Tables in the Dresden Codex

These tables from the Dresden Codex are imperfect owing to wear and to mistakes made by the Maya scribe, probably through hasy copying of a yet earlier original.  The computations are reproduced in figures 1 and 2, obliterated material being restored in dotted outline, and corresponding numbers in Arabic figures being added.  Readings and restorations do not differ materially from those made by other students.

On page 32a (fig.1) the calculations, based on the day 13 Akbal, start at the bottom right corner of the right upper half and read Maya fashion right to left, and after four places pass to the right of the upper horizontal row, then to the bottom right corner of the left half of the page.  They first advance at intervals of 91 days (one quarter of 364) up to 1001 (91 x 11).  The fourth and eighth multiples (364 and 728) were omitted from this first part of the table and placed instead at the start of the second half of the table.  This second part (the left half of the page) carries forward the count at intervals of 364 days, as far as 364 x 5.  It then jumps to 364 x 10 (half the next highest order in the vigesimal system), and next follow units of the second order.  The three last numbers (top row) of the table are entirely obliterated except for the zeros, and their restoration is open to question, although in view of what precedes, the system is pretty obvious.  The calculations as restored and corrected read as follows:

Days Days
91 (91 x 1) 364 (364 x 1)
182 (91 x 2) 728 (364 x 2)
273 (91 x 3) 1456 (364 x 4)
455 (91 x 5) 1820 (364 x 5)
546 (91 x 6) 3640 (364 x 10)
637 (91 x 7) 7280 (364 x 20 or 1 unit of second order)
819 (91 x 9) 14560 (364 x 40 or 2 units of second order)
910 (91 x 10) 21840 (364 x 60 or 3 units of second order)
1001 (91 x 11) 29120? (364 x 80 or 4 units of second order)
  36400? (364 x 100 or 5 units of second order)
72800? (364 x 200 or 10 units of second order)

Beneath the right-hand side of the table are given the glyphs of the 20 days arranged to read from right to left and top to bottom in five horizontal rows of four glyphs each (fig.1).  The series starts with 13 Ix, which is 91 days after 13 Akbal (the base of the count) and proceeds horizontally through 13 Chicchan, 13 Cib, etc.  (i.e. always at intervals of 91 days), but if read vertically the intervals within each column are always 364 days (e.i. 104 days plus one round of the 260-day count).  On reaching the end of a horizontal line one passes to the right of the line below.  In counting vertially one passes from the bottom to the top of the same column, and since the counts are re-entering, the last glyph reached horizontally, 13 Akbal, is the start of a new round.

These glyphs are to be used with the table of multiples given above.  The beauty of the system is that every multiple of 91 is divisible by 13, and the day sign coefficient therefore does not alter.

Example 1.  From your base 13 Akbal you wish to count forward 819 days.  The table shows this to be 91 x 9.  To get the day sign count 9 places horizontally from the starting point (always one place for each 91 days).  Nine places counted forward will bring you to the first (right) glyph of the third column, the day 13 Ik.  Answer:  13 Akbal + 819 = 13 Ik.

Example 2.  From your base 13 Akbal you wish to calculate forward 13 computing years.  This is not given directly in the table, but is obtained by adding 10 and 3 computing years.  As years are involved the day sign calculation must be made vertically from top to bottom.  Count 13 places in the left vertical column (i.e. the one in which 13 Akbal occurs), thereby passing twice through the column and reaching the third place, 13 Men.  Answer:  13 Akbal + 4732 = 13 Men.

Example 3.  From any desired base, say 4 Ahau, go forward 11 computing years and 91 days.  Count 11 glyphs vertically from Ahau in the Ahau column (for the 11 computing years) and one place horizontally to the left for the 91 days, thereby reaching by the first step 4 Kan, by the second step 4 Men.  Answer:  4 Ahau + 4-095 days = 4 Men.

