An Introduction to the Study of the Maya Hieroglyphs/Chapter 6
Chapter VI
THE CODICES
The present chapter will treat of the application of the material presented in Chapters III and IV to texts drawn from the codices, or hieroglyphic manuscripts; and since these deal in great part with the tonalamatl, or sacred year of 260 days, as we have seen (p. 31), this subject will be taken up first.
Texts Recording Tonalamatls
The tonalamatl, or 260day period, as represented in the codices is usually divided into five parts of 52 days each, although tonalamatls of four parts, each containing 65 days, and tonalamatls of ten parts, each containing 26 days, are not at all uncommon. These divisions are further subdivided, usually into unequal parts, all the divisions in one tonalamatl, however, having subdivisions of the same length.
So far as its calendric side is concerned,^{[1]} the tonalamatl may be considered as having three essential parts, as follows:
1. A column of day signs.
2. Red numbers, which are the coefficients of the day signs.
3. Black numbers, which show the distances between the days designated by (1) and (2).
The number of the day signs in (1), usually 4, 5, or 10, shows the number of parts into which the tonalamatl is divided. Every red number in (2) is used once with every day sign in (1) to designate a day which is reached in counting one of the black numbers in (3) forward from another of the days recorded by (1) and (2). The most important point for the student to grasp in studying the Maya tonalamatl is the fundamental difference between the use of the red numbers and the black numbers. The former are used only as day coefficients, and together with the day signs show the days which begin the divisions and subdivisions of the tonalamatl. The black numbers, on the other hand, are exclusively time counters, which show only the distances between the dates indicated by the day signs and their corresponding coefficients among the red numbers. They show in effect the lengths of the periods and subperiods into which the tonalamatl is divided.
Most of the numbers, that is (2) and (3), in the tonalamatl are presented in a horizontal row across the page or pages^{[2]} of the manuscript, the red alternating with the black. In some instances, however, the numbers appear in a vertical column or pair of columns, though in this case also the same alternation in color is to be observed. More rarely the numbers are scattered over the page indiscriminately, seemingly without fixed order or arrangement.
It will be noticed in each of the tonalamatls given in the following examples that the record is greatly abbreviated or skeletonized. In the first place, we see no month signs, and consequently the days recorded are not shown to have had any fixed positions in the year. Furthermore, since the year positions of the days are not fixed, any day could recur at intervals of every 260 days, or, in other words, any tonalamatl with the divisions peculiar to it could be used in endless repetition throughout time, commencing anew every 260 days, regardless of the positions of these days in succeeding years. Nor is this omission the only abbreviation noticed in the presentation of the tonalamatl. Although every tonalamatl contained 260 days, only the days commencing its divisions and subdivisions appear in the record; and even these are represented in an abbreviated form. For example, instead of repeating the numerical coefficients with each of the day signs in (1), the coefficient was written once above the column of day signs, and in this position was regarded as belonging to each of the different day signs in turn. It follows from this fact that all the main divisions of the tonalamatl begin with days the coefficients of which are the same. Concerning the beginning days of the subdivisions, a still greater abbreviation is to be noted. The day signs are not shown at all, and only their numerical coefficients appear in the record. The economy of space resulting from the above abbreviations in writing the days will appear very clearly in the texts to follow.
In reading tonalamatls the first point to be determined is the name of the day with which the tonalamatl began. This will be found thus:
Rule 1. To find the beginning day of a tonalamatl, prefix the first red number, which will usually be found immediately above the column of the day signs, to the uppermost^{[3]} day sign in the column.
From this day as a starting point, the first black number in the text is to be counted forward; and the coefficient of the day reached will be the second red number in the text. As stated above, the day signs of the beginning days of the subdivisions are always omitted. From the second red number, which, as we have seen, is the coefficient of the beginning day of the second subdivision of the first division, the second black number is to be counted forward in order to reach the third red number, which is the coefficient of the day beginning the third subdivision of the first division. This operation is continued until the last black number has been counted forward from the red number just preceding it and the last red number has been reached.
This last red number will be found to be the same as the first red number, and the day which the count will have reached will be shown by the first red number (or the last, since the two are identical) used with the second day sign in the column. And this latter day will be the beginning day of the second division of the tonalamatl. From this day the count proceeds as before. The black numbers are added to the red numbers immediately preceding them in each case, until the last red number is reached, which, together with the third day sign in the column, forms the beginning day of the third division of the tonalamatl. After this operation has been repeated until the last red number in the last division of the tonalamatl has been reached—that is, the 260th day—the count will be found to have reentered itself, or in other words, the day reached by counting forward the last black number of the last division will be the same as the beginning day of the tonalamatl.
It follows from the foregoing that the sum of all the black numbers multiplied by the number of day signs in the column—the number of main divisions in the tonalamatl—will equal exactly 260. If any tonalamatl fails to give 260 as the result of this test, it may be regarded as incorrect or irregular.
The foregoing material may be reduced to the following:
Rule 2. To find the coefficients of the beginning days of succeeding divisions and subdivisions of the tonalamatl, add the black numbers to the red numbers immediately preceding them in each case, and, after subtracting all the multiples of 13 possible, the resulting number will be the coefficient of the beginning day desired.
Rule 3. To find the day signs of the beginning days of the succeeding divisions and subdivisions of the tonalamatl, count forward in Table I the black number from the day sign of the beginning day of the preceding division or subdivision, and the day name reached in Table I will be the day sign desired. If it is at the beginning of one of the main divisions of the tonalamatl, the day sign reached will be found to be recorded in the column of day signs, but if at the beginning of a subdivision it will be unexpressed.
To these the test rule above given may be added:
Rule 4. The sum of all the black numbers multiplied by the number of day signs in the column of day signs will equal exactly 260 if the tonalamatl is perfectly regular and correct.
In plate 27 is figured page 12 of the Dresden Codex. It will be noted that this page is divided into three parts by red division lines; after the general practice these have been designated a, b, and c, a being applied to the upper part, b to the middle part, and c to the lower part. Thus "Dresden 12b" designates the middle part of page 12 of the Dresden Codex, and "Dresden 15c" the lower part of page 15 of the same manuscript. Some of the pages of the codices are divided into four parts, or again, into two, and some are not divided at all. The same description applies in all cases, the parts being lettered from top to bottom in the same manner throughout.
The first tonalamatl presented will be that shown in Dresden 12b (see the middle division in pl. 27). The student will readily recognize the three essential parts mentioned on page 251: (1) The column of day signs, (2) the red numbers, and (3) the black numbers. Since there are five day signs in the column at the left of the page, it is evident that this tonalamatl has five main divisions. The first point to establish is the day with which this tonalamatl commenced. According to rule 1 (p. 252) this will be found by prefixing the first red number to the topmost day sign in the column. The first red number in Dresden 12b stands in the regular position (above the column of day signs), and is very clearly 1, that is, one red dot. A comparison of the topmost day sign in this column with the forms of the day signs in figure 17 will show that the day sign here recorded is Ix (see fig. 17, t), and the opening day of this tonalamatl will be, therefore, 1 Ix. The next step is to find the beginning days of the succeeding subdivisions of the first main division of the tonalamatl, which, as we have just seen, commenced with the day 1 Ix. According to rule 2 (p. 253), the first black number—in this case 13, just to the right of and slightly below the day sign Ix—is to be added to the red number immediately preceding it—in this case 1—in order to give the coefficient of the day beginning the next subdivision, all 13s possible being first deducted from the resulting number. Furthermore, this coefficient will be the red number next following the black number.
Applying this rule to the present case, we have:
1 (first red number) + 13 (next black number) = 14. Deducting all the 13s possible, we have left 1 (14  13) as the coefficient of the day beginning the next subdivision of the tonalamatl. This number 1 will be found as the red number immediately following the first black number, 13. To find the corresponding day sign, we must turn to rule 3 (p. 253) and count forward in Table I this same black number, 13, from the preceding day sign, in this case Ix. The day sign reached will be Manik. But since this day begins only a subdivision in this tonalamatl, not one of the main divisions, its day sign will not be recorded, and we have, therefore, the day 1 Manik, of which the 1 is expressed by the second red number and the name part Manik only indicated by the calculations.
