As yet I do not understand sign 5 on page 49, which we seem to have met
before on page 22c.
Sign 7 on page 49 is the moon, which is very curious here.
I would like to call special attention to signs 5 to 8 on page 50. I
interpret the passage thus:--At the time of the summer solstice after the
reappearance of the Pleiades, the change of the Venus year takes place
(this time). I have already discussed the Venus sign in the sixth place and
the screech-owl so closely connected with the Moan (Pleiades) in the
seventh place. Sign 5 connects the sun (Kin) with the Ahau (lord) and the
cross-hatching on the left of it, which I have assigned to the tortoise and
thus to the summer solstice (Zur Entzifferung III, 3). Sign 8 is recognized
as very appropriate to the change of year; compare the first sign of the
middle section on pages 25-28. All this points to the day 18 Kayab, of one
of the Kan years, if, as I stated above, we base our computation on the
middle series of dates.
Now we have yet to examine the eight signs of the lower group, which we
will do in the following order:--
1 2 3 4
5 6 7 8.
Regarding the beginnings of these groups, I will venture a bold surmise,
which will, I hope, be improved upon by some one else. It concerns the
first sign of four of these five groups, which seem to me to refer to the
end of the Venus year, as those above refer to the beginning. This sign has
the following form:--
[Illustration]
I see in this the term of 73 days, which is the fifth part of the 365 days
of the solar year and the eighth part of the 584 days of the Venus year:--
It is combined with Chuen in all four cases (pages 46, 48, 49 and 50). But
I attribute the meaning of eight days to this Chuen sign, as I did on pages
25-28 and 42c-45c, though I am doubtful in these as in other cases.
Page 46 contains the sign for 73 with a Chuen under it, and a 1 prefixed to
each sign; _i.e._, 1 × 8 × 73 = expiration of the first Venus year.
On page 48 Chuen follows the sign for 73 and each sign has a 3 prefixed to
it; _i.e._, 3 × 8 × 73, expiration of the third Venus year.
On page 49 the two signs again stand side by side, but the prefix is a 7
instead of the expected 4. By an error this 4 has been added to the 3 of
the preceding page, but, for a wholly unintelligible reason, prefixed to
the crouching person below the Chuen, as if to correct the 7.
Page 50 again has the sign for 73 above and the Chuen below. A prefixed 5
would seem to be in order; instead of it, there is a 10, one 5 for the 73
and another 5 for the 8 days. In this connection let me say that I believe
I have found on page 27, top left, the year of 365 days divided into 5 ×
73.
Page 47 differs from the others. Above two oval bodies appears the
cross-hatched figure resembling a clamp, like the one in the middle group
of page 50 in the fifth place, which I ventured to refer to the summer
solstice. There is a 1 prefixed to it. Is this equivalent to a union of two
Venus revolutions?
Next we repeatedly meet here, as we did in the middle groups, with the Moan
sign and that of the screech-owl belonging with it; the former is the 6th
sign on page 46 and page 50, and the latter is the 3d and 7th on page 47,
the 7th on page 49 and finally the 2nd and 4th on page 50.
The moon is represented in the 5th sign on page 48 and in the 3d on page 49
and indistinctly in the 4th on page 48.
The cardinal points occur here several times. The 3d and 7th signs on page
46 have at least the superfix of the south as a prefix; the 8th on page 47
apparently has the east, but with the familiar cross-hatched sign prefixed;
the 7th on page 48 plainly has the east, the 3d on page 50 the prefix of
the north prefixed to the cross _b_, and the 8th on page 50 the west, thus
approximating the usual order and distribution.
Of the gods I note the Akbal head, perhaps intended for D, in the 4th place
on page 46, also in the 3d on page 48, and lastly in the 5th on page 49,
the first two times with the Ben-Ik superfix, and in the 2nd place on page
47 the sign for A.
In the 4th place on page 47 we have the tortoise as the sign of the month
Kayab or of the summer solstice, in the 6th on page 47 the lightning-beast
or the month Kankin with a Ben-Ik superfix; the beast itself is pictured
below, and the same hieroglyph also with the Ben-Ik superfix is the 8th
sign on page 49.
It is hard to decide whether the sign 4 on page 49 represents the god F
owing to the line through the eye, or a female by reason of the prefixed
lock.
