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Maya Eclipses: Modern Data, The Triple Tritos And The Double Maya Eclipses: Modern Data, The Triple Tritos And The Double

Tzolkin Tzolkin

William Earl Beck University of Central Florida

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STARS Citation STARS Citation Beck, William Earl, "Maya Eclipses: Modern Data, The Triple Tritos And The Double Tzolkin" (2007). Electronic Theses and Dissertations, 2004-2019. 3078. https://stars.library.ucf.edu/etd/3078

MAYA ECLIPSES: MODERN ASTRONOMICAL DATA, THE TRIPLE TRITOS AND THE

DOUBLE-ZTOLKIN

by

WILLIAM E. BECK B.A. University of Central Florida, 2001

A thesis submitted in partial fulfillment of the requirements for the degree of Masters of Arts

in the Department of Liberal Studies in the College of Graduate Studies at the University of Central Florida

Orlando, Florida

Fall Term 2007

© William E. Beck

ii

ABSTRACT The Eclipse Table on pages 51-58 of the Dresden Codex has long fascinated Maya scholars.

Researchers use the mean-value method of 173.3 days to determine nodal passage that is the

place where eclipses can occur. These studies rely on Oppolzer’s Eclipse Canon and Schram’s

Moon Phase Tables to verify eclipse occurrences. The newer canons of Jean Meeus and Bao-Lin

Liu use decimal accuracy. What would be the effect of modern astronomical data on the previous

studies and the Maya Eclipse Table?

The study utilizes a general view of eclipses that includes eclipses not visible to the Maya.

Lunar eclipses are also included. This inquiry differs from previous studies by calculating the

Maya dates of eclipses instead of nodal passage. The eclipse dates are analyzed using the three

eclipse seasons, of the 520 days, which is the Double Tzolkin or twice the Sacred Calendar of the

Maya. A simulation of the Eclipse Table, using the 59-day calendar, is created to test modern

data against the Dresden Table. The length of the Table is the Triple Tritos of 405 lunations. The

use of the Tritos instead of the Saros suggests the Table is independent of Western Astronomy.

Advanced Astronomy is not needed to produce this Table; a list of eclipses could produce this

Table.

The result of this inquiry will be to create a facsimile of the Eclipse Table, which can be

compared to the Eclipse Table to test the structure, function and purpose of the Table.

iii

In the loving memory of my parents

Harold and Virginia Beck

iv

ACKNOWLEDGMENTS I wish to thank my committee members, Drs. Arlen Chase, Diane Chase and Elayne Zorn, for

their help and advice. I also wish thank the staff of the University of Central Florida Library for

their tireless work in finding the resources for this project and the Office of Instructional

Resources for their help with formatting the thesis and images in this study.

v

TABLE OF CONTENTS CHAPTER ONE: BACKGROUND............................................................................................... 1

Introduction................................................................................................................................. 1 Dresden Codex............................................................................................................................ 2 Science ...................................................................................................................................... 15

CHAPTER TWO: DISCUSSION................................................................................................. 26 Calendars and Maya Math ........................................................................................................ 26 Eclipse Data .............................................................................................................................. 31 Eclipse Periods.......................................................................................................................... 38 Simulation ................................................................................................................................. 51

Chapter Three: Conclusion ........................................................................................................... 57 Appendix A Meeus Lunar Data .................................................................................................. 60 Appendix B Liu Lunar Data ....................................................................................................... 63 Appendix C Oppolzer Lunar Data .............................................................................................. 66 Appendix D Meeus Solar Data ................................................................................................... 69 Appendix E Oppolzer Solar Data ............................................................................................... 72 Appendix F Lunar-Solar Data..................................................................................................... 75 Appendix G Meeus Lunar Season Distribution .......................................................................... 79 Appendix H Meeus Solar Season Distribution ........................................................................... 81 Appendix I Teeple Season Distribution...................................................................................... 83 Appendix J Solar-Lunar Season Distribution ............................................................................. 85 Appendix K Table Simulation .................................................................................................... 87 Appendix L Glossary .................................................................................................................. 90 LIST OF REFERENCES.............................................................................................................. 95

vi

LIST OF FIGURES Figure 1 Solar and Lunar Eclipse Glyphs Pages 53a and 58b of the Dresden Codex after Thompson (1972)............................................................................................................................ 6 Figure 2 “Bookend Gods” Page 68 of the Dresden Codex after Thompson (1972).................... 18 Figure 3 Serpent Images Pages 56b and 57b of the Dresden Codex after Thompson (1972) ..... 19 Figure 4 Ah Tzul Ahau Page 58b of the Dresden Codex after Thompson (1972) ...................... 21 Figure 5 Teeple Arc after Teeple (1930:89) ................................................................................. 34 Figure 6 Meeus Arc following Teeple (1930:89) ......................................................................... 37

vii

LIST OF TABLES Table 1 Picture Intervals after Guthe (1978:11) ........................................................................... 13 Table 2 Lunar Semesters after Guthe (1932:275)......................................................................... 43 Table 3 Eclipse Periods after Table 4 Van Den Berg (1955:28) ................................................. 50 Table 4 Guthe's Semesters after Guthe (1932:275) ..................................................................... 53 Table 5 Tritos Simulation of Table 2............................................................................................ 54

viii

CHAPTER ONE: BACKGROUND

Introduction

The purpose of this study is to compare modern astronomical data against the Dresden Eclipse

Table on pages 51-58 of the Dresden Codex. The Dresden Codex is 405 lunations or 46 Tzolkins

in length. A Tzolkin is the 260 day calendar used by the Maya. This duration of time is three

times the Tritos eclipse period of 135 lunations or a Triple Tritos. The Dresden Codex is one of

three surviving Maya texts that display charts believed to contain astronomical data. The

Dresden Eclipse Table is one of the most studied and least understood pieces of Maya

Iconography. Is it a lunar calendar; a solar warning table; a sysygy, (a list of actual eclipses) or

just a list of potential eclipses? For this reason the chart on pages 51-58 will be referred to as the

Eclipse Table in this thesis.

This inquiry differs from earlier studies by computing the Maya Date of modern eclipses

instead of nodal passage, the area where eclipses are most likely to occur. This method will allow

for the elimination of the use of the mean-value method of 173.31 days, prominent in previous

studies of Maya eclipses. Three times the mean-value is a close approximation to the 520-day

period, the Double Tzolkin. This study will also investigate other eclipse periods and not just the

Saros period used in the earlier studies. The Saros is very prominent in Western Astronomy but

not in Maya astronomy.

A secondary purpose is to compare modern astronomical data against the astronomical data

used in the previous studies of the Eclipse Table. This data is Oppolzer’s Eclipse Canon (1887)

and Schram’s Moon Phase Tables (1908). To test the validity of these texts, two newer eclipse

1

canons, one by Bao-Lin Liu (1992) and another by Jean Meeus (1966, 1979) will be compared to

Oppolzer’s Canon. These newer canons are more accurate due to decimal approximations but the

Maya did not use decimal numbers. Would this increased accuracy be of any importance to the

Eclipse Table’s structure or function?

This study uses a general approach, including eclipses observed in the Maya area and those not

visible in this area. Other questions open to investigation are; are the eclipses observed or

computed; and does the Table list solar or lunar eclipses?

The first chapter introduces the history and structure Eclipse Table. It investigates Maya beliefs

about eclipses and compares this to modern scientific knowledge of eclipses. The second chapter

investigates Maya and Western calendars. Calendars are important tools in developing eclipse

periodicity. Modern eclipse dates are converted into Maya dates using the Goodman-Martinez-

Thompson correlation constant of 584285 days, equating to November 12, 755. These dates are

evaluated using the three eclipse seasons of 520 days. A 59-day Lunar Count is used to create a

simulation of the Eclipse Table. A simulation is also created using the Meeus lunar data.

Simulations are needed because; the Codex is a one of a kind artifact. There is nothing to

compare the Table with, except dates in an ephemeris.

Dresden Codex

The background of the Dresden Codex is not fully known. Some facts have been presented by

J.E.S. Thompson (1972:15-19) and George Dicken Everson (1995:57-58). The Codex was

acquired by the Royal Public Library of Dresden from Vienna, Austria. It first appeared in a

1739 catalog of the Dresden Library, produced by Johann Christian Goetz, the Library’s

2

Director. In the catalog, the Codex was believed to be of Mexican (Aztec) origin because of

similarities to the Mexican codices in the Vatican, but its origins are Maya. The Codex is

believed to have been sent in the first Royal Fifth, the Spanish portion of the New World

treasure, around the date of 1519, according to (Everson 1995:57). In Thompson, this shipment is

referred to as Cortes’ Gift to Emperor Charles V. This act did save the Codex from the burning

of Maya books in July of 1562 by Friar Diego de Landa. The Habsburg Dynasty, to which Spain

belonged, did have a villa in Vienna, Austria. The Codex was observed by visitors to the Royal

Court and very well could have been brought to Vienna by the Royal Court.

In 1880, Ernst Förstemann, the Head Librarian of the Dresden Library, started his classic

translation of the Dresden Codex. He was the first researcher to identify pages 51-58 as being

eclipse related material. The Dresden Library was damaged in World War II by the fire bombing

of Dresden. The Codex received some water damage, but no fire damage. The Codex was

rescued by a Russian soldier, who was a student of languages, named Yuri Knorsov. Mr.

Knorsov later became Russia’s pre-eminent Mayanist and epigrapher.

The Dresden Codex may have had four different scribes recording information in it. The Codex

also may have contained additional pages (Thompson 1972:20). The date of the Codex is also

not known. The Codex contains eight, nine, and ten cycle dates. Some researchers place it in the

eleventh century. Thompson places it between 1200 and 1250 (Thompson 1972:15). Everson

places it in the thirteenth century (Everson 1995:4). Satterthwaite places it no earlier than A.D.

1345 (Thompson 1972:15). Some researchers have speculated that this is a copy of an earlier

work ca. A.D. 755. Any definite date in the Eclipse Table would help to understand some of the

questions and inconsistencies of the Codex.

3

The history of the Eclipse Table is covered by Thompson (1975:233-236). The study of Maya

eclipses parallels the history of the study of Maya epigraphy as described by Morley (1940). The

early study of epigraphy was focused on aligning the Maya calendar with the Western calendar.

The study of the Eclipse Table began with Ernst Förstemann (1967; see Thompson 1975:29,

1972:71-77). Bowditch (1910) brought Förstemann’s work to America. Guthe (1978), Willson

(1974; see Spinden 1969, 1928, 1928a) and Teeple (1930) are all prominent researchers in the

early investigations of the Codex and Eclipse Table. Villacorta and Villacorta (1976) and Gates

(1932) created reproductions of the Dresden Codex. Thompson (1972) continued Gates’ work.

Makemson (1943), Satterthwaite (1947), Lounsbury (1978, 1986),; Bricker and Bricker (1983),

Kelly (1976), Justeson (1986), Campbell (1992), Everson (1995) and Smither (1986) all

continued the research of the earlier scholars. Although all of the researchers believe that the

Eclipse Table is associated with eclipse periods, exactly which eclipses match those periods is

still in question. All of the newer researchers believe that the Dresden Table is solar in nature.

Smiley (1973:175, 1771975:248) calls the Lunar Table a Solar Eclipse Warning Table and

viewed the Venus Table as a Solar Prediction Table. Spinden states that “the Eclipse Table is a

calendar of the moon” (Spinden 1928a:148). A study of the lunar calendar is investigated in the

Simulation Section of Chapter Two. Guthe believed the Table served both for predicting eclipses

and for lunar reckonings (Thompson 1975:234). MacPherson (1987:443) considered the Table to

be a chart of observations of sunsets and moonsets.

“It is either a record of observed results over a thirty-three year period or a computation made

for a given thirty-three year period, but cannot possibly be a formal calendar for repeated reuse”

4

(Teeple 1930a:138). Carlson believes that it “is a canonical, quasi-astronomical calendar rather

than a true astronomical ephemeris” (Carlson 1984:241).

The Table is a table of solar sysygies, according to Teeple (1930a:137). A sysygy or ephemeris

is a list of dates when eclipses are likely to occur. The Maya Tables are tables of mean motion

rather than ephemerides, according to Kelly and Kerr (1973:182).

The Table is an Eclipse Warning Table, according to Schove (1982:241, 1984:304), Aveni

(1981:80), and Malmstrom (1977:147). Kelly (1976:43) stated that “it is a Prediction Table

rather than a record of past events.” Satterthwaite (1962:255) also called it an Eclipse Predicting

Table.

The Eclipse Table covers pages 51-58 of the Dresden Codex. It contains an upper register

designated “a” and a lower register designated “b.” This is the old pagination which is still used

because of all the past studies which use this system. One must remember to read the Table

starting with 51a-58a then 51b-58b. The Table can be divided into four sections: glyphic text;

pictures; numbers (bars and dots); and dates. The dates divide the numbers into a bottom section

of eclipse intervals and an upper section of totals. The totals, intervals, and dates make up the

heart of the Eclipse Table.

Kelly states that “our knowledge of the mechanism of various astronomical tables and of the

calendar is quite adequate but our understanding of the associated glyphs has lagged far behind”

(Kelly 1976:52).The eclipse glyphs in the codices (Figure 1) are the Kin (sun) and Uo (moon)

signs enclosed in white and black elbow elements.

5

Figure 1 Solar and Lunar Eclipse Glyphs Pages 53a and 58b of the Dresden Codex after Thompson (1972)

There is only one glyph from the monuments, identified by Juan Palacios at Santa Poco Unic.

It has a kin sign in brackets (Teeple 1930:115). The glyphs of eclipses are from the codices.

There are in the references monuments to the sun being in his house; could the bracket in the

eclipses glyphs signify his house? There is also a reference to New-Sun-at-horizon.

Coincidently, there occurred a nearly total eclipse at sunrise on December 11, 847 (Schove

1982:251). Schove thought this proved his theory of the Table. The sky-in-hand glyph (Figure 1)

also appears quite frequently with eclipses. These glyphs are repeated in a phrase which opens

and closes the Eclipse Table.

The Eclipse Table is 405 lunations or 11,960 days in length. This is equivalent to forty-six

tzolkins. There are sixty-nine eclipses in three sections of 3,986. This is a total of 11,958. Some

researchers also list an 11,959 total, giving a three day spread for the totals. The Table is further

divided into “six months” of thirty days and “six months” of twenty nine days for a total of 177

days. There are six parts of 177 to one of 148 days. There are nine parts of 177 to one of 178

days (Förstemann 1967:200-202). The big question has been where to place the 148 and 178-day

periods.

The Eclipse Table on pages 51a and 52a opens with an Introductory Section which lists the

starting date of the Maya Long Count, 4 Ahau 8 Cumhu. To this date eight days are added to

6

give the date 12 Lamat. Lamat is linked to the planet Venus. Some researchers believe that the

12 Lamat date is a ritual starting date for the Eclipse Table and have tried other dates involving

12 Lamat. Any change of ten tzolkins would return the Lamat date to the base date. The ten-

tzolkin interval is 1.31 days more than one of the subdivisions of the Saros, the 88-lunations

(Lounsbury 1992:204).

The major problem with the Table is that the eclipse dates change over time. The Table is

supposed to have a self-correction system, but no one has described precisely how it works.

These questions are complicated by the fact that only one version of the Eclipse Table has

survived. Another version of the Table would answer some of the questions about the base date

and whether the 12 Lamat date is a ritual date for the Table.

It is impossible to determine from the Table’s form whether it is used for solar or lunar eclipses

(Beyer 1933:305). The Table could be employed by both with the addition or subtraction of

fifteen days. The Introductory Section contains a series of five Maya dates repeated seven times.

Campbell (1992:51) speculated these “year bearer” dates became eclipse bearer dates. Saros

eclipses can be predicted by using the first full or new moon of the year

(http://www.astro.uu.nl/~strous/AA/en/saros.html). The Maya may also have developed a way to

predict eclipses with their year bearer date. Additional studies will have to be done to determine

the nature and function of this group of 15-day dates. Meinhausen suggested that pictures come

after the 148-day eclipse periods because when a solar eclipse occurs at an interval of 148 days,

then a lunar eclipse will follow fifteen days later (Everson 1995:160; Thompson 1972:72,

1975:233). Makemson states that “the fifteen days indicates pairs of solar eclipses” (Makemson

1943:206-207).

7

On page 52a, there is a series of thirteen thirteens. This creation number is also recorded on

Stela 1 at Coba, Mexico (Freidel, Schele and Parker 1993:62-63). The thirteen thirteens are also

listed on pages 23 and 24 of the Paris Codex (Willson 1974:19). The thirteens are Oxlahun-ti-Ku,

“Gods of the Thirteen Heavens” (Jakeman 1947:9). Oxlahun-ka’an-ub is “thirteen-sky-moon,” a

reference for a full moon (Marci 1996:285-286). In the Motul dictionary, Oxlahun-caan-u is

“thirteen heaven moon” (Campbell 1992:52). There are several instances of the thirteen gods and

the thirteen heavens. There are thirteen variations of the head-variant glyphs, which are seen as

having had a lunar significance (Marci 1996:278-279). A period of thirteen days is the average

number of days from visible first crescent until full moon (waxing moon). The Maya could

incorporate a seven day waning period to third quarter and a nine day period through invisibility.

There are nine moonless nights, signified by the Lords of the Night (Marci 1996:275, 278-279).

Thirteen times thirteen is 169 days, a close approximation to the 177-day eclipse cycle (Smiley

1975:248).

The Chilam Balam of Chumayel, the Maya Book of the Jaguar Priests, tells of the creation of

the unial, or thirteen entities added to seven making twenty (Jakeman 1947:8). A variant moon

sign Uo (Frog) is often used for the number twenty. With this change, the calendar became

divorced from observed phenomena, but had to be tracked by specialists (Marci 1996:286).

The Eclipse Table does contain numerous errors. These have been discussed by Förstemann

(1967), Bowditch (1910), Guthe (1921) and Thompson (1972) as “copying errors.” Most involve

dropped bars and dots. Some day signs are incorrect. Fortunately, the Maya used a redundant

system of distance numbers. Some of the dates are correct but the numbers are off, or the

8

numbers correct and the dates are wrong. Both situations are easily corrected and pose no

problem to this investigation.

There is also an eleven day error (Appendix F) between the lunar date of the Table and that of

the nearest lunar eclipse date. This error may be caused by the date being a “vague date.” The

Maya did not add a day for leap years. There is an expression that “in the thirteenth Ahau; Pop

was set in order” (Long 1921:37). Altar U, at Copán, contains two different 0 Pop dates (Carlson

1977:104-106). Kelly speculated on an eclipse date of 9.17.0.0.0 13 Ahau 18 Cumhu, is January

24, 771. The 9.17 date also occurs on Stela E at Quirigua (Kelly 1977:62; Teeple 1925a:115;

Closs 1986:236, 1989:232, 1992:140; Milbrath 1999:115. This date is believed to be an eclipse

date, but is in the next cycle of the Eclipse Table used in this study. This eclipse date may hold

clues to the ordering of Pop.

The introduction contains a series of numbers that are multiples of 11,960 (1.13.4.0 vigesimal).

This is the length of the Eclipse Table. The most prominent number is eighteen times the Eclipse

Table. This number is the date Satterthwaite uses in his study (1947). These multiples could have

been used for recycling the Eclipse Tables or predicting future eclipses. The dates in the Table

became obsolete over time due to the accumulated fractional part of the eclipse cycle. How the

Maya recycled the Table is still not fully known. How many times could the Table be used?

There have been suggestions that the Table could be used for some 800 years, but the Maya

could have redone the Table more frequently. Teeple (1930a:138) believed the Table was used

only once. Makemson (1943:194) stated the Table could be used four times from 1083 to 1214.

Spinden (1969:69), Bowditch (1910:224) and Thompson (1972:74) suggested eight times

(Satterthwaite 1947:77). The four times is due to the 1.6 day regression in the node. The reason

9

for eight times is that the Table of 405 lunations is 0.11 of a day shorter than the 11,960 days.

The .11 days shortfall uses the 11,959.889 day computation (Guthe 1978:3).

