Plate 31PAGE 24 OF THE DRESDEN CODEX, SHOWING INITIAL SERIES
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 TableXIVthe 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 Initial-series number here recorded may be reduced to units of the 1st order, as follows:
Deducting from this number all the Calendar Rounds possible, 72 (see TableXVI), 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 page143.
Applying rules 1, 2, and 3 (pp.139,140, and141) to the remainder, that is, 0, the terminal date of the Initial Series will be found to be4 Ahau 8 Cumhu, exactly the same as the starting point of Maya chronology. This must be true, since counting forward 0 from the date4 Ahau 8 Cumhu, the date4 Ahau 8 Cumhuwill be reached. Instead of recording this date immediately below the last period of its Initial-series 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, is4 Ahau 8 Cumhu, and it is recorded just to the left of its usual position in the lower left-hand 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 signAhauwith the corresponding form in figure17,c', d', and the month signCumhuwith the corresponding form in figure20,z-b'. The Initial Series here recorded is therefore 9.9.16.0.04 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:
Deducting from this number all the Calendar Rounds possible, 71 (see TableXVI), it may be reduced to 16,780. Applying to this number rules 1, 2, and 3 (pp.139,140, and141, respectively), its terminal date will be found to be1 Ahau 18 Kayab; this date is recorded just to the left below the kin place of theprecedingInitialSeries. Compare the day sign and month sign of this date with figures17,c', d', and20,x, y, respectively. This second Initial Series in plate31therefore reads 9.9.9.16.01 Ahau 18 Kayab. In connection with the first of these two Initial Series, 9.9.16.0.04 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 left-hand column.
It was explained on pages136-137that 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 figure64, and, as explained on page137, it usually is attached only to the lowest period. Returning once more to our text, in plate31we see this "backward" sign—a red circle surmounted by a knot—surrounding the 0 kins of this Secondary-series 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 be1 Ahau 18 Kayab. But since the date4 Ahau 8 Cumhuis stated in the text to have corresponded with the Initial-series value 9.9.16.0.0, by deducting 6.2.0 from this number we may work out the Initial-series value for this date as follows:
The accuracy of this last calculation is established by the fact that the Initial-series value 9.9.9.16.0 is recorded as the second Initial Series on the page above described, and corresponds to the date1 Ahau 18 Kayabas here.
It is difficult to say why the terminal dates of these two Initial Series and this Secondary Series should have been recorded to theleftof the numbers leading to them, and not justbelowthe numbers in each case. The only explanation the writer can offer is that the ancient scribe wished to have the starting point of his Secondary-series 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 Initial-series value of1 Ahau 18 Kayabalso 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 plate32is figured page 62 from the Dresden Codex. In the two right-hand 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 Initial-series 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 Initial-series 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:
Deducting from this number all the Calendar Rounds possible, 67 (see TableXVI), it may be reduced to 1,261. Applying rules 1, 2, and 3 (pp.139,140, and141, respectively) to this remainder, the terminal date reached will be4 Imix 9 Mol. This is not the terminal date recorded, however, nor is it the terminal date standing below the next Initial-series 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 Initial-series 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 Secondary-series number connected in some way with our Initial Series. The red circle surmounted by a knot which surrounds the 16 kins of this Secondary-series number (1.4.16) indicates that the whole number is to be countedbackwardfrom 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 be4 Imix 9 Molin this case. Assuming that this is the case here, we might count 1.4.16backwardfrom the date4 Imix 9 Mol.
Performing all the operations indicated in such cases, the terminal date reached will be found to be3 Chicchan 18 Zip; this is very close to the date which is actually recorded just above the Secondary-series number and just below the Initial-series number. The date here recorded is3 Chicchan 13 Zip, and it is not improbable that theancient scribe intended to write instead3 Chicchan 18 Zip, the date indicated by the calculations. We probably have here:
In these calculations the terminal date of the Initial Series,4 Imix 9 Mol, is suppressed, and the only date given is3 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 plate32. The Initial-series number 8.16.14.15.4 there recorded reduces to units of the first order as follows:
Deducting from this number all the Calendar Rounds possible, 67 (see TableXVI), it will be reduced to 884, and applying rules 1, 2, and 3 (pp.139,140, and141, respectively) to this remainder, the terminal date reached will be4 Kan 17 Yaxkin. This date is not recorded. There follows below, however, a Secondary-series 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 countedbackwardfrom 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 8.16.14.15.44 Kan 17 Yaxkin, the terminal date reached will be13 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 nameAkbalcould occupy only the first, sixth, eleventh, and sixteenth positions of a month. See TableVII. 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 justifiedin making the correction here suggested. We have recorded here therefore:
In these calculations the terminal date of the Initial Series,4 Kan 17 Yaxkin, is suppressed and the only date given is13 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[254]there are 27 of these altogether, distributed as follows:
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 backfor 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 [9.9.16.0.04 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 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 [9.9.16.0.04 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 4Ahau 8 Cumhuand 9.9.9.16.01 Ahau 18 Kayab] alluding to remarkable days in the future." He assigns to this date1 Ahau 18 Zipthe 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 calls1 Ahau 18 Zipactually reads1 Ahau 18 Uo, as he himself admits. The month sign he corrects toZipin spite of the fact that it is very clearlyUo. Compare this form with figure20,b, c. The date1 Ahau 18 Uooccurs 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 plate31, that is, 9.9.9.16.01 Ahau 18 Kayab: "My opinion is that the date 9.9.9.16.01 Ahau 18 Kayabis the present time with reference to the time of writing the codex and is the date from which the whole calculation starts."[262]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
Plate 32PAGE 62 OF THE DRESDEN CODEX, SHOWING THE SERPENT NUMBERS
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 plate32. 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 left-hand serpent in plate32, reads as follows: 4.6.7.12.4.10, which, reduced to units of the first order, reads:
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 date3 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 12,438,810backwardfrom3 Ix 7 Tzec. Performing this operation according to the rules laid down in such cases, the starting point reached will be9 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—
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 be9 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:
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 date9 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 plate32, the red number in the second serpent, reads, 4.6.1.9.15.0 and reduces as follows:
Assuming that the date below this number,3 Kan 17 Uo,[263]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 date9 Kan 12 Kayab, which appears above this last serpent.
