]_]_ 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.