The Table includes five Maya Long Count Dates. Two of the dates are questionable dates that

do not appear to be eclipse dates. Three of the dates have become vital to the current

understanding of the Eclipse Table. These dates are:

9.16.4.10.8 12 Lamat 1 Muan 9.16.4.11.3 1 Akbal 16 Muan 9.16.4.11.18 3 Eznab 11 Pax These dates are also fifteen days apart. One of the dates is a 12 Lamat date. It has been

designated as the Base Date of the Table. The date 12 Lamat is 177 days before the first date in

the Table on page 53a. Most of the researchers since Willson have assumed the 12 Lamat date to

be that of a solar eclipse. If that date is solar, the Table is a list of new moons. If they are new

moons, the Table is solar. This is the circular logic researchers use to determine that the Table is

solar. The solar aspect has not been proven, only accepted as fact.

The Base Date used in this study is the 9.16.4.10.8 date without regards to whether it is a solar

or lunar date. This date is November 12, 755, using the 584285 constant. The solar proponents

believe the 12 Lamat and 3 Eznab dates are solar eclipses with a lunar eclipse in between.

Satterthwaite (1962:256) mentions some studies of the solar-lunar-solar eclipses. Makemson

states “it makes no difference whether these dates are two lunar with a solar in the middle or two

solar with a lunar in the middle” (Makemson 1943:187). Appendix F combines the solar and

lunar data for the period 755. A cursory study of Appendix F does not appear to support this

hypothesis. This hypothesis has no relevance to this inquiry but should be further studied in

future research.

10

Additional studies are also required for the erroneous Long Count date 9.19.8.7.8 7 Lamat

(Milbrath 1999:115; Thompson 1972:71). The date does not agree with the base date of the

Table. Makemson (1943:189) stated that there was no 7 Lamat to be found among the day

names. This date is in conflict with the other 15-day periods of the solar-lunar-solar eclipses.

Other base dates have been proposed. Makemson (1943:194) utilized the date 10.12.16.14.8 12

Lamat 1 Chuen (Lounsbury 1992:203). This date using the 584285 constant is April 19, 1083.

Closs (1989: 234) mentions the date 10.19.6.1.8 12 Lamat 6 Cumhu, which is another tenth cycle

date with a 12 Lamat base. Satterthwaite used the date 11.6.2.10.8 12 Lamat 11 Zac, which is

eighteen times 405 lunations (Everson 1995:175). This is the date April 12, 1345. Most of the

dates are tenth cycle dates and are also 12 Lamat dates. Some base dates use other Ahau

constants; (Makemson 489,138; Owen 487,410; Smiley 482,699) Owen (1975:240-241). Smiley

(1975:256) states that the 9.16.4.10.8 date occurred at September 22, 477. These dates are fifth

century eclipses. These dates use a different Ahau correlation moving the date back in time.

These dates should be studied because of their high activity of eclipse occurrences.

The structure of the Table consists of sixty-nine eclipses divided into three parts of twenty-

three eclipses each, for a total of 405 lunations or 11,960 days. The numbers 11,958 and 11,959

are also mentioned as lengths of the Table, which are derived from different multiples of

lunations or mean values. One third of 11,958 days is 3,986 days. One third of 405 lunations are

135 lunations, a period called the Tritos. The 405 lunations are equal to forty-six rounds of Maya

tzolkin. The three parts are not the eclipse seasons. The seasons are every third interval of the

177 or 148 numbers.

11

Across the bottom of the Table are a series of Maya numbers: 8.17 (8 times 20 = 160 + 17 =

177) and 7.8 (7 times 20 = 140 + 8 = 148). These are the intervals that mark each eclipse and

project eclipses into the future and past. A lunation is 29.53 days. Six times the lunation equals

177.18 days, a very close approximation of the eclipse-half-year. Five times 29.53 equals 147.65,

a very close approximation of 148. There are seven groups of six lunations (177 days) and a

group of five lunations (148 days), and six groups of six lunations and a group of five lunations.

Some of the 177-day groups contain an additional day (178 days). The five lunation groups upset

the sequence. The five lunation groupings are important to the structure of the Table. In fact,

nine out of the ten pictures are located at these periods. The tenth picture is located at the end of

the Table. This may be a new base date if the chart is used more than once. Makemson

(1984:192) believed that the pictures were inserted at points where lunar eclipses would occur.

This may well be true, but it should be remembered that solar and lunar eclipse are separated by

fifteen days. Willson (1974:11, 16) tried to find eclipses in Oppolzer’s canon, which he aligned

with the pictures in the Table. He was not able to find a match.

The image in picture 10 wears the Venus symbol in the head band. Venus images in picture 3

and 8 indicate Venus is involved in the Eclipse Table (Makemson 1943:191-193). Most symbols

are solar in nature, but picture 3 depicts the Moon Goddess. Makemson ignored the Moon

Goddess because she believed that Venus’ importance to the sun proved the solar eclipses.

However, there is a Venus-Moon relationship (Satterthwaite 1962:258). The solar researchers

are not looking at the moon.

12

Förstemann (1967:205) listed the intervals between the pictures in the Eclipse Table. Bowditch

(1910:217) created a table of intervals (Table 1) used by most other researchers to find eclipse

periods.

Table 1 Picture Intervals after Guthe (1978:11)

Zero picture to the first 502 Fifth to Sixth 1034 First to second 1742 Sixth to Seventh 1210 Second to Third 1034 Seventh to Eighth 1565 Third to Fourth 1210 Eighth to Ninth 1211 Fourth to Fifth 1742 Ninth to end 708

The first column plus the last equals 1210 (Bowditch 1910:217; Kelly 1976:43).

Teeple (1930:63) believed the regularity of the twenty-nine and thirty day moons proved

computation, but thought that the irregularities in the intervals proved observation (Teeple

1930:91). The three parts (Tritos) are divided into sections; 1742 equals eight times 177 + 148 +

178; 1034 equals four times 177 + 148 + 178; and 1210 equals six times 177 + 148; for a total of

3986 days Förstemann (1967:202). This is the Maya Sariod (Willson 1974:15).

The bulk of the Table is made up of sets of three consecutive dates. These dates are 176, 177,

and 178 days from the preceding set of dates for the six lunation groups, and 147, 148, and 149

days for five lunation groups. This variation has caused some researchers to question whether the

dates are a sysygy, which is a list of eclipses for predictions or a warning period of when eclipses

are likely to occur. The three day spread does allow wiggle-room for the variations in eclipse

periods caused by the accumulation of the fractional part of the eclipse period.

Guthe (1978:27) believed that the one-day error in the calendar was corrected each pass

through the Table by moving down from the middle row to the lower row. The three dates were

first thought to be three ephemerides (Bowditch 1910:221-224). Eclipses start at the bottom

13

repeating for eight times (Satterthwaite 1947:77, 1962:270). The discrepancy of 11/100 of a day

creates, on the ninth time through the Table, a one day correction. The correction was

accomplished by moving up one row of dates (Lounsbury 1978:802).

Another theory explaining these dates is the Lunar Variation Theory (Satterthwaite 1947:77).

The rationale behind this theory is that the fraction part of the eclipse period accumulates;

resulting in a three-day range of dates (176-178). These variations become apparent in the Meeus

data (Appendix A).

The only part of eclipse periods not found in the Table are the one day lunations. These may be

the same periods proposed by Satterthwaite (1947:147). The researchers who have been

calculating nodal periods from mean values only accept a deviation of twenty-five days. Since

one lunation of 29.5 days exceeds this limit, the one lunation eclipses would be fictive or false

eclipses. These are like the 148-day eclipse, but only at the other end of the eclipse group. The

148-day eclipses are referred to as “pre-nodal eclipses.” The one-lunation eclipses are referred to

as “post-nodal eclipses.” These eclipses are in the Table, but are not as apparent as other

eclipses. Some produce the 148 day eclipses in the Table. Others simply recede into the 177- day

period and are not differentiated in the Table. The Table contains approximately six one-

lunation eclipses.

The three-day groupings in the seasons (Appendix G, H and I) are not readily apparent, but

there is a close association between eclipses in the group. The three groupings are controlled by

the Base Date of the Table. A change in Base Dates does change the sequence of the lower

numbers as well as the upper totals. Appendix J is a combination of solar and lunar eclipses

distributions, which shows that the three dates can handle either solar or lunar eclipses.

14

Above the three dates are Maya numerals, which are cumulative totals of the number at the

bottom and the preceding total. This is the same procedure as Meinhausen (1913:221-225) used

in his study on eclipse periods. These numbers are also similar to Table 2 of Schram’s Moon

Table (Morley 1977: 394; Schram 1908:358, and Willson 1974:10).

Science

The first investigations of Maya astronomical data sought to determine a correlation between

the Maya and Christian calendars. This is known as the astronomical approach to a correlation.

John Teeple’s (1930:36) work Maya Astronomy sought to determine if it was sufficient to

establish a correlation. Willson’s failure to find eclipses to match the pictures in the Eclipse

Table prompted him to say that “no correlation of the Mayan and Julian calendars could be found

from the Lunar Series alone” (1974:16). The newest area of research into the astronomical data

is the investigation of how much science the Maya actually utilized. David Freidel (1993) and

Dennis Tedlock (1992, 1996) have been investigating the links between the Popol Vuh and

astronomy. Charts of recurring astronomical and meteorological events serve as signs for the

mythic deeds of the gods (Tedlock 1992:249). The newest hypothesis is that the stories in the

Popol Vuh are based on actual astronomical events some 5,000 years ago. Quiche rites give

ample testimony to a long-standing Maya concern with actual astronomical events (Tedlock

1992:269). Aveni (1975, 1977, 1980, and 1992) has become a leading authority in the field of

ArchaeoAstronomy. He states that their literature – in all forms - is filled with celestial

knowledge (Aveni 1992:4). Kelly and Kerr (1993:179) state that “there is a considerable amount

15

of astronomical data in the inscriptions.” Unfortunately, the Maya did not leave texts of their

celestial knowledge: only charts which are reported to contain astronomical data.

The Maya practiced naked-eye astronomy, which focuses on the helical risings and settings of

celestial objects. The Maya did not have the telescope, but may have used sighting sticks during

observations. There is also speculation that Maya astronomers aligned stone monuments, called

stela, and Maya buildings, called E-groups, as lines of sight for celestial observation.

Maya astronomy is different than Western astronomy in that the Maya priests did not use

astronomy as an exact science (Thompson 1975:33). “The Maya were more concerned with

numerological commensurations within their calendar than with geometrical and mathematical

relationships of positional astronomy” (Carlson 1984:236). Time, not space, is the principal

medium of expression for all astronomy (Aveni 1981:85). Another difference in Western

astronomy is in the use of the Babylonian eclipse period, the Saros. The Maya use the Triple

Tritos, which is divisible by 260, whereas the Saros is not. The Maya would study the heavens

for divination purposes (Landa 1978:13). The celestial bodies exert direct control over the affairs

of man (Andrews 1940:150). As far as the Maya were concerned, astronomy was astrology

(Aveni 1981:85). As Thompson (1966:173) stated, “astronomy is the handmaid of astrology.”

Maya priests were obsessed with knowing time. Time is cyclical. Events in the past are the same

as events in the present or future. There are good days and bad. The Maya also believed that the

conditions in the heavens were a portent of situations on earth. The best way to cope with the bad

days was to keep records and search those records for similarities. That way, a proper ritual

could be followed to mitigate the bad effects. Careful observation, record keeping, and

16

experimentation are a major part of scientific investigation. The Maya Priest kept records of

celestial observation to make predictions.

Other almanacs may have been consulted by the priests making eclipse predictions. This thesis

focuses mainly on the Eclipse Table on pages 51-58 of the Dresden Codex. Other studies have

been conducted and more should be done on the other almanacs. Eclipse intervals are common

between some almanacs and many dates are repeated in more than one almanac. The Moon

Goddess Almanac on pages 16-23 of the Dresden Codex is one of the other almanacs. The Moon

Goddess does have death images suggesting eclipses (Hoflin and O’Neil 1992:102, 118-120).

The Agricultural Almanac on pages 38-41 of the Dresden Codex depicts meteorological and

agricultural activities. The almanac is 520 days in length, the Double Tzolkin. It does contain

eclipse glyphs (Bricker and Bricker 1986:29-30). The Seasonal Tables on pages 61-69 also have

eclipse glyphs on pages 66a and 68a (Figure 2) (Bricker and Bricker 1986a:232-235). Knowlton

(2003:294-298) has also studied the Seasonal Tables to determine if this Table is relevant to

eclipses occurring during the rainy season. Weather is always a factor in observing celestial

phenomena.

17

Figure 2 “Bookend Gods” Page 68 of the Dresden Codex after Thompson (1972)

The Venus Table on pages 48-50 of the Dresden Codex is another astronomical table located

directly in front of the Eclipse Table. Five Venus cycles of 584 days are equal to eight tropical

years of 365 days, producing a cycle of 2,920 days (Kelly 1977:58). The 584 days is divided into

sections of 236, 90, 250, and 8 days. Venus appears as the Morning star for about eight months

after inferior conjunction; it disappears for three months at superior conjunction and reappears as

the Evening Star for eight months, then it disappears for two weeks at inferior conjunction

(Teeple 1930:94). The Venus Table does have some connections with eclipses trough

moonphases and certain periods of time which Smiley has investigated

Earlier peoples believed that during eclipses the Sun God abandoned them or that a celestial

monster devoured the Sun or Moon. On pages 56b and 57b (Figure 3) of the Dresden Codex, a

serpent is attempting to devour a Maya solar eclipse symbol.

18

Figure 3 Serpent Images Pages 56b and 57b of the Dresden Codex after Thompson (1972)

The Maya believed that eclipses were caused by fights between the Sun and Moon (Thompson

1939:164, 1975:231). They also believed that eclipses are caused by an agent (that agent is the

ant or a jaguar) biting the Sun or Moon (Thompson 1939:164). Closs (1989:229-234) lists that

agent as the planet Venus: Venus as Evening Star. The ant and the jaguar are associated with

Venus. The Maya religion has a close relationship with the Sun, Moon, and Venus, the three

most brilliant objects in the heavens. “Venus and the Moon marched together” (Aveni 1992:15).

The Moon is a very important clock for the visibility of Venus (Romano 1999:558). D. Juan Pio

Perez states that the Sun, Moon, and Venus all have a prominent role in the Maya universe, while

19

the other planets and stars occupy a relatively minor position (Closs 1978:148). The eclipse

overtones of the Venus Table, noted by Spinden and Smiley, are no accident (Kelly and Kerr

1973:188; Schove 1984a:23). Some Venus periods are eclipse period,

To the Maya, eclipses were dreadful times. Both solar and lunar eclipses were portents of the

end of time (Closs 1989:234). On page 72 of the Dresden Codex, the flood scene depicts the

flood that destroyed the previous creation. At the top of the downpours are glyphs of solar and

lunar eclipses. Like another story from the Popol Vuh, total solar or lunar eclipses could cause all

of the domestic instruments to be transformed into living creatures that could kill their masters

(Closs 1986:392). Eclipses also caused illness and deformity. Pregnant women and their infants

were extremely susceptible to eclipse effects. Infants would get gastrointestinal problems and

pregnant women would have their infants born with dark splotches, called sun and moon bites

(Closs 1986:391).

The most dreaded part of eclipses was the monster that would descend to earth to devour

people when the sun became obscured. On Page 58b there is an image of the Diving God (Figure

4). This monster is similar to the Mexican Tzitzimime Monster.

20

Figure 4 Ah Tzul Ahau Page 58b of the Dresden Codex after Thompson (1972)

Closs (1978:161; 1986:405, 409; 1989:229; 1992:143) has studied the ethnographic details of

Maya eclipses. He calls the image in Figure 4 the Ah Tzul Ahau, or the Ant Lord or Dog/Spine

Lord (Figure 1). The image is associated with the Venus god, Lahun Cahn. To frighten the

eclipse and to defend the moon, the Maya would make noise, shoot arrows into the air, and pinch

their dog’s ear to make them howl. This practice is described in a letter by Alfonso Dáavila in

1531. He stated that an eclipse would have inspired fear in the spirit of the Spaniards (Closs

1986:390).

Venus has a period of 584 days, which is a close approximation to the synodic period of

Venus, which is 583.92 days. The planet Venus does not affect the celestial mechanics of

eclipses, but there is a strong association between Venus and the phases of the moon. If the

21

Moon is at first quarter at morning helical rising, it will be at the same phase when on its last day

as Morning Star. It will reappear as Evening Star at the opposite phase or last quarter (Aveni

1992a:89; Juteson 1986:94-95). The Venus Table on Pages 46-50 of the Dresden Codex is

directly in front of the Eclipse Table. It is based on a period of 2,920 days. Lines 14, 20 and 25

are at or near eclipse intervals. Lines 14 and 20 are 11,960 days apart, which is forty-six times

the Tzolkin. Lines 20 and 25 are 9,360 days apart, which is thirty-six times the Tzolkin (Aveni

1992a:88). Two prominent dates in the Venus Table are 1 Ahau 18 Kayeb and 1 Ahau 13 Mac.

These dates are 11,960 days apart, the length of the Dresden Eclipse Table. The Maya date 3 Xul

to any date 1 Ahau 18 Kayeb is 9360 days (Kelly 1977:58-59).

On page 24 of the Dresden Codex are periods of forty-six times 260 days, which are equal to

104 times 115 days, the synodic period of Mercury (Förstemann 1967:114).

Eclipses are not very important to modern astronomers. The celestial mechanics behind

eclipses is fairly well known. Solar eclipses are used to study the sun’s corona. Lunar eclipses

are studied to measure the effect of pollution in the earth’s atmosphere. With the aid of modern

computers, astronomers are able to improve the accuracy of recording eclipses periods. These

improvements create only small changes in the onset, duration, and area covered by the eclipse.

Solar eclipses are very special events. They happen in the daylight and demonstrate dynamic

changes in the environment. The levels of light and heat diminish rapidly. Solar eclipses are very

fleeting, only lasting up to 7 minutes and 31 seconds in duration as the path of the moon’s

shadow sweeps across the earth. An observer would need to travel at speeds greater than 1,000

miles per hour to keep up with the shadow for an hour or more of a solar eclipse (Smiley

1961:212-213). Total solar eclipses occur about every year and a half but they are only seen in

22

the same location on earth about every 300 years. In contrast, lunar eclipses happen at night, so

the effects of the eclipse are not as noticeable as solar eclipses. The Earth’s atmosphere acts like

a prism, which leaks light into the shadow of the earth; producing the rusty red color of lunar

eclipses. Due to size and distance, the eclipse window for solar eclipses is larger, thus producing

more solar than lunar eclipses. Although more frequent, solar eclipses can only be seen in areas

of the earth where the sun’s shadow passes. Lunar eclipses last for several hours and can be

observed by anyone living where the sky is clear and the moon is above the horizon. This gives

an observer a 50% chance of seeing a lunar eclipse, but only an 8% chance of observing a solar

eclipse during their life (Lounsbury 1978:798).

Willson (1974:11) neglects the study eclipses of the moon in the belief that they are

unimportant to the Maya, but he gives no reason for this declaration (Guthe 1932:272; Spinden

1928a:144). “Contemporary Quiches regard the full moon as a nocturnal equivalent of the Sun”

(Tedlock, B. 1992:31; Tedlock, D. 1996:43).

The Sun’s rays shine out into space, illuminating one side of an object in space and creating a

long shadow behind that object. On the earth this process creates day and night; on the moon it

creates the phases of the moon, as seen on earth. The shadow is measured from the center of the

shadow’s vertex (called centrality.) The shadow has two parts; a dark inner region called the

umbra and a less dark outer region (called the penumbra). In solar eclipses, each time the moon

touches the boundary of a shaded area it is said to bite the sun. If the moon fits into each shaded

segment, the eclipse will be total; if not, then partial. This creates the two types of eclipses; the

partial and total eclipses. Some astronomers divide total and partial eclipses into four types; total

umbra and total penumbra, and partial umbra and partial penumbra. A fifth type of eclipse in

23

solar eclipses is the annular eclipse. This eclipse is caused by the vertex of the sun’s shadow

falling before the Earth, due to the larger distance of the sun from the earth. The shadow does not

completely cover the solar disc, creating an outer ring. A sixth type of eclipse, the hybrid, is total

or annular depending on the time of day of the eclipse. In hybrid eclipses, the eclipses are total

around midday and annular in the early morning and late afternoon due to the curvature of the

earth’s surface.

There are two other circumstances that control eclipses, but that are not classifications of

eclipses. One is the borderline eclipse, which occurs at the border between zones of totality and

partiality. A zone of partiality is at each artic pole with a zone of totality around the equator.

