first example, it is seen by Table XVI that 1 Calendar Round, or 18,980
days, may be deducted from it; 31,741 - 18,980 = 12,761. In other words, we
can count the number 12,761 forward (or backward had the count been
backward in our example) from the starting point 4 Ahau 8 Cumhu, and reach
exactly the same terminal date as though we had counted forward 31,741, as
in the first case.
Mathematical proof of this point follows:
12,761 ÷ 13 = 981-8/13 12,761 ÷ 20 = 638-1/20 12,761 ÷ 365 = 34-351/365
The numerators of the fractions in these three quotients are 8, 1, and 351;
these are identical with the numerators of the fractions in the quotients
obtained by dividing 31,741 by the same divisors, those indicated in rules
1, 2, and 3, respectively. Consequently, if these three numerators be
counted forward from the corresponding parts of the starting point, 4 Ahau
8 Cumhu, the resulting terms together will form the corresponding parts of
the same terminal date, 12 Imix 14 Kayab.
Similarly it could be shown that 50,721 or 69,701 counted forward or
backward from any starting point would both reach this same terminal date,
since subtracting 2 Calendar Rounds, 37,960 (see Table XVI), from the
first, and 3 Calendar Rounds, 56,940 (see Table XVI), from the second,
there would remain in each case 12,761. The student will find his
calculations greatly facilitated if he will apply this rule whenever
possible. To familiarize the student with the working of these rules, it is
thought best to give several additional examples involving their use. {144}
TABLE XVI. 80 CALENDAR ROUNDS EXPRESSED IN ARABIC AND MAYA NOTATION
+--------+---------+-----------------+
|Calendar| Days| Cycles, Etc.|
| Rounds| | |
+--------+---------+-----------------+
| 1| 18,980| 2. 12. 13. 0|
+--------+---------+-----------------+
| 2| 37,960| 5. 5. 8. 0|
+--------+---------+-----------------+
| 3| 56,940| 7. 18. 3. 0|
+--------+---------+-----------------+
| 4| 75,920| 10. 10. 16. 0|
+--------+---------+-----------------+
| 5| 94,900| 13. 3. 11. 0|
+--------+---------+-----------------+
| 6| 113,880| 15. 16. 6. 0|
+--------+---------+-----------------+
| 7| 132,860| 18. 9. 1. 0|
+--------+---------+-----------------+
| 8| 151,840| 1. 1. 1. 14. 0|
+--------+---------+-----------------+
| 9| 170,820| 1. 3. 14. 9. 0|
+--------+---------+-----------------+
| 10| 189,800| 1. 6. 7. 4. 0|
+--------+---------+-----------------+
| 11| 208,780| 1. 8. 19. 17. 0|
+--------+---------+-----------------+
| 12| 227,760| 1. 11. 12. 12. 0|
+--------+---------+-----------------+
| 13| 246,740| 1. 14. 5. 7. 0|
+--------+---------+-----------------+
| 14| 265,720| 1. 16. 18. 2. 0|
+--------+---------+-----------------+
| 15| 284,700| 1. 19. 10. 15. 0|
+--------+---------+-----------------+
| 16| 303,680| 2. 2. 3. 10. 0|
+--------+---------+-----------------+
| 17| 322,660| 2. 4. 16. 5. 0|
+--------+---------+-----------------+
| 18| 341,640| 2. 7. 9. 0. 0|
+--------+---------+-----------------+
| 19| 360,620| 2. 10. 1. 13. 0|
+--------+---------+-----------------+
| 20| 379,600| 2. 12. 14. 8. 0|
+--------+---------+-----------------+
| 21| 398,580| 2. 15. 7. 3. 0|
+--------+---------+-----------------+
| 22| 417,560| 2. 17. 19. 16. 0|
+--------+---------+-----------------+
| 23| 436,540| 3. 0. 12. 11. 0|
+--------+---------+-----------------+
| 24| 455,520| 3. 3. 5. 6. 0|
+--------+---------+-----------------+
| 25| 474,500| 3. 5. 18. 1. 0|
+--------+---------+-----------------+
| 26| 493,480| 3. 8. 10. 14. 0|
+--------+---------+-----------------+
| 27| 512,460| 3. 11. 3. 9. 0|
+--------+---------+-----------------+
| 28| 531,440| 3. 13. 16. 4. 0|
+--------+---------+-----------------+
| 29| 550,420| 3. 16. 8. 17. 0|
+--------+---------+-----------------+
| 30| 569,400| 3. 19. 1. 12. 0|
+--------+---------+-----------------+
| 31| 588,380| 4. 1. 14. 7. 0|
+--------+---------+-----------------+
| 32| 607,360| 4. 4. 7. 2. 0|
+--------+---------+-----------------+
| 33| 626,340| 4. 6. 19. 15. 0|
+--------+---------+-----------------+
| 34| 645,320| 4. 9. 12. 10. 0|
+--------+---------+-----------------+
| 35| 664,300| 4. 12. 5. 5. 0|
+--------+---------+-----------------+
| 36| 683,280| 4. 14. 18. 0. 0|
+--------+---------+-----------------+
| 37| 702,260| 4. 17. 10. 13. 0|
+--------+---------+-----------------+
| 38| 721,240| 5. 0. 3. 8. 0|
+--------+---------+-----------------+
| 39| 740,220| 5. 2. 16. 3. 0|
+--------+---------+-----------------+
| 40| 759,200| 5. 5. 8. 16. 0|
+--------+---------+-----------------+
| 41| 778,180| 5. 8. 1. 11. 0|
+--------+---------+-----------------+
| 42| 797,160| 5. 10. 14. 6. 0|
+--------+---------+-----------------+
| 43| 816,140| 5. 13. 7. 1. 0|
+--------+---------+-----------------+
| 44| 835,120| 5. 15. 19. 14. 0|
+--------+---------+-----------------+
| 45| 854,100| 5. 18. 12. 9. 0|
+--------+---------+-----------------+
| 46| 873,080| 6. 1. 5. 4. 0|
+--------+---------+-----------------+
| 47| 892,060| 6. 3. 17. 17. 0|
+--------+---------+-----------------+
| 48| 911,040| 6. 6. 10. 12. 0|
+--------+---------+-----------------+
| 49| 930,020| 6. 9. 3. 7. 0|
+--------+---------+-----------------+
| 50| 949,000| 6. 11. 16. 2. 0|
+--------+---------+-----------------+
| 51| 967,980| 6. 14. 8. 15. 0|
+--------+---------+-----------------+
| 52| 986,960| 6. 17. 1. 10. 0|
+--------+---------+-----------------+
| 53|1,005,940| 6. 19. 14. 5. 0|
+--------+---------+-----------------+
| 54|1,024,920| 7. 2. 7. 0. 0|
+--------+---------+-----------------+
| 55|1,043,900| 7. 4. 19. 13. 0|
+--------+---------+-----------------+
| 56|1,062,880| 7. 7. 12. 8. 0|
+--------+---------+-----------------+
| 57|1,081,860| 7. 10. 5. 3. 0|
+--------+---------+-----------------+
| 58|1,100,840| 7. 12. 17. 16. 0|
+--------+---------+-----------------+
| 59|1,119,820| 7. 15. 10. 11. 0|
+--------+---------+-----------------+
| 60|1,138,800| 7. 18. 3. 6. 0|
+--------+---------+-----------------+
| 61|1,157,780| 8. 0. 16. 1. 0|
+--------+---------+-----------------+
| 62|1,176,760| 8. 3. 8. 14. 0|
+--------+---------+-----------------+
| 63|1,195,740| 8. 6. 1. 9. 0|
+--------+---------+-----------------+
| 64|1,214,720| 8. 8. 14. 4. 0|
+--------+---------+-----------------+
| 65|1,233,700| 8. 11. 6. 17. 0|
+--------+---------+-----------------+
| 66|1,252,680| 8. 13. 19. 12. 0|
+--------+---------+-----------------+
| 67|1,271,660| 8. 16. 12. 7. 0|
+--------+---------+-----------------+
| 68|1,290,640| 8. 19. 5. 2. 0|
+--------+---------+-----------------+
| 69|1,309,620| 9. 1. 17. 15. 0|
+--------+---------+-----------------+
| 70|1,328,600| 9. 4. 10. 10. 0|
+--------+---------+-----------------+
| 71|1,347,580| 9. 7. 3. 5. 0|
+--------+---------+-----------------+
| 72|1,366,560| 9. 9. 16. 0. 0|
+--------+---------+-----------------+
| 73|1,385,540| 9. 12. 8. 13. 0|
+--------+---------+-----------------+
| 74|1,404,520| 9. 15. 1. 8. 0|
+--------+---------+-----------------+
| 75|1,423,500| 9. 17. 14. 3. 0|
+--------+---------+-----------------+
| 76|1,442,480|10. 0. 6. 16. 0|
+--------+---------+-----------------+
| 77|1,461,460|10. 2. 19. 11. 0|
+--------+---------+-----------------+
| 78|1,480,440|10. 5. 12. 6. 0|
+--------+---------+-----------------+
| 79|1,499,420|10. 8. 5. 1. 0|
+--------+---------+-----------------+
| 80|1,518,400|10. 10. 17. 14. 0|
+--------+---------+-----------------+
{145}
Let us count forward the number 5,799 from the starting point 2 Kan 7 Tzec.
It is apparent at the outset that, since this number is less than 18,980,
or 1 Calendar Round, the preliminary rule given on page 143 does not apply
in this case. Therefore we may proceed with the first rule given on page
139, by means of which the new day coefficient may be determined. Dividing
the given number by 13 we have: 5,799 ÷ 13 = 446-1/13. Counting forward the
numerator of the fractional part of the resulting quotient (1) from the day
coefficient of the starting point (2), we reach 3 as the day coefficient of
the terminal date.
The second rule given on page 140 tells how to find the day sign of the
terminal date. Dividing the given number by 20, we have: 5,799 ÷ 20 =
289-19/20. Counting forward the numerator of the fractional part of the
resulting quotient (19) from the day sign of the starting point, Kan, in
the sequence of the twenty-day signs given in Table I, the day sign Akbal
will be reached, which will be the day sign of the terminal date. Therefore
the day of the terminal date will be 3 Akbal.
The third rule, given on page 141, tells how to find the position which the
day of the terminal date occupied in the 365-day year. Dividing the given
number by 365, we have: 5,799 ÷ 365 = 15-324/365. Counting forward the
numerator of the fractional part of the resulting quotient, 324, from the
year position of the starting date, 7 Tzec, in the sequence of the 365 year
positions given in Table XV, the position 6 Zip will be reached as the
position in the year of the day of the terminal date. The count by means of
which the position 6 Zip is determined is given in detail. After the year
position of the starting point, 7 Tzec, it requires 12 more positions (Nos.
8-19, inclusive) before the close of that month (see Table XV) will be
reached. And after the close of Tzec, 13 uinals and the xma kaba kin must
pass before the end of the year; 13 × 20 + 5 = 265, and 265 + 12 = 277.
This latter number subtracted from 324, the total number of positions to be
counted forward, will give the number of positions which remain to be
counted in the next year following: 324 - 277 = 47. Counting forward 47 in
the new year, we find that it will use up the months Pop and Uo (20 + 20 =
40) and extend 7 positions into the month Zip, or to 6 Zip. Therefore,
gathering together the values determined for the several parts of the
terminal date, we may say that in counting forward 5,799 from the starting
point 2 Kan 7 Tzec, the terminal date reached will be 3 Akbal 6 Zip.
For the next example let us select a much higher number, say 322,920, which
we will assume is to be counted forward from the starting point 13 Ik 0
Zip. Since this number is above 18,980, we may apply our preliminary rule
(p. 143) and deduct all the Calendar {146} Rounds possible. By turning to
Table XVI we see that 17 Calendar Rounds, or 322,660, may be deducted from
our number: 322,920 - 322,660 = 260. In other words, we can use 260 exactly
as though it were 322,920. Dividing by 13, we have 260 ÷ 13 = 20. Since
there is no fraction in the quotient, the numerator of the fraction will be
0, and counting 0 forward from the day coefficient of the starting point,
13, we have 13 as the day coefficient of the terminal date (rule 1, p.
139). Dividing by 20 we have 260 ÷ 20 = 13. Since there is no fraction in
the quotient, the numerator of the fraction will be 0, and counting forward
0 from the day sign of the starting point, Ik in Table I, the day sign Ik
will remain the day sign of the terminal date (rule 2, p. 140). Combining
the two values just determined, we see that the day of the terminal date
will be 13 Ik, or a day of the same name as the day of the starting point.
This follows also from the fact that there are only 260 differently named
days (see pp. 41-44) and any given day will have to recur, therefore, after
the lapse of 260 days.[101] Dividing by 365 we have: 260 ÷ 365 = 260/365.
Counting forward the numerator of the fraction, 260, from the year position
of the starting point, 0 Zip, in Table XV, the position in the year of the
day of the terminal date will be found to be 0 Pax. Since 260 days equal
just 13 uinals, we have only to count forward from 0 Zip 13 uinals in order
to reach the year position; that is, 0 Zotz is 1 uinal; to 0 Tzec 2 uinals,
to 0 Xul 3 uinals, and so on in Table XV to 0 Pax, which will complete the
last of the 13 uinals (rule 3, p. 141).
