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.

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.

Table XI.—THE ARRANGEMENT OF LUNAR MONTHS
IN THE DRESDEN TABLE
No. ofDays inDresden groupsDays inDresden groupsDays inDresden groups
monthmonth123month 123month123
130 30 30
229 29 29
33017717717730 17717717730177177177
42929 29
530 30 30
629 29 29
730 30 30
829 29 29
93017717717730 17717717730177177177
102929 29
1130 30 30
1229 29 29
1330 30 30
1429 29 29
153014814814830 17717717730148148148
1629 29 29
1730 30 30
1829 29 29
1930 30 30
202917717717729 14829177177177
213030177177 30
2229 29 29
2330 30 30
2429 29 29
2530 30 30
262917717717729 17729177177177
273030148148 30
2829 29 29
2930 30 30
3029 29 29
3130 30 30
*323017817817830 17817817829177177177
333030 30
3429 29 29
3530 30 30
3629 29 29
3730 30 30
382917717717729 17717717729178177

177

393030 30
4029 29 (30) 29 (178)
4130 30 -30
4229 29
4330 30
442917717717729 177177177
453030
4629 29
4730 30