Part 2

Finally there appears to be an error in the totals of the series, for the upper number series records as a total 11,958 days and the day series 11,959 days, although there is strong reason for believing that the series should record 11,960 days. This discrepancy in the totals will be referred to again.

In general, then, the apparent irregularities in the manuscript fall into two great classes, those which are corrected in the next column or are easily detected because of their disagreement with the record in the other two series, and those which are not obviously due to carelessness. The latter will be considered under the solutions. The former may be dismissed as clerical errors not affecting the solution. In this group are two of the irregularities in the lower numbers (columns 26 and 50), and all eight in the upper numbers, seven of which occur in the first third of the manuscript. The six errors in the day series, and the transposition of columns 6 and 7, also belong in this class.

By referring to Table II it will be noticed that the pictures occur after the 148-day groups in each case. The upper numbers immediately preceding the pictures are given in Table III (p. 11), together with the differences between them. By grouping these differences, it becomes apparent that the pictures may be divided into three large groups of 3986 days; two out of the three containing the same difference numbers, 1742, 1034, and 1210. If, in the last group, the number 10,039 were changed to 10,216 by adding 177, the differences for this group would also read as the others, when the end of the series and the beginning of the series are added together (708 + 502 = 1210), for the 10th picture is, in a sense, out of the grouping since it occurs after the last number in the series. The 148-day groups are arranged in the same order for they occur in the same columns as the numbers used above.

By applying the same process to the 178-day groups, it is found that they also can be divided into groups which contain 3986 days. In this case the second and third groups contain the same numbers, 2598 and 1388 (Table IV). If the number 1211 in the first group is changed to 1034 by subtracting 177, the last number of this group would be 1388; and the first number 2598 could be formed by adding the remainder at both ends of the series (1564 + 1034 = 2598).

It should be remembered at this point that the only column in which the lower numbers contained 178 is column 23, of which the upper number is 3986. This gives further grounds for dividing the series as it stands into three parts of 3986 days, each containing 23 columns.

TABLE III

UPPER NUMBERS OF 148-DAY GROUPS

Number Difference Group (0) 502 502 1742 \ 2244 | 1034 | 3986 3278 | 1210 / 4488 1742 \ 6230 | 1034 | 3986 7264 | 1210 / 8474 1565 \ 10039 | 1211 | 11250 | 3986 708 | (11958) | (502) /

TABLE IV

UPPER NUMBERS OF 178-DAY GROUPS

Number Difference Group (1564) \ (0) | 1211 | 3986 1211 | 1211 / 2422 2598 \ 5020 | 3986 1388 / 6408 2598 \ 9006 | 3986 1388 / 10394 1564 (11958)

The three parts are not exactly alike, however, as has already been pointed out in considering the probable errors. If the upper numbers and day numbers in column 6 should be altered, so that the difference 178 might occur in that column instead of column 7, and if, by the same process, the difference of 148 could occur in column 59 instead of 58, then the three parts of the series would be entirely alike. The three facts mentioned are, however, very strong evidence for supposing that the people who used this table considered it as consisting of three equal parts.

This series in the Dresden is very similar to other pages of the Dresden and other manuscripts, two examples of which are given as illustrations. One of the most interesting parallels is the series on pages 46-50 of this same manuscript. This series covers a period of 2920 days which is divided into 20 unequal subdivisions. On page 24, which just precedes page 46, this number is used as a unit in multiplication, that is, the numbers occurring on page 24 are separated from each other by 2920 or multiples of this number. On pages 44b and 45b the number 78 is divided into four unequal parts, and on pages 43b and 44b it is used as a unit in a series which finally reaches the number 1940 × 78.

SOLUTIONS

The first references to these pages in the manuscript were concerned chiefly with the reading of the numbers without any theories in regard to the probable meaning of the series.

Dr. Förstemann, in 1886, was probably the first to mention these pages specifically. At this time he corrected many of the errors in the series, and related the rows of days to the number series.[6] He had already recognized a close relation between the difference between the 1st and 9th pictures, i.e., 10,748, and the Saturn sidereal period of 10,753 days. Of course, in order to do this he had also identified the various signs in the “constellation bands,” assigning them to various planets.[7] These identifications are based on little more than the wish he had that they might be those planets, and for that reason they are seriously open to doubt.

[6] Förstemann, 1886, p. 34.

[7] Ibid., pp. 68-71.