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 27
PAGE 12 OF THE DRESDEN CODEX, SHOWING TONALAMATLS IN ALL THREE DIVISIONS
The beginning day of the next subdivision of the tonalamatl may now be calculated from the day 1 Manik by means of rules 2 and 3 (p. 253). Before proceeding with the calculation incident to this step it will be necessary first to examine the next black number in our tonalamatl. This will be found to be composed of this sign (
*) to which 6 (1 bar and 1 dot) has been affixed. It was explained on page 92 that in representing tonalamatls the Maya had to have a sign which by itself would signify the number 20, since numeration by position was impossible. This special character for the number 20 was given in figure 45, and a comparison of it with the sign here under discussion will show that the two are identical. But in the present example the number 6 is attached to this sign thus: (
**), and the whole number is to be read 20 + 6 = 26. This number, as we have seen in Chapter IV, would ordinarily have been written thus (
†): 1 unit of the second order (20 units of the first order) + 6 units of the first order = 26. As explained on page 92, however, numeration by position—that is, columns of units—was impossible in the tonalamatls, in which many of the numbers appear in a horizontal row, consequently some character had to be devised which by itself would stand for the number 20.
Returning to our text, we find that the "next black number" is 26 (20 + 6), and this is to be added to the red number 1 next preceding it, which, as we have seen, is an abbreviation for the day 1 Manik (see rule 2, p. 253). Adding 26 to 1 gives 27, and deducting all the 13s possible, namely, two, we have left 1 (27  26); this number 1, which is the coefficient of the beginning day of the next subdivision, will be found recorded just to the right of the black 26.
The day sign corresponding to this coefficient 1 will be found by counting forward 26 in Table I from the day name Manik. This will give the day name Ben, and 1 Ben will be, therefore, the beginning day of the next subdivision (the third subdivision of the first main division).
The next black number in our text is 13, and proceeding as before, this is to be added to the red number next preceding it, 1, the abbreviation for 1 Ben. Adding 13 to 1 we have 14, and deducting all the 23s possible, we obtain 1 again (14  13), which is recorded just to the right of the black 13 (rule 2, p. 253).^{[4]} Counting forward 13 in Table I from the day name Ben, the day name reached will be Cimi, and the day 1 Cimi will be the beginning day of the next part of the tonalamatl. But since 13 is the last black number, we should have reached in 1 Cimi the beginning day of the second main division of the tonalamatl (see p. 253), and this is found to be the case, since the day sign Cimi is the second in the column of day signs to the left. Compare this form with figure 17, i, j. The day recorded is therefore 1 Cimi.
The first division of the tonalamatl under discussion is subdivided, therefore, into three parts, the first part commencing with the day 1 Ix, containing 13 days; the second commencing with the day 1 Manik, containing 26 days; and the third commencing with the day 1 Ben, containing 13 days.
The second division of the tonalamatl commences with the day 1 Cimi, as we have seen above, and adding to this the first black number, 13, as before, according to rules 2 and 3 (p. 253), the beginning day of the next subdivision will be found to be 1 Cauac. Of this, however, only the 1 is declared (see to the right of the black 13). Adding the next black number, 26, to this day, according to the above rules the beginning day of the next subdivision will be found to be 1 Chicchan. Of this, however, the 1 again is the only part declared. Adding the next and last black number, 13, to this day, 1 Chicchan, according to the rules just mentioned the beginning day of the next, or third, main division will be found to be 1 Eznab. Compare the third day sign in the column of day signs with the form for Eznab in figure 17, z, a'. The second division of this tonalamatl contains, therefore, three parts: The first, commencing with the day 1 Cimi, containing 13 days; the second, commencing with the day 1 Cauac, containing 26 days; and the third, commencing with the day 1 Chicchan, containing 13 days.
Similarly the third division, commencing with the day 1 Eznab, could be shown to have three parts, of 13, 26, and 13 days each, commencing with the day 1 Eznab, 1 Chuen, and 1 Caban, respectively. It could be shown, also, that the fourth division commenced with the day 1 Oc (compare the fourth sign in the column of day signs with figure 17, o), and, further, that it had three subdivisions containing 13, 26, and 13 days each, commencing with the days 1 Oc, 1 Akbal, and 1 Muluc, respectively. Finally, the fifth and last division of the tonalamatl will commence with the day 1 Ik. Compare the last day sign in the column of day signs with figure 17, c, d; and its three subdivisions of 13, 26, and 13 days each with the days 1 Ik, 1 Men, and 1 Imix, respectively. The student will note also that when the last black number, 13, has been added to the beginning day of the last subdivision of the last division, the day reached will be 1 Ix, the day with which the tonalamatl commenced. This period is continuous, therefore, reentering itself immediately on its conclusion and commencing anew.
There follows below an outline^{[5]} of this particular tonalamatl:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 13 days, beginning with day 
1 Ix  1 Cimi  1 Eznab  1 Oc  1 Ik 
2d part, 26 days, beginning with day 
1 Manik  1 Cauac  1 Chuen  1 Akbal  1 Men 
3d part, 13 days, beginning with day 
1 Ben  1 Chicchan  1 Caban  1 Muluc  1 Imix 
Total number of days  52  52  52  52  52 
Next tonalamatl: 1st Division, 1st part, 13 days, beginning with the day 1 Ix, etc.
We may now apply rule 4 (p. 253) as a test to this tonalamatl. Multiplying the sum of all the black numbers, 13 + 26 + 13 = 52, by the number of day signs in the column of day signs, 5, we obtain 260 (52 × 5), which proves that this tonalamatl is regular and correct.
The student will note in the middle division of plate 27 that the pictures are so arranged that one picture stands under the first subdivisions of all the divisions, the second picture under the second subdivisions, and the third under the third subdivisions. It has been conjectured that these pictures represent the gods who were the patrons or guardians of the subdivisions of the tonalamatls, under which each appears. In the present case the first god pictured is the Death Deity, God A (see fig. 3). Note the fleshless lower jaw, the truncated nose, and the vertebræ. The second deity is unknown, but the third is again the Death God, having the same characteristics as the god in the first picture. The cloak worn by this deity in the third picture shows the crossbones, which would seem to have been an emblem of death among the Maya as among us. The glyphs above these pictures probably explain the nature of the periods to which they refer, or perhaps the ceremonies peculiar or appropriate to them. In many cases the name glyphs of the deities who appear below them are given; for example, in the present text, the second and sixth glyphs in the upper row^{[6]} record in each case the fact that the Death God is figured below.
The glyphs above the pictures offer one of the most promising problems in the Maya field. It seems probable, as just explained, that the four or six glyphs which stand above each of the pictures in a tonalamatl tell the meaning of the picture to which they are appended, and any advances made, looking toward their deciphering, will lead to farreaching results in the meaning of the nonnumerical and noncalendric signs. In part at least they show the name glyphs of the gods above which they occur, and it seems not unlikely that the remaining glyphs may refer to the actions of the deities who are portrayed; that is, to the ceremonies in which they are engaged. More extended researches along this line, however, must be made before this question can be answered.
The next tonalamatl to be examined is that shown in the lower division of plate 27, Dresden 12c. At first sight this would appear to be another tonalamatl of five divisions, like the preceding one, but a closer examination reveals the fact that the last day sign in the column of day signs is like the first, and that consequently there are only four different signs denoting four divisions. The last, or fifth sign, like the last red number to which it corresponds, merely indicates that after the 260th day the tonalamatl reenters itself and commences anew.
Prefixing the first red number, 13, to the first day sign, Chuen (see fig. 17, p, q), according to rule 1 (p. 252), the beginning day of the tonalamatl will be found to be 13 Chuen. Adding to this the first black number, 26, according to rules 2 and 3 (p. 253), the beginning day of the next subdivision will be found to be 13 Caban. Since this day begins only a subdivision of the tonalamatl, however, its name part Caban is omitted, and merely the coefficient 13 recorded. Commencing with the day 13 Caban and adding to it the next black number in the text, again 26, according to rules 2 and 3 (p. 253), the beginning day of the next subdivision will be found to be 13 Akbal, represented by its coefficient 13 only. Adding the last black number in the text, 13, to 13 Akbal, according to the rules just mentioned, the beginning day of the next part of the tonalamatl will be found to be 13 Cib. And since the black 13 which gave this new day is the last black number in the text, the new day 13 Cib will be the beginning day of the next or second division of the tonalamatl, and it will be recorded as the second sign in the column of day signs. Compare the second day sign in the column of day signs with figure 17, v, w.
