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Title: A Possible Solution of the Number Series on Pages 51 to 58 of the Dresden Codex
Author: Guthe, Carl E.
Language: English
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Copyright Status: Not copyrighted in the United States. If you live elsewhere check the laws of your country before downloading this ebook. See comments about copyright issues at end of book.

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  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.

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

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

TABLE IX

COMPARISON OF SYNODIC MONTHS AND ECLIPSES

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

TABLE X

148-DAY GROUPS

  Upper         Month
  number       number            Interval        Groups
    502           17
                                    59 \
   2244           76                   |         94(47 + 47)
                                    35 /
   3278          111
                                    41           41
   4488          152
                                    59 \
   6230          211                   |         94(47 + 47)
                                    35 /
   7264          246
                                    41           41
   8474          287
                                    53 \
  10039          340                   |         94(47 + 47)
                                    41 /
  11250          381

The next logical step is a comparison between the theoretical calendars
just described and the manuscript. A study of the manuscript reveals
that: (1) the series recorded represents 405 lunar months or three times
135 months, and that the series naturally falls into three great
subdivisions of 3986 days each; (2) each third consists of 23 columns
or unequal subdivisions; (3) the intervals between the 178-day groups
are 47 and 88 months; (4) the 148-day groups fall approximately at 47
and 41 month intervals (see Table X); (5) the first 178-day group in
each third occurs between the 30th and 35th month inclusive, and the
other 178-day group of the third comes 47 months later. Since the number
178 is composed of four 30-and two 29-day months, an extra day must have
been added, that is, a 30-day month was substituted for one of the
29-day months, if the manuscript represents a regular alternating
series.

The obvious conclusions to be drawn from these facts are: (1) that the
series was divided into three groups of 3986 days each in order to
associate the lunar calendar closely with the ecliptic cycle of the same
length; (2) that the 23 columns in each third may represent the
twenty-three eclipse seasons in each eclipse period of 3986 days; (3)
that groups of 47 and 41 months were used in some way in the series, for
the 178-day groups are separated by 47 and 88 months and 88 is composed
of 47 and 41, the two periods so closely associated with the recurrence
of eclipses; (4) that the six months period was changed to one of five
months of 148 days approximately every 47 and 41 months, which is the
method already advanced in the theoretical ecliptic lunar series for
keeping the synodical months and ecliptic seasons together; (5) that one
extra day was added to the alternating 29-and 30-day months, between the
30th and 35th month inclusive of each third, in accordance with the
theoretical necessity for so doing already brought out, and that another
of the three extra days was added 47 months later.

When the difference groups[25] are divided into months it is found that
it is an easy matter to arrange the months in an alternating series. The
group of 177 days is composed of three 30-and three 29-day months,
either of which when alternated can begin the group, which then ends
with the other, i.e., 29, 30, 29, 30, 29, 30, or 30, 29, 30, 29, 30, 29.
The group of 148 days consists of three 30-and two 29-day months,
necessitating that it begin and end with a 30-day month when alternated,
thus, 30, 29, 30, 29, 30. In the 178-day group one of the 29-day months
is replaced by a 30-day month, otherwise the group is exactly like that
of 177 days, which it exceeds by one day. It is evident that there will
always be three 30-day months in succession in the 178-day group, and
that care must be taken in choosing the right sequence of the 177-day
groups which fall near those of 148 days in order to avoid having two
30-day months in succession.

  [25] That is, the 177-day, 148-day and 178-day groups.

There remains simply the substitution of the six or five months, as the
case may be, in place of the difference groups in the manuscript.
However, if the Mayas considered each third of the table as a unit, it
is reasonable to assume that the sequence of the months in each third is
identical. Therefore it is necessary to arrange a sequence for only
one-third, that is, 135 months, and then, if the assumption is correct,
this sequence will fit the other two-thirds of the series.

