130 BUREAU OF AMERICAN ETHNOLOGY [bdll. 57 



with their distance from the bottom, according to the terms of the 

 vigesimal system shown in Table VIII. Tliis second method, or 

 " numeration by position," as it may be called, was a distinct advance 

 over the first, since it required for its expression only the signs for 

 the numerals to 19, inclusive, and did not involve the use of any 

 period glyphs, as did the first method. To its greater brevity, no 

 doubt, may be ascribed its use in the codices, where numerical calcu- 

 lations running into numbers of 5 and 6 terms form a large part of 

 the subject matter. It should be remembered that in numeration 

 by position only the normal forms of the numbers — bar and dot 

 numerals — are used. This probably results from the fact that head- 

 variant numerals never occur independently, but are always prefixed 

 to some other glyph, as period, day, or month signs (see p. 104). 

 Since no period glyphs are used in numeration by position, only 

 normal-form numerals, that is, bar and dot numerals, can appear. 



The numbers from 1 to 19, inclusive, are expressed in this method, as 

 shown in figure 39, and the number as shown in figure 46. As aU 

 of tliese numbers are below 20, they are expressed as units of the first 

 place or order, and consequently each should be regarded as having 

 been multiplied by 1 , the numerical value of the first or lowest position. 



The number 20 was expressed in two different ways: (1) By the 

 sign shown in figure 45; and (2) b\' the numeral in the bottom 

 place and the numeral 1 in the next place above it, as in figure 63, a. 

 The first of these had only a very restricted use in connection with 

 the tonalamatl, wherein numeration by position was impossible, and 

 therefore a special character for 20 (see fig. 45) was necessary. 

 See Chapter VI. 



The numbers from 21 to 359, inclusive, involved the use of two 

 places — the kin place and the uinal place— which, according to Table 

 VIII, we saw had numerical values of 1 and 20, respectively. For 

 example, the number 37 was expressed as shown in figure 63, h. The 

 17 in the kin place has a value of 17 (17x1) and the 1 in the uinal, or 

 second, place a value of 20 (1 (the numeral) X 20 (the fixed numerical 

 value of the second place)). The sum of these two products equals 

 37. Ato^'i^; 300 was written as in figure 63, c. The in the kin 

 place has the value (0x1), and the 15 in the second place has the 

 value of 300 (15x20), and the sum of these products equals 300. 



To express the numbers 360 to 7,199, inclusive, three places or 

 terms were necessary — kins, uinals, and tuns — of wliich the last had a 

 numerical value of 360. (See Table VIII.) For example, the number 

 300 is shown in figure 63, d. The in tlie lowest place indicates that 

 kins are involved, the in the second place indicates that uinals 

 or 20's are involved, while the 1 in tlie third place shows that there is 1 

 tun, or 360, kins recorded (1 (the numeral) X 360 (the fixed numerical 

 value of tlie third position)); the sum of these three products equals 

 360. Again, the number 7,113 is expressed as shown in figure 63, e. 