Following the above rules, the student will have no difficulty in working out the beginning days of the remaining divisions and subdivisions of this tonalamatl. These are given below, though the student is urged to work them out independently, using the following outline simply as a check on his work. Adding the last black number, 13, to the beginning day of the last subdivision of the last division, 13 Eznab, will bring the count back to the day 13 Chuen with which the tonalamatl began:

1st Division  2d Division  3d Division  4th Division 
1st part, 26 days, beginning with day 
13 Chuen  13 Cib  13 Imix  13 Cimi 
2d part, 26 days, beginning with day 
13 Caban  13 Ik  13 Manik  13 Eb 
3d part, 13 days, beginning with day 
13 Akbal  13 Lamat  13 Ben  13 Eznab 
Total number of days  65  65  65  65 
Next tonalamatl: 1st division, 1st part, 26 days, beginning with the day 13 Chuen, etc.
Applying the test rule to this tonalamatl (see rule 4, p. 253), we have: 26 + 26 + 13 = 65, the sum of the black numbers, and 4 the number of the day signs in the column of day signs,^{[7]} 65 × 4 = 260, the exact number of days in a tonalamatl.
The next tonalamatl (see the upper part of pl. 27, that is, Dresden 12a) occupies only the latter twothirds of the upper division, the black 12 and red 11 being the last black and red numbers, respectively, of another tonalamatl.
The presence of 10 day signs arranged in two parallel columns of five each would seem at first to indicate that this is a tonalamatl of 10 divisions, but it develops from the calculations that instead there are recorded here two tonalamatls of five divisions each, the first column of day signs designating one tonalamatl and the second another quite distinct therefrom.
The first red numeral is somewhat effaced, indeed all the red has disappeared and only the black outline of the glyph remains. Its position, however, above the column of day signs, seems to indicate its color and use, and we are reasonably safe in stating that the first of the two tonalamatls here recorded began with the day 8 Ahau. Adding to this the first black number, 27, the beginning day of the next subdivision will be found to be 9 Manik, neither the coefficient nor day sign of which appears in the text. Assuming that the calculation is correct, however, and adding the next black number, 25 (also out of place), to this day, 9 Manik, the beginning day of the next part will be 8 Eb. But since 25 is the last black number, 8 Eb will be the beginning day of the next main division and should appear as the second sign in the first column of day signs. Comparison of this form with figure 17, r, will show that Eb is recorded in this place. In this manner all of the beginning days could be worked out as below:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 27 days, beginning with day 
8 Ahau  8 Eb  8 Kan  8 Cib  8 Lamat 
2d part, 25 days, beginning with day 
9 Manik  9 Cauac  9 Chuen  9 Akbal  9 Men 
Total number of days  52  52  52  52  52 
The application of rule 4 (p. 253) to this tonalamatl gives: 5 × 52 = 260, the exact number of days in a tonalamatl. As previously explained, the second column of day signs belongs to another tonalamatl, which, however, utilized the same red 8 as the first and the same black 27 and 25 as the first. The outline of this tonalamatl, which began with the day 8 Oc, follows:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 27 days, beginning with day 
8 Oc  8 Ik  8 Ix  8 Cimi  8 Eznab 
2d part, 25 days, beginning with day 
9 Caban  9 Muluc  9 Imix  9 Ben  9 Chicchan 
Total number of days  52  52  52  52  52 
The application of rule 4 (p. 253) to this tonalamatl gives: 5 × 52 = 260, the exact number of days in a tonalamatl. It is interesting to note that the above tonalamatl, beginning with the day 8 Oc, commenced just 130 days later than the first tonalamatl, which began with the day 8 Ahau. In other words, the first of the two tonalamatls in Dresden 12a was just half completed when the second one commenced, and the second half of the first tonalamatl began with the same day as the first half of the second tonalamatl, and vice versa.
The tonalamatl in plate 28, upper division, is from Dresden 15a, and is interesting because it illustrates how certain missing parts may be filled in. The first red number is missing and we can only say that this tonalamatl began with some day Ahau. However, adding the first black number, 34, to this day ? Ahau, the day reached will be 13 Ix, of which only 13 is recorded. Since 13 Ix was reached by counting 34 forward from the day with which the count must have started, by counting back 34 from 13 Ix the starting point will be found to be 5 Ahau, and we may supply a red bar above the column of the day signs. Adding the next black number, 18, to this day 13 Ix, the beginning day of the next division will be found to be 5 Eb, which appears as the second day sign in the column of day signs.
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 28
PAGE 15 OF THE DRESDEN CODEX, SHOWING TONALAMATLS IN ALL THREE DIVISIONS
The last red number is 5, thus establishing as correct our restoration of a red 5 above the column of day signs. From here this tonalamatl presents no unusual features and it may be worked as follows:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 34 days, beginning with day 
5 Ahau  5 Eb  5 Kan  5 Cib  5 Lamat 
2d part, 18 days, beginning with day 
13 Ix  13 Cimi  13 Eznab  13 Oc  13 Ik 
Total number of days  52  52  52  52  52 
Applying rule 4 (p. 253), we have: 5 × 52 = 260, the exact number of days in a tonalamatl. The next tonalamatl (see lower part of pl. 28, that is, Dresden 15c) has 10 day signs arranged in two parallel columns of 5 each. This, at its face value, would seem to be divided into 10 divisions, and the calculations confirm the results of the preliminary inspection.
The tonalamatl opens with the day 3 Lamat. Adding to this the first black number, 12, the day reached will be 2 Ahau, of which only the 2 is recorded here. Adding to 2 Ahau the next black number, 14, the day reached will be 3 Ix. And since 14 is the last black number, this new day will be the beginning of the next division in the tonalamatl and will appear as the upper day sign in the second column.^{[8]} Commencing with 3 Ix and adding to it the first black number 12, the day reached will be 2 Cimi, and adding to this the next black number, 14, the day reached will be 3 Ahau, which appears as the second glyph in the first column. This same operation if carried throughout will give the following outline of this tonalamatl:

1st Division  2d Division  3d Division  4th Division  5th Division  6th Division  7th Division  8th Division  9th Division  10th Division 
1st part, 12 days, beginning with day 
3 Lamat  3 Ix  3 Ahau  3 Cimi  3 Eb  3 Eznab  3 Kan  3 Oc  3 Cib  3 Ik 
2d part, 14 days, beginning with day 
2 Ahau  2 Cimi  2 Eb  2 Eznab  2 Kan  2 Oc  2 Cib  2 Ik  2 Lamat  2 Ix 
Total number of days  26  26  26  26  26  26  26  26  26  26 
Applying rule 4 (p. 253) to this tonalamatl, we have: 10 × 26 = 260, the exact number of days in a tonalamatl.
The tonalamatl which appears in the middle part on plate 28—that is, Dresden 15b—extends over on page 16b, where there is a black 13 and a red 1. The student will have little difficulty in reaching the result which follows: The last day sign is the same as the first, and consequently this tonalamatl is divided into four, instead of five, divisions:

1st Division  2d Division  3d Division  4th Division 
1st part, 13 days, beginning with day 
1 Ik  1 Manik  1 Eb  1 Caban 
2d part, 31 days, beginning with day 
1 Men  1 Ahau  1 Chicchan  1 Oc 
3d part, 8 days, beginning with day 
6 Cimi  6 Chuen  6 Cib  6 Imix 
4th part, 13 days, beginning with day 
1 Ix  1 Cauac  1 Kan  1 Muluc 
Total number of days  65  65  65  65 
Applying rule 4 (p. 253) to this tonalamatl, we have: 4 × 65 = 260, the exact number of days in a tonalamatl. The tonalamatls heretofore presented have all been taken from the Dresden Codex. The following examples, however, have been selected from tonalamatls in the Codex TroCortesianus. The student will note that the workmanship in the latter manuscript is far inferior to that in the Dresden Codex. This is particularly true with respect to the execution of the glyphs.
