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The Project Gutenberg EBook of A Possible Solution of the Number Series on Pages 51 to 58 of the Dresden Codex, by Carl E. Guthe This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A Possible Solution of the Number Series on Pages 51 to 58 of the Dresden Codex Author: Carl E. Guthe Release Date: March 16, 2013 [EBook #42346] Language: English Character set encoding: UTF-8 *** START OF THIS PROJECT GUTENBERG EBOOK SOLUTION PAGES 51-58 DRESDEN CODEX *** Produced by Julia Miller, Fred Salzer and the Online Distributed Proofreading Team at http://www.pgdp.net Cover Please see Transcriber’s notes at the end of this document. PAPERS OF THE PEABODY MUSEUM OF AMERICAN ARCHAEOLOGY AND ETHNOLOGY, HARVARD UNIVERSITY Vol. VI.—No. 2 A POSSIBLE SOLUTION OF THE NUMBER SERIES ON PAGES 51 TO 58 OF THE DRESDEN CODEX BY CARL E. GUTHE Cambridge, Mass. Published by the Museum 1921 COPYRIGHT, 1921 BY THE PEABODY MUSEUM OF AMERICAN ARCHAEOLOGY AND ETHNOLOGY, HARVARD UNIVERSITY NOTE The solution set forth in this paper formed a part of a thesis for the degree of Doctor of Philosophy, taken in Anthropology at Harvard University. Thanks are due Dr. A. M. Tozzer of Harvard University, and Mr. S. G. Morley of The Carnegie Institution of Washington, for the interest they have taken in my work and the valuable aid they have given. I must thank especially Professor R. W. Willson of the Astronomical Department of Harvard for the inestimable help and unfailing patience he has contributed to my work. There are many statements throughout this paper which are the direct result of his teachings. I wish also to thank Mr. Charles P. Bowditch, without whose inspiration and aid this work would not have been accomplished. Carl E. Guthe Andover, Massachusetts July, 1919 A POSSIBLE SOLUTION OF THE NUMBER SERIES ON PAGES 51 TO 58 OF THE DRESDEN CODEX DESCRIPTION In the Dresden Codex, one of the three Maya manuscripts in existence, there is found a series of numbers covering eight pages, 51 to 58 (plate I). As early as 1886, Dr. Förstemann recognized this series as an important one, and one which probably referred to the moon in some way. Each page is divided into an upper and a lower half designated, respectively, “a” and “b.” Pages 51a and 52a form a unit in themselves, but are clearly associated with the remaining pages. The probable meaning of this group is still so doubtful that it has been deemed best to omit entirely a discussion of it at the present time. The remaining sections of these pages form one long series of numbers which should be read from left to right, beginning at 53a, reading to 58a, continuing on 51b, and ending the series at 58b. Each half-page is divided, horizontally, into four sections. The upper section consists of two rows of hieroglyphs. The section just below it contains a series of black numbers which increase in value from left to right. The third section consists of three rows of day glyphs with red numbers attached to them. The interval between the glyphs in successive rows can, of course, be mathematically obtained. The last, and bottom, division of the page is filled with a series of black numbers which are of three values only, namely, 177, 148, and 178, of which the first is the most frequent. At more or less regular intervals a vertical strip is run from the top of the half-page to the bottom. This strip contains, in the upper part, eight or ten glyphs. Below them in all but the first strip is a constellation band, and below that a figure of some kind. These strips divide the number series into groups, and are called “pictures,” occurring on ten of the fourteen half-pages. Considered vertically the pages are composed of columns. Each column contains, beginning at the top, two hieroglyphs, a long number, three day glyphs, and their numbers, and finally, at the bottom, a short number. The pictures occur between these columns. The series covers a period of 11,960 days, although the last number recorded in the upper series is only 11,958. By means of the columns this period of 11,960 days is divided into 69 unequal parts. Let columns 2, 3, and 4 on page 54b be taken as examples. Then each column in the series should be read in the following manner:[1] The lower number of column 3 is 8.17 or 177. Add this number to the upper number of column 2, which is 1. 