Commentary on the Maya Manuscript in the Royal Public Library of Dresden/5

Page 60.

This is the last page of the front of the second part and is divided into four sections:—at the top we find hieroglyphs, below these a picture, then hieroglyphs again and in the lowest section another picture.

The upper picture contains first a rectangular elevation like a platform. Enclosed in this rectangle is the picture of the animal resembling a dog lying down, which we have often met with, the last time on page 47. In front of the dog is a hieroglyph, which, I regret to say, is still unknown and which occurred six times as a heading on page 23b. On the platform two personages are fighting; one is in war-dress holding in his left hand the throwing-stick or atlatl, and in his right probably arrows; the other figure, whose back is somewhat indistinct owing to obliteration, is apparently unarmed and is making a defensive gesture with one hand. Beside the platform, and therefore on a lower level, is a second person walking behind the armed person as if to help him. He too is in war-dress and likewise holds an atlatl. A black 3 is set down between the two combatants, and there may also have been a red 2, which is indistinct owing to the red background of the picture.

Let us next examine the lower picture. A blindfolded personage is kneeling on the left. A serpent's head rises from the ground in front of him. A second serpent rises in several coils on the shoulders of the blindfolded personage and on the serpent's neck sits enthroned another personage, who is rather indistinct, holding a spear and a shield. On the right, opposite this group and facing it, is a second. A personage with arms bound and bowed head is sitting on the ground. There is a black ring around his eye. Behind him stands the victor in war-dress and again equipped with spear and shield. There is a red 11 and a black 2 between the two groups.

We see that the reference here is to combat, just as it was on the right side of pages 46-50. And since the subject of these pages like that of 46-59 is confined to the revolutions of the planets, it is natural that the pursuit of one by the other, their periodical disappearance, the crossing of their orbits and the variation in the length of their revolutions should be looked upon as a contest. Therefore, since the sun, the moon and the five planets have hitherto been treated of on these pages, I look for these seven heavenly bodies in the seven personages pictured here on page 60. I will attempt to explain them, hoping that my interpretation may be replaced by a better one.

The sun and moon stand on the platform in the upper picture; their combat is equivalent to the eclipses to which they at times succumb. The moon is the assailant and the sun makes only a proud defensive gesture. The person behind the moon must be Mars. The animal under the two persons is the embodiment of the eclipses, which the Aztecs interpreted as the act of being devoured by the jaguar. The hieroglyph in front looks very much like the meeting of two circles. Does it refer to the day Lamat (Aztec tochtli = rabbit)?

At the left, bottom, the powerful Venus triumphs over the weak Mercury. The two planets are real chronometers by reason of the regular alternation of their appearance as morning and evening stars, and also by their disappearance twice in each revolution and finally even in the variation in the length of the two periods of invisibility. Hence they are each accompanied by a serpent as the usual symbol of time.

On the right, on the other hand, Jupiter as the stronger has vanquished Saturn, whose bound arms symbolize his slowness of motion and the fact that he is confined to the same region of the sky. Should not the ring around his eye have a very special meaning? But we must guard against an excess of imagination. Jupiter and Saturn are the last to be represented, as they were of but secondary importance, on pages 51-58 and perhaps also in the 2200 on page 24.

I will not deny that yet another interpretation of this page is possible. The top picture may be Venus and the moon opposing one another, and the bottom picture may represent the sun as victor over Mercury. There are some things in favor of this point of view.

The correct order of the twenty-four hieroglyphs is the following, in my opinion, which is borne out by the different colors of the four groups:—

These signs can have no relation to mythology. There is not a hieroglyph of a god among them, for if sign 6 could be taken for B's hieroglyph, the resemblance to the sign of the fist, familiar from the inscriptions, as well as the Imix and the cross-hatching as a prefix, makes this doubtful. The latter component would rather suggest the summer solstice. If sign 12 were intended to denote the Bacab, then it would refer to chronology rather than to mythology. Also the Cimi in 17 might equally well mean the day as the god. Indeed several things refer here to chronology and astronomy, among them the unmistakable union of numbers and month signs, which occur here repeatedly. Thus from what remains of the almost obliterated signs 1 and 2, they might denote the normal date IV Ahau 8 Cumhu, which always occupies the first place. Signs 7 and 20 are plainly the same, 9 Xul (sixth month) and sign 14 is 10 Yaxkin (seventh month). Sign 5 might be Caban combined with Uo (second month) and a ten. In sign 19 we again see Yaxkin without a number. Signs 9 and 23 are Zec (fifth month) and signs 21 and 22 may be Kankin (fourteenth month). The days occur in the same manner as the months. It is true that Kin is only a part of hieroglyph 10, the rest of which is effaced, but the familiar compound of Caban and Muluc appears in 18 and 24 and Cimi is in 17, as we have seen. In sign 13, Ahau is combined with a red number, which must lie between X and XV. But this should not be regarded as forming a calendar date with the 10 Yaxkin near by, for Ahau is never the tenth day of a month. Can 16 be the sign of the twelfth month, Ceh, combined with that for 7200? Hieroglyphs 3 and 8 are effaced and I do not understand 4, 11 and 15.

There are no parallels in the kindred passages 46-50, unless it be 7 Zec on the bottom of page 49 and here in signs 9 and 23, but without a number. Cf. my paper on this page 60 in the "Weltall," year 6, pages 251-257.

Page 61—64.

On examining the reverse of the second part of our Manuscript, i.e., pages 61-74, we find an empty page on the left, the back of which is occupied by page 60. This may be explained by assuming that the scribe wrote pages 61-64 and possibly even pages 61-74 from right to left, the great series having occasioned such a proceeding, and that his material came to an end when he had finished page 61. Nevertheless, it is advisable to continue with the original numbering in order to avoid confusion.

Aside from the concluding (or beginning) page 74, this whole section of pages 61-74 consists of three parts:—61-64, 65-69 and 70-73. Let us first consider the first section, which I have already discussed in my treatise "Zur Erläuterung der Mayahandschriften II."

The basis of this section is a series, the beginning of which is on the bottom, right, of page 64. Its primary difference is always that which we found on pages 31-32, viz:—the Bacab period of 91 days, the quarter of the ritual year of 364 days = 7 weeks of 13 days each. It ascends by 91 until it reaches 1820, which number is a multiple of both 364 and 260 and is also divisible by 28, the number of weeks in a year. Just as on page 32 the series continues with the new difference 1820 as far as 7280, its fourth multiple, which then becomes the third difference. Indeed, I believe that even the partially effaced numbers could be so restored as to carry the series to the number 36,400 = 400 × 91, which would then become the fourth difference and the series would close at the top of page 63 with 145,600 = 1600 × 91, i.e., with the numbers 1. 0. 4. 8. 0. of which the 1 is entirely and the 0 half effaced. The series on pages 31-32, however, closed with 29,120 = 320 × 91, but there is still room for a higher series.

Under this largest number (1600 × 91) there is on page 63 a large red number consisting of 19. 0. 4. 4. which is crowded into a very small space between the figures of 1820. I can only understand it by replacing the first 4 by a 3, for then it is 136,864 = 1504 × 91 or by addition of a zero. We shall return to this number in the examination of the serpent numerals.

The series is accompanied in the regular way by five days. At the beginning of this series, page 64, right bottom, are the days III Cib, III Men, III Chicchan, III Caban and XIII Ix; the III is set down only with the first of these days and is to be supplied with the next three. Hence the actual zero point is to be found 91 days back in the days III Chicchan, III Kan, III Ix, III Cimi and XIII Akbal, the last of which is also the beginning of the corresponding series on page 32. From 1820 on, these last-named days, of course unchanged, accompany the numbers. The most important of these days are the first and last, but we shall see later in connection with the serpent numbers that the other three, which are separated from one another by 39, 130 and 52, i.e., 3 × 13, 10 × 13 and 4 × 13, are likewise not set down here by mere accident.

We come now to the five columns, three on page 63 and two on page 62, which join this series on the left. They contain the large numbers, which invariably accompany these series. Here there are six numbers, four of which, in my opinion, refer to the past and two to the future. Two of these numbers, the two largest, are set down together in the third column on page 63, one with red numbers and the other with black. Of these black numbers, I take the second from the top to be not 8 but 13, assuming that a line is omitted. The normal date IV Ahau 8 Cumhu from which, as the starting-point, all these numbers are to be computed, is set down below at the end of each of the five columns.

I now give the six numbers, first the two highest, then the other four from right to left, adding in each case the calendar date and the year in which they should be situated:—

The first, third and fifth numbers are already known from page 31a, and hence they need no further discussion here.

As these three numbers depend on the day XIII Akbal, so the other three all proceed from the day III Chicchan in the following positions, which are again suppressed in the Manuscript:—

The second date in the manuscript is 13 Kankin and the third is 13 Zip; hence there is one line too many in the former number and one too few in the latter. While on page 31a the origin of the numbers belonging to the day XIII Akbal seems to be quite clear, here their relation to one another is entirely concealed. I must, therefore, refrain from expressing any conjecture in regard to them.

