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                     MEMORANDA ON THE MAYA CALENDARS
                           USED IN THE BOOKS OF
                               CHILAN BALAM

                                    BY
                           CHARLES P. BOWDITCH

 (From the American Anthropologist (N. S.), Vol. 3, January-March, 1901)

                                 NEW YORK
                           G. P. PUTNAM'S SONS
                                   1901




    MEMORANDA ON THE MAYA CALENDARS USED IN THE BOOKS OF CHILAN BALAM

                         BY CHARLES P. BOWDITCH


Dr Brinton, in his _Maya Chronicles_, has translated the following
passages from the Book of Chilan Balam of Mani:

    ... in the thirteenth Ahau Ahpula died; for six years the count of
    the thirteenth Ahau will not be ended; the count of the year was
    toward the East, the month Pop began with (the day) fourth Kan; the
    eighteenth day of the month Zip (that is) 9 Ymix, was the day on
    which Ahpula died; and that the count may be known in numbers and
    years, it was the year 1536.

And again from the Book of Chilan Balam of Tizimin:

    The thirteenth Ahau; the death of Ahpulha took place; it was the
    sixth year when ended the count of the thirteenth Ahau,--the count
    of the year was from the east (the month) Pop passed on the fourth
    Kan; on the eighteenth of (the month) Zip, 9 Imix was the day
    Ahpulha died; it was the year 1536.

In his remarks on these books Dr Brinton says:

    According to the reckoning as it now stands, six complete great
    cycles were counted, and parts of two others, so that the native at
    the time of the Conquest would have had eight great cycles to
    distinguish apart.

    I have not found any clear explanation how this was accomplished.
    We do not even know what name was given to this great cycle,[1] nor
    whether the calendar was sufficiently perfected to prevent
    confusion in dates in the remote past.

  [1] It should be noted that the grand cycle, which Dr Brinton refers
  to, is the period of 13 × 7200 days = 93,600 days or 260 periods of
  360 days; while the grand cycle according to Goodman's method is 13
  × 144,000 days or 5200 periods of 360 days.

It would seem, however, as if the reckoning of time as given in these
books is very accurate, fixing a date which would not be duplicated
within a limit of thirty-five hundred or four thousand years.

The Books of Chilan Balam number the katuns on a different principle
from that used on the inscriptions or in the Dresden Codex, but the two
methods can be readily and usefully brought together, as the katun
itself remains the same in both methods. In the inscriptions the katuns
are numbered from 0 to 19, using Goodman's method though not his exact
nomenclature, and twenty of them equal one cycle. In the Chilan Balam
books, the katuns are named as Katun 13 Ahau, Katun 11 Ahau, etc., these
being the days with which they begin or with which the previous katun
ended; and as after thirteen katuns the same name is again given, this
nomenclature fixes a date within a period which equals 13 multiplied by
the number of days in a katun. There has been a difference of opinion as
to this number of days in a katun, but it is clear from the Books of
Chilan Balam that their reckoning was by terms of 20 × 360 days. The
followers of Perez, however, insist that the length of the katun was 24
× 365 days. Sr Perez has indeed made this assertion,[2] but he rests his
opinion to a great degree on the fact that the naming of the katuns
proceeded in the following order, taking their names from the day Ahau
with which they began, viz.:

    Katun 13 Ahau,
    Katun 11 Ahau,
    Katun  9 Ahau,
    Katun  7 Ahau, etc.,

and that by starting with a katun which begins with 13 Ahau and counting
forward a period of 24 × 365 days, we should reach another katun
beginning with 11 Ahau. But the same result is brought about by
considering the katun as a period of 20 × 360 days, as has been shown by
Dr Seler, among others; and since the Books of Chilan Balam state
distinctly that they reckon by so many scores of so-called years, and
as the initial dates of the inscriptions all reckon in the same way, it
is now generally considered that the katun consisted of 20 × 360 or 7200
days. An objection to considering a katun as 20 × 360 days may be raised
in that the Books of Chilan Balam use the word "año" or year, but this
can be easily explained by the fact that the Spanish "year" was the
period which most nearly agreed with their tun or 360-day period, and
that the Books did not pretend to speak with scientific accuracy.

