Code of the quipu: a study in media, mathematics, and culture 9780472093250, 9780472063253

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Code of the Quipu

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Code of the Qui A Study in Media, Mathematics, and Culture

Marcia Ascher and | Robert Ascher

Ann Arbor The University of Michigan Press

To our parents

and the people |

of the book

Copyright © by The University of Michigan 1981 All rights reserved

Published in the United States of America by :

The University of Michigan Press and simultaneously in Rexdale, Canada, by John Wiley & Sons Canada, Limited Manufactured in the United States of America

Library of Congress Cataloging in Publication Data

Ascher, Marcia, 1935-— | Code of the quipu. Includes bibliographical references.

1. Quipu. 2. Mathematics, Inca. I. Ascher, Robert, 1931- joint author. II. Title.

F3429.3.Q6A82 1981 985'.01 80—25409

ISBN 0—472-—09325-8 ISBN 0—472-06325—1 (pbk.)

This book is about an artifact that became the medium for expression

in a major civilization. The medium, its content and its context—the Preface quipu in the Inca state—is our focus. But we see our work in a broader frame. There are limitations on anyone’s capacity to understand what is going on in another contemporary culture. The limitations are compounded when trying to understand what went on in a culture that was destroyed centuries ago. The effort forced us to question many previously held notions. In particular, we had to rethink some of our ideas about mathematics, media, and culture. And we were forced to reexamine how we and others codify the universe that surrounds us. We learned from the effort, and here, for the reader, we convey what we learned. The book does not assume any specialized knowledge. It is intended for anyone who is curious and willing to read participatively. Exercises and discussive answers are included in some of the chapters. Notes are keyed to chapter sections. They include source references and places for further reading about a topic, as well as some comments that we wanted to make but are tangential to the flow of the text. We do not describe all quipus in all their details. However, the book gives a very good sense of them. There are photographs of eight quipus; thirty are explicitly discussed in the context of ideas; and four are used in the exercises. Our interest in the problems posed by quipus started about ten years ago. At first, we worked from quipu descriptions published by others. Then, we started to examine specimens close to home. Our early work taught us a few things. The colored cotton or wool strings from which quipus were made could decompose even under good conditions. (We had the experience of touching a quipu string only to withdraw quickly as it collapsed into a small heap of dust. Fortunately, all quipus are not in such poor condition.) The published descriptions were too few and too variable to permit us to follow up some of our initial ideas. So, prior to further attempts at interpretation, we set out to locate, and to describe in a uniform fashion, as many quipus as we could. We eventually located Specimens in museums and private collections in South America, Europe, and North America. We asked permission to study and describe them, and we received it in all but one instance. We are indebted to those people who permitted us to study and describe the quipus that are the primary basis for our study. They are: Jorge C. Muelle, Edward Versteylen, Alicia Gamarra, Ana Maria Soldi, and Yoshitaro Amano (Lima, Peru); Oscar Ntfez del Prado (Cuzco, Peru); A. Pezzia (Ica, Peru); Percy Dauelsberg (Arica, Chile); José Maria Vargas (Quito, Ecuador); Mireille Simon-Abbat (Paris, France); Daniel

Schoepf and A. Jeanneret (Geneva, Switzerland); Gerhard Baer (Basel,

| Switzerland); Jacques J. Buffort (Rotterdam, Netherlands); Hans Becher (Hanover, Germany); Imina von Schuler and Dieter Eisleb (Berlin, West Germany); Otto Zerries (Munich, Germany); J. M. Pugh (London,

England); Michael Kan, Frederick Docksteader, Junius Bird, and Craig Morris (New York, New York); Clifford Evans (Washington, D.C.); Thomas C. Greaves (Philadelphia, Pennsylvania); Frank A. Norick and Lawrence E. Dawson (Berkeley, California); Christopher B. Donnan (Los Angeles, California); and Gordon R. Willey (Cambridge, Massachusetts). The list is a testimony to international cooperation in scholarship. In most cases, we had to travel to where the quipus are stored. The Wenner-Gren Foundation for Anthropological Research provided partial

assistance for our foreign journey. We thank the foundation, and in

particular its director, Lita Osmundsen. . The descriptions we recorded covered more than a thousand typed pages. Carol Mitchell of the University of Michigan Press understood the importance of making primary information available to any interested

person. The descriptions are now published (see the Authors’ Note). We thank Ms. Mitchell for her understanding. In the course of the decade of intermittant work, many people provided assistance of one kind or another. We gratefully acknowledge their

help. They are: David H. Abrahams, William B. Bergmark, Marjorie

Ciaschi, Denise Everhart, Ann Fairchild, Steven Frankel, Stephen Hilbert, Billie-Jean Isbell, Ruth H. Johnson, Lawrence Kaplan, Margaret LeBoffe, Mercedes L6épez-Baralt, Duncan Mason, Dorothy Menzel, Elias Mujica, Gerald Munchel, John V. Murra, Joan Oltz, Dottie Owens, William Pinkham, John H. Rowe, Mary Saintangelo, Max Saltzman, Steven Sayardar, D. J. Struik, Gary Wayne, and R. L. Wilder.

| | We appreciate the use of the photographs and figures that are not our own. These are: plate 2.4, Department of Anthropology, Smithsonian National Museum, Washington, D.C.; plates 3.2, 5.1, 5.2, 5.3, and 5.4, Museum fiir Vélkerkunde, West Berlin: plate 3.6, and our redrawn plan

| of part of Inty Pata (plate 3.7), are from Paul Fejos, Archaeological Explorations in the Cordillera Vilcambamba Southeastern Peru, Viking

Fund Publications in Anthropology, No. 3, Copyright 1944 by the Wenner-Gren Foundation for Anthropological Research, Inc.; plates 3.3 and 3.4, Institute of Andean Research, and American Museum of Nat-

ural History, respectively; plate 3.1, Fratelli Alinari, Florence, Italy; plate 4.1 is from J. B. Nies, Ur Dynasty Tablets, 1919, J. C. Hinrichs’sche Buchhandlung, Leipzig; and plates 4.2 and 4.3 are from Felipe Guaman

vi , CODE OF THE QUIPU

Université de Paris. |

Poma de Ayala, Nueva Cronica y Buen Gobierno, Institut d’ Ethnologie,

We hope that you enjoy the book.

Seal Harbor, Maine 1979

AUTHORS’ NOTE : The interpretations in this book draw primarily upon our Code of the Quipu Databook. It was published in 1978 by the University of Michigan Press and is available on microfiche from University Microfilms International, Ann Arbor, Michigan.

The databook contains descriptions of 191 quipus. These were obtained by first-hand study of specimens in museums and private collections spread over three continents. The introduction includes information

needed to understand the databook descriptions, the locations of all known quipus, and a complete bibliography of previously published descriptions. The descriptions in the databook are written in such a way, and in sufficient detail, so that any interested person can formulate his or her independent interpretations.

PREFACE : vii

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Contents 1 1 Odyssey 13 2 How to Make a Quipu

37 3 Inca Insistence 59 4 The Quipumaker 81 5 Format, Category, and Summation

109 6 Hierarchy and Pattern 133 7 Arithmetical Ideas and Recurrent Numbers

157 8 A Piece of String

- owe | Odyss ey

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Se ee OR ae a : Be PARE EARRAESS SES RIGS ORS OSU aA RAE OSS SR SA IE SR SA Oe OSS S SSE MEG SS cre Biotec Eh gcc lub tana tton ent tO le ROE SCS EE cc re ge Se = EEee EE ee? Hay stivitaapenye gest yi re oe aed EE a eee cits Shae Pata Dea deed iad bebeaSegndl es 8 ek aan ee S oy # : ah nee CAE re era nian ee a s : x i & : es oe ue Oe ue a ue a . : ‘ Be :

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Le ee CO

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meee sapES neNe anne ORES ee ER LEE A TNE ES aiatidaM dap Case as AORN TEE ace“EO SESE Swee Berek ee ees Sos Ra eeIA i aceee veinEES FUSES NTS SAC ECAR DL sige sata SUA SLA CA ICS OE TLeibiawningjelisigiicinag STRLUDN OESeee eC oh fo chug ggasi tiai CECE ea > “nhs ;AS " : 19 eee ee : aeeea err Ee Si ceaeeage, eR Ms RIKEN UES BiEPE saat CIE MICA NOL Silesia sbi ipices guiness 0 on. Wah : Aes Bee OASIS TEAR Ee eeseals EO ES GN Rn emia niin SI I AU ce:tin, i:UO cen caine newer oe ee SLO LLLa EI eeCL Pe Leia a gg ae me Ce eeDe EYERO EL : “Sis eee a See CEO ae shabsotvinasolgi ont sigiecnstloctsocts tages a petopeerehgro st woe thy pea BEES ORIN LOSS SOIL SERIE OAS LEE CEN? RSPAS SE ERS boat SPL S ev sN CE ECUD CUO CRT SASS EASR ITE NCEEE COTE CRETE LS DO EIERILT PSE ANT AGES! NEG SCESU On! Hho gAESES AGAIN TISES EAA ABA SAILS ESCA ADK. SEESE SRR ERL APSE OP Ea SEE TEC. SOLE: TiS CRORE eee TEE LESLIE CREES,

LE LI aa eeee eee nyee ee eea) LR Gy eR RGae sane catia... ena a Us EEE ele ee ee | acces eaecgeaeh eee ee ae sosacnnuncnnemrme nO TT BESBANE p eT OTe ISIE "EEESERGESS AIRE ORIOLES PEP OSIM ORI cage 90 OODLE ELE SITES SEA EE TESUIELSS BET? AS GEILE Gp IS MONS ILCS PLATESZOE SELIM HIDaS NETENME LE Nar EE IED PTETSERB OH ca te Eee i es cios iesnega tate peer ORCS Sie UOTE em Ei SEAMEN ODOM EGOD LE ICT es IME DEM OOo: ESSER IVR70000 USEARATEISE VILA OIE ES isIE SCE SCUSESRESSISIIS LED LALA RESTA EUUSES ESSE 2: tre 12.Sysminsiaste von Spanish understood about quipus is increased only a little by going to Cieza’s times and a summary of his other early writings, for Cieza understood more than his contemporar- Oe, se inidines ie found in pecs on

1es, and many later writers simply copied from him. Leon, Algunas Observaciones Sobre.

° . . . e eru aqdrid: p1iolnloteca NOManica . . the work and life work of Cieza.