Now the numbers set down in the Manuscript are formed only by the addition of the encircled numbers also found there. The encircled number for the first expressed number is 51,419, which is the same number we found with the corresponding day XIII Akbal; the second has 235 and the third 456 = 260 + 196. The 51,419 was 197 × 260 + 199; but 199 is the interval from III Chicchan to VII Kan, just as it is from XIII Akbal to IV Ik. The 235 is the interval between III Chicchan and IV Ahau and the 196 that from III Chicchan and IV Imix.

By the addition of these differences, the numbers written out in the Manuscript are obtained:—

Keeping in mind what was said in reference to page 31a, let us now examine the six numbers and dates collectively.

The fact that the days IV Ahau and XIII Akbal occur here and consequently also III Chicchan is not surprising. Nor is the choice of VII Kan and IV Ik an accident, for the interval between these days is exactly the same as that between III Chicchan and XIII Akbal, viz:—218 days.

Hence the distance from III Chicchan to VII Kan is also exactly equal to that between XIII Akbal to IV Ik, viz:—199 days.

Finally, the distance from VII Kan to III Chicchan is exactly equal to that between IV Ik and XIII Akbal, viz:—61 days.

IV Imix and IV Kan are separated from the normal date IV Ahau by 3 × 13 = 39 and 8 × 13 = 104 days.

Regarding the encircled numbers, so far as they are independent of 260, I would note the following:—

In addition let me remark that 36 = VII Kan to IV Ahau, 39 = IV Imix to IV Ahau and 104 = IV Ahau to IV Kan.

The following arrangement will prove that these numbers were as usual also employed to form the large numbers by multiplication:—

But the highest number, 1,538,342, was formed in a different way; it = 59,167 × 26; but the interval from IV Ahau to IV Ik = 182 = 7 × 26, and from IV Ik to IV Ahau = 78 = 3 × 26.

If in conclusion, we now examine the twelve numbers of seven figures given in this section, we will clearly see that by twos and twos they plainly belong together in pairs:—

The three pairs of numbers found by computation are as follows:—

Difference 3338 = 12 × 260 + 218 (VII Kan to IV Ik, III Chicchan to XIII Akbal).

Difference 34,538 = 132 × 260 + 218 (as above).

Difference 42 (which is 260 - 218); 42 = IV Kan to IV Imix.

On the other hand the three pairs specified in the Manuscript are as follows:—

Difference 3338 = 12 × 260 + 218 as above, by reason of the encircled number 51,419 which is common to both numbers.

Difference 34,320 = 132 × 260, on account of the same day.

Difference 117 = IV Kan to IV Imix; strictly speaking 377 = 260 + 117.

The upper part of the five columns just now under discussion still remains to be examined. Here are five vertical rows of hieroglyphs, the first four each containing seven, and the fifth only six owing to lack of space.

The two rows at the top are as usual much obliterated, which is the more to be deplored since they consisted of five calendar dates, which would have contributed materially to the comprehension of the entire section. Fortunately, however, one of these dates is preserved complete, and we are able to see in what relation it stands to the rest. Thus we find in the third column of page 63 the date XIII Imix 9 Uo. It comes in the year 12 Ix and represents the number 1,523,921 (or a number separated from it by a multiple of 18,980). Now 1,523,921 = 4175 × 365 + 46 and = 5861 × 260 + 61. This agrees with the lower number inserted in red:—1,538,342 = IV Ik 15 Zac (12 Muluc), which comes later by 14,421 = 39 × 365 + 186 and = 55 × 260 + 121. 121, however, is the difference between both XIII Imix and IV Ik and the days XIII Akbal and IV Kan in the last column of page 62. If we set down with these two numbers, those of the normal date just preceding and the normal date next following, we have

This is a period of 37,960 = 2 × 18,980 days. It is possible that at some future time an indication of such a transition from one Katun to the other will be found in the writings. Now these two top lines contain two dates; on page 62 we find 13 Ceh, and on page 63, 13 Xul, but nothing further is to be learned from this than that one or the other of the day-signs, 2, 7, 12, 17, must have been set down in the effaced indication of the position in the Tonalamatl. All else is obliterated. From the third to the seventh row of these five columns it is all extremely simple. The third row consists only of five signs for beginning, the fourth, of five for end, the sixth of B's sign five times and the seventh of the elongated head q four times. But in the fifth, two deities alternate, one is apparently male and the other female; the god is in columns 1, 3 and 5 and the goddess in 2 and 4; the god probably belonging to the days III Chicchan and the goddess to XIII Akbal.

If we look upon this series as the first story of a structure and the large numbers just now discussed as the second, then we find the third story here, as we shall find it again on page 69. In the passage on page 31, which is so closely related to the present one, a timid attempt has already been made with the number 2,804,100 to erect a third story of this kind, which however barely attained to a quarter of the height of the one which now engages our attention. If the numbers hitherto examined refer to a time not very far from the present, we now come to numbers which lie in so remote a future that they can hardly suggest anything else than the destruction of the world or a sort of twilight of the gods. Nevertheless the starting-point of the whole, the series, which is built up with the number 91, i.e., the Bacab period or the quarter of a ritual year, continually comes to view. Indeed, the number of serpents is suggestive of this.

There are four large serpents, which fill most of the space on the left half of page 62 and the right of page 61. The two outer ones are bluish and the two inner ones white. They rise in several coils, their tails below and their heads above. A deity is represented above the gaping jaws of each of the four serpents, having apparently been vomited up. Above the first and third serpents B is represented in a fashion very similar to that which we have already seen on pages 33-35. Above the first serpent B has the pouch hanging from his neck and his hatchet is held downward; above the third he wears the pouch and the gala mantle and his hatchet is raised. Above the second fourth serpents, on the other hand, there are four-footed animals, but of a species not represented elsewhere. They might suggest a (four-footed?) walrus and a bear. We have here a double contrast, apparently referring to the four cardinal points.

The veil enveloping this representation would be lifted to a considerable extent, if all the eight hieroglyphs written above each serpent, were still legible. But, unfortunately, the second group is wholly and the third almost wholly effaced, while the first is partially effaced and only the fourth is preserved in its entirety. I read these groups in the following order:—

Of these 7 and 8 in the first and fourth groups form the date IX Kan 12 Kayab, which is in the year 4 Ix; this same date probably occurred also in the other two groups. That it is of special importance here, is shown by the two columns of hieroglyphs on the left side of page 61, where this date occurs again in the lowest place. The last three large numbers are not computed from the normal date IV Ahau 8 Cumhu, but from this very date and the other five from a similar one. The sixth hieroglyph in the first group seems to correspond to the fifth in the fourth, since both contain the elongated head q, though with different accompaniments.

In the first group the fourth hieroglyph is the Bacab sign familiar to us from pages 51-58, suggesting that the series here is closely connected with the one which had the difference 91. The fifth sign of the same group is that for beginning, probably to confirm the fact that this section begins here. The third sign of the first group is probably an Imix, as it is in the first and fourth of the fourth group, combined here with the woman's head, which we saw repeated on pages 62 and 63 at the top; and over it in the second place of the fourth group is B's hieroglyph, which is also repeated on pages 62 and 63 at the top. The third place of the fourth group is occupied by a head, which may be C's and which is distinguished by the same kind of circle which on page 9b surrounded the Ahau.

Eight complete dates are set down below the serpents, among which are the XIII Akbal already found with the previous large numbers, and III Chicchan (repeated three times), and then III Kan (twice), forming the beginning and end of the series (page 64), and also III Cimi and III Ix. As we shall see directly these are the end dates of the large numbers, and Xul = end repeated eight times at the extreme bottom corresponds with this. On the other hand, the starting-points must be found by computation, with the exception of the date IX Kan 12 Kayab, which is actually written down and is the point of departure for three of the numbers.

I will designate the black numbers by a and the red by b. Seven of the eight numbers are undoubtedly absolutely correct; but I must alter the number 1b, the red number belonging to the first serpent. I assume that a line is wanting in the lowest figure, i.e., it should be 8 instead of 3, and that the conspicuously large 1 further down on the page serves also as the red number, which belongs here. Only one slight change is necessary in the dates on the bottom of the pages, which were mentioned above. To the 16 in the date 4b I add a dot, and read it 17.

I will now give a table of the numbers, the starting-points of the periods obtained by computation, and the ends of the latter which are indicated below the serpents:—

See my treatise, "Die Schlangenzahlen in der Dresdener Mayahandschrift" (Weltall, year 5, pages 199-203).

Several details show how this number-structure forms a definite, closely connected whole.

1. The beginning day in each case is the day Kan, which thereby indicates its position as the first.

2. The last three starting-points are the same; the first three end dates, at least, are the same in the Tonalamatl, though not in the year.

3. The two numbers 2b and 4b are exactly the same.

4. The first three numbers are each divisible without a remainder by 17, the interval from XIII Akbal to IV Ahau, which was true also of the 1,268,540 in the second column on page 63, although only this last number has anything to do with these important days, of which the other three numbers are independent.

