[1] For those unacquainted with Maya arithmetic the following points will explain matters: the Mayas used the vigesimal system of enumeration; they counted by twenties instead of tens. A bar represented five, and a dot stood for one. They represented numbers larger than twenty by position, just as we do. However, instead of having the smallest denomination at the right and the largest at the left of a horizontal series of figures, they had the smallest at the bottom and the largest at the top of a column of numbers. Instead of each unit in a given position representing ten times the value of that of the preceding position, it represented twenty times the value, except in the third position where it was only eighteen times as great. Thus each unit of the bottom number represented one (Kin), that of the number above it twenty (Uinal), that of the third number 20 × 18 or 360 (Tun), that of the fourth position 20 × 360 or 7200 (Katun), etc. For ease in handling, these numbers are written in our script with arabic numerals, the bottom number on the right, and separated by periods. Thus in column three, page 53a, the upper number is 1. 7. 2, which means that the kin of this group is 2, the Uinal 7, (7 × 20) and the Tun 1 (1 × 360), making in all 2 + 140 + 360 or 502. The Maya calendar, like ours, consisted of a series of numbers and a series of names for each day, each series repeating itself constantly, irrespective of the other. There were twenty different day names, which remained in an unchangeable order, and thirteen numbers. In the pages under discussion these day names appear as glyphs preceded by the necessary number. For further details consult S. G. Morley, _An Introduction to Maya Hieroglyphs_, Bulletin 57, Bureau of American Ethnology, Washington, D. C., 1915, and C. P. Bowditch, 1910. In short, then, the ideal arrangement of the series is as follows: Each upper number is the sum of all the lower numbers of the preceding columns and its own column. Each lower number expresses the difference between the upper number of its own column and that of the column immediately preceding it. The day names and numbers are three horizontal series, each starting a day later than the one above it, and recording three sets of day names and numbers which would fit the series formed by the upper numbers. It should be noticed that the mathematical interpretation of the series does not appear to depend in any way upon the hieroglyphs appearing at the top of the columns, or upon the pictures.