Combining the above values, we find that in counting forward 322,920 (or
260) from the starting point 13 Ik 0 Zip, the terminal date reached is 13
Ik 0 Pax.
In order to illustrate the method of procedure when the count is
_backward_, let us assume an example of this kind. Suppose we count
backward the number 9,663 from the starting point 3 Imix 4 Uayeb. Since
this number is below 18,980, no Calendar Round can be deducted from it.
Dividing the given number by 13, we have: 9,663 ÷ 13 =743-4/13. Counting
the numerator of the fractional part of this quotient, 4, _backward_ from
the day coefficient of the starting point, 3, we reach 12 as the day
coefficient of the terminal date, that is, 2, 1, 13, 12 (rule 1, p. 139).
Dividing the given number by 20, we have: 9,663 ÷ 20 = 483-3/20. Counting
the numerator of the fractional part of this quotient, 3, _backward_ from
the day sign of the starting point, Imix, in Table I, we reach Eznab as the
day sign of the terminal date (Ahau, Cauac, Eznab); consequently the day
reached in the count will be 12 Eznab. Dividing the given number by 365, we
have {147} 9,663 ÷ 365 = 26-173/365. Counting _backward_ the numerator of
the fractional part of this quotient, 173, from the year position of the
starting point, 4 Uayeb, the year position of the terminal date will be
found to be 11 Yax. Before position 4 Uayeb (see Table XV) there are 4
positions in that division of the year (3, 2, 1, 0). Counting these
_backward_ to the end of the month Cumhu (see Table XV), we have left 169
positions (173 - 4 = 169); this equals 8 uinals and 9 days extra.
Therefore, beginning with the end of Cumhu, we may count _backward_ 8 whole
uinals, namely: Cumhu, Kayab, Pax, Muan, Kankin, Mac, Ceh, and Zac, which
will bring us to the end of Yax (since we are counting backward). As we
have left still 9 days out of our original 173, these must be counted
backward from position 0 Zac, that is, beginning with position 19 Yax: 19,
18, 17, 16, 15, 14, 13, 12, 11; so 11 Yax is the position in the year of
the day of the terminal date. Assembling the above values, we find that in
counting the number 9,663 _backward_ from the starting point, 2 Imix 4
Uayeb, the terminal date is 12 Eznab 11 Yax. Whether the count be forward
or backward, the method is the same, the only difference being in the
direction of the counting.
This concludes the discussion of the actual arithmetical processes involved
in counting forward or backward any given number from any given date;
however, before explaining the fifth and final step in deciphering the Maya
numbers, it is first necessary to show how this method of counting was
applied to the Long Count.
The numbers used above in connection with dates merely express the
difference in time between starting points and terminal dates, without
assigning either set of dates to their proper positions in Maya chronology;
that is, in the Long Count. Consequently, since any Maya date recurred at
successive intervals of 52 years, by the time their historic period had
been reached, more than 3,000 years after the starting point of their
chronology, the Maya had upward of 70 distinct dates of exactly the same
name to distinguish from one another.
It was stated on page 61 that the 0, or starting point of Maya chronology,
was the date 4 Ahau 8 Cumhu, from which all subsequent dates were reckoned;
and further, on page 63, that by recording the number of cycles, katuns,
tuns, uinals, and kins which had elapsed in each case between this date and
any subsequent dates in the Long Count, subsequent dates of the same name
could be readily distinguished from one another and assigned at the same
time to their proper positions in Maya chronology. This method of fixing a
date in the Long Count has been designated Initial-series dating.
The generally accepted method of writing Initial Series is as follows:
9.0.0.0.0. 8 Ahau 13 Ceh
The particular Initial-Series written here is to be interpreted thus:
"Counting forward 9 cycles, 0 katuns, 0 tuns, 0 uinals, and 0 kins {148}
from 4 Ahau 8 Cumhu, the starting point of Maya chronology (always
unexpressed in Initial Series), the terminal date reached will be 8 Ahau 13
Ceh."[102] Or again:
9.14.13.4.17. 12 Caban 5 Kayab
This Inital Series reads thus: "Counting forward 9 cycles, 14 katuns, 13
tuns, 4 uinals, and 17 kins from 4 Ahau 8 Cumhu, the starting point of Maya
chronology (unexpressed), the terminal date reached will be 12 Caban 5
Kayab."
The time which separates any date from 4 Ahau 8 Cumhu may be called that
date's Initial-series value. For example, in the first of the above cases
the number 9.0.0.0.0 is the Initial-series value of the date 8 Ahau 13 Ceh,
and in the second the number 9.14.13.4.17 is the Initial-series value of
the date 12 Caban 5 Kayab. It is clear from the foregoing that although the
date 8 Ahau 13 Ceh, for example, had recurred upward of 70 times since the
beginning of their chronology, the Maya were able to distinguish any
particular 8 Ahau 13 Ceh from all the others merely by recording its
distance from the starting point; in other words, giving thereto its
particular Initial-series value, as 9.0.0.0.0. in the present case.
Similarly, any particular 12 Caban 5 Kayab, by the addition of its
corresponding Initial-series value, as 9.14.13.4.17 in the case above
cited, was absolutely fixed in the Long Count--that is, in a period of
374,400 years.
Returning now to the question of how the counting of numbers was applied to
the Long Count, it is evident that _every date in Maya chronology, starting
points as well as terminal dates, had its own particular Initial-series
value_, though in many cases these values are not recorded. However, in
most of the cases in which the Initial-series values of dates are not
recorded, they may be calculated by means of their distances from other
dates, whose Initial-series values are known. This adding and subtracting
of numbers to and from Initial Series[103] constitutes the application of
the above-described arithmetical processes to the Long Count. Several
examples of this use are given below.
Let us assume for the first case that the number 2.5.6.1 is to be counted
forward from the Initial Series 9.0.0.0.0 8 Ahau 13 Ceh. By multiplying the
values of the katuns, tuns, uinals, and kins given in Table XIII by their
corresponding coefficients, in this case 2, 5, 6, and 1, respectively, and
adding the resulting products together, we find that 2.5.6.1 reduces to
16,321 units of the first order.
Counting this forward from 8 Ahau 13 Ceh as indicated by the rules on pages
138-143, the terminal date 1 Imix 9 Yaxkin will be reached. {149} Moreover,
since the Initial-series value of the starting point 8 Ahau 13 Ceh was
9.0.0.0.0, the Initial-series value of 1 Imix 9 Yaxkin, the terminal date,
may be calculated by adding its distance from 8 Ahau 13 Ceh to the
Initial-series value of that date:
9.0.0.0.0 (Initial-series value of starting point) 8 Ahau 13 Ceh
2.5.6.1 (distance from 8 Ahau 13 Ceh to 1 Imix 9 Yaxkin)
9.2.5.6.1 (Initial-series value of terminal date) 1 Imix 9 Yaxkin
That is, by calculation we have determined the Initial-series value of the
particular 1 Imix 9 Yaxkin, which was distant 2.5.6.1 from 9.0.0.0.0 8 Ahau
13 Ceh, to be 9.2.5.6.1, notwithstanding that this fact was not recorded.
The student may prove the accuracy of this calculation by treating
9.2.5.6.1 1 Imix 9 Yaxkin as a new Initial Series and counting forward
9.2.5.6.1 from 4 Ahau 8 Cumhu, the starting point of all Initial Series
known except two. If our calculations are correct, the former date will be
reached just as if we had counted forward only 2.5.6.1 from 9.0.0.0.0 8
Ahau 13 Ceh.
In the above example the distance number 2.5.6.1 and the date 1 Imix 9
Yaxkin to which it reaches, together are called a Secondary Series. This
method of dating already described (see pp. 74-76 et seq.) seems to have
been used to avoid the repetition of the Initial-series values for all the
dates in an inscription. For example, in the accompanying text--
9.12. 2. 0.16 5 Cib 14 Yaxkin
12. 9.15
[9.12.14.10.11][104] 9 Chuen 9 Kankin
5
[9.12.14.10.16] 1 Cib 14 Kankin
1. 0. 2. 5
[9.13.14.13. 1] 5 Imix 19 Zac
the only parts actually recorded are the Initial Series 9.12.2.0.16 {150} 5
Cib 14 Yaxkin, and the Secondary Series 12.9.15 leading to 9 Chuen 9
Kankin; the Secondary Series 5 leading to 1 Cib 14 Kankin; and the
Secondary Series 1.0.2.5 leading to 5 Imix 19 Zac. The Initial-series
values: 9.12.14.10.11; 9.12.14.10.16; and 9.13.14.13.1, belonging to the
three dates of the Secondary Series, respectively, do not appear in the
text at all (a fact indicated by the brackets), but are found only by
calculation. Moreover, the student should note that in a succession of
interdependent series like the ones just given the terminal date reached by
one number, as 9 Chuen 9 Kankin, becomes the starting point for the next
number, 5. Again, the terminal date reached by counting 5 from 9 Chuen 9
Kankin, that is, 1 Cib 14 Kankin, becomes the starting point from which the
next number, 1.0.2.5, is counted. In other words, these terms are only
relative, since the terminal date of one number will be the starting point
of the next.
Let us assume for the next example, that the number 3.2 is to be counted
forward from the Initial Series 9.12.3.14.0 5 Ahau 8 Uo. Reducing 3 uinals
and 2 kins to kins, we have 62 units of the first order. Counting forward
62 from 5 Ahau 8 Uo, as indicated by the rules on pages 138-143, it is
found that the terminal date will be 2 Ik 10 Tzec. Since the Initial-series
value of the starting point 5 Ahau 8 Uo is known, namely, 9.12.3.14.0, the
Initial Series corresponding to the terminal date may be calculated from it
as before:
9.12.3.14.0 (Initial-series value of the starting point) 5 Ahau 8 Uo
3.2 (distance from 5 Ahau 8 Uo forward to 2 Ik 10 Tzec)
[9.12.3.17.2] (Initial-series value of the terminal date) 2 Ik 10 Tzec
The bracketed 9.12.3.17.2 in the Initial-series value corresponding to the
date 2 Ik 10 Tzec does not appear in the record but was reached by
calculation. The student may prove the accuracy of this result by treating
9.12.3.17.2 2 Ik 10 Tzec as a new Initial Series, and counting forward
9.12.3.17.2 from 4 Ahau 8 Cumhu (the starting point of Maya chronology,
unexpressed in Initial Series). If our calculations are correct, the same
date, 2 Ik 10 Tzec, will be reached, as though we had counted only 3.2
forward from the Initial Series 9.12.3.14.0 5 Ahau 8 Uo.
One more example presenting a "backward count" will suffice to illustrate
this method. Let us count the number 14.13.4.17 _backward_ from the Initial
Series 9.14.13.4.17 12 Caban 5 Kayab. Reducing 14.13.4.17 to units of the
1st order, we have 105,577. Counting this number _backward_ from 12 Caban 5
Kayab, as indicated in the rules on pages 138-143, we find that the
terminal date will be 8 Ahau 13 Ceh. Moreover, since the Initial-series
value of the starting point 12 Caban 5 Kayab is known, namely,
9.14.13.4.17, the Initial-series value of {151} the terminal date may be
calculated by _subtracting_ the distance number 14.13.4.17 from the Initial
Series of the starting point:
9.14.13.4.17 (Initial-series value of the starting point)
12 Caban 5 Kayab
14.13.4.17 (distance from 12 Caban 5 Kayab backward to 8 Ahau 13 Ceh)
[9. 0. 0.0. 0] (Initial-series value of the terminal date) 8 Ahau 13 Ceh
The bracketed parts are not expressed. We have seen elsewhere that the
Initial Series 9.0.0.0.0 has for its terminal date 8 Ahau 13 Ceh; therefore
our calculation proves itself.
The foregoing examples make it sufficiently clear that the distance numbers
of Secondary Series may be used to determine the Initial-series values of
Secondary-series dates, either by their addition to or subtraction from
known Initial-series dates.
We have come now to the final step in the consideration of Maya numbers,
namely, the identification of the terminal dates determined by the
calculations given under the fourth step, pages 138-143. This step may be
summed up as follows:
FIFTH STEP IN SOLVING MAYA NUMBERS
Find the terminal date to which the number leads.
As explained under the fourth step (pp. 138-143), the terminal date may be
found by calculation. The above direction, however, refers to the actual
finding of the terminal dates in the texts; that is, where to look for
them. It may be said at the outset in this connection that terminal dates
in the great majority of cases follow immediately the numbers which lead to
them. Indeed, the connection between distance numbers and their
corresponding terminal dates is far closer than between distance numbers
and their corresponding starting points. This probably results from the
fact that the closing dates of Maya periods were of far more importance
than their opening dates. Time was measured by elapsed periods and recorded
in terms of the ending days of such periods. The great emphasis on the
closing date of a period in comparison with its opening date probably
caused the suppression and omission of the date 4 Ahau 8 Cumhu, the
starting point of Maya chronology, in all Initial Series. To the same cause
also may probably be attributed the great uniformity in the positions of
almost all terminal dates, i.e., immediately after the numbers leading to
them.
We may formulate, therefore, the following general rule, which the student
will do well to apply in every case, since exceptions to it are very rare:
_Rule._ The terminal date reached by a number or series almost invariably
follows immediately the last term of the number or series leading to it.
{152}
This applies equally to all terminal dates, whether in Initial Series,
Secondary Series, Calendar-round dating or Period-ending dating, though in
the case of Initial Series a peculiar division or partition of the terminal
date is to be noted.