Eclipses that should be partial are total in the partial zone, and partial in the total zone. Another

circumstance is the grazing eclipse. These eclipses are the polar eclipses that occur (or fail to

occur) due to the flattened nature of the Earth’s poles. Eclipses at the poles are partials. In the

artic region the partial eclipses could be seen as total but this eclipse is seen from the opposite

side of the world. The line of centrality does not touch the earth, but the shadow does. Can these

eclipses really be defined as an eclipse? This is why astronomers refer to these types of eclipses

as non-central and central eclipses, where the shadow’s center line touches the Earth (Meeus

1997:43-44).

When discussing eclipses by year, it is important to remember that not all eclipses are visible

from any given point on the Earth. The minimum number of eclipses per year is four; two solar

and two lunar. The maximum is seven; five solar and two lunar, four solar and three lunar, three

solar and four lunar or two solar and five lunar eclipses (Meeus 1997:45-49). One point to be

realized is that one year, the time from January 1st to December 31st, is just a convention of

24

society. January 1st has no astronomical meaning. Due to the leap year addition of one day, the

time changes; therefore, a 365 day cycle may have more than five eclipses compared to a period

from January 1st to December 31st. In one year no more than two solar eclipses can be total, but

three lunar eclipses can. Four consecutive lunar eclipses may all be total ones, called Tetrad

(Meeus 1979: xiii). This is why a total lunar eclipse is said to either precede or follow another

total eclipse. Two lunar eclipses can occur at one lunation, but both are almost always

penumbral. Two successive new moons can be eclipses but most are partial and visible in

opposite hemispheres (northern and southern). Clusters of eclipses can be generated during a

period of three centuries (293 years) followed by three centuries of few or none. This has been

shown by Schiaparelli to be a period of 586 years (Meeus 1979:100).

25

CHAPTER TWO: DISCUSSION

Calendars and Maya Math

The idea of accuracy in Maya data is complicated by the fact that the Maya utilize a different

numbering system than Western societies. The Maya use the vigesimal system with a base of

twenty instead of ten. The Maya system contains no decimals. The Maya were aware of parts of

a whole, but their main concern was with completeness. The Maya did not possess complex

mathematics. They were counters and were very good at it. The Maya used a unique form of the

vigesimal writing called the bar-dot system. A dot represented one and was used in the numbers

1-4 and 6-9. A bar was used to represent five. A shell was used to represent a zero. The numbers

were positional based on the powers of twenty. “They count by fives up to twenty, by twenty to a

hundred and by hundreds to four hundred” (Landa 1978:40). This statement blends Western and

Old World ideas; one hundred is decimal, not vigesimal. The blending of ideas is one of the

pitfalls of research. It occurs again in eclipse research with the use of the Western idea of the

Saros to explain the Eclipse Table. The math used for calendars is called a modified vigesimal

system, because the tun position has eighteen uinals, instead of twenty, which gives a 360 day

count. For a detailed understanding of the Maya calendar and mathematics, An Introduction to

the Study of the Maya Hieroglyphs by Sylvanus G. Morley (1915) or Maya Hieroglyphic Writing

by J. Eric Thompson (1975) should be consulted.

Calendars, like eclipses, come in lunar and solar varieties. Time periods have been developed

to measure the cyclic motion of the heavens. The motions are variable; there are no uniform

processes in nature. The measure of time is simply the aggregate or mean of certain observed

motions expressed in arbitrary terms (McGee 1892:331). These problems have caused the need

26

for many different calendars from the Numan calendar, corrected by Julius Caesar, to

aculmination in our present day Gregorian calendar. This calendar did not escape criticism of its

inaccuracy. The Gregorian calendar affords a highly satisfactory compromise between essential

accuracy and much desired simplicity (Moyer 1982:152). The calendar eliminated ten days to

keep March 21 as the vernal equinox at the same date as the First Council of Nicea in 325. The

Church was also interested in having the Paschal full moon fall at Easter. The Paschal full moon

created more criticism than the Gregorian calendar did, but the calendar remains viable after

many centuries.

A second calendar, called the Julian calendar, was developed about the same time as the

Gregorian calendar by Julius Scalinger. Backers of the Julian calendar felt that it was more

astronomical than the Gregorian calendar thus more accurate. The Julian calendar uses three

cycles to calculate the date. Those cycles are the twenty-eight year solar cycle, the nineteen year

lunar cycle, and the fifteen year civil cycle of the Romans. This creates a cycle of 7980 years

(twenty eight times nineteen times fifteen). The calendar starts on the date B.C January 1, 4713.

The Maya also possessed two calendars. There is a 260-day religious calendar made up of

thirteen numbers (Trecena) and twenty day names (Vientena). This Sacred Calendar is also

called the Tzolkin or “Count of Days.” The names of the tzolkin are Imix, Ik, Akbal, Kan,

Chicchan, Cimi, Manik, Lamat, Muluc Oc, Chuen, Eb, Ben, Ix, Men, Cib, Cuban, Eznab, Cauac

and Ahau. This calendar always ends on a day named “Ahau.”

The Maya also use a 365-day calendar called the Haab. The haab is made up of twenty

numbers (0-19) attached to eighteen month names, giving a 360-day solar year called a Tun. A

nineteenth month of five days (0-4) called Uayeb makes up the 365-day tropical calendar. The

27

Maya also call the Uayeb, the xma kaba kin, or the days without name. The translation “days not

counted” has caused much controversy in the theories of Maya dates. These days need to be

counted to keep the Long Count going. This debate caused some researchers to question whether

the Maya used the “haab” or the “tun” (Long 1925:575).

The Maya combined the tzolkin and haab to create the Calendar Round. The Calendar Round

repeats itself every fifty-two years (365 times 52 equals 18,980 days). This equates to seventy-

three revolutions of the tzolkin (260 times 73 equals 18,980). Since the Calendar Round repeats

every fifty-two years, the Maya needed a way to identify which Calendar Round was referred to.

This was accomplished in two ways. One method, starting around the tenth century, was the

Mexican Katun Ahau method, sometimes referred to as the Short Count. This method identified

the Calendar Round by the ending tzolkin date. If the date was 3 Ahau, the Calendar Round was

said to be a Katun 3 Ahau. This date repeats itself every 260 tuns or about 256 years.

The other uniquely Maya method was the creation of the Maya Long Count or Initial Series

date. From right to left there are five columns of numbers. The first position is the count of days

or Kins. The second column to the left is the Uinals, which are twenty kins. The third column is

the Tuns, which are eighteen uinals of twenty days, giving the 360-day solar calendar. The fourth

is the Katun, which equals twenty tuns. The fifth position is the Baktun, which is twenty katuns.

The baktuns are also called cycles. In the Long Count system the base date of the Dresden

Eclipse Table is 9.16.4.10.8 12 Lamat 1 Muan.

The Maya Date is a count of days from a starting point of 4 Ahau 8 Cumhu. The Maya did

possess other calendars with different starting dates. Maya mathematics also possesses a system

to add and subtract dates known as Distance Numbers. These dates are written in reverse order to

28

distinguish them from Initial Series Dates. They could be added to or subtracted from the starting

date to create the new starting date. The Maya scribes would make charts of multiples of

numbers to calculate dates. Another series of numbers, also in reverse order, is the

Supplementary Series.

J. T. Goodman was the first to call attention to the glyphs that contained lunar information, but

it is Charles P. Bowditch who gave the glyphs the name, “the Supplementary Series” (Andrews

1951; Morley 1940, 1977). This information is sometimes placed before and sometimes after the

Calendar Round date. The majority of times this data is placed between the tzolkin and haab

portion of the calendar round. Not all Long Count Dates contained lunar information. The

proximity to the Initial Series Date made it appear to supplement the Initial date. The glyphs are

lettered A through G in reverse order. Later researchers found new information that they

identified as X, Y and Z. The Glyphs F through A are called the Lunar Series and provide

information on the moons age. Linden (1986:123) lists the sequence of the series as G, F, Z, Y,

E, D, C, X, B, A.

For this study only Glyphs A and C will be used. Glyph A represents either 29 or 30 days,

signifying the length of the month. Glyph C is never higher than six, indicating the number of

lunations. A zero indicates one or the current lunation.

The Ahau Equation was first suggested by the astronomer Willson (1974:17). The Ahau

Equation is a quantity of days that have to be added to the Maya date to get the Julian date; JD

(Julian Date) = MD (Maya Date) + Ahau Equation. Satterthwaite (1962:253) called the Ahau

Equation the Correlation Constant. This equation backs the Maya date to the starting point of the

29

Julian Count. The Long Count date of the Eclipse Table is 9.16.4.10.8 12 Lamat 1 Muan. The

above date is 9 Baktun, 16 Katun, 4 Tuns, 10 Uinals and 8 Kins.

9 Baktuns are 9 X 144,000 days = 1,296,000 16 Katuns are 16 X 7200 = 115,200 4 Tuns are 4 X 360 days, 1440 days 10 Uinals are 10 X 8 days = 180 8 Kins are 8 X 1 day = 8 The Maya date is equal to 1,412,848 days. The JD (Julian date) = MD (Maya Date) + constant (584285). This is 1,412,848 plus 584,285

which equals 1,997,133 days or Nov 12, 755, the difference is the Ahau equation of 584285

days. This is the constant used in the Goodman-Martinez-Thompson (GMT) correlation.

Different constants move the date forward and back in time by changing the Julian Date.

Numerous constants have been tried for correlating the Maya and Julian Dates with little

success. The GMT constant (584285) is the most accepted. J. E. S. Thompson also came up with

a modification of the GMT, called the Thompson modified (584283). This constant is only one

of many in the GMT Family of constants (584281-584288). The following researchers have

suggested constants that place the eclipse dates in the fifth Century: Willson (438906); Smiley

(482699); Makemson (489138); and; Spinden (489384).

The Maya calendar did not add leap years. Several theories have been proposed as to how the

Maya handled the leap year problem. Without adding leap years, the days drop back one day

every four years and create a Vague Year.

30

Eclipse Data

To compare the eclipse data, Excel worksheets have been prepared for each of the three canons

Oppolzer (1962), Liu and Fiala (1992) and Meeus (1966, 1979) for both lunar eclipses

(Appendix A – C) and solar eclipses (Appendix D, E). Liu and Fiala’s Canon does not have a

solar eclipse. Meeus’ Solar Canon only extends back to the year 1898. These Canons utilize

different methods for calculating eclipse occurrences. Oppolzer uses the mean-value method. Liu

and Fiala use the 1/50th Rule, which adds 1/50th the radius of the Earth to the eclipse

computation. Meeus uses the French Method, which compensates for perceived errors in the

1/50th rule. Small penumbral eclipses, found by the 1/50th rule for the enlargement of the

penumbra, do, in fact, not exist (Meeus 1979: x). Besides the different methods of computation,

the cannons also use different time measurements. Oppolzer’s Canon uses Universal Time,

whereas Liu and Fiala and Meeus use Ephemeris Time (Meeus, Grosjean and Vandreleen

1966:1; Sadler 1966:1121). Ephemeris time is an astronomical measurement of time not

dependent on movements of the sun. The difference between the two time periods is no more

than six hours and is listed in the ephemeris as ∆T. Fredrick Martin (1993:74-83) has studied

solar/lunar eclipse pairs. His chart uses U.S. Naval Observatory Ephemerides for the years 1970-

1992. This data appears compatible with the other Canons in this study (Martin 1993:86-92).

Debate has arisen about whether the Dresden Table is solar or lunar in nature. Willson

(1974:11) was one of the first to say that the Eclipse Table was solar. Most researchers since

have followed his lead. Martinez, Pogo, and Spinden thought the Table to be lunar.

Pogo (1937:159) made the comment that 33 years is sufficient to create an eclipse table from

observed lunar eclipses. Even the solar proponents agree that the Table must have been created

31

from lunar eclipse data. Visible solar eclipses are extremely rare and would require an extremely

long period of time to collect enough data to construct a Table. This question will not be fully

explored until observed eclipses are isolated for the general eclipse data. Nuclear physicist

Robert Smither has studied this question and concluded it could have been done in a single

lifetime (Campbell 1992:47). Justeson (1986:84) states that two or three decades of observation

and recording are necessary and sufficient to produce a model for the timing of eclipses so

complete that a system for anticipating all eclipse-possible dates would be revealed. This debate

about solar and lunar eclipses is the reason that both solar and lunar eclipses are analyzed.

These worksheets contain the Gregorian and Julian Dates of modern eclipses. The Gregorian

Dates of the canons are converted into Maya Dates utilizing the calculator

http://www.pauahtun.org/Calendar/tools.html. This site uses the Goodman-Martinez-Thompson

(GMT) constant of 584285 (an explanation of Maya Dates is given in the Calendars Section of

Chapter 2).

A column in the worksheets calculates the difference in days between Julian Dates. In this

column, the numbers 29, 30, 148, 176, 177 and 178 appear repeatedly. All of these dates, except

for the twenty-nine and thirty days, are major time periods of the Eclipse Table. Lounsbury

(1978:791; Satterthwaite 1947:147) mentions the instances of two solar eclipses one month

apart. The twenty-nine and thirty days are not readily apparent in the Table; however, the

eclipses are there and will be explained later. Another column in the worksheets calculates the

sum of the differences. These sums are similar to the totals in Schram’s Table. (Schram’s Table

is the repeated sums of 29.5.) This method of creating sums is also the method that Meinhausen

(1913) uses; however his totals are of 177 days. Meinhausen’s totals are used by other

32

researchers to identify eclipse periods. Except for the Long Count Dates, these worksheets are

similar to Guthe’s Table II (1921:6-7), which is used by all researchers of Maya eclipses.

The three charts of modern eclipse data are quite similar, with only slight variations due to the

differing methods of computing eclipse occurrences. The solar charts of Meeus and Oppolzer are

extremely similar, despite the decimal accuracy of Meeus. There is only one date that is different

between the charts. That date is September 1, 1997 in Meeus, and September 2, 1997 in

Oppolzer. The lunar charts have the most variance. Oppolzer’s chart contains forty-seven instead

of seventy-five eclipses. Oppolzer’s lunar charts have gaps where the non-visible eclipses are

located. Eclipses that are less than .07 in magnitude are not visible to the human eye. Magnitude

is a mathematical designation and cannot be detected by the naked-eye. These gaps are not a

problem when searching for observed eclipses but may be statistically valuable in other studies

of eclipses. Van den Berg (1955:20, 169) states that “Oppolzer’s Canon at once proves its

validity.” He even creates a version of his Eclipse Panorama using Oppolzer’s data. “Oppolzer’s

monumental work remains excellent for historical research” (Meeus 1979: xi).

There are four dates in Liu and Fiala that are different than Meeus. Those dates are December

10, 1973, December 20, 1983, April 4, 1996, and November 9, 2003. These differences are

caused by the increases in the Earth’s shadow due to the 1/50th Rule. This study will utilize

Meeus’ Canon for comparison with Maya data.

The Maya Tzolkin dates from the conversions are charted on a wheel similar to one Teeple

uses in his classic work, Maya Astronomy (1930). Teeple (1930:89, 1930a:138) was the first to

demonstrate the link between the three Eclipse Seasons and the Double Tzolkin. Spinden and

Ludendorf investigated this phenomenon, but did not complete the whole table, focusing instead

33

on the date 1 Imix. Three times the mean value of 173.31 days is 519.93 days, a close

approximation to the Double Tzolkin of 520 days and the three eclipse seasons.

Figure 5 Teeple Arc after Teeple (1930:89)

Teeple’s chart lists the dates from the Eclipse Table. These are not dates of actual eclipses but

rather date when eclipses could occur. The dates cluster around three groups which Teeple calls

arcs. Lounsbury refers to them as eclipse seasons and Bricker refers to them as danger windows

(Bricker and Bricker 1983:7). These clusters produce fail-safe areas where eclipses would not

occur. Not all the eclipses predicted would be visible in the Maya area, thus producing false-

alarms. The false-alarms cause the debate between the predictions versus the warning aspects of

the Table (Bricker and Bricker 1983:7-8).

34

Every other dangerous period in a single tzolkin is passed over, thus creating a Double Tzolkin

(Satterthwaite 1947:144-145). Teeple believed the eclipses distributed themselves around the

mean value of the seasons (the inner spokes in figure 5.). One of his papers mentions 166, 339,

or 512 as the dates of the mean-value (Teeple 1930a:138). His famous paper on Maya astronomy

places the dates one day later at 167 (11 Manik), 340 (2 Ahau), and 514 (7 Ix) (Teeple 1925:546-

548, 1928:547, 1930:90-91). Thompson (1975:234) provides the days numbers and dates of 168

(12 Lamat); 341 (3 Imix), and 514 (7 Ix), which would return to 12 Lamat after the next eclipse

half-year. Theses dates of nodes are nearly stationary. These nodes recede with each pass

through the Table. A regression of 1.61 days occurs in the Table (Teeple 1930:90, 1930a:138).

The eclipses would occur eighteen days on either side of the mean-value. Lunar eclipse would be

within a narrower limit of thirteen days (Teeple 1925:547). Bricker and Bricker (1983:6) believe

the first half of the eclipse arcs contain null predictions. The true predictions are in the last half

of the arcs.

Another Excel chart of the Double Tzolkin was used to sort the dates created by the conversion

of modern eclipse data. Entering dates into this chart works so well that it is possible to identify

errors in the conversions. The data has been rechecked on several occasions to insure its

correctness. While placing data in the chart, sequentially, the distribution of eclipses within the

three groups of seasons became apparent. (This process can be observed by following the eclipse

number (Ecl) in Meeus’ lunar season chart in Appendix G.) This process works equally well for

either solar or lunar eclipses. The only difference is the dates involved. By merging the two

season charts of lunar and solar eclipses (Appendix J), the same grouping of dates emerges. This

35

demonstrates that the three dates in the Eclipse Table could be used for either lunar or solar

eclipses.

The first trial of the Excel chart contained thirty-three years of eclipses. This is slightly larger

than the thirty-two and three-quarters years of the Maya Eclipse Table. The additional eclipses

have no effect on the three eclipse periods. The fewer number of eclipses in the Oppolzer Canon

also have no effect on these periods. The only differences in the seasons are the number of

eclipses in each group. A chart was also made of one half the previous Excel chart (appendix A).

This was done to simulate the effects of observed eclipses. The basic structure of the three

seasons remains fairly intact. There are noticeable changes in the outer boundaries of the seasons

and gaps where the missing eclipses should have been. These gaps would have been present in

the development of the Chaldean Table from visible eclipses (Pannekok 1961:60-62).

The copy of the chart is modified to the 32 and ¾ years for comparison against Teeple’s work

and the Dresden Eclipse Table. A chart was also made of the dates in the Dresden Eclipse Table

from Table II of Guthe (1978:6-7) to retest Teeple’s original work. These dates are found Table

8 of Maya Astronomy (Teeple 1930:87-88).

36

Figure 6 Meeus Arc following Teeple (1930:89)

The eclipses do distribute themselves in the seasons: not randomly, but sequentially by the

177/78 and 148 values plus the accumulated fractions of the eclipse period. The first feature that

becomes apparent is that the 148-eclipses are distributed at the beginning of the season. This is

the area of the “fictive” or “pre-nodal” eclipses. The 29 and 30 eclipses are distributed at the end

of the season, in the area of post-nodal eclipses.

Some of the 148-day events are followed by an eclipse one lunation later. This is at perigee, the

point in the earth’s orbit nearest the sun. The 29 and 30 eclipses are not readily apparent in the

Dresden Table. Some are hidden in the 177/178-day eclipse periods. There are only about six

37

eclipses of this type in the 33-year period. It is a problem of correlation studies that the

information that could specifically identify time periods is not available in the extant record.

Some of these early eclipses are not 148-day eclipses and eclipses at the end of the season are not

29 and 30 days. Some from time to time naturally distribute themselves at the beginning or end

of the season. Some of these eclipses do have a secondary relationship to the 148-day eclipses.

They either proceed or follow the 148-day eclipses.

Eclipse Periods

Sadler (1966:1119) stated that there are two methods to calculate eclipses. One method is to

precisely predict eclipse occurrences by using the theories of motions for the sun and moon. The

other is to use the mean periods derived from past observations. The former method of celestial

mechanics may well be needed to finally clarify the Eclipse Table, but further study of the mean

periods will aid the understanding of the how the Maya astronomers created the Table without an

understanding of celestial mechanics.