On the other hand, a notable difference between the first serpent and the other three is, that the day XI Kan is the starting-point of the first and IX Kan of the others. There are, however, 80 days between IX Kan and XI Kan. Hence the numbers 2a and 1b are separated from each other by 66,640 = 256 × 260 + 80, although they have the same end III Chicchan.

Further it is to be noted that the largest of the eight numbers, 12,489,781, is separated from the lowest, 12,388,121, i.e., the black number from the red one of the first serpent, by only 101,660, i.e., by not a full one per cent of the entire magnitude. 101,660 = 5 × 18,980 + 26 × 260 or 391 × 260 or 7820 × 13.

It is to be noted also that the differences between the black and red numbers in the second and third serpents (60,021 and 28,132) are divisible by 13 (4617 × 13 and 2164 × 13). They must be, since all six numbers refer to the day III.

Finally the question naturally arises, how did the computer obtain these values, i.e., how was the whole structure built up? On page 63 we found a 136,864 (not 136,884) set down in strikingly small characters and crowded between the other numbers, which would remain a mystery unless one assumed that it was reserved there for this structure; it is 91 × 1504. At first I thought it possible that this 136,864 had been again multiplied by 91, the real basal number of this section; for we had found a second power once before (on pages 46-50) by computation, viz:—2 × 260 × 260. The result of multiplication in this case would be 12,454,624, and the differences between the eight numbers in the serpents would be as follows:—1a + 35,157, 1b 66,503, 2a + 137, 2b and 4b  60,884, 3a  17,814, 3b + 12,318, 4a  165. But these differences are doubtful, inasmuch as they bear no relation to the dates beginning and ending the serpent numbers.

On the other hand, another number contains the desired properties. I refer to the 12,412,920, i.e., it is 109 times the so-called Ahau-Katun of 113,880 days, and I believe I have found that the Ahau-Katun and its multiples were mostly used in the formation of the large numbers. In the following table I have placed this number beside each of the serpent numbers, have then found the difference between the two and have added to it the interval between the first and last day of each serpent number:—

Where the serpent number is less than 12,412,920, I have had to place the last day before the initial day.

The work of the Indian computer was, therefore, as follows:—

He took the days for granted. First he determined the differences between them; then he added to each difference a multiple of 260; and the choice of the multiple seems to have been quite arbitrary. The number thus obtained he added to 12,412,920, unless it was the smaller, in which case he subtracted it from 12,412,920, and the result he wrote down in the serpents.

We shall find the same process, only somewhat amplified, with the serpent on page 69.

Are the seven numbers intended to denote the destruction of the seven planets? I hope this question will be answered in the near future.

There now remains of the contents of these pages only the two columns on the left of page 61, which we will now examine and at the same time compare them with the corresponding column of page 69, the upper part of which is exactly the same, and the lower very nearly so. Each column consists of 18 hieroglyphs, which I count from the top downward, designating those of the first column by a and those of the second by b.

At the first glance these double columns remind one of the inscriptions in the temples and on the stelae, especially of their beginnings, the so-called initial series. Here, in the second column, we find the statement of the usual periods:—144,000, 7200, 360, 20, 1, but in the first column we find faces belonging to them. In his work "The Archaic Maya Inscriptions," 1897, which, on the whole, contains more of imagination than of science, J. T. Goodman unqualifiedly declares these faces to be numbers by which the periods indicated beside them are to be multiplied, and this theory has already found considerable recognition; we will therefore try to follow where he leads.

1a and 1b are effaced on page 61; they probably contained a sort of superscription as on the inscriptions. 2a is effaced on page 61, but the sign may be recognized from page 69 as that with which on page 46 the series of the twenty deities begins after 236 (4 × 59) days. On pages 61 and 69 it takes the place of a face, to which I am inclined to assign the numerical value 4. In 2b, which is C's head, I am inclined to look for the value 2,880,000 = 20 × 20 × 20 × 360 days, which is not at all inappropriate for C, as the sign of the north pole around which everything revolves. I therefore propose to read 2ab as 4 × 2,880,000 = 11,520,000. 3b, it seems to me, resembles the sign for 144,000, which I found in the inscriptions and which is repeated in 12a. It must, however, be left undecided by what this same number in 3a is to be multiplied; 3a is repeated besides in 8a and 13b. 4a contains the head of E, and 4b that of the Moan. 4a seems to refer to 5a, and 4b to 5b. But 5a and 5b are the same sign, which, inserted between the 144,000 and the 7200, can scarcely mean anything else than the so-called Ahau-Katun of 6 × 18,980 = 113,880 days. Have we two such periods here? Were they designated by consecutive numbers? Now comes the 7200 in 6a, and the number 8 with E's head and the inserted sign for 360 days in 6b (on page 69 without E's head), therefore 8 × 360 = 2880. Seler also thinks 7a has the numerical value 16 (Einiges mehr über die Monumente von Copan, etc., page 217); 7b belongs to 7a. 7b, a Kin with a I and a suffix and a leaf-shaped prefix, is inserted between the 360 and 20. What can it mean? Hardly the 260, for this is represented elsewhere (e.g., page 24) by the thirteenth month Mac. Or can it possibly refer to the month Yaxkin (days 120-140)?

8b is a Chuen sign, which, with its prefix (superfix on page 69) always denotes twenty days in the inscriptions. It is multiplied with the same unknown head in 8a, which we have already met with in 3a. 9a contains H's head, and 9b is an unknown head with inserted Kin; the two signs must of necessity indicate the single days still to be added to the period, though as yet we do not know how.

The normal date IV Ahau 8 Cumhu then follows in 10ab. If it refers to the signs just now discussed, then they must denote a number of about the same magnitude as the serpent numbers. 653 or 654 times 18,980 seems to suggest itself, but we shall have more to say later on this subject. My efforts to reach a definite result here have failed.

Nor does the lower part of the two columns lead me to the desired goal. As it seems to consist of several groups, I will first combine 11ab and 12ab. I look upon 11a as denoting 20, and with regard to 11b I have already expressed the surmise in the Zeitschrift für Ethnologie XXIII, page 153, that it may mean 8760 = 24 × 365, i.e., three Venus-solar periods. That would be 20 × 8760 = 480 × 365 = 175,200. The Moan in 12a may have the value 13, for this number is so often combined with the Moan. As we saw under page 51, 12b is = 18,980; 13 × 18,980 = 246,740. Accordingly the four signs taken together may mean 421,940 = 1156 × 365.

The second group, from 13a to 15b, refers, on the other hand, to the year of 360 days. First 13a = 144,000, having in 13b the unknown multiplier, which we have already seen in 3a and 8a. Then follows in 14a, 15 × 7200 = 108,000; in 14b, 9 × 360 = 3240; in 15a, a 20 with a prefixed 1 (21?); and in 15b, three days. It would be more correct to place the 1 beside the following 3. The whole sum would then end with the number 4, which would agree with the day Kan, the date specified below.

In the third group the 16a = 19 × 18,980 = 360,620, remains a mystery; an empty outline of a sign is added in 16b.

17ab also forms a group by itself. 17a contains a sign, which rather suggests the Bacab, upon whose period of 91 days the series belonging here is based. The Imix in 17b with a superfix is still unintelligible.

The columns end in 18 with the date IX Kan XII Kayab, the starting-point of the serpent numbers.

Pages 65—69.

I think it very likely that this section bears the same relation to pages 61-64 as pages 46-50 do to 24 and as 53-58 to 51-52. For here, too, a period of time forming the basis of the earlier section seems to be divided into smaller parts. On page 64 we recognize as the basis of the series the number 91, the quarter of the ritual year of 364 days; here we have to do with the fourfold division of 91 into 13 unequal parts. And the real starting-points on these pages, as on the previous ones, are the days III Chicchan and XIII Akbal.

The four series of numbers, the top one of which I have probably correctly restored from what still remains, are as follows:—

The first two lines, forming together a single period of 182 days, refer to a day III, as we see by the ending, and the last two to XIII, which undoubtedly refers to the III Chicchan and XIII Akbal, the days so significant in the preceding section. Hence an interval of 218 days (III Chicchan to XIII Akbal) is to be assumed between the second and third lines, with the addition of which interval each of the two periods extends over 400 days.

The first and fourth series have the same difference; and the second and third correspond with one another in this respect. In the first and fourth the differences follow a rule, viz:—as if one were walking in a ring having on its edge the numbers 1 to 13, and kept stepping backward four numbers. The differences of the second and third series apparently do not follow any rule. Hence I think that the fourth series follows the third by mistake and ought rightfully to precede it. Only the fifth member in the first and second series has the same day VIII and the day V in the third and fourth series, otherwise the week-days of each series differ from those of the others.

As I regard III Chicchan and XIII Akbal as unquestionably the starting-points, I will here give a table of the days on which each of the twenty-six members of each series must fall and at the same time I will indicate for each day its number from the beginning of the series. Accordingly the first 182 days present the following appearance:—

In the same way I will tabulate the second group of 182 days, but in this case I shall place the fourth line before the third, which is probably correct, and which shows for the first time parallelism of the two rows:—

It would be very essential now to know what place these days occupy in the year, and what year is meant; the answer to one of these questions would at the same time solve the other.