Sign 7 on page 50 represents the deity whose sign began the series of
twenty gods on the left of page 46 and which we have already met with
several times in the centre of the right side. We recognize the prefix as
having occurred in the middle group of the same page.
Sign 6 on page 48 is a Kin combined with an unfamiliar sign. Sign 5 on page
50 contains a Kin with a Yax and probably with 18 as a superfix (as on
pages 27 and 46 middle).
Sign 6 on page 49 contains a crouching person with a 4 which is probably
out of place here and to be regarded as a correction of the 7 above it.
Sign 5 on page 46 contains a Mac denoting the thirteenth Uinal or a
Tonalamatl, and having the sign _p_ as a superfix and a double Ik as a
prefix.
Sign 3 on page 46 merits special attention, because it contains the
duplication of the sign, which, at the end of the first part of the
Manuscript, pages 29-41, always began the groups of hieroglyphs on the
lower third of the pages.
I do not understand the second hieroglyph on page 46 and the 5th on page
47.
In conclusion I would call attention to the fact that the last hieroglyph
on page 48 is very peculiar. As on pages 51, 52, 61 and 69 it has the
meaning of 18,980 days and consists of an Imix with a comprehensive
superfix; its prefix is a 7.
But what is the meaning here of 7 × 18,980 = 132,860? When we recall the
statement made above that the whole section of pages 46-50 embraces 130,520
days, or, according to another calculation 135,200 days, it is a striking
fact that 132,860 is exactly the mean of the two numbers, being separated
from each by 2340 days = 9 × 260. Can it be an accident that on the next
page (page 49) the fourth Venus revolution is reached, for 4 × 584 = 2336,
_i.e._, almost 2340? The hieroglyph discussed here would not be so
extraordinary on page 50. I will not venture to assert as to the 511 in
132,860 = 511 × 260, that it is connected with the 511 which will appear as
the difference on page 58.
Before leaving these pages, I will give a brief survey of the two signs of
the screech-owl and the Moan (hieroglyph _c_ and the lower part of _d_)
which occur on these pages with such marked frequency.
In spite of obliteration, the first of these two signs is distinguishable
in the top groups on pages 47, 49 and 50, in the middle groups on pages 47,
48, 49 and twice on page 50, in the lower groups on page 46, twice on page
47, once on 49, twice again on 50, making 14 times in all. A few additional
cases might be added to these where the similar hieroglyph of the moon may
have been set down instead of the one in question.
On the other hand the second sign, always provided with the same prefix and
suffix as the first, occurs in the top groups on page 48 and 50, in the
middle of pages 46, 48, 49 and 50, and in the lowest on pages 46 and 50, 8
times in all.
Since the subject here is astronomical, it is suggestive less of a deity or
a sacrifice than of a period of time to which the allied page 24 has
already referred (see page 110 of this book). The inner meaning of these
pages is of course still enveloped in mystery.
Pages 51a--52a.
I shall begin the discussion of this very peculiar section with the
remarkable fourth column on page 52, which, very possibly, the scribe ought
to have placed at the beginning; for it looks like a repetition of the
section on pages 46-50, while everything else on the left and right of it,
apparently belongs together.
If we omit the two hieroglyphs at the top, which I regard as belonging to
the two rows of hieroglyphs extending over these two pages, we shall have
the following result, according to my point of view:--
1 5
Chuen 360
2 18,980.
Since, as is frequently the case, the Chuen will here have the value of 8
days and the 5 with the sign for 360 may be regarded as 365, this group
might denote 8 × 365 = 2920, but actually be 2 × 18,980 = 37,960. Both
numbers are the basis of the section included on pages 46-50. And in the
same way the 13 repeated 13 times seems to me to refer to the 13 series of
days on those pages, which begin with the 13th day of the Uinal.
The two rows of hieroglyphs are in the main destroyed. We can still
recognize in the second and third columns of page 51 the signs for end and
beginning, which we often find in the vicinity of numbers; in the second
and third columns of page 52, the sun and moon; in the fourth column, the 8
days of such significance here and in the fifth and sixth, the normal date
IV Ahau 8 Cumhu repeated twice.
As the problem on pages 46-50 was to bring into accord the solar year with
the Venus year and consequently also the Tonalamatl, _i.e._, to combine
365, 584 and 260, so the aim here is first of all to bring the Tonalamatl
into unison with the Mercury year (115). For this purpose the number 11,960
is employed. This is equal to 46 × 260 = 104 × 115, including, therefore
just as many Mercury years as there were solar years in the preceding
section. 11,960 is also 8 × 1495, and this 8 is significant here, for, as
we shall see directly, the day forming the basis of this calculation is XII
Lamat, which comes 8 days after the normal date IV Ahau.
