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: 1. The day coefficient, which must be one of the numerals 1 to 13, inclusive. 2. The day name, which must be one of the twenty given in Table I. 3. The position of the day in some division of the year, which must be one of the numerals 0 to 19, inclusive. 4. The name of the division of the year, which must be one of the nineteen given in Table III. These four unknown elements all have to be determined from (1) the starting date, and (2) the number which is to be counted from it. If the student will constantly bear in mind that all Maya sequences, whether the day coefficients, day signs, positions in the divisions of the year, or what not, are absolutely continuous, repeating themselves without any break or interruption whatsoever, he will better understand the calculations which follow. It was explained in the text (see pp. 41-44) and also shown graphically in the tonalamatl wheel (pl. 5) that after the day coefficients had reached the number 13 they returned to 1, following each other indefinitely in this order without interruption. It is clear, therefore, that the highest multiple of 13 which the given number contains may be subtracted from it without affecting in any way the value of the day coefficient of the date which the number will reach when counted from the starting point. This is true, because no matter what the day coefficient of the starting point may be, any multiple of 13 will always bring the count back to the same day coefficient. {139}