THE MAYAN CALENDAR

- WHY 260 DAYS?

by

Robert D. Peden

Foreword

The Mayan Ritual year of 260 days was successful for one major reason - after a cycle lasting 59 Ritual years, the tropical year and the Ritual year lock together in step. A period of 59 by 260 days equals a period of 42 tropical years, of 365.242 days.

MEngSc. BE. ( Melbourne University ) ARMTC TTTC. Formerly Principal Lecturer in the School of Sciences, Deakin University, Australia.

Copyright Robert D. Peden - 1981 and May 24, 2004. All rights reserved

Updated June 15, 2004.

The following publication is based on papers written some 20 years ago, but unpublished at that time. Most references are as of 1981 and have not been updated.


ABSTRACT.

 

Interlocking cycles of the form:

a x astronomical period = b x interlock period,

are derived for the tropical year, and the synodic periods of the moon, Venus, Mars and Jupiter, where a, b and the interlock period are whole numbers and the interlock period is in days. In particular it is concluded that the 260 day ritual period of Mesoamerica is the optimum interlock period for creating such interlocking cycles. If cycles with a common interlock period, having lengths less than 100 years and seeking accuracies better than one day in 100 years are required, there is no other choice.

 

The full set of derived interlocking cycles, having a 260 day interlock period are as follows,

42 TY = 59 x 260 [0.4]

405 LM = 46 x 260 [0.3]

61 VY = 137 x 260 [0.8]

1 MY = 3 x 260 [2.8]

88 JY = 135 x 260 [0.3]

Figures in square brackets give the cycle accuracies in days in 100 years. TY is the tropical year, LM is the lunar month, VY the Venus year, MY the Mars year and JY the Jupiter year.

It is further shown that:

The Long Count is a lunar calendar with an accuracy of 0.8 days in 100 years.

The Accounting year of 364 days might be considered a Venus Calendar with an accuracy of 0.3 days in 100 years.

104 years is the natural and optimum intercalation time to correct both the the solar and Venus calendars - and this fact could be considered the basis for the Mesoamerican Calendar Round. The solar intercalation is virtually an exact intercalation and the Venus intercalation has an accuracy of 0.07 days in 100 years.

 

It is proposed that these factual astronomical derivations are ipso facto sufficient to prove the astronomical base for the Mesoamerican calendrical system.

 


INTRODUCTION

 

Many people of ancient Mesoamerica, and the Maya in particular, used a whole number, positional and vigesimal system of arithmetic and counting that was largely devoted to the development of a calendar of divination and the associated problem of aligning the apparently incompatible periods of the tropical year, the lunar month and the synodic periods of the planets. The major difficulty is that the celestial cycles are non-commensurable because the natural counting unit, the solar day, does not divide without remainder into the astronomical periods. To complicate the problem Mesoamerican arithmetic is without fractions and hence whole number ratios are required to represent accurate astronomical relationships. The calendrical solution to this problem determined by such peoples as the Babylonians, Egyptians, Chinese, Hindus, Jews or Moslems differed in the particular method of intercalation but they all in common used a technique of separately intercalating the various cycles and then endeavouring by irregular corrections to move from one cycle to another ( Encyclopedia Britanica ). In contrast the Mesoamericans, according to (Coe 1971: 186; Aveni 1980: 170) did not obviously intercalate their calendars. ( But see Intercalation ). It seems that their method (Makemson 1943: 190) was to use a whole number ritual period, such that particular multiplies of the ritual period interlocked with particular multiples of the astronomical periods, or whole number approximations to them, making them individually commensurable. Maya interlocking cycles had the form:

a x astronomical period = b x ritual period,

where a, b and the ritual period are whole numbers.

In other words, they replaced a multiple of a non-commensurable astronomical period, which their arithmetic could not handle, with a corresponding commensurable whole number cycle.