Throughout the inscriptions, excepting in the case of Initial Series, the
month parts of the dates almost invariably follow immediately the days
whose positions in the year they designate, without any other glyphs
standing between; as, for example, 8 Ahau 13 Ceh, 12 Caban 5 Kayab, etc. In
Initial Series, on the other hand, the day parts of the dates, as 8 Ahau
and 12 Caban, in the above examples, are almost invariably separated from
their corresponding month parts, 13 Ceh or 5 Kayab, by several intervening
glyphs. The positions of the day parts in Initial-series terminal dates are
quite regular according to the terms of the above rule; that is, they
follow immediately the lowest period of the number which in each case shows
their distance from the unexpressed starting point, 4 Ahau 8 Cumhu. The
positions of the corresponding month parts are, on the other hand,
irregular. These, instead of standing immediately after the days whose
positions in the year they designate, follow at the close of some six or
seven intervening glyphs. These intervening glyphs have been called the
Supplementary Series, though the count which they record has not as yet
been deciphered.[105] The month glyph in the great majority of cases
follows immediately the closing[106] glyph of the Supplementary Series. The
form of this latter sign is always unmistakable (see fig. 65), and it is
further characterized by its numerical coefficient, which can never be
anything but 9 or 10.[107] See examples of this sign in the figure just
mentioned, where both normal forms _a, c, e, g,_ and _h_ and head variants
_b, d,_ and _f_ are included.
The student will find this glyph exceedingly helpful in locating the month
parts of Initial-series terminal dates in the inscriptions. For example,
let us suppose in deciphering the Initial Series 9.16.5.0.0 8 Ahau 8 Zotz
that the number 9.16.5.0.0 has been counted forward {153} from 4 Ahau 8
Cumhu (the unexpressed starting point), and has been found by calculation
to reach the terminal date 8 Ahau 8 Zotz; and further, let us suppose that
on inspecting the text the day part of this date (8 Ahau) has been found to
be recorded immediately after the 0 kins of the number 9.16.5.0.0. Now, if
the student will follow the next six or seven glyphs until he finds one
like any of the forms in figure 65, the glyph immediately following the
latter sign will be in all probability the month part, 8 Zotz in the above
example, of an Initial-series' terminal date. In other words, although the
meaning of the glyph shown in the last-mentioned figure is unknown, it is
important for the student to recognize its form, since it is almost
invariably the "indicator" of the month sign in Initial Series.
[Illustration: FIG. 65. Sign for the "month indicator": _a, c, e, g, h_,
Normal forms; _b, d, f_, head variants.]
In all other cases in the inscriptions, including also the exceptions to
the above rule, that is, where the month parts of Initial-series terminal
dates do not immediately follow the closing glyph of the Supplementary
Series, the month signs follow immediately the day signs whose positions in
the year they severally designate.
In the codices the month signs when recorded[108] usually follow
immediately the days signs to which they belong. The most notable
exception[109] to this general rule occurs in connection with the
Venus-solar periods represented on pages 46-50 of the Dresden Codex, where
one set of day signs is used with three different sets of month signs to
form three different sets of dates. For example, in one place the day 2
Ahau stands above three different month signs--3 Cumhu, 3 Zotz, and 13
Yax--with each of which it is used to form a {154} different date--2 Ahau 3
Cumhu, 2 Ahau 3 Zotz, and 2 Ahau 13 Yax. In these pages the month signs,
with a few exceptions, do not follow immediately the days to which they
belong, but on the contrary they are separated from them by several
intervening glyphs. This abbreviation in the record of these dates was
doubtless prompted by the desire or necessity for economizing space. In the
above example, instead of repeating the 2 Ahau with each of the two lower
month signs, 3 Zotz and 13 Yax, by writing it once above the upper month
sign, 3 Cumhu, the scribe intended that it should be used in turn with each
one of the three month signs standing below it, to form three different
dates, saving by this abbreviation the space of two glyphs, that is, double
the space occupied by 2 Ahau.
With the exception of the Initial-series dates in the inscriptions and the
Venus-Solar dates on pages 46-50 of the Dresden Codex, we may say that the
regular position of the month glyphs in Maya writing was immediately
following the day glyphs whose positions in the year they severally
designated.
In closing the presentation of this last step in the process of deciphering
numbers in the texts, the great value of the terminal date as a final check
for all the calculations involved under steps 1-4 (pp. 134-151) should be
pointed out. If after having worked out the terminal date of a given number
according to these rules the terminal date thus found should differ from
that actually recorded under step 5, we must accept one of the following
alternatives:
1. There is an error in our own calculations; or
2. There is an error in the original text; or
3. The case in point lies without the operation of our rules.
It is always safe for the beginner to proceed on the assumption that the
first of the above alternatives is the cause of the error; in other words,
that his own calculations are at fault. If the terminal date as calculated
does not agree with the terminal-date as recorded, the student should
repeat his calculations several times, checking up each operation in order
to eliminate the possibility of a purely arithmetical error, as a mistake
in multiplication. After all attempts to reach the recorded terminal date
by counting the given number from the starting point have failed, the
process should be reversed and the attempt made to reach the starting point
by counting backward the given number from its recorded terminal date.
Sometimes this reverse process will work out correctly, showing that there
must be some arithmetical error in our original calculations which we have
failed to detect. However, when both processes have failed several times to
connect the starting point with the recorded terminal date by use of the
given number, there remains the possibility that either the starting point
or the terminal date, or perhaps both, do not belong to the given number.
The rules for determining this fact {155} have been given under step 2,
page 135, and step 4, page 138. If after applying these to the case in
point it seems certain that the starting point and terminal date used in
the calculations both belong to the given number, we have to fall back on
the second of the above alternatives, that is, that there is an error in
the original text.
Although very unusual, particularly in the inscriptions, errors in the
original texts are by no means entirely unknown. These seem to be
restricted chiefly to errors in numerals, as the record of 7 for 8, or 7
for 12 or 17, that is, the omission or insertion of one or more bars or
dots. In a very few instances there seem to be errors in the month glyph.
Such errors usually are obvious, as will be pointed out in connection with
the texts in which they are found (see Chapters V and VI).
If both of the above alternatives are found not to apply, that is, if both
our calculations and the original texts are free from error, we are obliged
to accept the third alternative as the source of trouble, namely, that the
case in point lies without the operation of our rules. In such cases it is
obviously impossible to go further in the present state of our knowledge.
Special conditions presented by glyphs whose meanings are unknown may
govern such cases. At all events, the failure of the rules under 1-4 to
reach the terminal dates recorded as under 5 introduces a new phase of
glyph study--the meaning of unknown forms with which the beginner has no
concern. Consequently, when a text falls without the operation of the rules
given in this chapter--a very rare contingency--the beginner should turn
his attention elsewhere. {156}
CHAPTER V
THE INSCRIPTIONS
The present chapter will be devoted to the interpretation of texts drawn
from monuments, a process which consists briefly in the application to the
inscriptions[110] of the material presented in Chapters III and IV.
[Illustration: FIG. 66. Diagram showing the method of designating
particular glyphs in a text.]
Before proceeding with this discussion it will first be necessary to
explain the method followed in designating particular glyphs in a text. We
have seen (p. 23) that the Maya glyphs were presented in parallel columns,
which are to be read two columns at a time, the order of the individual
glyph-blocks[111] in each pair of columns being from left to right and from
top to bottom. For convenience in referring to particular glyphs in the
texts, the vertical columns of glyph-blocks are lettered from left to
right, thus, A, B, C, D, etc., and the horizontal rows numbered from top to
bottom, thus, 1, 2, 3, 4, etc. For example, in figure 66 the glyph-blocks
in columns A and B are read together from left to right and top to bottom,
thus, A1 B1, A2 B2, A3 B3, etc. When glyph-block B10 is reached the next in
order is C1, which is followed by D1, C2 D2, C3 D3, etc. Again, when D10 is
reached the next in order is E1, which is followed by F1, E2 F2, E3 F3,
etc. In this way the order of reading proceeds from left to right and from
top to bottom, in pairs of columns, that is, G H, I J, K L, and M N
throughout the inscription, and usually closes with the glyph-block in the
lower right-hand corner, as N10 in figure 66. By this simple system of
coordinates any particular glyph in a text may be readily referred to when
the need arises. Thus, for example, in figure 66 glyph [alpha] is referred
to as D3; glyph [beta] as F6; glyph [gamma] as K4; glyph [delta] as N10. In
a few texts the glyph-blocks are so irregularly placed that it is
impracticable to designate them by the above coordinates. In such cases the
order of the glyph-blocks will be indicated by numerals, 1, 2, 3, etc. In
two Copan texts, Altar S (fig. 81) and Stela J (pl. 15), made from the
drawings of Mr. Maudslay, his numeration of the glyphs has been followed.
This numeration appears in these two figures.
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR
AND DOT NUMERALS AND NORMAL-FORM PERIOD GLYPHS]
{157}
TEXTS RECORDING INITIAL SERIES
Because of the fundamental importance of Initial Series in the Maya system
of chronology, the first class of texts represented will illustrate this
method of dating. Moreover, since the normal forms for the numerals and the
period glyphs will be more easily recognised by the beginner than the
corresponding head variants, the first Initial Series given will be found
to have all the numerals and period glyphs expressed by normal forms.[112]
In plate 6 is figured the drawing of the Initial Series[113] from Zoömorph
P at Quirigua, a monument which is said to be the finest piece of
aboriginal sculpture in the western hemisphere. Our text opens with one
large glyph, which occupies the space of four glyph-blocks, A1-B2.[114]
Analysis of this form shows that it possesses all the elements mentioned on
page 65 as belonging to the so-called Initial-series introducing glyph,
without which Initial Series never seem to have been recorded in the
inscriptions. These elements are: (1) the trinal {158} superfix, (2) the
pair of comblike lateral appendages, (3) the normal form of the tun sign,
(4) the trinal subfix, and (5) the variable central element. As stated
above, all these appear in the large glyph A1-B2. Moreover, a comparison of
A1-B2 with the introducing glyphs given in figure 24 shows that these forms
are variants of one and the same sign. Consequently, in A1-B2 we have
recorded an Initial-series introducing glyph. The use of this sign is so
highly specialized that, on the basis of its occurrence alone in a text,
the student is perfectly justified in assuming that an Initial Series will
immediately follow.[115] Exceptions to this rule are so very rare (see p.
67) that the beginner will do well to disregard them altogether.
The next glyph after the introducing glyph in an Initial Series is the
cycle sign, the highest period ever found in this kind of count[116]. The
cycle sign in the present example appears in A3 with the coefficient 9 (1
bar and 4 dots). Although the period glyph is partially effaced in the
original enough remains to trace its resemblance to the normal form of the
cycle sign shown in figure 25, _a-c_. The outline of the repeated Cauac
sign appears in both places. We have then, in this glyph, the record of 9
cycles[117]. The glyph following the cycle sign in an Initial Series is
always the katun sign, and this should appear in B3, the glyph next in
order. This glyph is quite clearly the normal form of the katun sign, as a
comparison of it with figure 27, _a, b_, the normal form for the katun,
will show. It has the normal-form numeral 18 (3 bars and 3 dots) prefixed
to it, and this whole glyph therefore signifies 18 katuns. The next glyph
should record the tuns, and a comparison of the glyph in A4 with the normal
form of the tun sign in figure 29, _a, b_, shows this to be the case. The
numeral 5 (1 bar prefixed to the tun sign) shows that this period is to be
used 5 times; that is, multiplied by 5. The next glyph (B4) should be the
uinal sign, and a comparison of B4 with figure 31, _a-c_, the normal form
of the uinal sign, shows the identity of these two glyphs. The coefficient
of the uinal sign contains as its most conspicuous element the clasped
hand, which suggests that we may have 0 uinals recorded in B4. A comparison
of this coefficient with the sign for zero in figure 54 proves this to be
the case. The next glyph (A5) should be the kin sign, the lowest period
involved in recording Initial Series. A comparison of A5 with the normal
form of the kin sign in figure 34, _a_, shows that these two forms are
identical. The coefficient of A5 is, moreover, exactly like the coefficient
of B4, which, we have seen, meant zero, hence glyph A5 stands for 0 kins.
Summarizing the above, we may say that glyphs A3-A5 record an
Initial-series number consisting of 6 cycles, 18 katuns, 5 tuns, 0 uinals,
and 0 kins, which we may write thus: 9.18.5.0.0 (see p. 138, footnote 1).
{159}
Now let us turn to Chapter IV and apply the several steps there given, by
means of which Maya numbers may be solved. The first step on page 134 was
to reduce the given number, in this case 9.18.5.0.0, to units of the first
order; this may be done by multiplying the recorded coefficients by the
numerical values of the periods to which they are respectively attached.
These values are given in Table XIII, and the sum of the products arising
from their multiplication by the coefficients recorded in the Initial
Series in plate 6, A are given below:
A3 = 9 × 144,000 = 1,296,000
B3 = 18 × 7,200 = 129,600
A4 = 5 × 360 = 1,800
B4 = 0 × 20 = 0
A5 = 0 × 1 = 0
----------
1,427,400
Therefore 1,427,400 will be the number used in the following calculations.
