No. Days Days Days
of in Dresden groups in Dresden groups in Dresden groups
month month 1 2 3 month 1 2 3 month 1 2 3
1 30 \ 30 \ 30 \
2 29 | 29 | 29 |
3 30 | 30 | 30 |
4 29 | 177 177 177 29 | 177 177 177 29 | 177 177 177
5 30 | 30 | 30 |
6 29 / 29 / 29 /
7 30 \ 30 \ 30 \
8 29 | 29 | 29 |
9 30 | 30 | 30 |
10 29 | 177 177 177 29 | 177 177 177 29 | 177 177 177
11 30 | 30 | 30 |
12 29 / 29 / 29 /
13 30 \ 30 \ 30 \
14 29 | 29 | 29 |
15 30 | 148 148 148 30 | 30 | 148 148 148
16 29 | 29 | 177 177 177 29 |
17 30 / 30 | 30 /
18 29 \ 29 / 29 \
19 30 | 30 \ \ 30 |
20 29 | 29 | | 148 29 |
21 30 | 177 177 177 30 | | 177 177 30 | 177 177 177
22 29 | 29 | | 29 |
23 30 / 30 | / 30 /
24 29 \ 29 / \ 29 \
25 30 | 30 \ | 30 |
26 29 | 29 | | 177 29 |
27 30 | 177 177 177 30 | | 148 148 30 | 177 177 177
28 29 | 29 | | 29 |
29 30 / 30 / / 30 |
30 29 \ 29 \ 29 /
31 30 | 30 | 30 \
*32 30 | 30 | 29 |
33 30 | 178 178 178 30 | 178 178 178 30 | 177 177 177
34 29 | 29 | 29 |
35 30 / 30 / 30 /
36 29 \ 29 \ 29 \
37 30 | 30 | 30 |
38 29 | 177 177 177 29 | 177 177 177 29 | 178 177 177
39 30 | 30 | 30 | (178)
40 29 | 29 | (30)29 |
41 30 / 30 / -30 /
42 29 \ 29 \
43 30 | 30 |
44 29 | 29 |
45 30 | 177 177 177 30 | 177 177 177
46 29 | 29 |
47 30 / 30 /
The great bulk of the difference groups as expressed by the lower number
series are 177, the only departure from these being the designation of
the 148-day groups and the extra 178-day group at the end of the first
third. The complete disregard of all of the six normal 178-day groups by
the lower numbers seems to imply that no attempt was made to have the
latter agree with the actual differences in the upper numbers, a
conclusion which is strengthened by the fact that none of the lower
numbers shows evidence of the clerical errors in the differences of the
upper numbers. It seems most probable that the lower number series was
intended merely as a guide to indicate the position of the five month
periods and to place emphasis on the extra intercalary day added in the
23d column, without attempting to have this series accurate.
The presence of the 178-day group in column 7 instead of 6 has been
discussed at some length under the description of the errors. The
scribe, realizing that in neglecting to put in this 178-day group, the
first one of the series, a serious error had been committed, may have
attempted to erase the incorrect record in column 6; then, realizing
that four numbers and three glyphs would have to be altered, decided to
correct this mistake—although it was of more importance than the other
two errors—as he had the former ones, i.e., by making the correction in
the next column.
The very similar irregularity in the last third of the manuscript, the
placing of the 148-day group one column ahead of its expected position,
cannot be explained in the same manner. It is very evident that this
column has been deliberately placed where it is. That it does not have
to do with the month sequence is evident, since it does not affect it.
It must then affect the ecliptic part of the series, for it causes a
short season to occur six months earlier than expected. Upon comparison
of Tables VI and X, it will be seen that all of the dates of the 148-day
group occur during one of the eclipse groups given in Schram’s table.
However, had the 148-day group under discussion been placed in the 59th
column, as uniformity demands, this number, 10,216, would not have
fallen in one of the eclipse groups given in Table VI. This tends to
show that there was some reason other than regulating the difference
groups to agree with the eclipse seasons, for the position of the
148-day groups. This reason, as yet undetermined, is possibly associated
with the pictures, which immediately follow the 148-day groups.
Finally there remain only the totals of the series to be considered. The
total of the upper number series records 11,958 days. Sixty-nine eclipse
seasons complete 11,958.39 days, less than half a day more than the
recorded number. This close agreement and the failure to add the extra
intercalary day to the upper number series at the end of the first
third, give rise to the belief that the upper number series is a
calendar in itself, and records a means by which dates of probable
eclipses may be reckoned. The units of the count were eclipse seasons
expressed as lunar months, 69 of which are represented in the calendar
recorded on these pages.
The Mayas undoubtedly knew the relation of the eclipses to the moon, at
least in a vague way, and felt that it was necessary to associate this
eclipse calendar in some way with the lunar calendar, composed of 29-and
30-day months. Therefore the day series is found immediately below the
upper number series. This series of days constitutes a lunar calendar
which coincides as closely as possible with the eclipse calendar. It may
be the formal lunar calendar of the Mayas, but it may also be an
adaptation of the formal calendar to the eclipse periods. The day series
varies from the eclipse series in two places only. At the end of the
first third of the series, it was necessary to add one day to the lunar
calendar, an addition strongly pointed out in the record, but not to the
eclipse calendar, because of the increasing error between the
revolutions of the moon and the calendrical lunar months. Therefore,
throughout the remaining two-thirds of the series, the lunar calendar
was one day in advance of the eclipse calendar. At the end of the
series, since 405 of the moon’s revolutions complete 11,959.89 days, and
the day series only 11,959, one more day should be added, in order to
keep the error as small as possible. This was accomplished by changing
from the middle to the lower line of days.
On page 52a, immediately preceding the calendar, are four day signs with
numbers. One of these, 12 Lamat, is the zero day of the day series, but
is associated with the middle line of day glyphs and not the upper line,
as might be expected. The series of days which come, calendrically
speaking, just before and after the actual series, may have been placed
in the record to show that slight variations from the average were to be
expected. The entire record is based on the middle line of days until
the end of the series. Here the day just below the last day of the
middle line is 12 Lamat, the end of 46 _tonalamatls_ (260-day cycle),
and the zero day of the recorded series. The _tonalamatl_ was probably
as easily used by the Mayas as “60 days” and “90 days” are used now. The
entire calendrical system of the Mayas is based on the cycle principle.
The series recorded in these pages was probably also a cycle, and in
order to repeat it, 12 Lamat must again be used as the zero date. If to
these arguments is added the fact that an additional day is necessary to
keep the calendar in accord with the synodic revolution of the moon,
there remains little doubt but that the users of this calendar added the
extra day by going from the middle to the lower line of day glyphs,
thereby keeping the error between the moon and the calendar as low as
possible, completing the 46th _tonalamatl_, and at the same time making
it possible to repeat the recorded series as a cycle. If the series is
repeated once, at the end of 810 months, or about 66-1/2 years, the
eclipse calendar will be behind the average eclipse season .78 days, and
the lunar calendar will be in advance of the synodical revolutions of
the moon only .22 days.
In general, then, the irregularities in the calendar recorded on these
pages fall into two groups, those which are clerical errors of the
scribe and do not therefore affect the solution advanced, and those
which do not appear to be of the clerical type. In the light of the
solution advanced, it has been shown that there are perfectly logical
reasons for the latter group of apparent irregularities.