The writings of Cieza seized and held our attention from the start of J ean o de Man “ eee 4 la e onica our studies. There are minor reasons and one major reason for this. Hispanica, 1973). This book also conCieza was in Inca territory only fifteen years after the conquest; this tains a rather complete bibliography on alone commends his work. He saw things that others who followed For English language readers who cannot have seen, and he spoke with people who were adults at the want to pursue what chroniclers other apogee of Inca power. In addition, Cieza is an able writer. His choice nan nlncc is the etn rote secon at the of vocabulary, his ability to put things concisely, his conscientious attempt the back of Leland L. Locke, The

. . . Cw YOrK: merican useum O

to weigh evidence, and his respect for the intelligence of the reader, set en Cue or Per uvian Knot Necore

him apart as much as does his being there before others. Natural History, 1923). This section

But the major appeal of Cieza centers on what leads us to use the contains translations from about fifteen word odyssey to describe his work. At one level, an Odyssey simply length from a few lines to several means a journey; that clearly applies to Cieza. In a broader sense, an hundred lines of print. Some other odyssey is a quest or a search that may or may not involve travel through i vole See for cxamole H Thiaboe. real space. It is important that the term odyssey in this second sense Quellen zur Kulturgeschichte des Prakoalso applies to Cieza. In literature, the odyssey is often a self-conscious fom S Sheoder Verlag. 1936, John V. search for the self. Cieza’s quest is different: he is not looking for him- Murra, “‘An Aymara Kingdom in

dod d ‘be hi kA 1 | d “mol early Spanish sources. They vary in

. . . 6s presents data al 18 Sal O have veen

self, but rather for the substance of the victims of conquest. That Cieza oe etary 0 G68): oe 1,

. we +3 umbers an

thought of the Incas as victims is clear; as he says, ‘*wherever the Span- originally recorded on a quipu; our ish have passed, conquering and discovering, it is as though a fire had cscussion of nis possible quip. is in gone, destroying everything it passed.’” Although he was a party to Andean Quipus,”” Archive for the Hisconquest, the victor is of less interest to him than the vanquished. tory of Exact Sciences 8 (1972): 288-320.

ODYSSEY 3

. *.?.**. #*

* e s e s when * *an sr iS j i

In most odysseys, the route is selected by the traveler. Not in this 3

case. Cieza was a common soldier, and as such he was more or less told en and where to move. The latter part of his journey interests us. It

, began in April, 1547, when he walked into the northern extreme of the , area once ruled by the Incas. Cieza was then twenty-nine years old. He

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moved along good roads, ones which, he says, were superior to the

| Roman roads he knew as a boy in Spain. Day after day, however wea, ried, he paused and recorded some of the things he heard and saw. He

stopped writing in September, 1550. As we follow Cieza on his quest for the Incas, keep in mind that he lived over four centuries ago and that he describes something that was very different from what he knew. We convey the sense of his vision as

we understand it. Even under the best circumstances, another culture is blurred as if seen through heavy gauze. Or, as Cieza put it, ““Peru

, and the rest of the Indies are so many leagues from Spain and there are ee so many seas between.

ee And as the traveler trudges through all this sand and glimpses the

Lr ee EEE eenOo ONO EOS in oO h d lhrf 5h y. ee Sd—*Fr OO an é 5rturave Hl Il RY lZfeant [éhRY ED CI ——— eee In the morning, Cieza set out from a place where he was soaking wet; SS asnightfall, es faa ee :where . ee a bd ee eee by he wasSe in a region it never rained.Ld To dobd this, he | traveled west and down fr om the mountains and then stopped when he -— . .. |.CO valleyEE evenSS though his heart rejoices ee Se GEfrom EEafar, NOSE SS : bdespecially hd . hgif.he ts OPE SORRENTO hos aN RR SR SEES SS FEE SSS ESS TE

Coe ea ea ee, So a oe Oe reached the eee ca Be desert nssands. eeeIn thehlatehafternoon h dof]another d day he went

Oe ee ee sout t rough t e esert e A ternately, but still hea ing south, he stayed

et, (ee resting nnev ee, PP ee 1 dde th O h the mount d d ; th n lo e de oes

oo ee Soe ee with mountains or the plains. ine roads that he followed can be visu-

ee ee eae zea aS a ladder ti 2Z ne enly upo the ground One side of the ee, a eno adder runsatthroug and 1s raised, lower side of ee through the desert the edgeains of the Pacific Ocean.the The crossbars

oe a a i the ladder go down through the valleys connecting the mountains with 3 Cl ae ee ee the desert and the sea. At its greatest, a side of the ladder, as Cieza TS pak SR CR SEUSS QU TORE FR ANRC ce! a CNG : Bc eae

ES NO Se€da ae eegee ount d t.(adoul 1 2005, |Mmues), b t 3 600 i] Th b M d ee rn ees ¢C it, was 1, cagues Lhe Crossbars vane aCS NS ee a ae okrom eeasaelittle esaseeoneeehundred ca fontwenty ae Sc eStofas] long 1 h asd three d il hundre hdd ee, ope miles

oe ee‘ .*: +. .‘ a a es VN rule

ee a. miles. The roads, built by the Incas, roughly defined the extent of their They observed the customs of their own people and dressed after

Ld *. ° * s

, the fashion of their own land, so that if there were a hundred thousand men, they could easily be recognized by the insignia

| | they wore about their heads. .

Cieza, the quester, was impressed by the diversity he found as he moved

along the road. He noticed changes in animals, crops, weather, and landscape. Mostly, he was struck by the differences in groups of people. He might anticipate some of these: for example, people in rainy moun-

tainous regions usually built houses of fieldstone; in the desert, clay bricks dried in the sun were used for the same purpose. There were also marked differences in customs, languages, myths, and sexual practices. Regarding the latter, Cieza often inquired about the diabolical sin of sodomy; he was very disturbed when he was told that it existed in one place or another. Beyond these observations, Cieza noted that peo-

ple differed in the directness of their speech, their looks, and their attitudes. The quality was somehow different. For example, two groups living in the mountains might prefer wool for making clothing, but the bearing of the people, and hence the way their clothing fell about them, was not the same in both groups. Originally, the Incas were a group of

people living in the mountains to the south. The ruling Inca family, the | head of which we call Sapa Inca, started with three mythical brothers, whose thoughts, according to Cieza, soared high. The Incas differed from their neighbors just as their neighbors differed from others living close by. In our terms, the Incas were a culture, and each of the other

groups were separate, more or less self-contained cultures. He sent messengers to these people with great gifts urging them not to fight him for he wanted only peace with honorable conditions and they would always find help in him as they had in his father and he wished to take nothing from them but to give them what he brought.

At almost every step along the way, Cieza noted the presence of one culture in the midst of all others. There was no puzzle in this. Going step by step, and increasing the pace in the three generations before Cieza, the Incas moved upon their neighbors. Doing this, they upset the solitude of the cultures in western South America. The Incas moved upon a group as if they were the bearers of important gifts. A deity bringing better times, or a method to make the land more fruitful, or food if the need for it existed, are examples of the gifts. If the gifts were accepted, there was no need for violence; if not, force was applied. In any case, the gifts were delivered, and thus, selected parts of Inca culture were everywhere superimposed. According to Cieza, the Incas did not want to destroy and replace the cultures already there. For example, an Inca deity was added to, not substituted for, the local gods. And

ODYSSEY 5

locally important people continued to be important, even if they now had to take an interest in what the Incas wanted done. There were even provinces where, when the natives alleged that they were unable to pay their tribute, the Inca ordered that each inhabitant should be obliged to turn in every four months a large quill full of live lice, which was the Inca’s way of teaching and accustoming them to pay tribute.

Cieza did not go far in any direction before he came upon compelling physical reminders of Inca rule. This took the form of a complex of buildings that were put up soon after the Incas took over. The components were mostly the same, regardless of region. There was a temple to the sun, storehouses, and lodgings. The buildings were now empty, their furnishings removed, and their elaborate decorations torn away. The bureaucratic officials, religious functionaries, and military personnel who peopled the buildings were gone, of course. When the buildings were in use, control had come from Cuzco, the principal Inca city, but it was hundreds of miles away, and particulars had to be administered locally. For example, tribute was fixed in Cuzco after an investigation to determine the form the tribute should take. The collection of tribute, however, was in the hands of local people. Another function of regional supervision was associated with the Inca policy of moving, en masse, thousands of people from one place to another. After the colonists arrived at their new place, it was the people in the Inca buildings who arranged the details of resettlement. Cieza thought that the tribute was levied fairly. He also thought well of the removal policy, pointing out that the

, colonists were often taught new trades. When the great dances were held the square of Cuzco was roped off with a cable of gold which he had ordered made of the stores of the metal which the regions paid as tribute, whose size I have already told, and an even greater display of statues and relics. “T recall,’ writes Cieza, ‘“‘that when I was in Cuzco last year, 1550, in the month of August, after they had harvested their crops, the Indians

and their wives entered the city making a great noise, carrying their plows in their hands, and straw and corn, to hold a feast that was only singing and relating how in the past they used to celebrate festivals.’’ In the past and during Cieza’s visit, festivals were held in the central square

, : of the city. Literally and figuratively, the four highways which partitioned the world of the Incas into four main divisions met at the center of the square. In the same square, the men who held leadership powers were

6 CODE OF THE QUIPU

confirmed in their authority amid theatrical displays of their inheritance rights. The remains of their ancestors were paraded about. The Incas founded Cuzco, resided in it, and ruled from it; yet, the people in the

city included non-Incas. The crowd at festivals was made up of men from other cultures dressed in their traditional garb. They were there to keep up the connection between the Incas and the people whom they ruled. And they were there to learn the ways of the Incas. Cieza thought that Cuzco was a glorious place, with large buildings, wide streets, and lavish displays of wealth. It had, he says, ‘‘an air of nobility.’’ He was

just as certain that with so many different people in Cuzco, wizards, witches, and idolators were plentiful, and the devil, having insinuated himself, held sway.

Not a day went by that posts did not arrive, not one or a few but many, from Cuzco, the Colla, Chile, and all his kingdom.