The series given here is based, therefore, on 11,960 and consists entirely
of multiples of this number, which, it is true, are recorded with the usual
irregularity. The members of this series, representing the greatest values,
which are set down in red numbers among the black, are the 31st and 39th
multiples of 11,960, which are separated from each other by 8 × 11,960,
viz:--370,760 and 466,440. All these numbers, of course, denote the day IV
Ahau.
The day XII Lamat as the actual starting-point of the Mercury revolution is
not introduced until we come to the dates placed below the series. Here we
find the days XII Lamat, I Akbal, III Ezanab, V Ben and VII Lamat written
one below the other, and repeated seven times. Each of these days is
separated from the next by 15, and the last of one row and the first of the
next on the left are 200 days apart, hence the whole is equal to 7 × 260 =
1820 days. From XII Lamat begins also the Peresianus, pages 21-22.
Now these dates are connected with the four large numbers, which we find on
page 52, but between the third and fourth, one number corresponding to the
day V Ben is omitted for lack of space.
These four numbers, to which I have added the corresponding dates, are as
follows:--
1,412,848 = XII Lamat I Muan (6 Muluc).
1,412,863 = I Akbal 16 Muan (6 Muluc).
1,412,878 = III Ezanab 11 Pax (6 Muluc).
1,434,748 = VII Lamat I Muan (1 Muluc).
It is curious that while the first three are separated from each other by
15, between the 3d and 4th, or rather between the missing 4th and 5th, 84 ×
260 days are inserted in excess of the required 15, _i.e._, 21,855. This,
however, is not accidental, but is due to the fact that between the first
number and the last exactly 21,900 = 60 × 365 days have elapsed. This
number is, however, = 18,980 + 2920, i. e., the sum of two very important
numbers, in the first of which the Tonalamatl and the solar year accord,
while both the solar and Venus years occur in the second.
I must here call attention to the fact that the four numbers are not
obtained without slight corrections, since in the 20-place of the third, I
have put a 11 instead of 10, while in the 360-place of the fourth, I have
omitted the three dots, _i.e._, set down a 5 instead of the 8.
Of these four dates, which were doubtless not far removed from the time of
the scribe, the three last are only the result of the first. Day XII Lamat
is the most important. As the beginning of a Mercury period it should be
regarded in the same way as I Ahau of the Venus period and IV Ahau of the
solar period; and the very next day, XIII Muluc, will subsequently be seen
to be the beginning day of the Mars period.
The four dates XII Lamat, I Akbal, III Ezanab and VII Lamat are set down in
the Manuscript directly below the numbers.
Now in the first column on page 51 we again find a day XII Lamat, as is
expressly stated beneath it. It has the number 1,578,988 and the
corresponding date is XII Lamat 6 Cumhu (6 Kan). This day, however, is
separated from the same day on page 52 (1,412,848 = XII Lamat I Muan 6
Muluc) by 166,140 days, that is by 8 × 18,980 + 14,300 = 639 × 260, _i.e._,
by 8 so-called Katuns increased by 55 Tonalamatls. Here 58 × 260 = 15,080
seems to have been added to 252 (XII Lamat - IV Ahau) and the sum
subtracted from 14 Ahau-Katuns = 1,594, 320. I could obtain this number
only by substituting 1 for 0 in the 20-place.
In the Manuscript the sign XII Lamat is set down above and below this
number. I must leave undetermined whether the 8 directly above the number
and combined with Kin and the Katun sign refers only to the 8 Katuns or at
the same time also to the 8 days from IV Ahau to XII Lamat.
It is also to be noted here that once before on page 24 of this Manuscript
(which forms the basis of this section) 8 × 18,980 = 151,840 days was found
to be the difference between 185,120 and 33,280, and that there, too, if my
restoration is correct, it was the highest term of the series = 4 × 37,960.
Finally, in the first column of page 51, we have the complete normal date 4
Ahau 8 Cumhu (9 Ix). But below this, between red numerals denoting the
1,578,988 mentioned above, there is set down in black the number 1,268,800.
