Thus Ik occupies 0 position the first year, 5, the second year, 10 the third, 15 the fourth, and 0 the fifth. While Manik that belongs to the same set has position 5 the first year, 10 the second, etc. It will be noted that Imix, the first day of the formal permutation of the tzolkin is never the first day of a month.
The Calendar Round.
But this assignment of particular day names to particular places in the month does not close the problem. Each day name is associated in the tzolkin, or permutation, with a day number. While it is true that each day can occupy only four month positions in as many years, it must be remembered that the day numbers associated with these names can run the whole gamut of 13 changes. Thus, although Ik must always occupy the fifth position in the months during a certain year, nevertheless it will have numbers which fall in the sequence 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 13, 1, etc. The complete cycle of variations must run through the least common multiple of 260 (the permutation) and 365 (the conventional year) or 18,980 days. This cycle is commonly known as the Calendar Round. A Mayan day fixed in a month, or let us say a calendar round date, has four parts to its name, thus, 11 Ahau 18 Mac. We describe a day as Tuesday, July 4, meaning “Tuesday the third day of the seven day week occupies the fourth position in the month of July.” Similarly the Mayan date 11 Ahau 18 Mac may be read “the day named Ahau as eleventh day in a thirteen day week occupies the eighteenth position in the month Mac.” Owing to leap year corrections the European date given above does not recur at regular intervals, but a Mayan day recurs infallibly in 52 calendar years, never sooner, never later.
So far we have considered two kinds of Mayan dates, first the tzolkin date, recurring every 260 days, secondly the calendar round date recurring every 18,980 days. Before we can understand a third and much more important kind of date, namely a date which states, in addition to the calendar round designation, the total number of days since a beginning day called 4 Ahau 8 Cumhu, located far in the past, we must direct our attention to the matter of numbers and notation.
Mayan Numbers.
The three most common numerical systems in use in the world are all derived from man’s anatomy. The quinary system is based on counting the fingers of one hand, the decimal system on counting those of both hands and the vigesimal system, which prevailed in Central America, is based on counting all the fingers and all the toes. The vigesimal system is seen in imperfect form in our count of scores, where seventy years are three score and ten.
The Mayan name for one was hun: they had simple names to 9 and composite ones from 10 to 19, much as in English, and twenty was hun kal, one score. The ascending values in the vigesimal scale were as follows:—
| Mayan Numbers | Arabic Equivalents | ||
|---|---|---|---|
| hun | 1 | ||
| 20 hun | = 1 | kal | 20 |
| 20 kal | = 1 | bak | 400 |
| 20 bak | = 1 | pic | 8,000 |
| 20 pic | = 1 | cabal | 160,000 |
| 20 cabal | = 1 | kinchil | 3,200,000 |
| 20 kinchil | = 1 | alau | 64,000,000 |
| 20 alau | = 1 | hablat | 1,280,000,000 |
They invented signs for zero and discovered the principle of “local value” in the writing down of numbers centuries before these ideas (which are fundamental to higher mathematics) were known in the Old World. The notation of numbers had its simpler and more complicated phase. In the simpler phase 1 was represented by a dot, 2 by two dots, 5 by a bar, 6 by a bar and dot, 15 by three bars, etc. The commonest sign for zero was a shell while a picture of the moon stood for twenty. In the more elaborate notation a series of twenty faces of gods represented the numerals from 0 to 19.