Now I think I come nearer to the solution of this problem by assuming that the pictures and hieroglyphs refer here only to the more important of the two days, XIII Akbal, and that III Chicchan is represented only by the numbers of the series. Thus both the pictures and the hieroglyphs of the two sections connect without the interval of 218 days, which must be assumed in the case of the numbers.

Here, as is usually the case of series, we have to begin at the bottom. Now the first group of the lower half of page 65 contains the sign 9 Kan. If, as it seems, this actually denotes the year, then the day XIII Akbal must be the first of the eleventh month, i.e., the 201st day of the year. Hence I will again set down the twenty-six dates, but add to them the position in the year.

Let us now prove the correctness of my theory by an examination of groups 22 and 23. In 22 the 160th day of this period, the 361st day of the year is reached, i.e., the first of the five Uayeyab days. The year 9 Kan is ended and the year 10 Muluc is not yet reached. In the corresponding picture we see B occupied in conveying in a bag the image of God K to whom belongs the next year. B is armed with the official staff and the bag also contains water (rain). In the 23d group the 166th day has passed and the second of the year 10 Muluc is reached, which gives the name to this year. The first hieroglyph shows two personages sitting back to back. This representation is repeated on a larger scale below in the Janus picture of B who is sitting on signs of planets. The second hieroglyph, with equal fitness, represents a clamp, which is intended for fastening two objects together, and which is repeated twice over the Janus picture, black in one case and white in the other. Rain is pouring over the second half of the picture, for it has long been known that Muluc and rain belong together, and in our examination of page 7a we saw that K is the ruler of the day Muluc (6).

Now, before I begin the examination of the separate pictures and the groups of six hieroglyphs belonging to each picture, I wish to mention three things which are often repeated here.

First, B's picture, which appears in all the twenty-six pictures with the exception of 20, 24 and 25, and represents the god in the most varied positions and activities. These pictures are very similar to those on pages 29-46 and we shall therefore make frequent reference to the section there represented.

Second, the first hieroglyph in groups 1 to 13, strange to say, is not found in the second half. It is hieroglyph f, which appears in exactly the same way in close combination with B in two sections, which differ from each other but are placed side by side on pages 30c-39c. In the present passage it has a distinct prefix resembling the beak of a bird or tortoise, but in the former passage it has rather a stunted appearance. It seems to refer to the eagle in B's hands in group 13.

Third, the head with no underjaw, which is the sixth hieroglyph in groups 1 to 13, but does not occur in groups 14 to 26. It is repeated in a very similar fashion in the last hieroglyph but one on page 23b. I propose to attribute to it the meaning of fasting.

Now for the single groups:—

1. B is seated rowing in a boat, as he is represented also on pages 29c, 36b, 40c and 43c. A creature is swimming beneath him, which may be a crocodile. The fifth hieroglyph is the important 9 Kan already discussed, the fourth is a and the second the cross b combined with Caban. The day is the 210th of the year.

2. B is walking with the atlatl in his hand, and armed with javelins. Hieroglyph 5, Manik, denotes the chase, but has a prefix, which often seems to have the meaning of 20. 2 is the elongated head q with the prefix of the east belonging to the Kan years. 4 is a Moan sign (c) with the leaf-shaped prefix. Does this perhaps denote the slaying of game in the forest? It is remarkable that B's feet are hidden, as if he were walking in sand or in a bog.

3. B is walking, carrying a large stick like that for tilling the field, as on pages 38b and 39b, and he bears a carrying-frame; there are footprints below him. Hieroglyph 2 is the compound of the signs for south and east, 4 (r) may denote rain, and 5 is two elongated heads with an unknown prefix.

4. B, is seated on astronomical signs as on page 37c. The copal pouch is hanging from his neck and he is brandishing his hatchet. Sign 2 is b, 4 is a and 5 is r, but all three signs have unusual prefixes; the first of these prefixes appears again in the tenth group, 41 days later.

5. B is seated on a head, probably that of D, which, however, is peculiar owing to the ornaments resembling bunches of grapes in place of both the eye and the ear (compare pages 39c and 41a). I do not venture to decide what he holds in his hand nor what are the other objects which he carries. Sign 2 is r with a prefix, 4 is Imix perhaps with a knife as a prefix, 5 is the skeleton which sometimes belongs to the lightning beast, but also to the 14th month; its prefix is unknown.

6. B is seated on a support, which contains two cross-bones, down to which he points with his right hand, while his left hand holds the hatchet on his knee. Sign 2 is the crouching naked personage, with the cross b prefixed, 4 is the elongated head with a prefixed Yax, and 5 is Kan with a vessel as a prefix (instead of Imix) from which steam or froth is rising. The day is the 234th of the year, i.e., the end of a week of 18 × 13 days.

7. B is sitting on a tree at the root of which his own head appears (compare with this the representations on pages 31c, 33c, and especially 40a, and also 41b and 42b). The second sign is Yax with a prefix; 4 is Kin within which there is a 1, as is several times the case, for example, on pages 61 and 69. The fifth sign is still a mystery to me. The day here is V Manik. Do the hieroglyphs suggest that the interval from the day IX Kan, which gives the name to the year, to V Manik is exactly the same as that from the normal date IV Ahau to the true starting-point of our passage, the day XIII Akbal? Both intervals are 243.

8. B is seated in a house, on the roof, wall and floor of which are several Caban signs, just as on page 30a; he seems to be pointing forward. Sign 2 is Caban with a prefix, the 4th and also the 5th is Kan with two unusual prefixes.

9. Water is pictured at the bottom of this picture, and in it are a fish, a mussel and a snail (possibly page 37b may be compared with this). There seems to be a suggestion of footprints on the margin of the water, back of which B is walking, his legs hidden as far as the knees. He holds the hatchet uplifted in his left hand and his right holds what may be a long-stemmed aquatic plant (compare page 42b). Sign 2 is composed of b, Imix, the mouth and nose of C and the object which apparently is a beak, previously met with in sign 1. 4 is Kan-Imix, and 5 is Kan with prefix and suffix.

10. B is seated in an expectant attitude, his hands resting on his knees. We see a very similar representation of him on page 38a, where he faces himself, and in general the remaining pictures of that passage furnish a striking parallel to the present one. Sign 2 is a head (E's?) with a call seemingly issuing from its mouth. 4 is the elongated head q with the Ben-Ik superfix and an unusual prefix, which we found on page 66c prefixed to the cross b; 5 is Kan with the same prefix, which I regarded as denoting a call in sign 2, and which is probably answered here by an affirmative cry.

11. The expectation has been fulfilled. B is seated on a mat holding a woman in the same position as on page 38a. Sign 2 is the cross b with the prefixed beak as in 1, and also with another prefix, which seems sometimes to denote the number 20. 4 is exactly the same Kin with 1 and the leaf-shaped prefix, which occurred in the same place with the seventh picture. 5 contains the sign for 73 days; a new period of this length begins here on the 74th day.

12. As in the parallel passage on page 38 B seems to be offering a Kan, so here his gift consists of a kind of wreath, like the one in the fifth picture; he is seated on astronomical signs, which contain the cross b twice as does also hieroglyph 2. 4 is Kin-Akbal, and 5 is a Kan with the prefix which generally belongs to the south as a superfix.

13. B is seated on the elongated head q with a superfix and a prefix, exactly as on pages 37c and 40a, and this sign is repeated in the hieroglyphs (in 2) just as it is in the two former places. He holds the eagle on his lap and we see him connected with the same bird in a different way on page 43c. Is B represented here as the preventer of evil? Hieroglyph 4 is a, while 5 is Kan, apparently with the sign of the south as a prefix. A Bacab period of 91 days ends here. We come now to the upper series of pictures.

14. B is walking in the rain, with the copal pouch around his neck and the hatchet uplifted in his left hand. An unknown object, possibly held in his right hand, is hanging in front of his legs. Hieroglyphs 1 and 3 are effaced, 2 is indistinct, 5 seems to be a Xul (end, close) and 6 is E's head.

15. B is walking, brandishing the hatchet in his left hand, and holding in his right an object resembling a cornucopia filled with fruit; below this hangs what appears to be a flower. The god wears the copal pouch. Hieroglyph 1 is a hand holding K's head; it is curious that this sign should also occur in the next group as an indication of the approaching Muluc year. 3 is a sign still undetermined; but the prefix is the crouching naked personage with dots suggesting stars around its head. I have often thought that similar figures represented Mercury; it is remarkable that exactly the 115th day of this section is reached here, corresponding with the apparent revolution of Mercury = 115 days. Similarly sign 2 invites computation; it is a face resembling an Ahau sign, with a 3 as a superfix and a 9 as a prefix; compare the other places containing the same face, with 33c. After the fashion of the inscriptions this would denote 9 + 3 × 20 = 69, which by the way is three fifths of the Mercury revolution. 5 is a compound of Akbal and Imix and 6 a compound of a Moan sign (c) with a.

