Hieroglyphs.
1 17 29 151,840 113,880 75,920 37,960
2 18 30 (4 × 37,960) (3 × 37,960) (2 × 37,960) (13 × 2920)
3 19 31 I Ahau I Ahau I Ahau I Ahau
4 20 32 185,120 68,900 33,280 9100
5 21 33 I Ahau I Ahau I Ahau I Ahau
6 22 34 35,040 32,120 29,200 26,280
7 23 35 (12 × 2920) (11 × 2920) (10 × 2920) (9 × 2920)
8 24 36 VI Ahau XI Ahau III Ahau VIII Ahau
9 25 37 23,360 20,440 17,520 14,600
10 26 38 (8 × 2920) (7 × 2920) (6 × 2920) (5 × 2920)
11 27 39 XIII Ahau V Ahau X Ahau II Ahau
12 28 40
13 11,680 8,760 5,840 2920
14 (4 × 2920) (3 × 2920) (2 × 2920)
VII Ahau XII Ahau IV Ahau IX Ahau
15
16 2,200 1,366,560 1,364,360
IV Ahau I Ahau I Ahau
8 Cumhu 18 Kayab 18 Zip.
First let me observe that I have restored the four large numbers at the
top, which are almost entirely effaced, as follows:--
1 15 10 5
1 16 10 5
1 6 16 8
14 0 0 0.
0
And furthermore, at the right, bottom, I have substituted the third month
for the second of the Manuscript, which proceeding will be justified later
on.
The least difficult portion of the contents of this page is the first
series consisting of 16 members, each being a multiple of 2920. It begins
with the date I Ahau (which is always concealed in these series), regularly
stops at the month day Ahau (since 2920 = 146 × 20), but necessarily
advances in the week days by 8 days each (since 2920 = 224 × 13 + 8), until
37,960 is reached, when the day I Ahau again appears (since 37,960 = 146 ×
260).
According to my method of filling in the numbers, the top row of the page
consists only of multiples of 37,960.
On the other hand, the four numbers of the second row from the top are more
difficult. They are, it is true, all divisible without remainder by 260,
but otherwise they seem to be without rule, and they give one somewhat the
impression of a subsidiary computation such as one might jot down on a slip
of paper in the course of some important mathematical work.
Nevertheless, the following remarkable results are obtained when the first
and third and the second and fourth numbers are combined by addition or
subtraction:--
1) 185,120 + 33,280 = 218,400, which is just 600 years of 13 × 28 = 364
days, 280 Mars years of 780 days, 840 Tonalamatls of 260 days or 7800
months of 28 days.
2) 185,120 - 33,280 = 151,840, _i.e._, precisely the highest number of the
top row, = 416 solar years of 365 days each or 260 Venus years of 584 days
each, _i.e._, the product of the days of the Tonalamatl multiplied by the
Venus years. We shall again find the 151,840 on page 51, and Seler
("Quetzalcoatl and Kukulcan," p. 400) finds this same period on a relief of
Chichen Itza.
3) 68,900 + 9100 = 78,000, _i.e._, 100 Mars years or 300 Tonalamatls. The
half of this number, or 39,000, we shall find again on pages 69-73 by
computation; also the whole 78,000.
4) 68,900 - 9100 = 59,800, _i.e._, 520 Mercury years of 115 days, or 230
Tonalamatls, or five times the period of 11,960 days, in which these two
periods are united. By computation again we find the 59,800 on page 58.
This period of 11,960 days is, however, to the period of 37,960 in the
proportion of 23:73, _i.e._, 23 × 520:73 × 520. 23 is the fifth part of the
apparent Mercury year, as 73 is of the solar year.
Let us now turn to the numbers, which form the bottom of my transcription,
but only the left hand lower corner in the Manuscript. Here, in the latter,
we find the following (with the correction already mentioned of the second
to the third month):--
2200 1,366,560 1,364,360
IV Ahau I Ahau I Ahau
8 Cumhu 18 Kayab 18 Zip.
The first thing to be done is to arrange and fill out these numbers to suit
our purpose.
The 2200 is clearly nothing more than the difference between the two high
numbers. We can therefore dispense with it.
Further, we find by the usual computation, that the second number belongs
to the first date and the third to the second. Hence the number
corresponding to the third date is wanting from lack of space. This number
can be calculated from that date; it is 1,352,400. It would suit this date
equally well if the number were higher or lower by 18,980 or a multiple of
18,980; but it will be seen directly that it agrees with the other two
numbers only at the value given above.
Now, if we add to this passage the years in which the dates must lie, they
are in the case of the date on the left, the year 9 Ix, in the case of the
middle date, the year 3 Kan, and of that on the right hand, the year 10
Kan.
Then if we arrange the three numbers with the dates and years belonging to
them, according to the value of the first, this part of the page will run
as follows:--
1,352,400 1,364,360 1,366,560
I Ahau I Ahau IV Ahau
18 Zip 18 Kayab 8 Cumhu
10 Kan 3 Kan 9 Ix.
Let us now consider the properties of the three numbers individually.