2. 11. 9 or 8149. The result is 8326 which is expressed correctly as 1. 3. 2. 6 in the upper number of column 3. The lower number should also be added to the upper day glyph of column 2, which is 10 Caban, giving 5 Ix, which is the day glyph and number appearing as the first in column 3. The second day glyph and number is that of the day following 5 Ix, namely 6 Men. Similarly, 7 Cib is the day after 6 Men. Going through the same process for column 4, 148, that is, the lower number 7. 8 of that column, should be added to 8326 to obtain 8474, which is expressed in the upper number of column 4 as 1. 3. 9. 14. Likewise, 148 days after 5 Ix comes 10 Ik, which is the upper day glyph of column 4, and below which are found the two days immediately following, namely, 11 Akbal and 12 Kan. In short, then, the ideal arrangement of the series is as follows: Each upper number is the sum of all the lower numbers of the preceding columns and its own column. Each lower number expresses the difference between the upper number of its own column and that of the column immediately preceding it. The day names and numbers are three horizontal series, each starting a day later than the one above it, and recording three sets of day names and numbers which would fit the series formed by the upper numbers. It should be noticed that the mathematical interpretation of the series does not appear to depend in any way upon the hieroglyphs appearing at the top of the columns, or upon the pictures. This series deals quite clearly with synodical revolutions of the moon. The entire series records 11,960 days, although the last number in the upper series is only 11,958, a condition that will be explained later. Four hundred and five synodical revolutions of the moon consume, according to modern astronomy, 11,959.889 days, or about .11 of a day less than the length of the series. Moreover, the difference groups 148, 177, and 178 which separate the upper numbers, also record synodical months, for five months consume 147.65 days, and six months 177.18 days. In fact the correspondence between the numbers in the series and the synodical months is so exact, that nowhere does an error of more than one day exist.[2] Unfortunately the ideal arrangement given above is not followed exactly. The actual series as it occurs in the manuscript appears to be full of errors, a list of which will be found in Table I, p. 4. Most of these errors have been pointed out and discussed repeatedly.[3] There still exists some doubt as to which numbers should be considered errors of the original writer and which should be taken at their face value. For this reason the errors are here discussed in some detail, for in some cases the errors, or supposed errors, affect theories in regard to the series. Table I APPARENT IRREGULARITIES Lower number series: Absence of all 178’s that occur in upper series. Column 23. Presence of 178. Column 26. 177 instead of 148. Column 50. 157 “ “ 177. Upper number series: Column 1. 157 instead of 177. [1] [2] [3] Column 2. 353 “ “ 354. Column 4. 674 “ “ 679. Column 10. 1748 “ “ 1742. Column 12. 2016 “ “ 2096. Column 14. 3142 “ “ 2422. Column 15. 2598 “ “ 2599. Column 24. 4164 “ “ 4163. Day series: Column 5. 4 Chicchan instead of 11 Chicchan. Column 11. Omission of 1/2 tonalamatl. Column 17. 1 Ik instead of 2 Ik. Column 36. 4 Ben instead of 4 Ahau. Column 47. 10 Eznab instead of 11 Eznab. Column 49. 