16. B is in a half sitting position and holds a strange object before himself. On top of his own head is K's, which is repeated in sign 2. I do not know how to explain 1, unless it is the bat-god; 3 is a Xul = end (but of what?) combined with Imix, and 5 is the usual Kan-Imix. 6 is a Kin with an 8 back of it (as 36b, 37b, 67a, 68a) and over it is a hand pointing to the right, just like those in groups 20 and 25. This looks as if we ought to count forward 8 days, but what can be the purpose of doing so?

17. B is walking armed with spear and shield. Sign 1 is b, 2 the face resembling an Ahau, which occupied the second place in group 15, 3 is probably Xul again, but with an effaced prefix; of 5 also only an Imix remains; while 6 is the usual compound of Muluc-Caban.

18. We have now reached the day 16 Kayab, a day very close to the day 18 Kayab, which on page 24 we recognized as an especially important day, while in my article "Zur Entzifferung III" I regarded it as the day of the summer solstice. Computed from the normal date IV Ahau 8 Cumhu it may also have denoted the end of a lunar year, as on pages 51-58 where it is the basis of the series. The picture here agrees with this. B is sitting in the pouring rain of the rainy season and gazing upward at the planets, as on page 36c and particularly on 39c; the sun and moon are also represented, but below the planets. The hieroglyphs likewise contain the sun and moon in 1 and 2, in 3, Ahau and Xul with a prefix, as if this were the end of the increase of the sun's power; 5 is Kin-Akbal, day and night, and 6 is Caban with the cross b.

19. B is walking armed with hatchet and shield. He holds a serpent in his hand as on page 40c, but here with the head downward. Hieroglyphs 1 and 2 are destroyed, 3 is the cross b with a suffix and the horse-shoe prefix e, known to us from pages 5 and 6. 5 is Imix combined with Chuen and probably with Yax, and 6 is E's head.

20. This is the old red woman with the tiger claws, whom we saw on pages 39b and 43b and shall see again on page 74; she reinforces the water falling from the planets by pouring a stream from her jug. The first three hieroglyphs are effaced, 4 is the elongated head q, 5 is Kin-Akbal, 6, as in group 16, is again the enigmatical 8 with a hand pointing to the right.

21. B is walking and bears pouch, spear and shield. Hieroglyph 1 is a hand holding the sign of the rising Moan, just as in 15 a hand holds the head of K; 2 is again K, whose sign is probably effaced several times in the last groups of this series. 3 is E with the sign of the east; 5 is compounded of Imix, Chuen and b, and 6 is Kin with the sign of the north. Here the day of the normal date is reached, but this may be significant only for the year 9 Ix.

22. We come now to the representation of the change of the year, which we have already mentioned. Hieroglyph 1 is curious, consisting of the moon with a stripe running around it like a strap; 3 and 5 are not clear to me and are doubtless closely connected with one another; 3 also contains a trace of K and is perhaps a determinative of the same. 6 is again E, and suggests the tilling of the fields.

23. This picture as well as the first two hieroglyphs have already been discussed above. The crouching personage, repeated again in 3 as a prefix to the cross b, is curious. 5 is again E and 6 is Imix, referring to grain and honey.

24. The picture and three of the hieroglyphs plainly correspond. The grain deity E holds food and drink in his hand. Rain is pouring from the planets, and the wind-beast plunges down, as on pages 44 and 45. Sign 3 is E's hieroglyph, 6 is Kan-Imix and 2 is the wind-beast. B is superfluously added in 4 and the same is true of the cross b in 1, while Kin-Akbal in 5 seems to fit almost everywhere. Pages 29a, 30a and 45c show the lightning-beast in a different form.

25. As is usually the case, rain is pouring from the stars and below them are the sun and moon as before. This time C is sitting in the rain, clad in the gala mantle and holding Kan. Hieroglyphs 1, 2 and 4, the latter apparently representing C, are effaced. The other three are enigmatical, 3 is again Xul with a prefixed 9, 5 a Caban, but with an unintelligible prefix, and 6 is again the mysterious 8.

26. B is sitting on a tree or sacrificial stone, which is colored half blue and half red, and may denote the ceasing of the rainy season; he is brandishing his hatchet. Hieroglyphs 1, 2 and 3 are effaced; 4 is B's sign, 5 might be Xul and 6 is a with c added and thus referring to the Moan. And here the half of the ritual year ends with the 182nd day, which is XIII Chicchan 18 Pop (10 Muluc); and it is left to the reader to imagine or to find hieroglyphs and pictures for the other two series of numeral signs.

I am troubled about the five naked crouching figures of this section, which I am inclined to regard as the sign for Mercury with its apparent revolution of 115 days, which, however, seems sometimes (as on pages 54, 56 and 58 in the upper sections) to be raised to the value of half a Tonalamatl = 130 days. This may be explained by the fact that it is difficult to determine exactly the length of the revolution of Mercury. In group 15 this figure appears exactly on the 115th day of this section, but in group 6 on the 234th day of the year, i.e., approximately at the expiration of two Mercury periods after the beginning of the year. But now for group 23. Here there are three of these crouching figures. The two upper ones leaning back to back must serve the purpose of indicating the change in the year. But they would hardly do so, if the third personage were not added, which may indicate that the solar year consists approximately of three Mercury periods. I look upon this view of the matter merely as the first attempt at an explanation.

Pages 69—73.

The chief subject of the last great section of this Manuscript is two of the usual series, from which large numbers are developed in the usual way and the largest of all is finally recorded in a serpent. This section thus forms a parallel to the contents of pages 61-64, but is somewhat more composite.

Before I begin the discussion of these series, I wish to examine two passages, which I think are not connected with these series, but are independent, like the instance on pages 51-58, where the hieroglyphs were found to be quite independent of the numerals. The Mayas took advantage of space wherever it presented itself, which is admissible in ideographic writing.

The first of these two passages is at the top of pages 71-73. Here there are four horizontal rows of twelve hieroglyphs each. Since, however, the top row is entirely effaced and none of the other three are perfectly preserved, it is quite impossible at present to judge of the interconnection of the whole. But I must point out a certain resemblance to the passage on pages 44b-45b, where a period of 78 days is considered with reference to the wind-deities. The first and sixth columns of pages 71-72 likewise contain the signs for wind and the pierced ears. The fact that the Bacab sign occurs in the eighth column, and in no other, must attract 'attention; if we knew it to be effaced in the first column, then each column might refer to 13 days, though 12 × 13, it is true, does not form a natural whole. C's sign is the only hieroglyph of a god to be found in both passages. E also occurs on pages 44b-45b and may be one of the effaced signs on pages 71-72. There is no trace left of the others. The fact that some hieroglyphs occur in both the passages referred to proves nothing with regard to signs in frequent use and I can find no cases of correspondence among those occurring more rarely. Hence this passage must be left for the present as an almost complete mystery.

I have discussed the second passage in detail in my article "Zur Entzifferung der Mayahandschriften V," of the year 1895, and from it I will borrow the following. This second passage fills the middle and lower thirds of pages 71-73, occupying the same space as the first passage in the upper third, and offering far more reliable material than the latter.

That these hieroglyphs are not connected with the numerals above and below, can be deduced from the fact that the numbers follow one another from right to left and the hieroglyphs in the reversed order. This is proved by the hand pointing to the right, which occurs here at least eight times like the one occurring twenty times on pages 46-50.

But the scribe, misled by the direction of the numeral series, began on page 71 to write the first of these hieroglyphs from the right instead of from the left, but after the first four groups he corrected his mistake. Hence I read the groups of three hieroglyphs each, in the following order:—

The number 28 shows that we have to do here with 28 weeks of 13 days each, i.e., with a ritual year of 364 days, as was the case on pages 31-32, 63-64 and 65-69. This year, however, is divided into four parts of 7 x 13 = 91 days, i.e., into four so-called Bacab periods. This is very plainly indicated here, for groups 4, 11, 18 and 25, i.e., those separated by seven groups each, are exactly alike, but in group 4b (I will designate the three hieroglyphs of each group from top to bottom by a, b and c) there is a prefixed 4 which refers to the four Bacabs as does the same 4 prefixed to the Bacab sign at the top of page 72.

Now the question arises as to when this ritual year began. Undoubtedly its beginning day was very different from that of the civil year (360 days) and from that of the astronomical year (365 days).

In this matter I follow Mrs. Zelia Nuttall, who has rendered such estimable service to Aztec science. At the Congress of Americanists at Stockholm in 1894, she submitted an article entitled "Note on the Ancient Mexican Calendar System," in which with keen discernment she pointed out a year beginning with the spring equinox and including in its centre the sacred Tonalamatl, i.e., 260 days, which were preceded and followed by 52 days. I recognize this ritual year also in the present passage of the "Dresdensis," as the one current in the Maya country. It probably began about the 10th of March, at that period about the time of the vernal equinox, according to the Julian Calendar.

