MORLEY] INTRODUCTION TO STUDY OF MAYA HIEROGLYPHS 133 



It is clear that in numeration by position the order of the units 

 could not be reversed as in the first method without seriously affecting 

 their numerical values. This must be true, sin-ce in the second method 

 the numerical values of the numerals depend entirely on their position — 

 that is, on their distance above the bottom or first term. In the first 

 method, the multiplicands — the period glyphs, each of which had a 

 fixed numerical value — are always expressed * with their correspond- 

 ing multiphers — the numerals to 19, inclusive; in other words, the 

 period glyphs themselves show whether the series is an ascending or 

 a descending one. But in the second method the multipUcands are 

 not expressed. Consequently, since there is nothing about a column 

 of bar and dot numerals which in itself indicates whether the series 

 is an ascending or a descending one, and since in numeration by 

 position a fixed starting point is absolutely essential, in their sec- 

 ond method the Maya were obliged not only to fix arbitrarily the 

 direction of reading, as from bottom to top, but also to confine them- 

 selves exclusively to the presentation of one kind of series only — that 

 is, ascending series. Only by means of these two arbitrary rules was 

 confusion obviated in numeration by position. 



However dissimilar these two methods of representing the numbers 

 may appear at first sight, fundamentally they are the same, since 

 both have as their basis the same vigesimal system of numeration. 

 Indeed, it can not be too strongly emphasized that throughout the 

 range of the Maya writings, codices, inscriptions, or Books of Chilam 

 Balam ^ the several methods of counting time and recording events 

 found in each are all derived from the same source, and all are expres- 

 sions of the same numerical system. 



That the student may better grasp the points of difference between 

 the two methods they are here contrasted : 



Table XII. COMPARISON OF THE TWO METHODS OF NUMERATION 



FIRST METHOD 



1 . Use confined almost exclusively to the 



inscriptions. 



2. Numerals represented by both normal 



forms and head variants. 



3. Numbers expressed by using the num- 



erals to 19, inclusive, as multipliers 

 with the period glyphs as multipli- 

 cands. 



4. Numbers presented as ascending or de- 



scendiag series. 



5. Direction of reading either from bot- 



tom to top, or vice versa. 



SECOND METHOD 



1. Use confined exclusively to the co- 



dices. 



2. Numerals represented by normal forms 



exclusively. 



3. Numbers expressed by using the nu- 



merals to 19, inclusive, as multi- 

 pliers in certain positions the fixed 

 numerical values of which served as 

 multiplicands. 



4. Numbers presented as ascending series 



exclusively. 



5. Direction of reading from bottom to top 



exclusively. 



> A few exceptions to this have been noted on pp. 127, 128. 



2 The Books of Chilan Balam have been included here as they are also expressions of the native Maya 

 mind. 



