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.