Read Ebook: A Possible Solution of the Number Series on Pages 51 to 58 of the Dresden Codex by Guthe Carl E Carl Eugen

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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.

UPPER NUMBERS OF 148-DAY GROUPS

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

TABLE IV

UPPER NUMBERS OF 178-DAY GROUPS

Number Difference Group | 1211 | 3986 1211 | 1211 / 2422 2598 5020 | 3986 1388 / 6408 2598 9006 | 3986 1388 / 10394 1564

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.

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. F?rstemann, 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. 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. 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.

F?rstemann, 1886, p. 34.

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 F?rstemann. 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 , we must add two days to complete the series, which ends with 12 Lamat."

Thomas, 1888, p, 325.

F?rstemann, 1898.

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 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." 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.

F?rstemann, 1901, p. 121.

Dr. F?rstemann 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. 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.

Ibid., p. 123.

This detailed discussion by Dr. F?rstemann 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, discussed these pages and their relation to the astronomical knowledge of the Mayas. He divided the series into the same groups as Dr. F?rstemann, basing his division upon the pictures which occur in every case immediately after the number 148. 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. 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, and then discussed the relation of this series to the Saturn and Mercury periods, disagreeing with F?rstemann on several points.

Bowditch, 1910, pp. 211-231.

Ibid., p. 218.

Bowditch, 1910, pp. 222, 223.

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. F?rstemann'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. "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."

Ibid., pp. 229, 230.

Ibid., p. 231.

Dr. F?rstemann 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. 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." 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," 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.

Meinshausen, 1913, pp. 221-227.

Ibid., p. 225.

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."

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. This apparently invalidates Meinshausen's theory.

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 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.

Schram, 1908, pp. 358, 359.

This remarkable agreement between the 178-day groups in the Dresden and the occurrences of eclipses may have several meanings. 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. It may be that the numbers refer to dates of prophesied eclipses which the Mayas had learned occurred at more or less regular intervals. Since this table has a place in the calendar of the Mayas , 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

Number Month

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. 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.

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

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