According to Liu and Fiala (1992:6-7) and Sadler (1966:1119-1120), there are three

requirements for an eclipse to occur; the moon must be in the same phase; the moon must be in

the same place with respect to the node, and the sun and moon must be at the same relative

distance. The distance is controlled by the orbit of the earth around the sun. The rotation of the

earth controls the time of day and the location of the eclipse.

The three periods that satisfy these requirements for eclipses are called the synodic month,

draconic month, and eclipse year. Another period involved in eclipses is the anomalistic month

of 27.55455 days. This is the period required for the moon to move from perigee to perigee, the

38

nearest point of the earth’s orbit around the sun. It is this cycle that determines if the central

eclipse will be a total or annular eclipse. (In an annular eclipse the shadow does not cover the

entire sun).

The smallest period for eclipses is the month or lunation. The sidereal cycle is the time it takes

for the moon to revolve around the earth relative to the stars. The sidereal month is 27.321661

days. Due to the fractional part of the moon’s orbit, the moon returns to the same place but at a

different time of day. In three sidereal cycles the moon returns to the same constellation at the

same time of day. The sidereal month does not meet the requirement of the same phase, but the

synodic month does.

The synodic month is the period of the moon’s revolution relative to the same phase of the

moon. The synodic month is 29.53058877315 days (Spinden 1928a:141). It is slightly larger than

the sidereal month. To the Maya, the Young Moon Goddess and an Old Moon Goddess

represented the waxing and waning moon (Thompson 1975:231). Eclipses can only occur at new

moon for solar and at full moon for lunar. A big debate has lingered over which phase of the

moon starts the Maya month (Guthe 1932:272). A month could be counted from full moon to full

moon or from new moon to new moon. The new moon can create problems because some

societies use the last visible crescent moon; some use the astronomer’s new moon at conjunction,

while other societies use the first visible crescent to start a lunar cycle. This creates about a three-

day spread for the lunar count. Barbra Tedlock (1992:30) has stated that the full moon is used by

Maya midwives because of the comparative ease of observation. Teeple (1930a:137) and

Spinden (1930:63-66) debated whether the Maya used the new or full moon. Landa (1978:59)

stated that “the Maya counted from the rising of new moon.” Teeple (1928:396, 1930:46, 49)

39

followed Landa’s lead. The Maya keep track of the moon’s age with glyphs D and E (Weitzel

1935:14). D and E show the age of the moon counted from the last new moon (Roys 1933:411).

Most scholars do agree that the Maya used the new moon for astronomy, but there is still much

debate about whether they use disappearance, conjunction, or first crescent for new moon for the

beginning of the cycle. Satterthwaite (1949:230, 1962:254) called the conjunction, “the Dark

Days” or the “Dark Phase.” This spread also creates debate about whether Maya astronomy is

based on observation or computation (Guthe 1932:272). Both observation and computation have

their own inherent problems. However, both may not be mutually exclusive.

Teeple’s (1925a:111-114) Table 1 of the Supplementary Series of moon ages shows complete

agreement. Roys’ (1969:165, 169) Table 2 charts moon ages by five tun intervals. Guthe

(1932:276-277) believed that the Supplementary Series was a computed record instead of one

based on observation. Weitzel (1935: 23) stated that the moon glyphs did not constitute an

observational record of new moons. Lounsbury (1978:774) “concluded that at many Maya sites

that moon age was reckoned from the first visibility of new crescent.” Shove (1984a:21-22) lists

deviations between the recorded and predicted moon ages. Satterthwaite (1951:142, 143, 152)

noted that some sites have double dates for moon ages. There is also a seven-day range of moon

ages that is excessive whether observed or calculated. He assumed that incorrect ages were

sometimes recorded for esoteric reasons (Satterthwaite 1951:142). Roys believed that certain

days were taboo as new moons and that an alternate had to be found. Teeple believed that the

errors are records of observation not of computation (Gibbs 1977:31).

The second requirement for eclipse periods is the node or the place where the moon crosses the

ecliptic. The ecliptic is the plane where the earth orbits the sun. If the moon were in the same

40

plane as the earth and sun, there would be a solar eclipse at new moon (conjunction) and a lunar

eclipse at full moon (opposition) every month. The moon actually orbits the earth at an angle of

5° 8´ and only intersects the plane of the ecliptic at two points: one node for lunar eclipses and

one for solar eclipses.

The draconic month is the period the moon takes to return to the same position relative to the

nodes of its orbit. Draconic refers to the dragon on pages 56 and 57, which eats the sun. The

draconic month is 27.212220 days: slightly less than the sidereal month due to the westerly drift

of the node. The nodes are not static. This drift is one of the reasons for the 148-day period of

eclipses. The nodes are classified by the orbit of the moon as it passes the node. If the moon is

going up in the orbit as it crosses the ecliptic, it is said to be an ascending node; if going down

the node is descending. Eclipses are measured from the ascending node. The shifting nodes

create problems in identifying which node is active. There is a solar node and a lunar node that

controls which type of eclipse occurs. The Saros period is such that the node does not shift. At a

half-Saros, the node will shift to the lunar node from a solar eclipse. In some other periods the

shift of nodes does not always change the eclipse from lunar to solar. The change in nodes is

between the Northern and Southern hemispheres.

It is only at or near these nodes that the sun, moon, and earth are aligned so their shadows

create eclipses. The nearness is called an eclipse window and is measured in angular distance.

Eclipses occur at an angle of 11° to 18°. From 11° to 14° the eclipses are total. Larger than 14°,

the eclipses are partial. Because of the size and distances involved, the solar eclipse window is

slightly larger than the lunar window. This means there is a statistical advantage for solar

eclipses over lunar. One would think lunar eclipses would be more prevalent, since the lunar

41

eclipse is visible to anyone when the moon is above the horizon. The solar eclipse is only visible

to the area of the earth that is covered by the Moon’s shadow. Due to the seasons when eclipses

occur, there is also a slight statistical advantage in the Northern Hemisphere.

The third requirement, which is distance, is controlled by the yearly revolution of the earth

around the sun. The eclipse year is 354 days, which is slightly shorter than a tropical year of 365

days. This allows an occasional three eclipses in one tropical year. Although eclipses can occur

after one lunation, the main periods for eclipses is the semester or eclipse half-year (Berlin

1943:156). The eclipse half-year could, as the ancient Chaldean astronomers pointed out,

function as a means of eclipse warning (Aveni 1981:80; Pannekoek 1961:57).

The semester consists of six lunations for a total of 177 or 178 days (8.17 or 8.18 vigesimal).

The semester also has an occasional period of five lunations or 148 days (7.8 vigesimal). The

148-day period was noted by Bowditch (1910:213) on page 53a of the Table (Pannekoek

1961:60). There must be one five-month season for each 6.623 six-month seasons (MacPherson

1987:444). Where to place the 178 and 148-days has been a problem for researchers. The 148-

day semester has been viewed as an adjustment and not as a part of the eclipse cycle. The mean-

value method has ignored this period. These periods are at the end of the seasons and are

considered fictive or potential eclipses. The mean-value would work on a static node system,

however; in a static system most 148-day semesters would not exist. The semester is six months

of alternating twenty-nine and thirty days. Table 2 demonstrates the semester alternations. The

twenty-nine and thirty days could be in any order. There is no right way or wrong way, but there

usually is a fixed way. The semester is set up this way because of the Supplementary Series

glyphs. If glyph C (1-6) is odd then glyph A (29 or 30) is even. If Glyph C is even then glyph A

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is odd. An occasional extra day is added to the last position of the semester to create two periods

of 30 days. This gives an occasional Glyph C and Glyph A, which are both even. This is

displayed on line 5 of Table 2.

Table 2 Lunar Semesters after Guthe (1932:275)

1 2 3 4 5 6 1 30 29 30 29 30 29 1772 30 29 30 29 30 29 1773 30 29 30 29 30 29 1774 30 29 30 29 30 29 1775 30 29 30 29 30 30 1786 30 29 30 29 30 148

Glyph C = 1- 6 Glyph A = 29 or 30

Linden (1986:125) suggested an eighteen month calendar was used with Glyph X, which has a

strong association with Glyph C. The Glyph X cycle is keyed directly to the moon number

(Justeson 1986:91). This relationship to the moon should be studied further.

The semesters can be added together to derive other periods of eclipse occurrences. Since the

Maya do not have decimals, they needed to find an integral number of moons in order to

calculate eclipses (Teeple 1930:65).

Meinhausen (1913) was the German astronomer who proved that the Dresden Codex Table

contained eclipse cycles. His data used the dates A.D. 1775 to A.D. 1808. He calculated the

difference between dates and the sum of those differences. These periods (sums) have become

the standard of eclipse periodicity for other researchers of Maya eclipses. He also noted a sum of

502 days (1.7.2. vigesimal). This sum is also prominent in the Oppolzer canon. The 502 days is

the sum of 177, 177 and 148. The 502-day period is prominent on page 53 of the Dresden Codex

43

(Bowditch 1910:213). The 502-day cycle is also the amount of time needed for the eclipses to

travel though the polar region. There is also a 325-day period in Oppolzer, which is 177 + 148

days. These are the hidden 148-day eclipses in Oppolzer’s canon. There are other periods of time

when eclipses can occur. Most eclipse cycles start without names. They are identified by the

lunations of the eclipse cycle or an approximation of the years in the cycle (Table 3). It should be

remembered that all eclipse periods have distinct advantages and disadvantages. The

disadvantages do not necessarily make these periods incorrect, but rather inappropriate for the

situation.

Because of its simplicity and accuracy, the Saros has become the preeminent eclipse reckoning

period. The Saros is used because this is the cycle that not only answers the when question of

eclipses but also informs us as to where an eclipse will occur. If an eclipse occurs, another will

occur 6585 1/3 days later, 120° to the west. The Saros cycle is 223 lunations (29.530588 times

223 equals 6585.321124). This is a period of 18 years 11 and 1/3 days. This third of a day is

what causes the next occurrence of the eclipse to be about 120° to the west. It is derived from the

Babylonians, but Chinese astronomers knew about it much earlier. The Saros is also involved in

the Numan cycle of the ancient Roman lunar calendar (Magini 2001:73). It takes three Saros

cycles to return to the same longitude. The Babylonians also knew the importance of the triple or

Mega-Saros of 54 years and one month.

The Saros is made up of six lunations. Some periods can be five lunations (Pannekok 1961:57-

60). Two consecutive new moons can each give rise to partial eclipses called a Nova (Van Den

Berg 1955:10). Babylonian science developed the Saros-Canon by noticing eclipses in

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succession (Pannekok 1961:60-62). Researchers believe the Maya astronomers did the same with

their Table.

Saros eclipses are given numbers to denote families or series of eclipses. Families are eclipses

separated by one Saros period. There are thirty-eight different families active at any given time.

There can be 69 to 86 eclipses in each family. Odd number eclipses are at ascending nodes and

even number eclipses are at descending nodes. It is important not to confuse Saros families and

the Saros period. Families of Saros eclipses are born, can live for 1226 to 1532 years, and then

die. They disappear and are replaced by a new family. Families are separated by a 29-year

period. The Saros period is such that it does not produce a node shift as in other periods. The

node shift causes the eclipses to jump from the northern to southern hemisphere. A new Saros

series will be born on July 1, 2011 (Meeus 1997:49-51). It will contain sixty-nine eclipses. This

is the same number of eclipses as in the Eclipse Table. The 38th eclipse in the Table is the Saros.

It is the 41st in Appendix A. The 61st eclipse is the length of the Schram Table or 10,571 days

(Makemson 1943:190). Researchers were glad to find the Saros period in the Table. The other

periods are also there, but no one actually looked for them.

In the first section of the Eclipse Table (page 52b, column D) is the number 18.5.5 vigesimal.

This is 6585 days in length: the same period as the Saros (Milbrath 1999:114). The Saros is not a

popular system with the Maya because the Saros is not divisible by 260. The Table can only

express one Saros because the Table is only 33 years in length and two Saros periods are 36

years. The Numan Cycle has been compared to the Dresden Eclipse Table in Table 11 (Magini

2001:106). There is a period of one half of the Saros cycle. At this point there is a shift in nodes,

45

from solar to lunar eclipses. This is how one creates a lunar eclipse table from solar eclipse data.

In like manner, the Maya could make a solar eclipse table from a lunar table.

Stockwell (1901:185) lists an unnamed cycle of seven years or eighty-eight lunations for 2598

days. Van Den Berg (1955:28) calls this period the anonymous. The seven-year cycle is good for

predicting eclipses over short periods of time, but loses accuracy after about 250 years. The

nodes change so that 14 ½ years are required for eclipses in the same area. Robert Smither

(1986:104) has studied the 88-month cycle to predict periods of minimal lunar activity indicating

solar eclipses. Campbell (1992:53) has also studied the relationship between lunar activity and

solar eclipses. Low lunar activity is believed to be where the five-lunation periods are located.

The 88-lunations period is made up of 41 and 47 lunations: a missing part of the Saros eclipse

period (Pannekoek 1961:58; Smither 1986:99-111). The Table is forty-one lunations short of two

Saros periods (Satterthwaite 1978:799). It is not known if the Maya were aware of this 88-

lunation eclipse period, but these are the dates in the Season Tables (Appendix G, H, and I)

which have duplicate occurrences.

After 19 years and 11 days the moon returns to the same position in the sky. This cycle is also

referred to as the Metonic cycle or the Meton. It is named for the Greek philosopher Meton, but

all societies that have lunar calendars were aware of this cycle. The Chinese called it the Tchang

(Mcgee 1892:329). The Metonic cycle is 235 lunations or 6,940 (19-5-0 vigesimal) days

(Spinden 1930:49). The Meton is an easy way of predicting phases of the moon, but it does not

take into account the drift of the nodal line. The five- lunation period creates some dates that

should be eclipse dates, but the actual eclipse has occurred one lunation earlier. Not all full

moons are eclipse full moons; since the Meton only predicts 9 out of 10 eclipses, some have

46

called it the so-called eclipse period (Carlson 1984:236). The Meton is the basis of the Golden

Number of the Greeks. The Golden Number is one more than the remainder of the year divided

by nineteen (Pannekoek 1961:218). The Meton is the nineteen-year cycle of the Julian calendar.

The Maya may also have known about the Meton, but Lounsbury (1978:804) states “the Metonic

cycle appears to have attracted no particular attention.”

Teeple (1928:392-394, 1930:56-69, 1930a:137) researched the Lunar Series dates and noticed

a Period of Uniformity in lunar data. Prior to Uniformity, sometime around the Maya date

9.12.15.0.0 - 9.13.0.0.0, there was a period when each area had its own system for moon

numbering. During Uniformity, between 9.13.0.0.0 and 9.16.0.0.0, all areas used the same

system for moon the numbering system. The system was called the “Palenque System.” The

system uses an 81-moon count (6.11.12 equals 2392 days) to calculate moon numbering (Beyer

1935:66). This ratio of 81 moons to 2,392 days gives a mean lunar month of 25.530864 days

Lounsbury 1978:775). This system counted lunar half-years of six moons each

(Justeson1986:86-91; Satterthwaite 1959:200). Berlin (1943:156) has doubted some of Teeple’s

finding about Glyph C and the 6 moon groupings because two dates did not match what should

have been expected. The 81 lunations are one-fifth of the Eclipse Table of 405 lunations. This

period is not an eclipse period, but the nine lunations may have a connection with Venus (Aveni

1986:315).

The Palenque System fell to a new system instituted at Copán. This system uses 149 moons

(12.4.0) equaling 4,400 days. The ratio of 149 moons to 4,400 days gives a mean lunar month of

29.530201 (Lounsbury 1978:775). This is the number some people use to claim the increased

accuracy of the Maya calendar over the Gregorian calendar. At Copán, the Meton period shows

47

up in the monument dates. The Meton is one katun minus one tzolkin, which equal 6,940 days

(Spinden 1928:49, 1930:49). Stela A has the date 9.14.19.5.0 4 Ahau 18 Muan. This is 19.5.0

after the katun ending date of 9.14.0.0.0. This date is linked to the date 9.11.19.5.0 10 Ahau 13

Ceh on Stela I. This date is 3 Katuns earlier. The period 19.5.0 is the Meton (Chambers

1965:350-351; Milbrath 1999:106; Morley 1920:178, 222; and Teeple 1930:71). This period is

also associated with Stela C and H and with altar U (Spinden 1928a:145). Stela H has the date

9.14.19.5.0 4 Ahau 18 Muan (Baudez 1994:59).

On Copán’s Altar Q on the Maya date 9.16.12.5.17 6 Caban 10 Mol, a supposed astronomical

congress took place at Copán, on the Gregorian date July 2, 763 (Carlson 1977:101). The ruler is

interpreted as New-Sun-at-Horizon. On December 11, 847, there was a nearly total eclipse at

sunrise (Schove 1982:251). Schove claimed confirmation of his method because of these dates

and the “at Horizon” clause. The major problem of this theory concerning Altar Q is that the altar

has no dates of astronomical significance (Baudez 1994:97).

The cycle of 135 lunations is the Tritos (Stockwell 1901:186). This is a period of 3,986 2/3

days. Willson (1974:15) calls this period the Maya Saroid (Spinden 1969:71; 1928a:145;

1930:52). The Maya appear to have used the Tritos (Smiley 1973:179). The Dresden Codex is

made up of three sections of Tritos for a period 11,960 days or 405 lunations. This period is

given the name Maya (Van Den Berg 1955:24).

Van Den Bergh (1955) studied a twenty-nine year period that is described by Stockwell

(1901:186) and Crommelin (1901:380), but did not have a name at that time. Oppolzer knew

about this period, but did not use it. Van Den Berg calls this period the Inex because it was the

time that an eclipse enters or exits an eclipse zone (In-Exit). This period is twenty-nine years

48

minus twenty days or 358 lunations which equals 6,940 days. The period consists of no fewer

than seventy families co-exist at one time. Each family has 780 members. The lifetime of each

family is 22,600 years (780 times 29 years). The family member enters at one of the poles,

entering a zone of partiality. After 140 partial eclipses, the family enters a zone of centrality at

the tropical area of the earth. After 250 eclipses, the family reaches the equator. After 250 more

central eclipses, the family enters the other zone of partiality at the opposite pole. The eclipses

alternate between the north and south partial zones due to the alternation of ascending and

descending nodes (as opposed to the Saros, which does not change nodes). The Central eclipses

alternate between the northern and southern hemispheres. Since multiple families are active at

the same time, it is possible for one family that enters a zone later than another family to produce

an eclipse sooner than another family. Also, there is a gap of time from when one family leaves a

zone and another family enters that zone. The Inex has an advantage over the Saros over long

periods, but has unfavorable results with the anomalistic month. All eclipse periods have

strengths and weakness. That is why researchers should look at more periods than just the Saros.

It is not that the Inex should replace the Saros, but that both can be used to create new periods of

eclipses.

One last fact about the Inex is the Inex law, which states that when a family enters a zone, it

stays in that zone until it exits it. One situation that seems to disprove that law is the border

eclipses. Although the family has not entered the other zone, circumstances can cause a partial

eclipse in the central zone or total eclipse in a partial zone. This creates undulations or clusters in

the number of eclipses. These clusters seem to follow a cycle of 358 years. Van Den Bergh

(1955) created a Panorama of eclipses, by organizing eclipses in rows and columns. The

49

columns are Tritos periods and the rows are Saros periods. The Inex is the diagonal, one Saros

plus one Tritos (Meeus, Grosjean and Vandreleen 1966:41). With this panorama of eclipses Van

Den Bergh was able to develop a formula, T = mS + nI, to find the time intervals between

eclipses. The Inex minus the Saros is 358-223 equals 135 lunations, or a Tritos. A Saros minus

Tritos (223 – 135) equals eighty-eight lunations called an Anonymous. With this formula, any

eclipse period could be found since all are combinations of Inex and Saros periods.