The first tonalamatl figured from the Codex TroCortesianus (see pl. 29) extends across the middle part of two pages (TroCor. 10b, 11b). The four day signs at the left indicate that it is divided into four divisions, of which the first begins with the day 13 Ik.^{[9]} Adding to this the first black number 9, the day 9 Chuen is reached, and proceeding in this manner the tonalamatl may be outlined as follows:
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 29
MIDDLE DIVISIONS OF PAGES 10 AND 11 OF THE CODEX TROCORTESIANO, SHOWING ONE TONALAMATL EXTENDING ACROSS THE TWO PAGES
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 30
PAGE 102 OF THE CODEX TROCORTESIANO, SHOWING TONALAMATLS IN THE LOWER THREE SECTIONS

1st Division  2d Division  3d Division  4th Division 
1st part, 9 days, beginning with day 
13 Ik  13 Manik  13 Eb  13 Caban 
2d part, 9 days, beginning with day 
9 Chuen  9 Cib  9 Imix  9 Cimi 
3d part, 10 days, beginning with day 
5 Ahau  5 Chicchan  5 Oc  5 Men 
4th part, 6 days, beginning with day 
2 Oc  2 Men  2 Ahau  2 Chicchan 
5th part, 2 days, beginning with day 
8 Cib  8 Imix  8 Cimi  8 Chuen 
6th part, 10 days, beginning with day 
10 Eznab  10 Akbal  10 Lamat  10 Ben 
7th part, 5 days, beginning with day 
7 Lamat  7 Ben  7 Eznab  7 Akbal 
8th part, 7 days, beginning with day 
12 Ben  12 Eznab  12 Akbal  12 Lamat 
9th part, 7 days, beginning with day 
6 Ahau^{[10]}  6 Chicchan^{[10]}  6 Oc^{[10]}  6 Men^{[10]} 
Total number of days  65  65  65  65 
Applying rule 4 (p. 253) to this tonalamatl, we have: 4 × 65 = 260, the exact number of days in a tonalamatl.
Another set of interesting tonalamatls is figured in plate 30, TroCor., 102.^{[11]} The first one on this page appears in the second division, 102b, and is divided into five parts, as the column of five day signs shows. The order of reading is from left to right in the pair of number columns, as will appear in the following outline of this tonalamatl:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 2 days, beginning with day 
4 Manik  4 Cauac  4 Chuen  4 Akbal  4 Men 
2d part, 7 days, beginning with day 
6 Muluc  6 Imix  6 Ben  6 Chicchan  6 Caban 
3d part, 2 days, beginning with day 
13 Cib  13 Lamat  13 Ahau  13 Eb  13 Kan 
4th part, 10 days, beginning with day 
2 Eznab  2 Oc  2 Ik  2 Ix  2 Cimi 
5th part, 9 days, beginning with day 
12 Lamat  12 Ahau  12 Eb  12 Kan  12 Cib 
6th part, 22 days, beginning with day 
8 Caban  8 Muluc  8 Imix  8 Ben  8 Chicchan 
Total number of days  52  52  52  52  52 
Applying rule 4 (p. 253) to this tonalamatl, we have: 5 × 52 = 260, the exact number of days in a tonalamatl. The next tonalamatl on this page (see third division in pl. 29, that is, TroCor., 102c) is interesting chiefly because of the fact that the pictures which went with the third and fourth parts of the five divisions are omitted for want of space. The outline of this tonalamatl follows:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 17 days, beginning with day 
4 Ahau  4 Eb  4 Kan  4 Cib  4 Lamat 
2d part, 13 days, beginning with day 
8 Caban  8 Muluc  8 Imix  8 Ben  8 Chicchan 
3d part, 10 days, beginning with day 
8 Oc  8 Ik  8 Ix  8 Cimi  8 Eznab 
4th part, 12 days, beginning with day 
5 Ahau  5 Eb  5 Kan  5 Cib  5 Lamat 
Total number of days  52  52  52  52  52 
Applying rule 4 (p. 253) to this tonalamatl, we have: 5 × 52 = 260, the exact number of days in a tonalamatl. The last tonalamatl in plate 29, TroCor., 102d, commences with the same day, 4 Ahau, as the preceding tonalamatl and, like it, has five divisions, each of which begins with the same day as the corresponding division in the tonalamatl just given, 4 Ahau, 4 Eb, 4 Kan, 4 Cib, and 4 Lamat. TroCor. 102d differs from TroCor. 102c in the number and length of the parts into which its divisions are divided.
Adding the first black number, 29, to the beginning day, 4 Ahau, the day reached will be 7 Muluc, of which only the 7 appears in the text. Adding to this the next black number, 24, the day reached will be 5 Ben. An examination of the text shows, however, that the day actually recorded is 4 Eb, the last red number with the second day sign. This latter day is just the day before 5 Ben, and since the sum of the black numbers in this case does not equal any factor of 260 (29 + 24 = 53), and since changing the last black number from 24 to 23 would make the sum of the black numbers equal to a factor of 260 (29 + 23 = 52), and would bring the count to 4 Eb, the day actually recorded, we are justified in assuming that there is an error in our original text, and that 23 should have been written here instead of 24. The outline of this tonalamatl, corrected as suggested, follows:

1st Division  2d Division  3d Division  4th Division  5th Division 
1st part, 29 days, beginning with day 
4 Ahau  4 Eb  4 Kan  4 Cib  4 Lamat 
2d part, 23^{[12]} days, beginning with day 
7 Muluc  7 Imix  7 Ben  7 Chicchan  7 Caban 
Total number of days  52  52  52  52  52 
Applying rule 4 (p. 253) to this tonalamatl, we have: 52 × 5 = 260, the exact number of days in a tonalamatl.
The foregoing tonalamatls have been taken from the pages of the Dresden Codex or those of the Codex TroCortesiano. Unfortunately, in the Codex Peresianus no complete tonalamatls remain, though one or two fragmentary ones have been noted.
No matter how they are divided or with what days they begin, all tonalamatls seem to be composed of the same essentials:
1. The calendric parts, made up, as we have seen on page 251, of (a) the column of day signs; (b) the red numbers; (c) the black numbers.
2. The pictures of anthropomorphic figures and animals engaged in a variety of pursuits, and
3. The groups of four or six glyphs above each of the pictures.
The relation of these parts to the tonalamatl as a whole is practically determined. The first is the calendric background, the chronological framework, as it were, of the period. The second and third parts amplify this and give the special meaning and significance to the subdivisions. The pictures represent in all probability the deities who presided over the several subdivisions of the tonalamatls in which they appear, and the glyphs above them probably set forth their names, as well as the ceremonies connected with, or the prognostications for, the corresponding periods.
It will be seen, therefore, that in its larger sense the meaning of the tonalamatl is no longer a sealed book, and while there remains a vast amount of detail yet to be worked out the foundation has been laid upon which future investigators may build with confidence.
In closing this discussion of the tonalamatl it may not be out of place to mention here those whose names stand as pioneers in this particular field of glyphic research. To the investigations of Prof. Ernst Förstemann we owe the elucidation of the calendric part of the tonalamatl, and to Dr. Paul Schellhas the identification of the gods and their corresponding name glyphs in parts (2) and (3), above. As pointed out at the beginning of this chapter, the most promising line of research in the codices is the groups of glyphs above the pictures, and from their decipherment will probably come the determination of the meaning of this interesting and unusual period.
Texts Recording Initial Series
Initial Series in the codices are unusual and indeed have been found, up to the present time, in only one of the three known Maya manuscripts, namely, the Dresden Codex. As represented in this manuscript, they differ considerably from the Initial Series heretofore described, all of which have been drawn from the inscriptions. This difference, however, is confined to unessentials, and the system of counting and measuring time in the Initial Series from the inscriptions is identical with that in the Initial Series from the codices.
The most conspicuous difference between the two is that in the codices the Initial Series are expressed by the second method, given on page 129, that is, numeration by position, while in the inscriptions, as we have seen, the period glyphs are used, that is, the first method, on page 105. Although this causes the two kinds of texts to appear very dissimilar, the difference is only superficial.