Cyrus Thomas, two years later, also discussed this series at some length, but confined his considerations entirely to the mathematical side of the work. He also pointed out most of the errors, agreeing in the main with Förstemann. He considered that the series contained 11,960 days. In his conclusion he said “the sum of the series as shown by the numbers over the second column of Plate 58b is 33 years, 3 months, and 18 days. As this includes only the top day of this column (10 Cimi), we must add two days to complete the series, which ends with 12 Lamat.”[8]

[8] Thomas, 1888, p, 325.

During the following years, Dr. Förstemann repeatedly referred to these pages in his publications and, in 1898, published an article devoted to these pages alone.[9] The most detailed as well as the final discussion of these pages is that given in his book on the Dresden Codex.[10] In pages 53-58, and 51b and 52b he recognizes the similarity to pages 46-50, and remarks that the Mayas not only combined the _tonalamatl_ and the Mercury year, but also attempted to bring the lunar revolution into accord with these two. In other words, Förstemann seems to imply that the primary purpose of the series was the counting of the Mercury years, and that the lunar part of the problem was secondary.

[9] Förstemann, 1898.

[10] Ibid., 1901, pp. 118-133.

He explains the number 11,958 as the result of attempts to make the lunar count agree with 11,960. “They [the Mayas] found that 405 lunar revolutions amounted approximately to 11,958 days, which is, in fact, the largest number on the second half page of page 58.”[11] This will not stand at all as the reason for the 11,958 since 405 lunar revolutions come to 11,959.889 days, and if the Mayas knew the revolutions accurately enough to know when to intercalate a day, they most certainly would not have intentionally formed the number 11,958, when they were perfectly well aware of the fact that the time was more than 11,959 days. He recognizes in the numbers 177, 148 and 178 multiples of lunar months of 29 and 30 days.

[11] Förstemann, 1901, p. 121.

Dr. Förstemann at this time divides the series into the three equal divisions in which it has since been considered. These are of 3986 days, thus causing the intercalated days to come at the same time in all three.[12] He also divided each of these three divisions into three unequal groups of 1742, 1034, and 1210 days each. He advances theories, based on the positions of the pictures in the series, to show that the series also referred to the siderial periods of Saturn and Jupiter, and discusses the meaning of the glyphs found on these pages.

[12] Ibid., p. 123.

This detailed discussion by Dr. Förstemann of pages 51-58 of the Dresden has been used as a foundation by many in further studies of these pages. It is highly probable, however, that a careful study of his interpretations will have to be made, in which the proved assumptions must be clearly differentiated from those in which the “wish is father to the thought.”

Mr. Bowditch, in 1910,[13] discussed these pages and their relation to the astronomical knowledge of the Mayas. He divided the series into the same groups as Dr. Förstemann, basing his division upon the pictures which occur in every case immediately after the number 148.[14] Mr. Bowditch brought out very clearly that this series is a lunar series, by means of a table which compares the numbers recorded in the manuscript and the multiples of true lunations.[15] There can be no question on this point, for the difference between the recorded days and the true lunations is never more than .9 of a day. He also pointed out a way in which this series could be used over and over again in the form of a cycle,[16] and then discussed the relation of this series to the Saturn and Mercury periods, disagreeing with Förstemann on several points.

[13] Bowditch, 1910, pp. 211-231.

[14] Ibid., p. 218.

[15] Bowditch, 1910, pp. 222, 223.

[16] Ibid., p. 224.

Mr. Bowditch also pointed out a peculiar coincidence between the synodical revolutions of Jupiter and the numbers in the series, but based his argument on quite different material from the similar theory of Dr. Förstemann’s. The important fact brought out is that the three parts of the series under discussion are almost exactly equal to 10 revolutions of Jupiter, for one revolution of Jupiter consumes 398.867 days.[17] “This would give a reason for the selection of 11,958 to 11,960 days or 405 revolutions, and for the division of this number into three sections of 3986 days each.”[18]

[17] Ibid., pp. 229, 230.

[18] Ibid., p. 231.

Dr. Förstemann and Mr. Bowditch differ in regard to some of the corrections which should be made in the manuscript, but on the whole the two discussions of these pages supplement one another. The general conclusion to be drawn from them is that these pages of the Dresden are closely associated with the synodical lunar month, and possibly, with the synodical revolution of Jupiter.

