characters (hieroglyphics), and immediately under the latter three
horizontal lines of black numerals, with here and there a red, diamond
shaped symbol inserted. As these numerals stand directly in the vertical
lines of the day columns, it is possible the two have some connection
with each other, a supposition somewhat strengthened by what has been
observed in regard to the red numerals at the bottom of the plates. To
test this and also for the reason that we propose to discuss their
relations and their use, we give here the bottom line of days of each of
the five series (or plates), together with their week numbers attached;
also, the numbers of the three lines of black numerals mentioned, taking
them in the order of the paging as here shown:
_Plate_ 46:
VIII Cib. VII Cimi. X Cib. V Kan.
1 1
11 16 10 11
16 6 16 4
_Plate_ 47:
VII Ahau. VI Oc. IX Ahau. IV Lamat.
2 2 3 3
5 9 4 4
0 10 0 8
_Plate_ 48:
VI Kan. V Ix. VIII Kan. III Eb.
3 4 4 4
16 2 15 15
3(?) 14 4 12
_Plate_ 49:
V Lamat. IV Ezanab. VII Lamat. II Cib.
5 5 6 6
9 13 8 8
8 18 8 16
_Plate_ 50:
IV Eb. III Ik. VI Eb. 1 Ahau.
7 7 8 8
3 7 1 2
12 2 12 0
In considering these horizontal lines it is to be understood that the
series runs through the five pages, 46-50.
Let us proceed upon the supposition that the figures of the lowest of the
three lines denote days of the month, the numbers of the middle line
months, and those of the upper line years. As already shown, the interval
between 8 Cib and 7 Cimi is 4 months and 10 days; adding 4 months and 10
days to 11 months and 16 days (bearing in mind that 20 days make a month
and 18 months a year), the sum is found to be 16 months and 6 days,
precisely the figures under 7 Cimi. As already ascertained, the interval
between 7 Cimi and 10 Cib is 12 months and 10 days; this added to 16
months and 6 days gives 1 year, 10 months, 16 days, precisely the figures
under 10 Cib. The interval between 10 Cib and 5 Kan is 8 days; this added
to the 1 year, 10 months, and 16 days gives 1 year, 11 months, and 4
days, the figures under 5 Kan. The interval between 5 Kan and 7 Ahau is
11 months, 16 days, which, added to the preceding, gives 2 years, 5
months, 0 day, agreeing with the figures under 7 Ahau, if the symbol
represented by 0 signifies nought. That this rule holds good throughout
the entire series, by making one correction, is shown by the following
additions:
Years. Months. Days.
11 16 Under VIII Cib, Plate 46.
4 10
-- --
16 6 Under VII Cimi, Plate 46.
12 10
-- --
1 10 16 Under X Cib, Plate 46.
8
-- -- --
1 11 4 Under V Kan, Plate 46.
11 16
-- -- --
2 5 0 Under VII Ahau, Plate 47.
4 10
-- -- --
2 9 10 Under VI Oc, Plate 47.
12 10
-- -- --
3 4 0 Under IX Ahau, Plate 47.
8
-- -- --
3 4 8 Under IV Lamat, Plate 47.
11 16
-- -- --
3 16 4[300-1] Under VI Kan, Plate 48.
4 10
-- -- --
4 2 14 Under V Ix, Plate 48.
12 10
-- -- --
4 15 4 Under VIII Kan, Plate 48.
8
-- -- --
4 15 12 Under III Eb, Plate 48.
11 16
-- -- --
5 9 8 Under V Lamat, Plate 49.
4 10
-- -- --
5 13 18 Under IV Ezanab, Plate 49.
12 10
-- -- --
6 8 8 Under VII Lamat, Plate 49.
8
-- -- --
6 8 16 Under II Cib, Plate 49.
11 16
-- -- --
7 2 12 Under IV Eb, Plate 50.
4 10
-- -- --
7 7 2 Under III Ik, Plate 50.
12 10
-- -- --
8 1 12 Under VI Eb, Plate 50.
8
-- -- --
8 2 0 Under I Ahau, Plate 50.
The proof of the correctness of the theory advanced may, therefore, be
considered conclusive, as it amounts, in fact, to a mathematical
demonstration.
Dr. Foerstemann, who considers these lines of black numbers, standing one
above another, as representing different grades of units--thus, the
lowest, single units; the second, units twenty-fold the lower; the third,
eighteen-fold the second; the fourth, twenty-fold the third, &c.--has
found the correct intervals of the series, which he states are 236, 90,
250, and 8 days, agreeing with our 11 months, 16 days; 4 months, 10 days;
12 months, 10 days, and 8 days.
As all the discoveries mentioned herein were made previous to the receipt
of Dr. Foerstemann's work, I give them according to my own method,
acknowledging any modification due to his work. Although I shall compare
special results from time to time, an explanation of Dr. Foerstemann's
method is reserved for a future paper, as his work was not received until
I was revising my notes for publication.
The foregoing explanation of the series shows it to be very simple and
makes it clear that it relates to the day columns at the top of the
pages. Still, there is one point somewhat difficult to understand. Are
the numbers of the third or lowest line intended to denote the positions
in the month of the days in the columns above? If so, the month must have
commenced with Ymix, as can readily be shown in the following manner:
TABLE III.
1. Ymix.
2. Ik.
3. Akbal.
4. Kan.
5. Chicchan.
6. Cimi.
7. Manik.
8. Lamat.
9. Muluc.
10. Oc.
11. Chuen.
12. Eb.
13. Been.
14. Ix.
15. Men.
16. Cib.
17. Caban.
18. Ezanab.
19. Cauac.
20. Ahau.
If we write in a column in proper order the 20 days of the Maya month,
commencing with Ymix, and number them consecutively, as in Table III, we
shall find by comparison that the numbers in the lower line indicate the
position, in this column, of the days directly over them. Take, for
example, the lower line of black numerals on Plate 46, writing over them
the respective days of the columns, thus:
Cib. Cimi. Cib. Kan.
16 6 16 4
Referring to Table III we see that Cib is the sixteenth day, Cimi the
sixth, and Kan the fourth.
The days and numbers of Plate 47 are:
Ahau. Oc. Ahau. Lamat.
0 10 0 8
Ahau is the twentieth day--here is the diamond shaped symbol--Oc is the
tenth, and Lamat the eighth, and so on to the end of the series on Plate
50.
It may be justly argued that such relation to some given day of the month
would necessarily follow in any series of this kind made up by adding
together intervals of days and months. Still it is not at all likely that
these series were made up without reference to fitted and determinable
dates. If so, the months given must be months of certain determinable
years, and the days denoted must be days of particular months. In other
words, if we had the proper starting point we should be able to determine
the position in the calendar of any day or month mentioned in the series.
First. It is easily seen by reference to the calendar (Table II) that Cib
is not the sixteenth day of the month of any of the four years, nor is
Cimi the sixth nor Kan the fourth. The idea that the figures of this
lower line represent the days of the month must, therefore, be given up
unless we assume that the year commenced with Ymix. It may be worthy of
notice at this point that the list of days on the so-called "title page"
of the Manuscript Troano begins with Ymix. It is also true that the
remarkable quadruple series in the Codex Cortesianus on Plates 13-18
commences with Ymix; as this is evidently some kind of a calendar table,
its bearing on the question now before us is important.
Second. It can easily be shown that the months referred to in the series,
if the numbers given denote specific months, are not those of the Kan
years. The first, 8 Cib, if in the eleventh month, must be in the year 4
Kan; counting forward from this 4 months and 10 days to 7 Cimi brings us
into the sixteenth month of the year 4 Kan; this agrees with our figures
on Plate 46. Counting forward 12 months and 10 days to 10 Cib, we reach
the tenth month of the next year; 8 days more carry us to the eleventh
month, which still agrees with the figures in the codex. Counting 11
months and 16 days more to 7 Ahau, we reach but do not pass the fourth
month of the next year; hence the result does not correspond with the
series, which has at this point a 5 in the middle line. The same will be
found true in regard to the other years as given in our calendar (Table
II). This result, as a matter of course, must follow if the figures in
the lower line of the series do not denote the month days of some one of
the year series as usually given.
Another fact also becomes apparent here, viz, that the 5 supplemental
days of the year are not brought into the count, the year consisting
throughout of 360 days. There is, in fact, nothing here indicating the
four year series as given in the authorities and as represented in our
calendar table; yet this ought to appear wherever a series extends over
more than one year.
Dr. Foerstemann says that this entire series of black numerals covers
2,920 days, or 8 years of 365 days. This is true, but the concluding
figures show that it is given by the writer of the codex as 8 years and 2
months, which would also be 2,920 days, counting the years at 360 days
each and the months 20 days each; moreover, the members of the series are
based throughout upon the year of 360 days. His theory that the intervals
of the series relate to the movements of the planet Venus is, as yet, a
mere hypothesis, which needs further proof before it can demand
acceptance; but his discovery of the methods of identifying the month
symbols on the five plates now under consideration is important. Although
I had noticed that most of the characters which he mentions are month
symbols, I did not succeed in identifying all of them.
According to his conclusion, which appears to be justified not only by
the evidence he gives but by an additional fact that I shall, presently
mention, there are four of these symbols in the upper row of the middle
group of written characters on each plate and four in the upper and lower
lines of the lower group on each plate (see, for example, Fig. 362). Each
of these symbols (except three or four) has a black number attached to it
which denotes the day of the month represented by the symbol.
