Pages 46--74.
The first glance at the form and contents of the second part of the
Manuscript shows that it is very different from the first. The pages are no
longer divided into the usual three parts and there are fewer pictures. The
Tonalamatls, which form the principal contents of the first part, disappear
wholly, and with them both the vertical columns of day-signs and the
horizontal lines of numerals alternating between red and black. On the
other hand, the large number series as well as the high numbers
significantly increase and we note the appearance of the large vertical
columns of hieroglyphs, which were impossible in the triple division of the
earlier pages. We also find a large number of hieroglyphs which did not
occur in the first part. The contents are essentially astronomical.
And yet the two parts are so closely connected with one another that the
idea of two independent Manuscripts must be dismissed. Especially the front
side of the second part as far as page 60 is nothing more than an
amplification of page 24. The contents of pages 61-74 are of a more
independent nature, but special attention should be called to the relation
of 31a-32a to 62-63.
Pages 46--50.
The entire contents of these pages must be represented as a unit, for what
is in the main true of page 24 is also true of these pages, namely that
they treat exclusively of the period of 2920 days, in which five Venus
years of 584 days each are brought into accord with eight solar or
terrestrial years of 365 days each. Each page is a direct continuation of
the preceding one. Each period of 2920 days is taken 13 times, the result
being 37,960 days, which are equal to 146 Tonalamatls.
I will give here first a reproduction, as it were, of the left side of the
five pages, omitting for greater clearness a few indifferent matters, which
are intended only to fill the blank spaces, viz:--
1. The twenty hands pointing to the right, with a knife placed over them,
in the middle of the pages, which mean nothing more than that these parts
of the Venus year are to be read from left to right.
2. The Venus hieroglyph three lines below, repeated twenty times with the
sign of the knife, to denote the _division_ of the Venus revolution.
3. The _Akbal_ sign occurring further down, four times on each page, except
on pages 46 and 47. This is the last of the day-signs, again counting
forward from the day Kan, and means only that henceforth the _close_ of the
four periods of the Venus year is indicated below, as the beginning is
indicated above.
4. The sixteen Venus signs also occurring below, except on page 48. This
sign likewise occurs in a very similar form on Altar R of Copan.
With these omissions, the left side of these pages presents the following
appearance:--
Page 46.
III Cib II Cimi V Cib XIII Kan
XI Cib X Cimi XIII Cib VIII Kan
VI Cib V Cimi VIII Cib III Kan
I Cib XIII Cimi III Cib XI Kan
IX Cib VIII Cimi XI Cib VI Kan
IV Cib III Cimi VI Cib I Kan
XII Cib XI Cimi I Cib IX Kan
VII Cib VI Cimi IX Cib IV Kan
II Cib I Cimi IV Cib XII Kan
X Cib IX Cimi XII Cib VII Kan
V Cib IV Cimi VII Cib II Kan
XIII Cib XII Cimi II Cib X Kan
VIII Cib VII Cimi X Cib V Kan
4 Yaxkin 14 Zac 19 Zec 7 Xul
North. West South East
Gods.
236 326 576 584
9[1] Zac 19[1] Muan 4 Yax 12 Yax
Gods
East North West South
19 Kayab 4 Zotz 14 Pax 2 Kayab
236 90 250 8
Page 47.
II Ahau I Oc IV Ahau XII Lamat
X Ahau IX Oc XII Ahau VII Lamat
V Ahau IV Oc VII Ahau II Lamat
XIII Ahau XII Oc II Ahau X Lamat
VIII Ahau VII Oc X Ahau V Lamat
III Ahau II Oc V Ahau XIII Lamat
XI Ahau X Oc XIII Ahau VIII Lamat
VI Ahau V Oc VIII Ahau III Lamat
I Ahau XIII Oc III Ahau XI Lamat
IX Ahau VIII Oc XI Ahau VI Lamat
IV Ahau III Oc VI Ahau I Lamat
XII Ahau XI Oc I Ahau IX Lamat
VII Ahau VI Oc IX Ahau IV Lamat
3 Cumhu 8 Zotz 18 Pax 6 Kayab
North West South East
Gods.
