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Dresden 59: 780-Day Multiplication Table. (05/01/01)

On page 59 of the Dresden Codex the Maya preserved a long series of multiples of 780 days. Many scholars assume the table somehow relates to the synodic period of Mars, since that planet's average period is equal to 779.94 days. The introductory material for the table, which includes data establishing several Long Count notations for its possible use, however, was placed on page 58 of the Codex and, by virtue of simple juxtaposition, may instead suggest the table of multiples was used in conjunction with Maya eclipse prediction technology since it follows immediately after the conclusion of the day-name positions listed by them for eclipse occurrences. The Eclipse Table itself occupies pages 51a through 58a (top half of the page) and 51b through 58b (bottom half of the page). The first four columns on pages 58a and 58b belong to the Eclipse Table. There are four stations on the top half of the page (27, 28, 29, and 30) and two on the bottom half (68 and 69). Two additional columns on page 58b contain the tenth "picture" in the text, which probably signals the termination point for the eclipse sequence, since the last recorded station (#69) lists the tzolkin day-names 10 Cimi, 11 Manik, and 12 Lamat, where the base-day for the Table itself is 9.16.4.10.8 12 Lamat 1 Muan. Since only four columns (a, b, c, d) were used on page 58 for the Eclipse Table itself, the final two columns (e and f) on both the top and bottom half of the page were used to write the introductory material for the 780-day multiplication table. While this physical structure does not prove a connection between the two tables, the juxtaposition certainly suggests that there might be one.

On the bottom portion of page 58e, in the introductory material for the 780-day table, two ring-numbers were recorded, as 1.7.11 (511 days) and 12.11 (251 days), which were used to count back from the zero base-day of the LC notation at 13.0.0.0.0 4 Ahau 8 Cumku. On the bottom half of page 58f, two distance numbers were recorded, as 9.18.2.2.0 and 9.12.11.11.0, which were then added to the back-counted dates designated by the ring-numbers in order to reach the base-days of the 780-day multiplication table. The second distance number, and the one considered in what follows, produces two dates separated by 260 days (511 - 251 = 260) at 9.12.10.3.9 13 Muluc 7 Mol and 9.12.10.16.9 13 Muluc 2 Zip. In the correlation used in this analysis, set at Julian Day #563334 for the zero base-day of the LC notation, the first date occurred on March 5, 625 A. D. (JD#1949403) and the second one fell 260 days later on November 20, 625 A. D. (JD#1949663). That there might be, and probably is, a connection to the synodic period of Mars in the table's structure can be inferred from the fact that 7 days prior to the first base-day, at 9.12.10.3.2 6 Ik 0 Mol on February 26, 625 A. D. (JD#1949396), Mars reached 12.3* of elongation from the sun in the evening sky on its last day of visibility before its conjunction with the sun. Two other facts which may help explain the astronomy of the table's base-day are apparent in that vernal equinox occurred 12 days after 13 Muluc 7 Mol at 9.12.10.4.1 12 Imix 19 Mol (March 17, 625 A. D.-Julian Day #1949415) when the sun's setting declination reached -00*03'02" at 18:18 PM local time at Palenque and crossed the celestial equator into the northern sky about three hours later.

Also true is the fact that the following day, recorded at 9.12.10.4.2 13 Ik 0 Ch'en, marked the 73^rd Calendar Round anniversary of the day on which the World Tree was raised into the sky, a date recorded at Palenque in the Temple of the Cross (TC:C9-D9) as 0.0.1.9.2 13 Ik 0 Ch'en. Calendar Round anniversaries in the Maya calendrical system were highly significant because they mark points in time when the day-names in the tzolkin (260 days) and haab (365 days) repeat in juxtaposition to each other. This happens only after 52 Maya years have passed. In this instance, and on the day before 13 Ik 0 Ch'en in the ancient sequence, at 0.0.1.9.1 12 Imix 19 Mol (October 22, 3170 B. C.-Julian Day #563875), Venus reached a declination of -00*00'01" at 12:31 PM as it crossed the celestial equator moving from northern to southern sky. It was invisible at the time with an elongation of 2.7* in the morning sky as it moved toward superior conjunction with the sun which occurred eleven days later. At the same time, Saturn reached 3.5* of elongation in the morning sky three days after its solar conjunction. Since both planets crossed the eastern horizon together (at 5:54 AM local time at Palenque), they were in conjunction with each other on the day before the World Tree was raised. A final point of interest in comparing the parallels in the astronomy here concerns the fact that Saturn reached its final day of visibility before solar conjunction at 0.0.1.8.2 6 Ik 0 Mol (October 3, 3170 B. C.-Julian Day #653856) when it reached an elongation of 14.7* in the evening sky sixteen days before solar conjunction and twenty days before the World Tree was raised. The "parallel" here, of course, is that Saturn and Mars switch places at 6 Ik 0 Mol after 73 Calendar Rounds have been counted. What this might imply is that these events were linked to each other across this interval of time because the day-names that mark their occurrence are identical. Finally, 0.0.1.8.9 13 Muluc 7 Mol in the ancient sequence does not mark a celestial event. Data for these events in 625 A. D. are given below:

Base Date A

9.12.10.3.2 6 Ik 0 Mol February 26, 625 A. D. JD#1949396

Mars 12.3* Esky; Last Day visible before solar conjunction.