Three years after Mr. Bowditch’s discussion, Mr. Meinshausen published an article in which the relation of this series to eclipses was first brought out.[19] He compared, by means of two tables, recorded eclipses of the 18th and 19th centuries with the numbers in the Dresden Codex. Out of the 69 dates in the manuscript all but 15 dates agreed with the first case, and, in the second, all but 13, due to the fact that all the eclipses are not visible at one place on the earth’s surface. “Another indication that the numbers in the codex have arisen from the observation of eclipses lies in the fact that the exact grouping of the numbers which is induced by the insertion of pictures in the number periods is also possible in lunar eclipses which are visible at one particular point.”[20] In the table given to uphold this statement, the numbers, to be sure, can be grouped in the manner which he suggests; but they can also be grouped in other series. In his opinion the reason for the grouping “lies in the close proximity of a solar eclipse to a lunar eclipse,”[21] that is, that at the date at which the pictures are inserted a solar eclipse occurred 15 days either before or after a lunar eclipse. There are two facts which tend to uphold this theory. One is the occurrence of the sun and the moon in shields over nearly all pictures, which he interprets as “signs of solar and lunar eclipses”; the other is the series of dates on pages 51a and 52a, which are 15 days apart. In a table of recorded eclipses proof is given that such double eclipses can occur at the intervals which separate the pictures in the manuscript. Since these intervals vary a great deal, Meinshausen believes that they will form the means of identifying the specific eclipses recorded in the manuscript.

[19] Meinshausen, 1913, pp. 221-227.

[20] Ibid., p. 225.

[21] Meinshausen, 1913, p. 225.

His general conclusion is that “the material advanced will prove sufficiently that these numbers are associated in some way with solar and lunar eclipses, and this explanation must remain standing at least until other numbers, corresponding equally remarkably, are found.”[22]

[22] Ibid., pp. 226, 227.

Professor R. W. Willson of the Astronomical Department of Harvard University, working on a similar theory at about the same time, had found, however, that no series of solar eclipses corresponding to the intervals of the pictures in the text was visible in Yucatan between the Christian era and the time of the Spanish conquest.[23] This apparently invalidates Meinshausen’s theory.

[23] Professor Willson’s work on the Dresden manuscript has not yet been published. It is referred to here only through his kind permission.

Professor Willson believes that the table in the manuscript indicates the days of ecliptic conjunction (that is, New Moon occurring so near the moon’s node that eclipses _may_ occur) and, as Mr. Bowditch has shown, with a high degree of accuracy. Sufficient proof of this, in Professor Willson’s opinion, is the close correspondence of the intervals of the codex with the intervals of Schram’s lunar table.[24]

[24] Schram, 1908, pp. 358, 359.

The similarity between the numbers in the Dresden and Schram’s table is so remarkable that it seems advisable to point out some of the most outstanding features. In addition to giving the days of multiples of the lunar synodic months, this table also gives the time of possible occurrences of both solar and lunar eclipses. Eclipses occur in cycles, the best known of which is the Saros, although there are also smaller cycles which are not so accurate. Table V (p. 17) gives the occurrences of central solar eclipses according to Schram. It should be noticed that they occur in groups of threes and fours, each set being separated from the preceding one by 29 synodical months. The numbers in each group are only six months apart. Table VI (p. 17) is a corresponding series of lunar eclipses, which also occur in a grouping similar to that of the solar eclipses. It should be noticed in passing that the first numbers of these groups, in both the solar and lunar eclipses are separated by 47 and 41 lunations, the latter occurring after every third group in Table V.

Table VII (p. 17) contains the numbers which are in the same columns as the 178-day groups in the Dresden. By comparing Table V and Table VII, it will be found that the numbers in the Dresden are the same as the first numbers in groups 1, 2, 4, 5, 7 and 8 of the solar eclipses. In the last two numbers there is a difference of one day, which is explained by recalling the addition of an extra day in the day series but not in the upper numbers of the Dresden. If 679 days are added to each number in Table VII, which amounts to the same thing as advancing the Dresden table 679 days with respect to Schram’s table, it will be found that these numbers will also agree with the first numbers in groups 2, 3, 5, 6 and 8 and with the second number in group 9 of the lunar eclipses, in Table VI. A similar agreement may be observed for the 148-day groups (see Table III).

This remarkable agreement between the 178-day groups in the Dresden and the occurrences of eclipses may have several meanings. (1) One possibility, and one which should always be kept in mind, is that this agreement is simply another coincidence, of which there are always many in chronological work. (2) It may be that the numbers refer to dates of prophesied eclipses which the Mayas had learned occurred at more or less regular intervals. (3) Since this table has a place in the calendar of the Mayas (for a date probably occurs on page 52a), it may be that these numbers refer to definite historical eclipses. If they do, they will afford a means by which an absolute correlation between the Maya and the Julian calendars may be obtained. Professor Willson is at present working on this problem.