The second step (see step 2, p. 135) is to determine the starting point
from which this number is counted. According to rule 2, page 136, if the
number is an Initial Series the starting point, although never recorded, is
practically always the date 4 Ahau 8 Cumhu. Exceptions to this rule are so
very rare that they may be disregarded by the beginner, and it may be taken
for granted, therefore, in the present case, that our number 1,427,400 is
to be counted from the date 4 Ahau 8 Cumhu.
The third step (see step 3, p. £136) is to determine the direction of the
count, whether forward or backward. In this connection it was stated that
the general practice is to count forward, and that the student should
always proceed upon this assumption. However, in the present case there is
no room for uncertainty, since the direction of the count in an Initial
Series is governed by an invariable rule. In Initial Series, according to
the rule on page 137, the count is always forward, consequently 1,427,400
is to be counted _forward_ from 4 Ahau 8 Cumhu.
The fourth step (see step 4, p. 138) is to count the given number from its
starting point; and the rules governing this process will be found on pages
139-143. Since our given number (1,427,400) is greater than 18,980, or 1
Calendar Round, the preliminary rule on page 143 applies in the present
case, and we may therefore subtract from 1,427,400 all the Calendar Rounds
possible before proceeding to count it from the starting point. By
referring to Table XVI, it appears that 1,427,400 contains 75 complete
Calendar Rounds, or 1,423,500; hence, the latter number may be subtracted
{160} from 1,427,400 without affecting the value of the resulting terminal
date: 1,427,400 - 1,423,500 = 3,900. In other words, in counting forward
3,900 from 4 Ahau 8 Cumhu, the same terminal date will be reached as though
we had counted forward 1,427,400.[118]
In order to find the coefficient of the day of the terminal date, it is
necessary, by rule 1, page 139, to divide the given number or its
equivalent by 13; 3,900 ÷ 13 = 300. Now since there is no fractional part
in the resulting quotient, the numerator of an assumed fractional part will
be 0; counting forward 0 from the coefficient of the day of the starting
point, 4 (that is, 4 Ahau 8 Cumhu), we reach 4 as the coefficient of the
day of the terminal date.
In order to find the day sign of the terminal date, it is necessary, under
rule 2, page 140, to divide the given number or its equivalent by 20; 3,900
÷ 20 = 195. Since there is no fractional part in the resulting quotient,
the numerator of an assumed fractional part will be 0; counting forward 0
in Table I, from Ahau, the day sign of the starting point (4 Ahau 8 Cumhu),
we reach Ahau as the day sign of the terminal date. In other words, in
counting forward either 3,900 or 1,427,400 from 4 Ahau 8 Cumhu, the day
reached will be 4 Ahau. It remains to show what position in the year this
day 4 Ahau distant 1,427,400 from the date 4 Ahau 8 Cumhu, occupied.
In order to find the position in the year which the day of the terminal
date occupied, it is necessary, under rule 3, page 141, to divide the given
number or its equivalent by 365; 3,900 ÷ 365 = 10-250/365. Since the
numerator of the fractional part of the resulting quotient is 250, to reach
the year position of the day of the terminal date desired it is necessary
to count 250 forward from 8 Cumhu, the year position of the day of the
starting point 4 Ahau 8 Cumhu. It appears from Table XV, in which the 365
positions of the year are given, that after position 8 Cumhu there are only
16 positions in the year--11 more in Cumhu and 5 in Uayeb. These must be
subtracted, therefore, from 250 in order to bring the count to the end of
the year; 250 - 16 = 234, so 234 is the number of positions we must count
forward in the new year. It is clear that the first 11 uinals in the year
will use up exactly 220 of our 234 positions (11 × 20 = 220), and that 14
positions will be left, which must be counted in the next uinal, the 12th.
But the 12th uinal of the year is Ceh (see Table XV); counting forward 14
positions in Ceh, we reach 13 Ceh, which is, therefore, the month glyph of
our terminal date. In other words, counting 250 forward from 8 Cumhu,
position 13 Ceh is reached. Assembling the above values, we find that by
calculation we have determined the terminal date of the Initial Series in
plate 6, _A_, to be 4 Ahau 13 Ceh. {161}
At this point there are several checks which the student may apply to his
result in order to test the accuracy of his calculations; for instance, in
the present example if 115, the difference between 365 and 250 (115 + 250 =
365) is counted forward from position 13 Ceh, position 8 Cumhu will be
reached if our calculations were correct. This is true because there are
only 365 positions in the year, and having reached 13 Ceh in counting
forward 250 from 8 Cumhu, counting the remaining 115 days forward from day
reached by 250, that is, 13 Ceh, we should reach our starting point (8
Cumhu) again. Another good check in the present case would be to count
_backward_ 250 from 13 Ceh; if our calculations have been correct, the
starting point 8 Cumhu will be reached. Still another check, which may be
applied is the following: From Table VII it is clear that the day sign Ahau
can occupy only positions 3, 8, 13, or 18 in the divisions of the
year;[119] hence, if in the above case the coefficient of Ceh had been any
other number but one of these four, our calculations would have been
incorrect.
We come now to the final step (see step 5, p. 151), the actual finding of
the glyphs in our text which represent the two parts of the terminal
date--the day and its corresponding position in the year. If we have made
no arithmetical errors in calculations and if the text itself presents no
irregular and unusual features, the terminal date recorded should agree
with the terminal date obtained by calculation.
It was explained on page 152 that the two parts of an Initial-series
terminal date are usually separated from each other by several intervening
glyphs, and further that, although the day part follows immediately the
last period glyph of the number (the kin glyph), the month part is not
recorded until after the close of the Supplementary Series, usually a
matter of six or seven glyphs. Returning to our text (pl. 6, _A_), we find
that the kins are recorded in A5, therefore the day part of the terminal
date should appear in B5. The glyph in B5 quite clearly records the day 4
Ahau by means of 4 dots prefixed to the sign shown in figure 16, _e'-g'_,
which is the form for the day name Ahau, thereby agreeing with the value of
the day part of the terminal date as determined by calculation. So far then
we have read our text correctly. Following along the next six or seven
glyphs, A6-C1a, which record the Supplementary Series,[120] we reach in C1a
a sign similar to the forms shown in figure 65. This glyph, which always
has a coefficient of 9 or 10, was designated on page 152 the month-sign
"indicator," since it usually immediately precedes the month sign in
Initial-series terminal dates. In C1a it has the coefficient 9 (4 dots and
1 bar) and is followed in C1b by the month part {162} of the terminal date,
13 Ceh. The bar and dot numeral 13 appears very clearly above the month
sign, which, though partially effaced, yet bears sufficient resemblance to
the sign for Ceh in figure 19, _u, v,_ to enable us to identify it as such.
Our complete Initial Series, therefore, reads: 9.18.5.0.0 4 Ahau 13 Ceh,
and since the terminal date recorded in B5, C1b agrees with the terminal
date determined by calculation, we may conclude that this text is without
error and, furthermore, that it records a date, 4 Ahau 13 Ceh, which was
distant 9.18.5.0.0 from the starting point of Maya chronology. The writer
interprets this text as signifying that 9.18.5.0.0 4 Ahau 13 Ceh was the
date on which Zoömorph P at Quirigua was formally consecrated or dedicated
as a time-marker, or in other words, that Zoömorph P was the monument set
up to mark the hotun, or 5-tun period, which came to a close on the date
9.18.5.0.0 4 Ahau 13 Ceh of Maya chronology.[121]
In plate 6, _B_, is figured a drawing of the Initial Series on Stela 22 at
Naranjo.[122] The text opens in A1 with the Initial-series introducing
glyph, which is followed in B1 B3 by the Initial-series number
9.12.15.13.7. The five period glyphs are all expressed by their
corresponding normal forms, and the student will have no difficulty in
identifying them and reading the number, as above recorded.
By means of Table XIII this number may be reduced to units of the 1st
order, in which form it may be more conveniently used. This reduction,
which forms the first step in the process of solving Maya numbers (see step
1, p. 134), follows:
B1 = 9 × 144,000 = 1,296,000
A2 = 12 × 7,200 = 86,400
B2 = 15 × 360 = 5,400
A3 = 13 × 20 = 260
B3 = 7 × 1 = 7
---------
1,388,067
And 1,388,067 will be the number used in the following calculations.
The next step is to find the starting point from which 1,388,067 is counted
(see step 2, p. 135). Since this number is an Initial Series, in all
probability its starting point will be the date 4 Ahau 8 Cumhu; at least it
is perfectly safe to proceed on that assumption.
The next step is to find the direction of the count (see step 3, p. 136);
since our number is an Initial Series, the count can only be forward (see
rule 2, p. 137).[123] {163}
Having determined the number to be counted, the starting point from which
the count commences, and the direction of the count, we may now proceed
with the actual process of counting (see step 4, p. 138).
Since 1,388,067 is greater than 18,980 (1 Calendar Round), we may deduct
from the former number all the Calendar Rounds possible (see preliminary
rule, page 143). According to Table XVI it appears that 1,388,067 contains
73 Calendar Rounds, or 1,385,540; after deducting this from the given
number we have left 2,527 (1,388,067 - 1,385,540), a far more convenient
number to handle than 1,388,067.
Applying rule 1 (p. 139) to 2,527, we have: 2,527 ÷ 13 = 194-5/13, and
counting forward 5, the numerator of the fractional part of the quotient,
from 4, the day coefficient of the starting point, 4 Ahau 8 Cumhu, we reach
9 as the day coefficient of the terminal date.
Applying rule 2 (p. 140) to 2,527, we have: 2,527 ÷ 20 = 126-7/20; and
counting forward 7, the numerator of the fractional part of the quotient,
from Ahau, the day sign of our starting point, 4 Ahau 8 Cumhu, in Table I,
we reach Manik as the day sign of the terminal date. Therefore, the day of
the terminal date will be 9 Manik.
Applying rule 3 (p. 141) to 2,527, we have: 2,527 ÷ 365 = 6-337/365; and
counting forward 337, the numerator of the fractional part of the quotient,
from 8 Cumhu, the year position of the starting point, 4 Ahau 8 Cumhu, in
Table XV, we reach 0 Kayab as the year position of the terminal date. The
calculations by means of which 0 Kayab is reached are as follows: After 8
Cumhu there are 16 positions in the year, which we must subtract from 337;
337 - 16 = 321, which is to be counted forward in the new year. This number
contains just 1 more than 16 uinals, that is, 321 = (16 × 20) + 1; hence it
will reach through the first 16 uinals in Table XV and to the first
position in the 17th uinal, 0 Kayab. Combining this with the day obtained
above, we have for our terminal date determined by calculation, 9 Manik 0
Kayab.
The next and last step (see step 5, p. 151) is to find the above date in
the text. In Initial Series (see p. 152) the two parts of the terminal date
are generally separated, the day part usually following immediately the
last period glyph and the month part the closing glyph of the Supplementary
Series. In plate 6, _B_, the last period glyph, as we have seen, is
recorded in B3; therefore the day should appear in A4. Comparing the glyph
in A4 with the sign for Manik in figure 16, _j_, the two forms are seen to
be identical. Moreover, A4 has the bar and dot coefficient 9 attached to
it, that is, 4 dots and 1 bar; consequently it is clear that in A4 we have
recorded the day 9 Manik, the same day as reached by calculation. For some
unknown reason, at Naranjo the month glyphs of the Initial-series terminal
dates do not regularly follow the closing glyphs of the Supplementary
Series; {164} indeed, in the text here under discussion, so far as we can
judge from the badly effaced glyphs, no Supplementary Series seems to have
been recorded. However, reversing our operation, we know by calculation
that the month part should be 0 Kayab, and by referring to figure 49 we
find the only form which can be used to express the 0 position with the
month signs--the so-called "spectacles" glyph--which must be recorded
somewhere in this text to express the idea 0 with the month sign Kayab.
Further, by referring to figure 19, _d'-f'_, we may fix in our minds the
sign for the month Kayab, which should also appear in the text with one of
the forms shown in figure 49.
Returning to our text once more and following along the glyphs after the
day in A4, we pass over B4, A5, and B5 without finding a glyph resembling
one of the forms in figure 49 joined to figure 19, _d'-f'_; that is, 0
Kayab. However, in A6 such a glyph is reached, and the student will have no
difficulty in identifying the month sign with _d'-f'_ in the above figure.
Consequently, we have recorded in A4, A6 the same terminal date, 9 Manik 0
Kayab, as determined by calculation, and may conclude, therefore, that our
text records without error the date 9.12.15.13.7 9 Manik 0 Kayab[124] of
Maya chronology.
The next text presented (pl. 6, C) shows the Initial Series from Stela I at
Quirigua.[125] Again, as in plate 6, A, the introducing glyph occupies the
space of four glyph-blocks, namely, A1-B2. Immediately after this, in
A3-A4, is recorded the Initial-series number 9.18.10.0.0, all the period
glyphs and coefficients of which are expressed by normal forms. The
student's attention is called to the form for 0 used with the uinal and kin
signs in A4a and A4b, respectively, which differs from the form for 0
recorded with the uinal and kin signs in plate 6, A, B4, and A5,
respectively. In the latter text the 0 uinals and 0 kins were expressed by
the hand and curl form for zero shown in figure 54; in the present text,
however, the 0 uinals and 0 kins are expressed by the form for 0 shown in
figure 47, a new feature.