+7 = 9.12.10.3.9 13 Muluc 7 Mol March 5, 625 A. D. JD#1949403

Base-Day 780-day Multiplication Table Dresden 59

+ 12 (after 13 Muluc 7 Mol) = 9.12.10.4.1 12 Imix 19 Mol March 17, 625 A. D. JD#1949415

Vernal equinox Sun dcl -00*03'02" at 18:18 PM

+ 13 = 9.12.10.4.2 13 Ik 0 Ch'en March 18, 625 A. D. JD#1949416

73 CR after raising of World Tree

The second date and base-day in this structure, 260 days later, at 9.12.10.16.9 13 Muluc 2 Zip, fell on November 20, 625 A. D. (Julian Day #1949663) and occupied the middle day of station 54 in the Dresden Codex eclipse table (55b D) but occurred 3 x 11,960 days prior to the actual position recorded in the Codex itself after the base-day at 9.16.4.10.8 12 Lamat 1 Muan (June 29, 698 A. D.-Julian Day #1976182). The days of record in the Codex are listed as 9.17.10.10.8 12 Lamat 11 Mol, 9.17.10.10.9 13 Muluc 12 Mol, and 9.17.10.10.10 1 Oc 13 Mol. The middle day recorded a lunar eclipse (umbra magnitude .717 at 6:28 PM local time at Palenque) on February 14, 724 A. D. (Julian Day #1985543). The position at the base-day of the 780-day multiplication table, at 13 Muluc 2 Zip, also recorded a lunar eclipse (total) with an umbra magnitude of 1.624 at 6:02 PM, which was visible at Palenque because its greatest magnitude occurred 35 minutes after sunset on November 20, 625 A. D. The Maya number recorded above the tzolkin day-names in the Dresden text at station 54 is 1.6.0.0, which converts to 9,360 days in Arabic numerals. That value is significant for several reasons. When counting forward from the base-day of the Dresden eclipse table that strikes 13 Muluc 2 Zip as the middle day of station 54, 9,360 days of the table have been counted. The next base-day of the table then occurs 2,600 days later (11,960 - 9,360 = 2,600). From that point forwward, exactly two turns of 11,960 are required to reach the base-day recorded in the Dresden Table at 9.16.4.10.8. The total interval, therefore, from 13 Muluc 2 Zip, as the base-day of the 780-day multiplication table, to 12 Lamat 1 Muan, as the base-day of the Maya eclipse table, is one day less than 2,600 + 2 x 11,960 = 26,520 days. This value is equal to 34 x 780, a value that was not recorded in the multiplication table on page 59 of the Dresden Codex, but one that was certainly accessible to the astronomers at the time. The reason this value was not recorded in the table concerns the fact that prior to its expected position in the text the sequence was shifted from single multiples of 780 to ones that were multiples of 19 x 780 (14,820 days). Hence, 38 x 780 (2 x 19) was recorded in the text, where 34 x 780 was not listed. No clear reason has been proposed to explain why the Maya shifted the sequence of multiples at this particular point in the table's development.