The most important Mesoamerican ritual period was the Tzolkin, a calendar of great age in Mesoamerica, with a period of 260 days; made up of a repeating sequence of the numbers one to 13, and 20 day names. Thus each day was individually identified with a unique combination of name and number. The count was basic to Maya ritual and sacred life. Up to now no satisfactory explanation of the choice of 260 days has been given, although the period has been linked by Malstrom (1973) with a 260 day interval between transits of the zenithal sun at Copan, at a latitude of 15 degrees North. There are various other possibilities discussed by Aveni (1980: 148-150), such as the the average length of the human gestation period (278 days) , the duration of the visibility of Venus in the sky (263 days) , or that 2 x 260 days is closely equivalent to three eclipse half years (519.93 days) . An inscription from the Zapotec region of Mont Alban, in the highlands of central Oaxaca, dating between 700 and 500 BC, is the oldest evidence known for the existence of a 260 day calendar (Marcus 1980: 49) . The 260 day calendar is still in use today in the highland region of Guatemala (Aveni 1980: 148) .

The question is raised could a calendar in constant use for 2500 years survive on ritual significance alone or must there be some underlying astronomical reason that would account for such a long retention? In 1966 Coe stated:

How such a period of time ever came into being remains an enigma, but the use to which it was put is clear. Every single day had its own omens and associations, and the inexorable march of the twenty days acted as a kind of fortune-telling machine guiding the destinies of the Maya and all peoples of Mexico (Coe 1971: 66) .

And as Thomson (1974: 86) said, "every astronomical mechanism, just like everything else in Maya life, had to be related to the 260-day sacred almanac."

In parallel with the 260 day count the classical Maya used the Long Count, a positional vigesimal system based on a 360 day period called the Tun. Dated Maya classical monuments were imbedded within the Long Count from a reference date generally calculated as 3113 BC. No clear reason for the choice of 360 days is known other than it is the closest multiple of 20 to the tropical year.

KNOWN CYCLES

The known Maya interlocking cycles are as follows.

52 x 365 = 73 x 260 [24.2] - (1)

 

65 x 584 = 146 x 260 [4.9] - (2)

or 5 x 584 = 8 x 365

 

149 LM = 11 x 400 [0.5] - (3)

= 20 x 220

= 22 x 200

 

405 LM = 46 x 260 [0.3] - (4)

or 81 LM = 46 x 52

 

61 VY = 137 x 260 [0.8] - (5)

 

301 VY = 676 x 260 [0.1] -(6)

 

1 MY = 3 x 260 [2.8]- (7)

 

Cycle (1) is called the Calendar Round. It has a length of 52 Vague years of 365 days, so called because the actual length of a tropical year is 365.242 days. At the end of 100 years the tropical year would have advanced on the Calendar Round by 24.2 days Despite this continuous slippage, with respect to the seasons, the Calendar Round was of great ritual significance to the Mesoamericans.

In cycle (2) 584 is the approximate synodic period of Venus. The cycle is two Calendar Rounds long and hence cycles (1) and (2) combined, link together the periods of Venus, the Vague year and the Tzolkin. These cycles are significant because they only involve whole numbers, hence they are obviously idealized calendars which continuously slip in time with reality.

Cycle (3) to (7) however, reflect reality. In these cases the Mesoamericans have accurately replaced a multiple of a non-commensurate astronomical period with a multiple of a ritual period. Cycle (4) is a lunar calculating cycle which is the length of an eclipse table in the Dresden Codex, a book of divination, the contents from around AD 682. Cycle (3) is a lunar calculating cycle. There is some debate about its use (Chambers 1965: 350) but if we accept Coe (1971: 188) it was used in Copan in AD 682 and was subsequently adopted by almost all Maya centres. Cycles (5) and (6) link the Venus year to the Tzolkin and can be found in the Venus tables of the Dresden codex. The tables are based on cycle (2) . It seems (Thomson) 1974: 86-88) that corrections were progressively made to cycle (2) resulting in more accurate relationships of cycles (5) and (6) . Cycle (7) is from a multiplication table in the Dresden Codex involving the number 780 and is usually assumed to relate to Mars. On thing is clear, the interlocking cycles (3), (4) and (5) demonstrate that the Maya could measure astronomical cycles to a high degree of accuracy. We can suppose that that the length of the tropical year was known (Aveni 1980: 1972) and it is probable that the periods of planets such as Jupiter might have also been measured.