Another difference the student will note is the absence from the codices of the socalled Initialseries "introducing glyph." In a few cases there seems to be a sign occupying the position of the introducing glyph, but its identification as the Initialseries "introducing glyph" is by no means sure, and, moreover, as stated above, it does not occur in all cases in which there are Initial Series.
Another difference is the entire absence from the codices of Supplementary Series; this count seems to be confined exclusively to the monuments. Aside from these points the Initial Series from the two sources differ but little. All proceed from identically the same starting point, the date 4 Ahau 8 Cumhu, and all have their terminal dates or related Secondaryseries dates recorded immediately after them.
The first example of an Initial Series from the codices will be found in plate 31 (Dresden 24), in the lower lefthand corner, in the second column to the right. The Initialseries number here recorded is 9.9.16.0.0, of which the zero in the 2d place (uinals) and the zero in the 1st place (kins) are expressed by red numbers. This use of red numbers in the last two places is due to the fact that the zero sign in the codices is always red.
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 31
PAGE 24 OF THE DRESDEN CODEX, SHOWING INITIAL SERIES
The student will note the absence of all period glyphs from this Initial Series and will observe that the multiplicands of the cycle, katun, tun, uinal, and kin are fixed by the positions of each of the corresponding multipliers. By referring to Table XIV the values of the several positions in the second method of writing the numbers will be found, and using these with their corresponding coefficients in each case the Initialseries number here recorded may be reduced to units of the 1st order, as follows:
9 ×  144,000 =  1,296,000 
9 ×  7,200 =  64,800 
16 ×  360 =  5,760 
0 ×  20 =  0 
0 ×  1 =  0 
————  
1,366,560 
Deducting from this number all the Calendar Rounds possible, 72 (see Table XVI), it may be reduced to zero, since 72 Calendar Rounds contain exactly 1,366,560 units of the first order. See the preliminary rule on page 143.
Applying rules 1, 2, and 3 (pp. 139, 140, and 141) to the remainder, that is, 0, the terminal date of the Initial Series will be found to be 4 Ahau 8 Cumhu, exactly the same as the starting point of Maya chronology. This must be true, since counting forward 0 from the date 4 Ahau 8 Cumhu, the date 4 Ahau 8 Cumhu will be reached. Instead of recording this date immediately below the last period of its Initialseries number, that is, the 0 kins, it was written below the number just to the left. The terminal date of the Initial Series we are discussing, therefore, is 4 Ahau 8 Cumhu, and it is recorded just to the left of its usual position in the lower lefthand corner of plate 31. The coefficient of the day sign, 4, is effaced but the remaining parts of the date are perfectly clear. Compare the day sign Ahau with the corresponding form in figure 17, c', d', and the month sign Cumhu with the corresponding form in figure 20, zb'. The Initial Series here recorded is therefore 9.9.16.0.0 4 Ahau 8 Cumhu. Just to the right of this Initial Series is another, the number part of which the student will readily read as follows: 9.9.9.16.0. Treating this in the usual way, it may be reduced thus:
9 ×  144,000 =  1,296,000 
9 ×  7,200 =  64,800 
9 ×  360 =  3,240 
16 ×  20 =  320 
0 ×  1 =  0 
————  
1,364,360 
Deducting from this number all the Calendar Rounds possible, 71 (see Table XVI), it may be reduced to 16,780. Applying to this number rules 1, 2, and 3 (pp. 139, 140, and 141, respectively), its terminal date will be found to be 1 Ahau 18 Kayab; this date is recorded just to the left below the kin place of the preceding Initial Series. Compare the day sign and month sign of this date with figures 17, c', d', and 20, x, y, respectively. This second Initial Series in plate 31 therefore reads 9.9.9.16.0 1 Ahau 18 Kayab. In connection with the first of these two Initial Series, 9.9.16.0.0 4 Ahau 8 Cumhu, there is recorded a Secondary Series. This consists of 6 tuns, 2 uinals, and 0 kins (6.2.0) and is recorded just to the left of the first Initial Series from which it is counted, that is, in the lefthand column.
It was explained on pages 136137 that the almost universal direction of counting was forward, but that when the count was backward in the codices, this fact was indicated by a special sign or symbol, which gave to the number it modified the significance of "backward" or "minus." This sign is shown in figure 64, and, as explained on page 137, it usually is attached only to the lowest period. Returning once more to our text, in plate 31 we see this "backward" sign—a red circle surmounted by a knot—surrounding the 0 kins of this Secondaryseries number 6.2.0, and we are to conclude, therefore, that this number is to be counted backward from some date.
Counting it backward from the date which stands nearest it in our text, 4 Ahau 8 Cumhu, the date reached will be 1 Ahau 18 Kayab. But since the date 4 Ahau 8 Cumhu is stated in the text to have corresponded with the Initialseries value 9.9.16.0.0, by deducting 6.2.0 from this number we may work out the Initialseries value for this date as follows:
9.  9.  16.  0.  0  4 Ahau 8 Cumhu 
6.  2.  0  Backward  
9.  9.  9.  16.  0  1 Ahau 18 Kayab 
The accuracy of this last calculation is established by the fact that the Initialseries value 9.9.9.16.0 is recorded as the second Initial Series on the page above described, and corresponds to the date 1 Ahau 18 Kayab as here.
It is difficult to say why the terminal dates of these two Initial Series and this Secondary Series should have been recorded to the left of the numbers leading to them, and not just below the numbers in each case. The only explanation the writer can offer is that the ancient scribe wished to have the starting point of his Secondaryseries number, 4 Ahau 8 Cumhu, recorded as near that number as possible, that is, just below it, and consequently the Initial Series leading to this date had to stand to the right. This caused a displacement of the corresponding terminal date of his Secondary Series, 1 Ahau 18 Kayab, which was written under the Initial Series 9.9.16.0.0; and since the Initialseries value of 1 Ahau 18 Kayab also appears to the right of 9.9.16.0.0 as 9.9.9.16.0, this causes a displacement in its terminal date likewise.
Two other Initial Series will suffice to exemplify this kind of count in the codices. In plate 32 is figured page 62 from the Dresden Codex. In the two righthand columns appear two black numbers. The first of these reads quite clearly 8.16.15.16.1, which the student is perfectly justified in assuming is an Initialseries number consisting of 8 cycles, 16 katuns, 15 tuns, 16 uinals, and 1 kin. Moreover, above the 8 cycles is a glyph which bears considerable resemblance to the Initialseries introducing glyph (see fig. 24, f). Note in particular the trinal superfix. At all events, whether it is an Initial Series or not, the first step in deciphering it will be to reduce this number to units of the first order:
8 ×  144,000 =  1,152,000 
16 ×  7,200 =  115,200 
15 ×  360 =  5,400 
16 ×  20 =  320 
1 ×  1 =  1 
————  
1,272,921 
Deducting from this number all the Calendar Rounds possible, 67 (see Table XVI), it may be reduced to 1,261. Applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively) to this remainder, the terminal date reached will be 4 Imix 9 Mol. This is not the terminal date recorded, however, nor is it the terminal date standing below the next Initialseries number to the right, 8.16.14.15.4. It would seem then that there must be some mistake or unusual feature about this Initial Series.
Immediately below the date which stands under the Initialseries number we are considering, 8.16.15.16.1, is another number consisting of 1 tun, 4 uinals, and 16 kins (1.4.16). It is not improbable that this is a Secondaryseries number connected in some way with our Initial Series. The red circle surmounted by a knot which surrounds the 16 kins of this Secondaryseries number (1.4.16) indicates that the whole number is to be counted backward from some date. Ordinarily, the first Secondary Series in a text is to be counted from the terminal date of the Initial Series, which we have found by calculation (if not by record) to be 4 Imix 9 Mol in this case. Assuming that this is the case here, we might count 1.4.16 backward from the date 4 Imix 9 Mol.