This corresponds to the date IV Ahau 3 Zip (2 Cauac). It may have been
formed by adding 16,120 = 62 × 260 to 11 Ahau-Katuns = 1,252,680. It is,
however, not only equal to 4880 × 260, but also to 158,600 × 8, therefore
also divisible by the interval between IV Ahau XII Lamat, as well as by 104
= 8 × 13, while on the contrary it is not as we should expect, divisible by
11,960. I have changed the 11, in the 20 × 11, to 8 by omitting one line
and adding two dots, for otherwise the result would not be the one
required.
The magnitude of the number recalls the one on page 31, which is only 260
less, and that on page 62.
Finally it should be noted that the two large numbers on page 51 are
separated from one another by 310,188 days = 849 years and 303 days, which
corresponds exactly to the dates given for each. One may be situated as far
in the future as the other is in the past, but this does not necessarily
mean that the present coincides exactly with 1,423,894.
Pages 51--58.
Thus far we have examined only the upper halves of pages 51 and 52 and have
still to consider the lower, but not until we have finished the upper parts
of pages 53-58 of which the former are the continuation. We have first to
consider the series, then the pictures and lastly the hieroglyphs.
As on page 24 we found multiples of the number 2920 (= 8 × 365 = 5 × 584),
while on pages 46-50 it was divided into four unequal parts, so on pages
51-52 we find multiples of the number 11,960 (104 × 115 = 46 × 260) while
on pages 53-58 it is divided into 69 unequal parts. On pages 51-52 it was
the aim to combine only the Mercury course with the Tonalamatl, but here we
are confronted with the additional problem of bringing the lunar revolution
into accord with these two.
The lunar revolution, which we assume to be 29.53 days, of course requires
fractional computation, of which the Mayas either were ignorant or which
they timorously avoided; like the ancient Egyptians, who were acquainted
only with fractions having 1 as numerator, or beyond these at most with 2/3
(see Hultsch, "Die Elemente der ägyptischen Teilungsrechnung," 1895, page
16).
Now the Mayas had determined the lunar revolution so exactly that they
perceived the incompatibility of the period of 11,960 days with a multiple
of lunar revolutions. They found that 405 lunar revolutions amounted
approximately to 11,958 days, which is, in fact, the largest number on the
second half of page 58. In order not to drop the significant 11,960
altogether, they made use of a very shrewd artifice. They took as the
starting-point the day XII Lamat, corresponding to the number 11,960, and
set down XI Manik before it and XIII Muluc after it. Now if the count began
with XIII Muluc and ended with XI Manik, it actually resulted in 11,958.
Therefore what the Manuscript presents here is, in the first place, the
series, which is this time to be read from left to right. Below it are the
three days belonging to each member of the series and then a number for
each member stating the interval between it and the preceding one. The
members, the days and the differences must correspond with one another. It
is, therefore, no longer necessary to pay especial attention to the two
latter. They will serve merely to control and to correct the manifold
errors.
The entire period of 11,958 days was doubtless first divided into three
equal periods of 3986 days. And in order still further to subdivide these
shorter periods, the term of 177 days was employed as far as it would go;
177, however, is the half of a lunar year of 354 days, made up of 6 months
of 30 days and 6 of 29 days, thus allowing 29.5 days in round numbers for
each month.
Now 177 is = 3 × 29 + 3 × 30. The average, 29.5, however, is too short for
the length of the lunar revolution. In order to raise it as nearly as
possible to the exact time, two other numbers were introduced at certain
points of the series, viz:--148 = 2 × 29 + 3 × 30, 178 = 2 × 29 + 4 × 30.
148 = 5 months of 29.6 days, while 178 = 6 months of 29-2/3 days. Now let
us see in what _proportion_ these 148 and 178 days were distributed among
the periods of 177.
First we see that the period of 3986 days (_i.e._, a third of the whole)
was divided into 3 sections of 1742, 1034 and 1210 days, as follows:--
1742 = 8 × 177 + 148 + 178
1034 = 4 × 177 + 148 + 178
1210 = 6 × 177 + 148
----------------------------
3986 = 18 × 177 + 3 × 148 + 2 × 178.
This is equal to 135 months of 29.526 days each. Now the question arises
how did the Mayas express this fraction?
Perhaps some time in the future it will be found, that following their
vigesimal system, they designated it approximately thus:--
29 + ½ + 1/40 + 1/800.