Some pertinent questions arise.

Why is there no interlocking cycle linked to the tropical year? Is the arithmetic such that there is no useful cycle? Or as most of the Maya writings were destroyed by the Spanish, is it that the evidence for such a cycle has also been destroyed?

Frequently glyphs giving lunar information are inscribed together with Long Count dates in monuments. Why have no Mesoamerican cycles been recorded linking lunations to the Long count?

Did the Maya use an interlocking cycle for Jupiter?

The simple Mars relationship of Cycle (7) has a rather large error of 2.8 days in 100 years. Is there a more accurate choice of cycle?

Finally, if as it were the Mesoamericans could start over again, in terms of forming interlocking cycles, is there a better choice for a ritual period than 260 days?

 

INTERLOCKING CYCLE SEARCH

To help answer these questions a search was made for possible interlocking cycles of the form

a x astronomical cycle = b x interlock period,

where a, b and the interlock period are whole numbers.

To put limits on the search the cycle lengths were limited to 100 tropical years and the cycles were required to have accuracies better than one day in 100 years; both reasonable limits in terms of the known cycles.

Because of the tidal friction (Jeffreys 1976: 316-363; Munk and MacDonald 1975: 175-249) the period of rotation of the Earth has increased since the inception of the Mesoamerican calendars. Although the difference is small the period used for the tropical year was corrected for 500 BC using an acceleration of 1".55 T^-2 for the Earth, where T is in Julian centuries. The periods used were as follows:

TY = 365.242,3 days

LM = 29.530,590 days

VY = 583.922 days

MY = 779.936 days

JY = 398.867 days.

 

The results are presented in three parts. The first part gives results for interlock periods which are multiples of 20 and which lie in the range 200 to 360. These initial limits are chosen because 20 is the Mesoamerican counting base and a period which is longer than half a year but shorter than a full year can be related to the tropical year without confusion. As well any interlock period shorter than half a year can always be expressed as an appropriate multiple. The second part is a search for the shortest length cycles with interlock periods in the range 1 to 365. The third part is a search for any interlock periods in the range 182 to 365 that are jointly common to all five interlocking cycles.

 

RESULTS PART 1

Results for interlock periods which are multiples of 20 and which lie in the range 200 to 360.

 

TROPICAL YEAR

42 TY = 59 X 260 [0.4] - (8)

LUNAR MONTH

149 LM = 20 x 220 [0.5] - (9)

=22 x 200

256 LM = 21 x 360 [0.8] - (10)

= 27 x 280

405 LM = 46 x 260 [0.3] - (11)

VENUS YEAR

46 VY = 79 x 340 [0.6] - (12)

56 VY = 109 x 300 [0.4] - (13)

61 VY = 137 x 260 [0.8] - (14)

MARS YEAR

1 MY = 3 x 260 [2.8] - (15)

Note that there are no solutions for Mars with an accuracy better than one day in 100 years.

JUPITER YEAR

71 JY = 118 X 240 [0.6] - (16)

88 JY = 135 x 260 [0.3] - (17)

= 117 x 300

 

RESULTS PART 2

Shortest length cycles with interlock periods in the range 1 to 365.

 

TROPICAL YEAR

21 TY = 26 x 295 [0.4] - (18)

= 59 x 130

= 65 x 118

29 TY = 32 x 331 [0.1] - (19)

37 TY = 58 x 233 [0.1] - (20)

 

LUNAR MONTH

32 LM = 3 x 315 [0.8] - (21)

= 5 x 189

= 7 x 135

= 9 x 105

= 15 x 63

= 21 x 45

= 27 x 35

 

81 LM = 8 x 299 [0.3] - (22)

= 13 x 184

= 23 x 104

= 26 x 92

= 46 x 52

 

83 LM = 19 x 129 [0.6] - (23)

= 43 x 57

 

VENUS YEAR

12 VY = 49 x 143 [0.3] - (24)

= 77 x 91

 

14 VY = 25 x 327 [0.4] - (25)

= 75 x 109

 