The experienced traveler, and that Cieza was, is not gullible. He preferred to omit a story if it was not reinforced by the testimony of several people, or better yet, by his own observations. In the case of the communication system devised by the Incas for their own use, Cieza writes: ‘“‘Nowhere in the world does one read that such an invention existed ... 380 he was cautious. But he found corroboration for what he heard in the shape of small posthouses along the highways. The system was simple. A runner carried a message from the posthouse where he was stationed to the next posthouse up the road. As he approached, he called out for attention and then passed the message on; a new runner took the message the next few miles. This was repeated by successive runners

until the message reached its destination. With a certainty unusual for him, Cieza says that the messenger system was superior to one dependent on horses and mules, animals he was familiar with. Only runners could negotiate the perilous mountain passes, suspension bridges, and rocky wastes overgrown with briars and thorns. Through this system, simple though it was, the Inca bureaucracy continuously monitored the areas under its control. The day of a Sapa Inca, according to Cieza, was taken up in large part with receiving messages and sending instructions. The Incas thought of something else to do about communication. They insisted that everyone learn Quechua, the Inca language, along with their native language. In time, many people could speak Quechua, and official business was carried on in that language. There have been occasions when I stopped beside one of the canals, and before I had time to pitch my tent, the ditch was dry and the water had been diverted elsewhere.

ODYSSEY 7

The lowly potato was first described in writing by Cieza. Potatoes are a basic food, akin to rice and wheat. For the benefit of his European readers, he explained that potatoes were like truffles, a food they knew.

On his trip into the southern Andes, Cieza found that it was common .

7 practice to dry potatoes in the sun, put them away in that condition, and eat them the following year. In Cieza’s time, the benefactors of this

, planning were often limited to the people who grew the potatoes. It was different under Inca rule. Recall that storehouses were part of the regional building complex. They held various items, including food. The Incas

responded to a food shortage by taking food from a storehouse and sending it to where it was needed, regardless of who grew it. If food was in surplus, it was distributed to the poor and aged. The benefits of already existent planning schemes, such as stored, dried potatoes, were thus extended across ecological and cultural boundaries and reached those who were not party to the plan. The case of the dried potato can stand as a model for Inca planning and control. For example, the model fits the Inca solution to the water problem. As with dried potatoes, the problem was to move a resource from where it was abundant to where it was needed. The solution was irrigation and terracing. Irrigation, like the cultivation and storage of potatoes, predates the Inca expansion. The

, Incas redeveloped old systems and ordered the construction of new ones. In an agrarian society, nothing is as important as water, except perhaps the sun; and the Incas and the sun had a special relationship if the number of temples dedicated to that star is any guide. Cieza was | struck by the beauty the irrigation systems brought: ‘‘All this irrigation,”’ he says, ‘‘makes it a pleasure to cross the valleys, because it is as though one were walking amidst gardens and cool groves.”’ And little boys, who if you saw them you would not think knew how to talk yet, understood how to do these things.

On the slope of a hill overlooking Cuzco, Cieza took the measure of an enormous stone and found that it was 270 hand spans in circumference. It was also very high. Quarry marks on the stone lent credence to the story that it was hauled up the hill to become part of a massive stronghold. The stone that Cieza measured had not reached the top of the hill,

but others did, and were there to be seen, as they are today. Large numbers were needed to express the sizes of the stones, the years the project was underway, the dimensions of the overall structure, and the people needed to do the various jobs. Impressed as he was by the scale of Inca projects, Cieza did not neglect to write about the quality of the undertakings. In this instance, he dwelt on the craftsmanship of the

8 CODE OF THE QUIPU

e

,: *.e°

m. Skill in dealing with very large 1. Forsmall. example, the , Cieza wrote about goblets

things also applied in del b d ith th se of two or three stones and a few

; e . bd . *

.

i He admired the cloth made by women using a simple e

e *. *.

dh lied the term skill to undertakings other than making J I how of statecraft, a Sapa Inca dressed himself in the native b of h vill he moved upon, and in this way, says Cieza, won

ss

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es

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**.

the people over to his service.

[ d it ce what Hho longer is and y Ww at it is, we

i 1 e | ¥ . * * * . * e behavior o ; wer. planting ofe crops can Juage wnat it was.

me toward an understanding of it, the word wou ave .

Ci ed the word in connection with almost everything, including the I e

*

. ° Way: tion € eiavorate e€ poin :W wnen € anls

H t 3 f: to s y that order saved the Incas from total dest uc*

.

*

houses; those who paid out the most, and were in refor r; therefore trouble e

were resupplied from the storehouses oe. 18

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SOEs STS ESS RASTA UAE ERT RN SETUSSR LUE CPSU CUS RLU UT TREC EEROERRTEN ERT RET CLAUS CE CCC RIE CEASA OER CERIRUN RRO See Beas GENES Sie eens USSUne R EEO SOBs ste Jat AABN Coca . Sy SRR NAC URC OSCR SEAN ALISNSTGAD ROSA SREN TUG AN EINEOS ETRE SUNTAN RESURES RUS AS ENI Mey SS Bese SEVERN S Lene Hes SASSI SBE Sra SAR OA UR oe iA : nicual ATRUNEN CANE SONSEAGER aC NAN OUEST NOREEN S.C RUAN BRAC AS CARSON REN SARISERA NS SEIU SOEUR CAE NES USS OC IR ASRO AU A ares crraise SLE EUS GsEeMa Seusc : nae esas f fs Oe EeeCRS SeeCASES SAC G UU SRST ERENT SREAU CENCE SRR OS TEARG REECARTERET SU TEES RARE RTS RRRUUSE CTE REET EET REUSE EOS SAN CN Cas CHS Cc ue EeRS PSS SCRE 8 ER PURE GS CEIACRCL CREAR ECC GSI NSEC BAU CONUN OCS NROR oRCGT RCC USO UN GGBOEEUORS St ee ee: HOES eeeeee : Ho eee : : REO AAS NUNC CT NEC SOBRREC ANALG ERG CEC AEH CURTCS ULCER EOL astSSE cum MUSG AD RNG ARSE SGU) ee ARUN

ees ASSES STENOSES ESSEC PR Ge SERRA, RS RCO CRe LUE UNS UNUM HUANG SENSO LNaota ass SES aRR BARES ACES SOREN SA EAE SERIA ae SRR OSS ars UCU eae GRU Cy iyi SUS 03eos SESE ceases ReGOES EN ceeAa eeaeaeran re ee HERS ae CONES NOOSA UU SURLCE RCRA CRS NEAT NN EEE BE SE SUR SOLEUS SANIT ICE RU Ceeterathig IASI ORES RNODUM AN RESO SUN VCC cece rt AS SERRE Unita SeBe SS ee 2 : is Sa PASe RE UREA LCSESSION SNC GUERRA BES RSNA RE GeO SCOUT CEE NUE SATORU MUU aR ee Ray accu a Byprae arNe ESSM ES RU ae B :oe ce SUTRAS CNSR CERNE ERC OSECEURERUU RS SL EUAN GEC ESTRUS CRCUERL USST SAAT ORLA UCOTEAON OO WAP acSeeo UG DUC nudes) SOUS eRe :i: AS HARES AES DAZE ARSENATE URAC URS SUE RETOSES SARC UU CEG SRT ES EEE REUSE SESS USNS SESEGG: CNGSKU RCSA SSSSELF Oe CETUS) COCR UE TES LUGAU ECT ens URSIN ete OPES Eeeniacin ee ee BESS nay ia ONE Ge ce zg eee PRE SNE RGR NETRUNNER REELS RIRS GSS STSSOEUSOSESCOaCee Se ere ee eee SARS SISOS SAAS CRSUTRUOA CULAR ARE ERE E COUSIN EOE ERE RTT ONCE SURES Cea OILS Melee een CERNE CACTUS in atone iusGUUS DeneEO Nnigmnce oc ERC ere cma Se SES RE AARNE SERS ARERR SA CEES LANCE SERPSREECE Saas (OREN NOUR SURE COL ORERRE Sar aRECTUS EROS PACSUN SN ir RaAU eighteen SCRA CRU CRu Ee viae We Sa REIN PAW Ou SO Ea UR EO OSI TS ereR Ee egy a z

ESSE he ASS RUSE SSE EASES RGEC CU USEC ENGR GLU SUG UOT G0 ee rr—s—S RRC E EOS CRESS RUSAUN ECGOCE RRLENS SUEUR CE CRC EGU EE AES SOUL CEE Se SOs GOO OSCURT NCO GSEEN EUOEE eeS28 ceAUS ae SEE : &CL2 BOURSES URGE SURES SEER URC CARCCUSC SG RSI SEAN SSSRRCS UCU CORRIENTES CE TLD ORTEGS USSU UGR EES ES CET EROS SCF EE ds Hesae ee as . GE RS GRU RRR SORE EASES EASON RRC AS RSCG ULE ECGS TGCSESE CUSE NSU Wea CoH SECS vas CAMO DAUR UEC aSEieeu ente ACCOR ARN GES eso ESSE ASRS COS ey SRR USSR CESETHE IES SECS TECSA RECS UE SEUSS SEEGER EE SEER ULC EGS CEC SUC nae ee aeVe ee :PEE SeEURANE NSS ERE STE CSE SEE SEUNR Pe ON ScEE eSEEE Pes eA OT oe UO BO REO ECCT ANU GCE HURTS ECC UE CAN CESS CAGE UE UT CO CLAN UCB NCU C U UU BU NC POSER SUN ren isa nue fuer euch AN CMM ere ees SS OE

ANS SSS ee Seo eee cee ROUTE US Go UN SUSU SPOS RCO LEAS CERN SCRE ESE EERE SRLS GLUES EES ORET IN) UAT EU SEINE ACUSEAS REESE ROU Se REG Sea tare SSBC AG SU HOLDING SLUG ROR EPOTO OsIS US aE Hos LE DL eR Te eOCEn em ee 3

LO SEER USERS NSEC USERS RG USUI E ENE REE ATS IRSS Ea E STRE EEO AER SERRE ROIS SUES EEEED NU EN AUREUS SURES CSCC RRND SUNRISE AEA C IEE UN OR TEY OF See ROS SIERRA ECR AAR eC e ee o eoeaae RE z es

Beare a SESE OCR RRE NURI RRC OAR COE PRUE GEERT SECO EECEC SEAR RRCU SG RETR SESE ay RY CRONE SBR NRESELECHEAA SESSA AATCC FLEECE ETL AUST ieee ae eee AIREY Gta Ee I SEN Ue gens : as

we

accomplishments of the new leader’s predecessors. Quipus probably predate the coming to power of the Incas. But under the Incas, they became a part of statecraft. Cieza, who attributed much to the action of kings, concluded his chapter on quipus this way: ‘‘Their orderly system in Peru is the work of the Lord-Incas who rule it and in every way brought it so high, as those of us here see from this and other greater things. With this, let us proceed.”’ His travels at an end, Cieza returned to Spain in 1551 at age thirtythree. A year later, the first part of his odyssey was in the hands of a printer. Another year went by, and after approval by the Holy Order of the Inquisition, the King’s Council, and the Council of the Indies, it was finally issued in 500 copies. The next year, 1554, Cieza was dead. From his last will and testament, we know that at the time of his death, Cieza had in his possession about half of the copies of the first printing of his book. In the portion of the will devoted to the saying of masses, he requests hundreds of them for himself, his family, and others.