Table 3 Eclipse Periods after Table 4 Van Den Berg (1955:28)

Name Lunations Time Formula 1 1 month 5 5 months Semester 6 6 month 5T-3S or 5I-8S Anonymous 88 7 years Saros - Tritos Tritos 135 11 years - 1 month Inex - Saros Saros 223 18 months + 11 days Meton 235 19 months Inex 358 29 years Saros + Tritos

Astronomer Charles Smiley was not impressed with George Van Den Bergh’s formula for

eclipse periods. He believed that the statement amounted to nothing more than stating that a solar

eclipse occurs at new moon (Smiley 1975a:133). Jean Meeus (1975) and Charles H. Smiley

(1975) debated each other in articles about the Saros–Inex eclipse periods. Smiley (1973, 1975,

and 1975a) introduced two periods of solar observation coordinated with the Maya sacred

calendar of 260 days. These periods he called the Thix and the Fox. The Thix is THirty-sIX times

260 days and equals 9360 days. The Fox is FOrty-siX times 260 and equals 11,960 days. These

are the periods (11,960 and 9360) mentioned by Aveni in the Venus section of this thesis (Kelly

50

1977:59, 1992:88). This is why Smiley (1973:177-178) called the Venus Table a Solar Eclipse

Prediction Table. In Konnen and Meeus (1976:81), the authors state that the Maya Thix period

(317 lunations) can be expressed as 4I – 5S as well as 227I – 363S. The second equation

produces a shift in the node.

Smiley also describes a period of forty-one times the 260 days, equaling 10,660 days. He

called this period the Fone (pronounced phony) because it is not a solar eclipse interval (Smiley

1975:255; 1975a:134). The Thix minus the Fox is equal to ten tzolkin. This interval is 1.31 days

more than one subdivision of the Saros, or the 88 lunations. Subtracting 2600 days from any 12

Lamat date will locate another 12 Lamat date, 1 day prior to another new moon (Lounsbury

1992:204).

Simulation

The origins of the Eclipse Table point to a lunar calendar as the origin of the Table. Belmont

(1935:147) called it “a Lunar Eclipse Count.” Spinden (1930:42) called it the so-called Lunar

Calendar of the Maya. Miles (1949:275) found evidence of a survival of a lunar count of fifty-

nine days in Chiapas Mexico. The lunar calendar is 354 days, or twice the length of the eclipse

half-year of 177 days. Lunar calendars also have the fifty-nine day periods, as those in Table 2.

Lunar calendars are tied to the seasons rather than to eclipses. However, studying the eclipse

half-years, a pattern of eclipses would appear.

Morley (1977:395) states that “the next point to be determined is the sequence of the twenty-nine

and thirty-day lunar months as they actually occurred.” Many researchers have studied the

twenty-nine and thirty day periods. Table VIII of Guthe (1978:19) lists the days in a thirtyfour

51

month period. Table XI of Guthe (1978:25) arranges the periods in the 177 and 148 day patterns.

Table I of Belmont (1934:145-146) arranges the Table by the sixty-nine eclipses. Beyer’s Table I

(1935:71) creates the eclipse half year periods without the 148-lunation periods. Beyer’s,

(1937:78-80) Table II does list the 148 period, but Tables III and IV only expresses the 177 and

178-day periods. Lounsbury’s (1978:800) study of the twenty-nine and thirty-day eclipses found

that 11,960 days minus the 405 lunations times 29.5 are 12.5 days short of the total.

Satterthwaite’s (1947:71; 1948:61) study of the twenty-nine and thirty-day groupings is made

up of sixty 6-moon groups (177 days) and nine 5-moon groups (148 days). Merrill (1946:40)

creates a chart of the twenty-nine and thirty days, which contains no 148-day periods. The

occasional double thirty-day grouping is a correction to the chart. This created a debate between

Satterthwaite (1948:61) and Merrill (1949:228) about half-day and full-day corrections. Do you

add a half or subtract one half? Does the half day matter? A high degree of accuracy was never

sought and not required for the Maya problem (Satterthwaite 1948:62).

Guthe (1932:274-275) did look into the Tritos, but later rejected it because of the problems

with the 59-day cycle and the five lunations. Western astronomy is based on the Saros and the

six lunations, not five.

52

Table 4 Guthe's Semesters after Guthe (1932:275)

1 2 3 4 5 6 30 29 30 29 30 29 17730 29 30 29 30 29 17730 29 30 29 30 30 17830 29 30 29 30 29 17730 29 30 29 30 30 17830 29 30 29 30 30* 14829 30 29 30 29 30 17729 30 29 30 29 148

Glyph C = 1- 6 Glyph A = 29 or 30

Moon groups are arranged in six lunation sets, while five lunation groups upset the parallelism

(Beyer 1933:309). At the 148-day group, the relationship between the Glyph A and C reverses

from odd to even (line 7 of Table 4). The five lunations also upset the belief in the six lunations

of the Saros. There is a five-lunation period in the Saros called a Nova (Van Den Berg 1955:10).

Guthe (1932:276) stated “the manuscript table can not be applied to the record of the inscriptions

as it stands because of the existence of the five-month groups which were not in use during the

Period of Uniformity.”

The six lunation groupings are compatible with glyph X of the Supplementary Series. Linden

(1896:125) created an eighteen-month calendar based on three lunar semesters. Although not

associated with eclipses, further studies need to be done in relation to Glyph X and the moon.

In computing a series of lunations in a Tritos fashion, more than 11,960 days (11,981)

appeared. The 23rd eclipse was at 4,013, not at 3,986, days. This was done by using seven

grouping of 177 days to one of 148. The problem is that the eclipses alternate between six and

seven groups of the 177 days. The key to the Tritos structure is the five lunation groupings.

53

Properly adding the 148-groups gives a total slightly smaller than the required number. This is

due to the missing 178-day corrections. This discrepancy was amajor question that early

researchers faced; where to add the 178 and 148-groupings? This also provides a clue about one

of the questions that has not been fully explained. Is the Eclipse Table computed or is it created

by observation of eclipses? At some point, computations would have to be checked against

actual observations, but computation and observation are not mutually exclusive.

Table 5 Tritos Simulation of Table 2

1 1 30 29 30 29 30 29 177 1772 2 30 29 30 29 30 29 177 3543 3 30 29 30 29 30 29 177 5314 4 30 29 30 29 30 29 177 7085 5 30 29 30 29 30 29 177 8856 6 30 29 30 29 30 29 177 10627 7 30 29 30 29 30 29 177 12398 8 30 29 30 29 30 148 13879 9 30 29 30 29 30 29 177 1564

10 10 30 29 30 29 30 29 177 174111 11 30 29 30 29 30 29 177 191812 12 30 29 30 29 30 29 177 209513 13 30 29 30 29 30 29 177 227214 14 30 29 30 29 30 29 177 244915 15 30 29 30 29 30 29 177 262616 16 30 29 30 29 30 148 277417 17 30 29 30 29 30 29 177 295118 18 30 29 30 29 30 29 177 312819 19 30 29 30 29 30 29 177 330520 20 30 29 30 29 30 29 177 348221 21 30 29 30 29 30 29 177 365922 22 30 29 30 29 30 29 177 383623 23 30 29 30 29 30 148 3984

54

24 1 30 29 30 29 30 29 177 416125 2 30 29 30 29 30 29 177 433826 3 30 29 30 29 30 29 177 451527 4 30 29 30 29 30 29 177 469228 5 30 29 30 29 30 29 177 486929 6 30 29 30 29 30 29 177 504630 7 30 29 30 29 30 148 519431 8 30 29 30 29 30 29 177 537132 9 30 29 30 29 30 29 177 554833 10 30 29 30 29 30 29 177 572534 11 30 29 30 29 30 29 177 590235 12 30 29 30 29 30 29 177 607936 13 30 29 30 29 30 29 177 625637 14 30 29 30 29 30 29 177 643338 15 30 29 30 29 30 148 658139 16 30 29 30 29 30 29 177 675840 17 30 29 30 29 30 29 177 693541 18 30 29 30 29 30 29 177 711242 19 30 29 30 29 30 29 177 728943 20 30 29 30 29 30 29 177 746644 21 30 29 30 29 30 29 177 764345 22 30 29 30 29 30 29 177 782046 23 30 29 30 29 30 148 7968

47 1 30 29 30 29 30 29 177 814548 2 30 29 30 29 30 29 177 832249 3 30 29 30 29 30 29 177 849950 4 30 29 30 29 30 29 177 867651 5 30 29 30 29 30 29 177 885352 6 30 29 30 29 30 29 177 903053 7 30 29 30 29 30 29 148 917854 8 30 29 30 29 30 177 935555 9 30 29 30 29 30 29 177 953256 10 30 29 30 29 30 29 177 970957 11 30 29 30 29 30 29 177 988658 12 30 29 30 29 30 29 177 1006359 13 30 29 30 29 30 29 177 1024060 14 30 29 30 29 30 148 1038861 15 30 29 30 29 30 29 177 1056562 16 30 29 30 29 30 29 177 1074263 17 30 29 30 29 30 29 177 1091964 18 30 29 30 29 30 29 177 1109665 19 30 29 30 29 30 29 177 1127366 20 30 29 30 29 30 29 177 1145067 21 30 29 30 29 30 29 177 1162768 22 30 29 30 29 30 148 1177569 23 30 29 30 29 30 29 177 11952

55

After creating Table 5, the next step is to separate the Meeus Solar data into the three Tritos

groupings at 3,986 and 7,972 days. This task is rather simple; the tricky part is in matching the

177 and 148 lunations with those found in the Dresden Table. The problem is that the Meeus

chart is not aligned with the Table’s sequence. The problem is created by the one-lunation (29.5)

eclipses. The 29.5 lunation is used to differentiate these twenty-nine and thirty-day lunations, but

it should be remembered that the 29.5 can be either of these lunations. The 29.5 lunations are

hidden within the 177-day period. It is this group that causes the alternations between the six and

seven groupings of the 177-day period. The 29.5 period occurs between two 148-day periods.

With two 148-day periods, which 148-day grouping is pictured in the Table?

When comparing the Dresden Eclipse Table with the Meeus data (Appendix K), other periods

than the six and seven groupings appear in the Eclipse Table. The Dresden Eclipse Table is not

laid out in a six-seven groupings of 177 days. This would assume that observation played a big

part in the 148-day periods placement, as Teeple (1930:91) believes. Combining the one-lunation

groups with a 148- day period reduces seventy-five eclipses to sixty-seven. This would suggest

that at least two eclipses in the Eclipse Table are at the 29.5 day lunations; the others are hidden

in the 177-day groups. The Lunar eclipses of Meeus (Appendix K) appear to be a closer fit to the

Table, but like Willson’s study are not perfect matches. This is as far as general data will be

useful for studying the Maya eclipses. A further investigation of observed eclipses may better

establish some benchmarks to aid this investigation.

56

CHAPTER THREE: CONCLUSION The Eclipse Table has great flexibility. It can be a sysygy, a lunar calendar, and a warning

table, all at once. The Table can easily be made by a list of eclipses, either solar or lunar. It

would be very valuable to have the source material that the Maya used to create this Table. The

other unfortunate problem is that the Dresden Codex contains the only copy of a Maya Eclipse

Table. Another copy would answer questions about the Base Date and the ritual nature of day

Lamat, in the Table.

The origins of the Table cannot be determined by general data, but this data is compatible with

a lunar origin for the Eclipse Table. As Pogo and Smither have suggested thirty years worth of

data is sufficient to create the Table, but how long would it take to acquire a list of thirty years

worth of eclipses by observation, in order to fill in the gaps. This method was the way the

Chaldean Tablets revealed the Saros period. McGee (1892) has noticed that all societies which

observed the moon have recognized different periods associated with the moon. Some are more

relevant and are given names. The three groupings of the Eclipse Table are the Maya Sariod of

3896 days, the Tritos. The difference between the Saros and Tritos suggests an independent

development of the Maya calendar and astronomy.

Observation, record keeping, and experimentation make eclipse prediction possible. Increased

accuracy is not needed for eclipse predictions. Oppolzer’s data serves well for most studies. With

the many different methods of eclipse approximation, the key to eclipse predictions is the new

moons and full moons. Not all new moons and full moons produce eclipses.

The three sections of 69 eclipses are not divided by the seasons, but by the 135-lunation period

Tritos. The Table is not a list of eclipse dates, but rather dates when eclipses could occur. The

57

Table does chart all but six eclipses. These are the eclipses at one month intervals. They may be

picked up by the 15-day dates. These dates need further study. What are their importance,

function and relationship to the 148-day period? These dates may help determine the relationship

between solar and lunar eclipse. It is curious that the Maya, who had no interest in lunar eclipses,

would place a lunar eclipse glyph along side the solar eclipse glyph suspended from sky-band

images in the flood scene elsewhere in the Dresden Code (Figures 2 and 4).

In summary, this thesis expands the examination of eclipse data beyond the mean-value of

173.31 days. This will further the investigation of eclipse dates instead of nodal passage. The

148-day and 29.5 day periods, ignored by earlier researchers, may hold clues to proper date

alignment of the Eclipse Table. These periods have been overlooked by earlier researchers; their

only importance has been related to the location of the pictures in the Table. The 148-day

eclipses are caused by the rotation of the nodes. They are also produced by the sun at perigee,

which allows the eclipse window to remain open at one lunation, producing another 148 and 29

day eclipse from the 177-day period. This is the reason for the six and seven groups of 177-day

eclipses found in the eclipse data. Unfortunately, the Eclipse Table does not have the six and

seven groupings. The task now is to find specific eclipses visible in the Maya area that will

match the computations of eclipse periods.

More attention is required concerning the three sequential dates in the Table. Meeus’ data

(Appendix A) does appear to support the Lunar Variation Theory. The dates have been

overlooked in earlier studies. The emphasis has been on the numbers and totals.

The fifth century eclipse need further study since they are at the time of the other correlations

that have been suggested. What is the function of the multiples of the Dresden Eclipse Table?

58

How many times could the Table be used? Even Glyph X with its six lunation periods should be

investigated thoroughly.

A deeper understanding of the role Venus plays in eclipse prediction should be investigated. As

Satterthwaite (1962:258) states, the Maya were interested in the relationship of Venus to solar

eclipses; the Venus Table implies an interest in the Venus-moon relationships. The Thix and Fox

are eclipse periods in the Venus Table. More studies are needed into the other charts of the

Dresden Codex and their relationship to eclipse predictions.

As Guthe said, “The Indigenous records of the Maya lunar count still contain many interesting,

unsolved problems” (1932:277).

59

APPENDIX A MEEUS LUNAR DATA

60

Lunar 1971 to 2003 Meeus P 1Year MM DD Julian Diff Sum Tzolkin Haab

1 1971 8 6 2441170.32 177.5 177.5 12 17 18 0 5 8 Chicchan 13 Xul2 1972 1 30 2441346.95 176.63 354.13 12 17 18 9 2 3 Ik 10 Muan3 1972 7 26 2441524.8 177.85 531.98 12 17 19 0 0 12 Ahau 3 Xul4 1973 1 18 2441701.39 176.59 708.57 N 12 17 19 8 16 6 Cib 19 Kankin5 1973 6 15 2441849.37 147.98 856.55 N 12 17 19 16 4 11 Kan 2 Zotz6 1973 7 15 2441878.99 29.62 886.17 N 12 17 19 17 14 2 Ix 12 Tzec7 1973 12 10 2442026.57 147.58 1033.8 X* 12 18 0 7 2 7 Ik 0 Mak8 1974 6 4 2442203.43 176.86 1210.6 * 12 18 0 15 18 1 Eznab 11 Zip9 1974 11 29 2442381.13 177.7 1388.3 12 18 1 6 16 10 Cib 9 Ceh

10 1975 5 25 2442557.74 176.61 1564.9 12 18 1 15 13 5 Ben 1 Zip11 1975 11 18 2442735.43 177.69 1742.6 * 12 18 2 6 10 13 Oc 18 Zac12 1976 5 13 2442912.33 176.9 1919.5 12 18 2 15 7 8 Manik 10 Uo13 1976 11 6 2443089.46 177.13 2096.6 N* 12 18 3 6 4 3 Kan 7 Zac14 1977 4 4 2443237.68 148.22 2244.9 12 18 3 13 13 9 Ben 16 Cumhu15 1977 9 27 2443413.85 176.17 2421 N 12 18 4 4 9 3 Muluc 7 Chen16 1978 3 24 2443592.18 178.33 2599.4 12 18 4 13 7 12 Manik 5 Cumhu17 1978 9 16 2443768.29 176.11 2775.5 12 18 5 4 3 6 Akbal 16 Mol18 1979 3 13 2443946.38 178.09 2953.6 * 12 18 5 13 1 2 Imix 14 Kayeb19 1979 9 6 2444122.95 176.57 3130.1 12 18 6 3 18 10 Eznab 6 Mol20 1980 3 1 2444300.37 177.42 3307.6 N 12 18 6 12 15 5 Men 3 Kayeb21 1980 7 27 2444448.3 147.93 3455.5 N 12 18 7 2 3 10 Akbal 6 Xul22 1980 8 26 2444477.65 29.35 3484.8 N 12 18 7 3 13 1 Ben 16 Yaxkin23 1981 1 20 2444624.83 147.18 3632 N 12 18 7 11 0 5 Ahau 3 Muan24 1981 7 17 2444802.7 177.87 3809.9 12 18 8 1 18 1 Eznab 16 Tzec25 1982 1 9 2444979.33 176.63 3986.5 12 18 8 10 14 8 Ix 12 Kankin26 1982 7 6 2445156.81 177.48 4164 12 18 9 1 12 4 Eb 5 Tzec27 1982 12 30 2445333.98 177.17 4341.2 12 18 9 10 9 12 Muluc 2 Kankin28 1983 6 25 2445510.85 176.87 4518 * 12 18 10 1 6 7 Cimi 14 Zotz29 1983 12 20 2445688.58 177.73 4695.8 XN* 12 18 10 10 4 3 Kan 12 Mac30 1984 5 15 2445835.7 147.12 4842.9 N 12 18 10 17 11 7 Chuen 14 Uo31 1984 6 13 2445865.1 29.4 4872.3 N 12 18 11 1 0 10 Ahau 3 Zotz32 1984 11 8 2446013.25 148.15 5020.4 N 12 18 11 8 8 2 Lamat 11 Zac33 1985 5 4 2446190.33 177.08 5197.5 12 18 11 17 5 10 Chicchan 3 Uo34 1985 10 28 2446367.24 176.91 5374.4 12 18 12 8 2 5 Ik 0 Zac35 1986 4 24 2446545.03 177.79 5552.2 12 18 12 17 0 1 Ahau 13 Pop36 1986 10 17 2446721.3 176.27 5728.5 12 18 13 7 16 8 Cib 9 Yax37 1987 4 14 2446899.6 178.3 5906.8 N 12 18 13 16 15 5 Men 3 Pop38 1987 10 7 2447075.67 176.07 6082.9 12 18 14 7 11 12 Chuen 19 Chen39 1988 3 3 2447224.18 148.51 6231.4 N 12 18 14 14 19 4 Cauac 7 Kayeb40 1988 8 27 2447400.96 176.78 6408.1 12 18 15 5 16 12 Cib 19 Yaxkin