  [2] Stephens, _Incidents of Travel in Yucatan_, p. 441 et seq.

Besides the above count, it is well known that the Mayas had a
year-and-month count. This consisted in naming each one of the twenty
days and in attaching to each of these days one of the numbers 1 to 13.
Besides this, each day so numbered was declared to be a given day of a
given month and to occur in a year marked by one of the year bearers--as
for instance in the Book of Chilan Balam, already quoted, where the day
is given as 9 Ymix 18 Zip in the year 4 Kan. Now this day and this year
could recur only after the lapse of fifty-two years or 18,980 days.

It should be noted here that in the inscriptions and in the Dresden
Codex, the day Ymix was always the day 4, 9, 14, or 19 of any month,
showing that the day 1 of the month was Eznab, Akbal, Lamat, or Ben;
while in Landa and the Books of Chilan Balam the day Ymix was the day 3,
8, 13, or 18, showing that the day 1 of the month was Cauac, Kan, Muluc,
or Ix. That is, the months in modern times began with the day which
followed the day with which the months began in more ancient times. As
the tables are calculated for the inscriptions, it will be well, in
order to facilitate our calculations, to call the day on which Ahpula
died the nineteenth of the month Zip, instead of the eighteenth of that
month.

Given that the katun consisted of 7200 days, a Katun 13 Ahau could not
recur until after the lapse of 13 × 7200 or 93,600 days, and the
recurrence of any day marked by the year-and-month count, and occupying
any particular place in a given katun, could not occur until after the
lapse of a period which is found by finding the least common multiple of
the two numbers 93,600 and 18,980. This is 6,832,800 days, which is a
period of 360 calendar rounds of 18,980 days or of 52 years each. This
is equal to 18,720 years, and, in the method of reckoning shown in the
initial dates of the inscriptions, would equal 3 grand cycles, 8 cycles,
and 9 katuns, or, to use the method of Goodman, 3.8.9.0.0.0.

I have said that a day marked by the year-and-month count, and occupying
any particular place in a given katun, could not recur until the lapse
of this long period. This would be true if the day was specified as
being a given day in a given tun in a given katun, or even if the day
was stated as falling in a given uinal of a given tun in a given katun.
But in the case before us the death of Ahpula is said to have taken
place in the Katun 13 Ahau when six tuns or years of that katun remained
unexpired. Even with this rather loose designation such a day would not
recur within a period of 3500 or 4000 years.

The day 4 Ahau 8 Cumhu seems to have been regarded as the beginning day
of the beginning cycle of some grand cycle. From this day all the
initial series of the inscriptions of Copan and Quirigua, of Piedras
Negras and Tikal, so far as we know them, count, except one where this
day 4 Ahau 8 Cumhu is itself given. In this place (on Stela C of
Quirigua) 4 Ahau 8 Cumhu is reckoned thus: "Grand cycle glyph
.13.0.0.0.0.", while in the Temple of the Cross it is declared to be a
thirteenth cycle. As this was the beginning date, there is reason to
believe that the beginning cycle of a great cycle received the number
13.

I give here the first and last terms of a list of the beginning days of
the Katuns 13 Ahau in a complete round of 18,720 years occurring after
the beginning of the grand cycle called by Goodman Grand Cycle 54, which
began with 4 Ahau 8 Cumhu. It is of little consequence what particular
number is given to the grand cycle, as the whole series forms a
continuous count, and I shall therefore follow Goodman, who gives the
number 54 to the grand cycle glyphs common to Copan, Quirigua, etc.