23 VY = 79 x 170 [0.6] - (26)

= 85 x 158

 

MARS YEAR

12 MY = 49 x191 [0.9] - (39)

14 MY = 61 x 179 [0.3] - (27)

18 MY = 101 x 139 [0.4] - (28)

19 MY = 73 x 203 [0.5] - (29)

 

JUPITER YEAR

7 JY = 8 x 349 [0.9] - (30)

15 JY = 31 x 193 [0.03] - (31)

22 JY = 25 x 325 [0.3] - (32)

= 27 x 351

=39 x 225

= 45 x 195

= 65 x 135

= 75 x 117

 

RESULTS PART 3

For cycle lengths less than 100 years and for accuracies better than one day in 100 years, a search for interlock periods in the range 182 to 365 that are jointly common to all five interlocking cycles,

Interlock periods are:

Common to the solar year and Venus are: 191, 316, 260, and 311.

Common to Mars and Venus are: 191, 314 and 218.

Common to Mars, Venus the lunar month and the solar year is: 191.

Common to Mars, Venus and Jupiter is 314.

Common to Venus, the lunar month, Jupiter and the solar year is: 260.

With the accuracy relaxed to less than 3 days in 100 years for Mars the only interlock period common to all five astronomical cycles is 260.

 

 

DISCUSSION OF RESULTS

TROPICAL YEAR CYCLE

Within the limits stated, cycle (8), 42 TY = 59 x 260, is the only solution for an interlocking cycle linking the tropical year with an interlock period which is a multiple of 20. It is a multiplied up version of cycle (18) , 21 TY = 59 x 130, the shortest possible cycle with an error of less than one day in 100 years. Note that 59 is the the sum of 29 and 30, which is the Mesoamerican whole number way of expressing an approximate lunar month of 29.5 days. Hence this interlocking cycle links the tropical year, and the Maya approximation to the lunar month, with a 260 day interlock period. This cycle could track the tropical year for 200 to 300 years before a one day error accumulated and a correction would be necessary. It follows that if this cycle is in fact a previously unknown Mesoamerican cycle, then the 260 day Tzolkin could be considered as an accurate solar calendar. ( See a web paper "18-Rabbit" by Frederick Martin which refers to this cycle. )

 

LUNAR MONTH

Cycles (9) and (11) correspond to the known Maya cycles (3) and (4). Cycle (10) , 256 LM = 21 x 360 = 7560 days, is a new cycle. However it can be derived from cycles (9) and (11) by simple subtraction, an easy task for Maya arithmetic, so in that sense it can be assumed that the Maya knew of it. It is a multiplied up version of cycle (21) the shortest possible lunar cycle with an error less than one day in 100 years. As 256 = 2^8 and 21 x 360 have as factors all the numbers between 1 and 10 it would have been a very attractive cycle for the application of Maya whole number arithmetic. The special numbers of this interlocking cycle cycle can be used to carry out lunar computations. 256 progressively divided by two forms a binary geometric progression. If, with two whole number corrections, the right hand side of the new cycle is also divided, (i.e. 7560, 3780, 1890, 945, 472, 236, 118, 59,30 ) , then other lunar cycles can be accurately calculated by the primitive method of multiplication by repeated addition, such as was the multiplication method of ancient Egypt (Neugebauer 1969: 73-74) .

What seems not to have been recognized is that, given the Maya used certain astronomical interlocking cycles, any cycles derived by addition or subtraction of these cycles, or their multiples or sub-multiples, can also be considered Maya cycles. The subtraction of cycles (3) and (4) follows:

405 LM = 47 x 260 = 11960 days [0.3]

less 149 LM = 11 x 400 = 4400 days [0.5]

gives 256 LM = 21 x 360 = 7560 days [0.8] - (10)

= 2x2x2x3x3x3x5x7

The two known lunar interlocking cycles have concealed a third interlocking cycle linking the lunar to the 360 day tun, and hence the Long Count. It is again emphasized that this cycle should be considered just as much a Maya cycle as the two cycles from which it was derived.