61

Lunar 1971 to 2003 Meeus P 2Year MM DD Julian Diff Sum Tzolkin Haab

41 1989 2 20 2447578.15 177.19 6585.3 12 18 15 14 13 7 Ben 16 Pax42 1989 8 17 2447755.63 177.48 6762.8 12 18 16 5 11 3 Chuen 9 Yaxkin43 1990 2 9 2447932.3 176.67 6939.5 12 18 16 14 7 10 Manik 5 Pax44 1990 8 6 2448110.09 177.79 7117.3 12 18 17 5 5 6 Chicchan 18 Xul45 1991 1 30 2448286.75 176.66 7293.9 N 12 18 17 14 2 1 Ik 15 Muan46 1991 6 27 2448434.64 147.89 7441.8 N 12 18 18 3 10 6 Oc 18 Zotz47 1991 7 26 2448464.26 29.62 7471.4 N 12 18 18 4 19 9 Cauac 7 Xul48 1991 12 21 2448611.94 147.68 7619.1 12 18 18 12 7 1 Manik 15 Mak49 1992 6 15 2448788.71 176.77 7795.9 12 18 19 3 4 9 Kan 7 Zotz50 1992 12 9 2448966.49 177.78 7973.7 * 12 18 19 12 1 4 Imix 4 Mac51 1993 6 4 2449143.04 176.55 8150.2 12 19 0 2 18 12 Eznab 16 Zip52 1993 11 29 2449320.77 177.73 8328 12 19 0 11 16 8 Cib 14 Ceh53 1994 5 25 2449497.65 176.88 8504.8 12 19 1 2 13 3 Ben 6 Zip54 1994 11 18 2449674.78 177.13 8682 N 12 19 1 11 10 11 Oc 3 Ceh55 1995 4 15 2449823.01 148.23 8830.2 12 19 2 0 18 3 Eznab 6 Pop56 1995 10 8 2449999.17 176.16 9006.4 N 12 19 2 9 14 10 Ix 2 Yax57 1996 4 4 2450177.51 178.34 9184.7 X* 12 19 3 0 13 7 Ben 1 Uayeb58 1996 9 27 2450353.62 176.11 9360.8 12 19 3 9 9 1 Muluc 12 Chen59 1997 3 24 2450531.69 178.07 9538.9 12 19 4 0 7 10 Manik 10 Cumhu60 1997 9 16 2450708.28 176.59 9715.5 12 19 4 9 3 4 Akbal 1 Chen61 1998 3 13 2450885.68 177.4 9892.9 N 12 19 5 0 1 13 Imix 19 Kayeb62 1998 8 8 2451033.6 147.92 10041 N 12 19 5 7 9 5 Muluc 2 Yaxkin63 1998 9 6 2451062.97 29.37 10070 N 12 19 5 8 18 8 Eznab 11 Mol64 1999 1 31 2451210.18 147.21 10217 N 12 19 5 16 5 12 Chicchan 18 Muan65 1999 7 28 2451387.98 177.8 10395 12 19 6 7 3 8 Akbal 11 Xul66 2000 1 21 2451564.7 176.72 10572 12 19 6 16 0 3 Ahau 8 Muan67 2000 7 16 2451742.08 177.38 10749 12 19 7 6 17 11 Caban 0 Xul68 2001 1 9 2451919.35 177.27 10927 12 19 7 15 14 6 Ix 17 Kankin69 2001 7 5 2452096.12 176.77 11103 12 19 8 6 11 1 Chuen 9 Tzec70 2001 12 30 2452273.94 177.82 11281 N 12 19 8 15 9 10 Muluc 7 Kankin71 2002 5 26 2452421 147.06 11428 N 12 19 9 4 16 1 Cib 9 Zip72 2002 6 24 2452450.39 29.39 11458 N 12 19 9 6 5 4 Chicchan 18 Zotz73 2002 11 20 2452598.57 148.18 11606 N* 12 19 9 13 13 9 Ben 6 Ceh74 2003 5 16 2452775.65 177.08 11783 12 19 10 4 11 5 Chuen 19 Uo75 2003 11 9 2452952.56 176.91 11960 X* 12 19 10 13 8 13 Lamat 16 Zac

Night Eclipse *Not Liu XNot Oppolzer N

62

APPENDIX B

LIU LUNAR DATA

63

Lunar 1971 to 2003 Liu P 1Year MM DD Julian Diff Sum Tzolkin Haab

1 1971 8 6 2441170 177 177 12 17 18 0 5 8 Chicchan 13 Xul2 1972 1 30 2441347 177 354 12 17 18 9 2 3 Ik 10 Muan3 1972 7 26 2441525 178 532 12 17 19 0 0 12 Ahau 3 Xul4 1973 1 18 2441701 176 708 N 12 17 19 8 16 6 Cib 19 Kankin5 1973 6 15 2441849 148 856 N 12 17 19 16 4 11 Kan 2 Zotz6 1973 7 15 2441879 30 886 N 12 17 19 17 14 2 Ix 12 Tzec7 1973 12 9 2442026 147 1033 X* 12 18 0 7 1 6 Imix 19 Ceh8 1974 6 4 2442203 177 1210 * 12 18 0 15 18 1 Eznab 11 Zip9 1974 11 29 2442381 178 1388 12 18 1 6 16 10 Cib 9 Ceh

10 1975 5 25 2442558 177 1565 12 18 1 15 13 5 Ben 1 Zip11 1975 11 18 2442735 177 1742 * 12 18 2 6 10 13 Oc 18 Zac12 1976 5 13 2442912 177 1919 12 18 2 15 7 8 Manik 10 Uo13 1976 11 6 2443089 177 2096 N* 12 18 3 6 4 3 Kan 7 Zac14 1977 4 4 2443238 149 2245 12 18 3 13 13 9 Ben 16 Cumhu15 1977 9 27 2443414 176 2421 N 12 18 4 4 9 3 Muluc 7 Chen16 1978 3 24 2443592 178 2599 12 18 4 13 7 12 Manik 5 Cumhu17 1978 9 16 2443768 176 2775 12 18 5 4 3 6 Akbal 16 Mol18 1979 3 13 2443946 178 2953 * 12 18 5 13 1 2 Imix 14 Kayeb19 1979 9 6 2444123 177 3130 12 18 6 3 18 10 Eznab 6 Mol20 1980 3 1 2444300 177 3307 N 12 18 6 12 15 5 Men 3 Kayeb21 1980 7 27 2444448 148 3455 N 12 18 7 2 3 10 Akbal 6 Xul22 1980 8 26 2444478 30 3485 N 12 18 7 3 13 1 Ben 16 Yaxkin23 1981 1 20 2444625 147 3632 N 12 18 7 11 0 5 Ahau 3 Muan24 1981 7 17 2444803 178 3810 12 18 8 1 18 1 Eznab 16 Tzec25 1982 1 9 2444979 176 3986 12 18 8 10 14 8 Ix 12 Kankin26 1982 7 6 2445157 178 4164 12 18 9 1 12 4 Eb 5 Tzec27 1982 12 30 2445334 177 4341 12 18 9 10 9 12 Muluc 2 Kankin28 1983 6 25 2445511 177 4518 12 18 10 1 6 7 Cimi 14 Zotz29 1983 12 19 2445688 177 4695 XN 12 18 10 10 3 2 Akbal 11 Mac30 1984 5 15 2445836 148 4843 N 12 18 10 17 11 7 Chuen 14 Uo31 1984 6 13 2445865 29 4872 N 12 18 11 1 0 10 Ahau 3 Zotz32 1984 11 8 2446013 148 5020 N 12 18 11 8 8 2 Lamat 11 Zac33 1985 5 4 2446190 177 5197 12 18 11 17 5 10 Chicchan 3 Uo34 1985 10 28 2446367 177 5374 12 18 12 8 2 5 Ik 0 Zac35 1986 4 24 2446545 178 5552 12 18 12 17 0 1 Ahau 13 Pop36 1986 10 17 2446721 176 5728 12 18 13 7 16 8 Cib 9 Yax37 1987 4 14 2446900 179 5907 N 12 18 13 16 15 5 Men 3 Pop38 1987 10 7 2447076 176 6083 12 18 14 7 11 12 Chuen 19 Chen39 1988 3 3 2447224 148 6231 N 12 18 14 14 19 4 Cauac 7 Kayeb40 1988 8 27 2447401 177 6408 12 18 15 5 16 12 Cib 19 Yaxkin41 1989 2 20 2447578 177 6585 12 18 15 14 13 7 Ben 16 Pax42 1989 8 17 2447756 178 6763 12 18 16 5 11 3 Chuen 9 Yaxkin43 1990 2 9 2447932 176 6939 12 18 16 14 7 10 Manik 5 Pax

64

Lunar 1971 to 2003 Liu P 2

Year MM DD Julian Diff Sum Tzolkin Haab

44 1990 8 6 2448110 178 7117 12 18 17 5 5 6 Chicchan 18 Xul45 1991 1 30 2448287 177 7294 N 12 18 17 14 2 1 Ik 15 Muan46 1991 6 27 2448435 148 7442 N 12 18 18 3 10 6 Oc 18 Zotz47 1991 7 26 2448464 29 7471 N 12 18 18 4 19 9 Cauac 7 Xul48 1991 12 21 2448612 148 7619 12 18 18 12 7 1 Manik 15 Mak50 1992 12 9 2448966 177 7973 * 12 18 19 12 1 4 Imix 4 Mak51 1993 6 4 2449143 177 8150 12 19 0 2 18 12 Eznab 16 Zip52 1993 11 29 2449321 178 8328 12 19 0 11 16 8 Cib 14 Ceh53 1994 5 25 2449498 177 8505 12 19 1 2 13 3 Ben 6 Zip54 1994 11 18 2449675 177 8682 N 12 19 1 11 10 11 Oc 3 Ceh55 1995 4 15 2449823 148 8830 12 19 2 0 18 3 Eznab 6 Pop56 1995 10 8 2449999 176 9006 N 12 19 2 9 14 10 Ix 2 Yax57 1996 4 3 2450177 178 9184 X* 12 19 3 0 12 6 Eb 0 Uayeb58 1996 9 27 2450354 177 9361 12 19 3 9 9 1 Muluc 12 Chen59 1997 3 24 2450532 178 9539 12 19 4 0 7 10 Manik 10 Cumhu60 1997 9 16 2450708 176 9715 12 19 4 9 3 4 Akbal 1 Chen61 1998 3 13 2450886 178 9893 N 12 19 5 0 1 13 Imix 19 Kayeb62 1998 8 8 2451034 148 10041 N 12 19 5 7 9 5 Muluc 2 Yaxkin63 1998 9 6 2451063 29 10070 N 12 19 5 8 18 8 Eznab 11 Mol64 1999 1 31 2451210 147 10217 N 12 19 5 16 5 12 Chicchan 18 Muan65 1999 7 28 2451388 178 10395 12 19 6 7 3 8 Akbal 11 Xul66 2000 1 21 2451565 177 10572 12 19 6 16 0 3 Ahau 8 Muan67 2000 7 16 2451742 177 10749 12 19 7 6 17 11 Caban 0 Xul68 2001 1 9 2451919 177 10926 12 19 7 15 14 6 Ix 17 Kankin69 2001 7 5 2452096 177 11103 12 19 8 6 11 1 Chuen 9 Tzec70 2001 12 30 2452274 178 11281 N 12 19 8 15 9 10 Muluc 7 Kankin71 2002 5 26 2452421 147 11428 N 12 19 9 4 16 1 Cib 9 Zip72 2002 6 24 2452450 29 11457 N 12 19 9 6 5 4 Chicchan 18 Zotz73 2002 11 19 2452598 148 11605 N* 12 19 9 13 13 9 Ben 6 Ceh74 2003 5 16 2452776 178 11783 12 19 10 4 11 5 Chuen 19 Uo75 2003 11 8 2452952 176 11959 X* 12 19 10 13 7 12 Manik 15 Zac

Night Eclipse *Not Muess XNot Oppolzer N

65

APPENDIX C

OPPOLZER LUNAR DATA

66

Lunar 1971 to 2003 Oppolzer P 1Year MM DD Julian Diff Sum Tzolkin Haab

1 1971 8 6 2441170 177 177 12 17 18 0 5 8 Chicchan 13 Xul2 1972 1 30 2441347 177 354 12 17 18 9 2 3 Ik 10 Muan3 1972 7 26 2441525 178 532 12 17 19 0 0 12 Ahau 3 Xul4 1973 12 10 2442027 502 1034 X* 12 18 0 7 2 7 Ik 0 Mak5 1974 6 4 2442203 176 1210 * 12 18 0 15 18 1 Eznab 11 Zip6 1974 11 29 2442381 178 1388 12 18 1 6 16 10 Cib 9 Ceh7 1975 5 25 2442558 177 1565 12 18 1 15 13 5 Ben 1 Zip8 1975 11 18 2442735 177 1742 * 12 18 2 6 10 13 Oc 18 Zac9 1976 5 13 2442912 177 1919 12 18 2 15 7 8 Manik 10 Uo

10 1977 4 4 2443238 326 2245 12 18 3 13 13 9 Ben 16 Cumhu11 1978 3 24 2443592 354 2599 12 18 4 13 7 12 Manik 5 Cumhu12 1978 9 16 2443768 176 2775 12 18 5 4 3 6 Akbal 16 Mol13 1979 3 13 2443946 178 2953 * 12 18 5 13 1 2 Imix 14 Kayeb14 1979 9 6 2444123 177 3130 12 18 6 3 18 10 Eznab 6 Mol15 1981 7 17 2444803 680 3810 12 18 8 1 18 1 Eznab 16 Tzec16 1982 1 9 2444979 176 3986 12 18 8 10 14 8 Ix 12 Kankin17 1982 7 6 2445157 178 4164 12 18 9 1 12 4 Eb 5 Tzec18 1982 12 30 2445334 177 4341 12 18 9 10 9 12 Muluc 2 Kankin19 1983 6 25 2445511 177 4518 * 12 18 10 1 6 7 Cimi 14 Zotz20 1985 5 4 2446190 679 5197 12 18 11 17 5 10 Chicchan 3 Uo21 1985 10 28 2446367 177 5374 12 18 12 8 2 5 Ik 0 Zac22 1986 4 24 2446545 178 5552 12 18 12 17 0 1 Ahau 13 Pop23 1986 10 17 2446721 176 5728 12 18 13 7 16 8 Cib 9 Yax24 1987 10 7 2447076 355 6083 12 18 14 7 11 12 Chuen 19 Chen25 1988 8 27 2447401 325 6408 12 18 15 5 16 12 Cib 19 Yaxkin26 1989 2 20 2447578 177 6585 12 18 15 14 13 7 Ben 16 Pax27 1989 8 17 2447756 178 6763 12 18 16 5 11 3 Chuen 9 Yaxkin28 1990 2 9 2447932 176 6939 12 18 16 14 7 10 Manik 5 Pax29 1990 8 6 2448110 178 7117 12 18 17 5 5 6 Chicchan 18 Xul30 1991 12 21 2448612 502 7619 12 18 18 12 7 1 Manik 15 Mak31 1992 6 15 2448789 177 7796 12 18 19 3 4 9 Kan 7 Zotz32 1992 12 9 2448966 177 7973 * 12 18 19 12 1 4 Imix 4 Mak33 1993 6 4 2449143 177 8150 12 19 0 2 18 12 Eznab 16 Zip34 1993 11 29 2449321 178 8328 12 19 0 11 16 8 Cib 14 Ceh35 1994 5 25 2449498 177 8505 12 19 1 2 13 3 Ben 6 Zip

67

Lunar 1971 to 2003 Oppolzer P 2

Year MM DD Julian Diff Sum Tzolkin Haab

36 1995 4 15 2449823 325 8830 12 19 2 0 18 3 Eznab 6 Pop37 1996 4 4 2450178 355 9185 X* 12 19 3 0 13 7 Ben 1 Uayeb38 1996 9 27 2450354 176 9361 12 19 3 9 9 1 Muluc 12 Chen39 1997 3 24 2450532 178 9539 12 19 4 0 7 10 Manik 10 Cumhu40 1997 9 16 2450708 176 9715 12 19 4 9 3 4 Akbal 1 Chen41 1999 7 28 2451388 680 10395 12 19 6 7 3 8 Akbal 11 Xul42 2000 1 21 2451565 177 10572 12 19 6 16 0 3 Ahau 8 Muan43 2000 7 16 2451742 177 10749 12 19 7 6 17 11 Caban 0 Xul44 2001 1 9 2451919 177 10926 12 19 7 15 14 6 Ix 17 Kankin45 2001 7 5 2452096 177 11103 12 19 8 6 11 1 Chuen 9 Tzec46 2003 5 16 2452776 680 11783 12 19 10 4 11 5 Chuen 19 Uo47 2003 11 9 2452953 177 11960 X* 12 19 10 13 8 13 Lamat 16 Zac

Night Eclipse *Not Liu X

68

APPENDIX D

MEEUS SOLAR DATA

69

Solar 1971 to 2003 Meeus P 1

Year MM DD Julian Diff Sum Tzolkin Haab

1 1971 7 22 2441154.9 147 147 12 17 17 17 10 6 Oc 18 Tzec2 1971 8 20 2441184.44 29.54 176.54 12 17 18 0 19 9 Cauac 7 Yaxkin3 1972 1 16 2441332.96 148.52 325.06 12 17 18 8 8 2 Lamat 16 Kankin4 1972 7 10 2441509.32 176.36 501.42 12 17 18 17 4 9 Kan 7 Tzec5 1973 1 4 2441687.16 177.84 679.26 12 17 19 8 2 5 Ik 5 Kankin6 1973 6 30 2441863.99 176.83 856.09 12 17 19 16 19 13 Cauac 17 Zotz7 1973 12 24 2442041.13 177.14 1033.2 12 18 0 7 16 8 Cib 14 Mac8 1974 6 20 2442218.7 177.57 1210.8 12 18 0 16 14 4 Ix 7 Zotz9 1974 12 13 2442395.18 176.48 1387.3 12 18 1 7 10 11 Oc 3 Mac

10 1975 5 11 2442543.8 148.62 1535.9 12 18 1 14 19 4 Cauac 7 Uo11 1975 11 3 2442720.05 176.25 1712.1 12 18 2 5 15 11 Men 3 Zac12 1976 4 29 2442897.93 177.88 1890 12 18 2 14 13 7 Ben 16 Pop13 1976 10 23 2443074.72 176.79 2066.8 12 18 3 5 10 2 Oc 13 Yax14 1977 4 18 2443251.94 177.22 2244 12 18 3 14 7 10 Manik 5 Pop15 1977 10 12 2443429.35 177.41 2421.5 12 18 4 5 4 5 Kan 2 Yax16 1978 4 7 2443606.13 176.78 2598.2 12 18 4 14 1 13 Imix 19 Cumhu17 1978 10 2 2443783.77 177.64 2775.9 12 18 5 4 19 9 Cauac 12 Chen18 1979 2 26 2443931.2 147.43 2923.3 12 18 5 12 6 13 Cimi 19 Pax 19 1979 8 22 2444108.22 177.02 3100.3 12 18 6 3 3 8 Akbal 11 Yaxkin20 1980 2 16 2444285.87 177.65 3278 12 18 6 12 1 4 Imix 9 Pax 21 1980 8 10 2444462.3 176.43 3454.4 12 18 7 2 17 11 Caban 0 Yaxkin22 1981 2 4 2444640.42 178.12 3632.5 12 18 7 11 15 7 Men 18 Muan23 1981 7 31 2444816.66 176.24 3808.8 12 18 8 2 12 2 Eb 10 Xul24 1982 1 25 2444994.7 178.04 3986.8 12 18 8 11 10 11 Oc 8 Muan25 1982 6 21 2445142 147.3 4134.1 12 18 9 0 17 2 Caban 10 Zotz26 1982 7 20 2445171.28 29.28 4163.4 12 18 9 2 6 5 Cimi 19 Tzec27 1982 12 15 2445318.9 147.62 4311 12 18 9 9 14 10 Ix 7 Mac28 1983 6 11 2445496.7 177.8 4488.8 12 18 10 0 12 6 Eb 0 Zotz29 1983 12 4 2445673.02 176.32 4665.1 12 18 10 9 8 13 Lamat 16 Ceh30 1984 5 30 2445851.2 178.18 4843.3 12 18 11 0 6 9 Cimi 9 Zip 31 1984 11 22 2446027.45 176.25 5019.6 12 18 11 9 2 3 Ik 5 Ceh32 1985 5 19 2446205.4 177.95 5197.5 12 18 12 0 0 12 Ahau 18 Uo33 1985 11 12 2446382.09 176.69 5374.2 12 18 12 8 17 7 Caban 15 Zac34 1986 4 9 2446529.76 147.67 5521.9 12 18 12 16 5 12 Chicchan 3 Uayeb35 1986 10 3 2446707.3 177.54 5699.4 12 18 13 7 2 7 Ik 15 Chen36 1987 3 29 2446884.03 176.73 5876.1 12 18 13 15 19 2 Cauac 12 Cumhu37 1987 9 23 2447061.63 177.6 6053.7 12 18 14 6 17 11 Caban 5 Chen38 1988 3 18 2447238.58 176.95 6230.7 12 18 14 15 14 6 Ix 2 Cumhu39 1988 9 11 2447415.7 177.12 6407.8 12 18 15 6 11 1 Chuen 14 Mol40 1989 3 7 2447593.26 177.56 6585.4 12 18 15 15 8 9 Lamat 11 Kayeb