Fig. 85Fig.85. Example of first method of numeration in the codices (part of page 69 of the Dresden Codex).
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 date9 Kan 12 Kayab, a date actually recorded in the text just above them. The all-important question of course is, What position did the date9 Kan 12 Kayaboccupy in the Long Count? The page (62) of the Dresden Codex weare 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 date9 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 date4 Ahau 8 Cumhuappears just above in the text, the first supposition is that 15.9.4.4 is a Secondary-series number which, if counted forward from4 Ahau 8 Cumhu, the starting point of Maya chronology, will reach9 Kan 12 Kayab, the date recorded immediately after it. Proceeding on this assumption and performing the operations indicated, the terminal date reached will be9 Kan 7 Cumhu, not9 Kan 12 Kayab, as recorded. The most plausible explanation for this number and date the writer can offer is that the whole constitutes a Period-ending date. On the west side of Stela C at Quirigua, as explained on page226, is a Period-ending date almost exactly like this (see pl.21,H). On this monument 17.5.0.06 Ahau 13 Kayabis 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 17.5.0.06 Ahau 13 Kayabwas a Period-ending date, declaring that Tun 5 of Katun 17 (of Cycle 9, unexpressed) ended on the date6 Ahau 13 Kayab.
Interpreting in the same way the glyphs in figure85, 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 date9 Kan 12 Kayab. Changing this Period-ending date into its corresponding Initial Series and solving for its terminal date, the latter date will be found to be13 Kan 12 Ceh, instead of9 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 date13 Kan 12 Cehis just 5 uinals earlier than9 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 date9 Kan 12 Kayab. On the above grounds the writer is inclined to believe that the last three Serpent numbers on plate32, which were shown to have proceeded from a date9 Kan 12 Kayab, were counted from the date 9.15.9.9.49 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 right-hand corner of plate31a series of this kind commences with the day9 Ahau.[264]Of this series the number 8.2.0 just above the9 Ahauis the first term, and the day9 Ahauthe first terminal date. As usual in Maya texts, the starting point is not expressed; by calculation, however, it can be shown to be1 Ahau[265]in this particular case.
Counting forward then 8.2.0 from1 Ahau, the unexpressed starting point, the first terminal date,9 Ahau, will be reached. See the lower right-hand corner in the following outline, in which the Maya numbers have all been reduced to units of the first order:
In the above outline each number represents the total distance of the day just below it from the unexpressed starting point,1 Ahau,notthe 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 from9 Ahauin order to reach its terminal date,4 Ahau, but from the unexpressed starting point of the whole series, the day1 Ahau. Similarly the third number, 8,760 (1.4.6.0), is not to be counted forward from4 Ahauin order to reach12 Ahau, but from1 Ahauinstead, and so on throughout the series.
Beginning with the number 2,920 and the starting point1 Ahau, the first twelve terms, that is, the numbers in the three lowest rows, are the first 12 multiples of 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 right-hand 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 left-hand number of the third row. Under the thirteenth number is set down the day1 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.
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 day1 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 so-called Venus-Solar 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. 46-50), 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 ofdays treated of in the upper divisions of pages 46-50 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 Venus-Solar periods on pages 46-50 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:
It will be noted in plate31that the concealed starting point of this series is the day1 Ahau, and that just to the left on the same plate are two dates,1 Ahau 18 Kayaband1 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 day1 Ahauwith which this series commences was1 Ahau 18 Kayab, which in turn was 9.9.9.16.01 Ahau 18 Kayabof the Long Count. This is rendered somewhat probable by the fact that the second division of 13 Venus-Solar periods on pages 46-50 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.