If 54.13.0.0.0.0. or the beginning of the grand cycle, called Grand
Cycle 54 by Goodman, begins with 4 Ahau 8 Cumhu, a Katun 13 Ahau will
appear two katuns after this or with the count of

    54.13.2.0.0.0.     13 Ahau 8 Mol       Year 10 Ix,

and other Katuns 13 Ahau will follow at intervals of 13 katuns as here
given:

    54.13.15.0.0.0.    13 Ahau  8 Pax      Year  6 Ix.
        1. 8.              "    3 Xul            3 Cauac.
        2. 1.              "    3 Kankin        12  "

     .    .    .    .    .    .    .    .    .   .   .

    57.5.19.0.0.0.     13 Ahau 18 Ceh           11 Kan.
       6.12                    13 Uo             8 Muluc.
       7. 5.                   13 Yax            4  "
         18.                   13 Cumhu         13  "
    57.8.11.0.0.0.     13 Ahau  8 Mol           10 Ix.

But we are seeking a Katun 13 Ahau in which 14 tuns have elapsed and of
which 6 tuns still remain unexpired. We must, therefore, add 14 tuns or
14 × 360 days = 5040 days to each of the dates given and we shall then
have the following complete list of the beginning days of Tun 14 of
Katun 13 Ahau for the term of 18,720 years:

    54.13. 2.14.0.0.    9 Ahau 18 Zotz          11 Kan.
          15.                  18 Ceh            7 Kan.
        1. 8.                  13 Uo             4 Muluc.
        2. 1.                  13 Yax           13 Muluc.
          14.                  13 Cumhu          9 Muluc.
        3. 7.                   8 Mol            6 Ix.
        4. 0.                   8 Pax            2 Ix.
          13.                   3 Xul           12 Cauac.
        5. 6.                   3 Kankin         8 Cauac.
          19.                  18 Zip            5 Kan.
        6.12.                  18 Zac            1 Kan.
        7. 5.                  13 Pop           11 Muluc.
          18.                  13 Chen           7 Muluc.
    54. 8.11.14.0.0.           13 Kayab          3 Muluc.
        9. 4.                   8 Yaxkin        13 Ix.
          17.                   8 Muan           9 Ix.
       10.10.                   3 Tzec           6 Cauac.
       11. 3.                   3 Mac            2 Cauac.
          16.                  18 Uo            12 Kan.
       12. 9.                  18 Yax            8 Kan.
    55.13. 2.14.0.0.           18 Cumhu          4 Kan.
                                                --------
          15.                  13 Mol            1 Muluc.
        1. 8.                  13 Pax           10 Muluc.
        2. 1.                   8 Xul            7 Ix.
          14.                   8 Kankin         3 Ix.
        3. 7.                   3 Zotz          13 Cauac.
        4. 0.                   3 Ceh            9 Cauac.
          13.                  18 Pop            6 Kan.
        5. 6.                  18 Chen           2 Kan.
        5.19.                  18 Kayab         11 Kan.
        6.12.                  13 Yaxkin         8 Muluc.
        7. 5.                  13 Muan           4 Muluc.
          18.                   8 Tzec           1 Ix.
        8.11.                   8 Mac           10 Ix.
        9. 4.                   3 Zip            7 Cauac.
          17.                   3 Zac            3 Cauac.
                                                --------
       10.10.                   3 Uayeb         12 Cauac.
       11.03.                  18 Mol            9 Kan.
          16.                  18 Pax            5 Kan.
       12. 9.                  13 Xul            2 Muluc.
    56.13. 2.14.0.0.           13 Kankin        11 Muluc.
          15.                   8 Zotz           8 Ix.
        1. 8.                   8 Ceh            4 Ix.
        2. 1.                   3 Uo             1 Cauac.
          14.                   3 Yax           10 Cauac.
        3. 7.                   3 Cumhu          6 Cauac.
        4. 0.                  18 Yaxkin         3 Kan.
          13.                  18 Muan          12 Kan.
        5. 6.                  13 Tzec           9 Muluc.
        5.19.                  13 Mac            5 Muluc.
        6.12.                   8 Zip            2 Ix.
         7.5.                   8 Zac           11 Ix.
          18.                   3 Pop            8 Cauac.
        8.11.                   3 Chen           4 Cauac.
        9. 4.                   3 Kayab         13 Cauac.
          17.                  18 Xul           10 Kan.
       10.10.                  18 Kankin         6 Kan.
       11.03.                  13 Zotz           3 Muluc.
          16.                  13 Ceh            12 Muluc.
       12. 9.                   8 Uo             9 Ix.
    57.13. 2.14.0.0.            8 Yax            5 Ix.
          15.                   8 Cumhu          1 Ix.
        1. 8.                   3 Mol           11 Cauac.
        2. 1.                   3 Pax            7 Cauac.
          14.                  18 Tzec           4 Kan.
                                                --------
        3. 7.                  18 Mac           13 Kan.
        4. 0.                  13 Zip           10 Muluc.
        4.13.                  13 Zac            6 Muluc.
        5. 6.                   8 Pop            3 Ix.
        5.19.                   8 Chen          12 Ix.
        6.12.                   8 Kayab          8 Ix.
        7. 5.                   3 Yaxkin         5 Cauac.
          18.                   3 Muan           1 Cauac.
        8.11.                  18 Zotz          11 Kan.

