]_]_

The Peruvian Qyipu

LIND MAE DIANA

c.7\ll"AN, having curiosity as one of his basic Peruvian mathematics or the place of the lncan. characteristics, has long desired to unlock the se­ quipu (a mnemonic device used to keep record of crets of those who have gone before him in time. computations) within that system. All that the Thus, he has embarked on archeological expedi­ inquirer finds are generalities, which do not pin­ tions, has studied history books, has consulted point specific information. ancient sources of information, and has generally However, using some of the hints given to us set about looking for the answers to his questions. by the aforementioned chroniclers, research stud­ Mathematicians, possessing that same curios­ ies have been carried out by several men in an ity, have been no different from their fellow be­ effort to discover what mathematical knowledge ings. They, too, have searched and traveled, and was known to the Incas and what importance the from their efforts have emerged hundreds ofbooks Peruvian quipu enjoyed in that mathematics. Two wrirten on the historical foundations of modem such scientists are Leland Locke and Erland mathematics. Especially has the greatest effort been Nordenskiold. concentrated in the eastern hemisphere (i.e., Eu­ The remainder of this discussion will be cen­ rope, Africa, and Asia). The logical question, then tered around the results of these studies. The reader, concerns the apparent lack of books containing however, should note that many of the conclusions information as to the nature of the mathematics of reached by these men (especially by Nordenskiold) the Americas-that is, of the original inhabitants are nothing more than beliefs. There is no definite of North and South America. The answer is two­ way to prove the truth of them, even though they fold. First, there really was a lack of specialized are based on long and rigorous studies. mathematics in these geographical areas; and sec­ Before discussing the intricacies of the lncan ondly, many of the secrets of ancient American mathematical system, it is necessary to note that, mathematics have been, and are being, carefully in comparison with other cultural achievements of guarded. the civilization, the field of mathematics was rela­ Such is apparendy the situation when one in­ tively barren. For example, the Incas lacked the quires into the nature of the mathematical system complex computational system of the Mayans of of the Incas of Peru. Many allusions to the subject Mexico; being somewhat like the ancient have been made by a great number of the Spanish Babylonians, theywere primarily interested in keep­ chroniclers whose works are still preserved today. ing records, rather than developing elaborate meth­ Yet we find that no distinct literary work has been ods of computation. completely dedicated to the subjects of ancient Thus, we arrive at the subject of our discus­ sion: the Peruvian quipu. The quipu was the de­ vice used by the Incas to record results of various Reprinted from Mathematics Teacher 60 (Oct., 1967): 623-28; with permission of the National Council of Teachers of kinds of mathematical problems. We have proof Mathematics. that qui pus were used not only in Peru but also in

80 THE PERUVIAN QUIPU

other areas of South America.' Most of those that squares that composed the instrument yielded the we now use for study were found in dry graves· total. For example, a pebble in a small square was along the coastal areas of Peru. This is especially one unit; when it was put into a rectangular one, true of those used by Mr. Locke and Mr. its value was doubled; and when it was put into the Nordenskiold. central octagonal area, its value was tripled. The The Incas were not especially adroit in ad­ value of the pebble would be multiplied by six if vanced mathematical computation. They were put on the first of the upperlevels; if it were placed much more interested in recording the results of on the uppermost level, its value would be multi­ the yearly census and in keeping an account of the plied by twelve. The color of the pebbles used nwnber of sheep they had in a herd. Their math­ indicated the nature of the objects being counted. ematical achievements were not great, but their Since this method of computation was efficient in method of keeping these statistics, which they arriving at results but did not yield a permanent apparendy valued so highly, was ingenious and record, the Incas (who most likely borrowed this still intrigues many twentieth-century scholars. abacus from the Canaris of Ecuador) felt a need But exacdy how did they compute the results for another instrument, which could be used not as they recorded on the quipus? An abacus was used, a calculating device but as a numerical record but not of the same type as came into such wide­ keeper.' spread use, during the time of the Roman Empire, The quipu fulfilled this need. A first look at a in the Mediterranean region of the world. Instead quipu might yield the impression that it is nothing of the wires and beads of the more conventional more than a knotted cord with no special signifi­ one, the lncan abacus com;isted of a rectangular cance, but this is not the case. It is true that a quipu slab of stone into which were cut a number of is a collection of cords in which knots have been rectangular and square compartments so that a tied, but these are definitely and systematically free octagonal space was left in the middle, and arranged to have special signifiqmce to whoever two opposite comer rectangles were raised. An­ embarks on the task of deciphering it. other two sections were mounted on the originally The quipu consists of a main cord to which raised portion such that there were now three branches (other cords) are attached like a fringe. levels represented. An overhead view would thus be depicted by Figure 12 with the darkened areas FIGURE 2 Quipu denoting raised portions. Pebbles were used to keep the accounts, and Main Cord their positions within the various rectangles and