Included in the list of others are eight masses to be said for an Indian woman called Ana, and another ten for ‘‘. . . the souls of Indian men and women in purgatory who came from the lands and places where I travelled in the Indies.”’

ODYSSEY 11

«2 Howto Make a Quipu

1 In building a house, making a sweater, or writing a symphony, guiding concepts are formed into specific images which always involve knowledge of the materials to be used. Thus, these creations depend, first of all, on knowing the shapes and stresses that can be taken by bricks and mortar, or knowing about yarn and what it forms when knitting or purling, or knowing about musical notes and the sounds they lead to when

interpreted on different instruments. Similarly, the first step toward making a quipu is knowing what materials are used and the significance of different manipulations of the material. The basic materials are colored cords. The basic manipulations are connecting cords together and tying knots in the individual cords. We begin with the preparation of the cords and how to connect them to each other. Then we discuss the significance of the color of the cords and the

significance of the knots that are tied in the individual cords. Elements |

of the symbolic system, therefore, are introduced along with steps in the construction process. In some instances, the step-by-step construction process is only instructional and not necessarily that followed by

an Inca quipumaker.

By the end of the chapter, there will not be anything like a completed

house or sweater or symphony. There will be the ideas necessary to | design and construct a quipulike object and to express some contemporary messages using elements from the quipumaker’s symbolic system. But these are only the basics; they are prerequisite to an understanding of the concepts used by the quipumaker.

2 Individually constructed cords are the basic units of a quipu. The

cords are distinctive in that each is at least two-ply and one end is looped ) while the other is tapered and finished with a small knot.

Fig. 2.1 | HOW TO MAKE A QUIPU 13

2 Preparation of Cords clockwise direction is called a Z-twist variation of the method just described.

(i) A drop spindle is necessary equip- and twisting in a counterclockwise The process uses one piece of yarn ment. It can be created with any direction is called an S-twist. Cut two twice the intended finished length. The straight stick as a shaft and any flat pieces of yarn of the same length. yarn is to be attached to the lead topped round weight as a whorl. A About 40 cm. is a convenient size with through a small loop. To form the loop

knitting needle and half of a potato or a which to start. Test a piece of yarn to in the lead cord, hold the top of the pencil and half of an apple are ideal. see which direction tightens its twist. If shaft in your left hand; pass the free For example, insert a knitting needle it is Z-twisted yarn, use Z-twisting to end of the lead over your index finger; through the center of half of a potato. double the ply; and if it is S-twisted, and secure it to the lead just below the use S-twisting. Now, place the two half-hitch with another half-hitch.

Fig. N.1 pieces of yarn parallel to each other

and knot one end of the pair. Tie the Fig. N.4

Sa

lead from the spindle to the pair just above the knot. Hold the other end of the pair firmly in your left hand. Drop the spindle and start twisting the yarn

IE with your right hand just above the top

, Cy of the shaft. Fig. N.3

Next, cut a piece of yarn about twice

the length of the shaft to be used as _ a lead. Tie the end of the lead around SS the shaft just above the whorl and —TR

secure it in place by wrapping it around Tr ,

the shaft three or four times. Bring the D Now, double the yarn to be twisted lead outside and under the of whorl: Y after one end through the . wrap it once around the stem the [ Ab lead’sinserting loop. Tighten the loop by pulling shaft; and bring it over the other side of the free end of the lead. Keep the the whorl up to the top of the shaft. end of the lead directed downward so it Attach the lead about 4 cm. from the does not tangle with the yarn as it is top of the shaft with a half-hitch. The twisted. With the pair of free ends spindle and lead are now complete and of the yarn held firmly in your left can be put aside while the yarn to be A \\ hand, drop the spindle and twist as

plied is readied. ii before. When the twisting is done, the tr lead loop can be pared opened and the prequipu cord slipped off the lead.

Fig. N.2 CH ey (‘v) Each quipu usually has one cord Netn)(the cord) that is much thicker thanmain the others. To make thicker cords, CES the ply of the yarn can be increased using the first method before finally

Move your right hand up and down the doubling the cords using the modified

yarn as you twist. To keep the spindle method.

(2 (= spinning constantly, assist (v) is possible to S-twist Z-twisted the whorl with youroccasionally right hand. When yarnIt and vice versa, thus creating =; the pair is tightly twisted together, differently textured cords. Color variaretaining a hold on the top end, grasp tion can be achieved by combining \wie, , theleft endhand. connected to the the leadlead. with(Snipyour different color yarns by differentyarns twist-of Disconnect ings. If, for example, S-twisted (SI ping the yarn and then reknotting two different colors are combined by

the end may be the easiest way.) Knot S-twisting, a candy cane pattern (ii) To double the ply of yarn, two the top end of the yarn and pull from results. Doubling this using Z-twisting,

pieces are twisted together. It is the both ends to straighten. gives a cord with a mottled pattern.

interlocking of the spurs on the yarn (ii) For a quipu, each cord has to be Later, in the discussion of color, it will that keep the pieces bound together. at least two-ply and only one end is be noted that the quipumakers were Commercially available hanks of wool knotted while the other end is looped. very inventive in producing many disgive better results than commercially So, as a last step in preparing quipu tinctly different looking cords from few available cotton cord. Twisting in a cords, the ply is doubled but by a slight basic colors.

14 CODE OF THE QUIPU

The looped ends are used for connecting cords together. When cotton (vi) For simple discussions of spinor wool is spun into cord, or if pieces of yarn are cut from longer cords, ing, ane Aah Strine ew York: ioe a special final spinning step is needed to prepare it as a quipu cord. As Publications, 1974): (b) Christine

. . . (Santa Rosa, Calif.: Thresh Publica-

the last step, the cord has to be bent double and twisted. Thresh, Spinning with a Drop Spindle With a few such cords at hand, it would be easier to follow the tions, 1971).

sae . , 3 Detailed descriptions of specific

description of how the cords are connected together. But such cords are _ .

not familiar or readily available. We suspect that even the Inca quipu- quipus that have the connections dismaker finished the cords himself or had them specially finished for him. cussed in this section can be found

. . . . in Marcia Ascher and Robert Ascher,

In the notes for this chapter, there are detailed instructions for prepa- Code of the Quipu Databook (Ann ration of cords by a simple method using commercially available yarn Arbor: University of Michigan Press, and a drop spindle made from some common items. The method is not Univereity Microftl microniche itom descriptive of Inca spinning techniques. It should, however, enable you Each quipu description in the datato produce facsimiles of quipu cords. Carrying out the instructions, or peor uncles ans enen and cover : at least reading them through, should also underscore that a quipu is, the cords, the type of nots in each

. . . of each Cord, € relative posiuions O

among other things, a handcrafted item. knot cluster, and the positions of the clusters on the cords. Also included are

3 . the current holder of the quipu, the . . information on its provenance or acquiIndividual cords connected together form a quipu. One cord serves holder’s designation for it, and any as the main cord. Other cords are suspended from it. The pendant cords sition. We use a tag system to refer to

can be attached as follows: published descriptions of quipus. Each quipu is tagged with a letter (or letters)

Fig. 2.2 and a number. The letter is derived

from the name of the person who published a specimen and the number

Pend refers to the order of publication. endantdcor For example, AS1—AS9 were published by us in ‘‘Numbers and Relations from Ancient Andean Quipus,” Archive for History of Exact Sciences 8 (1972): 288-320. Later, AS10—AS200 were

published in the databook. The databook also contains the tags and

c. The knotted end of the references for sixty-two quipus pub-

, cord is pulled downward lished by others. until the loop is tight around Top cords of the first type are on, for

cord. example, AS10, AS67, and AS115. Main cord Top cords the ofmain the second type are on,

for example, AS44, AS51, and AS199. Examples of quipus with dangle end cords are AS33, AS100, AS103, AS122,

a. The looped end of a cord b. The knotted end of the a e and As 103, Woven ba 1S an les iS pried open. With the main _ cord is passed over the main of decorative finishings. Examples ce

cord ring on a Nat surface ; cord and through the loop. these are on AS32, AS57, AS98, and

the IS faid AS195. Some AS136, completedand quipus, such © opene main 1core. asDenn AS106, AS124, AS140, are suspended from carved wooden bars.

HOW TO MAKE A QUIPU 15

| The main cord is generally much thicker than the pendant cords. Most cords are attached to the main cord in the way just described, and so the simplest blank quipu looks like this:

Fig. 2.3 TTT Main cord Pendant cords

Fig. 2.4 There are, however, a few other ways that cords are attached to the main cord. These will be seen to have special uses. If the knotted ends

as: of the cords are pointed downward when first laid behind the main cord, SSeS oN they point upward when passed over the main cord and pulled tight. Yini\ ‘| downward UU Such cords arependant referred cords. to as top cords toway distinguish facing A second that topthem cordsfrom are the at| tached causes them still to be directed upward but also to unite several pendant cords. For this, some pendant cords are attached to the main cord close together but the last step is not completed. That is, they are not pulled tight. Another cord is then passed, horizontally, looped end first, through the open loops of the pendants and back over them. This cord is secured by its knotted end being inserted through its looped end

\ (fig. 2.4). Lastly, they are all pulled tight. The results look like this: Fig. 2.5

Top ___ Top cords cords Main cord c——> —§—_—__)— Main cord

, Pendant cords Pendant cords

16 CODE OF THE QUIPU

A special cord can also be connected to the end of the main cord. A. Fig. 2.6 GLE cord is passed through the looped end of the main cord and then through if Main cord its own looped end and tightened. The effect is that this cord dangles EX

from the end of the main cord (fig. 2.6). fs Cords canare beattached attachedtoto pendant cords or top cords cords. They their host cord in the same way or thatdangle pen- J end CA AY dants are attached to the main cord. Because these hang from cords h other than the main cord, they are referred to as subsidiary cords. And, i additional cords can be attached to them so that subsidiaries are attached "

to a subsidiary of a subsidiary of a subsidiary, and so on. p

Combining some or all of the cord types forms a blank quipu. The Dangle end cord number of cords on a quipu varies. There can be as few as three cords R

and as many as several thousand. A quipu could look like this: iN

Fig. 2.7 Top cord

ivan

Main cord

Dangle

end cord

Pendants VY Pendants -—~—~ Pendants '—— Subsidiaries ———

4 Several extremely important properties of quipus are inherent in the

foregoing construction information. First of all, quipus can be assigned , horizontal direction. When seeing a film, there usually are credits at one end and the word END at the other. Even if the meaning of these were

not understood, they could still be used when faced with a jumble of film strips. With them, all the film strips could be oriented in the same

direction. All viewing and analysis would be based on the same running . direction. Therefore, terms like before and after could be applied. Sim-

ilarly, a main cord has direction. Quipumakers knew which end was which; we will assume that they start at the looped ends and proceed to the knotted ends. Quipus also can be assigned vertical direction. Pendant cords and top cords are vertically opposite to each other with pendant

cords considered to go downward and top cords upward. Terms like above and below, therefore, also became applicable. Quipus have levels.

