Fig. 44. Normal forms of numerals 1 to 13, inclusive, in the Books of Chilan Balam.
The use of these purely ornamental elements, to fill the empty spaces in the normal forms of the numerals 1, 2, 6, 7, 11, 12, 16, and 17, is a fruitful source of error to the student of the inscriptions. Slight weathering of an inscription is often sufficient to make ornamental crescents look exactly like numerical dots, and consequently the numerals 1, 2, 3 are frequently mistaken for one another, as are also 6, 7, and 8; 11, 12, and 13; and 16, 17, and 18. The student must exercise the greatest caution at all times in identifying these
numerals in the inscriptions, or otherwise he will quickly find himself involved in a tangle from which there seems to be no egress. Probably more errors in reading the inscriptions have been made through the incorrect identification of these numerals than through any other one cause, and the student is urged to be continually on his guard if he would avoid making this capital blunder.
Although the early Spanish authorities make no mention of the fact that the Maya expressed their numbers by bars and dots, native testimony is not lacking on this point. Doctor Brinton (1882 b: p. 48) gives this extract, accompanied by the drawing shown in figure [44], from a native writer of the eighteenth century who clearly describes this system of writing numbers:
They [our ancestors] used [for numerals in their calendars] dots and lines [i. e., bars] back of them; one dot for one year, two dots for two years, three dots for three years, four dots for four, and so on; in addition to these they used a line; one line meant five years, two lines meant ten years; if one line and above it one dot, six years; if two dots above the line, seven years; if three dots above, eight years; if four dots above the line, nine; a dot above two lines, eleven; if two dots, twelve; if three dots, thirteen.
This description is so clear, and the values therein assigned to the several combinations of bars and dots have been verified so extensively throughout both the inscriptions and the codices, that we are justified in identifying the bar and dot as the signs for five and one, respectively, wherever they occur, whether they are found by themselves or in varying combinations.
In the codices, as will appear in Chapter VI, the bar and dot numerals were painted in two colors, black and red. These colors were used to distinguish one set of numerals from another, each of which has a different use. In such cases, however, bars of one color are never used with dots of the other color, each number being either all red or all black (see p. [93], footnote 1, for the single exception to this rule).
By the development of a special character to represent the number 5 the Maya had far surpassed the Aztec in the science of mathematics; indeed, the latter seem to have had but one numerical sign, the dot, and they were obliged to resort to the clumsy makeshift of repeating this in order to represent all numbers above 1. It is clearly seen that such a system of notation has very definite limitations, which must have seriously retarded mathematical progress among the Aztec.
In the Maya system of numeration, which was vigesimal, there was no need for a special character to represent the number 20,[[61]] because
(1) as we have seen in Table [VIII], 20 units of any order (except the 2d, in which only 18 were required) were equal to 1 unit of the order next higher, and consequently 20 could not be attached to any period-glyph, since this number of periods (with the above exception) was always recorded as 1 period of the order next higher; and (2) although there were 20 positions in each period except the uinal, as 20 kins in each uinal, 20 tuns in each katun, 20 katuns in each cycle, these positions were numbered not from 1 to 20, but on the contrary from 0 to 19, a system which eliminated the need for a character expressing 20.