The only places where a year 4 Kan appears are at the dates

    55.13. 2.14.0.0.[3]     9 Ahau 18 Cumhu     Year 4 Kan, and
    57. 2.14.14.0.0.        9 Ahau 18 Tzec      Year 4 Kan.

But as the words used are that 6 years (or tuns) remained before the end
of the katun, and as a slightly longer time than just 6 tuns may have
remained, and as the month Zip in which the death of Ahpula occurred is
the third month of the year and so is near the beginning of the year 4
Kan, it is quite possible that the beginning of the Tun 14 may have been
in the latter part of the preceding year, in which case, in addition to
the preceding dates, the following date might be the one which we are
seeking:

    55. 9.17.14.0.0.        9 Ahau 3 Zac        Year 3 Cauac.

  [3] It is necessary to remember that, by Goodman's methods, these
  figures represent periods of past time. Thus the number 2 of the katun
  means that 2 katuns have passed, and that the current katun is what we
  should call the third; and that 0.0 means that a full count of uinals
  and kins has occurred and that the current uinal and kin are what we
  should call the first.

As 9 Ymix 19 Zip is said to be in the year 4 Kan, we shall find this
date before the dates of the beginning of Tun 14 in the first two cases
and after the beginning of Tun 14 in the last case. This date of 9 Ymix
19 Zip will then be numbered thus, placing the three dates in
consecutive order:

    1) 55.13. 2.13. 3. 1.      6 tuns 299 days to end of Katun 13 Ahau.
    2) 55. 9.17.14.11. 1.      5  "   139   "      "       "     "
    3) 57. 2.14.13.16. 1.      6  "    39   "      "       "     "

In no one of the cases is the date 9 Ymix 19 Zip exactly 6 tuns before
the end of the Katun 13 Ahau, but it is possible that the annalist took
no account of fractions of tuns, either in excess of the 6 tuns or
otherwise. Thus in the first and last cases of the three, as first
given, he may have said to himself, "There are but 6 whole tuns
remaining of the katun and I will call it 6," or in the second case he
may have said: "There are 5 tuns remaining and 139 days besides; I will
call it 6 tuns." Whichever was the plan he followed, we can have at
present no means of ascertaining except from the results which we obtain
by calculation.

