"Gallatin explained them as the numerical characters of the days "Ahau" following the first day of each year called Cauac; Dr. Valentini thinks they refer to the numbers of the various idols worshiped in the different Ahaus; Professor Thomas that they are the number of the year (in the indiction of 52 years) on which the Ahau begins. Each of these statements is true in itself, but each fails to show any practical use of the series; and of the last mentioned it is to be observed that the objection applies to it that at the commencement of an Ahau Katun the numbers would run 1, 12, 10, 8, etc., whereas we know positively that the numbers of the Ahaus began with 13 and continued 11, 9, 7, 5, etc. "The explanation which I offer is that the number of the Ahau was taken from the last day Cauac preceding the Kan with which the first year of each Ahau began--for, as 24 is divisible by 4, the first year of each Ahau necessarily began with the day Kan. This number was the "ruling number" of the Ahau, and not for any mystical or ceremonial purpose, but for the practical one of at once and easily converting any year designated in the Ahau into its equivalent in the current Kin Katun, or 52 year cycle. All that is necessary to do this is, to _add the number of the year in the Ahau to the number of the year Cauac corresponding to this "ruling number." When the sum exceeds 52, subtract that number._ "Take an example: To what year in the Kin Katun does 10 Ahau XI (the 10th year of the 11th Ahau) correspond? "On referring to a table, or, as the Mayas did, to a 'Katun wheel,' we find the 11th Cauac to be the 24th year of the cycle; add ten to this and we have 34 as the number of the year in the cycle to which 10 Ahau XI corresponds. The great simplicity and convenience of this will be evident without further discussion." The important question remains, how closely, by these cycles, did the Mayas approximate to preserving the exact date of an event? To answer this fairly, we should be sure that we have a perfectly authentic translation of their hieroglyphic annals. It is doubtful that we have. Those I present in this volume are the most perfect, so far as I know, but they certainly do not agree among themselves. Can their discrepancies be explained? I think they can in a measure (1) by the differing length of the katuns, (2) by the era assumed as the commencement of the reckoning. It must be remembered that there was apparently no common era adopted by the Mayas; each province may have selected its own; and it is quite erroneous to condemn the annals off-hand for inaccuracy because they conflict between themselves. § 8. _Ancient Hieroglyphic Books._