Performing all the operations indicated in such cases, the terminal date reached will be found to be 3 Chicchan 18 Zip; this is very close to the date which is actually recorded just above the Secondaryseries number and just below the Initialseries number. The date here recorded is 3 Chicchan 13 Zip, and it is not improbable that the ancient scribe intended to write instead 3 Chicchan 18 Zip, the date indicated by the calculations. We probably have here:
8.  16.  15.  16.  1  (4 Imix 9 Mol) 
1.  4.  16  Backward  
8.  16.  14.  11.  5  3 Chicchan 18^{[13]} Zip 
In these calculations the terminal date of the Initial Series, 4 Imix 9 Mol, is suppressed, and the only date given is 3 Chicchan 18 Zip, the terminal date of the Secondary Series.
Another Initial Series of this same kind, one in which the terminal date is not recorded, is shown just to the right of the preceding in plate 32. The Initialseries number 8.16.14.15.4 there recorded reduces to units of the first order as follows:
8 ×  144,000 =  1,152,000 
16 ×  7,200 =  115,200 
14 ×  360 =  5,040 
15 ×  20 =  300 
4 ×  1 =  4 
————  
1,272,921 
Deducting from this number all the Calendar Rounds possible, 67 (see Table XVI), it will be reduced to 884, and applying rules 1, 2, and 3 (pp. 139, 140, and 141, respectively) to this remainder, the terminal date reached will be 4 Kan 17 Yaxkin. This date is not recorded. There follows below, however, a Secondaryseries number consisting of 6 uinals and 1 kin (6.1). The red circle around the lower term of this (the 1 kin) indicates that the whole number, 6.1, is to be counted backward from some date, probably, as in the preceding case, from the terminal date of the Initial Series above it. Assuming that this is the case, and counting 6.1 backward from 8.16.14.15.4 4 Kan 17 Yaxkin, the terminal date reached will be 13 Akbal 16 Pop, again very close to the date recorded immediately above, 13 Akbal 15 Pop. Indeed, the date as recorded, 13 Akbal 15 Pop, represents an impossible condition from the Maya point of view, since the day name Akbal could occupy only the first, sixth, eleventh, and sixteenth positions of a month. See Table VII. Consequently, through lack of space or carelessness the ancient scribe who painted this book failed to add one dot to the three bars of the month sign's coefficient, thus making it 16 instead of the 15 actually recorded. We are obliged to make some correction in this coefficient, since, as explained above, it is obviously incorrect as it stands. Since the addition of a single dot brings the whole date into harmony with the date determined by calculation, we are probably justified in making the correction here suggested. We have recorded here therefore:
8.  16.  14.  15.  4  (4 Kan 17 Yaxkin) 
6.  1  Backward  
8.  16.  14.  9.  3  13 Akbal 16^{[14]} Pop 
In these calculations the terminal date of the Initial Series, 4 Kan 17 Yaxkin, is suppressed and the only date given is 13 Akbal 16 Pop, the terminal date of the Secondary Series.
The above will suffice to show the use of Initial Series in the codices, but before leaving this subject it seems best to discuss briefly the dates recorded by these Initial Series in relation to the Initial Series on the monuments. According to Professor Förstemann^{[15]} there are 27 of these altogether, distributed as follows:
Page 24:  9.  9.  16.  0.  0  ^{[16]}  Page 58:  9.  12.  11.  11.  0  
Page 24:  9.  9.  9.  16.  0  Page 62:  8.  16.  15.  16.  1  
Page 31:  8.  16.  14.  15.  4  Page 62:  8.  16.  14.  15.  4  
Page 31:  8.  16.  3.  13.  0  Page 63:  8.  11.  8.  7.  0  
Page 31:  10.  13.  13.  3.  2  ^{[17]}  Page 63:  8.  16.  3.  13.  0  
Page 43:  9.  19.  8.  15.  0  Page 63:  10.  13.  3.  16.  4  ^{[18]}  
Page 45:  8.  17.  11.  3.  0  Page 63:  10.  13.  13.  3.  2  
Page 51:  8.  16.  4.  8.  0  ^{[19]}  Page 70:  9.  13.  12.  10.  0  
Page 51:  10.  19.  6.  1.  8  ^{[20]}  Page 70:  9.  19.  11.  13.  0  
Page 52:  9.  16.  4.  11.  18  ^{[21]}  Page 70:  10.  17.  13.  12.  12  
Page 52:  9.  19.  5.  7.  8  ^{[22]}  Page 70:  10.  11.  3.  18.  14  
Page 52:  9.  16.  4.  10.  8  Page 70:  8.  6.  16.  12.  0  
Page 52:  9.  16.  4.  11.  3  Page 70:  8.  16.  19.  10.  0  
Page 58:  9.  18.  2.  2.  0 
There is a wide range of time covered by these Initial Series; indeed, from the earliest 8.6.16.12.0 (on p. 70) to the latest, 10.19.6.1.8 (on p. 51) there elapsed more than a thousand years. Where the difference between the earliest and the latest dates is so great, it is a matter of vital importance to determine the contemporaneous date of the manuscript. If the closing date 10.19.6.1.8 represents the time at which the manuscript was made, then the preceding dates reach back for more than a thousand years. On the other hand, if 8.6.16.12.0 records the present time of the manuscript, then all the following dates are prophetic. It is a difficult question to answer, and the best authorities have seemed disposed to take a middle course, assigning as the contemporaneous date of the codex a date about the middle of Cycle 9. Says Professor Förstemann (Bulletin 28, p. 402) on the subject:
 In my opinion my demonstration also definitely proves that these large numbers [the Initial Series] do not proceed from the future to the past, but from the past, through the present, to the future. Unless I am quite mistaken, the highest numbers among them seem actually to reach into the future, and thus to have a prophetic meaning. Here the question arises, At what point in this series of numbers does the present lie? or, Has the writer in different portions of his work adopted different points of time as the present? If I may venture to express my conjecture, it seems to me that the first large number in the whole manuscript, the 1,366,560 in the second column of page 24 [9.9.16.0.0 4 Ahau 8 Cumhu, the first Initial Series figured in plate 31], has the greatest claim to be interpreted as the present point of time.
In a later article (Bulletin 28, p. 437) Professor Förstemann says: "But I think it is more probable that the date farthest to the right (1 Ahau, 18 Zip ...) denotes the present, the other two [namely, 9.9.16.0.0 4 Ahau 8 Cumhu and 9.9.9.16.0 1 Ahau 18 Kayab] alluding to remarkable days in the future." He assigns to this date 1 Ahau 18 Zip the position of 9.7.16.12.0 in the Long Count.
The writer believes this theory to be untenable because it involves a correction in the original text. The date which Professor Förstemann calls 1 Ahau 18 Zip actually reads 1 Ahau 18 Uo, as he himself admits. The month sign he corrects to Zip in spite of the fact that it is very clearly Uo. Compare this form with figure 20, b, c. The date 1 Ahau 18 Uo occurs at 9.8.16.16.0, but the writer sees no reason for believing that this date or the reading suggested by Professor Förstemann indicates the contemporaneous time of this manuscript.
Mr. Bowditch assigns the manuscript to approximately the same period, selecting the second Initial Series in plate 31, that is, 9.9.9.16.0 1 Ahau 18 Kayab: "My opinion is that the date 9.9.9.16.0 1 Ahau 18 Kayab is the present time with reference to the time of writing the codex and is the date from which the whole calculation starts."^{[23]} The reasons which have led Mr. Bowditch to this conclusion are very convincing and will make for the general acceptance of his hypothesis.
Although the writer has no better suggestion to offer at the present time, he is inclined to believe that both of these dates are far too early for this manuscript and that it is to be ascribed to a very much later period, perhaps to the centuries following immediately the colonization of Yucatan. There can be no doubt that very early dates appear in the Dresden Codex, but rather than accept one so early as 9.9.9.16.0 or 9.9.16.0.0 as the contemporaneous date of the manuscript the writer would prefer to believe, on historical grounds, that the manuscript now known as the Dresden Codex is a copy of an earlier manuscript and that the present copy dates from the later Maya period in Yucatan, though sometime before either Nahuatl or Castilian acculturation had begun.