These months and days as given by Dr. Foerstemann are as follows, the
positions of the lines as here given corresponding with those of the
plates:
TABLE IV.--_Table showing months and days._
_____________________________________________________________________
| Month. Day. | Month. Day. | Month. Day. | Month. Day.
|--------------+--------------+--------------+--------------
Plate 46 | 7 4 | 11 14 | 5 19 | 6 7
| 11 8 | 15 18 | 10 4 | 10 12
| 1 14 | 6 4 | 18 14 | 1 2
Plate 47 | 18 3 | 4 8 | 16 18 | 17 6
| 4 3 | 8 13 | 2 18 | 3(not 2) 6
| 10 10 | 15 3 | 9 8 | 9 16
Plate 48 | 10 17 | 15 7 | 9 12 | 10 20
| 15 2 | 1 7 | 13 17 | 14 5
| 3 7 | 7 17 | 2 2 | 2 10
Plate 49 | 3 11 | 8 1 | 2 6 | 2 14
| 7 16 | 12 6 | 6 11 | 6 19
| 14 6 | 18 16 | 13 1 | 13 9
Plate 50 | 14 10 | 18 20 | 13 5 | 13 13
| 18 15 | 5 20 | 17 10 | 17 18
| 6 20 | 11 10 | 5 15 | 6 3
---------------------------------------------------------------------
An examination of the plates will show that Dr. Foerstemann has filled out
the following obliterated or wanting day numbers, to wit, the first of
the upper line of Plate 46, the fourth of the upper line of Plate 47, and
the second of the middle line and first of the lower line of Plate 50. He
has also ventured to change the first day number of the lower line of
Plate 46 from 16 to 14. Where the number 20 is found in his list there is
no corresponding number in the codex, the month symbol only being given.
It is evident he has proceeded in these cases upon the theory that the
absence of a number indicated that the month was completed. Although
probably correct in this conclusion, the question will arise, Does the
symbol in such cases denote the _month completed_ or the _month reached?_
The intervals between these dates are as follows, the left hand column
being those between the first and second columns of Foerstemann's list
(our Table IV), the second column those between the second and third
columns of his list, the third column those between the third and fourth
columns of his list, and the fourth column those between the last date of
one plate and the first of the next:
TABLE V.--_Table showing intervals between dates._
_____________________________________________________________________
| Month. Day. | Month. Day. | Month. Day. | Month. Day.
|--------------+--------------+--------------+--------------
| | | |
Plate 46 | 4 10 | 12 5 | 0 8 | 11 16
| 4 10 | 12 6_b_| 0 8 | 11 11
| 4 10 | 12 10 | 0 8 | 9 8_d_
Plate 47 | 4 5 | 12 10 | 0 8 | 11 11
| 4 10 | 12 5 | 0 8_c_| 11 16_e_
| 4 13_a_| 12 5 | 0 8 | 11 11
Plate 48 | 4 10 | 12 5 | 8 | 11 11
| 4 5 | 12 10 | 0 8 | 11 11
| 4 10 | 12 5 | 0 8 | 11 16
Plate 49 | 4 10 | 12 5 | 0 8 | 11 16
| 4 10 | 12 5 | 0 8 | 11 16
| 4 10 | 12 5 | 0 8 | 11 11
Plate 50 | 4 10 | 12 5 | 0 8 | 11 11
| 4 5 | 12 10 | 0 8 | 11 10
| 4 10 | 12 5 | 0 8 | 12 11_g_
---------------------------------------------------------------------
Although it is apparent that the variations from the intervals of the
black numeral and day series above them are too numerous and too uniform
to be considered mistakes, yet there is little reason to doubt that these
month numbers are connected with and depend upon the day series given in
the columns above.
That there are some errors is quite clear; for instance, the variation at
_a_ arises from the fact that Dr. Foerstemann gives the date here as 10
months, 10 days, whereas the codex has it 10 months, 13 days. Making this
correction the interval will be 4 months, 10 days. The correction will
make the interval at _d_ 9, 11, instead of 9, 8. Still there is a
variation of two months from the usual interval, which, if corrected on
the supposition that Dr. Foerstemann has mistaken the month, would
necessitate a change of the remainder of the series given in this line.
The interval at _c_, according to the figure given by Dr. Foerstemann,
would be retrograde, that is, minus 12. This arises from the fact that he
gives the last date in the middle line on Plate 47 as 2 months, 6 days,
whereas the symbol is very distinctly that of the third month, and the
eight day series is unbroken if this correction is made.
When these evident errors are corrected the series of intervals show
very clearly a system and periodicity depending on the day column series
in the upper part of the pages. In the first column (Table V) the
interval is usually 4 months, 10 days, precisely the same as between the
first and second day columns, but occasionally it is 4 months, 5 days,
which will still bring it to one of the four day series, including the
day indicated by the date--4 months, 10 days. This will be understood by
examining our calendar (Table II). The corresponding days in the four
year columns were, by the Maya system, necessarily brought together in
the calendar; for example, they are arranged in the series pictured on
Plates 13-18 of the Cortesian Codex precisely as given in our Table II.
This skip of five days is also apparent in the second and fourth columns
of differences (Table V). Whether Dr. Foerstemann is correct in all his
identifications of months among the symbols on the five plates now under
consideration is a question I feel unqualified to answer without a much
more careful comparison and study of these characters than I have given
them.
Running through the upper division of Plates 53 to 58 and continued
through the lower division of Plates 51 to 58--that is to say, commencing
in the upper division of 53 and running into 58, then back to the lower
division of 51 and ending in 58--is a remarkable compound series. It
consists, first, of a three line series of black numerals standing above;
second, a middle series of short, three day columns, or columns each of
three day symbols, with red numerals attached; and, third, below, a two
line series of numerals, those of the upper line red and of the lower
black numbers.
As this series is a very important one in the study of the relations of
the numerals to one another and to the days indicated, an exact copy of
it is given in Figs. 363-370, each figure representing a page and the
whole standing in the same order as in the original. The red numerals and
red symbols are, as usual, given in outline as an indication of their
color.
[Illustration: FIG. 363. Copy of Plate 51, Dresden Codex.]
[Illustration: FIG. 364. Copy of Plate 52, Dresden Codex.]
[Illustration: FIG. 365. Copy of Plate 55, Dresden Codex.]
[Illustration: FIG. 366. Copy of Plate 54, Dresden Codex.]
[Illustration: FIG. 367. Copy of Plate 55, Dresden Codex.]
[Illustration: FIG. 368. Copy of Plate 56, Dresden Codex.]
[Illustration: FIG. 369. Copy of Plate 57, Dresden Codex.]
[Illustration: FIG. 370. Copy of Plate 58, Dresden Codex.]
In order to assist those not familiar with the numeral and day symbols,
the entire series is given in the following tables in names and Arabic
and Roman numerals, as usual. The obliterated symbols and numbers are
restored.
TABLE VI.--_Table of numeral and day symbols._ (Plate 51_b_.)
______________________________________________________________________
14 | 15 | 15 | 16 | 16 | 17
16 | 7 | 16 | 7 | 16 | 5
14 | 11 | 8 | 5 | 2 | 10
IV Ik. | XII Cauac.| VII Cib. | II Been.| X Oc. | II Ezanab.
V Akbal.|XIII Ahau. |VIII Caban. |III Ix | XI Chuen|III Cauac.
VI Kan. | I Ymix. | IX Ezanab.| IV Men. |XII Eb. | IV Ahau.[VI-1]
VIII | VIII | VIII | VIII | VIII | VII
17 | 17 | 17 | 17 | 17 | 8
----------------------------------------------------------------------
[VI-1] The symbol in this case is that of Been, but this is a manifest
error, as Ahau follows Cauac.
TABLE VII.--_Table of numeral and day symbols._ (Plate 52_b_.)
____________________________________________________________
| 17 | 18 | 18 | 19
| 14 | 5 | 14 | 4
| 8 | 5 | 2 | 19
| XI Cib. | VI Been.| I Oc. | IX Manik.
[Picture.] | XII Caban. | VII Ix. | II Chuen.| X Lamat.
| XIII Ezanab. |VIII Men. |III Eb. | XI Muluc.
| VIII | VIII | VIII | VIII
|17? (18)[VII-1]| 17 | 17 | 17
------------------------------------------------------------
[VII-1] The variation from the rule found here is explained a little
further on.
TABLE VIII.--_Table of numeral and day symbols._ (Plate 53_a_.)
______________________________________________________________________________________
| 1 | 1 | | 2 | 2 |
7 | 17 | 7 | | 15 | 6 | 15
17 | 18 | 2 | |14?(19)[VIII-1]| 16 | 13
VI Kan. | I Ymix. | VI Muluc.|[Picture.]| I Cimi. |IX Akbal. |IV Ahau.
VII Chicchan.| II Ik. | VII Oc. | | II Manik. | X Kan. | V Ymix.
VIII Cimi. |III Akbal.|VIII Chuen.| | III Lamat. |XI Chicchan.|VI Ik.
