THE MAYAN CIVILIZATION 97 
contained evenly in either the month or the year, nor 
do these larger time measures show any simple relation 
to each other as regards length. The history of the 
calendar is one of compromise and correction. 
The Mayan calendars were made possible by: first, 
the knowledge of astronomical time periods; second, the 
possession of a suitable notation system; third, the dis- 
covery of a permutation system of names and numbers. 
Elements of the Day Count. There is reason to 
believe that the Mayas had first a lunar calendar of 
twelve months of thirty days each, making a year of 
360 days, and that they modified this so that the num- 
ber of days in the “‘month”’ became twenty instead of 
thirty to agree with the value of the second digit in 
their notation system and that the number of months 
was increased from twelve to eighteen, still making up a 
total of 360 days which agreed with the value of the 
third digit. With a truer knowledge of the length of 
the year an extra five day month was added to make 
a year of 365 days. Beyond this the “leap year” error 
was calculated but not interpolated. As proof that the 
lunar month of thirty days preceded the notation 
month of twenty days, it need only be pointed out that 
the name for this period, uenal, seems to be connected 
with the name for moon, u, and that the hieroglyph 
for moon has the value, twenty, in the inscriptions 
and ancient books. 
Before entering into a fuller discussion of the astro- 
nomical and notational facts let us turn for a moment 
to the third fact, the permutation system. ‘The origin 
of the cycle! known by the Nahuan name tonalamatl, 
1 The word cycle is applied in this book to re-entering series, or 
wheels, of days. These all contain the fonalamatl without a remainder. 
The word period is applied to fixed numbers that do not contain the 
tonalamatl. Unfortunately, the word cycleisalso applied to the fifth digit 
or position in the notationsystem and the word period is used asa general 
designation for any digit or position as well as for other fixed values. 