The table at the top of page 45 of the Dresden Codex handles only multiples of the computing year, again obtained not by multiplication but by addition (fig. 2).  The top line of multiples is almost completely gone, but the pattern of the table makes the restoration reasonably certain.  The higher numbers are built up as follows:

Days Multiples of 364
364 (364 x 1)
728 (364 x 2)
1092 (364 x 3)
1456 (364 x 4)
1820 (364 x 5)
3640 (364 x 10)
5460 (364 x 15)
7280 (364 x 20 or 1 unit of second order)
14560 (364 x 40 or 2 units of second order)
21840 (364 x 60 or 3 units of second order)
29120 (364 x 80 or 4 units of second order)
36400? (364 x 100 or 5 units of second order)
72800? (364 x 200 or 10 units of second order)
109200? (364 x 300 or 15 units of second order)
145600? (364 x 400 or 1 unit of third order)

The last number represents the third order in the vigesimal count of computing years corresponding to the cycle in the count of approximate years.

At the bottom of the table are (right to left) the following glyphs:  13 Eznab, 13 Ik, 13 Cimi, 13 Oc.  On the left is an Initial Series leading to 13 Oc, indicating that that day is the base of the table.  Beneath 364 x 2 stands 13 Eznab, which is actually 728 days from the base 13 Oc.  Beneath 364 x 13 stands 13 Ik, which is in fact the difference (364 days) forward from 13 Eznab.  There follows 13 Cimi beneath 364 x 4 which again represents the advance of 364 days from 13 Ik.  Finally 13 Oc occurs beneath 364 x 5, again reached by counting 364 days from 13 Cimi.  This is a repetition of the starting point of the count since 364 x 5 is divisible by 260, the number of days in the 260-day sacred almanac.  13 Ix, reached by an addition of 364 x 1, is omitted at the right of the row because for some unknown reason 364 days is not in the bottom line of the table but on the right of the second line.

A multiplication table, formed in the same way by additions, occupies the right half of page 63 and all page 64 of the Dresden Codex.  It reaches 400 computing years; that is, the start of the third order in the vigesimal system.  This being more simply arranged and very much better preserved, it is not here reproduced, but may be studied in any copy of the Dresden Codex.  It is arranged in typical Maya fashion since it starts at the bottom right numerical column of page 64, and continues horizontally to the bottom left numerical column on the right half of page 63.  It then passes to the extreme right of the row above on page 64, continues to the left until the left column of the right half of page 63 is reached, and then similarly traverses the top horizontal row from right to left to end at the top left corner of the right half of page 63.  The sequence is:

Days Multiples of 91
91 (91 x 1)
182 (91 x 2)
273 (91 x 3)
364 (91 x 4 or 1 computing year)
455 (91 x 5)
546 (91 x 6)
637 (91 x 7)
728 (91 x 8 or 2 computing years)
819 (91 x 9)
910 (91 x 10)
1001 (91 x 11)
1092 (91 x 12 or 3 computing years)
1183 (91 x 13)
1274 (91 x 14)
1355 (91 x 15)
1456 (91 x 16 or 4 computing years)
1547 (91 x 17)
1638 (91 x 18)
1729 (91 x 19)
1820 (91 x 20 or 5 computing years)
3640? (10 computing years)
5460 (15 computing years)
7280 (1 20-computing-year period)
14560 (2 20-computing-year periods)
21840 (3 20-computing-year periods)
29120 (4 20-computing-year periods)
36400? (5 20-computing-year periods)
72800? (10 20-computing-year periods)
109200 (15 20-computing-year periods)
145600 (1 400-computing-year periods)

Beneath the calculations are five sets of glyphs, spaced (right to left) at intervals of 91 days from one another as far as beneath the column giving five computing years.  Beyond this the days repeat themselves, since five computing years and its multiples are divisible by 260, the number of days in the sacred almanac.

The starting points of the calculations are 3 Chicchan, 3 Kan, 3 Ix, 3 Cimi, and 13 Akbal.  These are precisely the days reached by the so-called serpent numbers on pages 61 and 62 of this codex.  The extremely long distances placed in the coils of these four serpents contain several mistakes.  Three have been rectified by Cyrus Thomas.  Dr. Hermann Beyer, who seems to have missed Thomas’ writings on this subject, offers six lots of corrections, three of which are those given by Thomas, and all of which must be accepted.11  Why such lengthy distance numbers (the least of the eight exceeds 12,000,000 days) should have been used is a mystery.  None appears to be related to the computing year.  There are, in addition, six Initial Series, two of which ( and are also given on page 31 of this codex.  The remainder are not divisible by the mathematical year.

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