820 910 1160 1168
3 Zotz 13 Mol 18 Uo 6 Zip
Gods
East North West South
13 Yax 3 Pax 8 Chen 16 Chen
236 90 250 8
Page 48.
I Kan XIII Ix III Kan XI Eb
IX Kan VIII Ix XI Kan VI Eb
IV Kan III Ix VI Kan I Eb
XII Kan XI Ix I Kan IX Eb
VII Kan VI Ix IX Kan IV Eb
II Kan I Ix IV Kan XII Eb
X Kan IX Ix XII Kan VII Eb
V Kan IV Ix VII Kan II Eb
XIII Kan XII Ix II Kan X Eb
VIII Kan VII Ix X Kan V Eb
III Kan II Ix V Kan XIII Eb
XI Kan X Ix XIII Kan VIII Eb
VI Kan V Ix XIII Kan III Eb
17 Yax 7 Pax 12 Chen 0 Yax[2]
North West South East
Gods
1404 1494 1744 1752
2 Pax 7 Pop 17 Mac 5 Kankin
Gods
East North West South
7 Zip 17 Yaxkin 2 Uo 10 Uo
236 90 250 8
Page 49.
XIII Lamat XII Ezanab II Lamat X Cib
VIII Lamat VII Ezanab X Lamat V Cib
III Lamat II Ezanab V Lamat XIII Cib
XI Lamat X Ezanab XIII Lamat VII Cib
VI Lamat V Ezanab VIII Lamat III Cib
I Lamat XIII Ezanab III Lamat XI Cib
IX Lamat VIII Ezanab XI Lamat VI Cib
IV Lamat III Ezanab VI Lamat I Cib
XII Lamat XI Ezanab I Lamat IX Cib
VII Lamat VI Ezanab IX Lamat IV Cib
II Lamat I Ezanab IV Lamat XII Cib
X Lamat IX Ezanab XII Lamat VII Cib
V Lamat IV Ezanab VII Lamat II Cib
11 Zip 1 Mol 6 Uo 14 Uo
North West South East
Gods
1988 2078 2328 2336
16 Yaxkin 6 Ceh 11 Xul 19 Xul
Gods
East North West South
6 Kankin 16 Cumhu 1 Mac 9 Mac
236 90 250 8
Page 50.
XII Eb XI Ik I Eb IX Ahau
VII Eb VI Ik IX Eb IV Ahau
II Eb I Ik IV Eb XII Ahau
X Eb IX Ik XII Eb VII Ahau
V Eb IV Ik VII Eb II Ahau
XIII Eb XII Ik II Eb X Ahau
VIII Eb VII Ik X Eb V Ahau
III Eb II Ik V Eb XIII Ahau
XI Eb X Ik XIII Eb VIII Ahau
VI Eb V Ik VIII Eb III Ahau
I Eb XIII Ik III Eb XI Ahau
IX Eb VIII Ik XI Eb VI Ahau
IV Eb III Ik VI Eb I Ahau
10 Kankin 20 Cumhu[3] 5 Mac 13 Mac
North West South East
Gods
2572 2662 2912 2920
15 Cumhu 0 Zec[4] 10 Kayab 18 Kayab
Gods
East North West South
20 Xul 10 Zac 15 Zec 3 Xul
236 90 250 8
Let us first examine the numbers which are regularly repeated in the lowest
line:--236, 90, 250, and 8, and we shall find that the 584 days of the
apparent Venus revolution are divided into these four periods.
The number 236 denotes the time of the western elongation, when Venus is
the morning star, 90 the time of the invisibility of the planet, during its
superior conjunction, 250 that of its eastern elongation, when Venus is the
evening star, and 8 the time of its invisibility during inferior
conjunction. The disproportion between 236 and 250 is somewhat striking.
These periods which need not of course be exactly equal are usually
computed at 243 days. The short period of eight days is only calculated for
very sharp eyes; we actually find in the Anales del Museo Nacional de
Mexico II, 341 (Mex. 1882), that the Aztecs calculated only eight days for
the invisibility of Venus, and this period is also mentioned in the Anales
de Quauhtitlan. The repetition of the cardinal points in the 15th and 20th
lines of the extract given above refer to these periods; in the upper line
to their beginning and in the lower to their close. Hence in the lower line
the cardinal points must advance one place and the gods belonging to them
in the 16th and 19th lines must follow the same course.