The date found on Stela 9 of Copan, which is the earliest date of these
stelæ of that place, in which the numbers preceding the period glyphs
are given by the line-and-dot method, is 54.9.6.10.0.0. This precedes
the above dates by the following periods:

    1) 0.3.16.3. 3.1. =   548,341 days =  1,502 years 111 days.
    2) 1.0.11.4.11.1  = 1,952,861  "   =  5,350   "    14  "
    3) 2.6. 8.3.16.1  = 4,667,001  "   = 12,786   "   111  "

If, now, we accept the first date of 55.13.2.13.3.1. as the date of
Ahpula's death, we shall have the date of Stela 9 of Copan as A.D. 34,
since the death occurred in 1536. If we accept the second date,
55.9.17.14.11.1., as the true one, Stela 9 must represent a date of B.C.
3814, and in the case of the third date, 57.2.14.13.16.1. in which the
period to elapse to the end of Katun 13 Ahau is the nearest to an exact
6 tuns, we should throw back Copan to B.C. 11,250. It is not probable,
however, that either of the last two dates is correct, both because of
the immense time which would have elapsed and because the monuments show
signs of no such age. We are therefore left to the date A.D. 34 as the
probable date of the earliest stela of Copan which we know of at
present.

The following table gives the earliest and latest dates in Copan and
Quirigua as far as we know them, together with the dates of our calendar
corresponding thereto, on the supposition that the above date is rightly
deciphered:

 Copan:   Stela 9, 9. 6.10.0.0                                  A.D.  34.
            "   N, 9.16.10.0.0 = 197  years later than A.D. 34  A.D. 231.
 Quirigua:  "   C, 9. 1. 0.0.0 = 108 +  "  earlier  "   "  say  B.C.  75.
            "   K, 9.18.15.0.0 = 241 +  "   later   "   "       A.D. 275.

If this is correct, Copan lasted, so far as the erection of stelæ is
concerned, for about 200 years, and Quirigua for about 350 years, though
of course this may be only a small part of the period of their
existence.

The above calculations have been made on the supposition that the
initial dates record the date of the erection of the stelæ, and on the
further supposition, as has been stated, that the same principle of
calculating time has been continued from the earliest ages. There is,
however, some evidence that a change has been made, at least in detail.
It has already been seen that the beginning day of the month has been
shifted from the Eznab, Akbal series to the Cauac, Kan series of days.
What difference this would have made in the relation of the
year-and-month count with the long count it is impossible to say without
knowing the means used to effect the change; but it is quite likely that
this relation was not affected. In the Book of Chilan Balam of Mani is
the entry: "The Thirteenth Ahau; then Pop was counted in order." And in
the Book of Chilan Balam of Chumayel we find, "The Thirteenth Ahau; Pop
was set in order." This statement occurs in the early part of the
chronicle, and the calculation of the Ahaus goes on after it in exactly
the same way as before it. This setting in order of Pop would not then
seem to have made any difference in the long count. At least it is very
probable that it means merely that the seasons and the calendar were
made to agree.

Dr Brinton (_Maya Chronicles_, p. 85) also gives a translation of a part
of the Codice Perez, which refers to the "Doubling of the Katuns." The
statement is very obscure, but only tends to show that while the
counting of the katuns was carried on as in the Books of Chilan Balam,
the first of the series was called Katun 8 Ahau instead of Katun 13
Ahau, while the last of the series was Katun 10 Ahau. This would not
necessarily change the consecutive order of the katuns, but might merely
give a new starting-point.

While, therefore, it is impossible to say what change, if any, was made
in the reckoning of time, it may be said that there is no evidence at
present to show that the old relation of the long count to the
year-and-month count and to the count of the Books of Chilan Balam did
not continue to the time of the arrival of the Spaniards. Moreover, the
date of A.D. 34 for the monuments of Copan and Quirigua is by no means
unlikely to be the true one. At all events the above discussion of the
reckoning will not be useless if it succeeds in bringing out new facts,
and no one will be more ready to recognize any new evidence than I shall
be, even if the above deductions shall be shown to be erroneous.




       *       *       *       *       *




Transcriber's note:

In general every effort has been made to replicate the original text as
faithfully as possible, including some instances of inconsistencies of
spelling (Ahpula/Ahpulha; Ymix/Imix) and possible irregularities in the
use of commas and periods in Mayan dates.