VIII | VIII | VII | | VIII | VIII | VIII
17 | 17 | 8 | | 17 | 17 | 17
--------------------------------------------------------------------------------------
[VIII-1] The 14 here is manifestly an error, one of the lines in the
number symbol having been omitted; it should be 19.
TABLE IX.--_Table of numeral and day symbols._ (Plate 53_b_.)
_____________________________________________________________________
| 1 | | 1 | 1 | 1
19 | 0 | | 0 | 1 | 1
13 | 3 | | 12 | 2 | 11
16 | 4 | | 1 | 18 | 15
IV Kan. |IX Eb. |[Picture.]|IV Muluc.| XII Cimi. | VII Akbal.
V Chicchan.| X Been.| | V Oc. |XIII Manik.|VIII Kan.
VI Cimi. |XI Ix. | |VI Chuen.| I Lamat.| IX Chicchan.
VIII | VII | | VIII | VIII | VIII
17 | 8 | | 17 | 17 | 17
---------------------------------------------------------------------
TABLE X.--_Table of numeral and day symbols._ (Plate 54_a_.)
____________________________________________________________________________________
3 | 3 | 4 | 4 | 5 | 5 | 6
6 | 15 | 6 | 15 | 5 | 10 | 4
11 | 8 | 5 | 5 | 19 | 16 | 4
XIII Ezanab.|VIII Men. |III Eb. | XI Muluc.| VI Cib. | I Akbal. | VI Chuen.
I Cauac. | IX Cib. | IV Been.| XII Oc. | VII Caban. | II Kan. | VII Eb.
II Ahau. | X Caban.| V Ix. |XIII Chuen.|VIII Ezanab.|III Chicchan.|VIII Been.
VIII | VIII | VIII | VIII | VIII | VIII | VII
17 | 17 | 17 | 17 | 17 | 17 | 8
------------------------------------------------------------------------------------
TABLE XI.--_Table of numeral and day symbols._ (Plate 54_b_.)
_______________________________________________________________
1 | 1 | 1 | 1 | | 1
2 | 2 | 3 | 3 | | 4
2 | 11 | 2 | 9 | | 0[XI-1]
12 | 9 | 6 | 14 | | 11
II Ahau.| X Caban. | V Ix. | X Ik. |[Picture]| V Cauac.
III Ymix.| XI Ezanab.| VI Men.| XI Akbal.| | VI Ahau.
IV Ik. |XII Cauac. |VII Cib.|XII Kan. | |VII Ymix.
VIII | VIII | VIII | VII | | VII[XI-2]
17 | 17 | 17 | 8 | | 17
---------------------------------------------------------------
[XI-1] The 0 inserted at various points in these tables denotes as usual
the red, diamond shaped symbol, which apparently signifies "nought."
[XI-2] The numeral symbol in this case, both in Kingsborough's copy and
in the photograph, is VII, one dot having been omitted by a mistake of
the original artist.
TABLE XII.--_Table of numeral and day symbols._ (Plate 55_a_.)
____________________________________________________________________________
| 8 | 7 | 7 | 8 | 8
| 13 | 3 | 12 | 3 | 12
| 2 | 18 | 16 | 13 | 10
| II Muluc.[XII-1]| X Cimi. | V Akbal. |XIII Ahau.|VIII Caban.
[Picture]|III Oc. | XI Manik.| VI Kan. | I Ymix.| IX Ezanab.
| IV Chuen. |XII Lamat.|VII Chicchan.| II Ik. | X Cauac.
| VIII | VIII | VIII | VIII | VIII
| 17 | 17 | 17 | 17 | 17
----------------------------------------------------------------------------
[XII-1] In Kingsborough's work the symbol in this case is that of Been,
but should be Muluc, as it is in the photograph.
TABLE XIII.--_Table of numeral and day symbols._ (Plate 55_b_.)
__________________________________________________________________________________________
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
4 | 5 | 5 | 6 | 6 | 6 | 7 | 7
9 | 0 | 9 | 0 | 8 | 17 | 8 | 15
8 | 6 | 3 | 0 | 17 | 14 | 11 | 19
XIII Cib. |IX Ix. |IV Chuen.| XII Lamat.| VII Chicchan.| II Ik. | X Cauac.| II Manik.
I Caban. | X Men.| V Eb. |XIII Muluc.|VIII Cimi. |III Akbal.| XI Ahau. |III Lamat.
II Ezanab.|XI Cib.|VI Been. | I Oc. | IX Manik. | IV Kan. |XII Ymix. | IV Muluc.
VIII | VIII | VIII | VIII | VIII | VIII | VIII | VII
17 |17?(18)| 17 | 17 | 17 | 17 | 17 | 8
------------------------------------------------------------------------------------------
TABLE XIV.--_Table of numeral and day symbols._ (Plate 56_a_.)
____________________________________________________________
9 | | 9 | 10 | 10
1 | | 10 | 1 | 10
18 | | 15 | 12 | 9
XIII Chicchan.| |VIII Ik. |III Cauac.| XI Cib.
I Cimi. |[Picture]| IX Akbal.| IV Ahau. | XII Caban.
II Manik. | | X Kan. | V Ymix. |XIII Ezanab.
VII | | VIII | VIII | VIII
8 | | 17 | 17 | 17
------------------------------------------------------------
TABLE XV.--_Table of numeral and day symbols._ (Plate 56_b_.)
_________________________________________________________
| 1 | 1 | 1 | 1
| 8 | 8 | 9 | 9
| 6 | 15 | 6 | 15
| 16 | 14 | 11 | 8
[Picture]| X Kan. | VI Ik. | I Cauac.|IX Cib.
| XI Chicchan.| VII Akbal.| II Ahau. | X Caban.
|XII Cimi. |VIII Kan. |III Ymix. |XI Ezanab.
| VIII | VIII | VIII | VIII
| 17 | 17?(8) | 17 | 17
---------------------------------------------------------
TABLE XVI.--_Table of numeral and day symbols._ (Plate 57_a_.)
______________________________________________________
11 | 11 | 12 | 12 |
1 | 10 | 1 | 8 |
6 | 4 | 0 | 8 |
VII Ix. | II Chuen.| X Lamat.| II Cib. |
VIII Men.|III Eb. | XI Muluc.|III Caban. |[Picture]
IX Cib.| IV Been. |XII Oc. | IV Ezanab. |
VIII | VIII | VIII | VIII[XVI-1]|
17 | 17 | 17 | 17[XVI-2]|
------------------------------------------------------
[XVI-1] This should be VII.
[XVI-2] This should be 8.
TABLE XVII.--_Table of numeral and day symbols._ (Plate 57_b_.)
__________________________________________________________________
1 | 1 | 1 | | 1 | 1
10 | 10 | 11 | | 11 | 12
6 | 15 | 4 | | 13 | 4
5 | 2 | 10 | | 7 | 4
IV Been.| XII Oc. |IV Ezanab.|[Picture]| XII Men. | VII Eb.
V Ix. |XIII Chuen.| V Cauac. | |XIII Cib. |VIII Been.
VI Men. | I Eb. |VI Ahau. | | I Caban.| IX Ix.
VIII | VIII | VII | | VIII | VIII
17 | 17 | 8 | | 17 | 17
------------------------------------------------------------------
TABLE XVIII.--_Table of numeral and day symbols._ (Plate 58_a_.)
____________________________________________
12 | 13 | 13 | 14
17 | 8 | 17 | 7
5 | 2 | 0 | 17
X Been.| V Oc. | I Lamat.|II Chicchan.
XI Ix. | VI Chuen.| II Muluc.| X Cimi.
XII Men. |VII Eb. |III Oc. |XI Manik.
VIII | VIII | VIII | VIII
17 | 17 | 17 | 17
--------------------------------------------
TABLE XIX.--_Table of numeral and day symbols._ (Plate 58_b_.)
________________________________
1 | 1 |
12 | 13 |
13 | 3 |
1 | 18 |
II Muluc.| X Cimi. |[Picture.]
III Oc. | XI Manik.|
IV Chuen.|XII Lamat.|
VIII | VIII |
17 | 17 |
--------------------------------
The spaces in the lists indicate the positions of the pictures of persons
and curtain-like ornaments inserted here and there, as seen in Figs.
363-370.
In order to explain this series, we commence with that portion of it
found in the lower division of Plate 51 (Fig. 363).
Omitting any reference for the present to the black numbers over the day
columns, we call attention first to the days and to the red numerals
attached to them. Those in the division selected as an illustration are
as follows:
IV Ik. XII Cauac. VII Cib. II Been. X Oc. II Ezanab.
V Akbal. XIII Ahau. VIII Caban. III Ix. XI Chuen. III Cauac.
VI Kan. I Ymix. IX Ezanab. IV Men. XII Eb. IV Ahau.[317-1]
It will be observed that the week numbers of the days in each single
column follow one another in regular arithmetical order, thus: in the
first column, 4, 5, 6; in the second, 12, 13, 1; in the third, 7, 8, 9;
and so on throughout the entire series. The interval, therefore, between
the successive days of a column is 1; or, in other words, the days follow
one another in regular order, as in the month series, so that having the
first day of a column given we know at once the other two. It is
apparent, therefore, that the intervals between the three correspondingly
opposite days of any two associate columns are the same; that is to say,
the interval between 5 Akbal and 13 Ahau, in the first two columns given
above is the same as that between 4 Ik and 12 Cauac, and also as that
between 6 Kan and 1 Ymix. This is also true if the attached week numbers
are omitted; for instance, the interval between Ik and Cauac, counting on
the list of days forming the month, is 17 days, and it is the same
between Kan and Ymix. Taking the second and third columns we find here
the same interval. This holds good in that part of the series above given
until we reach the last two columns; here the interval between Oc and
Ezanab is 8 days and it is the same between the other days of these two
columns.
