80 BUREAU OF AMERICAN ETHNOLOGY [bull. 57 



which they occupied in the cycle, as Katun 14, for example, but by 

 the different days Ahau with which they ended, as Katiui 2 Ahau, 

 Katun la Ahau, etc. See Table IX. 



Table IX.— SEQUENCE OF KATUNS IN U KAHLAY KATUNOB 



Katun 2 Aliau Katun 8 Ahau 



Katun 13 Ahau Katun 6 Ahau 



Katun 11 Ahau Katiin 4 Ahau 



Katun 9 Ahau Katun 2 Ahau 



Katun 7 Ahau Katun 13 Ahau 



Katun 5 Ahau Katun 11 Ahau 



Katun 3 Ahau Katun 9 Ahau 



Katun 1 Ahau Katun 7 Ahau 



Katun 12 Ahau Katun 5 Ahau 



Katun 10 Ahau Katun 3 Ahau, etc. 



The peculiar retrograding sequence of the numerical coefficients in 

 Table IX, decreasing by 2 from katun to katun, as 2, 13, 11, 9, 7, 

 5, 3, 1, 12, etc., results directly from the number of days which the 

 katun contains. Since the 13 possible numerical coefficients, 1 to 

 13, mclusive, succeed each other in endless repetition, 1 following 

 immediately after 13, it is clear that in counting forward any given 

 number from any given numerical coefficient, the resulting numerical 

 coefficient will not be affected if we first deduct all the 13s possible 

 from the number to be counted forward. The mathematical dem- 

 onstration of this fact follows. If we count forward 14 from any 

 given coefficient, the same coefficient will be reached as if we had 

 counted forward but 1. This is true because, (1) there are only 13 

 numerical coefficients, and (2) these follow each other without inter- 

 ruption, 1 following immediately after 13; hence, when 13 has 

 been reached, the next coefficient is 1, not 14; therefore 13 or any 

 multiple thereof may be counted forward or backward from any one 

 of the 13 numerical coefficients without changing its value. This 

 truth enables us to formulate the following rule for finding numerical 

 coefficients: Deduct all the multiples of 13 possible from the number 

 to be counted forward, and then count forward the remainder from 

 the known coefficient, subtracting 13 if the resulting number is above 

 13, since 13 is the highest possible number which can be attached to 

 a day sign. If we apply this rule to the sequence of the numerical 

 coefficients in Table IX, we shall find that it accounts for the retro- 

 grading sequence there observed. The first katun m Table IX, 

 Katun 2 Ahau, is named after its ending day, 2 Ahau. Now let us 

 see whether the application of this rule will give us 13 Ahau as the 

 ending day of the next katun. The number to be counted forward 

 from 2 Ahau is 7,200, the number of days m one katun; therefore we 

 must first deduct from 7,200 all the 13s possible. 7,200 ^ 13 = 553H- 

 In other words, after we liave deducted all the 13's possible, that is, 