FIGURE I Inc an Abacus

H Cords

'

I BCords

81 PART n/IN THE BEGINNING ... I These branches have been named H cords by FIGURE 3 Quipu I Locke. They are fastened to the main cord in Main Cord groups, and many times these groups contain equal numbers of H cords. To the H cords are attached still other cords, these being called B cords. Any additional ones are labeled B cords of the second I order (see Fig. 2). H The knots themselves are seldom present on Cords I the main cord. Instead, they are concentrated on the Hand B cords. As to the nature of the knots, it has been noted by Mr. Locke that there are three main types of knots. These represent different values, depending on the particular knot used in a has; on the second, three overhand knots would particular position on the cord in question. The tally the number ofllamas; the third would boast a Incas used a decimal arithmetical system much long knot with three twists for the 3 dogs; and the like ours of today. It is interesting to note that, like fourth cord would have one overhand knot tied in ours, their system was based on ten and was a place it and placed in the appropriate spot to denote 100 system. Thus, on the quipu each type of knot that goats. The fifth cord is the totalizer cord, and it was used had a specific decimal value. There were would contain knots representing the total num­ so-called overhand knots, simple single knots in ber of animals in the herd, 143.4 the cord. These represented tens, hundreds, thou­ The colors of the cords are significant and sands, and ten thousands. The Incas rarely used indicate that the 10 registered on the quipu stands numbers greater than ten thousand. Then there for sheep and not llamas. Thus, each thing to be were "Flemish knots," S-shaped configurations used counted was assigned a color to help avoid c'onfu­ to denote "one." All other integers were depicted sion. But what was done when the Inca ran out of by "long knots" in which the specific value could colors? It is evident that colors had to be reused be determined by counting the number of times but with different significance. As a result, the the cord was wound before tying the knots. "Long deciphering of the quipu became an intricate art, knots" were also used at times to represent tens and a specially trained individual was needed to do and hundreds. In such cases the position of the the job. He was the Quipucamayoc (keeper of the knots on the cord would denote the value. The qui pus), and his job was the construction, preser­ units were placed nearest the bottom, the tens vation, and decoding of the quipu.' being placed immediately above them, then the The purposes of the quipu were numerous, hundreds, thousands, and ten thousands. With authorities agreeing and disagreeing vehe­ Let us illustrate exactly how the Incas used the mendy concerning specific uses. quipu by the following example. A native desires Mr. Mason asserts that the Incas used the to keep an account of the number of animals in his quipu to keep an account of their history and herds. He uses his abacus and arrives at the fact traditions. In short, he says, " ... they never devel­ that he has 10 sheep, 30 llamas, 3 dogs, and 100 oped any system of writing-pictographic, ideo­ goats, giving him a total of 143 animals. To con­ graphic, hieroglyphic, or alphabetic."' Their his­ struct his quipu he would begin with a horizontal tory was not written down; it was tied into the cord (main cord) from which he would suspend many knots that composed the quipu. five H cords (see Fig. 3). On the first (left) cord he In accord with this view of the purpose of the would place one overhand knot for the 10 sheep he quipu, we find an interesting reference in the