This being ascertained, the next step is to determine the true interval
between the first days of these columns, taking the numbers attached to
them into consideration. Referring to our calendar (Table II) and (for
reasons which will be given hereafter) using the Muluc column and
counting from 4 Ik, as heretofore explained, we find the interval between
this and 12 Cauac to be 8 months and 17 days; counting in the same way
from 12 Cauac, 8 months and 17 days more bring us to 7 Cib; 8 months and
17 days more to 10 Oc. So far the intervals have been the same; but at
this point we find a variation from the rule, as the interval between 10
Oc and 2 Ezanab (first of the next column) is 7 months and 8 days.
These intervals furnish the explanation of the red and black numerals
below the day columns.
These numerals, as the reader will observe by reference to Fig. 363 or
the written interpretation thereof in Table VI, are 8 and 17 under the
first five columns, but 7 and 8 under the sixth column, the red (8 under
the first five and 7 under the sixth) indicating the months and the black
(17 under the first five and 8 under the sixth) the days of the
intervals. This holds good throughout all that portion of the series
running through the lower divisions of Plates 51 to 58, with three
exceptions, which will now be pointed out.
In order to do this it will be necessary to repeat here a part of the
series on Plate 51_b_ and part of that on Plate 52_b_; that is, the two
right hand columns of the former and the two left hand columns of the
latter, between which is the singular picture shown in the _lower left
hand corner_ of our Fig. 364:
___________________________________________________________________
Plate 51_b_. || Plate 52_b_.
--------------------------++---------------------------------------
X Oc. | II Ezanab. || | XI Cib. | VI Been.
XI Chuen. | III Cauac. || | XII Caban. | VII Ix.
XII Eb. | IV Ahau. || [Picture.] | XIII Ezanab. | VIII Men.
VIII | VII || | VIII | VIII
17 | 8 || | 17 | 17
-------------------------------------------------------------------
As before stated, the interval between 10 Oc and 2 Ezanab is 7 months and
8 days, as indicated by the red and black numerals under the latter.
According to the red and black numbers under the column commencing with
11 Cib, the interval between 2 Ezanab and 11 Cib should be 8 months and
17 days, the usual difference, when, in fact, as we see by counting on
the calendar, it is 8 months and 18 days. That this variation cannot be
attributed to a mistake on the part of the author or of the artist is
evident from the fact that the interval between 11 Cib and 6 Been (first
of the next column) is 8 months and 17 days and that the difference
throughout the rest of the series follows the rule given; that is to say,
each is 8 months and 17 days, except at two other points where this
variation is found and at the regular intervals where the difference of
7 months and 8 days occurs.[319-1] Precisely the same variation occurs on
Plate 55_b_ in passing from the first to the second column and on Plate
56_b_ between columns 1 and 2.
Why these singular exceptions? It is difficult, if not impossible, for
us, with our still imperfect knowledge of the calendar system formerly in
vogue among the Mayas, to give a satisfactory answer to this question.
But we reserve further notice of it until other parts of the series have
been explained.
Reference will now be made to the three lines of black numerals
immediately above the day columns. Still confining our examinations to
the lower divisions, the reader's attention is directed to these lines,
as given in Tables VI, VII, IX, XI, XIII, XV, XVII, and XIX. As there are
three numbers in each short column we take for granted, judging by what
has been shown in regard to the series on Plates 46-50, that the lowest
of the three denotes days, the middle months, and the upper years, and
that the intervals are the same between these columns as between the day
columns under them. The correctness of this supposition is shown by the
following additions: Starting with the first or left hand column on Plate
51_b_, we add successively the differences indicated by the corresponding
red and black numbers under the day columns. If this gives in each case
(save the two or three exceptions heretofore referred to) the numbers in
the next column to the right throughout the series, the demonstration
will be complete.
Years. Months. Days.
14 16 14 First column on Plate 51_b_.
8 17
-- -- --
15 7 11 Second column on Plate 51_b_.
8 17
-- -- --
15 16 8 Third column on Plate 51_b_.
8 17
-- -- --
16 7 5 Fourth column on Plate 51_b_.
8 17
-- -- --
16 16 2 Fifth column on Plate 51_b_.
7 8
-- -- --
17 5 10 Sixth column on Plate 51_b_.
8 18[319-1]
-- -- --
17 14 8 First column on Plate 52_b_.
8 17
-- -- --
18 5 5 Second column on Plate 52_b_.
8 17
-- -- --
18 14 2 Third column on Plate 52_b_.
8 17
-- -- --
19 4 19 Fourth column on Plate 52_b_.
8 17
-- -- --
19 13 16 First column on Plate 53_b_.
7 8
-- -- --
20 3 4 Second column on Plate 53_b_.
At this point in the original, instead of 20 in the year series, we find
a diamond shaped symbol, represented by 0 in our tables, with one black
dot over it. From this it would seem that when this codex was written the
Maya method of counting years was by periods of 20 each, as in the case
of the month days. Whether there is any reference here to the ahaues is
uncertain. I am inclined to think with Dr. Foerstemann that it was rather
in consequence of the use of the vigesimal system in representing
numbers. It would have been very inconvenient and cumbersome to represent
high numbers by means of dots and lines; hence a more practicable method
was devised. It is evident, from the picture inserted at this point in
the series, that some important chronological event is indicated. Here
also in the written characters over this picture is the symbol for 20.
The last number given in the above addition may therefore, in order to
correspond with the method of the codex, be written as follows:
Twenty year periods. Years. Months. Days.
1 0 3 4
Continuing the addition in this way the result is as follows:
Twenty year periods. Years. Months. Days.
1 0 3 4
8 17
-- -- -- --
1 0 12 1 Third column on Plate 53_b_.
8 17
-- -- -- --
1 1 2 18 Fourth column on Plate 53_b_.
8 17
-- -- -- --
1 1 11 15 Fifth column on Plate 53_b_.
8 17
-- -- -- --
1 2 2 12 First column on Plate 54_b_.
8 17
-- -- -- --
1 2 11 9 Second column on Plate 54_b_.
8 17
-- -- -- --
1 3 2 6 Third column on Plate 54_b_.
7 8
-- -- -- --
1 3 9 14 Fourth column on Plate 54_b_.
8 17
-- -- -- --
1 4 0 11 Fifth column on Plate 54_b_.
8 17
-- -- -- --
1 4 0 8 First column on Plate 55_b_.
8 18[321-1]
-- -- -- --
1 5 0 6 Second column on Plate 55_b_.
8 17
-- -- -- --
1 5 9 3 Third column on Plate 55_b_.
8 17
-- -- -- --
1 6 0 0 Fourth column on Plate 55_b_.
8 17
-- -- -- --
1 6 8 17 Fifth column on Plate 55_b_.
8 17
-- -- -- --
1 6 17 14 Sixth column on Plate 55_b_.
8 17
-- -- -- --
1 7 8 11 Seventh column on Plate 55_b_.
7 8
-- -- -- --
1 7 15 19 Eighth column on Plate 55_b_.
8 17
-- -- -- --
1 8 6 16 First column on Plate 56_b_.
8 18[321-2]
-- -- -- --
1 8 15 14 Second column on Plate 56_b_.
8 17
-- -- -- --
1 6 6 11 Third column on Plate 56_b_.
8 17
-- -- -- --
1 9 15 8 Fourth column on Plate 56_b_.
8 17
-- -- -- --
1 10 6 5 First column on Plate 57_b_.
7 8
-- -- -- --
1 10 15 2 Second column on Plate 57_b_.
7 8
-- -- -- --
1 11 4 10 Third column on Plate 57_b_.
8 17
-- -- -- --
1 11 13 7 Fourth column on Plate 57_b_.
8 17
-- -- -- --
1 12 13 1 Fifth column on Plate 57_b_.
8 17
-- -- -- --
1 12 13 1 First column on Plate 58_b_.
8 17
-- -- -- --
1 13 3 18 Second column on Plate 58_b_.
The proof, therefore, that the theory advanced in regard to the order and
the plan of the series is correct seems to be conclusive. This probably
would have been conceded without the repeated additions given, but these
were deemed necessary because of several irregularities found in that
portion running through Plates 53_a_-58_a_, which constitutes the first
half of the series.
Turning back to our Table VIII, representing that part of the series on
Plate 53_a_, we will consider the three lines of black numerals above the
day columns, discussing the irregularities as we proceed.
The numbers in the first column are 7/17,[TN-6] or, according to the
explanation given, 7 months and 17 days. There is apparently a mistake
here, the correct numbers being 8 months and 17 days, as it is the usual
custom of the codex to commence numeral series with the prevailing
interval; moreover this correction, which has also been made by Dr.
Foerstemann, is necessary in order to connect rightly with what follows;
the counters under this first column require this correction, as they are
8 months, 17 days. Making this change we proceed with the addition.
