The Vigesimal System


As is well known, the Maya used the vigesimal system.  The unit of Maya calendrical calculations was the tun (approximate year of 360 days), the ascending units being the katun (360 days x 20), the cycle or baktun (360 days x 400) and the great cycle (360 days x 8000).  There is some evidence that time periods of 360 days x 160,000 and 360 days x 3,200,000 were occasionally used.  Remainders of less than a tun were expressed by the kin (day), and the uinal (20 days), but because the tun had to approximate in length the solar year, only 18, and not 20, uinals formed the tun.  The Quiche and Cakchiquel, however, maintained a straight vigesimal count from the day up, their approximate year being 400 days (20 months of 20 days), in contrast to the katun of 7200 days (20 approximate years of 18 months of 20 days) of the lowland Maya.

In Aztec picture writing, where the vigesimal system was also employed, symbols for the orders were a flag for 20, a conventionalized representation of hair or a tree for 400, and a copal pouch for 8000, these symbols being attached to the article enumerated.  Thus, 9245 bundles of quetzal feathers would be shown by drawings of eight bundles, one with a drawing of a copal pouch attached (8000), one with drawings of three hairs or trees (1200), one with two little flags (40) and five without any symbol attached (5).  THe corresponding Maya symbols are not known for certain, but a bundle of tobacco leaves is the name for 20 in the Guatemala highlands, and the 20-year period of the Cakchiquel and Quiche was called Mat, which has that meaning.  The word tun of the lowland Maya means in Yucatec jade or precious stone, and to a lesser extent, by extension, stone in general.  The number 1 is occasionally expressed by a drawing of a thumb, hinting that the numbers 1-20 were counted on fingers and toes.  The Maya, however, having evolved place notation and zero, could, and often did, dispense with symbols for their orders.  On the highly ritualistic stone monuments of the Old Empire they were usually retained,5 but in the codices, with rare exceptions, they are omitted, the orders being clearly indicated by their positions in ascending columns.  Presumably, when they counted “on the ground or on something level,” they did not emply symbols for their orders but utilized positional values, ascending, however, from bottom to top, not, as a rule, from right to left as in our Arabic orders.

Counters may have been grains of maize, beans, or pebbles.  Beans and maize were used in divination by numbers among both the Maya and Aztec,6 which suggests they may have been employed also in straight calculation.  Conceivably the numerical bar evolved from a bean pod.  In fact, grains of maize, beans or stones are still the aids to solving arithmetical problems among the Maya of the Guatemala highlands.  Dr. Sol Tax has kindly placed at my disposal notes made by his assistant, Rosales, on methods of counting at Panajachel on the shores of Lake Atitlan.  In a covering letter he adds:  “I have no reason to believe that Panajachel is unique in counting methods, but I do not have comparable information for other towns.  I recall that when a shaman of Chichicastenango was preparing to recite the calendar into the microphone for Andrade, he picked up 20 stones to aid him.  I have never heard of the abacus in Guatemala, but counting with both maize and beans seems to be very common.”  From the Panajachel notes the following is extracted:  “The Indians do their operations of adding and subtracting only with kernels of maize, beans, or with small stones if they do not have either of the former (this, however, is rare in an Indian house).  They always put groups of five kernels or beans in a row until they have twenty groups, or 100, and then they count the next hundred.  It is known that formerly groups were made up to 80.  If there were great quantities to be added, so many were taken from each groupd, and then they were all added together.  To subtract they first take the whole quantity and then take away the desired part.  Some use groups of ten.  To count days the people formerly used maize, and they say they were very exact because they made groups by weeks.  In January they could already say on what day a fiesta in which they were interested would fall at any time of the year, be it in July or any other month.  The record of their accounts was in keeping the counted kernels in little boxes, gourds or bags.”

In Peru also arithmetical calculations were made by means of grains of maize, as is attested to by Joseph Acosta.  He has left the following description, which doubtlessly would apply equally well to the Maya:  “It is an enchanting thing to see another kind of quipu which they work with grains of maize, since for a very difficult count, for which a good calculator would need pen and ink ... the Indians will take their grains [of maize].  They will place one here, three there, eight I don’t know where.  They will move one grain from here, bring three from there, and actually finish with their count absolutely correct without the smallest error.  And they are much better at calculating what every one should pay or give than we with pen and ink.”7

Henry Wassen, in an extremely interesting study, has uncovered evidence for the existence of a primitive abacus in ancient Peru.8  It is not improbable that the Maya also had a somewhat similar device.  Indeed, the system to be discussed approaches closely that of the abacus, and were only one kind of counter employed, eliminating the equivalent of the bar with value of 5, the Maya mathematicians could hardly have failed to evolve either the groove, rod, or Peruvian type.

As already noted, there are a considerable number of modern methods of calculating dates.  I myself first became interested in ancient methods of calculation through a combination of baggage phobia and dislike of drudgery.  In going into the field where inscriptions were to be expected, I was loath to carry with me a set of Goodman’s tables or books containing tables based on other systems.9  Shunning the tedium of copying such tables into a field notebook, I devised a method of rapic calculation without tables, which appeared to be the most foolproof and simple method that could be found.

As is well known, tables of 364 days are given on pages 32a, 45a, and 63-64 of the Maya Dresden Codex.  These have been interpreted as ritualistic,10 and it was not until recently when I rearranged my own system to eliminate all calculations other than those based on the 364-day period (henceforth called the computing year) that I realized how the Maya might well have utilized a similar system.  A re-examination of the tables in the Dresden Codex, and an analysis of mistakes in arithmetic in the inscriptions confirmed this belief.

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