An important point to take note of here is that these two tables, if they were being used somehow in tandem during the Classic period, would have been employed to count back in time from the base-day of the eclipse table (9.16.4.10.8 12 Lamat 1 Muan) to the base-day of the multiplication table (9.12.10.16.9 13 Muluc 2 Zip), rather than the other way around, for the simple reason that 13 Muluc 2 Zip precedes 12 Lamat 1 Muan by nearly 73 years. One reason this structure might have been preserved can be seen in the fact that the base-day in the 780-day multiplication table strikes a position in close proximity to the death date of Palenque's most widely known ruler during the Classic period, Pacal II. His death was recorded at 9.12.11.5.18 6 Etz'nab 11 Yax on April 18, 626 A. D. (Julian Day #1949812), which fell just 149 days after the 13 Muluc 2 Zip base-day of the 780-day table. The interval of separation is significant in Maya eclipse prediction technology because they used an interval of 148 day to mark specific advances in the eclipse sequence just prior to the insertion points of "pictures" in the text of the Dresden table. The interval occurs a total of nine times in the structure (at 53a D and E, 55a A and B, 56a B and C, 57a E and F, 52b A and B, 53b C and D, 54b E and F, 56b A and B, 57b D and E). The tenth "picture" in the text (at 58b C and D), as noted above, does not mark an insertion point for the 148-day interval but, instead, signals the termination point of the table as a whole. In this particular case, the next lunar eclipse in the sequence, at station 55 (55b E) in the table, fell 177 days after 13 Muluc 2 Zip at 9.12.11.7.6 8 Cimi 19 Zac on May 16, 626 A. D. (Julian Day #1949840), 28 days after Pacal II's death. A full moon, and the one that preceded the lunar eclipse by one lunar synodic month, fell 147 days after 13 Muluc 2 Zip, and two days before Pacal's death, at 9.12.11.5.16 4 Cib 9 Yax on April 16, 626 A. D. (Julian Day #1949810). A solar eclipse, and the one paired with the lunar event at 8 Cimi 19 Zac, occurred 13 days after Pacal II's death, at 9.12.11.6.11 6 Chuen 4 Zac on May 1, 626 A. D. (Julian Day #1949825). On this same day, Venus reached 3.1* of elongation from the sun in the morning sky on the day of its inferior conjunction with the sun. Also significant in this structure is the fact that a conjunction between Mars (at 76.5* of elongation in the morning sky) and Saturn (at 76.4* of elongation) occurred five days after the 13 Muluc 2 Zip base-day of the multiplication table at 9.12.10.16.14 5 Ix 7 Zip on November 25, 625 A. D. (Julian Day #1949668). This event may point back to the shifting positions of Mars and Saturn on their last days of visibility before solar conjunction on 6 Ik 0 Mol, with Mars holding that position seven days before the first base-day (at 9.12.10.3.2), where Saturn held it 73 Calendar Rounds earlier at 0.0.1.8.2 in the ancient sequence. While it might not be reasonable to argue the Maya knew the ancient astronomy, it is certainly clear they would have seen the connection between the contemporary events and their day-names and the date in the Temple of the Cross at Palenque that recorded the day the World Tree entered the sky (0.0.1.9.2 13 Ik 0 Ch'en). This is probably inescapable, since Chan-Bahlum II, Pacal's son, was the person responsible for the inscription where the date was recorded. Data relevant to this discussion appear below:

Base Date B

12 Lamat 1 Zip

9.12.10.16.9 13 Muluc 2 Zip November 20, 625 A. D. JD#1949663 Lunar eclipse

1 Oc 3 Zip

+ 5 = 9.12.10.16.14 5 Ix 7 Zip November 25, 625 A. D. JD#1949668

Mars 76.5* Msky rise 00:58 AM/ Saturn 76.4* Msky rise 00:59 AM

+ 147 = 4 Cib 9 Yax Full moon

+ 149 = 9.12.11.5.18 6 Etz'nab 11 Yax April 18, 626 A. D. JD#1949812

Pacal II Death at Palenque.

5 Oc 3 Zac

+162 = 9.12.11.6.11 6 Chuen 4 Zac May 1, 626 A. D. JD#1949825

Venus 3.1* Msky at IC; Solar eclipse (+ 13 after Pacal II's Death)

7 Eb 5 Zac

What seems apparent here, though perhaps not definitively so, is that the 780-day multiplication table on page 59 of the Dresden Codex was involved in the technology Classic period astronomers developed and used to predict eclipse occurrences in the eclipse table that precedes it in the Codex. Venus, Mars, and Saturn, since they also appear at significant points in their motion relative to the sun and to each other, also seem to be involved in the articulation of the base-days for the table at 13 Muluc 7 Mol and at 13 Muluc 2 Zip. The proximity of the base-days to the death date of Pacal II at Palenque, while proving nothing with certainty, may suggest that this technology evolved during his reign.

One possible function of the 19 x 780-day interval (at 14,820 days), which the Maya inscribed in the structure of the table for some reason, can be inferred from its application to a potential early base-day in the Dresden eclipse table that occurred 6 x 11,960 days prior to 9.16.4.10.8 12 Lamat 1 Muan. The position in question, at 9.6.5.4.8 12 Lamat 1 Zotz on January 9, 502 A. D. (Julian Day #1904422), was the day of a lunar eclipse (umbra magnitude .693 at 4:20 AM local time at Palenque), which certainly qualifies the position as a base-day in the Maya system. Counting forward from this position in the Maya LC notation by 2 x 14,820 days, a second position designated by 9.10.7.10.8 12 Lamat 16 Yaxkin is reached on March 5, 583 A. D. (Julian Day #1934062). This position is significant for a number of reasons. For instance, counting forward from 12 Lamat 16 Yaxkin by an interval equal to 59 x 260 days (15,340), a third 12 Lamat (6 Mol) is reached on March 4, 625 A. D. (Julian Day #1949402), which is, of course, the day before 9.12.10.3.9 13 Muluc 7 Mol, the first base-day in the 780-day multiplication table on page 59 of the Dresden Codex. Now, the second interval of extension here, at 15,340 days, is one in the Maya calendrical system that counts the exact value of 42 tropical years at 365.2422 days each (42 x 365.2422 = 15,340.172 days). What this means to the structure here is that the day of vernal equinox, 13 days after 12 Lamat 16 Yaxkin, at 9.10.7.11.1 12 Imix 9 Mol on March 18, 583 A. D. (Julian Day #1934075), shares the same tzolkin day-name with the one that occurred 12 days after the first base-day in the 780-day table. The advance in the haab position, as it always does in this structure, covers ten days and moves forward from 9 Mol to 19 Mol. The two dates, then, are designated by 9.10.7.11.1 12 Imix 9 Mol and 9.12.10.4.1 12 Imix 19 Mol (March 17, 625 A. D.-Julian Day #1949415) in the Maya calendrical system.