BUREAU OF AMERICAN ETHNOLOGYBULLETIN 57 PLATE 32
PAGE 62 OF THE DRESDEN CODEX, SHOWING THE SERPENT NUMBERS
Texts Recording Serpent Numbers
The Dresden Codex contains another class of numbers which, so far as known, occur nowhere else. These have been called the Serpent numbers because their various orders of units are depicted between the coils of serpents. Two of these serpents appear in plate 32. The coils of each serpent inclose two different numbers, one in red and the other in black. Every one of the Serpent numbers has six terms, and they represent by far the highest numbers to be found in the codices. The black number in the first, or lefthand serpent in plate 32, reads as follows: 4.6.7.12.4.10, which, reduced to units of the first order, reads:
4 ×  2,880,000 =  11,520,000 
6 ×  144,000 =  864,000 
7 ×  7,200 =  50,400 
12 ×  360 =  4,320 
4 ×  20 =  80 
10 ×  1 =  10 
—————  
12,438,810 
The next question which arises is, What is the starting point from which this number is counted? Just below it the student will note the date 3 Ix 7 Tzec, which from its position would seem almost surely to be either the starting point or the terminal date, more probably the latter. Assuming that this date is the terminal date, the starting point may be calculated by counting 12,438,810 backward from 3 Ix 7 Tzec. Performing this operation according to the rules laid down in such cases, the starting point reached will be 9 Kan 12 Xul, but this date is not found in the text.
The red number in the first serpent is 4.6.11.10.7.2, which reduces to—
4 ×  2,880,000 =  11,520,000 
6 ×  144,000 =  864,000 
11 ×  7,200 =  79,200 
10 ×  360 =  3,600 
7 ×  20 =  140 
2 ×  1 =  2 
—————  
12,466,942 
Assuming that the date below this number, 3 Cimi 14 Kayab, was its terminal date, the starting point can be reached by counting backward. This will be found to be 9 Kan 12 Kayab, a date actually found on this page (see pl. 32), just above the animal figure emerging from the second serpent's mouth.
The black number in the second serpent reads 4.6.9.15.12.19, which reduces as follows:
4 ×  2,880,000 =  11,520,000 
6 ×  144,000 =  864,000 
9 ×  7,200 =  64,800 
15 ×  360 =  5,400 
12 ×  20 =  240 
19 ×  1 =  19 
—————  
12,454,459 
Assuming that the date below this number, 13 Akbal 1 Kankin, was the terminal date, its starting point can be shown by calculation to be just the same as the starting point for the previous number, that is, the date 9 Kan 12 Kayab, and as mentioned above, this date appears above the animal figure emerging from the mouth of this serpent.
The last Serpent number in plate 32, the red number in the second serpent, reads, 4.6.1.9.15.0 and reduces as follows:
4 ×  2,880,000 =  11,520,000 
6 ×  144,000 =  864,000 
1 ×  7,200 =  7,200 
9 ×  360 =  3,240 
15 ×  20 =  300 
0 ×  1 =  0 
—————  
12,394,740 
Assuming that the date below this number, 3 Kan 17 Uo,^{[24]} was its terminal date, its starting point can be shown by calculation to be just the same as the starting point of the two preceding numbers, namely, the date 9 Kan 12 Kayab, which appears above this last serpent.
Fig. 85. Example of first method of numeration in the codices (part of page 69 of the Dresden Codex).
It will be seen from the foregoing that three of the four Serpent dates above described are counted from the date 9 Kan 12 Kayab, a date actually recorded in the text just above them. The allimportant question of course is, What position did the date 9 Kan 12 Kayab occupy in the Long Count? The page (62) of the Dresden Codex we are discussing sheds no light on this question. There are, however, two other pages in this Codex (61 and 69) on which Serpent numbers appear presenting this date, 9 Kan 12 Kayab, under conditions which may shed light on the position it held in the Long Count. On page 69 there are recorded 15 katuns, 9 tuns, 4 uinals, and 4 kins (see fig. 85); these are immediately followed by the date 9 Kan 12 Kayab. It is important to note in this connection that, unlike almost every other number in this codex, this number is expressed by the first method, the one in which the period glyphs are used. As the date 4 Ahau 8 Cumhu appears just above in the text, the first supposition is that 15.9.4.4 is a Secondaryseries number which, if counted forward from 4 Ahau 8 Cumhu, the starting point of Maya chronology, will reach 9 Kan 12 Kayab, the date recorded immediately after it. Proceeding on this assumption and performing the operations indicated, the terminal date reached will be 9 Kan 7 Cumhu, not 9 Kan 12 Kayab, as recorded. The most plausible explanation for this number and date the writer can offer is that the whole constitutes a Periodending date. On the west side of Stela C at Quirigua, as explained on page 226, is a Periodending date almost exactly like this (see pl. 21, H). On this monument 17.5.0.0 6 Ahau 13 Kayab is recorded, and it was proved by calculation that 9.17.5.0.0 would lead to this date if counted forward from the starting point of Maya chronology. In effect, then, this 17.5.0.0 6 Ahau 13 Kayab was a Periodending date, declaring that Tun 5 of Katun 17 (of Cycle 9, unexpressed) ended on the date 6 Ahau 13 Kayab.
Interpreting in the same way the glyphs in figure 85, we have the record that Kin 4 of Uinal 4 of Tun 9 of Katun 15 (of Cycle 9, unexpressed) fell (or ended) on the date 9 Kan 12 Kayab. Changing this Periodending date into its corresponding Initial Series and solving for its terminal date, the latter date will be found to be 13 Kan 12 Ceh, instead of 9 Kan 12 Kayab. At first this would appear to be even farther from the mark than our preceding attempt, but if the reader will admit a slight correction, the above number can be made to reach the date recorded. The date 13 Kan 12 Ceh is just 5 uinals earlier than 9 Kan 12 Kayab, and if we add one bar to the four dots of the uinal coefficient, this passage can be explained in the above manner, and yet agree in all particulars. This is true since 9.15.9.9.4 reaches the date 9 Kan 12 Kayab. On the above grounds the writer is inclined to believe that the last three Serpent numbers on plate 32, which were shown to have proceeded from a date 9 Kan 12 Kayab, were counted from the date 9.15.9.9.4 9 Kan 12 Kayab.
Texts Recording Ascending Series
There remains one other class of numbers which should be described before closing this chapter on the codices. The writer refers to the series of related numbers which cover so many pages of the Dresden Codex. These commence at the bottom of the page and increase toward the top, every other number in the series being a multiple of the first, or beginning number. One example of this class will suffice to illustrate all the others.
In the lower righthand corner of plate 31 a series of this kind commences with the day 9 Ahau.^{[25]} Of this series the number 8.2.0 just above the 9 Ahau is the first term, and the day 9 Ahau the first terminal date. As usual in Maya texts, the starting point is not expressed; by calculation, however, it can be shown to be 1 Ahau^{[26]} in this particular case.
Counting forward then 8.2.0 from 1 Ahau, the unexpressed starting point, the first terminal date, 9 Ahau, will be reached. See the lower righthand corner in the following outline, in which the Maya numbers have all been reduced to units of the first order:
151,840^{[27]}  113,880^{[27]}  75,920^{[27]}  37,960^{[27]} 
1 Ahau  1 Ahau  1 Ahau  1 Ahau 
185,120  68,900  33,280  9,100 
1 Ahau  1 Ahau  1 Ahau  1 Ahau 
35,040  32,120  29,200  26,280 
6 Ahau  11 Ahau  3 Ahau  8 Ahau 
23,360  20,440  17,520  14,600 
13 Ahau  5 Ahau  10 Ahau  2 Ahau 
11,680^{[28]}  8,760  5,840  2,920 
7 Ahau  12 Ahau  4 Ahau  9 Ahau 
(Unexpressed starting point, 1 Ahau.) 
In the above outline each number represents the total distance of the day just below it from the unexpressed starting point, 1 Ahau, not the distance from the date immediately preceding it in the series. For example, the second number, 5,840 (16.4.0), is not to be counted forward from 9 Ahau in order to reach its terminal date, 4 Ahau, but from the unexpressed starting point of the whole series, the day 1 Ahau. Similarly the third number, 8,760 (1.4.6.0), is not to be counted forward from 4 Ahau in order to reach 12 Ahau, but from 1 Ahau instead, and so on throughout the series.