Reducing the above number to units of the 1st order by means of Table XIII,
we have:
A3 = 9 × 144,000 = 1,296,000
B3a = 18 × 7,200 = 129,600
B3b = 10 × 360 = 3,600
A4a = 0 × 20 = 0
A4b = 0 × 1 = 0
---------
1,429,200
Deducting from this number all the Calendar Rounds possible, 75 {165} (see
Table XVI), it may be reduced to 5,700 without affecting its value in the
present connection.
Applying rules 1 and 2 (pp. 139 and 140, respectively) to this number, the
day reached will be found to be 10 Ahau; and by applying rule 3 (p. 141),
the position of this day in the year will be found to be 8 Zac. Therefore,
by calculation we have determined that the terminal date reached by this
Initial Series is 10 Ahau 8 Zac. It remains to find this date in the text.
The regular position for the day in Initial-series terminal dates is
immediately following the last period glyph, which, as we have seen above,
was in A4b. Therefore the day glyph should be B4a. An inspection of this
latter glyph will show that it records the day 10 Ahau, both the day sign
and the coefficient being unusually clear, and practically unmistakable.
Compare B4a with figure 16, _e'-g'_, the sign for the day name Ahau.
Consequently the day recorded agrees with the day determined by
calculation. The month glyph in this text, as mentioned on page 157,
footnote 1, occurs out of its regular position, following immediately the
day of the terminal date.
As mentioned on page 153, when the month glyph in Initial-series terminal
dates is _not_ to be found in its usual position, it will be found in the
regular position for the month glyphs in all other kinds of dates in the
inscriptions, namely, immediately following the day glyph to which it
belongs. In the present text we found that the day, 10 Ahau, was recorded
in B4a; hence, since the month glyph was not recorded in its regular
position, it must be in B4b, immediately following the day glyph. By
comparing the glyph in B4b with the month signs in figure 19, it will be
found exactly like the month sign for Zac (_s-t_), and we may therefore
conclude that this is our month glyph and that it is Zac. The coefficient
of B4b is quite clearly 8 and the month part therefore reads, 8 Zac.
Combining this with the day recorded in B4a, we have the date 10 Ahau 8
Zac, which corresponds with the terminal date determined by calculation.
The whole text therefore reads 9.18.10.0.0 10 Ahau 8 Zac.
[Illustration: FIG. 67. Signs representing the hotun, or 5-tun, period.]
It will be noted that this date 9.18.10.0.0 10 Ahau 8 Zac is just 5.0.0 (5
tuns) later than the date recorded by the Initial Series on Zoömorph P at
Quirigua (see pl. 6, _A_). As explained in Chapter II (pp. 33-34), the
interval between succeeding monuments at Quirigua is in every case 1,800
days, or 5 tuns. Therefore, it would seem probable that at Quirigua at
least this period was the unit used for marking the lapse of time. As each
5-tun period was completed, its close was marked by the erection of a
monument, on which was recorded its ending date. Thus the writer believes
Zoömorph P marked the close of the 5-tun period ending 9.18.5.0.0 4 Ahau 13
Ceh, and Stela I, the 5-tun period next following, that ending 9.18.10.0.0
{166} 10 Ahau 8 Zac. In other words, Zoömorph P and Stela I were two
successive time-markers, or "period stones," in the chronological record at
Quirigua. For this 5-tun period so conspicuously recorded in the
inscriptions from the older Maya cities the writer would suggest the name
_hotun_, _ho_ meaning 5 in Maya and _tun_ being the name of the 360-day
period. This word has an etymological parallel in the Maya word for the
20-tun period, _katun_, which we have seen may have been named directly
from its numerical value, _kal_ being the word for 20 in Maya and _kaltun_
contracted to katun, thus meaning 20 tuns. Although no glyph for the
_hotun_ has as yet been identified,[126] the writer is inclined to believe
that the sign in figure 67, _a, b_, which is frequently encountered in the
texts, will be found to represent this time period. The bar at the top in
both _a_ and _b_, figure 67, surely signifies 5; therefore the glyph itself
must mean "1 tun." This form recalls the very unusual variant of the tun
from Palenque (see fig. 29, _h_). Both have the wing and the () element.
The next Initial Series presented (see pl. 6, _D_) is from Stela 24 at
Naranjo.[127] The text opens with the introducing glyph, which is in the
same relative position as the introducing glyph in the other Naranjo text
(pl. 6, _B_) at A1. Then follows regularly in B1-B3 the number
9.12.10.5.12, the numbers and period glyphs of which are all expressed by
normal forms. By this time the student should have no difficulty in
recognizing these and in determining the number as given above. Reducing
this according to rule 1, page 134, the following result should be
obtained:
B1 = 9 × 144,000 = 1,296,000
A2 = 12 × 7,200 = 86,400
B2 = 10 × 360 = 3,600
A3 = 5 × 20 = 100
B3 = 12 × 1 = 12
---------
1,386,112
Deducting[128] from this number all the Calendar Rounds possible, 73 (see
preliminary rule, p. 143, and Table XVI), we may reduce it to 572 without
affecting its value in so far as the present calculations are concerned
(1,386,112 - 1,385,540). First applying rule 1, page 139, and next rule 2,
page 140, to this number (572), the student will find the day reached to be
4 Eb. And applying rule 3, page 141, he will find that the year position
reached will be 10 Yax;[129] hence, the terminal date as determined by
calculation will be 4 Eb 10 Yax.
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR
AND DOT NUMERALS AND HEAD-VARIANT PERIOD GLYPHS]
{167}
Turning again to the text (pl. 6, _D_), the next step (see step 5, p. 151)
is to find the glyphs representing the above terminal date. In this
connection it should be remembered that the day part of an Initial-series
terminal date usually follows immediately the last period glyph of the
number. The glyph in A4, therefore, should record the day reached.
Comparing this form with the several day signs in figure 16, it appears
that A4 more closely resembles the sign for Eb (fig. 16, _s-u_) than any of
the others, hence the student may accept Eb as the day sign recorded in A4.
The 4 dots prefixed to this sign show that the day 4 Eb is here indicated.
The month sign, as stated on page 152, usually follows the last glyph of
the Supplementary Series; passing over B4, A5, B5, and A6, we reach the
latter glyph in B6. Compare the left half of B6 with the forms given in
figure 65. The coefficient 9 or 10 is expressed by a considerably effaced
head numeral. Immediately following the month-sign "indicator" is the month
sign itself in A7. The student will have little difficulty in tracing its
resemblance to the month Yax in figure 19, _q, r_, although in A7 the Yax
element itself appears as the prefix instead of as the superfix, as in _q_
and _r_, just cited. This difference, however, is immaterial. The month
coefficient is quite clearly 10,[130] and the whole terminal date recorded
will read 4 Eb 10 Yax, which corresponds exactly with the terminal date
determined by calculation. We may accept this text, therefore, as recording
the Initial-series date 9.12.10.5.12 4 Eb 10 Yax of Maya chronology.
In the foregoing examples nothing but normal-form period glyphs have been
presented, in order that the first exercises in deciphering the
inscriptions may be as easy as possible. By this time, however, the student
should be sufficiently familiar with the normal forms of the period glyphs
to be able to recognize them when they are present in the text, and the
next Initial Series figured will have its period glyphs expressed by head
variants.
In A, plate 7, is figured the Initial Series from Stela B at Copan.[131]
The introducing glyph appears at the head of the inscription in A1 {168}
and is followed by a head-variant glyph in A2, to which is prefixed a bar
and dot coefficient of 9. By its position, immediately following the
introducing glyph, we are justified in assuming that A2 records 9 cycles,
and after comparing it with _d-f_, figure 25, where the head variant of the
cycle sign is shown, this assumption becomes a certainty. Both heads have
the same clasped hand in the same position, across the lower part of the
face, which, as explained on page 68, is the essential element of the cycle
head; therefore, A2 records 9 cycles. The next glyph, A3, should be the
katun sign, and a comparison of this form with the head variant for katun
in _e-h_, figure 27, shows this to be the case. The determining
characteristic (see p. 69) is probably the oval in the top of the head,
which appears in both of these forms for the katun. The katun coefficient
is 15 (3 bars). The next glyph, A4, should record the tuns, and by
comparing this form with the head variant for the tun sign in _e-g_, figure
29, this also is found to be the case. Both heads show the same essential
characteristic--the fleshless lower jaw (see p. 70). The coefficient is 0
(compare fig. 47). The uinal head in A5 is equally unmistakable. Note the
large curl protruding from the back part of the mouth, which was said (p.
71) to be the essential element of this sign. Compare figure 31, _d-f_,
where the head variant for the uinal is given. The coefficient of A5 is
like the coefficient of A4 (0), and we have recorded, therefore, 0 uinals.
The closing period glyph of the Initial Series in A6 is the head variant
for the kin sign. Compare this form with figure 34, _e-g_, where the kin
head is figured. The determining characteristic of this head is the
subfixial element, which appears also in the normal form for the kin sign
(see fig. 34, _a_). Again, the coefficient of A6 is like the coefficient of
A4 and A5, hence we have recorded here 0 kins.
The number recorded by the head-variant period glyphs and normal-form
numerals in A2-A6 is therefore 9.15.0.0.0; reducing this by means of Table
XIII, we have:
A2 = 9 × 144,000 = 1,296,000
A3 = 15 × 7,200 = 108,000
A4 = 0 × 360 = 0
A5 = 0 × 20 = 0
A6 = 0 × 1 = 0
---------
1,404,000
Deducting from this number all the Calendar Rounds possible, 73 (see Table
XVI), it may be reduced to 18,460. Applying to this number rules 1 and 2
(pp. 139 and 140, respectively), the day reached will be found to be 4
Ahau. Applying rule 3 (p. 141), the position of 4 Ahau in the year will be
found to be 13 Yax. Therefore the terminal date determined by calculation
will be 4 Ahau 13 Yax. {169}
According to step 5 (p. 151), the day reached should follow immediately the
last period glyph, which in this case was in A6; hence the day should be
recorded in A7. This glyph has a coefficient 4, but the glyph does not
resemble either of the forms for Ahau shown in B5, plate 6, _A_, or in B4a,
_C_ of the same plate. However, by comparing this glyph with the second
variant for the day sign Ahau in figure 16, _h'-i'_, the two forms will be
found to be identical, and we may accept A7 as recording the day 4 Ahau.
Immediately following in A8 is the month sign, again out of its usual place
as in plate 6, _C_. Comparing it with the month signs in figure 19, it will
be found to exactly correspond with the sign for Yax in _q-r_. The
coefficient is 13. Therefore the terminal date recorded, 4 Ahau 13 Yax,
agrees with the terminal date reached by calculation, and the whole Initial
Series reads 9.15.0.0.0 4 Ahau 13 Yax. This date marks the close not only
of a hotun in the Long Count, but of a katun as well.
In _B_, plate 7, is figured the Initial Series from Stela A at Copan.[132]
The introducing glyph appears in A1 B1, and is followed by the
Initial-series number in A2-A4. The student will have no difficulty in
picking out the clasped hand in A2, the oval in the top of the head in B2,
the fleshless lower jaw in A3, the large mouth curl in B3, and the flaring
subfix in A4, which are the essential elements of the head variants for the
cycle, katun, tun, uinal, and kin, respectively. Compare these glyphs with
figures 25, _d-f_, 27, _e-h_, 29, _e-g_, 31, _d-f_, and 34, _e-g_,
respectively. The coefficients of these period glyphs are all normal forms
and the student will have no difficulty in reading this number as
9.14.19.8.0.[133]
Reducing this by means of Table XIII to units of the 1st order, we have:
A2 = 9 × 144,000 = 1,296,000
B2 = 14 × 7,200 = 100,800
A3 = 19 × 360 = 6,840
B3 = 8 × 20 = 160
A4 = 0 × 1 = 0
---------
1,403,800
Deducting from this all the Calendar Rounds possible, 73 (see Table XVI),
and applying rules 1 and 2 (pp. 139 and 140, respectively), to the
remainder, the day reached will be 12 Ahau. And applying rule 3 (p. 141),
the month reached will be 18 Cumhu, giving for the terminal date as reached
by calculation 12 Ahau 18 Cumhu. The day should be recorded in B4, and an
examination of this glyph shows that its coefficient is 12, the day
coefficient reached by calculation. The glyph itself, however, is unlike
the forms for Ahau previously encountered in plate 6, _A_, B5 and _C_, B4b,
and in plate 7, _A_, A7. Turning {170} now to the forms for the day sign
Ahau in figure 16, it is seen that the form in A4 resembles the third
variant _j_' or _k'_, the grotesque head, and it is clear that the day 12
Ahau is here recorded. At first sight the student might think that the
month glyph follows in A5, but a closer inspection of this form shows that
this is not the case. In the first place, since the day sign is Ahau the
month coefficient must be either 3, 8, 13, or 18, not 7, as recorded (see
Table VII), and, in the second place, the glyph itself in A5 bears no
resemblance whatsoever to any of the month signs in figure 19. Consequently
the month part of the Initial-series terminal date of this text should
follow the closing glyph of the Supplementary Series. Following along the
glyphs next in order, we reach in A9 a glyph with a coefficient 9, although
the sign itself bears no resemblance to the month-glyph "indicators"
heretofore encountered (see fig. 65).
The glyph following, however, in A9b is quite clearly 18 Cumhu (see fig.