Glyphs giving lunar information are frequently inscribed with Long Count dates on Mesoamerican stelae. It could be that the inscriptions can be considered in part, as a measurement over long periods of time of the correlation between the moon and the Long Count. On and on through time and endless stretch of days were counted, with significant totals recorded in the inscriptions on the paraded stelae; a giant, distributed, lithic laboratory note book. Manipulating the interlocking lunar cycle is of the utmost simplicity. Successively dividing the equation by two produces the following table with one day corrections at (a) and (b).

256 LM = 7560 days

128 LM = 3780

64 LM = 1890

32 LM = 945 (a)

16 LM = 472

8 LM = 236

4 LM = 118

2 LM = 59 (b)

1 LM = 30

From a table like this the Maya could have generated luna interlocking cycles at will, either for accuracy, arithmetic convenience or ritual significance. As an example, a 361 lunar cycle discussed by Thomson (1974:90) will be calculated.

361 LM = 256 + 64 + 32 + 8 + 1

= 7560 + 1890 + + 945 + 236 + 30

= 10661 days

The exact figure is 10660.54 days.

It is conjectured that this recovered interlocking cycle is the basis for the Mesoamerican choice of the 360 day tun as the counting unit of the Long Count. If this cycle is in fact a previously unknown Mesoamerican interlocking cycle then the Long Count, based on the 360 day Tun, could be considered as a lunar calendar which would track the lunar month for over 100 years before it needed correction. The uinal of 20 days, the length of each of the 18 months of the Long Count, is commonly represented by a moon glyph. Much discussion has occurred to reconcile this seemingly odd name for a 20 day period (Aveni 1980:142; Thomson 1950:143 ). This now makes sense if this count of tuns is recognized as a count of moons.

 

VENUS YEAR

Cycle (12) equates with cycle (26). Cycle (13) equates with cycle (25). From the point of view of Venus alone 300 would probably be the best choice of an interlock period, followed by 340 and then 260. However 143 and 91 of cycle (24) are multiples of 13 and hence have some attraction.

MARS YEAR

No accurate solution, which is a multiple of 20, exists. The best solution is cycle (15), 1 MY = 3 x 260, which equates with the known cycle (7), having an error of 2.8 days in 100 years. Cycles (39), (27), (28), and (29), with interlock periods 191, 119, 139 and 203, have little to recommend them. It seems now that the Dresden 780 multiplication table can be more confidently considered as a Mars table because if an interlock period which is a multiple of 20 is required, 260 is the best choice despite the lower accuracy.

JUPITER YEAR

Cycle (17) , a multiple of 20 interlock period, equates with cycle (32. Interlock periods of 349 and 193 of cycles (30) and (31) have little to recommend them. The best choice is 260 or 300 of cycle (17) , which is more accurate than cycle (16) .

 

CHOICE OF RITUAL PERIOD

 

The results of part 1 show that 260 days is the only multiple of a 20 interlock period, common to all cycles. If interlocking cycles with accuracies better than one day in 100 years are required, there is no other choice. Coupled with the the fact that the related cycle (18) is the shortest interlock cycle available for the tropical year and that cycle (22) is the second shortest available for the moon then the conclusion is reached that 260 days is the optimum choice for a multiple of 20 interlock period.

COMMON INTERLOCK PERIODS

Part 3 shows that in a general search for any interlock periods jointly common to interlocking cycles for the tropical year, Venus year, lunar month, Mars year and the Jupiter year, 260 is the only solution in the range 182 to 365.

The full set of derived interlocking cycles, having a 260 day interlock period are as follows,

42 TY = 59 x 260 [0.4]

405 LM = 46 x 260 [0.3]

61 VY = 137 x 260 [0.8]

1 MY = 3 x 260 [2.8]

88 JY = 135 x 260 [0.3]

 

VENUS- ACCOUNTING YEAR INTERLOCKING CYCLE

The subtraction of equations (5) and (6) lead to what must also be recognized as a known Maya cycle. The cycle is;

301 VY = 676 x 260

less 61 VY = 137 x 260

gives 240 VY = 539 x 260 [0.3] - (33)

