John, May 10, 2012, Ancient Time: Earliest Mayan Astronomical Calendar Unearthed in Guatemala Ruins.
Saturno, William A., Explorers Journal.
Saturno, William A., David Stuart, Anthony F. Aveni, and Franco Rossi, 2012, Ancient Maya Astronomical Tables
from Xultun, Guatemala, Science 336:714717.
Thompson, Helen, May 10, 2012, Murals offer glimpse of Mayan astronomy.
Zender, Marc and Joel Skidmore, 2012,
Unearthing the Heavens: Classic Maya Murals and Astronomical Tables at Xultun, Guatemala. Photos and Captions. 