Part 1

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.

[1] For those unacquainted with Maya arithmetic the following points will explain matters: the Mayas used the vigesimal system of enumeration; they counted by twenties instead of tens. A bar represented five, and a dot stood for one. They represented numbers larger than twenty by position, just as we do. However, instead of having the smallest denomination at the right and the largest at the left of a horizontal series of figures, they had the smallest at the bottom and the largest at the top of a column of numbers. Instead of each unit in a given position representing ten times the value of that of the preceding position, it represented twenty times the value, except in the third position where it was only eighteen times as great. Thus each unit of the bottom number represented one (Kin), that of the number above it twenty (Uinal), that of the third number 20 × 18 or 360 (Tun), that of the fourth position 20 × 360 or 7200 (Katun), etc. For ease in handling, these numbers are written in our script with arabic numerals, the bottom number on the right, and separated by periods. Thus in column three, page 53a, the upper number is 1. 7. 2, which means that the kin of this group is 2, the Uinal 7, (7 × 20) and the Tun 1 (1 × 360), making in all 2 + 140 + 360 or 502.

The Maya calendar, like ours, consisted of a series of numbers and a series of names for each day, each series repeating itself constantly, irrespective of the other. There were twenty different day names, which remained in an unchangeable order, and thirteen numbers. In the pages under discussion these day names appear as glyphs preceded by the necessary number.

For further details consult S. G. Morley, _An Introduction to Maya Hieroglyphs_, Bulletin 57, Bureau of American Ethnology, Washington, D. C., 1915, and C. P. Bowditch, 1910.

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]

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

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.

[3] By Dr. Förstemann, Dr. Thomas, and Mr. Bowditch.

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. 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 Corrected Uncorrected and upper upper Lower Uncorrected Corrected Column number number number day signs 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) 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) 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]

[4] Förstemann, 1901, p. 123.

[5] Bowditch, 1910, p. 217.

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.