= 2x2x5x7x7x11x13

or 48 VY = 77 x 364

or 12 VY = 77 x 91

or 1 VY = 584 - 1/12 - (34)

 

Revealed in equation (33) is an interlocking cycle accurate to 0.3 days in 100 years and capable of factorization by both 260 and 364. The cycle has factors of 2,5,7,11 and 13, which can be combined to give multiples of 364, 260,91, 65 and 52. Extensive, unexplained multiplication tables are given in the Dresden Codex involving these particular numbers (Thomson 1950:252-258). This suggests that the Dresden multiplication tables are aids in calculating the Venus year. On this evidence it is conjectured that perhaps the Accounting year of 364 days might be considered the base of the Mesoamerican Venus Calendar.

The Dresden Codex also includes tables for 54 and 78. It is possible that the 54 times table is connected to the Luna-tun cycle of equation (10) by noting that that the cycle can be expressed as,

256 LM = 140 x 54.

It is proposed that the remaining 78 times table relates to Mars.

 

INTERCALATION

To use the Calendar Round as an actual calendar, correction days need to be calculated so that the true position of an astronomical period are known. A regular rule for correcting a calendar, by inserting or deleting days, is known as intercalation. Two methods are usually adopted. A gross intercalation takes the form:

A = V +or- a/b

where A is the actual astronomical period to be approximated, V is a whole number approximation to A called the Vague period an a and b are integers that, as a fraction, approximates the fractional part of A as closely as possible.

The second method of intercalation where the calendrical error must never exceed one day takes the form:

A = V +or- 1/c +or- 1/cd +or- 1/cde .... and so on,

where c, d, and e are integers that together approximate as closely as possible the fractional part of A. The Gregorian system forces the the corrections to coincide with the centuries. A natural intercalation, however, uses the denominators which best fit, regardless of their value, so as to attain the highest intercalation accuracy using the minimum of terms.

 

NATURAL VENUS INTERCALATION

 

The Venus Year is 583.922 = 584 - 1/13 [0.07] - (35)

= 583.923 days.

 

This natural intercalation for Venus produces a virtually perfect correction when an intercalation day is subtracted every 13 Venus years.

The equation may be expressed as,

13 VY = ( 13 x 584 ) - 1 day,

or 65 VY = ( 65 x 584 ) - 5 days

= ( 146 x 260 ) -5

= ( 104 x 365 ) -5

= two Calendar Rounds less 5 days.

 

The completion of two Calendar Rounds is seen to be the optimal time for a 5 day correction of the Venus calendar with an accuracy of 0.07 days in 100 tropical years. The correction is discussed by Aveni (1980:189) and Closs ( 1977:89 ) without realizing the exactness of the correction. Notice that the recovered interlocking Venus cycle of equation (34) uses 1/12 instead of the virtually exact 1/13 of equation (35)

 

NATURAL SOLAR INTERCALATION

 

365.2423 = 365 + 63/260 [0.00] - (36)

= 365 + 1/4 - 1/130 - (37)

= 365 + 1/4 - 1/104 + 1/520 - (38)

With the tropical year given to four decimal places this is an exact intercalation. The best correction time for the natural solar intercalation, coincides with multiples of the 52 year Calendar Round and the ubiquitous number 260. Equation (36) shows that a correction of 63 days every 260 Vague Years or five Calendar Rounds, exactly corrects the Solar Calendar. Equation (38) reads as, add a day every four years with a pause at the end of every two Calendar Rounds, except at the completion of ten Calendar Rounds.

It is conjectured that this coincidence - that 104 years is the natural and optimum time to correct both the the solar and Venus calendars - is the basis for the Mesoamerican Calendar Round. The ritual significance of the 52 year Calendar Round and the ceremonial binding of the years is now explained, because it was at these times that calendrical corrections were noted and the solar and Venus calendars, within the Calendar Round, were again exactly locked together.

