THE PERUVIAN QUIPU Author(s): Lind Mae Diana Source: The Mathematics Teacher, Vol. 60, No. 6 (OCTOBER 1967), pp. 623-628 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27957641 Accessed: 01-11-2015 15:03 UTC

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions HISTORICALLY SPEAKING?

Edited byHoward Eves, University ofMaine, Orono, Maine

THEPERUVIAN QUIPU

By Lind Mae Diana, Philadelphia, Pennsylvania

mne MAN, having curiosity as one of his or the place of the Incan quipu (a basic characteristics, has long desired to monic device used to keep record of com unlock the secrets of those who have gone putations) within that system. All that the before him in time. Thus, he has embarked inquirer finds are generalities, which do on archeological expeditions, has studied not pinpoint specific information. history books, has consulted ancient However, using some of the hints given sources of information, and has generally to us by the aforementioned chroniclers, set about looking for the answers to his research studies have been carried out by men an to discover what questions. several in effort Mathematicians, possessing that same mathematical knowledge was known to curiosity, have been no different from their the Incas and what importance the Peru fellow beings. They, too, have searched vian quipu enjoyed in that mathematics. and traveled, and from their efforts have Two such scientists are Leland Locke and emerged hundreds of books written on the Erland Nordenski?ld. historical foundations of modern mathe The remainder of this discussion will be matics. Especially has the greatest effort centered around the results of these stud been concentrated in the eastern hemi ies. The reader, however, should note that sphere (i.e., Europe, Africa, and Asia). many of the conclusions reached by these are The logical question, then, concerns the men (especially by Nordenski?ld) is no apparent lack of books containing infor nothing more than beliefs. There mation as to the nature of the mathemat definite way to prove the truth of them, ics of the Americas?that is, of the original even though they are based on long and inhabitants of North and South America. rigorous studies. The answer is twofold. First, there really Before discussing the intricacies of the neces was a lack of specialized mathematics in Incan mathematical system, it is with these geographical areas; and secondly, sary to note that, in comparison many of the secrets of ancient American other cultural achievements of the civili mathematics have been, and are being, zation, the field of mathematics was rela carefully guarded. tively barren. For example, the Incas Such is apparently the situation when lacked the complex computational system somewhat one inquires into the nature of the mathe of theMayans ofMexico ;being were matical system of the Incas of Peru. like the ancient Babylonians, they Many allusions to the subject have been primarily interested in keeping records, methods made by a great number of the Spanish rather than developing elaborate chroniclers whose works are still preserved of computation. we at the of our today. Yet we find that no distinct literary Thus, arrive subject The work has been completely dedicated to the discussion: the Peruvian quipu. quipu Incas to record subjects of ancient Peruvian mathematics was the device used by the

Historically speaking? 623

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions results of various kinds of mathematical into the central octagonal area, its value problems. We have proof that quipus were was tripled. The value of the pebble used not only in Peru but also in other would be multiplied by six if put on the areas of South America.1 Most of those first of the upper levels; if it were placed that we now use for study were found in on the uppermost level, its Value would dry graves along the coastal areas of Peru. be multiplied by twelve. The color of the This is especially true of those used by pebbles used indicated the nature of the Mr. Locke and Mr. Nordenski?ld. objects being counted. Since this method The Incas were not especially adroit in of computation was efficient in arriving advanced mathematical computation. They at results but did not yield a permanent were much more interested in recording record, the Incas (who most likely bor the results of the yearly census and in rowed this abacus from the Canaris of keeping an account of the number of sheep Ecuador) felt a need for another instru they had in a herd. Their mathematical ment, which could be used not as a cal achievements were not great, but their culating device but as a numerical record method of keeping these statistics, which keeper.3 they apparently valued so highly, was The quipu fulfilled this need. A first ingenious and still intrigues many twen look at a quipu might yield the impres tieth-century scholars. sion that it is nothing more than a knot But exactly how did they compute the results they recorded on the quipus? An abacus was used, but not of the same type as came into such widespread use, during the time of the Roman Empire, in the Mediterranean region of the world. In stead of the wires and beads of the more conventional one, the Incan abacus con sisted of a rectangular slab of stone into which were cut a number of rectangular and square compartments so that a free space was left in the middle, octagonal Fig. 1.?Incan Abacus and two opposite corner rectangles were were raised. Another two sections mounted ted cord with no special significance, but on the originally raised portion such that this is not the case. It is true that a quipu there were now three levels represented. is a collection of cords in which knots An overhead view would thus be depicted have been tied, but these are definitely areas by Figure l2 with the darkened de and systematically arranged to have spe noting raised portions. cial significance to whoever embarks on Pebbles were used to keep the accounts, the task of deciphering it. rec and their positions within the various The quipu consists of a main cord to tangles and squares that composed the in which branches (other cords) are attached strument yielded the total. For example, like a fringe. These branches have been was one a pebble in a small square unit; named H cords by Locke. They are when itwas put into a rectangular one, its fastened to the main cord in groups, and value was doubled; and when it was put many times these groups contain equal numbers of H cords. To the H cords are attached still other cords, these being 1Louis A Socialist The Incas Baudin, Empire: called cords. Any additional ones are of Peru (New York: D. Van Nostrand Co., 1961), p. 124. 2 3 Ibid., p. 125. Ibid., pp. 124-25.

