Upper Month
number number Interval Groups
502 17
59 \
2244 76 | 94(47 + 47)
35 /
3278 111
41 41
4488 152
59 \
6230 211 | 94(47 + 47)
35 /
7264 246
41 41
8474 287
53 \
10039 340 | 94(47 + 47)
41 /
11250 381
The next logical step is a comparison between the theoretical calendars
just described and the manuscript. A study of the manuscript reveals
that: (1) the series recorded represents 405 lunar months or three times
135 months, and that the series naturally falls into three great
subdivisions of 3986 days each; (2) each third consists of 23 columns
or unequal subdivisions; (3) the intervals between the 178-day groups
are 47 and 88 months; (4) the 148-day groups fall approximately at 47
and 41 month intervals (see Table X); (5) the first 178-day group in
each third occurs between the 30th and 35th month inclusive, and the
other 178-day group of the third comes 47 months later. Since the number
178 is composed of four 30-and two 29-day months, an extra day must have
been added, that is, a 30-day month was substituted for one of the
29-day months, if the manuscript represents a regular alternating
series.
The obvious conclusions to be drawn from these facts are: (1) that the
series was divided into three groups of 3986 days each in order to
associate the lunar calendar closely with the ecliptic cycle of the same
length; (2) that the 23 columns in each third may represent the
twenty-three eclipse seasons in each eclipse period of 3986 days; (3)
that groups of 47 and 41 months were used in some way in the series, for
the 178-day groups are separated by 47 and 88 months and 88 is composed
of 47 and 41, the two periods so closely associated with the recurrence
of eclipses; (4) that the six months period was changed to one of five
months of 148 days approximately every 47 and 41 months, which is the
method already advanced in the theoretical ecliptic lunar series for
keeping the synodical months and ecliptic seasons together; (5) that one
extra day was added to the alternating 29-and 30-day months, between the
30th and 35th month inclusive of each third, in accordance with the
theoretical necessity for so doing already brought out, and that another
of the three extra days was added 47 months later.
When the difference groups[25] are divided into months it is found that
it is an easy matter to arrange the months in an alternating series. The
group of 177 days is composed of three 30-and three 29-day months,
either of which when alternated can begin the group, which then ends
with the other, i.e., 29, 30, 29, 30, 29, 30, or 30, 29, 30, 29, 30, 29.
The group of 148 days consists of three 30-and two 29-day months,
necessitating that it begin and end with a 30-day month when alternated,
thus, 30, 29, 30, 29, 30. In the 178-day group one of the 29-day months
is replaced by a 30-day month, otherwise the group is exactly like that
of 177 days, which it exceeds by one day. It is evident that there will
always be three 30-day months in succession in the 178-day group, and
that care must be taken in choosing the right sequence of the 177-day
groups which fall near those of 148 days in order to avoid having two
30-day months in succession.
[25] That is, the 177-day, 148-day and 178-day groups.
There remains simply the substitution of the six or five months, as the
case may be, in place of the difference groups in the manuscript.
However, if the Mayas considered each third of the table as a unit, it
is reasonable to assume that the sequence of the months in each third is
identical. Therefore it is necessary to arrange a sequence for only
one-third, that is, 135 months, and then, if the assumption is correct,
this sequence will fit the other two-thirds of the series.
Each third of the table consists of 135 months covering three more days
than would be covered by a simple alternation of 30-and 29-day months.
These three intercalary days were inserted at definite intervals. A clue
to the position of two of them is given by the 178-day groups. One was
inserted between the 30th and 35th months, another 47 months later,
between the 77th and 82d months. Theoretically the extra day should be
inserted in the 34th month after the beginning of a series of
alternating 29-and 30-day months, for then the error between the
synodical revolution of the moon and the calendrical months becomes more
than one day. In the 29-day month preceding the 34th, namely the 32d
month, the error at the end is also practically one day, i.e., .98 days.
