We have seen in the foregoing pages the different characters which stood for the numerals 0 to 19, inclusive. The next point claiming our attention is, how were the higher numbers written, numbers which in the codices are in excess of 12,000,000, and in the inscriptions, in excess of 1,400,000? In short, how were numbers so large expressed by the foregoing twenty (0 to 19, inclusive) characters? The Maya expressed their higher numbers in two ways, in both of which the numbers rise by successive terms of the same vigesimal system: 1. By using the numbers 0 to 19, inclusive, as multipliers with the several periods of Table VIII (reduced in each case to units of the lowest order) as the multiplicands, and-- 2. By using the same numbers[71] in certain relative positions, each of which had a fixed numerical value of its own, like the positions to the right and left of the decimal point in our own numerical notation. {104} The first of these methods is rarely found outside of the inscriptions, while the second is confined exclusively to the codices. Moreover, although the first made use of both normal-form and head-variant numerals, the second could be expressed by normal forms only, that is, bar and dot numerals. This enables us to draw a comparison between these two forms of Maya numerals: [Illustration: FIG. 55. Examples of the use of head-variant numerals with period, day, or month signs. The translation of each glyph appears below it.] Head-variant numerals never occur independently, but are always prefixed to some period, day, or month sign. Bar and dot numerals, on the other hand, frequently stand by themselves in the codices unattached to other signs. In such cases, however, some sign was to be supplied mentally with the bar and dot numeral. {105} FIRST METHOD OF NUMERATION [Illustration: FIG. 56. Examples of the first method of numeration, used almost exclusively in the inscriptions.] In the first of the above methods the numbers 0 to 19, inclusive, were expressed by multiplying the kin sign by the numerals[72] 0 to 19 in turn. Thus, for example, 6 days was written as shown in figure 56, _a_, 12 days as shown in _b_, and 17 days as shown in _c_ of the same {106} figure. In other words, up to and including 19 the numbers were expressed by prefixing the sign for the number desired to the kin sign, that is, the sign for 1 day.[73]