11 Kan instead of 12 Kan. Columns: Columns 6 and 7 are reversed. Columns 58 and 59 are reversed. Totals: Upper number series totals 11,958 instead of 11,960. Day series totals 11,959 instead of 11,960. In Table II, pp. 6, 7, both the corrected and the uncorrected series are given. In the centre of the table are three columns containing the actual table. The third column contains the uncorrected upper number; the fourth the lower number; and the fifth the first day sign and its number. Since the other two day series agree, except in a very few cases which will be mentioned later, with the first series, they have been omitted from the table. The sixth column contains the day signs as they probably should occur, and the second contains the corrected upper number. The first column gives the pages of the manuscript and the number of the columns on each in order to facilitate reference to the manuscript. Each column of Table II, with the exception of the first and fourth, is composed of two series of numbers, since each interval between the numbers of the manuscript has been placed in parentheses after the last of the pair of numbers it deals with, in order to facilitate comparison with the lower numbers. The names and numbers in the fifth column which have parentheses have been obliterated in the manuscript, but are easily inferred from the other two rows of day signs and numbers. The most prominent irregularity is the absence of the number 178 in the lower numbers when the differences in both the day series and the upper numbers show that 178 should be the difference. This occurs in columns 7, 14, 29, 37, 52 and 60 of the manuscript. The only place in which 178 does occur in the lower number is in column 23, when it agrees with the difference in the day series, but not with that of the upper number. In other words, the six occurrences of the 178- day group in the upper numbers are neglected in the lower numbers, and the only occurrence of 178 in the lower numbers does not agree with the upper numbers. This implies that it is of deeper significance than a mere error. There is another disagreement between the upper and lower numbers which could very well be the result of carelessness. In column 26, the lower number is 177, while both the upper number and the day series give a difference of 148. This is the only case in which the differences of 148 are not found at the same place in all series, and, consequently, is probably an error of the scribe. Again in the lower number of column 50, the careless omission of one dot in the Uinal place has resulted in the record of 157 instead of the correct number, 177. With one exception all of the errors in the upper numbers occur in the first third of the series. That exception, i.e., the writing of 4164 for 4163 in the column 24, may be explained by the fact that the writer of the series had just added in column 23 the extra day to the day series, which threw it out of agreement with the upper numbers. For the moment this fact slipped his mind, but he corrected the mistake by subtracting one day from the difference between the upper numbers of columns 24 and 25. Table II (1) (2) (3) (4) (5) (6) Page and Column Corrected upper number Uncorrected upper number Lower number Uncorrected day signs Corrected day signs 0 11 Manik 53a 1 177 (177) 157 177 6 Kan 6 Kan (177) 2 354 (177) 353 (196) 177 1 Imix (177) 1 Imix (177) 3 502 (148) 502 (149) 148 6 Muluc (148) 6 Muluc (148) Picture 4 679 (177) 674 (172) 177 1 Cimi (177) 1 Cimi (177) 5 856 (177) 856 (182) 177 9 Akbal (177) 9 Akbal (177) 6 1034 (178) 1033 (177) 177 4 (Ahau) (177) 5 Imix (178) [4] [5] [6] 54a 7 1211 (177) 1211 (178) 177 (13) Enzab (178) 13 Enzab (177) 8 1388 (177) 1388 (177) 177 8 Men (177) 8 Men (177) 9 1565 (177) 1565 (177) 177 3 Eb (177) 3 Eb (177) 10 1742 (177) 1748 (183) 177 11 Muluc (177) 11 Muluc (177) 11 1919 (177) 1919 (171) 177 6 Cib ( 47) 6 Cimi (177) 12 2096 (177) 2016 ( 97) 177 1 Akbal (307) 1 Akbal (177) 13 2244 (148) 2244 (288) 148 6 Chuen (148) 6 Chuen (148) 55a Picture 14 2422 (178) 3142 (898) 177 2 Muluc (178) 2 Muluc (178) 15 2599 (177) 2598 (-544) 177 10 Cimi (177) 10 Cimi (177) 16 2776 (177) 2776 (178) 177 5 Akbal (177) 5 Akbal (177) 17 2953 (177) 2953 (177) 177 13 Ahau (177) 13 Ahau (177) 18 3130 (177) 3130 (177) 177 8 ? ? 