To support this conjecture, the Calendar Round solar intercalation can give an explanation of the change in year bearers. It seems that the year bearers may have changed over long time intervals. In the Classic period the inscribed year bearers were Ik, Manik, Eb, and Caban. In the Codex Peresionius, midway between the Classic and Conquest periods, they were Akbal, Lamat, Ben and Etznab. By the Spanish conquest they were Kan, Muluk, Ix and Cauac. It seems that two shifts may have occurred over some 12 centuries. If the year bearers are to be continually recovered then any gross calendrical corrections must occur in multiples of five days. Equation (38) can be written as,

1 TY = 365 + 25/104 + 1/520 [0.0003].

This process makes a correction of 25 days every two Calendar Rounds. The process repeats until an extra day is added, making a total of 26 days at the end of ten Calendar Rounds, causing an increment in the year bearers. Thus two shifts in the year bearers will occur in 1040 years.

 

 

CHOICE OR CHANCE

Did the Maya just happen by chance to choose the 260 day optimum interlock period? On the other hand it is possible that 260 days was an intentional choice based on long periods of astronomical observation and calculation. It is possible that the Long count, based on 360 days was a lunar calendar, tracking the moon with an interlocking cycle 256 LM = 21 x 360. As a 360 day interlock period cannot accurately and simultaneously track the tropical year, then cycle (8) , 42 TY = 59 x 260 may have been used as an accurate solar-lunar calendar. The use of 260 days may have then been reinforced when it was realized that 260 was more accurate than 360 days in tracking the moon, was able to satisfactorily track Venus and Mars, is the best choice to track Jupiter and is the only choice that can simultaneously track all five cycles.

 

CONCLUSIONS

The above possibilities aside, by whatever means the Mesoamericans arrived at their choice of 260 days, it is concluded that 260 days is optimum for creating interlocking cycles linking the tropical year and the Synodic periods of the moon, Venus, Mars and Jupiter. If cycles with a common interlock period, with cycle lengths less than 100 years and seeking accuracies better than one day in 100 years are required, there is no other choice.

The Long Count is a lunar calendar with an accuracy of 0.8 days in 100 years.

The Accounting year of 364 days might be considered a Venus Calendar with an accuracy of 0.3 days in 100 years.

104 years is the natural and optimum intercalation time to correct both the the solar and Venus calendars - and could be considered the basis for the Mesoamerican Calendar Round. The solar intercalation is virtually an exact intercalation and the Venus intercalation has an accuracy of 0.07 days in 100 years.

It is proposed that these factual astronomical derivations are ipso facto sufficient to demonstrate the astronomical base for the Mesoamerican calendrical system.

 


 

 

REFERENCES CITED

 

Aveni, A. F.

1980 Skywatchers of Ancient Mexico. - University of Texas press, Austin.

 

Chambers, D. W.

1965 Did the Maya Know the Metonic Cycle? - History of Science Society (ISIS) 56: 348-351.

 

Coe, M. D.

1971 The Maya - Penguin Books, UK

 

Jeffreys, Sir H.

1976 The Earth - Cambridge University Press, Cambridge.

 

Makemson, M. W.

1943 The Astronomical Tables of the Maya - Contributions to American Anthropology and History 42: 187-221.

 

Malstrom, V.H.

1973 Origin of the Mesoamerican 260-Day Calendar - Science 181: 939-941

 

Marcus, Joyce

1980 Zapotec Writing - Scientific American 242: 46-60.

 

Munk, W. H. and MacDonald, G. J.

1975 The Rotation of the Earth - Oxford University Press, Oxford.

 

Neugebauer, O.

1957 The Exact Sciences in Antiquity - Brown University Press, Providence.

 

Thompson, J.E.S.

1950 Maya Hierglyphic Writing, Introduction. Carnegie Institute of Washington, Pub. 589.

 

Thompson, J.E.S.

1974 Maya Astronomy. In The Place of Astronomy in the Ancient World. Oxford University Press, London.

 

Closs, M.P.

1977 Mechanism in the Venus Tables of the Dresden Codex. In Native American Astronomy. University of Texas Press, Austin

 

Martin, Frederick

Undated Web document. 18-Rabbit

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Copyright Robert D. Peden - 1981 and May 24, 2004. All rights reserved

Updated June 9, 2004; June 12, 2004; June 15, 2004; June 26, 2004.