Each third of the table consists of 135 months covering three more days
than would be covered by a simple alternation of 30-and 29-day months.
These three intercalary days were inserted at definite intervals. A clue
to the position of two of them is given by the 178-day groups. One was
inserted between the 30th and 35th months, another 47 months later,
between the 77th and 82d months. Theoretically the extra day should be
inserted in the 34th month after the beginning of a series of
alternating 29-and 30-day months, for then the error between the
synodical revolution of the moon and the calendrical months becomes more
than one day. In the 29-day month preceding the 34th, namely the 32d
month, the error at the end is also practically one day, i.e., .98 days.
The 29-day month most nearly the centre of the first 178-day group is
the 32d month of the series, the third in the group. The Mayas may have
chosen this month because of its position in the 178-day group, making
the sequence of the months 29, 30, 30, 30, 29, 30, if the 30th month was
a 29-day month as it would be by simple alternation.

The second time this intercalary day occurs in each third is 47 months
later. Obviously, this may be the recurrence of this intercalation in a
repetition of a smaller group of months than the 135-month group. If 47
months are subtracted from the 79th month which is the third in the
second 178-day group the result is 32, which implies that the smaller
division is 47 months. Two 47-month periods complete all but 41 of the
135 months in each third. Then, of necessity, if each third of the
manuscript is a unit, a 41-month group follows two 47-month groups, an
arrangement which also agrees with the eclipse groups in Tables V.

The two 178-day groups account for only two of the three intercalated
days, and since no 178-day group occurs in the 41-month division, the
addition of this day must have been accomplished in some more obscure
manner. Since both 47 and 41 are odd numbers, each group must contain at
least one more month of one kind than the other. Since two synodical
revolutions of the moon are slightly longer than two calendrical months
it is wisest to start and end each group with a 30-day month. If this is
done, the 47-month group will contain twenty-five 30-day and twenty-two
29-day months, and the 41-month group twenty-one 30-day and twenty
29-day months, making for the composition of the 135 months, seventy-one
30-day and sixty-four 29-day months, that is, seven more of the 30-day
months than of those of 29 days, showing that actually three of the
sixty-seven 29-day months expected in a normal repetition have become
30-day months. This is caused by the occurrence of two 30-day months in
succession at the end of one series and the beginning of the next. If
the 135 months in each third are numbered in succession it will be seen
that in the first 47-month group and in the 41-month group, the 30-day
months are the odd numbers. In the second 47-month group they are the
even numbers, of which there is one more in this division than odd
numbers, thus accounting for the additional one of the three days.

If the period of 3986 days were considered by itself, the arrangement
given would be sufficient. As soon, however, as this period is repeated
a number of times an error develops, since 135 synodical revolutions of
the moon are completed in 3986.63 days. Twice this number gives 7973.26,
or 1.26 days more than twice 3986. In order to keep the sequence of
months in the arrangement given above in accordance with the moon, it
becomes necessary to intercalate one more day every two repetitions of
the 3986 period. This may be done by changing the last 29-day month in
the 41-month group to a 30-day month, making the last 177-day group in
the third one of 178 days. The Mayas certainly did this in the first
third of the series given and arranged for it in the last third in a
manner which will be demonstrated later.

Tabulating the solution here advanced will form Table XI (p. 25), in
which the 30-and 29-day months in one-third of the manuscript have been
arranged in three columns, the first two of which represent the 47-month
groups and the last one the 41-month group. Before the first column of
months are numbers to facilitate locating any given month in the group.
The two kinds of months occur in direct alternation in each group, with
three exceptions. The 32d month in both of the 47-month groups is one of
30 days instead of 29, because of the addition of the intercalary day.
The 40th month in the 41-month group is given as one of 29 days with a
30 in parentheses before it, representing the fact that every other
third an extra intercalary day should be inserted in this month. To the
right of the month columns are three columns giving the difference
groups as found in the manuscript (see Table II), each column giving
those numbers found in each third of the manuscript, the first third
being the left one of the three. It should be noticed that the misplaced
(?) 148-day group in the last third does not interfere with the sequence
of the months.

