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 a.s.sumed, 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. .h.i.therto 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. a.s.suming 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 a.s.signed 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. Forstemann 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 pa.s.sing 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. Forstemann soon after he commenced the study of the Dresden ma.n.u.script.

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 ma.n.u.scripts 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.