19, _g'-h'_), which is the month part of the terminal date as reached by
calculation. Therefore, since A9a has the coefficient 9 it is probable that
it is a variant of the month-glyph "indicator";[134] and consequently that
the month glyph itself follows, as we have seen, in B9. In other words, the
terminal date recorded, 12 Ahau 18 Cumhu, agrees with the terminal date
reached by calculation, and the whole text, so far as it can be deciphered,
reads 9.14.19.8.0 12 Ahau 18 Cumhu. The student will note that this Initial
Series precedes the Initial Series in plate 7, _A_ by exactly 10 uinals, or
200 days. Compare _A_ and _B_, plate 7.
In plate 8, _A_, is figured the Initial Series from Stela 6 at Copan.[135]
The introducing glyph occupies the space of four glyph-blocks, A1-B2, and
there follows in A3-B4a the Initial-series number 9.12.10.0.0. The cycle
glyph in A3 is partially effaced; the clasped hand, however, the
determining characteristic of the cycle head, may still be distinguished.
The katun head in B3 is also unmistakable, as it has the same superfix as
in the normal form for the katun. At first sight the student might read the
bar and dot coefficient as 14, but the two middle crescents are purely
decorative and have no numerical value, and the numeral recorded here is 12
(see pp. 88-91). Although the tun and uinal period glyphs in A4a and
A4b,[136] respectively, are effaced, their coefficients may be
distinguished as 10 and 0, respectively. In such a case the student is
perfectly justified in assuming that the tun and uinal signs originally
stood here. In B4a the kin period glyph is expressed by its normal form and
the kin coefficient by a head-variant numeral, the clasped hand of which
indicates that it stands for 0 (see fig. 53, _s-w_).[137] The number here
recorded is 9.12.10.0.0.
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR
AND DOT NUMERALS AND HEAD-VARIANT PERIOD GLYPHS]
{171}
Reducing this to units of the 1st order by means of Table XIII, we have:
A3 = 9 × 144,000 = 1,296,000
B3 = 12 × 7,200 = 86,400
A4a = 10 × 360 = 3,600
A4b = 0 × 20 = 0
B4a = 0 × 1 = 0
---------
1,386,000
Deducting from this number all the Calendar Rounds possible, 73 (see Table
XVI), and applying to the remainder rules 1, 2, and 3 (pp. 139-141),
respectively, the date reached by the resulting calculations will be 9 Ahau
18 Zotz. Turning to our text again, the student will have little difficulty
in identifying B4b as 9 Ahau, the day of the above terminal date. The form
Ahau here recorded is the grotesque head, the third variant _j'_ or _k'_ in
figure 16. Following the next glyphs in order, A5-A6, the closing glyph of
the Supplementary Series is reached in B6a. Compare this glyph with the
forms in figure 65. The coefficient of B6a is again a head-variant numeral,
as in the case of the kin period glyph in B4a, above. The fleshless lower
jaw and other skull-like characteristics indicate that the numeral 10 is
here recorded. Compare B6a with figure 52, _m-r_. Since B6a is the last
glyph of the Supplementary Series, the next glyph B6b should represent the
month sign. By comparing the latter form with the month signs in figure 19
the student will readily recognize that the sign for Zotz in _e_ or _f_ is
the month sign here recorded. The coefficient 18 stands above.
Consequently, B4b and B6b represent the same terminal date, 9 Ahau 18 Zotz,
as reached by calculation. This whole Initial Series reads 9.12.10.0.0 9
Ahau 18 Zotz, and according to the writer's view, the monument upon which
it occurs (Stela 6 at Copan) was the period stone for the hotun which began
with the day 9.12.5.0.1 4 Imix 4 Xul[138] and ended with the day
9.12.10.0.0 9 Ahau 18 Zotz, here recorded.
In plate 8, _B_, is figured the Initial Series from Stela 9 at Copan.[139]
The introducing glyph stands in A1-B2 and is followed by the five period
glyphs in A3-A5. The cycle is very clearly recorded in A3, the clasped hand
being of a particularly realistic form. Although {172} the coefficient is
partially effaced, enough remains to show that it was above 5, having had
originally more than the one bar which remains, and less than 11, there
being space for only one more bar or row of dots. In all the previous
Initial Series the cycle coefficient was 9, consequently it is reasonable
to assume that 4 dots originally occupied the effaced part of this glyph.
If the use of 9 cycles in this number gives a terminal date which agrees
with the terminal date recorded, the above assumption becomes a certainty.
In B3 six katuns are recorded. Note the ornamental dotted ovals on each
side of the dot in the numeral 6. Although the head for the tun in A4 is
partially effaced, we are warranted in assuming that this was the period
originally recorded here. The coefficient 10 appears clearly. The uinal
head in B4 is totally unfamiliar and seems to have the fleshless lower jaw
properly belonging to the tun head; from its position, however, the 4th in
the number, we are justified in calling this glyph the uinal sign. Its
coefficient denotes that 0 uinals are recorded here. Although the period
glyph in A5 is also entirely effaced, the coefficient appears clearly as 0,
and from position again, 5th in the number, we are justified once more in
assuming that 0 kins were originally recorded, here. It seems at first
glance that the above reading of the number A3-A5 rests on several
assumptions:
1. That the cycle coefficient was originally 9.
2. That the effaced glyph in A4 was a tun head.
3. That the irregular head in B4 is a uinal head.
4. That the effaced glyph in A5 was a kin sign.
The last three are really certainties, since the Maya practice in recording
Initial Series demanded that the five period glyphs requisite--the cycle,
katun, tun, uinal, and kin--should follow each other in this order, and in
no other. Hence, although the 3d, 4th, and 5th glyphs are either irregular
or effaced, they must have been the tun, uinal, and kin signs,
respectively. Indeed, the only important assumption consisted in
arbitrarily designating the cycle coefficient 9, when, so far as the
appearance of A3 is concerned, it might have been either 6, 7, 8, 9, or 10.
The reason for choosing 9 rests on the overwhelming evidence of antecedent
probability. Moreover, as stated above, if the terminal date recorded
agrees with the terminal date determined by calculation, using the cycle
coefficient as 9, our assumption becomes a certainty. Designating the above
number as 9.6.10.0.0 then and reducing this by means of Table XIII, we
obtain:
A3 = 9 × 144,000 = 1,296,000
B3 = 6 × 7,200 = 43,200
A4 = 10 × 360 = 3,600
B4 = 0 × 20 = 0
A5 = 0 × 1 = 0
---------
1,342,800
{173} Deducting from this number all the Calendar Rounds possible, 70 (see
Table XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141,
respectively) to the remainder, the date determined by the resulting
calculations will be 8 Ahau 13 Pax. Turning to our text again, the student
will have little difficulty in recognizing the first part of this date, the
day 8 Ahau, in B5. The numeral 8 appears clearly, and the day sign is the
profile-head _h'_ or _i'_, the second variant for Ahau in figure 16. The
significance of the element standing between the numeral and the day sign
is unknown. Following along through A6, B6, A7, B7, the closing glyph of
the Supplementary Series is reached in A8. The glyph itself is on the left
and the coefficient, here expressed by a head variant, is on the right. The
student will have no difficulty in recognizing the glyph and its
coefficient by comparing the former with figure 65, and the latter with the
head variant for 10 in figure 52, _m-r_. Note the fleshless lower jaw in
the head numeral in both places. The following glyph, B8, is one of the
clearest in the entire text. The numeral is 13, and the month sign on
comparison with figure 19 unmistakably proves itself to be the sign for Pax
in _c'_. Therefore the terminal date recorded in B5, B8, namely, 8 Ahau 13
Pax, agrees with the terminal date determined by calculation; it follows,
further, that the effaced cycle coefficient in A3 must have been 9, the
value tentatively ascribed to it in the above calculations. The whole
Initial Series reads 9.6.10.0.0 8 Ahau 13 Pax.
Some of the peculiarities of the numerals and signs in this text are
doubtless due to its very great antiquity, for the monument presenting this
inscription, Stela 9, records the next to earliest Initial Series[140] yet
deciphered at Copan.[141] Evidences of antiquity appear in the glyphs in
several different ways. The bars denoting 5 have square ends and all show
considerable ornamentation. This type of bar was an early manifestation and
gave way in later times to more rounded forms. The dots also show this
greater ornamentation, which is reflected, too, by the signs themselves.
The head forms show greater attention to detail, giving the whole glyph a
more ornate appearance. All this embellishment gave way in later times to
more simplified forms, and we have represented in this text a stage in
glyph morphology before conventionalization had worn down the different
signs to little more than their essential elements. {174}
[Illustration: FIG. 68. Initial Series showing bar and dot numerals and
head-variant period glyphs: _A_, Stela C (west side), Quirigua; _B_, Stela
M, Copan.]
In figure 68, _A_, is figured the Initial Series on the west side of Stela
C at Quirigua.[142] The introducing glyph in A1-B2 is followed by the
number in A3-A5, which the student will have no difficulty in reading
except for the head-variant numeral attached to the kin sign in A5. The
clasped hand in this glyph, however, suggests that 0 kins are recorded
here, and a comparison of this form with figure 53, _s-w_, confirms the
suggestion. The number therefore reads 9.1.0.0.0. Reducing this number by
means of Table XIII to units of the 1st order, we obtain:
A3 = 9 × 144,000 = 1,296,000
B3 = 1 × 7,200 = 7,200
A4 = 0 × 360 = 0
B4 = 0 × 20 = 0
A5 = 0 × 1 = 0
---------
1,303,200
Deducting from this number all the Calendar Rounds possible, 68 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively)
to the remainder, we reach for the terminal date 6 Ahau 13 Yaxkin. Looking
for the day part of this date in B5, we find that the form there recorded
bears no resemblance to 6 Ahau, the day determined by calculation.
Moreover, comparison of it with the day signs in figure 16 shows that it is
unlike all of them; further, there is {175} no bar and dot coefficient.
These several points indicate that the day sign is not the glyph in B5,
also that the day sign is, therefore, out of its regular position. The next
glyph in the text, A6, instead of being one of the Supplementary Series is
the day glyph 6 Ahau, which should have been recorded in B5. The student
will readily make the same identification after comparing A6 with figure
16, _e'-g'_. A glance at the remainder of the text, will show that no
Supplementary Series is recorded, and consequently that the month glyph
will be found immediately following the day glyph in B6. The form in B6 has
a coefficient 13, one of the four (3, 8, 13, 18) which the month must have,
since the day sign is Ahau (see Table VII). A comparison of the form in B6
with the month signs in figure 19 shows that the month Yaxkin in _k_ or _l_
is the form here recorded; therefore the terminal date recorded agrees with
the terminal date reached by calculation, and the text reads 9.1.0.0.0 6
Ahau 13 Yaxkin.[143]
In figure 68, _B_, is shown the Initial Series on Stela M at Copan.[144]
The introducing glyph appears in A1 and the Initial-series number in
B1a-B2a. The student will note the use of both normal-form and head-variant
period glyphs in this text, the cycle, tun, and uinal in B1a, A2a, and A2b,
respectively, being expressed by the latter, and the katun and kin in B1b
and B2a, respectively, by the former. The number recorded is 9.16.5.0.0,
and this reduces to units of the first order, as follows (see Table XIII):
B1a = 9 × 144,000 = 1,296,000
B1b = 16 × 7,200 = 115,200
A2a = 5 × 360 = 1,800
A2b = 0 × 20 = 0
B2a = 0 × 1 = 0
---------
1,413,000
Deducting from this number all the Calendar Rounds possible, 74 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively)
to the remainder, the terminal date reached by the resulting calculations
will be 8 Ahau 8 Zotz. Turning to our text, the student will have no
difficulty in recognizing in B2b the day 8 Ahau. The month glyph in this
inscription irregularly follows immediately {176} the day glyph. Compare
the form in A3a with the month signs in figure 19 and it will be found to
be the sign for Zotz (see fig. 19, _e-f_). The coefficient is 8 and the
whole glyph represents the month part 8 Zotz, the same as determined by
calculation. This whole Initial Series reads 9.16.5.0.0 8 Ahau 8 Zotz.
The Maya texts presented up to this point have all been drawings of
originals, which are somewhat easier to make out than either photographs of
the originals or the originals themselves. However, in order to familiarize
the student with photographic reproductions of Maya texts a few will be
inserted here illustrating the use of bar and dot numerals with both
normal-form and head-variant period glyphs, with which the student should
be perfectly familiar by this time.
In plate 9, _A_, is figured a photograph of the Initial Series on the front
of Stela 11 at Yaxchilan.[145] The introducing glyph appears in A1 B1; 9
cycles in A2; 16 katuns in B2, 1 tun in A3, 0 uinals in B3, and 0 kins in
B4. The student will note the clasped hand in the cycle head, the oval in
the top of the katun head, the large mouth curl in the uinal head, and the
flaring postfix in the kin head. The tun is expressed by its normal form.
The number here recorded is 9.16.1.0.0, and reducing this to units of the
first order by means of Table XIII, we have:
A2 = 9 × 144,000 = 1,296,000
B2 = 16 × 7,200 = 115,200
A3 = 1 × 360 = 360
B3 = 0 × 20 = 0
A4 = 0 × 1 = 0
---------
1,411,560
Deducting from this number all the Calendar Rounds possible, 74 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively),
to the remainder, the terminal date reached by the resulting calculations
will be 11 Ahau 8 Tzec. The day part of this date is very clearly recorded
in B4 immediately after the last period glyph, and the student will readily
recognize the day 11 Ahau in this form. Following along the glyphs of the
Supplementary Series in C1 D1, C2 D2, the closing glyph is reached in C3b.