Beginning with this date, I will now attempt to tabulate the chronology of this passage. In the first column I will place the number of the group of hieroglyphs in question, in the second I will set down to what day of the Maya year each group refers; in the third, the corresponding day of our year, and finally in the fourth, the 20-day periods which agree in general with the dates.

In the following I will call attention to a few points by which this arrangement is justified.

Hieroglyph 1a admits of explanation. It consists of four parts:—the left top is Kin, meaning sun or day, the right top is the sign of the year, the right bottom is the knife as symbol of separation or division, and the left bottom, which is especially decisive, is the month Ceh. Hence I read 1a thus:—the day of the change of year in the month Ceh. The sign 1b is the familiar Kin-Akbal signifying either the beginning day or the day Akbal. If the year should be named from this sign, then this would mean a Kan year, as in the preceding section the beginning lay in the year 9 Kan. If the year in the latter section had been as equally divided as the one in question here, it would have furnished us with some very remarkable parallels.

Again the four groups:—4, 11, 18 and 25, which are alike, are important. The cross in sign a, combined with the three dotted lines passing from top to bottom, may refer to the wind and this meaning is further confirmed by the Ik sign (wind) in c. Further the sign b between them is that for the Bacab, the wind deity itself.

The most important events of the year are obviously the sowing and harvesting of the maize together with the beginning and end of the rainy season. Now we find the first two in connection with the god E, the maize-god, who is represented in 6c and 13c, 91 days apart, corresponding to the end of May and the beginning of August. Generally speaking, sixty days only were reckoned as the time between sowing and reaping, but here a quarter of a year may have been taken as a round number and it may also have reference to a more elevated region.

I am inclined to think that the beginning and end of the rainy season are referred to in signs 8c and 16c, where, as it seems to me, three lines of drops are falling from a rectangle denoting the sky (as is usual) like the representation of rain dropping from a cloud at the bottom of page 36 (second picture). The serpent 8b as symbol of water may also refer to the same thing, especially as it is combined with an Akbal (often denoting beginning). The sign, which I think denotes the rainy season, is very similar, but not the same as another one, which is common to the Dresdensis and Tro-Cortesianus, the significance of which is certainly very close to the idea of the week of 13 days.

I have some other ideas on this subject, which, however, are mere conjectures, advanced with some hesitation. If the Chuen sign in 7a is actually a serpent's jaw, then it might refer to the beginning of the astronomical year in May, since the serpent so often designates that time.

In 9b we find a crouching figure with the sign which is usually considered that of the death-bird. In another place (Zur Entzifferung IV, 12) I have regarded the naked human figure placed upside down on page 58 as the sign for Mercury, and on page 60 at the bottom, left, I also regarded the crouching figure as representing Mercury vanquished by Venus. But in 9b, which belongs to the 105th-117th days of the year, a 115 day revolution of Mercury is computed. A crouching figure, like that in 9b, likewise appears on page 65a in the second series of 91 days after 11 + 13 = 24 days of this series have elapsed, i.e., directly after the 115 days of the apparent revolution of Mercury.

In 10b, and it is the only place in this passage, we find the hieroglyph of B, the leading god of this Manuscript. This corresponds with the time of the greatest power of the sun and of the change in the civil year (July 16th). In Group 12, do a and c mean the year and is b the head with the Akbal eye, thus denoting the beginning of the civil year? It ought really to have formed group 11, but there was no room for it, since it was necessary that the signs for the period of 91 days should be set down there.

Signs 14a and the combined signs 15bc are almost alike and suggest 1a. Is it intended to designate here the ritual year, the time of the autumnal equinox (September 10th?). In 15a two hooks, turned in opposite directions proceed from one side of the sun-glyph. Do they signify two halves of the year and does the 3 in front of them signify the third quarter of the year?

20b is the sign of the death-god A, probably not placed accidentally here at the end of the month Xul, which denotes the end; but the end of what?

The hieroglyph in 23a is a black bird, with two hooks, one pointing up and the other down, projecting from its head. Usually these hooks belong to K, and by means of them this bird becomes the storm-bird; the year symbol is below. Does this hieroglyph signify the time of the shortest day, when darkness predominates?

A peculiarity of this passage is the striking frequency of the sign looked upon as that of the death-bird as well as of the cognate sign, which is commonly considered as that of the rising Moan. The first bird is in the 14th group, in the 9th it is combined with the apparent Mercury sign, and in the 17th with the year sign. The second bird with the prefixed Yax is in the 2nd group. But it is especially striking that several times both signs, and this is the case nowhere else, are combined into a single sign in groups 9, 13 and 26 and also probably in 19 where, however, the Moan sign seems to be effaced.

This is all I have to say at the present time in reference to this calendar. Some of my statements are positive and some are only conjectures. Compare my treatise "Zwei Hieroglyphenreihen in der Dresdener Mayahandschrift" (Zeitschrift für Ethnologie, 1905, 2 and 3).

Having disposed in this way of the two supplementary subjects of this section, I will now proceed to consider the principal theme, viz:—the two series and whatever is connected with them.

As with the other series, we begin here at the right, i.e., with page 73. There in the last column we find the superscription as it were. It is true that nothing positive can be gathered from the top part consisting of five hieroglyphs, which are mostly destroyed. The third hieroglyph seems to be the sign in group 2a discussed above. The fourth is an Akbal with a prefixed arm as on pages 8a, 36a, and the fifth is an Ik with a prefix.

Below these are three numbers:—14,040, 702 and 54, which are in the proportion of 260, 13 and 1, so that the 14,040 is a Tonalamatl, as it were, of 260 periods of 54 days each. The fact that 54 is chosen here as the difference of the following series is curious, because usually only parts of 260 or of 364 are selected. But 54 is probably only a secondary matter, while 14,040, with its marvellous property of divisibility into the most varied and important periods, is the chief subject.

There is a 9 in a red circle under the three numbers. It is meant to denote the starting-point of the series, the day IX Ix. Perhaps these two as well as the 54 are connected with the 9 "señores de las noches."

In passing on to the left, I shall not consider the hieroglyphs and numbers in the next two columns in the upper third, since they are only set down here in order to secure space for them. They will be discussed later.

The series itself begins in the upper third of page 71, in the next to the last column; it is continued on page 72 and on page 73 as far as the third column. The first twelve numbers are written from left to right contrary to the usual practice, doubtless occasioned by the passage above the series, which has already been discussed. And below, again contrary to rule, we find not the week and month days, but only the week days and they are in red circles. If written in the usual way, the series would have the following form (with the usual omission of the initial day IX Ix):—

The series must now continue with the 702 already specified on page 73, which it proceeds to do from right to left in the middle of page 71, and continues from there on with regularly added dates and with the 702 itself as the difference. At the same time, since 702 = 54 × 13, the week-days are forced to come to a standstill on the IX, while each of the month days ascends by two (702 = 35 × 20 + 2). The 4914 = 7 × 702 is obtained in the next to the last column of page 70. On page 71 the 702 is incorrectly set down as 1. 15. 2. instead of 1. 17. 2. The series continues on page 71 in the same way beyond the 702, until in 7020 a number is obtained which is also divisible by 260, so that now the accompanying day must be IX Ix. Now we ought to expect to see here the double of 7020, the very 14,040 abovementioned, but it is omitted just because it was set down on page 73. Nevertheless this very number forms the new difference with which the series returns from page 70 to the top line of page 71, where the numbers are mostly effaced, but enough remains to enable us to assume that the last number on page 71 is the 10th multiple of 14,040, and this may be followed by the 11th and 12th multiples, the last number being 168,480.

This series begins in the middle of page 73 with the day IV Caban, the zero-point therefore being IV Eb. It then advances to the left across 28 members, until on page 71 it reaches the number 1820 = 5 years of 364 days = 7 Tonalamatls. From there on, 1820 itself is the difference, and the accompanying day therefore remains IV Eb. Then, in the two lowest sections of pages 71 and 70, the fourth multiple of 1820, i.e., 7280, is the third difference and thus the series advances to 15 × 7280 = 109,200 on page 71, after which on page 70 the omitted 8 × 7280 = 58,240 is written out. Close beside this number are the figures 1. 0. 12. 3. and a 0 below the latter, which was not successfully erased; this would be the number 7443 of which I can make nothing at all.

The initial dates of the two series, IX Ix and IV Eb, are 138 days apart and reversely 122 days.

The transition, as it were, from the series to the large numbers is formed by a few groups of hieroglyphs.

The first of these groups is at the top of pages 69-70; its first top line is completely effaced. The remainder I will designate by the following numbers:—

The date IX Kan 12 Kayab, set down under 3 and 4 does not belong there but to the serpent below and will be discussed later.

I take sign 1 to be that of a Bacab, 2 I do not understand and it is half obliterated; it seems to occur again on page 73 in the column to the extreme right. 3 and 7 are the elongated head q with an unusual superfix, 4 and 8 correspond with one another, but I cannot explain them. 5, 10 and 14 denote the beginning, 6, 11 and 15, the end. 9 and 13 both designate the 8th day of the month Kayab and over them IV Ahau must have been set down twice. 12 and 16 are two heads of gods, 12 is probably D's with the sign for west and 16, B's with that of the east.

