D e d i c a t e d    t o

Part I

The 360 Day-Count

The numbering system of mayas consisted of three symbols: a shell-shaped figure that represented zero (); a dot or point for the number 1 ( ); and, a straight bar figure represented the number 5 ( ). These symbols were arranged vertically at distinct positional levels.

The positional levels determined specific values, whereby the given number at a particular level would be multiplied by that level's corresponding constant value. The following is generally cited as the basis of the system, although now it is known that there were more than five positional levels.

The maya system of the long count counted time-cycles in such a manner, that 20 kins (days) equaled 1 uinal (1 month); 20 uinals equal 1 tun (a year); and successively in this manner of multiples of twenty. We obviously have no commonday words in our language for an alautun, a time-cycle of 23,040,000,000 days, other than stating that that is a long time, over 63 million years.

For example, the number 1 (), in column one would have a distinct value depending upon its placement at each of the five levels. At positional level:

Such would be the translation of the constants into a specific number of years, given that 360 was the calendar count involved (360 days = 1 year, plus 5 extra days of the uayeb).

Similarly, regarding the number 5 (), the values would change as the number five would now be multiplied by the constants. Then, the number 5 would represent at positional level:

The maya long count, then, would be the cumulative sum of values generated in a specific column of figures. Let us examine a particular number example; the number 36,108 would be expressed in the following manner:

	=	5 times	7,200	=	36,000
	=	0 times	360	=	0
	=	5 times	20	=	100
	=	8 times	1	=	8
36,108

The creative nature of the maya numbering system becomes obvious at once. In fact, a similar counting system was used by other peoples of ancient Mexico (the mexicas and aztecs), while some authors consider that the original counting system was developed by the olmec culture.

Without entering into a discussion of who-did-what-first, one is concerned with the procedure of such extensive calculations with numbers running into the millions on the fifth positional level and beyond. For even though the ancient peoples of Mexico might have been able to visually represent the numerical result of 18 times 144,000 (i.e., 2,592,000), it becomes intriguing to enquire how they actually effected the multiplications. With that concern in mind, we have attempted to understand how these calculations and reckonings may have been achieved without the need for multiplication.

The enquiry began with the numbers 1 through 20 and the corresponding values for each positional level. The search has the purpose of understanding why those particular constant values were chosen as cut-off points on those levels.

The integer 360 is obtained by multiplying 18 x 20, which in calendaric terms represents 18 months of 20 days. The older calendar reckoning was based on 13 months of 20 days and yielded a day-count of 260. But, we shall first examine the 360 count, since that pertains to the values cited in the tun figure.

The following Table of Numbers is then implied in the procedure of multiplying the numbers 1-20 with each one of the five positional level's constants:

By carrying through this multiplication procedure with every combination cited, the following Table of Numbers obtains:

This Table of Numbers (360c) reveals many significant aspects which we shall now examine in detail. The positional level I involves the unit-numbers 1 through 20. We did carry out the positional level II multiplication to the 40th place, but proved to be unnecessary as numbers began to repeat themselves. Given that the number 20 is the significant integer of the system, carrying the calculation out to the 20th place is sufficient to comprehend the system.

The values of positional level I are 20 times less than those of level II; level II is 18 times less than level III; the values of level III are 20 times less than those of level IV; and, the values of level IV are 20 times less than those of positional level V. Hence, another way of expressing this is: level V is 144,000 times greater than level I, the values of level IV are 7,200 times greater than the values of level I; etc. In all cases, the different levels represent multiples of 20.

One should notice the placement of the zeros at each particular level, for this shall be meaningful in the remainder of the analysis regarding methods of calculations. For example, positional level V shows values with at least three zeros ending their terms; level IV with at least two zeros; and levels III and II with at least one zero at the end of each value.

The most significant aspect of the Table of Numbers is the cut-off point 360 at the positional level II, 18. This integer then becomes situated at positional level III, 1. The number 360, as mentioned, is the product of 18 x 20, which involves the relation months/days.

The relationship concerns the calendar count of 360 based on time-cycles.

Positional level III ends at the 20th integer with the value of 7,200, which begins also positional IV. It should be no surprise that positional level IV ends with the values of 144,000, the same constant that initiates positional level V.

The Table of Numbers, then, appears to arbitrarily be set at 360, since according to the logic of the integer 20, positional level III should begin with the number 400. However, the Table of Numbers reflects the logic of calendar reckoning, which in this case utilizes the earth's approximate time-cycle in seasonal years as a means for programming the entire set of values listed on the table. The positional levels III, IV, and V, then, become multiples of 360.

Now, let us analyze the numbers appearing in the table. The numbers situated at positional level II may also be obtained by doubling the numbers corresponding to positional level I (1 - 20), and then simply adding a zero;

In other words, multiplying by the integer 20 can also be achieved by simply doubling the number and adding a zero to the product. Observe the positional values at levels I and II on the table of numbers:

In each case, at each level and integer one can identify this possibility:

Now, observe that this same relationship does not exist between the values listed on positional levels II and III, since the difference between these two levels is 18; not 20. It would appear as though the calendar count has become interrupted, no longer following the integer 20 logic. It is not possible to simply double the number on level II and add a zero to it order to obtain the number on level III. By selecting the positional level to begin with 360, instead of the number 400 (20 x 20; the value on space 20), then the apparent logic has been broken.

However, as we examine levels III, IV and V, we now notice that the relationship of 20 does exist among these three levels. Again, instead of multiplying, one may revert to the procedure of doubling the corresponding number on the lower level to obtain the next higher level, and simply add a zero.

In order to make the translation from level II to level III, one would have to account for the two places (19 and 20) that were skipped, when the table was broken off at the value of 18 (360).

As a proposal for a method of calculation, let us run through a specific example.

For the sake of example, let us offer the case the number 15.

In this manner, one is able to obtain the corresponding numbers of each of the five positional levels on the Table of Numbers for the number 15:

The method of calculation, without multiplication, for obtaining the corresponding numbers on the Table of Numbers for the 360-count, for each positional level can be summarized as follows:

Choose a number 1 to 20:

Example:

In this manner one can obtain any number series on the Table of Numbers, thereby reconstructing the entire table without actually having it written down, and without performing any long method of multiplication. Calculations into the millions or billions, and beyond, may be obtained with relative ease.

In the following part of this essay we shall now review the 360c reckoning with the old calendar 260, in order to perceive their possible relationship.

Continued Part II

©1995-2003 Copyrighted by Charles William Johnson. All Rights Reserved
Reproduction prohibited without written consent of the author.