Beginning with the number 2,920 and the starting point 1 Ahau, the first twelve terms, that is, the numbers in the three lowest rows, are the first 12 multiples of 2,920.
2,920 = 1 × 2,920  20,440 =  7 × 2,920
5,840 = 2 × 2,920  23,360 =  8 × 2,920
8,760 = 3 × 2,920  26,280 =  9 × 2,920
11,680 = 4 × 2,920  29,200 = 10 × 2,920 
14,600 = 5 × 2,920  32,120 = 11 × 2,920 
17,520 = 6 × 2,920  35,040 = 12 × 2,920 
The days recorded under each of these numbers, as mentioned above, are the terminal dates of these distances from the starting point, 1 Ahau. Passing over the fourth row from the bottom, which, as will appear presently, is probably an interpolation of some kind, the thirteenth number—that is, the righthand one in the top row—is 37,960. But 37,960 is 13 × 2,920, a continuation of our series the twelfth term of which appeared in the lefthand number of the third row. Under the thirteenth number is set down the day 1 Ahau; in other words, not until the thirteenth multiple of 2,920 is reached is the terminal day the same as the starting point.
With this thirteenth term 2,920 ceases to be the unit of increase, and the thirteenth term itself (37,960) is used as a difference to reach the remaining three terms on this top line, all of which are multiples of 37,960.
37,960 = 1 × 37,960 or 13 × 2,920 
75,920 = 2 × 37,960 or 26 × 2,920 
113,880 = 3 × 37,960 or 39 × 2,920 
151,840 = 4 × 37,960 or 52 × 2,920 
Counting forward each one of these from the starting point of this entire series, 1 Ahau, each will be found to reach as its terminal day 1 Ahau, as recorded under each. The fourth line from the bottom is more difficult to understand, and the explanation offered by Professor Förstemann, that the first and third terms and the second and fourth are to be combined by addition or subtraction, leaves much to be desired. Omitting this row, however, the remaining numbers, those which are multiples of 2,920, admit of an easy explanation.
In the first place, the opening term 2,920, which serves as the unit of increase for the entire series up to and including the 13th term, is the socalled VenusSolar period, containing 8 Solar years of 365 days each and 5 Venus years of 584 days each. This important period is the subject of extended treatment elsewhere in the Dresden Codex (pp. 4650), in which it is repeated 39 times in all, divided into three equal divisions of 13 periods each. The 13th term of our series 37,960 is, as we have seen, 13 × 2,920, the exact number of days treated of in the upper divisions of pages 4650 of the Dresden Codex. The 14th term (75,920) is the exact number of days treated of in the first two divisions, and finally, the 15th, or next to the last term (113,880), is the exact number of days treated of in all three divisions of these pages.
This 13th term (37,960) is the first in which the tonalamatl of 260 days comes into harmony with the Venus and Solar years, and as such must have been of very great importance to the Maya. At the same time it represents two Calendar Rounds, another important chronological count. With the next to the last term (113,880) the Mars year of 780 days is brought into harmony with all the other periods named. This number, as just mentioned, represents the sum of all the 39 VenusSolar periods on pages 4650 of the Dresden Codex. This next to the last number seems to possess more remarkable properties than the last number (151,840), in which the Mars year is not contained without a remainder, and the reason for its record does not appear.
The next to the last term contains:
438  Tonalamatls of 260 days each 
312  Solar years of 365 days each 
195  Venus years of 584 days each 
146  Mars years of 780 days each 
39  VenusSolar periods of 2,920 days each 
6  Calendar Rounds of 18,980 days each 
It will be noted in plate 31 that the concealed starting point of this series is the day 1 Ahau, and that just to the left on the same plate are two dates, 1 Ahau 18 Kayab and 1 Ahau 18 Uo, both of which show this same day, and one of which, 1 Ahau 18 Kayab, is accompanied by its corresponding Initial Series 9.9.9.16.0. It seems not unlikely, therefore, that the day 1 Ahau with which this series commences was 1 Ahau 18 Kayab, which in turn was 9.9.9.16.0 1 Ahau 18 Kayab of the Long Count. This is rendered somewhat probable by the fact that the second division of 13 VenusSolar periods on pages 4650 of the Dresden Codex also has the same date, 1 Ahau 18 Kayab, as its terminal date. Hence, it is not improbable (more it would be unwise to say) that the series of numbers which we have been discussing was counted from the date 9.9.9.16.0. 1 Ahau 18 Kayab.
The foregoing examples cover, in a general way, the material presented in the codices; there is, however, much other matter which has not been explained here, as unfitted to the needs of the beginner. To the student who wishes to specialize in this field of the glyphic writing the writer recommends the treatises of Prof. Ernst Förstemann as the most valuable contribution to this subject.
 ↑ As explained on p. 31, tonalamatls were probably used by the priests in making prophecies or divinations. This, however, is a matter apart from their composition, that is, length, divisions, dates, and method of counting, which more particularly concerns us here.
 ↑ The codices are folded like a screen or fan, and when opened form a continuous strip sometimes several yards in length. As will appear later, in many cases one tonalamatl runs across several pages of the manuscript.
 ↑ If there should be two or more columns of day signs the topmost sign of the lefthand column is to be read first.
 ↑ In the original this last red dot has disappeared. The writer has inserted it here to avoid confusing the beginner in his first acquaintance with a tonalamatl.
 ↑ This and similar outlines which follow are to be read down in columns.
 ↑ The fifth sign in the lower row is also a sign of the Death God (see fig. 3). Note the eyelashes, suggesting the closed eyes of the dead.
 ↑ The last sign Chuen, as mentioned above, is only a repetition of the first sign, indicating that the tonalamatl has reentered itself.
 ↑ [Footnote 1: As previously stated, the order of reading the glyphs in columns is from left to right and top to bottom.
 ↑ The righthand dot of the 13 is effaced.
 ↑ ^{10.0} ^{10.1} ^{10.2} ^{10.3} The manuscript has incorrectly 7.
 ↑ In the title of plate 30 the page number should read 102 instead of 113.
 ↑ The manuscript incorrectly has 24.
 ↑ Incorrectly recorded as 13 in the text.
 ↑ Incorrectly recorded as 15 in the text.
 ↑ Bull. 28, Bur. Amer. Ethn., p. 400.
 ↑ The terminal dates reached have been omitted, since for comparative work the Initialseries numbers alone are sufficient to show the relative positions in the Long Count.
 ↑ The manuscript incorrectly reads 10.13.3.13.2; that is, reversing the position of the tun and uinal coefficients.
 ↑ The manuscript incorrectly reads 10.8.3.16.4. The katun coefficient is changed to 13, above. These corrections are all suggested by Professor Förstemann and are necessary if the calculations he suggests are correct, as seems probable.
 ↑ The manuscript incorrectly reads 8.16.4.11.0. The uinal coefficient is changed to an 8, above.
 ↑ The manuscript incorrectly reads 10.19.6.0.8. The uinal coefficient is changed to 1, above.
 ↑ The manuscript incorrectly reads 9.16.4.10.18. The uinal coefficient is changed to 11, above.
 ↑ The manuscript incorrectly reads 9.19.8.7.8. The tun coefficient is changed to 5, above.
 ↑ Bowditch, 1909: p. 279.
 ↑ The manuscript has incorrectly 16 Uo. It is obvious this can not be correct, since from Table VII Kan can occupy only the 2d, 7th, 12th, or 17th position in the months. The correct reading here, as we shall see, is probably 17 Uo. This reading requires only the addition of a single dot.
 ↑ In the text the coefficient appears to be 8, but in reality it is 9, the lower dot having been covered by the marginal line at the bottom.
 ↑ Counting backward 8.2.0 (2,920) from 9 Ahau, 1 Ahau is reached.
 ↑ ^{27.0} ^{27.1} ^{27.2} ^{27.3} Professor Förstemann restored the top terms of the four numbers in this row, so as to make them read as given above.
 ↑ The manuscript reads 1.12.5.0, which Professor Förstemann corrects to 1.12.8.0; in other words, changing the uinal from 5 to 8. This correction is fully justified in the above calculations.