On page 70, in the middle of the third and fourth columns, the day IX Ix occurs twice. In one case it ought to have been IV Eb and the scribe has really changed the IX to IV, but he omitted changing the Ix to Eb. Directly below these dates we find the second group, consisting of two rows of four hieroglyphs.

I think these eight hieroglyphs can be interpreted as follows:—

The following is to be noted in this connection:—

3 is really set down X Ahau, but an VIII is written above the Ahau by way of correction. The day VIII Ahau will presently prove to be important.

6, a compound of Imix and the superfix denoting multiplication, is the sign for 18,980, and its prefix seems to me to denote duplication. We have long known how important the 37,960 = 146 × 260 = 104 × 365 is, and, if my theory is correct, we shall see directly that it occurs again here.

8 seems really to be 1 Zec, but the composite prefixes demand further examination.

Impenetrable darkness still shrouds the meaning of the whole group. Though it is clear that in several cases certain days are specified according to their position in the year, their distance apart does not agree with the interval between days IV Eb, IX Ix and IV Ahau under discussion here.

If signs 3 and 4 ought to be read together as VIII Ahau 13 Yaxkin, then this date would come in the year 7 Muluc. In the Zeitschrift für Ethnologie I explained the five hieroglyphs in the third column at the bottom of page 70 (the third group) as civil years of 360 and astronomical years of 365 days:—

This, it is true, is a striking explanation and certainly a surprising one!

Now the date IX Ix 12 Kayab is at the very bottom of the fourth column. This, without apparent reason, would refer to the year 4 Kan. Should it not be read IX Kan 12 Kayab (4 Ix), thus indicating that the entire passage is only the preparation for the date from which the serpent numbers proceed? The scribe may have had in mind the IX Ix of the series.

The fourth and last group on page 73, above the two numbers 83,474 and 34,732, consists of four hieroglyphs. The two upper hieroglyphs on the left are effaced, and the top one on the right. I think it probable that the day VIII Ahau, which will be discussed later, may have stood in the top line, and possibly with a month date. Of the two remaining signs of the fourth group, the upper is the moon and the lower Imix, probably with the hieroglyph of the east as a prefix; but there is nothing to be done with it owing to the obliteration of the sign above it. In the Zeitschrift für Ethnologie, 1891, page 153, I have endeavored to explain these three signs on the right above 34,732, by suggesting for them the values

and calling special attention to the fact that between IV Eb and IV Ahau there are 208 days, and that the 34,732 placed below them in the Manuscript, increased by 208, is equal to 34,940. This group then seems really to belong to the day IV Eb and to the 65-series, while manifold problems are still to be encountered in interpreting the other groups.

The Manuscript offers material with which to work, beginning on page 70:—

This is followed at the right top of page 73 by

Two of the numbers and two of the dates are conjectural:—

I read the 1,202,240 as 8. 6. 19. 10. 0. while the Manuscript has 16 instead of 6. I read the 101,812 as 14. 2. 14. 12. the Manuscript has 16 instead of the second 14. And in two places in the third column of page 70, I have restored the day IV Eb, where the Manuscript incorrectly repeats the IX Ix, and does the same thing on page 73.

Let us now first consider the construction of those large numbers, which are connected with the day IX Ix and thus with the 54-series. These numbers are the two upper ones of columns 1 and 2 and the lower one of column 1 on page 70.

174 is the starting-point, the number of the day is IX Ix, which seems to have been chosen because it divides the Tonalamatl approximately in the proportion of 2 to 1. (IV Ahau IX Ix = 174.)

The 5359th, 5520th and 4619th multiples of 260 have been added to 174; why precisely these multiples were chosen remains a mystery. In this way were obtained the following numbers, which the Manuscript suppresses. I will give them with their corresponding dates:—

When we add to the above the three encircled numbers 606, 1,646 and 86, the resulting sums are the three numbers found in the Manuscript:—

I am placing the first two not far from the present and the third in the past.

As multiples of 260 these three numbers have the following form:—

Some curious facts come to light with regard to their magnitude and their mutual relation.

The two largest numbers are 165 × 260 = 660 × 65 apart; this recalls the 65-series. The third lowest number is 165 × 7280 and thus contains not only the 65 but = 165 × 65 × 112.

The ritual year (364) and its excess over the Tonalamatl (104) is likewise contained in these numbers, at least in the first and third:—

The three encircled numbers are connected with one another because the first = 2 × 260 + 86, the second = 6 × 260 + 86 and the third is 86 itself. The larger encircled numbers are, therefore, 1040 = 4 × 260 apart, and this is also the interval between the two numbers near the bottom. 1040, however, also = 5 × 208, and 208 is the interval from IV Eb to IV Ahau. Now it is curious that the two numbers below are 5775 × 208 and 5780 × 208, though the third belongs to day IX Ix and the fourth to IV Eb. One result of this is that 1,201,200 = 1155 × 1040 and 1,202,240 = 1156 × 1040.

As these three numbers relate to day IX Ix and the 54-series, so the fourth relates to IV Eb and the 65-series.

Here the starting-point is the number 52, which belongs to day IV Eb and this is separated from IV Ahau by 208 days i.e., it divides the Tonalamatl in the proportion of 1 to 4.

To the number 52 then, for unknown reasons was added 4623 × 260 = 1,201,980, and thus the number 1,202,032, suppressed in the Manuscript, was obtained for the day IV Eb. To this sum the encircled number 208 was then added and the result was 1,202,240, the number in the Manuscript.

The number = 23,120 × 52 = 4624 × 260, which is self-evident, but it also = 5780 × 208, i.e., it is a multiple of the encircled number. It consequently also = 11,560 × 104, and thus it is related to the first and third numbers just now discussed.

The position of this number is IV Ahau 18 Kankin (1 Kan) and the position of the suppressed number is IV Eb 10 Zotz (also 1 Kan).

We ought now to discuss the last two numbers of this section amounting to millions:—1,567,332 and 1,520,654, which are in the third and fourth columns at the top of page 70. But before going further, we must examine four other numbers, two of which, 111,554 and (with my correction) 101,812, are in column 4 on the lower part of page 70, and the other two, 83,474 and 34,732, are on the top of page 73. Although these four numbers are not ornamented with circles, they all have the significance of the numbers enclosed in circles and are designations of differences between suppressed and specified numbers.

Let us first of all examine their curious relation to one another:—

The Manuscript should have set down under these numbers the day IX Ix twice and IV Eb twice, from which days the numbers in question must be computed; but here the two errors already mentioned were made. 111,554 101,812 is 9742, the very same number which we shall afterward find as the difference of the serpent numbers on page 69.

83,474 - 34,732 = 48,742. If 9472 be subtracted from this, the remainder is exactly 39,000 = 150 Tonalamatls = 50 revolutions of Mars. I have already found this number on page 31a, and also the double of it, 78,000, on page 24, and this I found by using 68,900 + 9100 for my computation.

111,554 - 83,474 = 28,080, i.e., exactly the double of the important 14,040, which is recorded on page 73.

101,812 - 34,732 = 67,080, i.e., = 258 Tonalamatls or 86 revolutions of Mars.

111,554 - 34,732 = 76,822; if 122, the interval from IV Eb to IX Ix be subtracted from this, the remainder is 76,700 = 295 Tonalamatls.

101,812 - 83,474 = 18,338; if 138, the interval from IX Ix to IV Eb, be subtracted from 18,338, the remainder is 18,200 = 70 Tonalamatls = 50 ritual years of 364 days each, i.e., exactly the double of the 9100 specified on page 24.

Now we also have the following equations for the four numbers:—

A day VIII Ahau is 14 days back of the day IX Ix, and another VIII Ahau is 152 days back of IV Eb.

Thus a day VIII Ahau hitherto unmentioned is introduced into the computations. This day has no doubt been chosen, because it divides the Tonalamatl beginning with IV Ahau into two parts of 160 and 100 days, which are in the proportion of 8 to 5, i.e., the same proportion as the Venus year to the solar year.

This day VIII Ahau may also figure in the large numbers of the first two columns on page 70, for 1,394,120 and 1,201,200 are both divisible by 14, the interval between VIII Ahau and IX Ix.

Now I believe that the large numbers were constructed in the following twofold manner (I add the corresponding dates):—

The last record of the date of VIII Ahau seems to throw light on the date 13 Pax (page 70, column 3), which is directly above the date VIII Ahau, and which I have already mentioned in the discussion of the groups of hieroglyphs.

Indeed, it seems as if a day VIII Ahau occurred a fifth time in that passage, for in consequence of the correction made by the scribe we read here VIII Ahau 13 Yaxkin. This would point to a year 7 Muluc, the position of which between the other four is, of course, undetermined.

If the two large numbers in the Manuscript were treated in the same way as the other large numbers, they would not be recorded at all, but instead of them there would have been two numbers belonging to the day IV Ahau and under them would have been the encircled numbers 208 and 86, or these numbers increased by a multiple of 260. This passage would then read about as follows:—

These two numbers for IV Ahau are equal to 6029 and 5849 Tonalamatls. If 5549 × 260 be subtracted from these, the remainders are 480 and 300 Tonalamatls respectively, i.e., 124,800 and 78,000, and these are in the proportion of 8 to 5.