Finally it only remains to review the irregularities of the manuscript
in the light of the solution just advanced. Those irregularities which
are corrected immediately afterwards, or are at variance with the rest
of the column in which they occur, are, in all probability, errors on
the part of the writer of the series, such as might have been caused by
careless transcription from another copy of the table, and correction in
the following column to avoid the task of erasure. Eliminating these
irregularities, there remain three to be investigated, namely, (1) the
absence of the 178 numbers in the lower number series, with one
conspicuous exception; (2) the occurrence of 178 in column 7 instead of
6, and of 148 in column 58 instead of 59; and (3) the discrepancies in
the totals of the series.

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

   No.   Days                  Days                    Days
   of     in   Dresden groups   in    Dresden groups    in   Dresden groups
  month  month    1    2    3  month    1   2    3   month    1    2    3

    1     30 \                  30 \                   30 \
    2     29 |                  29 |                   29 |
    3     30 |                  30 |                   30 |
    4     29 |  177  177  177   29 |   177  177  177   29 |  177  177  177
    5     30 |                  30 |                   30 |
    6     29 /                  29 /                   29 /
    7     30 \                  30 \                   30 \
    8     29 |                  29 |                   29 |
    9     30 |                  30 |                   30 |
   10     29 |  177  177  177   29 |   177  177  177   29 |  177  177  177
   11     30 |                  30 |                   30 |
   12     29 /                  29 /                   29 /
   13     30 \                  30 \                   30 \
   14     29 |                  29 |                   29 |
   15     30 |  148  148  148   30 |                   30 |  148  148  148
   16     29 |                  29 |   177  177  177   29 |
   17     30 /                  30 |                   30 /
   18     29 \                  29 /                   29 \
   19     30 |                  30 \ \                 30 |
   20     29 |                  29 | |           148   29 |
   21     30 |  177  177  177   30 | | 177  177        30 |  177  177  177
   22     29 |                  29 | |                 29 |
   23     30 /                  30 | /                 30 /
   24     29 \                  29 / \                 29 \
   25     30 |                  30 \ |                 30 |
   26     29 |                  29 | |           177   29 |
   27     30 |  177  177  177   30 | | 148  148        30 |  177  177  177
   28     29 |                  29 | |                 29 |
   29     30 /                  30 / /                 30 |
   30     29 \                  29 \                   29 /
   31     30 |                  30 |                   30 \
  *32     30 |                  30 |                   29 |
   33     30 |  178  178  178   30 |   178  178  178   30 |  177  177  177
   34     29 |                  29 |                   29 |
   35     30 /                  30 /                   30 /
   36     29 \                  29 \                   29 \
   37     30 |                  30 |                   30 |
   38     29 |  177  177  177   29 |   177  177  177   29 |  178  177  177
   39     30 |                  30 |                   30 |           (178)
   40     29 |                  29 |               (30)29 |
   41     30 /                  30 /                  -30 /
   42     29 \                  29 \
   43     30 |                  30 |
   44     29 |                  29 |
   45     30 |  177  177  177   30 |   177  177  177
   46     29 |                  29 |
   47     30 /                  30 /

The great bulk of the difference groups as expressed by the lower number
series are 177, the only departure from these being the designation of
the 148-day groups and the extra 178-day group at the end of the first
third. The complete disregard of all of the six normal 178-day groups by
the lower numbers seems to imply that no attempt was made to have the
latter agree with the actual differences in the upper numbers, a
conclusion which is strengthened by the fact that none of the lower
numbers shows evidence of the clerical errors in the differences of the
upper numbers. It seems most probable that the lower number series was
intended merely as a guide to indicate the position of the five month
periods and to place emphasis on the extra intercalary day added in the
23d column, without attempting to have this series accurate.

The presence of the 178-day group in column 7 instead of 6 has been
discussed at some length under the description of the errors. The
scribe, realizing that in neglecting to put in this 178-day group, the
first one of the series, a serious error had been committed, may have
attempted to erase the incorrect record in column 6; then, realizing
that four numbers and three glyphs would have to be altered, decided to
correct this mistake—although it was of more importance than the other
two errors—as he had the former ones, i.e., by making the correction in
the next column.

