716 



SCIENCE 



[N. S. Vol. XLIV. No. 1142 



ratio of increase of successive units in this 

 and the other fully developed Maya systems 

 was not 10, as in the Hindu- Arabic system; 

 it was 20 in all positions except the third. 

 That is, 20 units of the lowest order (kins, or 

 days) make one unit of the next higher order 

 (uinal, or 20 days), 18 uinals make one unit 

 of the third order (tun, or 360 days), 20 tuns 

 make one unit of the fourth order (Jcatun, or 

 7,200 days), 20 katuns made one unit of the 

 fifth order (cycle, or 144,000 days), and 

 finally, 20 cycles make one great cycle of 2,- 

 880,000 days. It has been contended by some 

 archeologists that in Maya inscriptions, not 

 20, but 13, cycles constitute a great cycle, but 

 in the Maya codices all archeologists agree 

 that the only break in the vigesimal system 

 lies in the relation that 18 uinals equal 1 tun. 

 Proceeding now to the notation, as found in 

 the codices, we find symbols 1 to 19, both in- 

 clusive, expressed by bars and dots. Each bar 

 stands for five units, each dot for 1 unit. 

 For instance, 



19 



The values of the bars and dots are added in 

 each case. The zero, which plays a leading 

 part in the notations found on inscriptions 

 as well as those on codices, is represented in 

 the codices by a symbol that looks roughly like 

 a half-closed eye. This zero and the symbols 

 for 1 ■ — 19 in the Maya vigesimal notation 

 correspond to the symbols 0, 1, 2, ... 9 in 

 our decimal notation. In writing 20, in the 

 Maya codices, the principle of local value 

 enters for the first time. It is expressed by 

 a dot placed over the symbol for zero. The 

 numerals are written, not horizontally, but 

 vertically, the unit of lowest order or value 

 being assigned the lowest position. Accord- 

 ingly, 37 was expressed by the symbols for 17 

 (three bars and two dots) in the kin place 

 and one dot, representing 20, placed above the 

 17, in the uinal place. The number 300 is ex- 

 pressed by three bars drawn above the symbol 

 for zero (3X5X20 = 300). The largest 

 number which can be written by the use of 

 only two places or positions is 17 X 20 + 19 = 



359. To write 360, the Maya drew two zeros, 

 one above the other, with one dot higher up, 

 in third place. Using three places to repre- 

 sent kins, uinals and tuns, they could write 

 any number not larger than 7,199. Proceed- 

 ing in this way the Maya wrote numbers in 

 very compact form. The highest number 

 found in the codices is 12,489,781. It occurs 

 on page 61 of what is known as the " Dresden 

 Codex," a fiber-paper booklet that was repro- 

 duced facsimile by Professor E. Forstemann 

 in 1880 and 1892. The symbols representing 

 this number occupy six different places, one 

 above the other. Proceeding from bottom up, 

 the symbols in the six places are, respectively, 

 one dot, three bars, two bars and three dots, 

 two bars and four dots, one bar and one dot, 

 four dots. Thus the numerals in the six 

 places are, respectively, 1, 15, 13, 14, 6, 4. 

 Applying to these the principle of local value, 

 they represent altogether : 1 + 15 X 20 + 13 X 

 18 X 20 + 14 X 20 X 18 X 20 + 6 X 20 X 

 20 X 18 X 20 + 4 X 20 X 20 X 20 X 18 X 

 20 = 12,489,781. From these illustrations it 

 is seen that the Maya used the zero and the 

 principle of local value consistently in the 

 writing of numbers reaching into the millions. 



The second numeral notation that was fully 

 developed and employed by the Maya is found 

 in their inscriptions. It employs the zero, but 

 not the principle of local value. Special 

 glyphs are employed to designate the different 

 units. It is as if we were to write 1203 as 

 " 1 thousand, 2 hundred, tens, 3 ones." "We 

 omit a detailed description of the system. 

 The ratios of successive orders of units are the 

 same as in the preceding, with the exception, 

 perhaps, of the unit of the sixth order. In 

 this second notation, that unit may rest upon 

 the ratio 13, instead of 20, as we stated above. 



The numerals in the Maya codices appear 

 to the present writer to disclose traces of an 

 imperfect quinary system, as seen in the use 

 of the bar to represent 5. Similarly it seems 

 to the present writer that there is a trace of 

 an imperfect decimal system in Maya 

 numerals found in inscriptions, where 16 — 19 

 are represented by two symbols, one symbol 

 for 10 and the other for 6, 7, 8, 9, respectively. 



