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350 N o t e s o f M i d d l e A m e r i c a n A r c h a e o l o g y a n d E t h n o l o g y Carnegie Institution of Washington Division of Historical Research No. 88 March 31, 1948 Some Remarks on Maya Arithmetic R.C.E. Long reckoning has been quite lost in modern times and can be recovered only from the literature, for the practice was a commonplace in the sixteenth century. It is mentioned by Shakespeare and in many references to counting boards and to sets of jettons in inventories of property. When Landa said the Maya counted on the ground or on some flat surface he was referring to what was usual in Europe at the time he wrote. No abacus was used. In some systems, like that of the English Exchequer, the reckoning was done on a cloth marked with vertical and horizontal lines like a chessboard. But this was not essential, and in several systems the counters were merely arranged in columns separated by spaces, that is, the columns were separated from each other by vertical spaces and within the column the different orders of units were separated by horizontal spaces. The system was decimal (except, of course, where special units, as of money, were required). In all the systems cited by Barnard, as also in the abacus, whether Roman, Hindu, or Chinese, the system was quinary-decimal, not pure decimal. Each decimal column is in two parts, the lower for units up to five, the upper for fives up to ten. When the addition of more jettons made the lower portion of the column amount to five, then five units were taken from this and one jetton added to the upper portion; when the upper part amounted to ten, then two jettons were taken from it and one was put in the lower portion of the next higher decimal column , and so on. So in the case of units with another radix, such as twelve; then the lower portion of the column was for units up to six, and the upper portion for sixes up to twelve. In practice this is much superior J. Eric S. Thompson (1942) has set out very fully how the Maya might have used an abacus for calculating and he has given a diagram of such an abacus. Although I am in general agreement with his paper, I think that no abacus was used and that it is useful to consider the methods of reckoning used in Europe and very fully given by Francis P. S. Barnard (1916) with illustrative diagrams. He abstracts information given in ten arithmetical works of the sixteenth century . All give the method of calculation “by ciphers,” that is, our ordinary arithmetical rules, but they also give the method of calculation by counters. The counters were called “jettons,” and it will be convenient to use that term here. It was usual to have sets of jettons, which were made of silver or other metals, like coins or medals, and had ornamental designs, But as far as the theory of calculation is concerned, of course, any counters such as stones or grains of corn would do just as well. All the writers cited by Barnard consider the method by ciphers much superior to that by jettons. It was impossible to calculate “by ciphers” as long as the Roman numerals were used; the Arabic numerals came into general use only shortly before these works were written. One of the authors, Siliceus, who like Landa was a Spanish bishop of the sixteenth century, says “the reckoning by jettons is very useful to all persons deficient in a knowledge of written ciphering , as usually are merchants, money-changers, retail dealers, inn-keepers, and numerous others of low estate, besides such as through their own negligence or mental incapacity are unable to read, as is the case with many people of all conditions.” It is a striking instance of culture change that the memory of such Some Remarks on Maya Arithmetic 351 to a purely decimal arrangement, as it requires fewer jettons and there is less chance of error in counting them. Thompson’s abacus has only one column for each vigesimal order of units (as well as the special column for the 18 uinals of the tun). This is even more troublesome in a vigesimal than in a decimal...

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