The very similar irregularity in the last third of the manuscript, the
placing of the 148-day group one column ahead of its expected position,
cannot be explained in the same manner. It is very evident that this
column has been deliberately placed where it is. That it does not have
to do with the month sequence is evident, since it does not affect it.
It must then affect the ecliptic part of the series, for it causes a
short season to occur six months earlier than expected. Upon comparison
of Tables VI and X, it will be seen that all of the dates of the 148-day
group occur during one of the eclipse groups given in Schram’s table.
However, had the 148-day group under discussion been placed in the 59th
column, as uniformity demands, this number, 10,216, would not have
fallen in one of the eclipse groups given in Table VI. This tends to
show that there was some reason other than regulating the difference
groups to agree with the eclipse seasons, for the position of the
148-day groups. This reason, as yet undetermined, is possibly associated
with the pictures, which immediately follow the 148-day groups.

Finally there remain only the totals of the series to be considered. The
total of the upper number series records 11,958 days. Sixty-nine eclipse
seasons complete 11,958.39 days, less than half a day more than the
recorded number. This close agreement and the failure to add the extra
intercalary day to the upper number series at the end of the first
third, give rise to the belief that the upper number series is a
calendar in itself, and records a means by which dates of probable
eclipses may be reckoned. The units of the count were eclipse seasons
expressed as lunar months, 69 of which are represented in the calendar
recorded on these pages.

The Mayas undoubtedly knew the relation of the eclipses to the moon, at
least in a vague way, and felt that it was necessary to associate this
eclipse calendar in some way with the lunar calendar, composed of 29-and
30-day months. Therefore the day series is found immediately below the
upper number series. This series of days constitutes a lunar calendar
which coincides as closely as possible with the eclipse calendar. It may
be the formal lunar calendar of the Mayas, but it may also be an
adaptation of the formal calendar to the eclipse periods. The day series
varies from the eclipse series in two places only. At the end of the
first third of the series, it was necessary to add one day to the lunar
calendar, an addition strongly pointed out in the record, but not to the
eclipse calendar, because of the increasing error between the
revolutions of the moon and the calendrical lunar months. Therefore,
throughout the remaining two-thirds of the series, the lunar calendar
was one day in advance of the eclipse calendar. At the end of the
series, since 405 of the moon’s revolutions complete 11,959.89 days, and
the day series only 11,959, one more day should be added, in order to
keep the error as small as possible. This was accomplished by changing
from the middle to the lower line of days.

On page 52a, immediately preceding the calendar, are four day signs with
numbers. One of these, 12 Lamat, is the zero day of the day series, but
is associated with the middle line of day glyphs and not the upper line,
as might be expected. The series of days which come, calendrically
speaking, just before and after the actual series, may have been placed
in the record to show that slight variations from the average were to be
expected. The entire record is based on the middle line of days until
the end of the series. Here the day just below the last day of the
middle line is 12 Lamat, the end of 46 _tonalamatls_ (260-day cycle),
and the zero day of the recorded series. The _tonalamatl_ was probably
as easily used by the Mayas as “60 days” and “90 days” are used now. The
entire calendrical system of the Mayas is based on the cycle principle.
The series recorded in these pages was probably also a cycle, and in
order to repeat it, 12 Lamat must again be used as the zero date. If to
these arguments is added the fact that an additional day is necessary to
keep the calendar in accord with the synodic revolution of the moon,
there remains little doubt but that the users of this calendar added the
extra day by going from the middle to the lower line of day glyphs,
thereby keeping the error between the moon and the calendar as low as
possible, completing the 46th _tonalamatl_, and at the same time making
it possible to repeat the recorded series as a cycle. If the series is
repeated once, at the end of 810 months, or about 66-1/2 years, the
eclipse calendar will be behind the average eclipse season .78 days, and
the lunar calendar will be in advance of the synodical revolutions of
the moon only .22 days.

In general, then, the irregularities in the calendar recorded on these
pages fall into two groups, those which are clerical errors of the
scribe and do not therefore affect the solution advanced, and those
which do not appear to be of the clerical type. In the light of the
solution advanced, it has been shown that there are perfectly logical
reasons for the latter group of apparent irregularities.


