Evidence that the Maya used the system outlined above and the reverse
process mainly lies in the tables themselves in the Dresden Codex, but
some confirmation may be found in three decipherable long distance numbers
on page 31a of the Dresden Codex, which lead to the day 13 Akbal, the basis
of the table on page 32a. All three are supplied with Ring Numbers.12
Once proceeds from 13 Akbal to 13 Akbal; another from 13 Cauac to 13 Akbal;
the third, when emended, from 13 Imix to 13 Akbal. These distance
|18.104.22.168.4||(base 13 Cauac at 6.1 before 22.214.171.124.0.)|
|126.96.36.199.0||(base 13 Akbal at 17 before 188.8.131.52.0.)|
|184.108.40.206.2||(base 13 Imix at 7.0.19 before 220.127.116.11.0.)|
The last has been emended by changing 13 uinals to 10 uinals, and assuming that the partly obliterated cycle coefficient is 8. A change of some sort is necessary in order to make the number read 13 Akbal as recorded at the base of the column. The elimination of three dots in the uinal coefficient is the simplest, and therefore, the most logical correction. The Ring Number is also questionable. It reads 7, 2 within a zero sign, 19 (but the bottom bar is read indicating that the scribe corrected 14 to 19).
The first number (1272544 in Arabic notation) equals 3496 computing years. The second (1268540 in Arabic notation) equals 3485 computing years. The third, as emeded (1246882 in Arabic notation), resolves itself into 3425 computing years plus 182 days (a half computing year). THe fact that the first and second long numbers are integral mutliples of 364 is not chance, because the interval from a day in the 260-day sacred almanac to the same day at another position in the Long Count selected at random will by the law of averages be divisible by 364 only once in ever seven occurrences (e.g. a day 13 Akbal recurs every 260 days, but is found at the end of a computing year only after 1820 days). It is therefore quite probable these long distance numbers at the start of the table of computing years were formed by using the formulae of the computing year.
Mistakes in certain calculations in the inscriptions and codices tend to confirm, although, from their very nature, they cannot alone prove that the computing year (often in long distances in combination with the Calendar Round) was used by the Maya astrologers in working out dates. A number of such mistakes will now be discussed.
Case A. On the panel of the Temple of the Foliated Cross at Palenque
the following calculation is found:
|D14-C15||18.104.22.168.0||1 Ahau 13 Mac|
|L1-M1||(22.214.171.124.16)||2 Cib 14 Mol|
The long distance, however, leads not to 126.96.36.199.16, 2 Cib 14 Mol, but to 188.8.131.52.16, 2 Cib 14 Yax. One is immediately struck by the fact that the Calendar Round date is correctly reached so far as day sign and coefficient are concerned, and that the month coefficient is correct, but that the month reached is Yax instead of Mol. The position 184.108.40.206.16 in the Long Count is certain. That of the 220.127.116.11.0 for 1 Ahau 13 Mac is almost equally certain since it occurs as an Initial Series at the start of this text.
The Secondary Series must be changed to 18.104.22.168.16 in order to link
the dates. This involves an addition of 22.214.171.124, which in turn consists
of 2 Calendar Rounds and two 20-computing-year periods:
|126.96.36.199||(2 Calendar Rounds)|
|188.8.131.52||(364 x 20 x 2)|
To calculate this long distance by the computing-year formulae, the
Maya scribe should calculate as follows:
|184.108.40.206.0||1 Ahau 13 Mac|
|220.127.116.11.16||2 Cib 4 Tzec|
|Step 2.||18.104.22.168||(364 x 20 x 3) Subtract|
|22.214.171.124.16||2 Cib 4 Mol|
|Step 3.||10.2.0||(364 x 10) Subtract|
|126.96.36.199.16||2 Cib 4 Mol|
|Step 4.||188.8.131.52.0||(60 Calendar Rounds) Add|
|184.108.40.206.16||2 Cib 14 Mol|
Step 1 brings the day sign to the required 2 Cib, while causing the month to change to 4 Tzec. Step 2 involves the subtraction of three 20-computing-year periods to bring the month to the required Mol (but the coefficient is 4). Step 3 requires the subtraction of 10 computing years in order to convert 4 Mol to 14 Mol. Step 4 merely calls for the addition of Calendar Rounds (in this case 60) to reach the required Long Count position.
Our calculator, if he used this system, subtracted two too few 20-computing-year periods, and added two too few Calendar Rounds. No other method of calculation, so far as I can see, would as easily account for the error.