HOW TO MAKE A QUIPU 17

Cords attached to the main cord are on one level; their subsidiaries form a second level. Subsidiaries to these subsidiaries form a third level, and so on. Quipus are made up of cords and spaces between cords. Cords can easily be moved until the last step in their attachment when they

are fixed into position. Therefore, larger or smaller spaces between cords are an intentional part of the overall construction. The importance of these properties is that cords can be associated with different meanings depending on their vertical direction, on their level, on their relative positions along the main cord, and, if they are subsidiaries, on their relative positions within the same level. And, just as one suspects having missed the beginning of a film when walking in

on action rather than credits, quipu readers can doubt a specimen is complete if the main cord doesn’t have both a looped and a knotted end, and can surmise that suspended cords are incomplete if they lack knot-

. ted tapered ends.

system can be found on pp. 5-7 in . . . .

5 (a) Discussion of the resistor color 5 As well as having a particular placement, each cord has a color. Reference Data for Radio Engineers, Color 1S fundamental to the symbolic system of the quipu. Color coding, 6th ed. (New York: Howard W. Sams that is, using colors to represent something other than themselves, is a and Co., 1977). This book also contains familiar idea. But color systems are used in different ways. and a discussion of the various agencies The colors red and green used in traffic signals have a universal mean-

other color systems used in electronics . ; .

(b) The law quoted is Marine . . . . .

that define these systems. _ ing in Western culture. It is generally understood that red is stop and Resources Law—Title 12—Chap. 17, green is go. Moreover, this common understanding is incorporated into Subchap. 4, Art. 1, Secs. 4404 and 4405 the traffic regulations of Western governments. The color system is simas in Maine Marine Resources, ) d fi dno tainl f t on h own Lawsprinted and Regulations, revised to Octo- ple and specific, and certainly driver isdri free is to assign his hi or her

(c) Quipu AS29, for example, uses .

ber 24, 1977, pp. 121-24. meanings to these colors.

twenty-four cord colors. Six are from Several more elaborate color systems are used elsewhere in Western different dyes and eighteen more were culture, for example in the electronics field. The color system for resiscreated by spinning combinations of tors, espoused by the International Electrotechnical Commission, has the color combination retains the signif- been adopted as standard practice in many countries. Resistors are icance of both colors is AS71 (see ubiquitous in electrical equipment because the amount of electrical current in different parts of the circuitry can be regulated by their placement. In this international system, four bands of color appear on each resistor. Each of twelve colors is associated with a specific numerical value and each of the bands is associated with a particular meaning. The first two bands are read as digits (for example, violet = 7 and white = 9, so /violet/white/ = 79); the next band tells how many times to multiply by 10 (for example, red = 2, so /violet/white/red/ = 79 x 10 x 10); and the last describes the accuracy (for example, silver = 10 percent, so /violet/white/red/silver/ = 7,900 ohms plus or minus 10 percent). By combining meanings for colors with meanings for the positions, the information that can be represented has been greatly increased.

these. An example of a quipu on which ; ; ; ;

plate 2.3 on page 23). . . . . .

18 CODE OF THE QUIPU

Clearly, lettered signs for traffic messages and printed words on resistors would be less effective than colors. In the case of traffic messages, visibility from a distance and eliciting a rapid response are the important criteria. Locating and reading letters small enough to fit on a resistor when these components are intermingled with others in compact spaces would be difficult. Directing one’s fingers to the right component is what is important, and with color coding this can be more readily done. As useful as they are, these systems are inflexible. Some group, not the individual users, defines the system and, therefore, sets its limits. Consider another form of representation, the use of letters in physics formulas:

RT ds Vas V =7IR; Va

In these formulas, the letter V is shorthand for volume, or voltage, or velocity because the formulas come from three different contexts within physics. Their contexts are a discussion of gases, electricity, and motion respectively. What V stands for or what each formula means, of course, depends on a knowledge of the context. We are, however, free to change

the shorthand. In the first formula, which represents Boyle’s Law, instead of V, J PB. R, we could use, say, a = volume, b = temperature, c = pressure, and d = universal gas constant. But, because of the behavior of gases, we are not free to change the relationship between a, b, c, d to, say,

_ de

a= b . So, too, a color system increases in complexity as the number of contexts it describes increase and as statements of relationship become involved.

One more example, before moving to the quipu, is the color system |

currently in use among lobstermen in the state of Maine in the United States. Each lobster trap that is set in the water has a colored floating buoy attached to it. By the buoy, a lobsterman can locate his traps and distinguish them from the traps of others. While the license number of

the lobsterman could identify his buoy, it would probably necessitate

pulling the buoy into the boat-in order to read it. This would be much additional work for a lobsterman and also, if it were not his trap, could

lead to misunderstanding of his intentions. The use of color enables the lobsterman to spot his buoys from afar. Also from afar, the fishing warden has to be able to observe whether a lobsterman is picking up any

HOW TO MAKE A QUIPU 19

trap but his own. Having the colors stated on the license is inadequate for the warden’s task. And so the colors must appear on the boat of the

lobsterman as well as on his buoys. Color combinations have to be allowed because the number of lobstermen far exceeds the number of primary colors. Since colors can be of many different shades and combined through many different designs, legal definition or assignment of colors by the state government would limit the variety possible. And a lot of government mechanism would be needed to decide the color combinations available as different lobstermen leave or join the fleet. The

: solution satisfies the criteria for the system. Each lobsterman states his colors in words on the license application; he paints his buoys with those colors in whatever design pattern he chooses; and he has to mount “‘a buoy at least 12 inches in length with his color scheme thereon on said

boat so that it will be clearly visible from each side.”’ If a lobsterman decides that someone else who is lobstering in the same waters has colors too similar to his, one of them changes his colors. Thus, the statute continues: ‘‘Should a licensee change his buoy color design prior

to obtaining a new license, he shall so notify the department of that change on a form furnished by the commissioner.’”’

This color system relates the traps and their owner, distinguishes a lobsterman from others in his environment, and is clear but sufficiently flexible to include about five thousand individuals or more as needed. Notice that, as with the letters in Boyle’s Law, the colors for a particular

lobsterman can be changed but the relationship between license, trap buoys, and display buoy must be maintained. Also, as with the example of the letter V in the physics formulas, the same or similar colors can be used in different parts of the state as long as they are clear in a local context. In, for example, the resistor color system, there is no place for

, , aesthetic expression. Lobstermen’s buoys can be cheerful or dull and are even sold as decorative items to people who have no part in lobstering.

6 In the context of the traffic and resistor color systems, there is an answer to the question, What does red mean? But V has no fixed meaning

in physics and red is associated with no specific lobsterman in Maine. However, in their local context, be it a discussion of gases or a particular

port, and in association with other letters or colors, the meaning is sufficiently clear. The quipu color system, like the latter systems, is rich

| and flexible and of the type for which there is no one answer to such questions. Basically, the quipumaker designed each quipu using color coding to relate some cords together and to distinguish them from other cords. The number of colors on a particular quipu depends on the number of distinctions that are being made. The overall patterning of the

20 CODE OF THE QUIPU

colors exhibits the relationships that are being represented. The color coding of cords that are compactly connected together and likely to become intertwined, shares with the resistor color system the function of uniting the visual with the tactile. Also, recall that quipu cords can be on different levels, have different directions, and have relative positions. Another feature shared with the resistor color system is that meanings for color and meanings for positions are used in combination with each other. Yarns dyed different colors were available to the quipumaker. Additional cord colors were created by spinning the colored yarns together. Two solid colors twisted together gives a candy cane effect, two of these twisted together using the opposite twist direction gives a mottled effect,

and the two solid colors can be joined so that part of the cord is one color and the rest of it is another color. And the cord colors thus created can then be spun together creating new cord colors. With just three yarn

colors, say red, yellow, and blue, and the three operations of candy striping, mottling, and joining, consider the distinctly different cord colors that are possible. There is red alone, yellow alone, blue alone; red and yellow striped, red and blue striped, yellow and blue striped;

red and yellow mottled, red and blue mottled, yellow and blue mottled; : red above yellow, yellow above red, red above blue, blue above red, yellow above blue, and blue above yellow. Selecting from these fifteen and using the same operations on them, there are many more. In some cases, the quipumaker extended the subtlety of the color coding by having a two-color combination on one cord retain the significance of both colors rather than taking on a significance of its own. In these cases, a cord made of one color yarn had a small portion striped or mottled with a second color. Thus the overall cord color had one significance while the inserted color had another significance.

7 For the most part, cords had knots tied along them and the knots 7 Plates 2.2 and 2.3 are AS71. represented numbers. But we are certain that before knots were tied in the cords, the entire blank quipu was prepared. The overall planning and construction of the quipu was done first, including the types of cord connections, the relative placement of cords, the selection of cord colors, and even individual decorative finishings. In a few cases, quipus were found in groups mingled with other cords. Some of these quipu groups contain quipus in different stages of preparation from bundles of pre-

pared blank quipu cords, to completely constructed blank quipus, to completely constructed quipus with some or all cords knotted. Cords with knots tied in them are only found detached when they are evidently broken.