Years. Months. Days.
8 17 First column, Plate 53_a_ (corrected).
8 17
-- --
17 14 Second column. Plate 53_a_.
Here the author of the codex has made another mistake or varied from the
plan of the series. As several similar variations or errors occur in this
part of the series, it will be as well to discuss the point here as
elsewhere. Dr. Foerstemann, in discussing the series, takes it for granted
that these variations are errors of the aboriginal scribe; he remarks
that "It is seen here that the writer has corrected several of his
mistakes by compensation. For instance, the two first differences should
be 177 [8 months, 17 days] and 148 [7 months, 8 days], not 176 and 149,"
&c.
This is a strained hypothesis which I hesitate to adopt so long as any
other solution of the difficulty can be found. It is more likely that the
writer would have corrected his mistakes, if observed, than that he would
compensate them by corresponding errors.
Going back to that part of the series in the lower divisions which has
already been examined and commencing with Plate 51_b_ (see Table VI), we
observe that the numbers in the lowest of the three lines of black
numerals, immediately over the day columns, and the first day of these
columns are as follows (omitting the week days attached):
14 11 8 5 2 10
Ik. Cauac. Cib. Been. Oc. Ezanab.
Turning to the calendar (Table II) and using the Muluc column, we notice
that the figures of this third line of black numerals denote respectively
the month numbers of the days under them; that is to say, Ik is the
fourteenth day of the month in Muluc years, Cauac the eleventh, Cib the
eighth, Been the fifth, Oc the second, and Ezanab the tenth. This holds
good through Plates 52_b_ to 58_b_ without a single exception, provided
the diamond shaped symbol in the fourth column of Plate 55_b_ is counted
as 20. This test, therefore, presents fewer exceptions than are found in
counting the intervals as before explained; yet, after all, this would
necessarily result from the fact that the day Muluc was selected as the
commencement of the series, and hence may have no signification in
reference to or bearing on the question of the year series, especially as
the years counted are evidently of 360 days.
Returning now to our Table VIII, representing Plate 53_a_, we observe
that the number immediately over Kan in the first column is 17, whereas
Kan is the sixteenth day of the month. Is it not possible that the
intention was to designate as the ceremonial day Chicchan, standing
immediately below, which is the seventeenth day of the month in Muluc
years? Even though there is no reference to Muluc years, the intervals
may be given upon the same idea, that of reaching, for some particular
reason, the second or third day of the column instead of the first. This
would account for the compensation of which Dr. Foerstemann speaks,
without implying any mistake on the part of the writer. These
irregularities would then be intentional variations from the order of the
series, yet so as not to break the general plan.
The interval between 6 Kan of the first column (with the month number
corrected) and 1 Ymix of the second is 8 months and 17 days, as it should
be; between 6 Muluc and 1 Cimi, 8 months and 17 days; and between 1 Cimi
and 9 Akbal, 8 months and 17 days, thus conforming to the rule heretofore
given, a fact which holds good as a general rule throughout that portion
of the series in the upper division.
Continuing the addition as heretofore we note the variations.
Years. Months. Days. Column. Plate.
17 14 Second. 53_a_.
7 8
-- -- --
1 7 3 Third. 53_a_.
8 17
-- -- --
1 15 19[323-1] Fourth. 53_a_.
8 17
-- -- --
2 6 16 Fifth. 53_a_.
8 17
-- -- --
2 15 13 Sixth. 53_a_.
8 18[323-2]
-- -- --
3 6 11 First. 54_a_.
8 17
-- -- --
3 15 8 Second. 54_a_.
8 17
-- -- --
4 6 5 Third. 54_a_.
8 17
-- -- --
4 15 2[324-1] Fourth. 54_a_.
8 17
-- -- --
5 5 19 Fifth. 54_a_.
8 17
-- -- --
5 14[324-2] 16 Sixth. 54_a_.
7 8
-- -- --
6 4 4 Seventh. 54_a_.
8 18[324-3]
-- -- --
6[324-4] 13 2 First. 55_a_.
8 17
-- -- --
7 3 19[324-5] Second. 55_a_.
8 17
-- -- --
7 12 16 Third. 55_a_.
8 17
-- -- --
8 3 13 Fourth. 55_a_.
8 17
-- -- --
8 12 10 Fifth. 55_a_.
7 8
-- -- --
9 1 18 First. 56_a_.
8 17
-- -- --
9 10 15 Second. 56_a_.
8 17
-- -- --
10 1 12 Third. 56_a_.
8 17
-- -- --
10 10 9 Fourth. 56_a_.
8 17
-- -- --
11 1 6 First. 57_a_.
8 17
-- -- --
11 10 3 Second. 57_a_.
8 17
-- -- --
12 1 0 Third. 57_a_.
7 8[325-1]
-- -- --
12 8 8 Fourth. 57_a_.
8 17
-- -- --
12 17 5 First. 58_a_.
8 17
-- -- --
13 8 2 Second. 58_a_.
8 18[325-2]
-- -- --
13 17 0 Third. 58_a_.
8 17
-- -- --
14 7 17 Fourth. 58_a_.
8 17
-- -- --
14 16 14 First. 51_b_.[325-3]
We have in what has thus far been given a satisfactory explanation of the
meaning and use of the lines of numerals and also of their relation to
the day columns, but we still fall short of a complete interpretation,
inasmuch as we are unable to give the series a definite location in the
Maya calendar or in actual time. It is apparent, however, that the series
cannot by any possible explanation be made to agree with the calendar
system as usually accepted, as there is nothing in it indicating the four
series of years or the year of 365 days. It may be safely assumed, I
think, from what has been shown, that the year referred to in the series
is one of 360 days, with probably a periodic addition of one day, but the
reason of the addition is not yet apparent.
If the numbers in the lowest line of numerals over the day columns
indicate the days of the month, and those of the middle line the
respective months of the year, it is evident, as before stated, that
Muluc is the first day of the year throughout, a conclusion
irreconcilable with the Maya calendar as hitherto understood. It is
probable, however, that the month and day numbers do not refer to
particular months and days, but are used only as intervals of time
counted from a certain day, which must in this case have been Muluc.
The sum of the series as shown by the numbers over the second column of
Plate 58_b_ 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. This makes the sum of the entire
series 33 years, 4 months, or 11,960 days, precisely 46 cycles of 13
months, or 260 days each, the whole and also each cycle commencing with
13 Muluc and ending with 12 Lamat. It is also worthy of notice that in
the right hand column of characters (hieroglyphics) over the inverted
figure in Plate 58_b_ two numbers, 13 and 12, are found attached to
characters which appear to be abnormal forms of month symbols.
On Plates 63 and 64 are three series of ten day columns each and three
lines of numerals over each series. These are as follows, so far as they
can be made out, the numbers over the upper series being mostly
obliterated. The 0 denotes the red, diamond shaped symbol which is here
sometimes given in fanciful forms.
TABLE XX.--_Table showing series of day columns, with lines of numerals._
UPPER DIVISION.
__________________________________________________________________________
| |
Plate 63. | Plate 64. |
--------------------------------------------+-----------------------------|
4 | 3 | | | 0 |
8 | 6 | | 0 | 16 |
0 | 0 | 0 | 0 | 0 |
III Chicchan.| III Chicchan.| III Chicchan.| III Chicchan.| III Chicchan.|
Kan. | Kan. | Kan. | Kan. | Kan. |
Ix. | Ix. | Ix. | Ix. | Ix. |
Cimi. | Cimi. | Cimi. | Cimi. | Cimi. |
XIII Akbal. |XIII Akbal. |XIII Akbal. |XIII Akbal. |XIII Akbal. |
______________|______________|______________|______________|______________|
|
Plate 64. |
____________________________________________ _____________________________|
| | | | |
0 | 0 | | | |
12 | 8 | 3 | 3 | |
0 | 0 | 0 | 0 | 0 |
III Chicchan.| III Chicchan.| III Chicchan.| III Chicchan.| III Chicchan.|
Kan. | Kan. | Kan. | Kan. | Kan. |
Ix. | Ix. | Ix. | Ix. | Ix. |
Cimi. | Cimi. | Cimi. | Cimi. | Cimi. |
XIII Akbal. |XIII Akbal. |XIII Akbal. |XIII Akbal. |XIII Akbal. |
______________|______________|______________|______________|______________|
MIDDLE DIVISION.[TN-7]
___________________________________________________________________________
| | | | |
XIX 5 | 4 | 4 | 4 | 4 |
IV 1 | 14 | 9 | 5 | 0 |
IV 0 | 0 | 0 | 7 | 16 |
III Chicchan.| III Ix. | III Akbal. | III Eb. | III Ymix. |
Kan. | Been. | Ik. | Chuen. | Ahau. |
Ix. | Akbal. | Eb. | Ymix. | Oc. |
Cimi. | Men. | Kan. | Been. | Ik. |
XIII Akbal. |XIII Eb. |XIII Ymix. |XIII Oc. |XIII Cauac. |
______________|______________|______________|______________|______________|
| | | | |
3 | 3 | 3 | 3 | 2 |
14 | 9 | 5 | 0 | 14 |
5 | 14 | 3 | 12 | 1 |
III Oc. | III Cauac. | III Lamat. | III Caban. | III Cimi. |
Muluc. | Ezanab. | Manik. | Cib. | Chicchan.|
Cauac. | Lamat. | Caban. | Cimi. | Men. |
Chuen. | Ahau. | Muluc. | Ezanab. | Manik. |
XIII Lamat. |XIII Caban. |XIII Cimi. |XIII Men. |XIII Kan. |
______________|______________|______________|______________|______________|
LOWER DIVISION.[TN-7]
___________________________________________________________________________
| | | | |
2 | 2 | 2 | 1 | 1 |
9 | 4 | 0 | 13 | 9 |
10 | 19 | 8 | 17 | 6 |
III Men. | III Kan. | III Been. | III Ik. | III Chuen. |
Ix. | Akbal. | Eb. | Ymix. | Oc. |
Kan. | Been. | Ik. | Chuen. | Ahau. |
Cib. | Chicchan.| Ix. | Akbal. | Eb. |
XIII Been. |XIII Ik. |XIII Chuen. |XIII Ahau. |XIII Muluc. |
______________|______________|______________|______________|______________|
| | | | |
1 | 1 | | | |
4 | 0 | 13 | 9 | 4 |
15 | 4 | 13 | 2 | 11 |
III Ahau. | III Muluc. | III Ezanab. | III Manik. | III Cib. |
Cauac. | Lamat. | Caban. | Cimi. | Men. |
Muluc. | Ezanab. | Manik. | Cib. | Chicchan.|
Ymix. | Oc. | Cauac. | Lamat. | Caban. |
XIII Ezanab. |XIII Manik. |XIII Cib. |XIII Chicchan.|XIII Ix. |
______________|______________|______________|______________|______________|
By examining carefully the lines and columns of the middle and lower
divisions of the plates--those represented in Tables XXI and XXII--we
ascertain that the two together form one series; but, contrary to the
method which has prevailed in those examined, it is to be read from
_right_ to _left_, commencing with the right hand column of the lower and
ending with the left hand column of the middle division.