1) 1,352,400 = 28 × 48,300 and = 115 × 11,760, hence it is divisible by the
month days of the year of 364 days and by the Mercury year. At all events
this is the least important of the three numbers.
2) 1,364,360. This looks as if it referred particularly to the moon and to
Mercury; to the latter since it is equal to 115 × 11,864, and to the former
if we assume that the lunar revolution has been fixed at 29-2/3 days, in
which case this number is exactly equal to 46,000 such lunations. If this
last number be again divided by 115, the number of days required for a
revolution of Mercury, the quotient is 400, which is a round number in the
vigesimal system and which was therefore denoted by a single word, by Bák
in the Maya (according to Stoll) and by Huna in the Cakchiquel (according
to Seler). 1,364,360, therefore, is a Huna of lunar revolutions multiplied
by the number of days in the Mercury period. Later on we shall find the
lunar revolution fixed at 29-2/3 days.
3) 1,366,560. This is the most comprehensive number of the entire
Manuscript, for it is divisible into each of the following periods:--Those
of the Señores de la noche or Lords of the Cycle (9 × 151,840; this is,
however, the first number of the top row), the Tonalamatls (260 × 5256),
the old official years (360 × 3796), the solar years (365 × 3744), the
Venus years (584 × 2340), the Mars years (780 × 1752), the Venus-solar
periods (2920 × 468), the solar year-Tonalamatls (18,980 × 72), the Venus,
solar, Tonalamatl periods (37,960 × 36), and the periods which are
generally designated Ahau-Katuns (113,880 × 12).
We have next to consider the intervals which elapse between the three
dates.
1) From 1,352,400 to 1,364,360 is 11,960 days, which period we have already
found once on this page by computation. 11,960, however, is equal to 104 ×
115 and 46 × 260, _i.e._, the Mercury revolution and the Tonalamatl
combined. 11,960 is again equal to 32 × 365 + 280, and from the year 10 Kan
to 3 Kan it is actually 32 years, and from the date 18 Zip to 18 Kayab it
is, in fact, 280 days. The day I Ahau must be common to both dates.
2) From 1,364,360 to 1,366,560 is 2200 days, as the Manuscript expressly
states. 2200, however, is equal to 8 × 260+120, and the distance from the
day I Ahau to IV Ahau is in fact exactly 120 days. Further 2200=6 × 365+10;
from the year 3 Kan to 9 Ix it is 6 years and from the date 18 Kayab to 8
Cumhu it is 10 days.
3) From these two statements the third follows. The distance from 1,352,400
to 1,366,560 is 14,160. This contains first the 14040, in which both the
Tonalamatl and the old official year of 360 days meet, and second 120,
which is again the interval between I Ahau and IV Ahau. But 14,160 is also
equal to 38 × 365 + 290, and the interval between 10 Kan and 9 Ix is of
course 38 years, and from 18 Zip to 8 Cumhu it is 290 days.
The numbers with which we have had to do here will again occupy our
attention further on, especially the 2920 and the 37,960 on pages 46-50,
the 11,960 and 115 on pages 51-58, and the 14,040 on page 73.
That these computations are not confined to the Dresden Manuscript is
proved by the cross of Palenque, where we find in signs A B 16 precisely
the date I Ahau 18 Zotz, a Tonalamatl before 18 Kayab, in D 1 C 2 exactly
the difference 2200 and in D 3 C 4 the date IV Ahau 8 Cumhu. This is in
favor of the theory that our Manuscript did not originate far from
Palenque.
Now, the question finally arises as to what may, strictly speaking, be
considered the significance of these numbers, dates and differences.
In the first place, I would recall the fact that the dates of the monuments
of Copan and Quirigua, which doubtless refer to present time, are in the
neighborhood of 1,400,000. The high numbers of our Manuscript, so far as
they are in question here, form first a group, which extends from about
1,200,000 to 1,280,000, and then there is a blank, and next a large group
extending from about 1,350,000 to 1,480,000, then another blank and lastly
a group extending from about 1,520,000 to 1,580,000. If we assume that our
Manuscript belonged to about the same date as these inscriptions, then the
three numbers discussed here would extend over a past period lying about
160-170 years back, when a new period of importance had begun probably
dating from the immigration of the Aztecs into Mexico, which they placed in
the first half of the 14th century (see "Weltall," Vol. 5. pp. 374-377).
Now, however, the number 1,366,560 contains the statement that 3744 years
ago (each year having 365 days) an event must have occurred, which can
hardly be anything other (according to the belief of the Mayas) than the
creation of mankind. Hence all the _historical_ dates of the Mayas were
computed from this starting-point. But how did this event come to have the
date IV Ahau 8 Cumhu?
In my opinion this date is to be regarded only as the result of the far
more important date I Ahau 18 Kayab, lying 2200 days earlier. Day 17, Ahau,
belongs, without doubt, to the chief of the gods, and as the first week day
it must have been especially sacred. The prophecies of the Tonalamatl
preferably begin with the Ahau and with the I. The series on the page under
discussion, constructed with the difference 2920 as a basis, begins with I
Ahau, and the three series on pages 46-50 also have the same day as the
zero point of departure. I Ahau is therefore the starting-point of the
astronomical computations as IV Ahau is of the historical.
