52 BUEEAU OF AMEEICAN ETHNOLOGY [bdll. 57 



of the days and the haab the positions of these days m the divisions 

 of the year. The Calendar Round was the most important period in 

 Maya chronology, and a comprehension of its nature and of the prin- 

 ciples which governed its composition is therefore absolutely essential 

 to the understanding of the Maya system of counting time. 



It has been explained (see p. 41) that the complete designation 

 or name of any day in the tonalamatl consisted of two equally essen- 

 tial parts: (1) The name glyph, and (2) the numerical coefficient. 

 Disregarding the latter for the present, let us first see wMch of the 

 twenty names in Table I, that is, the name parts of the days, can 

 stand at the beginning of the Maya year. 



In applying any sequence of names or numbers to another there 

 are only three possibilities concerning the names or numbers which 

 can stand at the head of the resultmg sequence : 



1. When the sums of the units in each of the two sequences contain 

 no common factor, each one of the units in turn will stand at the 

 head of the resulting sequence. 



2. When the sum of the units in one of the two sequences is a 

 multiple of the sum of the units in the other, only the first unit 

 can stand at the head of the resulting sequence. 



3. When the sums of the units in the two sequences contain a 

 common factor (except in those cases which fall under (2), that is, 

 in which one is a multiple of the other) only certain units can stand at 

 the head of the sequence. 



Now, since our two numbers (the 20 names in Table I and the 365 

 days of the year) contain a common factor, and since neither is a 

 multiple of the other, it is clear that only the last of the three con- 

 tingencies just mentioned concerns us here; and we may therefore 

 dismiss the first two from further consideration. 



The Maya year, then, could begin only with certain of the days 

 in Table I, and the next task is to find out wliich of these twenty 

 names invariably stood at the beginnings of the years. 



When there is a sequence of 20 names in endless repetition, it is 

 evident that the 361st will be the same as the 1st, since 360 = 20 X 18. 

 Therefore the 362d will be the same as the 2d, the 363d as the 3d, 

 the 364th as the 4th, and the 365 as the 5th. But the 365th, or 

 5th, name is the name of the last day of the year, consequently the 

 1st day of the following year (the 366th from the begmning) mil 

 have the 6th name in the sequence. Following out this same idea, 

 it appears that the 361st day of the second year will have the same 

 name ps that with which it began, that is, the 6th name in the 

 sequence, the 362d day the 7th name, the 363d the 8th, the 364th 

 the 9th, and the 365th, or last day of the second year, the 10th name. 

 Therefore the 1st day of the tliird year (the 731st from the beginning) 

 will have the 1 1th name in the sequence. Similarly it could be shown 