The 29-day month most nearly the centre of the first 178-day group is
the 32d month of the series, the third in the group. The Mayas may have
chosen this month because of its position in the 178-day group, making
the sequence of the months 29, 30, 30, 30, 29, 30, if the 30th month was
a 29-day month as it would be by simple alternation.
The second time this intercalary day occurs in each third is 47 months
later. Obviously, this may be the recurrence of this intercalation in a
repetition of a smaller group of months than the 135-month group. If 47
months are subtracted from the 79th month which is the third in the
second 178-day group the result is 32, which implies that the smaller
division is 47 months. Two 47-month periods complete all but 41 of the
135 months in each third. Then, of necessity, if each third of the
manuscript is a unit, a 41-month group follows two 47-month groups, an
arrangement which also agrees with the eclipse groups in Tables V.
The two 178-day groups account for only two of the three intercalated
days, and since no 178-day group occurs in the 41-month division, the
addition of this day must have been accomplished in some more obscure
manner. Since both 47 and 41 are odd numbers, each group must contain at
least one more month of one kind than the other. Since two synodical
revolutions of the moon are slightly longer than two calendrical months
it is wisest to start and end each group with a 30-day month. If this is
done, the 47-month group will contain twenty-five 30-day and twenty-two
29-day months, and the 41-month group twenty-one 30-day and twenty
29-day months, making for the composition of the 135 months, seventy-one
30-day and sixty-four 29-day months, that is, seven more of the 30-day
months than of those of 29 days, showing that actually three of the
sixty-seven 29-day months expected in a normal repetition have become
30-day months. This is caused by the occurrence of two 30-day months in
succession at the end of one series and the beginning of the next. If
the 135 months in each third are numbered in succession it will be seen
that in the first 47-month group and in the 41-month group, the 30-day
months are the odd numbers. In the second 47-month group they are the
even numbers, of which there is one more in this division than odd
numbers, thus accounting for the additional one of the three days.
If the period of 3986 days were considered by itself, the arrangement
given would be sufficient. As soon, however, as this period is repeated
a number of times an error develops, since 135 synodical revolutions of
the moon are completed in 3986.63 days. Twice this number gives 7973.26,
or 1.26 days more than twice 3986. In order to keep the sequence of
months in the arrangement given above in accordance with the moon, it
becomes necessary to intercalate one more day every two repetitions of
the 3986 period. This may be done by changing the last 29-day month in
the 41-month group to a 30-day month, making the last 177-day group in
the third one of 178 days. The Mayas certainly did this in the first
third of the series given and arranged for it in the last third in a
manner which will be demonstrated later.
Tabulating the solution here advanced will form Table XI (p. 25), in
which the 30-and 29-day months in one-third of the manuscript have been
arranged in three columns, the first two of which represent the 47-month
groups and the last one the 41-month group. Before the first column of
months are numbers to facilitate locating any given month in the group.
The two kinds of months occur in direct alternation in each group, with
three exceptions. The 32d month in both of the 47-month groups is one of
30 days instead of 29, because of the addition of the intercalary day.
The 40th month in the 41-month group is given as one of 29 days with a
30 in parentheses before it, representing the fact that every other
third an extra intercalary day should be inserted in this month. To the
right of the month columns are three columns giving the difference
groups as found in the manuscript (see Table II), each column giving
those numbers found in each third of the manuscript, the first third
being the left one of the three. It should be noticed that the misplaced
(?) 148-day group in the last third does not interfere with the sequence
of the months.
Finally it only remains to review the irregularities of the manuscript
in the light of the solution just advanced. Those irregularities which
are corrected immediately afterwards, or are at variance with the rest
of the column in which they occur, are, in all probability, errors on
the part of the writer of the series, such as might have been caused by
careless transcription from another copy of the table, and correction in
the following column to avoid the task of erasure. Eliminating these
irregularities, there remain three to be investigated, namely, (1) the
absence of the 178 numbers in the lower number series, with one
conspicuous exception; (2) the occurrence of 178 in column 7 instead of
6, and of 148 in column 58 instead of 59; and (3) the discrepancies in
the totals of the series.