It is very clear and has a coefficient of 9. The glyph following (D3)
should record the month sign. A comparison of this form with the several
month signs in figure 19 shows that Tzec is the month here recorded.
Compare D3 with figure 19, _g-h_. The month coefficient is 8. The terminal
date, therefore, recorded in B4 and D3 (11 Ahau 8 Tzec) agrees with the
terminal date determined by calculation, and this whole text reads
9.16.1.0.0 11 Ahau 8 Tzec. The meaning of the element between the tun
coefficient and the tun sign in A3, which is repeated again in D3 between
the month coefficient and the month sign, is unknown.
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR
AND DOT NUMERALS AND HEAD-VARIANT PERIOD GLYPHS]
{177}
In plate 9, _B_, is figured the Initial Series on an altar in front of
Structure 44 at Yaxchilan.[146] The introducing glyph appears in A1 B1 and
is followed by the number in A2-A4. The period glyphs are all expressed as
head variants and the coefficients as bar and dot numerals. Excepting the
kin coefficient in A4, the number is quite easily read as 9.12.8.14.? An
inspection of our text shows that the coefficient must be 0, 1, 2, or 3.
Let us work out the terminal dates for all four of these values, commencing
with 0, and then see which of the resulting terminal days is the one
actually recorded in A4. Reducing the number 9.12.8.14.0 to units of the
first order by means of Table XIII, we have:
A2 = 9 × 144,000 = 1,296,000
B2 = 12 × 7,200 = 86,400
A3= 8 × 360 = 2,880
B3 = 14 × 20 = 280
A4= 0 × 1 = 0
---------
1,385,560
Deducting from this number all the Calendar Rounds possible, 73 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively),
to the remainder, the terminal day reached will be 11 Ahau 3 Pop. Therefore
the Initial-series numbers 9.12.8.14.1, 9.12.8.14.2, and 9.12.8.14.3 will
lead to the three days immediately following 9.12.8.14.0 11 Ahau 3 Pop.
Therefore our four possible terminal dates will be:
9.12.8.14.0 11 Ahau 3 Pop
9.12.8.14.1 12 Imix 4 Pop <------
9.12.8.14.2 13 Ik 5 Pop
9.12.8.14.3 1 Akbal 6 Pop
Now let us look for one of these four terminal dates in the text. The day
reached by an Initial Series is almost invariably recorded immediately
after the last period glyph; therefore, if this inscription is regular, the
day glyph should be B4. This glyph probably has the coefficient 12 (2 bars
and 2 numerical dots), the oblong element between probably being ornamental
only. This number must be either 11 or 12, since if it were 13 the 3 dots
would all be of the same size, which is not the case. An inspection of the
coefficient in B4 eliminates from consideration, therefore, the last two of
the above four possible terminal dates, and reduces the possible values for
the kin coefficient in A4 to 0 or 1. Comparing the glyph in B4 with the day
signs in figure 16, the form here recorded will be found to be identical
with the sign for Imix in figure 16, a. This eliminates the first terminal
date above and leaves the second, the day part of which {178} we have just
seen appears in B4. This further proves that the kin coefficient in A4 is
1. The final confirmation of this identification will come from the month
glyph, which must be 4 Pop if we have correctly identified the day as 12
Imix. If, on the other hand, the day were 11 Ahau, the month glyph would be
3 Pop. Passing over A5 B5, A6 B6, C1 D1, and C2, we, reach in D2a the
closing glyph of the Supplementary Series, here showing the coefficient 9.
Compare this form with figure 65. The month glyph, therefore, should appear
in D2b. The coefficient of this glyph is very clearly 4, thus confirming
our identification of B4 as 12 Imix. (See Table VII.) And finally, the
month glyph itself is Pop. Compare D2b with figure 19, a. The whole Initial
Series in plate 9, _B_, therefore reads 9.12.8.14.1 12 Imix 4 Pop.
In plate 10, is figured the Initial Series from Stela 3 at Tikal.[147] The
introducing glyph, though somewhat effaced, may still be recognized in A1.
The Initial-series number follows in B1-B3. The head-variant period glyphs
are too badly weathered to show the determining characteristic in each
case, except the uinal head in A3, the mouth curl of which appears clearly,
and their identification rests on their relative positions with reference
to the introducing glyph. The reliability of this basis of identification
for the period glyphs of Initial Series has been thoroughly tested in the
texts already presented and is further confirmed in this very inscription
by the uinal head. Even if the large mouth curl of the head in A3 had not
proved that the uinal was recorded here, we should have assumed this to be
the case because this glyph, A3, is the fourth from the introducing glyph.
The presence of the mouth curl therefore confirms the identification based
on position. The student will have no difficulty in reading the number
recorded in B1-B3 as 9.2.13.0.0.
Reducing this number by means of Table XIII to units of the first order, we
obtain:
B1 = 9 × 144,000 = 1,296,000
A2 = 2 × 7,200 = 14,400
B2 = 13 × 360 = 4,680
A3 = 0 × 20 = 0
B3 = 0 × 1 = 0
---------
1,315,080
Deducting all the Calendar Rounds possible from this number, 69 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively)
to the remainder, the terminal date reached will be 4 Ahau 13 Kayab. It
remains to find this date in the text. The glyph in A4, the proper position
for the day glyph, is somewhat effaced, though the profile of the human
head may yet be traced, thus enabling us to identify this form as the day
sign Ahau. Compare figure 16, _h', i'_. The coefficient of A4 is very
clearly 4 dots, that is, 4, and consequently this glyph agrees with the day
as determined by calculation, 4 Ahau. Passing over B4, A5, B5, and A6, we
reach in B6 the closing glyph of the Supplementary Series, here recorded
with a coefficient of 9. Compare B6 with figure 65. The month glyph follows
in A7 with the coefficient 13. Comparing this latter glyph with the month
signs in figure 19, it is evident that the month Kayab (fig. 19, _d'-f'_)
is recorded in A7, which reads, therefore, 13 Kayab. Hence the whole text
records the Initial Series 9.2.13.0.0 4 Ahau 13 Kayab.
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR
AND DOT NUMERALS AND HEAD-VARIANT PERIOD GLYPHS--STELA 3, TIKAL]
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF BAR
AND DOT NUMERALS AND HEAD-VARIANT PERIOD GLYPHS--STELA A (EAST SIDE),
QUIRIGUA]
{179}
This Initial Series is extremely important, because it records the earliest
contemporaneous[148] date yet found on a monument[149] in the Maya
territory.
In plate 11 is figured the Initial Series from the east side of Stela A at
Quirigua.[150] The introducing glyph appears in A1-B2 and the
Initial-series number in A3-A5. The student will have little difficulty in
picking out the clasped hand in A3, the oval in the top of the head in B3,
the fleshless lower jaw in A4, the mouth curl in B4, as the essential
characteristic of the cycle, katun, tun, and uinal heads, respectively. The
kin head in A5 is the banded-headdress variant (compare fig. 34, _i, j_),
and this completes the number, which is 9.17.5.0.0. Reducing this by means
of Table XIII to units of the first order, we have:
A3 = 9 × 144,000 = 1,296,000
B3 = 17 × 7,200 = 122,400
A4 = 5 × 360 = 1,800
B4 = 0 × 20 = 0
A5 = 0 × 0 = 0
---------
1,420,200
Deducting from this number all the Calendar Rounds possible, 73 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, {180}
respectively) to the remainder, the terminal day reached will be found to
be 6 Ahau 13 Kayab.
In B5 the profile variant of the day sign, Ahau, is clearly recorded (fig.
16, _h', i'_), and to it is attached a head-variant numeral. Comparing this
with the head-variant numerals in figures 51-53, the student will have
little difficulty in identifying it as the head for 6 (see fig. 51, _t-v_).
Note the so-called "hatchet eye" in A5, which is the determining
characteristic of the head for 6 (see p. 99). Passing over A6 B6, A7 B7, A8
B8, we reach in A9 the closing glyph of the Supplementary Series, here
showing the head-variant coefficient 10 (see fig. 52, _m-r_). In B9, the
next glyph, is recorded the month 13 Kayab (see fig. 19, _d'-f'_). The
whole Initial Series therefore reads 9.17.5.0.0 6 Ahau 13 Kayab.
All the Initial Series heretofore presented have had normal-form numerals
with the exception of an incidental head-variant number here and there. By
this time the student should have become thoroughly familiar with the use
of bar and dot numerals in the inscriptions and should be ready for the
presentation of texts showing head-variant numerals, a more difficult group
of glyphs to identify.
In plate 12, _A_, is figured the Initial Series on the tablet from the
Temple of the Foliated Cross at Palenque.[151] The introducing glyph
appears in A1 B2, and is followed by the Initial-series number in A3-B7.
The student will have little difficulty in identifying the heads in B3, B4,
B5, B6, and B7 as the head variants for the cycle, katun, tun, uinal, and
kin, respectively. The head in A3 prefixed to the cycle glyph in B3 has for
its determining characteristic the forehead ornament composed of _more than
one part_ (here, of two parts). As explained on page 97, this is the
essential element of the head for 1. Compare A3 with figure 51, _a-e_, and
the two glyphs will be found to be identical. We may conclude, therefore,
that in place of the usual 9 cycles heretofore encountered in Initial
Series, we have recorded in A3-B3 1 cycle.[152] The katun coefficient in A4
resembles closely the cycle coefficient except that its forehead ornament
is composed of but a single part, a large curl. As explained on page 97,
the heads for 1 and 8 are very similar, and are to be distinguished from
each other only by their forehead ornaments, the former having a forehead
ornament composed of more than one part, as in A3, and the latter a
forehead ornament composed of but one part, as here in A4. This head,
moreover, is very similar to the head for 8 in figure 52, _a-f_; indeed,
the only difference is that the former has a fleshless lower jaw. This is
the essential element of the head for 10 (see p. 100); when applied to the
head for any other numeral it increases the value of the resulting head by
10. Therefore we have recorded in A4 B4, 18 (8 + 10) katuns. The tun
coefficient in A5 has for its determining characteristic the tun headdress,
which, as explained on page 99, is the essential element of the head for 5
(see fig. 51, _n-s_). Therefore A5 represents 5, and A5 B5, 5 tuns. The
uinal coefficient in A6 has for its essential elements the large bulging
eye, square irid, and snaglike front tooth. As stated on page 98, these
characterize the head for 4, examples of which are given in figure 51,
_j-m_. Consequently, A6 B6 records 4 uinals. The kin coefficient in A7 is
quite clearly 0. The student will readily recognize the clasped hand, which
is the determining characteristic of the 0 head (see p. 101 and fig. 53,
_s-w_). The number recorded in A3-B7 is, therefore, 1.18.5.4.0. Reducing
this number to units of the 1st order by means of Table XIII, we obtain:
[Illustration: GLYPHS REPRESENTING INITIAL SERIES, SHOWING USE OF
HEAD-VARIANT NUMERALS AND PERIOD GLYPHS]
{181}
A3B3 = 1 × 144,000 = 144,000
A4B4 = 18 × 7,200 = 129,600
A5B5 = 5 × 360 = 1,800
A6B6 = 4 × 20 = 80
A7B7 = 0 × 1 = 0
-------
275,480
Deducting from this number all the Calendar Rounds possible, 14 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively),
the terminal date reached will be 1 Ahau 13 Mac. Of this date, the day
part, 1 Ahau, is recorded very clearly in A8 B8. Compare the head in A8
with the head in A3, which, we have seen, stood for 1 and also with figure
51, _a-e_, and the head in B8 with figure 16, _h', i'_, the profile head
for the day sign Ahau. This text is irregular in that the month glyph
follows immediately the day glyph, i.e., in A9. The glyph in A9 has a
coefficient 13, which agrees with the month coefficient determined by
calculation, and a comparison of B9 with the forms for the months in figure
19 shows that the month Mac (fig. 19, _w, x_) is here recorded. The whole
Initial Series therefore reads 1.18.5.4.0 1 Ahau 13 Mac.
In plate 12, _B_, is figured the Initial Series on the tablet from the
Temple of the Sun at Palenque.[153] The introducing glyph appears in A1-B2
and is followed by the Initial-series number in A3-B7. The student will
have no difficulty in identifying the period glyphs in B3, B4, B5, B6, and
B7; and the cycle, katun, and tun coefficients in A3, A4, and A5,
respectively, will be found to be exactly like the corresponding
coefficients in the preceding Initial Series (pl. 12, _A_, A3, A4, A5),
which, as we have seen, record the numbers 1, 18, and 5, respectively. The
uinal coefficient in A6, however, presents a new form. Here the determining
characteristic is the banded headdress, or fillet, which distinguishes the
head for 3, as explained on page 98 (see fig. 51 _h, i_). We have then in
A6 B6 record of 3 {182} uinals. The kin coefficient in A7 is very clearly
6. Note the "hatchet eye," which, as explained on page 99, is the essential
element of this head numeral, and also compare it with figure 51, _t-v_.