As proof of this we have only to note the fact that the series of black
numerals over the day columns ascends towards the left. Assuming the
lowest of the three lines to be days, the middle one months, and the
upper one years, the common difference is 4 months and 11 days. Numbering
the ten columns of each of our tables from left to right as usual and
adding successively the common difference, commencing with the tenth
column of the lowest division, of which Cib is the first day, the result
will be as follows:
Years. Months. Days.
4 11 Over tenth column, lower division.
4 11
-- --
9 2 Over ninth column, lower division.
4 11
-- --
13 13 Over eighth column, lower division.
4 11
-- --
1 0 4 Over seventh column, lower division.
4 11
-- -- --
1 4 15 Over sixth column, lower division.
4 11
-- -- --
1 9 6 Over fifth column, lower division.
4 11
-- -- --
1 13 17 Over fourth column, lower division.
4 11
-- -- --
2 0 8 Over third column, lower division.
4 11
-- -- --
3 4 19 Over second column, lower division.
4 11
-- -- --
2 9 10 Over first column, lower division.
4 11
-- -- --
2 14 1 Over tenth column, middle division.
4 11
-- -- --
3 0 12 Over ninth column, middle division.
4 11
-- -- --
3 5 3 Over eighth column, middle division.
4 11
-- -- --
3 9 14 Over seventh column, middle division.
4 11
-- -- --
3 14 5 Over sixth column, middle division.
4 11
-- -- --
4 0 16 Over fifth column, middle division.
4 11
-- -- --
4 5 7 Over fourth column, middle division.
4 11
-- -- --
4 9 18 Over third column, middle division.
4 11
-- -- --
4 14 9 Over second column, middle division.
4 11
-- -- --
5 1 0 Over first column, middle division.
The red numerals over the first column of the middle division, except the
lowest diamond shaped one, are omitted, as they do not appear to belong
to the series.
It must be borne in mind that the 4 months and 11 days form the common
difference between the corresponding days of the columns counting from
right to left; that is to say, counting 4 months and 11 days from the top
day of any column will bring us to the first or top day of the next
column to the left. The interval between the other corresponding days of
the columns is also the same if the same week numbers are assigned them.
This question arises here, Does the difference include the time embraced
in the entire column? That is to say, Is this interval of 4 months and 11
days (referring, for example, to the tenth and ninth columns of the lower
division, our table) the sum of the intervals between 3 Cib and Men; Men
and Chicchan; Chicchan and Caban; Caban and 13 Ix, and 13 Ix of the tenth
column and 3 Manik of the ninth column? If not, the columns do not form a
continuous series or must be taken in some other order.
Although Dr. Foerstemann discovered the order in which the series as a
whole was to be read, and also the common difference--given, as is his
custom, in days--he failed to furnish further explanation of the group.
In answer to the question presented I call attention to the following
facts:
Commencing again with the uppermost day, 3 Cib, of the tenth column,
lowest division, and counting on the calendar to 13 Ix of the same year,
the interval is found to be 10 months and 18 days, which is much more
than the interval between 3 Cib and 3 Manik (first of the ninth column),
and of course cannot be included in it.
Reversing the order in reading the columns, but counting forward on the
calendar as usual, we find the interval between 13 Ix and 3 Cib to be 2
months and 2 days, and, what is another necessary condition, the
intermediate days of the column are included in this period in the order
in which they stand, if read upwards. The interval between 3 Cib,
uppermost day of the tenth column, and 13 Chicchan, bottom day of the
ninth column, is 2 months and 9 days. The sum of these two intervals is 4
months and 11 days, as it should be on the supposition that the entire
columns follow one another in regular succession. This proves beyond
question that the columns are to be read from _bottom_ to _top_ and that
they follow one another from _right_ to _left_. This enables us to fix
the week numbers to the intermediate days and to determine the day to
which the entire series is referred as its starting point. The days and
their numbers of the tenth and ninth columns of the lower division,
writing them in reverse order, that is, from bottom to top, are as
follows: 13 Ix; 3 Caban; 11 Chicchan; 8 Men; 3 Cib; 13 Chicchan; 3 Lamat;
11 Cib; 8 Cimi; 3 Manik.
These numbers hold good throughout the series.
Commencing with 13 Ix, the lowest day of the tenth column, lower
division, but first day of the series, and ending with 13 Akbal, the
bottom of the first column, middle series, the time embraced is 5 years,
1 month, 0 day, less 4 months and 11 days--that is, 4 years, 14 months, 9
days (years of 360 days being understood). This is easily proved by
counting on the calendar 4 years, 14 months, and 9 days from 13 Ix, as it
brings us to 13 Akbal. If we add to this time 2 months and 2 days--the
interval between 13 Akbal and 3 Chicchan (top day of first column, middle
division)--we have, as the entire period embraced in the series as it
stands--from 13 Ix (first of the series) to 3 Chicchan (the last)--4
years, 16 months, 11 days. Add to this 4 months and 11 days, in order to
reach the day with which the count begins, and we have as the entire
period 5 years, 3 months, 2 days = 5 years, 1 month, 0 day + 2 months, 2
days. If we count back 4 months and 11 days from 13 Ix (first of the
series), we reach 1 Kan, the day to which the series is referred as its
starting point. Counting forward from this date 5 years, 3 months and 2
days brings us to 3 Chicchan, the last day of the series.
It is worthy of notice that, although this series appears to be referred
to Kan years, it is at variance with the idea of passing from one to the
other of the four year series, and is, moreover, based upon the year of
360 days. The order in which it is to be read, which is true also of some
other pages, indicates that these extracts pertain to a different
original codex than those to which we have heretofore alluded, a
conclusion reached by Dr. Foerstemann soon after he commenced the study of
the Dresden manuscript.
I was for a time inclined to believe there was a break between Plates 64
and 65, as there appeared to be no day columns with which the lines of
numerals running through Plates 65-69 could be connected, but the fact
that the sum of the black numbers in each is 91, precisely the interval
between the corresponding days of the columns in Plates 63 and 64, will
probably warrant the conclusion that they are connected with them. This
conclusion is strengthened, so far as those in the lower division are
concerned, by the fact that by taking the XIII attached to the lowest
days of the columns the numbers properly succeed one another and the
series conforms to the rule heretofore given. As proof of this I give
here the lower line of the lower division, prefixing the XIII, thus:
XIII; 9, IX; 5, I; 1, II; 10, XII; 6, V; 2, VII; 11, V; 7, XII; 3, II;
12, I; 8, IX; 4, XIII; 13, XIII.
Adding together the numbers and casting out the thirteens, thus, XIII + 9
- 13 = IX; IX + 5 - 13 = I, &c., the connection is seen to be regular.
The final red numeral is XIII, the same as that with which the series
begins, and the sum of the black numbers, 9, 5, 1, 10, 6, 2, 11, 7, 3,
12, 8, 4, 13, is 91, a multiple of 13. The middle line of numerals also
connects with the XIII attached to the bottom symbols of the day columns;
and the upper line of numerals connects with the III attached to the top
symbols of the day columns.
Plates 70 to 73 present some peculiarities difficult to account for. That
these pages belong to the same type as 62, 63, and 64 cannot be doubted,
and that as a general rule they are to be read from right to left is
easily proved; but this method does not seem to be adopted throughout,
the order being apparently reversed in a single series.