TABLE V

SOLAR ECLIPSES

Group Eclipse Month / 1034 35 1 | 1211 41 | 1388 47 \ 1565 53 / 2422 82 2 | 2599 88 | 2776 94 \ 2953 100 / 3632 123 3 | 3809 129 | 3987 135 \ 4164 141 / 5020 170 4 | 5197 176 | 5375 182 \ 5552 188 / 6408 217 5 | 6585 223 \ 6762 229 / 7619 258 6 | 7796 264 | 7973 270 \ 8150 276 / 9007 305 7 | 9184 311 | 9361 317 \ 9538 323 / 10395 352 8 | 10572 358 \ 10750 364 / 11606 393 9 | 11783 399 \ 11960 405

TABLE VI

LUNAR ECLIPSES

Group Eclipse Month / 502 17 1 | 679 23 \ 856 29 / 1713 58 2 | 1890 64 | 2067 70 \ 2244 76 / 3101 105 3 | 3278 111 \ 3455 117 / 4311 146 4 | 4489 152 | 4666 158 \ 4843 164 / 5699 193 5 | 5877 199 | 6054 205 \ 6231 211 / 7087 240 6 | 7264 246 \ 7442 252 / 8298 281 7 | 8475 287 | 8652 293 \ 8830 299 / 9686 328 8 | 9863 334 \ 10040 340 / 10896 369 9 | 11074 375 | 11251 381 \ 11428 387

TABLE VII

178-DAY GROUPS

Number Month

1034 35 2422 82 5020 170 6408 217 9006 305 10394 352

In order to determine the exact extent to which the eclipse seasons affect these pages in the Dresden Codex it is necessary to work out in as great detail as possible the calendar represented.

Modern astronomy shows that the synodical revolution of the moon consumes 29.53059 days, about .03 days more than 29-1/2 days. Since a calendar must be based on whole days the natural method of combining the months would be to alternate one of 29 days with one of 30 days. At the end of two months or 59 days the true synodical month would be in advance of the calendrical month by .06118 days. Every two months this error is doubled so that at the end of 34 months the calendar would have completed 1003 days and the synodical month 1004.04 days. (See Table VIII, p. 19.) One method of correcting this would be to make the last month a 30-day month instead of one of 29 days as it would be by simple alternation. This 34-month period could then be repeated as a cycle with an accumulating error of .04 days at every repetition.

Such a series utterly disregards, however, all other phenomena such as eclipses, seasons, etc. As soon as eclipses are considered the arrangement of the months must be altered in order to use the periodicity of eclipses in the calendar. Eclipses occur at regular seasons, approximately six months apart. The average interval between eclipse seasons is 173.310 days, 3.874 days less than six synodical lunar months. In Table IX (p. 20) the eclipse season is compared with the nearest synodical lunar month. It will be noticed that the difference increases between the two series until it is necessary to use five synodical months for one interval instead of six to keep the difference less than half a month. It is necessary to do this three times in 135 synodical months, or 3986.630 days, which exceed 23 eclipse seasons, or 3986.131 days, by practically one half-day. It would be most logical to drop these extra months out of the set of six, during that group in which the difference tends to become most nearly half a month. That would be just before the 23d, 70th, and 117th month, that is, 47 months apart, requiring 41 months to complete the 135-month period.

TABLE VIII

Number of Number of Elapsed days Elapsed days month days in month calendar month synodical month Error 1 30 30 29.53 -.47 2 29 59 59.06 .06 3 30 89 88.59 -.41 4 29 118 118.12 .12 5 30 148 147.65 -.35 6 29 177 177.18 .18 7 30 207 206.71 -.29 8 29 236 236.24 .24 9 30 266 265.78 -.22 10 29 295 295.31 .31 11 30 325 324.84 -.16 12 29 354 354.37 .37 13 30 384 383.90 -.10 14 29 413 413.43 .43 15 30 443 442.96 -.04 16 29 472 472.49 .49 17 30 502 502.02 .02 18 29 531 531.55 .55 19 30 561 561.08 .08 20 29 590 590.61 .61 21 30 620 620.14 .14 22 29 649 649.67 .67 23 30 679 679.20 .20 24 29 708 708.73 .73 25 30 738 738.26 .26 26 29 767 767.80 .80 27 30 797 797.33 .33 28 29 826 826.86 .86 29 30 856 856.39 .39 30 29 885 885.92 .92 31 30 915 915.45 .45 32 29 944 944.98 .98 33 30 974 974.51 .51 34 29 1003 1004.04 1.04