82 THE PERUVIAN QUIPU

Comentarios Reales written by El Inca Garcilaso de figures that corresponded to the yearly census. He la Vega. He suggested that the quipu was indeed based this presumption on the fact that primitive used as a replacement for writing because it formed peoples usually believe that entombing anything a basic part of the Incan postal system. He says, containing information about the living is likened to entombing the living themselves. As a result, he Otros recaudos !levaban, no de palabra, sino por escrito, digdmoslo asi, aunque hemos dicho que no tuvieron conjectured that the numbers recorded on the quipu letras, las cuales eran fiudos dados en diferentes hilos de might be connected with magic of some sort. This diversos co/ores. ... Los izudos y los co/ores de los hz1os conviction was further strengthened when he significaban el nrimero de gente, armas, o vestidos, o stumbled upon Bertonio's statement of the preoc­ bastimiento, o cualquiera otra cosa que se hubiese de cupation of the Incas with magic and counting. hacer, enviar o aprestar. A estos hilos afzudados llamaban los indios qu£pu ... 7 Adevinar to man do un pufzado de alga y contando despuis [Other messages were carried not by word but in los granos. Contar una almo;ada de papas para saber si 9 writing. We say thls although we have said that they el afio serd buena. Es gran supersticitfn. (the Incas) had no writing. This writing was com­ [Foretell future events by taking a handful of some­ po'sed ofknots of different colors on different cords. thing and counting the grains. Count a sack of ... The knots and the colors of the cords signified potatoes to know if it will be a good year or not.] the number of men, arms, dresses, supplies, or what­ ever that they had to make, send, or prepare. These knotted cords were called quipu ... ] Upon examination ofa group of qui pus, he did discover a predominance of the number 7, not as a Another interesting use of the quipu was pro­ number present on the cords themselves, but as a posed by Herrera. He says that after the conquest divisor of the totals registered on single cords and of the Incas by the Spaniards and1:he conversion to as the last digit of a number that was a divisor of Catholicism, there still remained a communica­ the totals registered on neighboring cords. Since tion problem between the two groups because of this number appeared so often in these divisors the difference in language. As a result, when an and since Bertonio stated that the Incas connected Inca wanted to confess to a Spanish priest, he did magic with counting, N ordenskiold concluded that so by means of a quipu on which were written ali the number 7 must have been considered as pos­ of his infringements of church law.' sessing mystical value.10 It has also been suggested that one of the The quipus studied also were shown to contain purposes of the quipu was the registering of astro­ astronomic~ numbers. An example is the first nomical and magic numbers. Erland Norden­ quipu Nordenskiold examined. It, like the others, skiOld's extensive research has resulted in the pro­ showed a predo-minance of numbers that contained mulgation of this theory. The quipus he used for 7 or numbers whose divisors ended in 7. But it also his research were solely those that came from dry contained a total equal to 1,172, which in turn coastal graves; and all were in excellent condition, equals 2 X 586. Nordenskiold reasoned that these the majoriry being perfect specimens. It should be numbers represented days, and that this total was noted before discussing the issue any further that to denote the period of two synodic revolutions of Nordenskiold's opinions are not wholly accepted Venus, which is equal to 2 x 584 days. The four­ by modem authorities, but they do plant seeds of day discrepancy was explained away as being added speculation in the reader's mind. so that the difference between three of our solar N ordenskiold believed, even before consulting years (3 X 365) and two synodical revolutions of the quipus he used for his study, that they were not Venus would be a magical number-in this case accounts of historical events or compilations of 77.