The numbers in the 17th line indicate to which day of the period of 2920
days the position has advanced.
But now we see that the indication of days in the lines 1-13, the
indication of months in lines 14, 18 and 21, and the numbers in line 17 are
separated from those directly to the right of them by a number of days
equal to the numbers given in the lowest line.
From this it follows that each day of the thirteen top lines is joined to
each of the month dates placed just below them, forming a complete calendar
date. Therefore from the III Cib on the left upper corner of page 46 a III
Cib 4 Yaxkin, a III Cib 8 Zac, a III Cib 19 Kayab must be formed.
All the 4 × 13 × 5 = 260 day indications combined with three month
indications each, show therefore that this whole passage is a huge
abbreviation for 780 calendar dates and that the whole refers to 3 × 37,960
days = 113,880 days. But 37,960 which we already found on page 24, is equal
to 146 × 260, 104 × 365, 65 × 584, 13 × 2920. I am inclined to think that I
also found 113,880 on page 24.
But the 3 × 37,960 = 113,880 days do not form the entire period treated of
here. For the three periods begin and end with the days:--
I Ahau 13 Mac (10 Muluc),
I Ahau 18 Kayab (3 Kan),
I Ahau 3 Xul (4 Cauac).
Hence these three dates, the second of which was found on page 24, prove
that the three periods of 37,960 are not consecutive, but that there is an
interval between them. Now between the first and second of the three dates
the interval is 19 years + 85 days = 7020 days, and between the second and
third, the interval is 26 years + 130 days = 9620 days. If these two
periods be added to the 113,880 days, the sum is the whole period treated
of here, viz:-- 130,520 = 502 × 260 days.
But a truly surprising result is obtained, if, as must often be the case
with series, we begin not with the upper of the three dates, but with the
lower.
From I Ahau 3 Xul (4 Cauac) to I Ahau 18 Kayab (3 Kan) there is a lapse of
9360 days or 12 apparent Mars years of 780 days, such as we shall find as
the principal subject of page 59. 9360, however, equals 25 × 365 + 235
days. We shall meet with this 235 again as a difference on page 63.
But from I Ahau 18 Kayab (3 Kan) to I Ahau 13 Mac (10 Muluc) there are
11,960 days, _i.e._, the 104 Mercury years, which we found on page 24, and
which we shall find again as the principal period on pages 51-58. But this
is equal to 32 years + 280 or 33 years - 85 days. Now if 113,880, 9360,
11,960 are added together, we have for the entire period under discussion
here, 135,200 days, and this is equal to 2 × 260 × 260 days. Thus the Mayas
seem actually to have had an idea of a second power.
Finally I would call attention to a singular double connection between the
numbers occurring here:--
37,960 - 11,960 = 26,000 = 100 Tonalamatls,
11,960 - 9,360 = 2600 = 10 Tonalamatls.
But if we subtract 2 × 11,960 = 23,920 from 37,960, the remainder is
14,040, _i.e._, an extraordinary number which often occurs and is equal to
54 × 260, 39 × 360 and 18 × 780.
In short, a Mars and a Mercury-lunar period are inserted in the two spaces
between the three solar-Venus periods.
Now, let us try to gain a clearer understanding of this subject by
approaching from another side.
As we have seen, the beginning of the middle one of the three equal periods
of 37,960 days, is the date I Ahau 18 Kayab (3 Kan). Now, however, page 24
furnished us with a day number for this date, 1,364,360, and from this the
beginnings of the other two periods may be computed in the following way:--
I Ahau 3 Xul (4 Cauac) = 1,317,040,
I Ahau 18 Kayab (3 Kan) = 1,364,360,
I Ahau 13 Mac (10 Muluc) = 1,414,280.
Between the first number and the second there are 47,320 days = 2^3 × 5 × 7
× 13^2, and between the second and third 49,920 days = 2^8 × 3 × 5 × 13
days.
But, according to what has been stated above, 47,320 = 37,960 + 9360, and
49,920 = 37,960 + 11,960.