The aboriginal artist has apparently made up these pages from two older
manuscripts or changed and added to his original. The last two columns of
Plate 70 and first five of 71 appear to have been thrust in here as an
afterthought or as a fragment from some other source, forming apparently
no legitimate connection with the series to either the right or to the
left of them. It is true, as will be shown, that there is some connection
with the lowest series on the right, but it would seem that advantage was
here taken of accidental correspondence rather than that this
correspondence was the result of a preconceived plan.
Commencing in the lower part of the middle division of Plate 73 and
running back (to the left) to the sixth column of 71 and returning to the
lower part of the lower division of 73 and ending with the sixth column
of 71, is the following series. The columns are given in the order in
which they stand on the respective plates, but the plates are taken in
reverse order:
TABLE XXIII.--_Table giving comparison between Plates 71, 72, and 73._
_______________________________________________________________________________
| First | Second | Third | Fourth | Fifth | |
| column. | column. |column. | column. | column. | |
---------+---------+---------+---------+---------+---------+---------+---------
Plate 73,| 16 | 13 | 9 | 6 | 3 | -- | --
middle | 5 | 0 | 15 | 10 | 5 | -- | --
division |IV Caban.| IV Eb. |IV Manik.| IV Ik. |IV Caban.| -- | --
-------------------------------------------------------------------------------
_______________________________________________________________________________
| First | Second | Third | Fourth | Fifth | Sixth |Seventh
| column. | column. | column. | column. | column. | column. |column.
---------+---------+---------+---------+---------+---------+---------+---------
Plate 72,| 2 | 1 | 1 | 1 | 1 | 1 | --
middle | 3 | 17 | 14 | 11 | 8 | 4 | 19
division | 0 | 15 | 10 | 5 | 0 | 15 | 10
| IV Eb. |IV Manik.| IV Ik. |IV Caban.| IV Eb. |IV Manik.|IV Ik.
-------------------------------------------------------------------------------
_______________________________________________________________________________
| | | | | | Sixth | Seventh
| | | | | | column. | column.
---------+---------+---------+---------+---------+---------+---------+---------
Plate 71,| -- | -- | -- | -- | -- | 2 | 2
middle | -- | -- | -- | -- | -- | 9 | 6
division | -- | -- | -- | -- | -- | 10 | 5
| -- | -- | -- | -- | -- | IV Ik. |IV Caban.
-------------------------------------------------------------------------------
_______________________________________________________________________________
| First | Second | Third | Fourth | Fifth | |
| column. | column. | column. | column. | column. | |
---------+---------+---------+---------+---------+---------+---------+---------
Plate 73,| 3 | 3 | 3 | 2 | 2 | -- | --
lower | 7 | 3 | 1 | 16 | 12 | -- | --
division | 15 | 10 | 5 | 0 | 15 | -- | --
|IV Manik.| IV Ik. |IV Caban.| IV Eb. |IV Manik.| -- | --
-------------------------------------------------------------------------------
_______________________________________________________________________________
| First | Second | Third | Fourth | Fifth | Sixth |Seventh
| column. | column. | column. | column. | column. | column. |column.
---------+---------+---------+---------+---------+---------+---------+---------
Plate 72,| 4 | 4 | 4 | 4 | 2 | 3 | 3
lower | 12 | 9 | 6 | 2 | 17 | 14 | 11
division | 10 | 5 | 0 | 15 | 10 | 5 | 0
| IV Ik. |IV Caban.| IV Eb. |IV Manik.| IV Ik. |IV Caban.| IV Eb.
-------------------------------------------------------------------------------
_______________________________________________________________________________
| | | | | | Sixth | Seventh
| | | | | | column. | column.
---------+---------+---------+---------+---------+---------+---------+---------
Plate 71,| -- | -- | -- | -- | -- | 5 | 4
lower | -- | -- | -- | -- | -- | 1 | 15
division | -- | -- | -- | -- | -- | 0 | 15
| -- | -- | -- | -- | -- | IV Eb. |IV Manik.
-------------------------------------------------------------------------------
The interval between the successive days, counting to the left, is in
each case 3 months and 5 days, corresponding with the numbers over IV
Caban, fifth column, middle division, Plate 73. Commencing with this
number and adding it successively, we obtain the numbers over the various
columns:
Years. Months. Days.
3 5 Over fifth column, middle division, Plate 73.
3 5
-- --
6 10 Over fourth column, middle division, Plate 73.
3 5
-- --
9 15 Over third column, middle division, Plate 73.
3 5
-- --
13 0 Over second column, middle division, Plate 73.
3 5
-- --
16 5 Over first column, middle division, Plate 73.
3 5
-- --
1 1[333-1] 10 Over seventh column, middle division, Plate 72.
3 5
-- -- --
1 4 15 Over sixth column, middle division, Plate 72.
3 5
-- -- --
1 8 0 Over fifth column, middle division, Plate 72.
3 5
-- -- --
1 11 5 Over fourth column, middle division, Plate 72.
3 5
-- -- --
1 14 10 Over third column, middle division, Plate 72.
3 5
-- -- --
1 17 15 Over second column, middle division, Plate 72.
3 5
-- -- --
2 3 0 Over first column, middle division, Plate 72.
3 5
-- -- --
2 6 5 Over seventh column, middle division, Plate 71.
3 5
-- -- --
2 9 10 Over sixth column, middle division, Plate 71.
3 5
-- -- --
2 12 15 Over fifth column, lower division, Plate 73.
3 5
-- -- --
2 16 0 Over fourth column, lower division, Plate 73.
3 5
-- -- --
3 1 5 Over third column, lower division, Plate 73.
3 5
-- -- --
3 4 10 Over second column, lower division, Plate 73.
3 5
-- -- --
3 7 15 Over first column, lower division, Plate 73.
3 5
-- -- --
3 11 0 Over seventh column, lower division, Plate 72.
3 5
-- -- --
3 14 5 Over sixth column, lower division, Plate 72.
3 5
-- -- --
3 17 10 Over fifth column, lower division, Plate 72.
3 5
-- -- --
4 2 15 Over fourth column, lower division, Plate 72.
3 5
-- -- --
4 6 0 Over third column, lower division, Plate 72.
3 5
-- -- --
4 9 5 Over second column, lower division, Plate 72.
3 5
-- -- --
4 12 10 Over first column, lower division, Plate 72.
3 5
-- -- --
4 15 15 Over seventh column, lower division, Plate 71.
3 5
-- -- --
5 1 0 Over sixth column, lower division, Plate 71.
It is worthy of notice that the sum of the series as expressed by the
final numbers is precisely that of the series on the middle and lower
divisions of Plates 63 and 64, heretofore given, and embraces seven
complete cycles of 13 months, or 260 days each. Counting back three
months and five days from 4 Caban (the day in the fifth column, middle
division, of Plate 73) we reach 5 Been as the starting point of the
series.
As there can be no doubt that the lines and days of the two divisions
form together one unbroken series, it is evident there is no connection
between that portion of it in the middle division and what lies to the
left of it in Plate 71; but there does appear to be, as before indicated,
some connection between the conclusion and what follows to the left in
the lower portion of 71. The series which lies to the left at this point
is as follows:
TABLE XXIV.--_Table showing relations of Plates 70 and 71._
________________________________________________________________
Plate 70. | Plate 71.
----------------+-----------------------------------------------
5th | 6th | 1st | 2d | 3d | 4th | 5th | 6th
column.|column. |column.|column.|column.|column.|column.|column.
6 | 5 | 4 | 3 | 2 | | |
1 | 1 | 0 | 0 | 0 | 15 | 10 | 5
6 | 2 | 16 | 12 | 8 | 3 | 2 | 1
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
IV Eb.| IV Eb. | IV Eb.| IV Eb.| IV Eb.| IV Eb.| IV Eb.| IV Eb.
---------------------------------------------------------------
For the purpose of assisting the reader to see the relation more clearly,
the last column of the preceding series--sixth of the lower division on
Plate 71--is added at the right as it stands in the original.
It is apparent that the figures in the fifth column of 71 are exactly
double those in the sixth column. This and the fact that the day IV Eb is
the same as those following are the only indications that there is any
connection between the series. Using the 5 years and 1 month as the
common difference and adding, the result is as follows:
Years. Months. Days.
5 1 0 Sixth column, lower division, Plate 71.
5 1 0
-- -- --
10 2 0 Fifth column, lower division, Plate 71.
5 1 0
-- -- --
15 3 0 Fourth column, lower division, Plate 71.
At this point another change occurs: the former difference is added to
the last figures and the sum is doubled.
Twenty year periods. Years. Months. Days.
15 3 0
5 1 0
-- -- -- --
1 0 4 0
2
-- -- -- --
2 0 8 0 Third column lower division,
Plate 71.
1 0 4 0
-- -- -- --
3 0 12 0 Second column, lower division,
Plate 71.
1 0 4 0
-- -- -- --
4 0 16 0 First column, lower division,
Plate 71.
1 0 4 0
-- -- -- --
5 1 2 0 Sixth column, lower division,
Plate 70.
1 0 4 0
-- -- -- --
6 1 6 0 Fifth column, lower division,
Plate 70.
This series does not end at this point, but is continued in the lines
immediately above, which are as follows:
TABLE XXV.--_Table showing relations between Plates 70 and 71._
________________________________________________________
Plate 70. | Plate 71.
