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.
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:—
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.
[Illustration: Fig. 42. Bar and Dot Numerals of the Mayas.]
The straight vigesimal system was doubtless used by the Mayas in ordinary counting, but in counting time a very important change was introduced in the third position. Also the names were modified: _hun_ was called _kin_ which means sun or day. In the second position _kal_ was called _uinal_ which means month and 18 of these were taken to form a _tun_, stone, which was the third unit. The _tun_ then had a value of 18 × 20 = 360 days, making a conventional year about five and a quarter days less than a true year. Twenty _tuns_ made a _kaltun_ or _katun_ and above this period the numeral system proceeded as before and in the ascending values the names already given were merely combined with _tun_, if Gates is right in his clever suggestion. For years it has been customary to speak of the fifth period as cycle for want of a native term: this will now be called _baktun_. One _hablatun_, the highest period with a name, has the astonishing value of 460,800,000,000 days. However, the highest numbers fall considerably short of this potential limit.