70

Solar 1971 to 2003 Meeus P 2

Year MM DD Julian Diff Sum Tzolkin Haab

41 1989 8 31 2447769.73 176.47 6761.8 12 18 16 6 5 4 Chicchan 3 Mol42 1990 1 26 2447918.31 148.58 6910.4 12 18 16 13 13 9 Ben 11 Muan43 1990 7 22 2448094.63 176.32 7086.7 12 18 17 4 10 4 Oc 3 Xul44 1991 1 15 2448272.5 177.87 7264.6 12 18 17 13 7 12 Manik 0 Muan45 1991 7 11 2448449.3 176.8 7441.4 12 18 18 4 4 7 Kan 12 Tzec46 1992 1 4 2448626.46 177.16 7618.6 12 18 18 13 1 2 Imix 9 Kankin47 1992 6 30 2448804.01 177.55 7796.1 12 18 19 3 19 11 Cauac 2 Zek48 1992 12 24 2448980.52 176.51 7972.6 12 18 19 12 16 6 Cib 19 Mac49 1993 5 21 2449129.1 148.58 8121.2 12 19 0 2 4 11 Kan 2 Zip 50 1993 11 13 2449305.41 176.31 8297.5 12 19 0 11 0 5 Ahau 18 Zac51 1994 5 10 2449483.22 177.81 8475.3 12 19 1 1 18 1 Eznab 11 Uo52 1994 11 3 2449660.07 176.85 8652.2 12 19 1 10 15 9 Men 8 Zac53 1995 4 29 2449837.23 177.16 8829.3 12 19 2 1 12 4 Eb 0 Uo54 1995 10 24 2450014.69 177.46 9006.8 12 19 2 10 10 13 Oc 18 Yax55 1996 4 17 2450191.44 176.75 9183.5 12 19 3 1 6 7 Cimi 9 Pop56 1996 10 12 2450369.09 177.65 9361.2 12 19 3 10 4 3 Kan 7 Yax57 1997 3 9 2450516.56 147.47 9508.7 12 19 3 17 12 8 Eb 15 Kayeb58 1997 9 2 2450693.5 176.94 9685.6 12 19 4 8 9 3 Muluc 7 Mol59 1998 2 26 2450871.23 177.73 9863.3 12 19 4 17 6 11 Cimi 4 Kayeb60 1998 8 22 2451047.59 176.36 10040 12 19 5 8 3 6 Akbal 16 Yaxkin61 1999 2 16 2451225.77 178.18 10218 12 19 5 17 1 2 Imix 14 Pax 62 1999 8 11 2451401.96 176.19 10394 12 19 6 7 17 9 Caban 5 Yaxkin63 2000 2 5 2451580.04 178.08 10572 12 19 6 16 15 5 Men 3 Pax 64 2000 7 1 2451727.31 147.27 10719 12 19 7 6 2 9 Ik 5 Tzec65 2000 7 31 2451756.59 29.28 10749 12 19 7 7 12 13 Eb 15 Xul66 2000 12 25 2451904.23 147.64 10896 12 19 7 14 19 4 Cauac 2 Kankin67 2001 6 21 2452082 177.77 11074 12 19 8 5 17 13 Caban 15 Zotz68 2001 12 14 2452258.37 176.37 11250 12 19 8 14 13 7 Ben 11 Mac69 2002 6 10 2452436.49 178.12 11429 12 19 9 5 11 3 Chuen 4 Zotz70 2002 12 4 2452612.81 176.32 11605 12 19 9 14 8 11 Lamat 1 Mac71 2003 5 31 2452790.67 177.86 11783 12 19 10 5 6 7 Cimi 14 Zip 72 2003 11 23 2452967.45 176.78 11960 12 19 10 14 2 1 Ik 10 Ceh

71

APPENDIX E

OPPOLZER SOLAR DATA

72

Solar 1971 to 2003 Oppolzer P1

Year MM DD Julian Diff Sum Tzolkin Haab

1 1971 7 22 2441155 147 147 12 17 17 17 10 6 Oc 18 Tzec2 1971 8 20 2441184 29 176 12 17 18 0 19 9 Cauac 7 Yaxkin3 1972 1 16 2441333 149 325 12 17 18 8 8 2 Lamat 16 Kankin4 1972 7 10 2441509 176 501 12 17 18 17 4 9 Kan 7 Tzec5 1973 1 4 2441687 178 679 12 17 19 8 2 5 Ik 5 Kankin6 1973 6 30 2441864 177 856 12 17 19 16 19 13 Cauac 17 Zotz7 1973 12 24 2442041 177 1033 12 18 0 7 16 8 Cib 14 Mac8 1974 6 20 2442219 178 1211 12 18 0 16 14 4 Ix 7 Zotz9 1974 12 13 2442395 176 1387 12 18 1 7 10 11 Oc 3 Mac

10 1975 5 11 2442544 149 1536 12 18 1 14 19 4 Cauac 7 Uo11 1975 11 3 2442720 176 1712 12 18 2 5 15 11 Men 3 Zac12 1976 4 29 2442898 178 1890 12 18 2 14 13 7 Ben 16 Pop13 1976 10 23 2443075 177 2067 12 18 3 5 10 2 Oc 13 Yax14 1977 4 18 2443252 177 2244 12 18 3 14 7 10 Manik 5 Pop15 1977 10 12 2443429 177 2421 12 18 4 5 4 5 Kan 2 Yax16 1978 4 7 2443606 177 2598 12 18 4 14 1 13 Imix 19 Cumhu17 1978 10 2 2443784 178 2776 12 18 5 4 19 9 Cauac 12 Chen18 1979 2 26 2443931 147 2923 12 18 5 12 6 13 Cimi 19 Pax 19 1979 8 22 2444108 177 3100 12 18 6 3 3 8 Akbal 11 Yaxkin20 1980 2 16 2444286 178 3278 12 18 6 12 1 4 Imix 9 Pax 21 1980 8 10 2444462 176 3454 12 18 7 2 17 11 Caban 0 Yaxkin22 1981 2 4 2444640 178 3632 12 18 7 11 15 7 Men 18 Muan23 1981 7 31 2444817 177 3809 12 18 8 2 12 2 Eb 10 Xul24 1982 1 25 2444995 178 3987 12 18 8 11 10 11 Oc 8 Muan25 1982 6 21 2445142 147 4134 12 18 9 0 17 2 Caban 10 Zotz26 1982 7 20 2445171 29 4163 12 18 9 2 6 5 Cimi 19 Tzec27 1982 12 15 2445319 148 4311 12 18 9 9 14 10 Ix 7 Mac28 1983 6 11 2445497 178 4489 12 18 10 0 12 6 Eb 0 Zotz29 1983 12 4 2445673 176 4665 12 18 10 9 8 13 Lamat 16 Ceh30 1984 5 30 2445851 178 4843 12 18 11 0 6 9 Cimi 9 Zip 31 1984 11 22 2446027 176 5019 12 18 11 9 2 3 Ik 5 Ceh32 1985 5 19 2446205 178 5197 12 18 12 0 0 12 Ahau 18 Uo33 1985 11 12 2446382 177 5374 12 18 12 8 17 7 Caban 15 Zac34 1986 4 9 2446530 148 5522 12 18 12 16 5 12 Chicchan 3 Uayeb35 1986 10 3 2446707 177 5699 12 18 13 7 2 7 Ik 15 Chen36 1987 3 29 2446884 177 5876 12 18 13 15 19 2 Cauac 12 Cumhu37 1987 9 23 2447062 178 6054 12 18 14 6 17 11 Caban 5 Chen38 1988 3 18 2447239 177 6231 12 18 14 15 14 6 Ix 2 Cumhu39 1988 9 11 2447416 177 6408 12 18 15 6 11 1 Chuen 14 Mol40 1989 3 7 2447593 177 6585 12 18 15 15 8 9 Lamat 11 Kayeb

73

Solar 1971 to 2003 Oppolzer P 2Year MM DD Julian Diff Sum Tzolkin Haab

41 1989 8 31 2447770 177 6762 12 18 16 6 5 4 Chicchan 3 Mol42 1990 1 26 2447918 148 6910 12 18 16 13 13 9 Ben 11 Muan43 1990 7 22 2448095 177 7087 12 18 17 4 10 4 Oc 3 Xul44 1991 1 15 2448272 177 7264 12 18 17 13 7 12 Manik 0 Muan45 1991 7 11 2448449 177 7441 12 18 18 4 4 7 Kan 12 Tzec46 1992 1 4 2448626 177 7618 12 18 18 13 1 2 Imix 9 Kankin47 1992 6 30 2448804 178 7796 12 18 19 3 19 11 Cauac 2 Zek48 1992 12 24 2448981 177 7973 12 18 19 12 16 6 Cib 19 Mac49 1993 5 21 2449129 148 8121 12 19 0 2 4 11 Kan 2 Zip 50 1993 11 13 2449305 176 8297 12 19 0 11 0 5 Ahau 18 Zac51 1994 5 10 2449483 178 8475 12 19 1 1 18 1 Eznab 11 Uo52 1994 11 3 2449660 177 8652 12 19 1 10 15 9 Men 8 Zac53 1995 4 29 2449837 177 8829 12 19 2 1 12 4 Eb 0 Uo54 1995 10 24 2450015 178 9007 12 19 2 10 10 13 Oc 18 Yax55 1996 4 17 2450191 176 9183 12 19 3 1 6 7 Cimi 9 Pop56 1996 10 12 2450369 178 9361 12 19 3 10 4 3 Kan 7 Yax57 1997 3 9 2450517 148 9509 12 19 3 17 12 8 Eb 15 Kayeb58 1997 9 1 2450693 176 9685 12 19 4 8 8 2 Lamat 6 Mol59 1998 2 26 2450871 178 9863 12 19 4 17 6 11 Cimi 4 Kayeb60 1998 8 22 2451048 177 10040 12 19 5 8 3 6 Akbal 16 Yaxkin61 1999 2 16 2451226 178 10218 12 19 5 17 1 2 Imix 14 Pax 62 1999 8 11 2451402 176 10394 12 19 6 7 17 9 Caban 5 Yaxkin63 2000 2 5 2451580 178 10572 12 19 6 16 15 5 Men 3 Pax 64 2000 7 1 2451727 147 10719 12 19 7 6 2 9 Ik 5 Tzec65 2000 7 31 2451757 30 10749 12 19 7 7 12 13 Eb 15 Xul66 2000 12 25 2451904 147 10896 12 19 7 14 19 4 Cauac 2 Kankin67 2001 6 21 2452082 178 11074 12 19 8 5 17 13 Caban 15 Zotz68 2001 12 14 2452258 176 11250 12 19 8 14 13 7 Ben 11 Mac69 2002 6 10 2452436 178 11428 12 19 9 5 11 3 Chuen 4 Zotz70 2002 12 4 2452613 177 11605 12 19 9 14 8 11 Lamat 1 Mac71 2003 5 31 2452791 178 11783 12 19 10 5 6 7 Cimi 14 Zip 72 2003 11 23 2452967 176 11959 12 19 10 14 2 1 Ik 10 Ceh

74

APPENDIX F LUNAR-SOLAR DATA

75

Lunar Eclipse 755 Oppolzer 755 P 1Year Month Day Julian Interval

725 1 19 1985883 178 9 14 13 5 14 3 Ix 2 Cumhu G6725 7 14 1986059 176 9 14 13 14 10 10 Oc 13 Mol G2

726 1 8 1986237 15 9 14 14 5 8 6 Lamat 11 Kayeb G9

726 7 4 1986414 15 9 14 14 14 5 1 Chicchan 3 Mol G6

726 12 28 1986591 15 9 14 15 5 2 9 Ik 0 Kayeb G3727 5 25 1986739 148 9 14 15 12 10 1 Oc 3 Xul G7

727 6 23 1986768 15 9 14 15 13 19 4 Cauac 12 Yaxkin G9727 11 17 1986915 147 9 14 16 3 6 8 Cimi 19 Kankin G3

728 5 13 1987093 162 9 14 16 12 4 4 Kan 12 Tzec G1

728 11 6 1987270 163 9 14 17 3 1 12 Imix 9 Kankin G7

729 5 2 1987447 14 9 14 17 11 18 7 Eznab 1 Tzec G4729 10 27 1987625 178 9 14 18 2 16 3 Cib 19 Mac G2

730 4 22 1987802 15 9 14 18 11 13 11 Ben 11 Zotz G8

730 10 16 1987979 15 9 14 19 2 10 6 Oc 8 Mac G5731 3 12 1988126 147 9 14 19 9 17 10 Caban 10 Uo G8

731 9 6 1988304 162 9 15 0 0 15 6 Men 8 Zac G6

732 3 1 1988481 163 9 15 0 9 12 1 Eb 0 Uo G3732 8 25 1988658 177 9 15 1 0 9 9 Muluc 17 Yax G9

733 2 19 1988836 16 9 15 1 9 7 5 Manik 10 Pop G7

733 8 14 1989012 14 9 15 2 0 3 12 Akbal 6 Yax G3734 1 10 1989161 149 9 15 2 7 12 5 Eb 15 Kayeb G8

734 2 8 1989190 15 9 15 2 9 1 8 Imix 4 Uayeb G1734 7 5 1989337 147 9 15 2 16 8 12 Lamat 6 Mol G4

15734 8 3 1989366 14 9 15 2 17 17 2 Caban 15 Chen G6734 12 30 1989515 149 9 15 3 7 6 8 Cimi 4 Kayeb G2

14735 6 25 1989692 163 9 15 3 16 3 3 Akbal 16 Yaxkin G8

14735 12 19 1989869 163 9 15 4 7 0 11 Ahau 13 Pax G5

725 12 24 1986222 163 9 14 14 4 13 4 Ben 16 Pax G3

726 6 19 1986399 162 9 14 14 13 10 12 Oc 8 Yaxkin G9

726 12 13 1986576 162 * 9 14 15 4 7 7 Manik 5 Pax G6

727 6 8 1986753 14 9 14 15 13 4 2 Kan 17 Xul G3

727 12 3 1986931 16 9 14 16 4 2 11 Ik 15 Muan G1

728 5 27 1987107 14 * 9 14 16 12 18 5 Eznab 6 Xul G6

729 4 18 1987433 163 9 14 17 11 4 6 Kan 7 Zotz G8

730 4 7 1987787 162 * 9 14 18 10 18 9 Eznab 16 Zip G2

730 10 1 1987964 162 X* 9 14 19 1 15 4 Men 13 Ceh G8

731 3 28 1988142 16 9 14 19 10 13 13 Ben 6 Zip G6

731 9 20 1988318 14 9 15 0 1 9 7 Muluc 2 Ceh G2

733 2 3 1988820 162 9 15 1 8 11 2 Chuen 19 Cumhu G9

733 7 31 1988998 162 9 15 1 17 9 11 Muluc 12 Chen G7

734 1 24 1989175 14 9 15 2 8 6 6 Cimi 9 Cumhu G4

734 7 20 1989352 9 15 2 17 3 1 Akbal 1 Chen G1

735 1 13 1989529 9 15 3 8 0 9 Ahau 18 Kayeb G7

735 7 9 1989706 9 15 3 16 17 4 Caban 10 Mol G4

76

Lunar Eclipse 755 Oppolzer 755 P 2Year Month Day Julian Interval

736 6 13 1990046 177 9 15 4 15 17 6 Caban 5 Yaxkin G2163

736 12 7 1990223 14 9 15 5 6 14 1 Ix 2 Pax G8162

737 6 3 1990401 16 9 15 5 15 12 10 Eb 15 Xul G6737 10 28 1990548 147 9 15 6 4 19 1 Cauac 2 Kankin G9

15737 11 26 1990577 14 9 15 6 6 8 4 Lamat 11 Muan G2738 4 23 1990725 148 9 15 6 13 16 9 Cib 14 Zotz G6

15738 10 18 1990903 163 9 15 7 4 14 5 Ix 12 Mac G4

14739 4 12 1991079 162 9 15 7 13 10 12 Oc 3 Zotz G9739 10 7 1991257 178 9 15 8 4 8 8 Lamat 1 Mac G7

163740 4 1 1991434 14 9 15 8 13 5 3 Chicchan 13 Zip G4

162740 9 25 1991611 15 9 15 9 4 2 11 Ik 10 Ceh G1741 2 20 1991759 148 9 15 9 11 10 3 Oc 13 Pop G5

15741 3 21 1991788 14 9 15 9 12 19 6 Cauac 2 Zip G7

163741 9 14 1991965 14 9 15 10 3 16 1 Cib 19 Zac G4742 2 10 1992114 149 9 15 10 11 5 7 Chicchan 3 Pop G9

14742 8 5 1992290 162 9 15 11 2 1 1 Imix 19 Chen G5

15743 1 30 1992468 163 9 15 11 10 19 10 Cauac 17 Cumhu G3743 7 25 1992644 176 9 15 12 1 15 4 Men 8 Chen G8

163744 1 19 1992822 15 9 15 12 10 13 13 Ben 6 Cumhu G6

162744 7 14 1992999 15 9 15 13 1 10 8 Oc 18 Mol G3

163745 1 7 1993176 14 9 15 13 10 7 3 Manik 15 Kayeb G9745 6 4 1993324 148 9 15 13 17 15 8 Men 8 Xul G4

14745 7 4 1993354 16 9 15 14 1 5 12 Chicchan 8 Mol G7745 11 28 1993501 147 9 15 14 8 12 3 Eb 15 Muan G1

15746 5 25 1993679 163 9 15 14 17 10 12 Oc 8 Xul G8

14746 11 17 1993855 162 9 15 15 8 6 6 Cimi 4 Muan G4

163

736 11 23 1990209 9 15 5 6 0 13 Ahau 8 Muan G3

737 5 18 1990385 * 9 15 5 14 16 7 Cib 19 Tzec G8

737 11 12 1990563 9 15 6 5 14 3 Ix 17 Kankin G6

738 5 8 1990740 9 15 6 14 11 11 Chuen 9 Tzec G3

738 11 1 1990917 9 15 7 5 8 6 Lamat 6 Kankin G9

740 3 18 1991420 9 15 8 12 11 2 Chuen 19 Uo G8

740 9 10 1991596 9 15 9 3 7 9 Manik 15 Zac G4

741 3 7 1991774 9 15 9 12 5 5 Chicchan 8 Uo G2

741 8 31 1991951 9 15 10 3 2 13 Ik 5 Zac G8

742 2 24 1992128 9 15 10 11 19 8 Cauac 17 Pop G5

742 8 20 1992305 9 15 11 2 16 3 Cib 14 Yax G2

744 1 4 1992807 9 15 12 9 18 11 Eznab 11 Kayeb G9

744 6 29 1992984 9 15 13 0 15 6 Men 3 Mol G6

744 12 24 1993162 9 15 13 9 13 2 Ben 1 Kayeb G4

745 6 18 1993338 9 15 14 0 9 9 Muluc 12 Yaxkin G9

745 12 13 1993516 9 15 14 9 7 5 Manik 10 Pax G7

746 6 8 1993693 9 15 15 0 4 13 Kan 2 Yaxkin G4

747 4 29 1994018 9 15 15 16 9 13 Muluc 2 Tzec G5

77

Lunar Eclipse 755 Oppolzer 755 P 3Year Month Day Julian Interval

748 4 18 1994373 9 15 16 16 4 4 Kan 12 Zotz G9

748 10 11 1994549 9 15 17 7 0 11 Ahau 8 Mak G5

749 4 7 1994727 9 15 17 15 18 7 Eznab 1 Zotz G3

749 9 30 1994903 * 9 15 18 6 14 1 Ix 17 Ceh G8

751 2 15 1995406 9 15 19 13 17 10 Caban 10 Pop G7

751 8 11 1995583 9 16 0 4 14 5 Ix 7 Yax G4

752 2 4 1995760 9 16 0 13 11 13 Chuen 4 Uayeb G1

752 7 31 1995938 X* 9 16 1 4 9 9 Muluc 17 Chen G8

753 1 24 1996115 X* 9 16 1 13 6 4 Cimi 14 Cumhu G5

753 7 20 1996292 X* 9 16 2 4 3 12 Akbal 6 Chen G2

754 12 4 1996794 9 16 3 11 5 7 Chicchan 3 Pax G9

755 5 30 1996971 9 16 4 2 2 2 Ik 15 Xul G6

755 11 23 1997148 9 16 4 10 19 10 Cauac 12 Muan G3

756 5 18 1997325 * 9 16 5 1 16 5 Cib 4 Xul G9

756 11 11 1997502 9 16 5 10 13 13 Ben 1 Muan G6

757 5 8 1997680 9 16 6 1 11 9 Chuen 14 Tzec G4

758 3 29 1998005 9 16 6 17 16 9 Cib 14 Zip G5

758 9 21 1998181 9 16 7 8 12 3 Eb 10 Ceh G1

747 5 14 1994033 15 9 15 15 17 4 2 Kan 17 Tzec G2747 11 7 1994210 177 9 15 16 8 1 10 Imix 14 Kankin G8

