Fig. 64. Figure showing the use of the "minus" or "backward" sign in the codices.
In the codices, moreover, when the count is backward, or contrary to the general practice, the fact is clearly indicated[[95]] by a special character. This character, although attached only to the lowest term[[96]] of the number which is to be counted backward, is to be interpreted as applying to all the other terms as well, its effect extending to the number as a whole. This "backward sign" (shown in fig. [64]) is a circle drawn in red around the lowest term of the number which it affects, and is surmounted by a knot of the same color. An example covering the use of this sign is given in figure [64]. Although the "backward sign" in this figure surrounds only the numeral in the first place, 0, it is to be interpreted, as we have seen, as applying to the 2 in the second place and the 6 in the third place. This number, expressed as 6 tuns, 2 uinals, and 0 kins, reduces to 2,200 units of the first place, and in this form may be more readily handled (first step). Since the starting point usually precedes the number counted from it and since in figure [64] the number is expressed by the second method, its starting point will be found standing below it. This follows from the fact that in numeration by position the order is from bottom to top. Therefore the starting point from which the 2,200 recorded in figure [64] is counted will be found to be below it, that is, the date 4 Ahau 8 Cumhu[[97]] (second step). Finally, the red circle and knot surrounding the lowest (0) term of this 2,200 indicates that this number is to be counted backward from its starting point, not forward (third step).
On the other hand, in the inscriptions no special character seems to have been used with a number to indicate that it was to be counted backward; at least no such sign has yet been discovered. In the inscriptions, therefore, with the single exception[[98]] mentioned below, the student can only apply the general rule given on page [136], that in the great majority of cases the count is forward. This rule will be found to apply to at least nine out of every ten numbers. The exception above noted, that is, where the practice is so uniform as to render possible the formulation of an unfailing rule, has to do with Initial Series. This rule, to which there are no known exceptions, may be stated as follows:
Rule 1. In Initial Series the count is always forward, and, in general throughout the inscriptions. The very few cases in which the count is backward, are confined chiefly to Secondary Series, and it is in
dealing with this kind of series that the student will find the greatest number of exceptions to the general rule.
Having determined the direction of the count, whether it is forward or backward, the next (fourth) step may be given.
Fourth Step in Solving Maya Numbers
To count the number from its starting point.
We have come now to a step that involves the consideration of actual arithmetical processes, which it is thought can be set forth much more clearly by the use of specific examples than by the statement of general rules. Hence, we will formulate our rules after the processes which they govern have been fully explained.
In counting any number, as 31,741, or 4.8.3.1 as it would be expressed in Maya notation,[[99]] from any date, as 4 Ahau 8 Cumhu, there are four unknown elements which have to be determined before we can write the date which the count reaches. These are: