The apparent error due to the addition of two dots in the Tun place in
the upper number of column 14 is more the result of an error than an
error in itself. This number shows a very clear case of erasure. The
writer of this section of the manuscript in copying from the older
source, at first overlooked column 14, and placed 7.3.18, the upper
number in column 15 in this place. Realizing his mistake he erased the
three dots in the Uinal place, but utilized two of the bars and the
three dots in the Kin place as the 13 needed in the Uinal in column 14,
and erased the lower bar of the original 18. This procedure of the
writer’s threw the upper number of column 14 out of alignment, for the
two dots of the Kin appear below the 13, somewhat below the line of Kins
of the other columns. The seven in the Tun place should have been a six,
so the scribe inserted an extra dot between the two of the original 7,
neglecting, however, to erase the other two dots. As a result the upper
number of column 14 records the number 3142, which is 720 greater than
it should be, namely, 2422.
In column 10, 1748 was recorded instead of 1742, for a bar and three
dots were written in the Kin place instead of only two dots. This is a
very peculiar and unexpected form of carelessness, which is, however,
corrected in the next column. The remaining irregularities in the upper
numbers are all due to the omission of a part of the number. In columns
2 and 15, one dot was omitted in the Kin place, thus recording 353
instead of 354 in the former, and 2598 instead of 2599 in the latter. In
column 4 one bar was omitted in the Kin place, making the number 674,
five less than it should be, namely, 679. In column 1, one dot was
omitted in the Uinal place, and in column 12, four dots of the same
denomination, recording, respectively, 157 and 2016 instead of 177 and
2096.
There is only one decided error in the day series. In column 11, 6 Cib,
7 Caban, 8 Eznab were written instead of 6 Cimi, 7 Manik, 8 Lamat. It
should be noticed that the number of the day was right. In fact just
one-half a tonalamatl, or 130 days, was dropped before the day series of
column 11, and added on again immediately afterwards. This is an
extremely curious error to make in calculating and may shed some light
on the way in which the Mayas reckoned.
The five remaining irregularities in the day series are of two kinds. In
column 5, the number preceding the third day, Chicchan, is 4 instead of
11. Apparently the writer of the manuscript forgot for the moment that
the day was added to the one above it and not the one to the left, and
wrote 4 because the number associated with the third day sign of column
4 was 3. The same mistake was made in the third day of column 36, except
that in this case it was the day sign and not the number which was
confused. Here, instead of writing Ahau, which followed the Cauac in
the second series, Ben was recorded because the sign to the left was Eb.
The other three irregularities are all due to carelessness in placing
sufficient dots in the number associated with the day sign, for in
columns 17, 47 and 49, 1 Ik, 10 Eznab and 11 Kan were recorded,
respectively, instead of the necessary 2 Ik, 11 Eznab and 12 Kan.
There are two places in which columns seem to be misplaced, although the
mathematics of the series at these points is correct as it stands. For
the sake of uniformity in the arrangement of the difference groups, the
178-day group of column 7 should occur in column 6, and for the same
reason, the 148-day group of column 58 should occur in column 59.
Professor Förstemann calls both of these variations errors, and arranges
his version of the table so that each part is just like the other two.
He gives no reason for his opinion other than the phrase “for the author
[of the manuscript] had confused the differences 178 and 148....”[4] Mr.
Bowditch, on the other hand, allows both of the variations to stand as
they appear in the manuscript, and quite rightly holds the opinion that,
“It may possibly be that these numbers thus placed are errors of the
scribe, but the mere plea for uniformity is not sufficient to lead us to
make these changes.”[5]
[4] Förstemann, 1901, p. 123.
[5] Bowditch, 1910, p. 217.
In Table II the apparent mistake in columns 58 and 59 remains as it
occurs in the manuscript, for the reason which Mr. Bowditch gives. In
the case of columns 6 and 7 there seems to be some evidence that there
actually was an error made. The last column on page 53a, which is the
one under discussion, contains no day glyph in the first day series. The
glyph should have been that of Ahau. There is distinct evidence, altho
very faint, that a glyph was once there. Moreover, the smooth coating
which covered the material of the manuscript page is not broken. There
are other obliterated glyphs in these pages of the manuscript, but few
in which the surface, although unbroken, still contains a faint, almost
continuous outline of a glyph. The glyph, then, was probably erased. The
writer of the manuscript had probably completely finished column 6 and
started column 7 before he detected the error. He began to erase the
part that was wrong, then realized what an amount of alteration would be
necessary, and finally compromised by making the difference come between
columns 6 and 7 instead of columns 5 and 6. This hypothesis in regard
to the manner in which the erasure was done may be wrong, but the
erasure still stands as a strong evidence to show that the 178 should
have occurred in column 6 rather than after it.
Finally there appears to be an error in the totals of the series, for
the upper number series records as a total 11,958 days and the day
series 11,959 days, although there is strong reason for believing that
the series should record 11,960 days. This discrepancy in the totals
will be referred to again.
In general, then, the apparent irregularities in the manuscript fall
into two great classes, those which are corrected in the next column or
are easily detected because of their disagreement with the record in the
other two series, and those which are not obviously due to carelessness.
The latter will be considered under the solutions. The former may be
dismissed as clerical errors not affecting the solution. In this group
are two of the irregularities in the lower numbers (columns 26 and 50),
and all eight in the upper numbers, seven of which occur in the first
third of the manuscript. The six errors in the day series, and the
transposition of columns 6 and 7, also belong in this class.
By referring to Table II it will be noticed that the pictures occur
after the 148-day groups in each case. The upper numbers immediately
preceding the pictures are given in Table III (p. 11), together with the
differences between them. By grouping these differences, it becomes
apparent that the pictures may be divided into three large groups of
3986 days; two out of the three containing the same difference numbers,
1742, 1034, and 1210. If, in the last group, the number 10,039 were
changed to 10,216 by adding 177, the differences for this group would
also read as the others, when the end of the series and the beginning of
the series are added together (708 + 502 = 1210), for the 10th picture
is, in a sense, out of the grouping since it occurs after the last
number in the series. The 148-day groups are arranged in the same order
for they occur in the same columns as the numbers used above.
By applying the same process to the 178-day groups, it is found that
they also can be divided into groups which contain 3986 days. In this
case the second and third groups contain the same numbers, 2598 and 1388
(Table IV). If the number 1211 in the first group is changed to 1034 by
subtracting 177, the last number of this group would be 1388; and the
first number 2598 could be formed by adding the remainder at both ends
of the series (1564 + 1034 = 2598).
It should be remembered at this point that the only column in which the
lower numbers contained 178 is column 23, of which the upper number is
3986. This gives further grounds for dividing the series as it stands
into three parts of 3986 days, each containing 23 columns.