8 Caban (177) 56a 19 3278 (148) 3278 (148) 148 ? Chicchan ? 13 Chicchan (148) Picture 20 3455 (177) 3455 (177) 177 8 Ik (177) 8 Ik (177) 21 3632 (177) 3632 (177) 177 3 Cauac (177) 3 Cauac (177) 22 3809 (177) 3809 (177) 177 11 Cib (177) 11 Cib (177) 57a 23 3986 (177) 3986 (177) 178 7 Ix (178) 7 Ix (178) 24 4163 (177) 4164 (178) 177 2 Chuen (177) 2 Chuen (177) 25 4340 (177) 4340 (176) 177 10 Lamat (177) 10 Lamat (177) 26 4488 (148) 4488 (148) 177 2 Cib (148) 2 Cib (148) Picture 58a 27 4665 (177) ? ? 177 10 Ben (177) 10 Ben (177) 28 4842 (177) 4842 ? 177 5 Oc (177) 5 Oc (177) 29 5020 (178) 5020 (178) 177 1 Lamat (178) 1 Lamat (178) 30 5197 (177) 5197 (177) 177 9 Chicchan (177) 9 Chicchan (177) 51b 31 5374 (177) 5374 (177) 177 4 Ik (177) 4 Ik (177) 32 5551 (177) 5551 (177) 177 12 Cauac (177) 12 Cauac (177) 33 5728 (177) 5728 (177) 177 7 Cib (177) 7 Cib (177) 34 5905 (177) 5905 (177) 177 2 Ben (177) 2 Ben (177) 35 6082 (177) 6082 (177) 177 10 Oc (177) 10 Oc (177) 36 6230 (148) 6230 (148) 148 2 Enzab (148) 2 Enzab (148) 52b Picture 37 6408 (178) 6408 (178) 177 11 Cib (178) 11 Cib (178) 38 6585 (177) 6585 (177) 177 6 Ben (177) 6 Ben (177) 39 6762 (177) 6762 (177) 177 1 Oc (177) 1 Oc (177) 40 6939 (177) 6939 (177) 177 9 Manik (177) 9 Manik (177) 53b 41 7116 (177) 7116 (177) 177 4 Kan (177) 4 Kan (177) 42 7264 (148) 7264 (148) 148 9 Eb (148) 9 Eb (148) Picture 43 7441 (177) 7441 (177) 177 4 Muluc (177) 4 Muluc (177) 44 7618 (177) 7618 (177) 177 12 Cimi (177) 12 Cimi (177) 45 7795 (177) 7795 (177) 177 7 Akbal (177) 7 Akbal (177) 54b 46 7972 (177) 7972 (177) 177 2 Ahau (177) 2 Ahau (177) 47 8149 (177) 8149 (177) 177 10 Caban (177) 10 Caban (177) 48 8326 (177) 8326 (177) 177 5 Ix (177) 5 Ix (177) 49 8474 (148) 8474 (148) 148 10 Ik (148) 10 Ik (148) Picture 50 8651 (177) 8651 (177) 157 5 Cauac (177) 5 Cauac (177) 55b 51 8828 (177) 8828 (177) 177 13 Cib (177) 13 Cib (177) 52 9006 (178) 9006 (178) 177 9 Ix (178) 9 Ix (178) 53 9183 (177) 9183 (177) 177 4 Chuen (177) 4 Chuen (177) 54 9360 (177) 9360 (177) 177 12 Lamat (177) 12 Lamat (177) 55 9537 (177) 9537 (177) 177 7 Chicchan (177) 7 Chicchan (177) 56 9714 (177) 9714 (177) 177 2 Ik (177) 2 Ik (177) 57 9891 (177) 9891 (177) 177 10 Cauac (177) 10 Cauac (177) 58 10039 (148) 10039 (148) 148 2 Manik (148) 2 Manik (148) 56b Picture 59 10216 (177) 10216 (177) 177 10 Kan (177) 10 Kan (177) 60 10394 (178) 10394 (178) 177 6 Ik (178) 6 Ik (178) 61 10571 (177) 10571 (177) 177 1 Cauac (177) 1 Cauac (177) 62 10748 (177) 10748 (177) 177 9 Cib (177) 9 Cib (177) 57b 63 10925 (177) 10925 (177) 177 4 Ben (177) 4 Ben (177) 64 11102 (177) 11102 (177) 177 12 Oc (177) 12 Oc (177) [7] 65 11250 (148) 11250 (148) 148 4 Eznab (148) 4 Eznab (148) Picture 66 11427 (177) 11427 (177) 177 12 Men (177) 12 Men (177) 67 11604 (177) 11604 (177) 177 7 Eb (177) 7 Eb (177) 58b 68 11781 (177) 11781 (177) 177 2 Muluc (177) 2 Muluc (177) 69 11958 (177) 11958 (177) 177 10 Cimi (177) 10 Cimi (177) Picture The apparent error due to the addition of two dots in the Tun place in the upper number of column 14 is more the result of an error than an error in itself. This number shows a very clear case of erasure. The writer of this section of the manuscript in copying from the older source, at first overlooked column 14, and placed 7.3.18, the upper number in column 15 in this place. Realizing his mistake he erased the three dots in the Uinal place, but utilized two of the bars and the three dots in the Kin place as the 13 needed in the Uinal in column 14, and erased the lower bar of the original 18. This procedure of the writer’s threw