Now the two large numbers have the difference 46,678 = 179 × 260 + 138; the latter is the interval from IX Ix to IV Eb.

The four numbers of the days VIII Ahau seem to stand in very irregular relation to one another and yet they show the following striking results, if the first and third and also the second and fourth numbers be combined (as I combined them under page 24):—

In the first case we see the following:—

While in the second case:—

As in the section occupying pages 61-64, the single series is crowned by four serpents with eight large numbers, so in this section the two series end in a single serpent with two numbers, one for each series, but both bear some obscure relation to the day VIII Ahau, which has made its appearance here. The two sections also correspond, inasmuch as the numbers in both are computed not from the normal date, but from the date IX Kan 12 Kayab (4 Ix).

The serpent pictured here is different from the previous ones, inasmuch as it is partly black. The god B is sitting on its opened jaws, and this time he, too, is painted black (as on page 31c); there is an animal's head upon the god's head, in which we again recognize that of the animal with the fourth serpent in the preceding section. The god is armed with spear and shield and recalls his picture at the bottom of page 74.

There are eight hieroglyphs above this picture, just as there are over each of the first four serpents. The two top hieroglyphs are obliterated. Of the legible hieroglyphs, the one at the left top is the Bacab sign, which also occurs over the first of the four serpents. In the third line are the same two hieroglyphs, which are in the third line of the first and second columns on page 70. The first of the two also occupies the same place on page 62 above the fourth serpent. But here at the bottom we find the date IX Kan 12 Kayab (4 Ix), the same date which we found over the fourth serpent, which is thus again brought into closer connection with the single serpent.

There can be no doubt here regarding the two numbers in the serpents, but notice should be taken of the fact that the figure 1 is barely visible in the red number.

The black number here has the figures 4. 5. 19. 13. 12. 8. and the red 4. 6. 1. 0. 13. 10. The black is therefore 12,381,728, and the red 12,391,470. The black number is somewhat less than the eight numbers in the four serpents, and the red is somewhat larger than the least of them.

The difference of the two is 9742 = 37 × 260 + 122; but 122 is the interval between days IV Eb and IX Ix. Now this is the same 9742 which we found on page 70, as the difference between 111,554 and 101,812.

In order not merely to examine these numbers, but also to understand them, we will again make use of 109 Ahau-Katuns = 12,412,920, as we did in the first four serpents, and we shall have the following:—

The date given for both numbers was the day IX Kan, which was likewise the starting-point for six of the eight numbers in the previous serpents.

Besides this the day IV Eb, the starting-point of the 65-series, is given for the black number, and therefore also the interval between IV Eb and IX Kan = 252.

To this 252 was added a multiple of 260, not an arbitrary choice, but one which combined with 252 resulted in a number divisible by 8, the interval from IX Kan to IV Eb. 31,192 = 3899 × 8 = 119 × 260 + 252 was thus obtained.

The subtraction of this number from 12,412,920 resulted in the serpent number 12,381,728.

In addition to all this the day IX Ix, the starting-point of the 54-series, is given for the red number; consequently also the interval between IX Ix and IX Kan = 130, which, at the same time, is reversely the interval from IX Kan to IX Ix.

To this 130 was added a multiple of 260, which must in every case be a multiple also of 130. Thus we obtain the 21,450 = 82 × 260 + 130.

The subtraction of this number from 12,412,920 results in the serpent number 12,391,470.

Reckoned from the starting-point IX Kan 12 Kayab (4 Ix) the black number corresponds to the date IV Eb 5 Chen (10 Muluc) and the red to IX Ix 12 Zip (11 Kan), and these two dates must certainly have been under the serpent; the months unfortunately are effaced.

It is self evident that the black number is exactly divisible by 8 and the red by 130.

The two events indicated by the two numbers must be to some extent coincident with the beginning of the seven events recorded in the previous four serpents. These large numbers pertaining to the destruction of the world are a reminder of the numbers, which on page 24 we believed were connected with the creation of the world. Thus here, too, we have the genesis and the apocalypse of all the mythologies.

The last portion of this section is formed by the two middle columns of hieroglyphs on page 69. They bear an extraordinary resemblance to those discussed under page 61 even in regard to the fact that each column contains 18 signs. Besides, the upper 10 lines, i.e., the upper 20 signs, are exactly alike on the two pages, aside from slight variations, and differ only in so far as the passage on page 69 is written on blue ground and the one on page 61 on white.

But also the lower part, with eight signs in each column, shows many points in common with page 61. Here as there the whole is divided into several groups.

With the four signs 11ab and 12ab, which formed the first group there, I can compose only the two signs 11ab here. In the cross 11a, as on pages 24 and 58 of the Manuscript, I see the sign for 20 with the prefixed 5 making 25. In 11b we find the sign for 18,980 days, which we have already met with several times. Hence 11ab would have the value of 25 × 18,980 = 474,500 days, as on page 61 the corresponding four signs seemed to form 421,940. And as the number there was 1156 × 365, so on page 69 we have 1300 × 365.

I believe there is a disarrangement in what follows, inasmuch as I assume that the two signs 12b and 13a ought to be placed before and not after 12a. Assuming that the two little crosses on either side of the 1 are meaningless, we should assign the value of 61 to the 3 Chuen, 1 Kin. Here, in the first place, the intention seems to be to establish some connection with the two days VII Kan and IV Ik specified with their numbers on page 63, column 3, as well as with the days most important there, III Chicchan and XIII Akbal, i.e., a connection with the previous section of the four serpents in general; for the interval from VII Kan to III Chicchan, as well as that between IV Ik and XIII Akbal is 61 and on pages 70-73 the two most important days, IV Eb and IX Ix, are 122 days apart, and 122 is the second multiple of 61. I can now put the 144,000 of 12a in the place of the 13a. Then, secondly, the four signs from 13a to 14b in the one section are exactly like those in the other section, and therefore need not be discussed here. Only 15ab differs from the signs in the other passages inasmuch as on page 69 we find 4 × 20 + 4 × 1. The last 4 agrees even better than it does there with the distance from IV Ahau to the day Kan with which the serpent numeral begins.

Nothing on page 69 corresponds to the signs in 16ab and 17ab of page 61. On the contrary, the initial date of the serpent IX Kan 12 Kayab, which on page 61 does not appear until 18ab is set down in 16ab. On the other hand on page 69 the four signs 17ab and 18ab are added, 17a being a sign as yet unknown with 13 as a superfix. I feel inclined, though with many misgivings, to treat 17ab like 5a and b of page 61 and to assign to them the value of an Ahau-Katun of 113,880 days. For then they would denote the 13th Ahau-Katun, which extends from the day 1,366,560 (page 24) to 1,480,440 and which contains the two large numbers on page 70, left, top, while the two lower numbers in the first and second columns of that page belong to the 12th Ahau-Katun, and the two in the third and fourth belong to the 14th Ahau-Katun. The 13th would be the present and the 12th and 14th the past and future; but all this could only be confirmed by further research. At all events, the signs for beginning in 17b and for end in 18a refer to past and future. Unfortunately, 18b is entirely effaced.

Page 74.

Besides the picture, this page contains only 15 hieroglyphs in three horizontal rows. Only about six of these signs are decipherable. The second, third and fourth of the lower line are three different heads; the middle is the familiar head of god B, the one on the left has the Akbal eye and the abbreviated sign for the south, which is repeated in the affix; the head on the right has the sign for the west as a prefix. Very little more is to be said of the other hieroglyphs than that the second and third of the second line have the sign for the east; the first of the second line, however, was the one which we found on pages 71-73 as the constant companion of the Bacabs and which suggested the wind. The last sign of the second line must have contained that for north, so that the four cardinal points all came together here.

The picture begins below these signs. Astronomical figures, apparently Venus, Mars, Mercury and Jupiter, end in the fore part of a crocodile. Below the astronomical signs are the signs for the sun and moon. Streams of water are falling from the jaws of the crocodile and also from the sun and moon. And a fourth stream is being poured from a jug by the old woman with the tiger claws, and with the serpent on her head, whom we saw on pages 39, 43 and 67 engaged in the same occupation. Cross-bones are represented on her skirt as the symbol of death. The sign of the ninth day, Eb, appears on the jug; this is the day which was avoided in the Tonalamatls, for not a single Tonalamatl begins with Eb in the Dresdensis, nor does one begin with the week-day IX; does Bolon meaning nine suggest Balam, the jaguar?

Still further down on the page sits a black god, who may be the same as the god on pages 7a and 16b, with a bird of prey on his head. There are two arrows in his right hand and his left hand holds what may be an atlatl, but it is very much longer than is usually the case; at the same time it can be regarded as a spear.

This page can denote nothing but the end of the world, for which the serpent numbers have prepared the way. Perhaps what looks like a zero above the sign Eb in the stream of water may likewise point to this calamity.