The number recorded in A3-B7 therefore is 1.18.5.3.6. Reducing this to
units of the first order by means of Table XIII, we obtain:
A3B3 = 1 × 144,000 = 144,000
A4B4 = 18 × 7,200 = 129,600
A5B5 = 5 × 360 = 1,800
A6B6 = 3 × 20 = 60
A7B7 = 6 × 1 = 6
-------
275,466
Deducting from this number all the Calendar Rounds possible, 14 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141), respectively,
to the remainder, the terminal date reached will be 13 Cimi 19 Ceh. If this
inscription is regular, the day part of the above date should follow in A8
B8, the former expressing the coefficient and the latter the day sign.
Comparing A8 with the head numerals in figures 51-53, it will be found to
be like the second variant for 13 in figure 52, _x-b'_, the essential
element of which seems to be the pendulous nose surmounted by a curl, the
protruding mouth fang, and the large bulging eye. Comparing the glyph in B8
with the day signs in figure 16, it will be seen that the form here
recorded is the day sign Cimi (fig. 16, _h, i_). Therefore A8 B8 expresses
the day 13 Cimi. The month glyph is recorded very irregularly in this text,
since it occurs neither immediately after the Supplementary Series or the
day sign, but the second glyph after the day sign, in B9. A comparison of
this form with figure 19, _u-v_, shows that the month Ceh is recorded here.
The coefficient is 19. Why the glyph in A9 should stand between the day and
its month glyph is unknown; this case constitutes one of the many unsolved
problems in the study of the Maya glyphs. This whole Initial Series reads
1.18.5.3.6 13 Cimi 19 Ceh.
The student will note that this Initial Series records a date 14 days
earlier than the preceding Initial Series (pl. 12, _A_). That two dates
should be recorded which were within 14 days of each other, and yet were
more than 3,000 years earlier than practically all other Maya dates, is a
puzzling problem. These two Initial Series from the Temple of the Sun and
that of the Foliated Cross at Palenque, together with a Secondary-series
date from the Temple of the Cross in the same city, have been thoroughly
reviewed by Mr. Bowditch (1906). The conclusions he reaches and the
explanation he offers to account for the occurrence of three dates so
remote as these are very reasonable, and, the writer believes, will be
generally accepted by Maya students. {183}
[Illustration: FIG. 69. Initial Series showing head-variant numerals and
period glyphs: _A_, House C of the Palace Group at Palenque; _B_, Stela P
at Copan.]
In figure 69, _A_, is shown the Initial Series inscribed on the rises and
treads of the stairway leading to House C in the Palace at Palenque.[154]
The introducing glyph is recorded in A1, and the Initial-series number
follows in B1-B3. The student will readily recognize the period glyphs in
B1b, A2b, B2b, A3b, and B3b. The head expressing the cycle coefficient in
B1a has for its essential element the dots centering around the corner of
the mouth. As explained on page 100, this characterizes the head for 9 (see
fig. 52, _g-l_, where variants for the 9 head are figured). In B1,
therefore, we have recorded 9 cycles, the number almost always found in
Initial Series as the cycle coefficient. The essential element of the katun
coefficient in A2a is the forehead ornament composed of a single part. This
denotes the head for 8 (see p. 100, and fig. 52, _a-f_; also compare A2a
with the heads denoting 18 in the two preceding examples, pl. 12, _A_, A4,
and pl. 12, _B_, A4, each of which shows the same forehead ornament). The
tun coefficient in B2a is exactly like the cycle coefficient just above it
in B1a; that is, 9, having the same dotting of the face near the corner of
the mouth. The uinal coefficient in A3a is 13. Compare this head numeral
with A8, plate 12, _B_, which also denotes 13, and also with figure 52,
_x-b'_. The essential elements (see p. 101) {184} are the large pendulous
nose surmounted by a curl, the bulging eye, and the mouth fang, the last
mentioned not appearing in this case. Since the kin coefficient in B3a is
somewhat effaced, let us call it 0 for the present[155] and proceed to
reduce our number 9.8.9.13.0 to units of the first order by means of Table
XIII:
B1 = 9 × 144,000 = 1,296,000
A2 = 8 × 7,200 = 57,600
B2 = 9 × 360 = 3,240
A3 = 13 × 20 = 260
B3 = 0 × 1 = 0
---------
1,357,100
Deducting from this number all the Calendar Rounds possible, 71 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively)
to the remainder, we reach as the terminal date 8 Ahau 13 Pop. Now let us
examine the text and see what is the terminal date actually recorded. In
A4b the student will have little difficulty in recognizing the profile
variant of the day sign Ahau (see fig. 16, _h', i'_). This at once gives us
the missing value for the kin coefficient in B3, for the day Ahau can never
be reached in an Initial Series if the kin coefficient is other than 0.
Similarly, the day Imix can never be reached in Initial Series if the kin
coefficient is other than 1, etc. Every one of the 20 possible kin
coefficients, 0 to 19, has a corresponding day to which it will always
lead, that is, Ahau to Cauac, respectively (see Table I). Thus, if the kin
coefficient in an Initial-series number were 5, for example, the day sign
of the resulting terminal date must be Chicchan, since Chicchan is the
fifth name after Ahau in Table I. Thus the day sign in Initial-series
terminal dates may be determined by inspection of the kin coefficient as
well as by rule 2 (p. 140), though, as the student will see, both are
applications of the same principle, that is, deducting all of the 20s
possible and counting forward only the remainder. Returning to our text, we
can now say without hesitation that our number is 9.8.9.13.0 and that the
day sign in A4b is Ahau. The day coefficient in A4a is just like the katun
coefficient in A2a, having the same determining characteristic, namely, the
forehead ornament composed of one part. A comparison of this ornament with
the ornament on the head for 8 in A2a will show that the two forms are
identical. The bifurcate ornament surmounting the head in A4a is a part of
the headdress, and as such should not be confused with the forehead
ornament. The failure to recognize this point might cause the student to
identify {185} A4a as the head for 1, that is, having a forehead ornament
composed of more than one part, instead of the head for 8. The month glyph,
which follows in B4b, is unfortunately effaced, though its coefficient in
B4a is clearly the head for 13. Compare B4a with the uinal coefficient in
A3a and with the heads for 13 in figure 52, _x-b'_. As recorded, therefore,
the terminal date reads 8 Ahau 13 ?, thus agreeing in every particular so
far as it goes with the terminal date reached by calculation, 8 Ahau 13
Pop. In all probability the effaced sign in B4b originally was the month
Pop. The whole Initial Series therefore reads 9.8.9.13.0 8 Ahau 13 Pop.
In figure 69, _B_, is shown the Initial Series from Stela P at Copan.[156]
The introducing glyph appears in A1-B2 and is followed by the
Initial-series number in A3-B4. The student will readily identify A3, B3,
and A4 as 9 cycles, 9 katuns, and 10 tuns, respectively. Note the beard on
the head representing the number 9 in both A3a and B3a. As explained on
page 100, this characteristic of the head for 9 is not always present (see
fig. 52, _g-i_). The uinal and kin glyphs have been crowded together into
one glyph-block, B4, the uinal appearing in B4a and the kin in B4b. Both
their coefficients are 0, which is expressed in each case by the form shown
in figure 47. The whole number recorded is 9.9.10.0.0; reducing this to
units of the first order by means of Table XIII, we obtain:
A3 = 9 × 144,000 = 1,296,000
B3 = 9 × 7,200 = 64,800
A4 = 10 × 360 = 3,600
B4a = 0 × 20 = 0
B4b = 0 × 1 = 0
---------
1,364,400
Deducting from this number all of the Calendar Rounds possible, 71 (see
Table XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141,
respectively) to the remainder, the terminal date reached will be 2 Ahau 13
Pop. In A5a the day 2 Ahau is very clearly recorded, the day sign being
expressed by the profile variant and the 2 by two dots (incorrectly shown
as one dot in the accompanying drawing).[157] Passing over A5b, B5, and A6
we reach in B6a the closing glyph of the Supplementary Series, and in the
following glyph, B6b, the month part of this terminal date. The coefficient
is 13, and comparing the sign itself with the month signs in figure 19, it
will be seen that the form in _a_ (Pop) is the month recorded here. The
whole Initial Series therefore reads 9.9.10.0.0 2 Ahau 13 Pop. {186}
[Illustration: FIG. 70. Initial Series, showing head-variant numerals and
period glyphs, from Zoömorph G at Quirigua.]
In figure 70 is illustrated the Initial Series from Zoömorph G at
Quirigua.[158] The introducing glyph appears in A1-B2 and is followed in
C1-H1 by the Initial-series number. Glyphs C1 D1 record 9 cycles. The dots
on the head for 9 in C1 are partially effaced. In C2 is the katun
coefficient and in D2 the katun sign. The determining characteristic of the
head for 7 appears in C2, namely, the scroll passing under the eye and
projecting upward and in front of the forehead. See page 100 and figure 51,
_w_. It would seem, then, at first sight that 7 katuns were recorded in C2
D2. That this was not the case, however, a closer examination of C2 will
show. Although the lower part of this glyph is somewhat weathered, enough
still remains to show that this head originally had a fleshless lower jaw,
a character increasing its value by 10. Consequently, instead of having 7
katuns in C2 D2 we have 17 (7 + 10) katuns. Compare C2 with figure 53,
_j-m_. In E1 F1, 15 tuns are recorded. The tun headdress in E1 gives the
value 5 to the head there depicted (see fig. 51, _n-s_) and the fleshless
lower jaw adds 10, making the value of E1 15. Compare figure 53, _b-e_,
where examples of the head for 15 are given. Glyphs E2 and F2 represent 0
uinals and G1 H1 0 kins; note the clasped hand in E2 and G1, which denotes
the 0 in each case. This whole number therefore reads 9.17.15.0.0. Reducing
this to units of the first order by means of Table XIII, we have:
C1 D1 = 9 × 144,000 = 1,296,000
C2 D2 = 17 × 7,200 = 122,400
E1 F1 = 15 × 360 = 5,400
E2 F2 = 0 × 20 = 0
G1 H1 = 0 × 1 = 0
---------
1,423,800
Deducting from this number all the Calendar Rounds possible, 75 (see Table
XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively),
to the remainder, the terminal day reached will be 5 Ahau 3 Muan. The day
is recorded in G2 H2. The day sign in H2 is quite clearly the grotesque
head variant for Ahau in figure 16, _j'-k'_. The presence of the tun
headdress in G2 indicates that the coefficient here recorded must have been
either 5 or 15, depending on whether or not the lower part of the head
originally had a fleshless lower jaw or not. In this particular case there
is no room for doubt, since the numeral in G2 is a day coefficient, and day
coefficients as stated in Chapter III, can never rise above 13.
Consequently the number 15 can not be recorded in G2, and this form must
stand for the number 5.
[Illustration: OLDEST INITIAL SERIES AT COPAN--STELA 15]
{187}
Passing over I1 J1, I2 J2, K1 Ll, K2 L2, we reach in M1 the closing glyph
of the Supplementary Series, here shown with a coefficient of 10, the head
having a fleshless lower jaw. The month sign follows in N1. The coefficient
is 3 and by comparing the sign itself with the month glyphs in figure 19,
it will be apparent that the sign for Muan in _a'_ or _b'_ is recorded
here. The Initial Series of this monument therefore is 9.17.15.0.0 5 Ahau 3
Muan.
In closing the presentation of Initial-series texts which show both
head-variant numerals and period glyphs, the writer has thought best to
figure the Initial Series on Stela 15 at Copan, because it is not only the
oldest Initial Series at Copan, but also the oldest one known in which
head-variant numerals are used[159] (see pl. 13). The introducing glyph
appears at A1-B2. There follows in A3 a number too much effaced to read,
but which, on the basis of all our previous experience, we are justified in
calling 9. Similarly B3 must be the head variant of the cycle sign. The
numeral 4 is clearly recorded in A4. Note the square irid, protruding fang,
and mouth curl. Compare A4 with figure 51, _j-m_. Although the glyph in B4
is too much effaced to read, we are justified in assuming that it is the
head variant of the katun sign. The glyph in A5 is the numeral 10. Note the
fleshless lower jaw and other characteristics of the death's-head. Again we
are justified in assuming that B5 must be the head variant of the tun sign.
The glyphs A6, B6 clearly record 0 uinals. Note the clasped hand denoting
zero in A6, and the curling mouth fang of the uinal period glyph in B6.
This latter glyph is the full-figure form of the uinal sign[160] (a frog).
Compare B6 with figure 33, which shows the uinal sign on Stela D at Copan.
The stela is broken off just below the uinal sign and its coefficient; and
therefore the kin coefficient and sign, the day coefficient and sign, and
the month coefficient and sign, are missing. Assembling the four periods
present, we have 9.4.10.0.?. Calling the missing kin coefficient 0, and
reducing this number to units of the first order by means of Table XIII, we
have:
A3 B3 = 9 × 144,000 = 1,296,000
A4 B4 = 4 × 7,200 = 28,800
A5 B5 = 10 × 360 = 3,600
A6 B6 = 0 × 20 = 0
0 × 1 = 0
---------
1,328,400
Deducting from this number all the Calendar Rounds possible, 69 {188} (see
Table XVI), and applying rules 1, 2, and 3 (pp. 139, 140, and 141,
respectively) to the remainder, the terminal date reached will be 12 Ahau 8
Mol. This date is reached on the assumption that the missing kin
coefficient was zero. This is a fairly safe assumption, since when the tun
coefficient is either 0, 5, 10, or 15 (as here) and the uinal coefficient
is 0 (as here), the kin coefficient is almost invariably zero. That is, the
close of an even hotun in the Long Count is recorded.