Chapter I: The Numerals in the Dresden Codex (4)

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.

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.