Part 3

This series of 135 lunar months, or 23 eclipse seasons, can be repeated almost indefinitely, alternating 3986 and 3987 days to the series and still keep the synodical month in accord with the eclipse season. But another factor must also be considered. Months of 29 and 30 days cannot be simply alternated and either conform with the true synodical month or complete the ecliptic series mentioned, for 3986 contains three more days than sixty-eight 30-day months, and sixty-seven 29-day months. Therefore in the 3986 series three of the 29-day months must be changed to 30-day months, and in the 3987 series four must be changed. The position of these changes is arbitrary. They can, for example, be the 34th, 68th, and 102d months, and when necessary, the 134th.

TABLE IX

COMPARISON OF SYNODIC MONTHS AND ECLIPSES

Eclipse season Synodic month Number Days Number Days Difference 1 173.310 6 177.184 3.874 2 346.620 12 354.367 7.747 3 519.930 18 531.551 11.621 4 693.240 23 679.204 -14.036 5 866.550 29 856.387 -10.163 6 1039.860 35 1033.571 -6.289 7 1213.170 41 1210.754 -2.416 8 1386.480 47 1387.938 1.458 9 1559.790 53 1565.121 5.331 10 1733.100 59 1742.305 9.205 11 1906.411 65 1919.489 13.078 12 2079.721 70 2067.141 -12.580 13 2253.031 76 2244.325 -8.706 14 2426.341 82 2421.508 -4.833 15 2599.651 88 2598.692 -.959 16 2772.961 94 2775.875 2.914 17 2946.271 100 2953.059 6.788 18 3119.581 106 3130.243 10.662 19 3292.891 112 3307.426 14.535 20 3466.201 117 3455.079 -11.122 21 3639.511 123 3632.263 -7.248 22 3812.821 129 3809.446 -3.375 23 3986.131 135 3986.630 .499

TABLE X

148-DAY GROUPS

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.

TABLE XI.—THE ARRANGEMENT OF LUNAR MONTHS IN THE DRESDEN TABLE

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.

CONCLUSION

On pages 51 to 58 of the Dresden Codex occurs a series of numbers, running continuously through all the pages except the upper halves of the first two. This series records a period of 11,960 days, divided by means of columns into sixty-nine unequal subdivisions, of 177, 148, and 178 days, of which the first is the most frequent.

There are three distinct series. One series of numbers is in the upper part of the record, and consists of totals increased step by step until the final total reached records 11,958 days. Just below this series are three series of day signs and numbers, the middle one of which is the actual series. These dates are separated by the same number of days as the upper number series, except in the 23d group, at which place one extra day is added to the day series and not to the upper number series, causing the former to be in advance of the latter one day throughout the remainder of the record. At the end of the day series another extra day is added by counting in the last day in the lower row of days, thus completing the 11,960-day period.

Below this day series is another number series no term of which exceeds 178. In a general way it records the differences between the dates appearing above each of its numbers. The agreement is however so inaccurate that this lower number series could, at best, have been used only as a general guide to the user of the manuscript, in that it calls attention to the intervals of unusual length.

The series recorded is composed of three equal parts, each composed of 23 subdivisions and covering 3986 days.