624 The Mathematics Teacher | October 1967

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions labeled cords of the second order (see immediately above them, then the hun Fig. 2). dreds, thousands, and ten thousands. Let us illustrate exactly how the Incas Mik) Cord used the quipu by the following example. A native desires to keep an account of the number of animals in his herds. He uses his abacus and arrives at the fact that he has 10 sheep, 30 llamas, 3 dogs, and 100 goats, giving him a total of 143 animals. To construct his quipu he would begin with a horizontal cord (main cord) from which he would suspend five H cords (see Fig. 3). On the first (left) cord he would place one overhand knot for the 10 sheep he has; on the second, three overhand knots would tally the number of llamas; the third would boast a long knot with three twists for the 3 dogs; and the fourth Fig. 2.?Quiptj cord would have one overhand knot tied The knots themselves are seldom pres in it and placed in the appropriate spot to ent on the main cord. Instead, they are denote 100 goats. The fifth cord is the concentrated on the and cords. As to totalizer cord, and itwould contain knots it has been noted the nature of the knots, Main Cord by Mr. Locke that there are three main types of knots. These represent different values, depending on the particular knot used in a particular position on the cord in question. The Incas used a decimal arithmetical system much like ours of to day. It is interesting to note that, like ours, their system was based on ten and was a place system. Thus, on the quipu each type of knot that was used had a Fig 3.?Quipu specific decimal value. There were so called overhand knots, simple single knots representing the total number of animals in the cord. These represented tens, hun in the herd, 143.4 dreds, thousands, and ten thousands. The The colors of the cords are significant Incas rarely used numbers greater than and indicate that the 10 registered on the were ten thousand. Then there "Flemish quipu stands for sheep and not llamas. to knots," S-shaped configurations used Thus, each thing to be counted was as were denote "one." All other integers de signed a color to help avoid confusion. picted by "long knots" in which the But what was done when the Inca ran out specific value could be determined by of colors? It is evident that colors had to counting the number of times the cord be reused but with different significance. was wound before tying the knots. "Long As a result, the deciphering of the quipu knots" were also used at times to represent became an intricate art, and a specially tens and hundreds. In such cases the trained individual was needed to do the on would position of the knots the cord job. He was the Quipucamayoc (keeper of denote the value. The units were placed * nearest the bottom, the tens being placed Ibid., p. 126.

Historically speaking? 625

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions the quipus), and his job was the con when an Inca wanted to confess to a struction, preservation, and decoding of Spanish priest, he did so by means of a the quipu.5 quipu on which were written all of his in The purposes of the quipu were nu fringements of church law.8 merous, with authorities agreeing and dis It has also been suggested that one of agreeing vehemently concerning specific the purposes of the quipu was the register uses. ing of astronomical and magic numbers. Mr. Mason asserts that the Incas used Erland Nordenski?ld's extensive research the quipu to keep an account of their his has resulted in the promulgation of this tory and traditions. In short, he says, theory. The quipus he used for his re "... they never developed any system of search were solely those that came from writing?pictographic, ideographic, hiero dry coastal graves; and all were in excel glyphic, or alphabetic."6 Their history was lent condition, the majority being perfect not written down; it was tied into the specimens. It should be noted before dis many knots that composed the quipu. cussing the issue any further that Norden In accord with this view of the purpose ski?ld's opinions are not wholly accepted of the quipu, we find an interesting refer by modern authorities, but they do plant ence in the Comentarios Reales written by seeds of speculation in the reader's mind. El Inca Garcilaso de la Vega. He sug Nordenski?ld believed, even before con gested that the quipu was indeed used as sulting the quipus he used for his study, a replacement for writing because it that they were not accounts of historical formed a basic part of the Incan postal events or compilations of figures that cor system. He says, responded to the yearly census. He based this on the fact that Otros recaudos llevaban, no de palabra, sino presumption prim por escrito, dig?moslo as?, aunque hemos dicho itive peoples usually believe that en no tuvieron las cuales eran ?udos dados que letras, tombing anything containing information en hilos de diversos colores. . . . Los diferentes about the is likened to ?udos y los colores de los hilos significaban el living entombing n?mero de gente, armas, o vestidos, o bastimiento, the living themselves. As a result, he con o otra cosa se hubiese de cualquiera que hacer, jectured that the numbers recorded on the enviar o aprestar. A estos hilos a?udados ... 7 be connected with of llamaban los indios quipu quipu might magic some sort. This conviction was further were carried not word [Other messages by when he stumbled Ber but in writing. We say this although we have strengthened upon said that they (the Incas) had no writing. This tonio's statement of the preoccupation of was of knots of different writing composed the Incas with magic and counting. colors on different cords. . . . The knots and the colors of the cords signified the number of men, Adevinar tomando un pu?ado de algo y con arms, dresses, supplies, or whatever that they tando despu?s los granos. Contar una almo?ada de had to make, send, or prepare. These knotted papas para saber si el a?o ser? buena. Es gran . . . cords were called quipu ] superstici?n.9