HOW TO MAKE A QUIPU 21

eos” aCe mai ¢ oe ee 3 Ae ae a rn i ee PS aes 5 he 8 Ps: + lee a dl)co agian 4 ite ‘Eve ser ieeenee soe he ee ineé Ee Be- ia. eeee cw~:"nf Bat: oh4fsers

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+ pao Ge ek, 2 Zoe hie Spr ae .erin 2 9g a: ‘-7 2 E an f pia og he a es 2. ak er toe Na ai te Ce d hie olka! and cr a. te ae Teh ete pes Ose rae ee Rae: te PR eat he eaue des aagile YS yA aeae ee)" Ne Oh Oe aa ;ht i+ ae 32 thsi UR ah lla la ees is) -ne +ene af ky As >. a, Me ‘> 2 Srey he ae ve : we ‘; ¥. r * * a> $ . ae nee ang ‘H Py ; oe Th, eno Cae ae ge Disa id : q RA, -s be 4x4 a, ye ha ng fee ee et ‘ te he ds is‘a Phat‘+, of ‘ .; elyis[ad A uJ awt eae 4 ‘: ipYi7»

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/raj tf:Ne|y,4.\‘My am/l'|h' RUEAS .

Pj Jrae 7 i?Bd : 7dy J AS | / iy . Ee » A > ’ ‘ +t 4 ~* é‘LL .

Plate 2.5. The quipu of plate 2.4 unrolled. (Jn the collection of the Plate 2.4. A quipu that has been completed and rolled. (Courtesy of the Smith- Smithsonian National Museum,

sonian National Museum, Washington, D.C.) Washington, D.C.)

HOW TO MAKE A QUIPU 33

';

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Plate 2.6. A completed quipu. Observe the details of cord and knot construction and placement. (/n the collection of the Museo National de Anthropologia y Arqueologia, Lima, Peru.)

34 CODE OF THE QUIPU

15 We can now complete the market tape example by including, on the schematic (fig. 2.16), the placement of knot clusters and the number and type of knots in each cluster. (There is no need to draw each knot.

_ We use one mark for the position of each cluster.) Fig. 2.16. Schematic for figure 2.8.

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R R R R R R B Y B B Y YY B G Y YY

HOW TO MAKE A QUIPU 35

chant 3 Inca Insistence —

l Walk into any museum devoted to the study of past cultures. There, 1 For a further discussion of artifacts

on display, you will see collections of things. Taken together, these things that fae on ee Te Can Archaeol document global achievement; they can be seen as alternative solutions ogy,” Historical Archaeology 8 (1974):

to panhuman problems such as crossing water, coming to terms with the TAS.

unknowable, and keeping oneself warm and protected. , In some ways, the boats, prayer shawls, and clothing in a museum room devoted to one culture will be similar to objects intended for the same purpose found in another room given over to a different culture. A boat is intended to stay afloat. To do so, it must be built within the constraints of physical law. Likewise, the biology of the human body puts limits on certain aspects of clothing; for example, a shirt must have a hole through which a person’s neck and head can protrude. Because similarly strong constraints are not involved for religious objects, more variety is found as one moves from room to room, from culture to culture. Another way to account for similarity across cultures is history. Often enough, a particular kind of boat, or shirt, or prayer shawl, or just

the idea of how to make one of these things, was brought from the people of one culture to the people of a different culture.

Within any one room, the artifacts of a particular culture will be

distinctive. Part of the distinction is due to factors such as climate; clearly, one needs bulkier clothing to live in the arctic than one needs in a tropical forest. Yet there is something more to it. For example, the objects in the display cases of one culture may tend toward the airy and light; elsewhere, in another room given over to a different culture, similar objects may tend toward the closed and massive. More often, the distinction is subtle; intuitively one senses being close to a particular way of doing things that, however difficult to define, seems to be present, strong, and quite real.

As one looks more closely at the things produced in a culture, an object may take on special significance. We may not know the process of thought that leads to this partially conscious settling on an object. It may even be difficult to state why one object, rather than another, becomes so important in our mind, or more precisely, in our imagery. But this experience is common, and often different people can agree that this thing, and not that one, can express the rest. Imagine a very large room filled with the things produced in our culture. The room contains advertisements, oil, and a bar of steel. In order to accommodate large things, there are scale models of shopping malls, billboards, and bridges.

A visitor to this room—a person from a very different culture—is at

first confused by the thousands of items. After several visits, he comes | away with the image of an automobile. If asked to explain, he might

INCA INSISTENCE 37

answer that the automobile stands for, sums up, or expresses his museum experience of the American branch of Western culture in the last quarter

of the twentieth century. Many of us can comprehend his answer and perhaps agree that his selection has merit. For the outsider, such objects are a key to the culture; for the insider, they are a door into larger areas of meaning. They can be common and replaceable like the automobile, fixed in place like the medieval cathedral, or impermanent and leave almost no trace, like the igloo. Such objects often sum up a culture in a way that words cannot. To sense that the objects of one culture are airy and light, while in another, similar objects are dense and massive, is a step toward the idea of insistence. The important thing about a culture is its particular insistence; in fact, that is what defines a culture. The quipu is an interesting and important object, worthy of attention in itself. But it also serves to express Inca insistence in one compelling and concrete image. We shall need to amplify the general idea of insistence. Then, what characterizes Inca insistence will be explored. How the quipu expresses Inca insistence will be understood at the end of the exposition. 2 In 1934-35, Gertrude Stein delivered 2 Gertrude Stein, the American novelist and essayist, is best known a series of lectures in the United States. for her comment on arose. A rose is a rose is arose, she said. Throughappears in the lecture entitled “Por- out her prose, Stein used thousands of such constructions. Critics who traits and Repetition,” Lectures in took her to task thought that her writing was repetitious. In a talk given

Her notion of insistence specifically . - . _

4915). ca 163 206 But how eh in 1934, Stein responded to the critics. What they took to be repetition employs insistence is found in all lec- was not that, but rather it was something she named insistence. At one

tures published in that book. place in her talk, she illustrated her point with the case of a frog. A frog hops many times but the hops are not repetitious: the hops are the frog’s way of insisting. Later in her talk, she recited several previously published word portraits or sketches of individuals. What makes one person

. different from another, according to Gertrude Stein, is their individual : insistence, so it is insistence that she captured in her word portraits. Then she went on to use the term in connection with civilization: ‘* |. when you first realize the history of various civilizations, that too

makes one realize repetition at the same time the difference of insistence. Every civilization insisted in its own way before it went away.”’ We rephrase Stein’s difficult prose this way: civilizations share much in common and hence they seem to repeat each other, but each civilization ‘“‘insisted in its own way.”’

When struck by insistence, say that of an individual, it is often hard to specify the source of the affect. Trying the analytic approach does not solve the problem. Break up what has been perceived into its constituent parts, examine each part independently, verbalize the results,

38 CODE OF THE QUIPU

and still the problem is not solved. Is it the way she walks, or moves her head, or how she uses her hands while making a point? No enu-

meration of features, however lengthy, can resolve what was rightly. per- | ceived as a whole. Mere addition gets no further. Surely the movements

of her feet, hands, and head, go into the making of her particular insistence, but taken independently or in sum, they are not it per se. However, we can say that her movements are some of the things that are characteristic of her insistence. _ In the case of a person, as in a culture, insistence often shows up in details. Two friends recognize each other from small cues that others miss. A fleeting gesture is sufficient for them, whule others, less familiar

with their particular insistences, need to know more. A person who is an expert in a different culture can often recognize that culture in a fragment of cloth. Occurrences like these suggest some generalizations. First of all, insistence is pervasive; it appears in the small as well as in the large. Second, the facility to identify it increases with knowledge of the whole, whether it be a person, a society, or a culture. Furthermore, every time it is successfully identified from a detail, the concept itself passes a test. There is nothing mysterious in the expert identifying a culture from a bit of cloth: he may not be able to spell out the processes that led to the identification, but no matter, it worked. The fact that it worked, and it works in a multitude of familiar daily instances, tells us that insistence truly exists in the world. The expert in the ways of a culture other than his own cannot always identify the culture from a fragmentary piece. It may be his failure, but more likely the distinctive insistence is weak or simply not in the fragment. Care must be taken to steer clear of the notion that insistence is monolithic; actually its strength varies from nothing to very forceful. In every culture there are alternative ways of doing things; there is always exception, variety, and idiosyncracy, and insistence behaves accordingly. There is also bound to be change through time, and as culture changes, so does the patterning of insistence. Differences in strength, then, vary with what is observed, and when the thing observed materialized in the history of the culture. Just one more general notion needs to be amplified. Newspapers devote separate pages or even entire sections to politics, cooking, sports, business, fashion, and so on. In spite of the separation, insistence may

surface under any number of these headings. That is to say, it does not ,

obey the boundaries of the categories into which a culture is ordinarily classified. In the example given earlier, the insistence of a woman was picked up from the movements of her feet, hands, and head. For these, substitute the way she wears a hat, laughs, and writes a note. Or mix

INCA INSISTENCE 39

them, taking for example, how she laughs, walks, and wears a hat, and the same holds true. In the case of a culture, insistence cuts across any

arbitrary category and in so doing provides a measure of cultural coherence.

3 The fresco shown in plate 3.1 is The 3 Thus far, the discussion has drawn largely upon an intuitive under-

iengelico. Submission, byofFra Our discussion how astanding of insistence. Now we turn to the problem of finding the par-

fifteenth-century Italian may have ticular insistence of a culture that no longer exists. Unlike insistence in viewed a painting draws upon the work the present which is accessible through immersion in a living culture, Experience in 15th Century Italy (New past insistence is perceived only through visual and tactile evidence that York: Oxford University Press, 1972). is lifeless and disjointed. To find what is characteristic of Inca insistence

of Michael Baxandall, Painting and i. ; . ; . .

The relative merits of artifacts and . . . . . . :

words is considered by Craig Gilborn in is especially difficult, as illustrated by the following comparative example.