Now, however, all the periods of 260 days end each time with I Ahau. Why is
precisely this day chosen here, which is the 18th day of the month Kayab,
therefore in the year 3 Kan, and lying 2200 days earlier than the
historical date?
Day 18 Kayab is our June 18th. In my treatise "Schildkröte und Schnecke in
der Mayaliteratur" (1892), I have sought to prove that the tortoise served
as symbol of the summer solstice, that the sign of Kayab was the head of a
tortoise, and that probably the 18th of June was regarded as the longest
day. The middle one of the three series on pages 46-50 begins with exactly
this date, I Ahau 18 Kayab.
But whence come the 2200 days? I will offer a suggestion which may serve
until a better theory is propounded. Let us assume that each of the five
principal planets had in succession regulated its time of revolution by
this astronomical starting-point, thus:--sun 365, moon 356, Mercury 115,
Venus 584, and Mars 780 days, these numbers added together give exactly
2200. It will scarcely excite surprise that I should set down the lunar
year at 356 days (and not at the usual 354 days) for there are 12 × 29-2/3
lunations in a year and we thought we had already found this period on this
page, while discussing the number 1,364,360; also on pages 51-58, in
addition to the half lunar year of 177 days, we shall find one of 178 days.
Were the planets therefore created 2200 days before the appearance of
mankind? Jupiter and Saturn, of course, with their 397 and 380 days are
probably not considered here, because their periods of revolution so nearly
correspond to that of the sun, and on pages 51-60 they are also treated as
of secondary importance.
I confess I am quite unable to discover what may have happened 11,960 days
before the creation of the stars--possibly the birth of one of the
principal deities. Perhaps one of my fellow-students may succeed in finding
an answer in one of the creation myths.
We come now to the 40 hieroglyphs on the left half of the page. These are
intended simply to familiarize the reader with those signs which are of
importance in the calendrical-astronomical portions of the Manuscript.
Since no phonetic system of writing existed, we cannot, of course, expect
that the scribe should have explained these signs.
Signs 1-4, which are mostly destroyed, can hardly denote anything other
than the four quarters of the globe, at least we can still recognize in 4
the sign for the east, which has also the fourth place in pages 46-50. They
stand thus together five times in the middle of the left side of pages
46-50, which pertain to this subject. 5 to 9 are the sign for Venus
repeated 5 times, probably denoting the four parts of its revolution as on
pages 46-50 and also the revolution as a whole. In connection with this
first appearance of the Venus sign, I would mention that the same
hieroglyph also occurs in the Tro-Cort., _e.g._, Cort. 25c, though this
Manuscript contains little else that is astronomical, yet it also has the
rectangular heavenly shields.
10. This is a well-known form of the Moan sign. In the Globus, Volume LXV,
1894, I sought to make it appear probable that the Moan also denoted the
Pleiades, with whose disappearance and reappearance the beginning of the
years seems to be connected. Likewise on page 50, where the 2920-period
ends, the Venus and Moan signs appear at the top on the right-hand side.
11 and 12 are the same sign, being that of the 13th Uinal (Mac), with which
260 days of the year end, and hence this sign is also used as the sign of
the Tonalamatl. The repetition seems to show, that not until the 73
Tonalamatls of the period of 18,980 days are doubled--thus obtaining the
number 37,960 of such importance here--are the sun and Venus periods
brought into unison (with the whole system).
13. The Kin sign (sun, day) with the superfix, which in all probability
expresses conjunction, union, and which, in my opinion, we also see on page
51, combined with Kin and Imix, as the sign for 18,980 days, is used here
after the two Tonalamatls to denote the doubling of this period.
14-18. If the preceding signs led us to the Venus-solar period and to the
continuation of this subject on pages 46-50, these five hieroglyphs bring
us to the Mercury-lunar period and later, on pages 51-58, which are devoted
to the same period, we shall find a parallel especially on the last page.
First comes 14, which, as has been acknowledged, is the sign for 20 × 360 =
7200 days. 15, a hand holding a rectangle divided by a cross into four
parts, is, I believe, the sign for the period of 20 days augmented to 21 by
the 1 in front of it. The much more distinct form of sign 16 on the middle
of page 58 and also at the top of page 53, should be compared with the sign
as given here. The top part is the familiar Ben-Ik sign denoting the 10th
and 19th days, and the bottom is the sign of the 14th division of 20 days,
which make up the year. Now, however, the 10th day, when it becomes the
19th of the next 20 days, is distant from the first 29 days. The prefix
consists of two parts:--First two small circles joined by a zigzag line,
which I think denotes the division of a day into halves; the sign would
then equal 29½ days, _i.e._, very nearly the true lunar month. Second, of
two vertical lines, which might denote a doubling. The whole would then be
equal to 2 × 29½ = 59. I admit that this interpretation is very artificial
and I should be very glad if a better explanation could be found. On the
other hand the 17th hieroglyph becomes quite clear when it is compared with
the parallel passage on page 58; it is 13 × 360 = 4680 days, a third of the
remarkable period of 14,040 days.