CONCLUSION

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

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

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

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

The number series on these pages record an eclipse calendar, that is, a
series of dates by means of which the occurrence of eclipses was
foretold. This calendar is composed of three identical parts, with the
exception of one 148-day group which occurs six months earlier in the
last third than in the other two. Each third is composed of 23 unequal
subdivisions which represent the twenty-three eclipse seasons, expressed
in lunar months, in 3986 days. The upper number series records this
calendar, and its total of 11,958 days is only .39 days less than 69
eclipse seasons.

In order to make it more intelligible this eclipse calendar is
accompanied by a probably more generally known lunar calendar, which may
have been altered slightly to conform to the requirements of the eclipse
calendar it accompanies. This lunar calendar is contained in the day
series just below the eclipse calendar. It also is recorded in three
divisions agreeing closely with the eclipse calendar. One hundred and
thirty-five lunar months of 30 and 29 days complete 3986 days, .63 days
less than 135 synodical revolutions of the moon. This error which
amounts to more than one day when repeated once, necessitates the
addition of an extra day in the lunar calendar every other third, which
was done in the manuscript in the first and last third, making the total
recorded by the lunar calendar 11,960, two days more than the eclipse
calendar, and .11 days more than 405 synodical revolutions of the moon.
This period of 11,960 days may have been used as a cycle, the zero day
of which is 12 Lamat.

Each third of the lunar calendar consists of 30-and 29-day months
arranged in alternating sequence, with intercalary days added by the
substitution of a 30-day for a 29-day month when the error arising from
the nonconformity of the moon’s revolution reaches more than one day. In
order that the lunar calendar might agree with the eclipse calendar more
closely, these months were recorded in groups of five and six.

The months in each third of the series were divided into three groups,
which are the same in each third. The first two groups contained 47
months each, and completed the first sixteen dates of the third. The
last group was one of 41 months, which was represented by the last seven
dates of the third. An intercalary day was added in the 32d month of
each of the 47-month groups to correct the accumulating error, thereby
causing the 6th and 14th subdivisions of the third to be of 178 days. In
the first and last third the 40th month of the 41-month group also
contained an intercalary day for the same reason, making the 23d
subdivision 178 days, but in the last column of the record this extra
day is added by going from the middle to the lower line of day signs.
Each of the 47-and 41-month divisions began and ended with a month of
thirty days.

The numerical series of these pages of the Dresden record, then, an
eclipse calendar which is referred to a lunar calendar. This solution
explains all the irregularities of the series except those which seem
clearly to be clerical errors of the scribe.

Only the numerical and calendrical series on these pages have been
considered. No attempt has been made to explain the hieroglyphs, the
pictures, or the first two pages, which, although showing a close
association to the long series, are nevertheless a unit in themselves.



BIBLIOGRAPHY

  BOWDITCH, C. P.
    1910. The Numeration, Calendar Systems and Astronomical Knowledge of
          the Mayas. Cambridge.

  FÖRSTEMANN, E.
    1886. Erläuterungen zur Mayahandschrift der Königlichen öffentlichen
          Bibliothek zu Dresden. Dresden.
    1898. Zur Entzifferung der Mayahandschrift, VII. Dresden. (Trans. in
          Bull. 28, Bur. Am. Ethnol., pp. 463-72.)
    1901. Commentar zur Mayahandschrift der Königlichen öffentlichen
          Bibliothek zu Dresden. Dresden. (Trans. in Papers of the
          Peabody Museum, Vol. IV, No. 2.)

  MEINSHAUSEN, MARTIN.
    1913.  Über Sonnen- und Mondfinsternisse in der Dresdener
           Mayahandschrift. Zeit. für Ethnol., Bd. XLV, pp. 221-227.

  SCHRAM, R.
    1908.  Kalendariographische und Chronologische Tafeln. Leipzig.

  THOMAS, CYRUS.
    1888.  Aids to the Study of the Maya Codices. 6th Ann. Rep., Bur.
           Am. Ethnol., pp. 253-371.



  PRINTED AT
  THE HARVARD UNIVERSITY PRESS
  CAMBRIDGE, MASS., U. S. A.



PEABODY MUSEUM PAPERS.

VOL. VI, NO. 2, PLATE 1


[Illustration: DRESDEN CODEX, PAGES 51 TO 58.]



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