future events a Another interesting use of the quipu [Foretell by taking handful of something and counting the grains. Count a was by Herrera. He says that proposed sack of potatoes to know if itwill be a good year after the conquest of the Incas by the or not.] Spaniards and the conversion to Catholi Upon examination of a group of quipus, cism, there still remained a communication he did discover a predominance of the problem between the two groups because number 7, not as a number present on the of the difference in language. As a result, 8 5 "Quipos," Enciclopedia Universal Ilustrada J. Alden Mason, The Ancient Civilizations of Europeo Americana (Madrid: Espasa-Calpe S.A., Peru 228. (Baltimore: Penguin Books, 1957), p. 1958), pp. 1417-18. ? p. 226. 9 Ibid., Nordenski?ld, The Secret of thePeruvian 7 Quipus Comentarios Reales (Buenos Aires: Espasa (Sweden: Comparative Ethnographical Studies, 1925), Calpe S.A., 1961), pp. 65-69. VI, 9.

626 The Mathematics Teacher | October 1967

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions lo en la memoria a los cords themselves, but as a divisor of the pasado y empap?ndolo av?an de maravilla se olvida totals on cords and as the que entrar, que por registered single ban cosa por peque?a que fuese ten?an en estos last digit of a number that was a divisor quipos que cassi son a modo de pavilos con que las re?an en nuestra salvo ser ramales of the totals registered on neighboring biejas Espa?a ten?an tanta en los luna cords. Since this number so often quenta a?os, messes, y appeared de tal suerte que no av?a lunar, luna, a?o, ni in these divisors and since Bertonio stated mes aunque no con tanta pulcia como despu?s a se?orear con that the Incas connected magic with que Ynga Yupanqui empe?o y esta hasta entonces los Nordenski?ld concluded that quistar tierra; porque counting, yngas no av?an salido de los alrededores del the number 7 must have been considered Cuzco. Como por la relacti?n que Va. Sa. Rma. este ?l as possessing mystical value.10 tiene, parece ynga fue primero que empe?? a poner y raz?n en todos las cossas y el The studied also were shown to quenta quipus que quito cultos y discultos y cerimonias que en contain astronomical numbers. An ex cada uno dellos hacen, porque no obstante que antes sus ten?an meses ample is the first quipu Nordenski?ld que reynasen antecesores, y a?os sus no se con tanto examined. like the showed a por quipos, reg?an It, others, concierto como despues que este fue se?or que se predominance of numbers that contained reg?an por los ynviernos y veranos.11 7 or numbers whose divisors ended in 7. [Notwithstanding the fact that they used in a But it also contained total equal to a very subtle manner some strings of wool with two in 1,172, which in turn equals 2 X 586. knots and that they put colored wool the knots that called used them Nordenski?ld reasoned that these num they quipus, having and using them now in acknowledging for more bers represented days and that this total than 500 years all of the things that have was to denote the period of two synodic happened in this land, some industrious and wise Indians had some quipus that they passed revolutions of Venus, which is equal to from generation to generation. These contained 2 X 584 days. The four-day discrepancy every little thing that had happened and were was explained away as being added so that such good accounts of the years, months, and moons that there was not a the difference between three of our solar single moon, year or month that was not included. But after two revo years (3 X 365) and synodical Ynga Yupanqui began to conquer the land, he lutions of Venus would be a magical began to construct accounts of everything. And even before he became number?in this case 77. though ruler, they had recorded months and years on their quipus, This is one the particular example of now they included winters and summers too.] reasons why experts believe Nordenski?ld Armed with the above information, it is has looked for relationships and therefore no wonder that Nordenski?ld felt certain forced a discovery of what he desired. His that some of the numbers registered on findings cannot be completely refuted, the he studied must have been though, because other quipus he studied quipus when he did en did show forth a prevalence of astronom astronomical?especially counter them too. ical numbers. himself, One studied by Nordenski?ld was The Spanish chroniclers themselves quipu of an odd circular It yielded num have stated the relationship between the shape. bers having to do with the types ofmonths quipus and the Incas' interest in heavenly the Incas used. He notes that the sum of bodies. the numbers on this specimen More exactly, Molina states: depicted equals 3,481 or 118 X 29.5. Also, the total No obstante que ussavan de una quenta muy on the cords of one subtil de unas ?bras de lana de dos ?udos y of the numbers all puesta lana de colores en los ?udos los quales color results in a number divisible by 30. llaman entend?anse y enti?ndense tanto quipos The total on the remaining colors is por esta quenta que dan raz?n de m?s de quinientos divisible 29.5. Over and over a?os de todas las cossas que en esta tierra en by again, este tiempo an passado: ten?an yndios yndustri examination of various other specimens ados y maestros de los dichos y y quipos quentas yields the same results: years of 365 days estos yban de generaci?n en generaci?n mostrando