the upper number of column 14 out of alignment, for the two dots of the Kin appear below the 13, somewhat below the line of Kins of the other columns. The seven in the Tun place should have been a six, so the scribe inserted an extra dot between the two of the original 7, neglecting, however, to erase the other two dots. As a result the upper number of column 14 records the number 3142, which is 720 greater than it should be, namely, 2422. In column 10, 1748 was recorded instead of 1742, for a bar and three dots were written in the Kin place instead of only two dots. This is a very peculiar and unexpected form of carelessness, which is, however, corrected in the next column. The remaining irregularities in the upper numbers are all due to the omission of a part of the number. In columns 2 and 15, one dot was omitted in the Kin place, thus recording 353 instead of 354 in the former, and 2598 instead of 2599 in the latter. In column 4 one bar was omitted in the Kin place, making the number 674, five less than it should be, namely, 679. In column 1, one dot was omitted in the Uinal place, and in column 12, four dots of the same denomination, recording, respectively, 157 and 2016 instead of 177 and 2096. There is only one decided error in the day series. In column 11, 6 Cib, 7 Caban, 8 Eznab were written instead of 6 Cimi, 7 Manik, 8 Lamat. It should be noticed that the number of the day was right. In fact just one-half a tonalamatl, or 130 days, was dropped before the day series of column 11, and added on again immediately afterwards. This is an extremely curious error to make in calculating and may shed some light on the way in which the Mayas reckoned. The five remaining irregularities in the day series are of two kinds. In column 5, the number preceding the third day, Chicchan, is 4 instead of 11. Apparently the writer of the manuscript forgot for the moment that the day was added to the one above it and not the one to the left, and wrote 4 because the number associated with the third day sign of column 4 was 3. The same mistake was made in the third day of column 36, except that in this case it was the day sign and not the number which was confused. Here, instead of writing Ahau, which followed the Cauac in the second series, Ben was recorded because the sign to the left was Eb. The other three irregularities are all due to carelessness in placing sufficient dots in the number associated with the day sign, for in columns 17, 47 and 49, 1 Ik, 10 Eznab and 11 Kan were recorded, respectively, instead of the necessary 2 Ik, 11 Eznab and 12 Kan. There are two places in which columns seem to be misplaced, although the mathematics of the series at these points is correct as it stands. For the sake of uniformity in the arrangement of the difference groups, the 178-day group of column 7 should occur in column 6, and for the same reason, the 148-day group of column 58 should occur in column 59. Professor Förstemann calls both of these variations errors, and arranges his version of the table so that each part is just like the other two. He gives no reason for his opinion other than the phrase “for the author [of the manuscript] had confused the differences 178 and 148....”[4] Mr. Bowditch, on the other hand, allows both of the variations to stand as they appear in the manuscript, and quite rightly holds the opinion that, “It may possibly be that these numbers thus placed are errors of the scribe, but the mere plea for uniformity is not sufficient to lead us to make these changes.”