83 PART II liN THE BEGINNING ..•

This particular example is one of the reasons after Ynga Yupanqui began to conquer the land, he why experts believe N ordenskiold has looked for began to construct accounts of everything. And even though before he became ruler, they had recorded relationships and therefore forced a discovery of months and years on their quipus, now they in­ what he desired. His findings cannot be com­ cluded winters and summers too.] pletely refuted, though, because other quipus he studied did show forth a prevalence of astronomi­ Armed with the above information, 1t IS no cal numbers. wonder that Nordenskiold felt certain that some The Spanish chroniclers themselves have stated of the numbers registered on the quipus he studied the relationship between the quipus and the Incas' must have been astronomical-especially when he interest in heavenly bodies. did encounter them himself, too. More exacdy, Molina states: One quipu studied by Nordenskiold was of an No obstante que ussavan de una quenta muy subtil de odd circular shape. It yielded numbers having to unas ebras de lana de dos fiudos y puesta lana de co/ores do with the types of months the Incas used. He en los fiudos los quales llaman quipos entendianse y notes that the sum of the numbers depicted on this entiindense tanto por esta quenta que dan raz:.On de mds specimen equals 3,481 or 118 X 29.5. Also, the de qut"nientos afios de todas las cossas que en esta tierra total of the numbers on ali the cords of one color en este tiempo an passado: fenian yndios yndustriados y maestros de los dichos quipos y quentas y estos yban de results in a number divisible },Y 30. The total on generaciOn en generaciOn mostrando lo pasado y the remaining colors is dlvisil:Jle by 29.5. Over and empapdndolo en Ia memoria a los que avian de entrar, over again, examination of various other speci­ que por maravilla se olv£daban cosa por pequefia que mens yields the same results: years of 365 days fuese fenian en estos quipos que cass£ son a modo de divided into months of either 29.5 or 30 days pavilos con que las biejas recan en nuestra Espana salvo 12 ser ramales fenian tanta quenta en los afios, messes, y each. luna de tal suerte que no avia lunar, luna, afio, ni mes Therefore, based on the previous discussion, aunque no con tanta pulc£a como despuis que Nordenskiold concluded that the quipus used by Ynga Yupanqui empe;o a deiiorear y conqu£star the Incas and examined by him contained: (1) the esta tierra; porque basta entonces los yngas no avian number 7 combined in various forms and often salida de los alrededores del Cuzco. Como par Ia relactiOn represented, (2) numbers that expressed days, and que Va. Sa. Rma. tiene, parece este ynga foe eJ primero que empe;6 aponer quenta y razon en todos las cossas y el (3) astronomical numbers. que quito cultos y discultos y cerimonias que en cada In conclusion, one should fully be aware of the uno dellos hacen, porque no obstante que antes que fuct that the secrets of the Peruvian quipus have reynasen sus antecesores, fenian meses y aiios por sus not yet been completely unlocked. This is clearly quipos, no se regian con tanto concierto como despues demonstrated when we note that Nordenskiold's que este foe seiior que se regian por, los ynviernos y veranos. 11 study occurred in 1925, and little' progress has been made since then. In summation, we do know [Notwithstanding the fact that they used in a very that the ln~as counted in groups of ten, used a· subtle manner some strings ofwool with two knots and that they put colored wool in the knots that decimal placement system, and figured on the aba­ they called quipus, having used them and using cus. The results of these computations were then them now in acknowledging for more than 500 transferred to the quipu by the Qyipucamayoc, years all of the things that have happened in this who was a specialist in his field. The purposes of land, some industrious and wise Indians had some this mnemonic device were varied: to record his­ quipus that they passed from generation to genera­ torical events, to keep statistical records, to aid in tion. These contained every little thing that had happened and were such good accounts of the years, the confessions of Inc an converts, to record magic months, and moons that there was not a single numbers, and perhaps even to record astronomical moon, year or month that was not included. But data.

84 r r.·.··. I THE PERUVIAN QUIPU

r

f Much additional research needs to be under­ Americana (Madrid: Espasa-Calpe S.A., 1985), taken. The possibilities of further discovery are pp. 1417-18. still present for tbose who are intrigued by tbe 9. N ordenskiold, The Secret of the Pernv£an Quip us mysteries that are tied into the quipu. (Sweden: Comparative Ethnographical Studies, 1925), VI, 9. 10. Ibid., pp. 5-37. I NOTES 11. Ibid., pp. 10-11. I 12. N ordenskiOld, Calculat£ons with Years and 1. Louis Baudin, A Socialist Empire: The Incas of Months in the Peruvian Quipus (Sweden: Comparative Peru (New York: D. Van Nostrand Co., 1961), Ethnographical Studies, 1925), II, Part 2, 5-34. p.124. 2. Ibid., p. 125. 3. Ibid., pp. 124-25. ADDITIONAL REFERENCES 4. Ibid., p. 126. Bushnell, G. H. S. Peru. New York: FrederickA Praeger, 5. J. Alden ~son, The Ancient Civilizations ofPeru 1960, pp. 125-28. (Baltimore: Penguin Books, 1957), p. 228. Cieza de Leon, Pedro De. The Incas. Norman, Okla.: 6. Ibid., p. 226. University of Oklahoma Press, 1959, pp. 39-40, 7. Comentarios Reales (Buenos Aires: Espasa-Calpe 105,163,172-75,177,187. S.A., 1961), pp. 65--69. Prescort, William H.Historyoftbe Conquest ofPeru Phila­ 8. "Quipos," Enclc!oped£a Un£versal Ilustrada Europeo delphia: J. B. Lippincort Co., 1847, pp. 122-26.

85