‘‘Words and Machines: The Denial of The example is drawn from Italy in the fifteenth century. In that Experience,” Museum News, Septem- . : ber 1968, pp. 25-29. A reconstruction century, the Incas flourished in another part of the world, so the same of the aftermath of the Spanish con- amount of time separates us from both cultures. The Incas survived for

aoe ened the pont 1. Wachtel, about one hundred years, and so did the Italian Renaissance; hence the

The Vision of The Vanquished: The cases are comparable in that respect also. .

jpn so New oe er a m ough The example is a fresco painting (plate 3.1). You may recognize the

Row, 1976). subject matter of the painting. If so, it is because it arises out of a religion that persists around you today. Nevertheless, the understandings and experiences that a contemporary man brings to the painting are very different from those brought by a fifteenth-century Italian. He saw many

paintings on the same subject, and he saw them inside churches, with their special aura, not in museums or books. Having been prepared by listening to sermons, he saw that the painting not only depicts the meeting of Gabriel and Mary, but also shows a particular well-defined stage in the spiritual changes in Mary as she spoke with the Angel Gabriel. We know this because sermons as well as pictures survived, and they were intended to complement each other. Acquaintance with Christian beliefs, such as the Annunciation, were further reinforced for the fifteenth-century Italian by hearing and watching religious processions, receiving the communion wafer, and by observing the ever present clerics in the marketplace. The colors of the fresco are now faded. We account for the change by the passage of time. The fifteenth-century person, seeing a faded painting, might pass a remark on the wealth of the person who paid for it. A picture was not created and then purchased; rather, a buyer would specify, in advance, what a picture should be about, perhaps how large it should be, and other details, including the quality of the paint. This is known from surviving commercial contracts. Many a buyer tried to insure in writing that the colors in his picture, especially the blue, did not fade or flake soon after the picture was completed. But sermons and contracts can take us just so far: look at how the Angel Gabriel and Mary hold their hands. There is no way

40 CODE OF THE QUIPU

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Plate 3.5. A girl at an Inca wall in Cuzco, 1975.

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84 hat match the Nn ,

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ae ge — ee ee

CiOS | n man S needs nd the natural rhy

eee eg fc Be ea oe DUE EPO aE pete BR Oe Es Se ee ae es, 8

MS BOEBe PRaphore Lee EpsPaes Lae Be AR:Le ee aes SE aes eg Oe oe PEON NS, "Weg Hee eeig aSING aesatecaONel pe Beg Be SRra ee Se ROE Se eter Pe Se ee|ee ARORA ER gag he ak uci ae a3Be Tee ee VEIN Seca see eG LSE BS MS SEN Esooo Sr UA Bd es Cie tae pe ON ER IGE REE SERB hig PR CN eeTee: aPIR eeZeeeBRR ee A a‘ESaas Le ei! aa,aMeas >eeaa i“ CIES : . Cd oe Lo ee CO ye ON ae NESSES G Ri 8 CeCe es as Cee ieee ee es Be Ja, tie Mania ana ee a eo is oo ee PEELE NEES SINE RS Le eeCO bo Lees SU aaoggke PRs BOM Pact iy Papen eres Pe enon SF Wie hs Seeeaegies AE ee nhs AanLG UES MEO Sis Baaged papFa! ae a uses eeWes Coe mes ee ee ee Pikes 8:Moa Aeg oe me em, IANS Bee ie Be SE OR GES SSRS ease Be eaeARDe Dosage Bee eR OTE gieLe: DiUigne sande Bile A CEN iia OO Bia ie kas tayot ss Chane US ORE GgMace, DSR) Pe ee. eRe ieSL Sie aaa SaaeGA A eg SSanaes cay ORO BREN RS aayMO aa ORR aRoh i ag ee iae yeee Lo ee 2 ee — BAHEO SEBS FOI SoG SERA a Depp STN TONIC Tras Pease ete, yaGR SOE GI ae,Pea PAIR SED ean eta Lene eas Oa aSieh Pee, ateLohner TOR Jiprecee Bags Dis: SH ia UGE Bes SiH Segoe NOs So ROM, 74 Cg i ee : eee

gee eeSieseee ORES LY ieaieBR eeeSEaRia LOSE SE es, ae2re ie ee er Bee fuss de Meee He itesGiese Se sos MSE cApoe Bees ieee eeeaa atSo Alone Gua Steace. Ne, Cae Figs ES ees gene outs PSeae Ge aiee) iae Me gdae agper aSe gage oS Bigs Reh cde eae EOE Uea DsOme es :ae a ee vy ay aessisIRE Pon ‘oe Bie cn EGE ee ee OUR, Es RE OS ORFORE RUSaiigBes ren ee ee 7ae tkee-ae

Ma, i Fae ga es ge os es LE eee e ee ee ee ee Sa ae pe ag Boe Cae ae | bts saiaeiietn CHE NRE Lge iA ENAE Es:BE es USS thie on: EY gstbo Bae aeST ip ui eg SeNE = Pa ee Ae ae BEM: ceeRpene ae: SSEe Be asaeeee aeeese Ee. aRiss ge 9 SSR agBee See os, aee ee OEEE OSES SOE NE aeDL ies iy BG HER Ie Ber Fk BE BOE Lee eeee. ee ES GI SSA Pr 6A eeSraa IRN a eeacee ee iesey BekREI ae aapROSA Oeesmee OSL: RE eo asfaes : a . oo reeCE ae ESR aS, Ae oy eyIS Dip ee AWASTAGE Ge ‘ eeNON ne 8sccsige BEES Bee Sie te Ree oeee! an,ae lee ie ee AR AEG AE Ripe tegen. SS Be ORS EOE Bis aeeReon: eesSe oe Bees aeBe ey,IE eee

ee .LO 7. A es is eee Saleh h iris a Sie aS eee OBE eH eSOO VaseGUe FeeBG Rigs ee so: opts Sig Bea EE SAS Be BG 5 eens BREE ETE one BeTGR RE 5RAN RII OS is 7AR AOA pose a hy in Bo is Nags Bags De Pre pete eee LATIN PARR, AS Sia eee Pe ae ae ieSe ae Lee Me Weeiitee PME ME Lope BEFfe85a I EE BSS PIGS MURAI SOE saatoe FileEE EDS SANE BEE, Phe MAM ys gee ie Nesd is eeeaye Bae ertyRes a eee REO eeNei oir ies ESSER NEE GEpelted 0 SE CAE ly PLease oe ee Pi Bg Day eer ae iaae Sse Pe RSaSER ey Gener asatePy es 6 Aas ln es Soo ae ay Te Cd ae eeBEES aPE as, ieSIN Ge aie PA he Ak Bigeye Ben tence See hes. eepena es iMn peRe Pei Me Wiss A aeae Say Soc Oe Gh tehas PTS oe Fee Sa Rinna Ne a ae TESOL OR My sei BSLERE REIL STARRY? SEDAN Pesos age Digoige a! sees aaaee Se ee aeRITE, ee GeSCE: LS Hees iyHS Sere eee aeteec antaRES arReehBee ee ee Weta a Reg je MeL 9 gages ence, Tye aOS a Ba Sea nenee” aan NBS a ie REUoeeBSR rosaPig a SLhe OeAes y Ba OEE EGE OD EEE OSE GES AsMe aT ars: eB Seas BitOO ERRORS Sst IScrBeare een kot 6 SRM Sah SS Si RES: Se ae fay LEESie Oean et Oe eee OS shia teNS iety AR AR Ee Geaea eeenag BM ONoAOR Re eR, Wan ceihies peer Seni cHass reno bieooRO

Dee OES NEP: Lear ee IDa,SENATE ee LES asthe Bier nen a OS IRB. SS IIAssSBE) OE RIG Ta PDOs teeSEEce eeean eae eePiSaeeNRE nesMEELIS ee eeBoBS) ee a eeaejgWee eaeeRS, a BO eseho ee ne SC, UR AOR ina op Le ISDS BEES agipcee eaeSME eee ia ae Buin HOH ES Bye E SURE av es seTe TEMS, 2b SEESeae PRED pia Bin Piste Migs ORS th ee aer eT acer eaten FE iaoe TE Iee ee 4 ee, Ce se * oe 8eieaHee ae TILE aeDERI: BG Mp Aeee ens roa PE Eas OER i BoASD DINE HIRE Ba a ee ene PessnEI a)-ORIAS DR Sa tae oie RAKEEES ET RaEG es aeEY SI NIG Beclin a aecise ieROR ieRG eee ee PeisBM SEL MS Sage seRR PIGahoS Sa ae ee Os SISAL ASI ED Rs SRR NEcovieges TY) a1)eae Sa BM Ie BE a asees Cc nePeone a oePRD cach ees: Saitease ec EAS! Ee me ee Re geda Me secler iC OBOR Ge ee CR Bs 7we be tar eae ass iea Le eae ee Ls ey GE Se Oke Sead ohaae fsPe me Bg eg (ise be: ate CI Se SS OIE Pyros iseedn a SORE US a La Up PORE TIS ie DOR | Cae Be CRON ic IS Gg ARES Saog Sieh ne Seay AB gp SES: AppaES Ae: UAE aseee en ae me SCAG Riera oes Fteg ae Oexgeraig er PD Byee igs oe eo ; aAbita ~“ay etae:2isHebe a) ORS es gli eeaAE aS ag og fkAa Nig SIE bee iOs Ce enaoeeas ee eecaeGere Gy Am ae ee Fe LISD lhede VR ip ee Boi Pain eeEg ee ee BOP aN DiSir ieuurairiig: toeBeiieLONG GS IN aa©SAY aVERE” «eg be we aAC ee VEEee etenRao pee) ee REC aS a6ieTee GERI RI SEE aes Se Le SOR eeEeeae eeag aoe RRC Te de ee ON oe SieII ERM ees Se ee ae ae SES, Wy cieshis Bice Be Soe eva y Fea fe areBAGS Lik SEs BleMe “SF, aces kana acne eieie aePe ee he ee ee eee Pee Fag oeSeRe oe a ees oe ae oa ‘eh aS ae ieMee . oo aEse re, ee age. SG TS BE ace NEES TE MRO ES reaeBag: es ewe Ag eeneOR, SES aBs ioey ae eee A EREwee SEER PERUSE BIST Wie ie ee cy bei Pg EER PE Se ia I ee BRR ee ae 3s COLE ue CAGES QS Sa er ee oe Tey IE UDRRN Oe Spree deSe Gee sec BOs SP GEN ap Nasik: eR aA) eR MAAC Ta ee SREay ae ss “Pe Be ESRes SNES RN GRC eer nee OSOR Oe SENG Sea OE BeOgRE at RON sa

i ae, OP ay on. Peat aha Le ae i Ce RSE ROU EG le Boe: ee eee a a eee cpa hang ee eee ee Ce nee COE gs | ae ‘a, aeaE Bae *Ee oauSDE aseRES teaSEDATE ee EeHe ORE ne Oe ey EEA Cea aeSPHee ges ETase yeGS ee Ree eyORS PeBEeeoy OR eee ASON a ae Ce ee eeOe ic ieABBne eeee eee eeCOUae Le Ca use a eeSA Oe Re, ee aa ce ee ee aPaa,eae aBP EBSCO UNE LE ae EE ee She ae ag ae 2 isee Be Ug AOS SsTe isa: i gs "af oe ee es Migs WS Seee RR RISE ae ae Bo YSae Uae ae aCo ig gE fi Be oe ee ae e iC ec UBER. Bap SEY Begs TTR AE ESSE CRASS FELL SEIS SEOSTEAS EOS aEE: REL iat Ra Pe ase ee Sie hae Tea NES ES aie ceilea tage LORIN ACL A a ee sf Og EON Cera Nang ERs DAME Se Nia ss ae Be BR EON GAGE od Sect paar Oe ae a Si Nee Bea eipdin: oe RSET RO SOR I a NEES Sorcrvcs AMS thee Be Eee ac er ee ROS A oe a,