[5] In Table II the apparent mistake in columns 58 and 59 remains as it occurs in the manuscript, for the reason which Mr. Bowditch gives. In the case of columns 6 and 7 there seems to be some evidence that there actually was an error made. The last column on page 53a, which is the one under discussion, contains no day glyph in the first day series. The glyph should have been that of Ahau. There is distinct evidence, altho very faint, that a glyph was once there. Moreover, the smooth coating which covered the material of the manuscript page is not broken. There are other obliterated glyphs in these pages of the manuscript, but few in which the surface, although unbroken, still contains a faint, almost continuous outline of a glyph. The glyph, then, was probably erased. The writer of the manuscript had probably completely finished column 6 and started column 7 before he detected the error. He began to erase the part that was wrong, then realized what an amount of alteration would be necessary, and finally compromised by making the difference come between columns 6 and 7 instead of columns 5 and 6. This hypothesis in regard to the manner in which the erasure was done may be wrong, but the erasure still stands as a strong evidence to show that the 178 should have occurred in column 6 rather than after it. 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 [8] [9] [10] 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 502 1742 1034 3986 1210 1742 1034 3986 1210 1565 1211 708 3986 (502) Table IV UPPER NUMBERS OF 178-DAY GROUPS Number Difference Group (1564) 1211 3986 1211 2598 3986 1388 2598 3986 1388 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. 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. 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 [11] (0) 502 2244 3278 4488 6230 7264 8474 10039 11250 (11958) (0) 1211 2422 5020 6408 9006 10394 (11958) [12] this column (10 Cimi), we must add two days to complete the series, which ends with 12 Lamat.”[8] 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. 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. 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. 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. 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] 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. 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] Professor R. W. Willson of the Astronomical Department of Harvard University, working on a similar theory at about [13] [14] [15] 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. 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] 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. [16] [17] 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 29½ 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. 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- [18] [19] 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 VIII Number of month Number of days in month Elapsed days calendar month Elapsed days synodical month Error 1 30 30 29.53 -0.47 2 29 59 59.06 0.06 3 30 89 88.59 -0.41 4 29 118 118.12 0.12 5 30 148 147.65 -0.35 6 29 177 177.18 0.18 7 30 207 206.71 -0.29 8 29 236 236.24 0.24 9 30 266 265.78 -0.22 10 29 295 295.31 0.31 11 30 325 324.84 -0.16 12 29 354 354.37 0.37 13 30 384 383.90 -0.10 14 29 413 413.43 0.43 15 30 443 442.96 -0.04 16 29 472 472.49 0.49 17 30 502 502.02 0.02 18 29 531 531.55 0.55 19 30 561 561.08 0.08 20 29 590 590.61 0.61 21 30 620 620.14 0.14 22 29 649 649.67 0.67 23 30 679 679.20 0.20 24 29 708 708.73 0.73 25 30 738 738.26 0.26 26 29 767 767.80 0.8 27 30 797 797.33 0.33 28 29 826 826.86 0.86 29 30 856 856.39 0.39 30 29 885 885.92 0.92 31 30 915 915.45 0.45 32 29 944 944.98 0.98 33 30 974 974.51 0.51 34 29 1003 1004.04 1.04 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 -0.959 16 2772.961 94 2775.875 2.914 17 2946.271 100 2953.059 6.788 18 3119.581 106 3130.243 10.662 [20] 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 0.499 Table X 148-DAY GROUPS Upper number Month number Interval Groups 502 17 2244 76 94 (47 + 47) 3278 111 4488 152 6230 211 94 (47 + 47) 7264 246 8474 287 10039 340 94 (47 + 47) 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. 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. 59 35 41 41 59 35 41 41 53 41 [21] [22]