Assuming for the moment that there were 13 cycles in a great cycle; it is clear that if this were the case 13 cycles could never be recorded in the inscriptions, for the reason that, being equal to 1 great cycle, they would have to be recorded in terms of a great cycle. This is true because no period in the inscriptions is ever expressed, so far as now known, as the full number of the periods of which it was composed. For example, 1 uinal never appears as 20 kins; 1 tun is never written as its equivalent, 18 uinals; 1 katun is never recorded as 20 tuns, etc. Consequently, if a great cycle composed of 13 cycles had come to its end with the end of a Cycle 13, which fell on a day 4 Ahau 8 Cumhu, such a Cycle 13 could never have been expressed, since in its place would have been recorded the end of the great cycle which fell on the same day. In other words, if there had been 13 cycles in a great cycle, the cycles would have been numbered from 0 to 12, inclusive, and the last, Cycle 13, would have been recorded instead as completing some great cycle. It is necessary to
admit this point or repudiate the numeration of all the other periods in the inscriptions. The writer believes, therefore, that, when the starting point of Maya chronology is declared to be a date 4 Ahau 8 Cumhu, which an "ending sign" and a Cycle 13 further declare fell at the close of a Cycle 13, this does not indicate that there were 13 cycles in a great cycle, but that it is to be interpreted as a Period-ending date, pure and simple. Indeed, where this date is found in the inscriptions it occurs with a Cycle 13, and an "ending sign" which is practically identical with other undoubted "ending signs." Moreover, if we interpret these places as indicating that there were only 13 cycles in a great cycle, we have equal grounds for saying that the great cycle contained only 10 cycles. For example, on Zoömorph G at Quirigua the date 7 Ahau 18 Zip is accompanied by an "ending sign" and Cycle 10, which on this basis of interpretation would signify that a great cycle had only 10 cycles. Similarly, it could be shown by such an interpretation that in some cases a cycle had 14 katuns, that is, where the end of a Katun 14 was recorded, or 17 katuns, where the end of a Katun 17 was recorded. All such places, including the date 4 Ahau 8 Cumhu, which closed a Cycle 13 at the starting point of Maya chronology, are only Period-ending dates, the writer believes, and have no reference to the number of periods which any higher period contains whatsoever. They record merely the end of a particular period in the Long Count as the end of a certain Cycle 13, or a certain Cycle 10, or a certain Katun 14, or a certain Katun 17, as the case may be, and contain no reference to the beginning or the end of the period next higher.
There can be no doubt, however, as stated above, that the cycles were numbered from 1 to 13, inclusive, and then began again with 1. This sequence strikingly recalls that of the numerical coefficients of the days, and in the parallel which this latter sequence affords, the writer believes, lies the true explanation of the misconception concerning the length of the great cycle in the inscriptions.
Table XI. SEQUENCE OF TWENTY CONSECUTIVE DATES IN THE MONTH POP
| 01 Ik | 0 Pop | 11 Eb | 10 Pop |
| 02 Akbal | 1 Pop | 12 Ben | 11 Pop |
| 03 Kan | 2 Pop | 13 Ix | 12 Pop |
| 04 Chicchan | 3 Pop | 01 Men | 13 Pop |
| 05 Cimi | 4 Pop | 02 Cib | 14 Pop |
| 06 Manik | 5 Pop | 03 Caban | 15 Pop |
| 07 Lamat | 6 Pop | 04 Eznab | 16 Pop |
| 08 Muluc | 7 Pop | 05 Cauac | 17 Pop |
| 09 Oc | 8 Pop | 06 Ahau | 18 Pop |
| 10 Chuen | 9 Pop | 07 Imix | 19 Pop |
The numerical coefficients of the days, as we have seen, were numbered from 1 to 13, inclusive, and then began again with 1. See
Table [XI], in which the 20 days of the month Pop are enumerated. Now it is evident from this table that, although the coefficients of the days themselves do not rise above 13, the numbers showing the positions of these days in the month continue up through 19. In other words, two different sets of numerals were used in describing the Maya days: (1) The numerals 1 to 13, inclusive, the coefficients of the days, and an integral part of their names; and (2) The numerals 0 to 19, inclusive, showing the positions of these days in the divisions of the year—the uinals, and the xma kaba kin. It is clear from the foregoing, moreover, that the number of possible day coefficients (13) has nothing whatever to do in determining the number of days in the period next higher. That is, although the coefficients of the days are numbered from 1 to 13, inclusive, it does not necessarily follow that the next higher period (the uinal) contained only 13 days. Similarly, the writer believes that while the cycles were undoubtedly numbered—that is, named—from 1 to 13, inclusive, like the coefficients of the days, it took 20 of them to make a great cycle, just as it took 20 kins to make a uinal. The two cases appear to be parallel. Confusion seems to have arisen through mistaking the name of the period for its position in the period next higher—two entirely different things, as we have seen.
A somewhat similar case is that of the katuns in the u kahlay katunob in Table [IX]. Assuming that a cycle commenced with the first katun there given, the name of this katun is Katun 2 Ahau, although it occupied the first position in the cycle. Again, the name of the second katun in the sequence is Katun 13 Ahau, although it occupied the second position in the cycle. In other words, the katuns of the u kahlay katunob were named quite independently of their position in the period next higher (the cycle), and their names do not indicate the corresponding positions of the katun in the period next higher.
Applying the foregoing explanation to those passages in the inscriptions which show that the enumeration of the cycles was from 1 to 13, inclusive, we may interpret them as follows: When we find the date 4 Ahau 8 Cumhu in the inscriptions, accompanied by an "ending sign" and a Cycle 13, that "Cycle 13," even granting that it stands at the end of some great cycle, does not signify that there were only 13 cycles in the great cycle of which it was a part. On the contrary, it records only the end of a particular Cycle 13, being a Period-ending date pure and simple. Such passages no more fix the length of the great cycle as containing 13 cycles than does the coefficient 13 of the day name 13 Ix in Table [XI] limit the number of days in a uinal to 13, or, again, the 13 of the katun name 13 Ahau in Table [IX] limit the number of katuns in a cycle to 13. This explanation not only accounts for the use of the 14 cycles or 17 cycles, as