10 " Ibid., pp. 5-37. Ibid., pp. 10-11.

Historically speaking? 627

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions divided into months of either 29.5 or 30 these computations were then transferred days each.12 to the quipu by the Quipucamayoc, who Therefore, based on the previous dis was a specialist in his field. The purposes cussion, Nordenski?ld concluded that the of this mnemonic device were varied: to quipus used by the Incas and examined by record historical events, to keep statistical him contained: (1) the number 7 com records, to aid in the confessions of Incan bined in various forms and often repre converts, to record magic numbers, and sented, (2) numbers that expressed days, perhaps even to record astronomical data. and (3) astronomical numbers. Much additional research needs to be In conclusion, one should fully be aware undertaken. The possibilities of further of the fact that the secrets of the Peruvian discovery are still present for those who quipus have not yet been completely un are intrigued by the mysteries that are locked. This is clearly demonstrated when tied into the quipu. we note that Nordenski?ld's study oc in little curred 1925, and progress has ADDITIONALREFERENCES been made since then. In summation, we do know that the Incas counted in groups Bushnell, G. H. S. Peru. New York: Frederick A. Praeger, 1960, pp. 125-28. of ten, used a decimal placement system, Cieza de Leon, Pedro De. The Incas. Norman, and on the abacus. The results of figured Okla.: University of Oklahoma Press, 1959, pp. 39-40, 105, 163, 172-75, 177, 187. 12 William H. the Nordenski?ld, Calculations with Years and Prescott, History of Conquest Months in thePeruvian Quipus (Sweden: Comparative of Peru. Philadelphia: J. B. Lippincott Co., Ethnographical Studies, 1925), II, Part 2, 5-34. 1847, pp. 122-26.

A TEXTON TRIGONOMETRY BYLEVI BEN GERSON (1288-1344)

By Pamela H. Espenshaole, Princeton Day School, Princeton, New Jersey

LEVI ben Gerson (1288-1344), a prom was elected May 7, 1342, it is probable inent Hebrew writer who lived in Pro that the work was undertaken forBenedict vence, is most noted for his religious and XIII (1334-1342).1 The Latin version of philosophical writings. Nonetheless, his the table of contents was published by astronomical works, although largely un Renan in 1843.2 studied, are quite imposing and volumi A section of these astronomical writings nous. His chief astronomical work, con was circulated separately under the title tained inWars of theLord, Book V, Part 1 of De Sinibus, Chor?is et Arcubus, item (136 chapters), was translated into Latin 1 by the Augustinian hermit Peter of Alex L. Thorndike, A History ofMagic and Experi mental Science, pp. 309-11. andria in 1342 and dedicated to Pope 2 E. Renan, "Les ?crivains juifs fran?ais du XIV Clement VI. Since, however, Clement VI si?cle," pp. 633 ff.

628 The Mathematics Teacher | October 1967

This content downloaded from 132.174.254.159 on Sun, 01 Nov 2015 15:03:34 UTC All use subject to JSTOR Terms and Conditions