A second aspect of the astronomy that surfaces here concerns the fact that nine days after 12 Lamat 16 Yaxkin, at 9.10.7.10.17 8 Caban 5 Mol on March 14, 583 A. D. (Julian Day #1934071), there was a lunar eclipse with an umbra magnitude of .473 at 11:16 PM local time at Palenque. This reinforces the notion that the 780-day table was involved in determining eclipse occurrences. This eclipse, furthermore, counts forward to the actual Dresden table position at 9.17.0.8.17 8 Caban 10 Ch'en on March 6, 714 A. D. (Julian Day #1981911). The lunar eclipse at this station (#33) had an umbra magnitude of .369 at 11:30 PM local time Palenque. This eclipse is significant because its LC notation, in the final two positions of its expression (8.17), preserves the primary interval of extension in the eclipse table itself (at 177 days) and is the first lunar eclipse after the Katun-ending position at 9.17.0.0.0 13 Ahau 18 Cumku.

The reason any of this matters is that 44 days after 12 Lamat 16 Yaxkin the death date of Lady Zac-Kuk was recorded (TISL: E42-E46) at Palenque as 9.10.7.13.5 4 Chicchan 13 Yax on May 1, 583 A. D. (Julian Day #1934119). Making this calendrical structure even more significant is the fact that Lady Zac-Kuk was Pacal II's mother, where the day-name for vernal equinox prior to the mother's death, at 12 Imix 9 Mol, fell 59 x 260 days prior to the one for her son's death, at 12 Imix 19 Mol, an interval that exactly preserves 42 tropical years of separation between them. Data for this structure are given below:

9.6.5.4.8 12 Lamat 1 Zotz January 9, 502 A. D. JD#1904422 (+ 6 x 11960 to 12 Lamat 1 Muan) Lunar eclipse: umbra magnitude .693 at 4:20 AM

+ 2 x 14,820 = JD#1934062

9.10.7.10.8 12 Lamat 16 Yaxkin March 5, 583 A. D.

+59 x 260 = 15,340 = JD#1949402

9.12.10.3.8 12 Lamat 6 Mol March 4, 625 A. D.

+ 1 = 9.12.10.3.9 13 Muluc 7 Mol Base-day Dresden 59.

12 Lamat 16 Yaxkin

+ 9 = 9.10.7.10.17 8 Caban 5 Mol March 14, 583 A. D. JD#1934071

Lunar eclipse: umbra magnitude .473 at 11:16 PM

+ 13 = 9.10.7.11.1 12 Imix 9 Mol March 18, 583 A. D. JD#1934075

Vernal equinox Sun dcl +00*00'01" at 17:19 PM

+ 44 = 9.10.7.13.5 4 Chicchan 13 Yax May 1, 583 A. D. JD#1934119

Lady Zac-Kuk Death at Palenque (Pacal II's mother).

While the structure here is more suggestive than it is definitive, the fact that death dates for two of Palenque's rulers, a mother and her son as it were, are separated by a calendrical interval in the tzolkin that insures positions of the sun in the tropical year will occur on the same day-names in the 260-day cycle, with a 10-day advance in their respective haab positions, implies that deliberate calculations based on the structure of the Dresden Codex eclipse table and the 780-day multiplication table played as much a role in establishing these dates as natural phenomena might have done. The fact that the day-names for vernal equinox in this structure point back to the day on which the World Tree was raised, at exactly 73 Calendar Rounds earlier, suggests a spiritual link between the death dates of mother and son and the day on which the Tree entered the sky. That eclipse occurrences, and the technology that delineates their periodicity in Classic period Maya astronomical practice, are directly linked as well to this spiritual ideology tells us that such events were seen as an essential part of the meaning of kingship in Maya civilization. What this final observation suggests, if it is valid, is that a primary objective of any correlation proposal must be that it resolve the issue of how, and even of whether or not, the day-name structure of the Dresden Codex eclipse table properly accounts for actual eclipse occurrences during the Classic period.