163748 5 2 1994387 14 9 15 16 16 18 5 Eznab 6 Tzec G5

162748 10 26 1994564 15 9 15 17 7 15 13 Men 3 Kankin G2749 3 23 1994712 148 9 15 17 15 3 5 Akbal 6 Zip G6

15749 9 16 1994889 162 9 15 18 6 0 13 Ahau 3 Ceh G3

14750 3 12 1995066 163 9 15 18 14 17 8 Caban 15 Uo G9750 9 5 1995243 177 9 15 19 5 14 3 Ix 17 Zac G6

163751 3 2 1995421 15 9 15 19 14 12 12 Eb 5 Uo G4

162751 8 25 1995597 14 9 16 0 5 8 6 Lamat 1 Zac G9752 1 21 1995746 149 9 16 0 12 17 12 Caban 10 Cumhu G5

14752 2 20 1995776 16 9 16 0 14 7 3 Manilk 15 Pop G8752 7 15 1995922 146 9 16 1 3 13 6 Ben 1 Chen G1

16752 8 14 1995952 14 9 16 1 5 3 10 Akbal 11 Yax G4753 1 9 1996100 148 9 16 1 12 11 2 Chuen 19 Kayeb G8

15753 7 5 1996277 162 9 16 2 3 8 10 Lamat 11 Mol G5

15753 12 29 1996454 162 9 16 2 12 5 5 Chicchan 8 Kayeb G2754 6 25 1996632 178 9 16 3 3 3 1 Akbal 1 Mol G9

162754 12 18 1996808 14 9 16 3 11 19 8 Cauac 17 Pax G5

163755 6 14 1996986 15 9 16 4 2 17 4 Caban 10 Yaxkin G3

162755 12 8 1997163 15 9 16 4 11 14 12 Ix 7 Pax G9756 5 4 1997311 148 9 16 5 1 2 4 Ik 10 Tzec G4

14756 10 28 1997488 163 9 16 5 9 19 12 Cauac 7 Kankin G1

14757 4 23 1997665 163 9 16 6 0 16 7 Cib 19 Zotz G7

15757 10 17 1997842 162 9 16 6 9 13 2 Ben 16 Mac G4

163758 4 12 1998019 14 9 16 7 0 10 10 Oc 8 Zotz G1

162758 10 7 1998197 16 9 16 7 9 8 6 Lamat 6 Mac G8

78

APPENDIX G MEEUS LUNAR SEASON DISTRIBUTION

79

Meeus Lunar 2003 P1

Season 1 Season 2 Season 3Ecl Num Day Tzolkin Ecl Num Day Tzolkin Ecl Num Day Tzolkin

1 62 109 5 Muluc l 21 283 10 Akbal l 71 456 1 Cib l2 46 110 6 Oc l 5 284 11 Kan l 457 2 Caban3 30 111 7 Chuen l 64 285 12 Chicchan l 55 458 3 Eznab l4 112 8 Eb 286 13 Cimi 39 459 4 Cauac l5 14/73 113 9 Ben ll 48 287 1 Manik l 23 460 5 Ahau l6 56 114 10 Ix l 32 288 2 Lamat l 461 6 Imix7 115 11 Men 15 289 3 Muluc l 7 462 7 Ik I8 40 116 12 Cib l 290 4 Oc 65 463 8 Akbal l9 117 13 Caban 74 291 5 Chuen l 49 464 9 Kan l

10 8/24 118 1 Eznab ll 292 6 Eb 33 465 10 Chicchan l11 119 2 Cauac 41/57 293 7 Ben lI 466 11 Cimi12 66 120 3 Ahau l 25 294 8 Ix l 16 467 12 Manik l13 50 121 4 Imix l 295 9 Men 75 468 13 Lamat I14 34 122 5 Ik l 9 296 10 Cib l 58 469 1 Muluc l15 17 123 6 Akbal l 67 297 11 Caban l 470 2 Oc16 124 7 Kan 51 298 12 Eznab l 42 471 3 Chuen l17 1 125 8 Chicchan l 299 13 Cauac 26 472 4 Eb l18 126 9 Cimi 35 300 1 Ahau l 10 473 5 Ben l19 43/59 127 10 Manik ll 18 301 2 Imix l 68 474 6 Ix l20 128 11 Lamat 2 302 3 Ik l 475 7 Men21 27 129 12 Muluc l 60 303 4 Akbal l 36/52 476 8 Cib ll22 11 130 13 Oc l 304 5 Kan 477 9 Caban23 69 131 1 Chuen l 44 305 6 Chicchan l 19 478 10 Eznab l24 132 2 Eb 28 306 7 Cimi l 479 11 Cauac25 53 133 3 Ben l 12 307 8 Manik l 3 480 12 Ahau l25 134 4 Ix 308 9 Lamat 61 481 13 Imix l27 20/37 135 5 Men ll 70 309 10 Muluc l 45 482 1 Ik l28 4 136 6 Cib l 54 310 11 Oc l 483 2 Akbal29 137 7 Caban 38 311 12 Chuen l 13/29 484 3 Kan ll30 63 138 8 Eznab l 312 13 Eb 72 485 4 Chicchan l31 47 139 9 Cauac l 22 313 1 Ben l32 31 140 10 Ahau l 6 314 2 Ix l

26 25 24

80

APPENDIX H

MEEUS SOLAR SEASON DISTRIBUTION

81

Meeus Solar 2003 P1Season 1 Season 2 Season 3

Num Day Tzolkin Num Day Tzolkin Num Day Tzolkin

1 110 6 Oc l 282 9 Ik l 457 2 Caban l2 111 7 Chuen 283 10 Akbal 458 3 Eznab3 112 8 Eb l 284 11 Kan l 459 4 Cauac ll4 113 9 Ben l 285 12 Chicchan l 460 5 Ahau l5 114 10 Ix l 286 13 Cimi l 461 6 Imix6 115 11 Men l 287 1 Manik 462 7 Ik l7 116 12 Cib 288 2 Lamat l 463 8 Akbal l8 117 13 Caban l 289 3 Muluc l 464 9 Kan l9 118 1 Eznab l 290 4 Oc l 465 10 Chicchan

10 119 2 Cauac l 291 5 Chuen 466 11 Cimi l11 120 3 Ahau 292 6 Eb l 467 12 Manik l12 121 4 Imix l 293 7 Ben ll 468 13 Lamat l13 122 5 Ik l 294 8 Ix 469 1 Muluc14 123 6 Akbal l 295 9 Men l 470 2 Oc l15 124 7 Kan l 296 10 Cib 471 3 Chuen l16 125 8 Chicchan 297 11 Caban ll 472 4 Eb l17 126 9 Cimi l 298 12 Eznab 473 5 Ben18 127 10 Manik l 299 13 Cauac l 474 6 Ix l19 128 11 Lamat l 300 1 Ahau 475 7 Men l20 129 12 Muluc 301 2 Imix ll 476 8 Cib l21 130 13 Oc l 302 3 Ik l 477 9 Caban l22 131 1 Chuen l 303 4 Akbal 478 10 Eznab23 132 2 Eb l 304 5 Kan l 479 11 Cauac l24 133 3 Ben 305 6 Chicchan 480 12 Ahau l25 134 4 Ix l 306 7 Cimi ll 481 13 Imix l26 135 5 Men l 307 8 Manik 482 1 Ik l27 136 6 Cib l 308 9 Lamat l 483 2 Akbal28 137 7 Caban l 309 10 Muluc 484 3 Kan l29 138 8 Eznab 310 11 Oc ll 485 4 Chicchan l30 139 9 Cauac ll 311 12 Chuen 486 5 Cimi l31 140 10 Ahau 312 13 Eb l 487 6 Manik

24 24 24

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APPENDIX I

TEEPLE SEASON DISTRIBUTION

83

Table 755 Teeple P1Season 1 Season 2 Season 3

Num Day Tzolkin Num Day Tzolkin Num Day Tzolkin

1 62 10 Ik l 236 2 Cib l 409 6 Muluc l2 63 11 Akbal 237 3 Caban 410 7 Oc3 64 12 Kan 238 4 Eznab l 411 8 Chuen4 65 13 Chicchan l 239 5 Cauac l 412 9 Eb ll5 66 1 Cimi l 240 6 Ahau 413 10 Ben l6 67 2 Manik l 241 7 Imix 414 11 Ix 7 68 3 Lamat 242 8 Ik l 415 12 Men l8 69 4 Muluc l 243 9 Akbal l 416 13 Cib l9 70 5 Oc l 244 10 Kan l 417 1 Caban

10 71 6 Chuen l 245 11 Chicchan 418 2 Eznab l11 72 7 Eb l 246 12 Cimi l 419 3 Cauac l12 73 8 Ben 247 13 Manik 420 4 Ahau l13 74 9 Ix l 248 1 Lamat l 421 5 Imix14 75 10 Men 249 2 Muluc ll 422 6 Ik l15 76 11 Cib ll 250 3 Oc 423 7 Akbal l16 77 12 Caban 251 4 Chuen l 424 8 Kan17 78 13 Eznab l 252 5 Eb 425 9 Chicchan l18 79 1 Cauac l 253 6 Ben l 426 10 Cimi ll19 80 2 Ahau l 254 7 Ix l 427 11 Manik20 81 3 Imix 255 8 Men l 428 12 Lamat l21 82 4 Ik l 256 9 Cib l 429 13 Muluc22 83 5 Akbal l 257 10 Caban l 430 1 Oc l23 84 6 Kan l 258 11 Eznab 431 2 Chuen l24 85 7 Chicchan l 259 12 Cauac l 432 3 Eb l25 86 8 Cimi 260 13 Ahau l 433 4 Ben l26 87 9 Manik l 261 1 Imix l 434 5 Ix l27 88 10 Lamat l 262 2 Ik l 435 6 Men28 89 11 Muluc l 263 3 Akbal 436 7 Cib l29 90 12 Oc l 264 4 Kan l 437 8 Caban l30 91 13 Chuen 265 5 Chicchan 438 9 Eznab31 92 1 Eb 266 6 Cimi l 439 10 Cauac l32 93 2 Ben l 267 7 Manik 440 11 Ahau33 94 3 Ix 268 8 Lamat 441 12 Imix34 269 9 Muluc l 442 13 Ik35 270 10 Oc l 443 1 Akbal l

22 24 24

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APPENDIX J

SOLAR-LUNAR SEASON DISTRIBUTION

85

Meeus Lunar AND Solar 2003

Season 1 Season 2 Seas 3

282 9 Ik S1 109 5 Muluc L 283 10 Akbal LS 456 1 Cib L2 110 6 Oc LS 284 11 Kan LS 457 2 Caban S3 111 7 Chuen L 285 12 Chicchan LS 458 3 Eznab L4 112 8 Eb S 286 13 Cimi S 459 4 Cauac LSS5 113 9 Ben LLS 287 1 Manik L 460 5 Ahau LS6 114 10 Ix LS 288 2 Lamat LS 461 6 Imix7 115 11 Men S 289 3 Muluc LS 462 7 Ik LS8 116 12 Cib L 290 4 Oc S 463 8 Akbal LS9 117 13 Caban S 291 5 Chuen L 464 9 Kan LS

10 118 1 Eznab LLS 292 6 Eb S 465 10 Chicchan L11 119 2 Cauac S 293 7 Ben LLSS 466 11 Cimi S12 120 3 Ahau L 294 8 Ix L 467 12 Manik LS13 121 4 Imix LS 295 9 Men S 468 13 Lamat LS14 122 5 Ik LS 296 10 Cib L 469 1 Muluc LS15 123 6 Akbal LS 297 11 Caban LSS 470 2 Oc LS16 124 7 Kan S 298 12 Eznab L 471 3 Chuen LS17 125 8 Chicchan L 299 13 Cauac S 472 4 Eb LS18 126 9 Cimi S 300 1 Ahau L 473 5 Ben L19 127 10 Manik LL 301 2 Imix LSS 474 6 Ix LS20 128 11 Lamat S 302 3 Ik LS 475 7 Men S21 129 12 Muluc L 303 4 Akbal L 476 8 Cib LS22 130 13 Oc LS 304 5 Kan S 477 9 Caban LS23 131 1 Chuen LS 305 6 Chicchan L 478 10 Eznab L24 132 2 Eb S 306 7 Cimi LSS 479 11 Cauac S25 133 3 Ben L 307 8 Manik L 480 12 Ahau LS26 134 4 Ix S 308 9 Lamat S 481 13 Imix LS27 135 5 Men LLS 309 10 Muluc L 482 1 Ik LS28 136 6 Cib LS 310 11 Oc LSS 483 2 Akbal29 137 7 Caban S 311 12 Chuen L 484 3 Kan LLS30 138 8 Eznab L 312 13 Eb S 485 4 Chicchan LS31 139 9 Cauac LSS 313 1 Ben L 486 5 Cimi S32 140 10 Ahau L 314 2 Ix L

on

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APPENDIX K

TABLE SIMULATION

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Lunar Eclipse 755 P 1Year Month Day Julian Interval Page 148 First Second

1 177 177.85 177.85 177.852 177 176.59 176.59 176.593 148 147.98 177.6 147.984 177 29.62 147.58 177.25 177 147.58 176.86 176.866 177 53 176.86 177.7 177.77 177 177.7 176.61 176.618 177 176.61 177.69 177.699 177 177.69 176.9 176.9

10 177 176.9 177.13 177.1311 177 177.13 148.22 148.2212 177 148.22 176.17 176.1713 148 54 176.17 178.33 178.3314 177 178.33 176.11 176.1115 177 176.11 178.09 178.0916 177 178.09 176.57 176.5717 177 176.57 177.42 177.4218 177 55 177.42 177.28 147.9319 148 147.93 147.18 176.5320 177 29.35 177.87 177.8721 177 147.18 176.63 176.6322 177 56 177.87 177.48 177.4823 178 176.63 177.17 177.1724 177 177.48 176.87 176.8725 177 177.17 177.73 177.7326 177 57 176.87 176.52 147.1227 177 177.73 148.15 177.5528 177 147.12 177.08 177.0829 177 29.4 176.91 176.9130 177 58 148.15 177.79 177.7931 177 177.08 176.27 176.2732 177 176.91 178.3 178.333 177 177.79 176.07 176.0734 177 176.27 148.51 148.5135 177 178.3 176.78 176.78

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Lunar Eclipse 755 P 2Month Day Julian Interval

36 148 51 176.07 177.19 177.1937 177 148.51 177.48 177.4838 177 176.78 176.67 176.6739 177 177.19 177.79 177.7940 177 52 177.48 176.66 176.6641 177 176.67 177.51 147.8942 148 177.79 147.68 177.343 177 176.66 176.77 176.7744 177 147.89 177.78 177.7845 177 53 29.62 176.55 176.5546 177 147.68 177.73 177.7347 177 176.77 176.88 176.8848 177 177.78 177.13 177.1349 148 176.55 148.23 148.2350 177 54 177.73 176.16 176.1651 177 176.88 178.34 178.3452 177 177.13 176.11 176.1153 177 148.23 178.07 178.0754 177 176.16 176.59 176.5955 177 178.34 177.4 177.456 177 176.11 177.29 147.9257 177 178.07 147.21 176.5858 148 55 176.59 177.8 177.859 177 177.4 176.72 176.7260 177 147.92 177.38 177.3861 177 29.37 177.27 177.2762 177 56 147.21 176.77 176.7763 177 177.8 177.82 177.8264 177 176.72 176.45 147.0665 148 177.38 148.18 177.5766 177 177.27 177.08 177.0867 177 57 176.77 176.91 176.9168 177 177.8269 177 147.06

29.39148.18177.08176.91

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APPENDIX L GLOSSARY

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Ah Tzul Ahau A name given to a cannibal monster that descended to earth during eclipses. This monster is similar to the Mexican Tzitzimine Monster. Ahau Constant A number that is added to the Maya Day Count that equals the Julian Date. Ahau Date The last day of the year is always Ahau. The number associated with this date identifies the Katun. If the date is 4 Ahau then the year is identified as a 4 Katun Ahau. Arcs Three areas where the eclipses dates group, when placed within the Double Tzolkin. Also called danger windows, and eclipse seasons. Baktun A period equal to 20 Katuns or 144,000 days. Calendar Round A period combining the Tzolkin and Haab. This is a period of 18,980 days. The period is 52 years or 73 Tzolkins. Copán Method A lunar period of 149 moons. The ratio of 149 moons to 4,400 days is the basis for the statement of the extreme accuracy of the Maya Calendar. Conjunction The place of new moon. Correlation An attempt to synchronize the Maya calendar with the Julian and Gregorian calendars. Draconic Month The time the moon reaches the same node again 27.212 days. It is less than the sidereal month due to the westerly drift of the node. Dresden Codex One of three surviving hieroglyphic texts believed to contain astronomical data. Eclipse The darkening of the sun or moon caused by the shadows produced by the alignment of the earth, sun and moon. Eclipse Table The chart on pages 51 – 58 of the Dresden Codex. The Table is 11,960 Days 1.13.4.0, 405 lunations, 3 Tritos periods or 46 tzolkins in length.

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Eclipse Window An area, measured in angular distance around the node where eclipses can occur. This window produces a 148-day period with a corresponding eclipse, one lunation later. Ecliptic The plane containing the earth and sun. Ephemeris An astronomical almanac or table of the predicted position of celestial bodies. Glyph A A part of the Supplementary Series denoting whether the month is twenty-nine or thirty days in length. Glyph C A part of the supplementary Series that denotes one of six lunar periods. If Glyph A is even, Glyph C is usually odd, if Glyph A is odd, then Glyph C is usually even. GMT (Goodman-Martinez-Thompson) The most widely accepted Ahau Constant of 584285 days. Haab The Maya calendar of 365 days produced by the combination of one of 18 numbers and 20 month names, plus the xma-kaba-kin, which are the five unlucky days at the end of the year. Inex An eclipse period of 358 lunations or 29 years or 10,561 days. This period separates Saros families. Julian Date A calendar produced by Joseph Scalinger that begins on the date B.C.. January 1, 4713. It is the result of the 12-year solar cycle, the 19-year lunar cycle and the 28-year civil cycle. Katun A period of 20 Tuns or 7,200 days. Kin A period equal to one day. Long Count Also called the Initial Series. This is a count of days from the Maya Calendar Round date of 4 Ahau 8 Cumhu. It is made up of the Baktun, Katun, Tun, Uinals, and Kins.

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Lunation A lunar period of 29.5 days. Maya Date The count of days produced by the Maya Long Count. Mean-value A period of 173.31 days that determines the Nodal Passage. Meton An eclipse period of 235 lunations or 19 years. Besides eclipses, the Meton is also used to calculate new and full moons. It is 6,940 (19.5.0) days. Nodal Passage The area in the mean-value system where the eclipses can occur. Nodes The two places were the moon crosses the ecliptic (one for solar and another for lunar eclipses). The nodes are not static but move producing the 148-day eclipses. Non-Central Eclipses Eclipses where the center line of the eclipse angle does not intersect the earth. These occur in the Polar Regions. Opposition The place of the full moon. Palenque Method A period of 81 moons to 2,396 (6.11.12) days. It is not an eclipse period but may be associated with Venus. Perigee The point in the earth’s orbit nearest the sun. Penumbra The light outer portion of the eclipse shadow. Period of Uniformity A brief 70-year period where all of the lunar data agrees at different sites. Popol Vuh The Maya book of creation. Saros A period of 223 lunations or 18 years. It is a period of 6,585 (18.5.5) days. Semester A period of six lunation of alternating twenty-nine and thirty days. Sideral Period A period of 27. 3216 days. This is the period of the moon requires to return to the same point in its orbit relative to the stars.

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Short Count A method of identifying the Calendar Round according to the ending Ahau date. Supplementary Series Also called the Lunar Series because of its lunar information. The glyphs are located between the Calendar Round date written in reverse order. Synodic Period A period of 29.5305 days. This is the period the moon requires to return to the same phase. Sysygy A list of eclipses. Tritos A eclipse period of 135 lunations or 3,986.days (11.1.6). Tzolkin Also known as the Sacred Calendar, A period of 260 days created by a combination of one of thirteen numbers and one of the twenty day names. Tun A period of eighteen Uinals or 360 days. Uayeb Another name for the five unlucky days. Uinal A period of 20 kins or days. Umbra The inner, darker portion of the eclipse shadow. Vigesimal A number system based on the number twenty. Xma-Kaba-Kin The five unlucky days at the end of the year.

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