Part 16

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:--

1,486,923, XIII Akbal. 1,483,585, III Chicchan.

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

1,268,523, XIII Akbal. 1,233,985, III Chicchan.

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

1,272,423, XIII Akbal. 1,272,465, III Chicchan.

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:--

1,538,342, IV Ik. 1,535,004, VII Kan.

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

1,268,540, IV Ahau. 1,234,220, IV Ahau.

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

1,272,544, IV Kan. 1,272,921, IV Imix.

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

1,518,400 = 80 × 18980. 1,523,921, 1,538,342, 1,556,360 = 82 × 18980.

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:--

1 2 3 4 5 6 7 8.

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:--

1a: 12,489,781; XI Kan 12 Kankin (1 Ix); III Chicchan 18 Xul (4 Muluc). 1b: 12,388,121; XI Kan 12 Muan (7 Ix); III Chicchan 13 Pax (4 Ix). 2a: 12,454,761; IX Kan 7 Kankin (4 Cauac); III Chicchan 13 Yaxkin (2 Ix). 2b: 12,394,740; IX Kan 2 Chen (5 Kan); III Kan 12 Ceh (7 Ix). 3a: 12,438,810; IX Kan 12 Xul (3 Ix); III Ix 7 Zac (9 Muluc). 3b: 12,466,942; IX Kan 12 Kayab (4 Ix); III Cimi 14 Kayab (9 Ix). 4a: 12,454,459; IX Kan 12 Kayab (4 Ix); XIII Akbal 1 Kankin (1 Kan). 4b: 12,394,740; IX Kan 12 Kayab (4 Ix); III Kan 17 Uo (7 Muluc).

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:--

1a) 12,489,781 1b) 12,388,121 12,412,920 12,412,920 ---------- ---------- 76,861 = 295 × 260 + 161 -24,799 = 95 × 260 + 99 XI Kan - III Chicchan = 161. III Chicchan - XI Kan = 99.

2a) 12,454,761 2b) 12,394,740 12,412,920 12,412,920 ---------- ---------- 41,841 = 160 × 260 + 241 -18,180 = 69 × 260 + 240 IX Kan - III Chicchan = 241. III Kan - IX Kan = 240.

3a) 12,438,810 3b) 12,466,942 12,412,920 12,412,920 ---------- ---------- 25,890 = 99 × 260 + 150 54,022 = 207 × 260 + 202 IX Kan - III Ix = 150. IX Kan - III Cimi = 202.

4a) 12,454,459 4b) = 2b 12,412,920 ---------- 41,539 = 159 × 260 + 199 IX Kan - XIII Akbal = 199.

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:--

9 XII, 5 IV, 1 V, 10 II, 6 VIII, 2 X, 11 VIII, 7 II, 3 V, 12 IV, 8 XII, 4 III, 13 III. 11 I, 13 I, 11 XII, 1 XIII, 8 VIII, 6 I, 4 V, 2 VII, 13 VII, 6 XIII, 6 VI, 8 I, 2 III. 11 XI, 13 XI, 11 IX, 1 X, 8 V, 6 XI, 4 II, 2 IV, 13 IV, 6 X, 6 III, 8 XI, 2 XIII. 9 IX, 5 I, 1 II, 10 XII, 6 V, 2 VII, 11 V, 7 XII, 3 II, 12 I, 8 IX, 4 XIII, 13 XIII.

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:--

III 2.

1. 9. XII Ix 2. 14. IV Cauac 3. 15. V Ahau 4. 25. II Oc 5. 31. VIII Cib 6. 33. X Eznab 7. 44. VIII Muluc 8. 51. II Cib 9. 54. V Cauac 10. 66. IV Chuen 11. 74. XII Cauac 12. 78. III Akbal 13. 91. III Cib 14. 102. I Manik 15. 115. I Ahau 16. 126. XII Chuen 17. 127. XIII Eb. 18. 135. VIII Ahau 19. 141. I Cimi 20. 145. V Oc 21. 147. VII Eb 22. 160. VII Chicchan 23. 166. XIII Chuen 24. 172. V Caban 25. 180. I Chicchan 26. 182. III Manik

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:--

XIII 20.

1. 9. IX Eb 2. 14. I Caban 3. 15. II Ezanab 4. 25. XII Lamat 5. 31. V Ix 6. 33. VII Cib 7. 44. V Manik 8. 51. XII Ix 9. 54. II Caban 10. 66. I Muluc 11. 74. IX Caban 12. 78. XIII Imix 13. 91. XIII Ix 14. 102. XI Chicchan 15. 115. XI Ezanab 16. 126. IX Muluc 17. 127. X Oc 18. 135. V Ezanab 19. 141. XI Kan 20. 145. II Lamat 21. 147. IV Oc 22. 160. IV Akbal 23. 166. X Muluc 24. 172. III Men 25. 180. XI Akbal 26. 182. XIII Chicchan

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.