Case B. The panel of the Temple of the Cross at Palenque has the
|DA16-B16||(12.1913.3.0||1 Ahau 18 Zotz|
|E9-F9||(220.127.116.11.2)||9 Ik 0 Zac|
Here the Long Count positions are certain, but 0 Zac is a mistake for
0 Yax. The Secondary Series resolves itself into:
|12.2||to reach 9 Ik 0 Kayab|
|18.104.22.168||(364 x 20 x 7) Add to reach 9 Ik 0 Yax|
|22.214.171.124.0||(13 Calendar Rounds)|
The mistake is best accounted for by supposing that in making the calculation the priest moved back the month indicator six places instead of the seven places necessary when seven 20-computing-year periods were added.
Case C. On Lintels 27, 28, and 59 at Mencho (Yaxchilan) one mistake
leads to another. The inscription, as it stands, reads:13
|(126.96.36.199.5)||6 Chicchan 8 Yax (mistake for 8 Zac)|
|(188.8.131.52.14)||6 Ix 12 Yaxkin|
|(184.108.40.206.14)||3 Ix 17 Zip (probable mistake for 10 Ix 17 Zotz|
|(220.127.116.11.3)||10 Akbal 16 Uo|
|(18.104.22.168.17)||6 Caban 10 Zac|
Here the month of the first date is one place short of the required Zac, and there is another (possibly a third) mistake in the second part of the inscription. It seems rather certain that the second date was used to calculate the first, since the former occurs in two other inscriptions at Menche, and its Long Count position must have been well known. If the priest used the computing year in his calculations, from 22.214.171.124.14 6 Ix 12 Yaxkin he would have subtracted one 10-computing-year period and 17 computing years, and then added the remainder of 1.19 to reach 126.96.36.199.5, but, forgetting to move the month indicator forward one month when he subtracted the 20-computing-year period, his count would come to 8 Yax instead of 8 Zac.
Then he may have calculated forward to reach 188.8.131.52.14 from his erroneous 8 Yax, but as that was one month under the mark, he reached 17 Zip instead of 17 Zotz (the next month), but calculated the day sign coefficient in agreement with the erroneous 17 Zip. Alternatively, 3 Ix 17 Zip is correct, and there are errors of one uinal in the Secondary Series on each side of that date. In any case the error or errors in the second part of the inscription almost certainly result from the original mistake of 8 Yax for 8 Zac.
Case D. Stela N at Copan records the Initial Series 184.108.40.206.0, 1 Ahau 8 Zip, 8 Zip being an error for 3 Zip. Here the error might be caused by forgetting to subtract five from the coefficient of the month sign when five computing years were added or the sculptor may have added the bar in copying the guide drawings.
Carelessness on the part of the sculptor must have been too dangerous to have been frequent. As it is known that a drummer who beat out of rhythm at an important ceremony might be sacrificed for his carelessness, we may assume that the sculptor who committed a more enduring error did not pay a lighter penalty unless he was indispensible. Furthermore, a careless sculptor might fail to copy a bar from the guide drawing, but he would scarcely put in one that never existed. It would therefore seem more probable that the error was made by the priest who calculated the date.
There is an error difficult of rectification in the extremely long distance number later in the inscription. Possibly, as is the case of Lintels 27, 28, and 59 at Menche, one error led to another.14
Case E. On page 58 of the Dresden Codex, Zac with an obliterated coefficient occurs where the calculations lead to 2 Mol. Here the astrologer may have forgotten to drop three places in the month column when adding 20-computing-year periods. Similar errors occur on other pages. Not all errors, however, could be caused by mistakes in the application of the computing-year formulae. Some are clearly due to carelessness. For example, one Initial Series is written 220.127.116.11.0 where 18.104.22.168.0 was meant. The computer, or more probably a copyist, dropped a dot from the cycle coefficient, but added one to the uinal coefficient. Again, in the case of two of the long distance serpent numbers which are intertwined, the katun coefficients are transposed, for the katun coefficient of 14 of the red distance number is drawn in black and the zero (always red) of the black distance number is placed on the wrong side so as to go with the red distance number.
A somewhat similar transposition of numbers on Altar Q at Copan has been discussed by Morley.15 Others are known. These mistakes that have been discussed involve only the month or its coefficient. Errors from calculations made in other systems (for example, additions of katuns, tuns, uinals, or kins) usually necessitate changes in day sign and month at every step, but by the formulae of the computing year the day part of the Calendar Round is subject to a minimum of changes, and is therefore less likely to be wrong. As day signs are very seldom wrong, and errors in both day and month even rarer, the evidence of mistakes, other than those due to carelessness, tends to support the evidence derived from the Dresden Codex that the 364-day year and its multiples were utilized for calculations.
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