----------------+---------------------------------------
5th 6th | 1st 2d 3d 4th 5th
column. column. |column. column. column. column. column.
1 |
0 8(?) | 15 13 10 9 7
| XII
12 1 | 3 2 2 2 1
| II
3 10 | 6 16 4 0 10
| XII
0 0 | 0 0 (?) 0 0
IV Eb. IV Eb. | IV Eb. IV Eb. IV Eb. IV Eb. IV Eb.
--------------------------------------------------------
Adding the difference, 1, 0, 4, 0, to the final result of the preceding
addition we obtain the figures of the right hand column (fifth column,
Plate 71) of this series:
6 1 6 0
1 0 4 0
-- -- -- --
7 1 10 0
To obtain the figures of the fourth column this difference must be
doubled, thus
7 1 10 0
2 0 8 0
-- -- -- --
9 2 0 0
To obtain the black numbers of the next (third) column, the lower cipher
symbol of which is wanting, we add the former difference:
9 2 0 0
1 0 4 0
-- -- -- --
10 2 4 0
This decrease in the difference is unusual and indicates some error. This
idea seems to be confirmed in the following way: In order to obtain the
numbers of the next (second) column it is necessary to add three times
the former difference, thus:
10 2 4 0
3 0 12 0
-- -- -- --
13 2 16 0 Second column, Plate 71.
If the increased difference, 2, 0, 8, 0, were retained after its
appearance the result would be as follows:
7 1 10 0 Fifth column, Plate 71.
2 0 8 0
-- -- -- --
9 2 0 0 Fourth column, Plate 71.
2 0 8 0
-- -- -- --
11 2 8 0 Third column, Plate 71.
2 0 8 0
-- -- -- --
13 2 16 0 Second column, Plate 71.
2 0 8 0
-- -- -- --
15 3 6 0 First column, Plate 71.
Adding the difference, 2, 0, 8, 0, to the third column, Plate 71, thus:
10 2 4 0
2 0 8 0
-- -- -- --
12 2 12 0
we obtain the red numerals inserted in the third column. It is probable
that the original or some subsequent scribe, observing an error at this
point, inserted these figures as a correction. If so, he failed to remedy
the confusion apparent in this portion of the series. The sum of the
entire series is 303 years (360 days each) and six months, equal to 420
cycles of 260 days.
I am strongly inclined to believe that this section and also pages 24 and
59 are interpolations by some aboriginal artist of a mathematical turn
and advanced ability in this direction, who has given these high series
more as curiosities than with reference to any specific dates or periods
of time.
[Illustration: FIG. 371. Specimens of ornamental loops from page 72,
Dresden Codex.]
Commencing in the sixth column of Plate 71_a_ and running through 72_a_
to the second column of 73_a_, is a numeral series which presents some
peculiarities that baffle all attempts at explanation. Contrary to the
rule which prevails in these pages it ascends from left to right and has
no day symbols connected with it. In addition to this, the numbers of its
lowest line are inclosed in loops of the form here shown (Fig. 371) and
have no apparent connection with the other lines of the series, but, on
the contrary, if taken from right to left, they present in the order
usually given the numbers of the ahaues or katunes.[337-1] It is as
follows:
1 1 1 1 1 1 1
2 5 8 10 13 16 0 3 6 9 11 14 17
II.
14 8 2 16 10 4 18 12 6 0 14 7(?) 2 XIV
[(11)][(13)][(2)][(4)][(6)][(8)][(10)][(12)][(1)][(3)][(5)][(7)][(9)]
The last (thirteenth) column of this series is not in a line with the
others, but is found in the lower part of the right hand column of Plate
73, and in connection with it we find the red numerals II and XIV,
denoting the difference between the columns, as is apparent from the
additions here given:
Years. Months. Days.
2 14 First or left hand column.
2 14
-- --
5 8 Second column.
2 14
-- --
8 2 Third column.
2 14
-- --
10 16 Fourth column.
2 14
-- --
13 10 Fifth column.
2 14
-- --
16 4 Sixth column[TN-8]
2 14
-- --
1 0 18 Seventh column.
2 14
-- -- --
1 3 12 Eighth column.
2 14
-- -- --
1 6 6 Ninth column.
2 14
-- -- --
1 9 0 Tenth column.
2 14
-- -- --
1 11 14 Eleventh column.
2 14
-- -- --
1 14 8[338-1] Twelfth column.[TN-9]
2 14
-- -- --
1 17 2 Thirteenth column.
FOOTNOTES:
[261-1] The work here referred to is entitled Die Mayahandschrift der
Koeniglichen oeffentlichen Bibliothek zu Dresden, herausgegeben von Prof.
Dr. E. Foerstemann, Hofrat und Oberbibliothekar. It contains, besides the
chromolithographs of the 74 plates, an introduction published at Leipzig,
1880, 4^o.
[269-1] A Study of the Manuscript Troano, by Cyrus Thomas, pp. 7-15.
[272-1] This method will be adopted throughout this paper where figures
containing numerals are introduced.
[273-1] In the representations of lines and columns of the codex Roman
numbers are necessarily used to distinguish the class of numerals, yet in
the text, as in this case, the Arabic numbers will be used as most
convenient.
[273-2] Strictly speaking, the interval between 11 Men and 13 Oc is
fourteen days, but throughout this paper, by "_interval between_" two
days, is to be understood the number of days to be counted _from_ one _to
and including_ the other. The one counted from is always _excluded_ and
the one reached or with which the interval terminates is always
_included_.
[273-3] Science, p. 459, April 11, 1884.
[277-1] Throughout this paper when the words "figure" and "character" are
used in reference to what appears in the codex, they are to be understood
as follows: "figure" refers to the picture, as of a person, animal, or
other object in the spaces; "character" refers to the hieroglyphics or
written symbols.
[278-1] Study of the Manuscript Troano, by Cyrus Thomas, Chapters II and
VII.
[278-2] Erlaeuterungen zur Mayahandschrift, p. 2.
[280-1] Erlaeuterungen zur Mayahandschrift, p. 16.
[280-2] Bureau of Eth., Third Ann. Rep., pp. 16 et seq.
[282-1] Study of the Manuscript Troano, by Cyrus Thomas, pp. 15, 16.
[282-2] Dechiffrement des ecritures calculiformes ou Mayas, par M. le
C^te H. de Charency, Alencon, 1849; also, Melanges, pp. 185-195.
[283-1] For an explanation of the principle upon which these day columns
were formed, see "Notes on certain Maya and Mexican manuscripts," by
Cyrus Thomas, published in the Third Annual Report of the Bureau of
Ethnology.
[290-1] The symbol for this day in Kingsborough resembles Lamat, but the
photographic copy makes it Ix, as it should be.
[290-2] Foerstemann, Erlaeuterungen zur Mayahandschrift, p. 42.
[291-1] Erlaeuterungen zur Mayahandschrift, p. 36.
[292-1] Erlaeuterungen zur Mayahandschrift, p. 60.
[293-1] Erlaeuterungen zur Mayahandschrift, p. 56.
[296-1] The bottom lines are selected because they are less injured in
the codex than the top lines, which are in most cases entirely
obliterated.
[300-1] 3 days in ms., should be 4.
[317-1] The third symbol in the last day column of Plate 51_b_ is Been in
the codex; but this is an evident mistake, as shown by the order of the
days, since Ahau, which has been substituted above, always follows Cauac.
This may be seen by reference to the middle column of 57_b_.
[319-1] This is one of the exceptional cases.
[321-1] Second exception.
[321-2] Third exception.
[323-1] One line has been omitted in the numeral symbol.
[323-2] Here we have again the added day.
[324-1] The 8 at this point in the codex is an evident error.
[324-2] Here is also an error in the original, this being 10.
[324-3] The symbols require an additional day here.
[324-4] The 8 in the year line in the original is a manifest error, as 6
precedes and 7 follows.
[324-5] The 18 in the day line at this point is also an error, as the
interval between 2 Muluc and 10 Cimi is 8 months and 17 days. Moreover,
the next day number being 16 requires this to be 19.
[325-1] The counters in the original at this point are certainly wrong,
for here should be 7 months and 8 days, whereas the symbols are those for
8 months and 17 days.
[325-2] Here we have again the additional day.
[325-3] Added to show connection with the lower series.
[333-1] Codex has 19, which is equivalent to 1 year and 1 month.
[337-1] While reading the final proof I fortunately discovered what may
prove to be the correct explanation of the numbers in the loops.
At the commencement of the series on Plate 71 and at its close on Plate
73 we observe the symbol of the day, 9 Ix. Starting from this date and
counting forward on the calendar two months and fourteen days, we reach
11 Lamat. This gives the number in the first loop of the series. Two
months and fourteen days more bring us to 13 Ik, the number in the second
loop; two months and fourteen days to 2 Cib, the number in the third
loop, and so on to the end. It is therefore probable that the numerals in
the loops indicate the week numbers of the days, though these are usually
expressed in red symbols.
[338-1] The 7 in the twelfth column is an error; it should be 8, as an
inspection shows the place of the missing dot. The additions make it
clear that the numbers of the second line refer to months, those of the
line below them to days, and those of the line above to years. The series
is, therefore, apparently complete without the numbers inclosed in the
loops.