SBI CRO MOSS 8 cessSp sf meee: ig CORE ae TE Tie 10 GeSEARO PS FEEtts OEE yeLES ee eePEI SLES E EGRESS 2 RRR Hebe Tes Aas ease: SO ast ICs OS ee bas Les SSIGS Chas ee27 NIRA OSosAR Ee SatasRR REG eeauiery. peek ci WR eit Rete mae Bienes. OE SR Ree a agae B ITIpes Ss ee “SN NRE, ET oe TERE: One EO Cg ee4aes ce NE PARR SNTEENSE PRS RAE Caeencnria UC HERE Bis 3a BO ae ise. eS pyre OOTE iio a Bee ate ei aes . o ce soBe : B3 ee : me Bi SDoe oeS382 ae ae eA et er ae ae OH ei LEIS SSIES, LOE TIS DSS RAN SRE GSS heath ble S SEN ae ee ichives, cnn ene aan flee, BOS Resa POaas SnCESERSSERISET eeNe ENN SS ieSe Oe 2sNT AREER RATEpee 18 EBS Slagey Rye bes 8": ae se ath RaiRe Me. TOR ae oe EES OS. steIRE Le TREE ga SPB 00) wee ay Se aayee se aa ye Pafae aonieTe eG rsSee scat aepe a"eG pieke ah La a ssi,SeeeCWB iyBE iDRA case RSs Pa eaaaSERN a NO: eee,soEES STE TOSS EISEE SLEe Be Seng aeSySSIES ee SREY OP sgt eet tga Pages: AS AS MGS RN CRIS Bee Re Hear Te .. caer EE I URES eesLS eee2 ereHE TDSsOO ERNST ae FEBS HERES ESESnnPer BEE Eee se eehsAe FEeee ie Cis TERT A EERO ane Pe BOehee PIRES Beas ORs: aes Tg Se iad orPee. an,eae See Sener osADeas os ee reFe : Ors lay +. OE LS Bes ks AOE EIRia EAE Rese ite Ly ee OOO NOM ESSei SLE NOae sc Bo eeSop SS Rg IDE aa Baye Sa BESO ESIila. IESae 276 SEE GES Daoes eS Hai SES 5 OS AS Bscyp cnttek. aeBe SE Cray a8Be eerSRE a,HO SIN! isk8" Bis Eig a REY Gu Ree CONS DENTE STOEL: DOB OS UE Sia OR ROE BEDI Pre BD Ae Se ORS IRE ISLET sas fgMagee PU gs Og Ceies NO eR ees BPO SN ME GP te aSoy aEeee ORL aS eaay, agte. Se Bi a aGSR ei ee eeeSaes Ca aeesoa TERI SeerME RRS OECen Se LE EN Ie ee LE Bier EO POP i area can: eben EkAean Pia eeae ee ee LES ESeee einSie ER Ge BOSS bes epgBEBE Peieak BEE ce CR Pa Mie SERED Wy SAE ROOO STSSg SSsAORN Bg ey Ree RN chee ee Me hag: Sg Boega sigMOSS eesro PRES SES. oy kes Bea ciegereee EEE erie EE: JR qs Neo BYeT sgeee a ay EE a ee hee Cee ss Re Ni cae Aa EFace PP ree: Agees SUD 98: RS tat Bg I SMPs ES Re feApe aOE TNoe ceeeetaePS OSE esMee esee rege. asShee Ramee. — eeDAS, 4 a)Sie: a es OSE SCALE gg: ne*aae acre | PER AoADE a Seite tenn Dae Baa tee, Fp OE a AION a BP Sate Bees Hee cllCan eSUE Beaetees OT oe Se. Be Gee1 Eo Riis BeI Ss BE eee COE Oe aa Be kag SE RES BO eaeBice Bee cers Sac, eoBs aeay’: paneer

rf fi. \

° 7 h‘iifig1 ifwm | Gi ] ON

(oe T ' OT! ‘ xe y A\ \ See ox , N re " T

| Sy pect at

‘ iY \ u Vig LG} Nw

OCA fea ty AE,| i | "} |\ey | H NUCH: NN

/| SA \Y HeSS.[ ort ce

V [425 a8

MTN Ch D548

| rtt—‘SCSN —eef _| Ae:

promi pw Sonal mint chau toy Duo mica agumagS SACRA»

Plate 4.2. Reprinted from Nueva Crénica y Buen Gobierno, by Felipe Guadman Poma de Ayala.

THE QUIPUMAKER 65

DEPOCITODELIVIGA ADMIVA TRADORDEPROVINC us

cue maven) es payngn, ¥ e ; - rai a A 6 a)

COLL CA a] SVINMIKGIATAPO

F ke Y a Wir 4 oe A, WN SGN , \ | ie LS | ; || : \ Si, a al | /\ if NN:

eva) | meh ¥ ad = \ y= iy YY oe We ae = lw} aieYY ==Z_\|

a;

: iY V Y; V = SA ) — A. Sepen tes of nga —jA p2 ~ a —— ee me PADORMATORITESORERC

FCIAC o WATT DAS\IC in’ | CAMPO } OV & DORCHAV.

8 ir_nina ==V/A N=

iPOCOV 2CAMA H VAIN QVIPOG Hauer

——————-_. ¥ SS Mia we: =

WW ae | lyBN WAN f;

—_ asa i N GAN LL Spray HW)

a) eh yf | qe \ cL Sa -& 4 ,berrel aw ® -— sien fea Fe 23 on beger 7 homes wen tedor Plate 4.3. Reprinted from Nueva Cronica y Buen Gobierno, by Felipe Gudadman | Poma de Ayala.

one section of his letter, there is a series of illustrations devoted to the Inca bureaucracy. Four of these show men with quipus (plate 4.3). The writing on the drawings, as on others in the text, follow a similar scheme. The top line is Spanish; below is the Quechua equivalent. The people in the drawings are often named. And at the bottom of the drawing, Poma usually has a caption. For example, in 4.34, DEPOCITODE-

LINGA and COLLCA, translate into Inca storehouse, and storehouse, } respectively; the script on the left names a particular Sapa Inca, topa ynca yupanqui; on the right, in mixed Spanish and Quechua, Poma has written administrador suyoyoc apo poma chaua. This identifies the man displaying the quipu as a very important functionary, and also a claimed ancestor of Guaman Poma. The caption on the bottom repeats the top line in the plural; that is, Inca storehouses.

The man shown with a quipu in each hand (4.3B) is also a suyoyoc, .

who was, says Poma, the administrator of a province. As such, he was responsible for many things, including the records of the communities under his charge. The position was hereditary, but not automatically so; to follow his father, a son had to show diligence and efficiency. In drawings 4.3C and 4.3D, the scene switches from the conquered territories to Cuzco. The person in 4.3C is no less than the secretary to the Inca and his council; and the man in 4.3D is the chief treasurer and accountant. In the texts accompanying both of these drawings, Poma points out that secretaries and treasurer-accountants were in each city, town, and village. And to help the king understand, in both cases Poma compares their work to what the Spanish do with pen and ink. Now in Poma’s visual vocabulary, objects held in the hands convey

unequivocal messages. Here the objects are quipus. But just about everyone in the 397 drawings is holding something. Priests hold a bible

in one hand and a crucifix in the other; soldiers grasp a spear and a shield or a club; and women are depicted spinning or holding flowers. For example, Guaman Poma holds a book in his hand; the king, his scepter. Even empty hands are busy: Poma and the king gesture with their free hand. The quipus in the drawings are not necessarily made by the people holding them—particularly so for the two suyoyocs—any more than the king made his scepter. Poma is visually associating high level functionaries with quipus: just as the soldier has his spear, the priest his crucifix, and the king his scepter, so bureaucrats have their quipus. In a sense, Poma’s bureaucracy is the same as Max Weber’s.

4 Thus far, the term the quipumaker has been used in the same way 4 (a) The map in plate 4.4 includes the

one might say the lawyer or the taxi driver meaning people who do the Me eer ae es Ne same kind of work. But individual quipumakers surely differed with locations of known quipus see Marcia

THE QUIPUMAKER 67

Ascher and Robert Ascher, Code of the respect to their work just as taxi drivers and lawyers do. The question Quipu Databook (Ann Arbor: Univer- : . . 9 . = aa sity of Michigan Press, 1978), available is, How did they differ? The only way to answer the question for quipu

on microfiche from University Micro- makers is to look at the products of their work. To do this, we first films International. The map, the have to be able to identify sets of quipus made by individuals. exclude cords with knots currently This is easier said than done. There are now about 400 known quipus used in the Andes. The contemporary in the world. This is a strikingly small sample. After all, a single quipin concept and cultural context. For umaker, producing say one quipu a week for eight years, would have further discussion o them, see Max made more than there are in the entire world sample. But remember suma, Bolivia,” Bulletin of the Free that quipus have come from graves. The quipus buried with a quipuMuseum of Science and Art 1 (Univer- maker are those few that just happened to be in his possession when he C.J. Mackey, Knot Records in Ancient died. So the first problem is that quipus that can be associated with an

databook, and our considerations . ; ; .

y p gie quip

cords with knots differ fundamentally k duci . k f oh

e, odern Quipu from Custu- . . . . .

sity of Pennsylvania, 1897): 51-63; . : . . . and Modern Peru (Ann Arbor: Univer- | sity Microfilms International, 1970);

Enrique de Guimaraes, ‘“‘Algo Sobre El Quipus,”’ Revista de Historica 2 (1907):

55-62; and Froilan Flores, ‘‘Los Kipus

Modernos de la Comunidad ® Santa Valley

de Laramarca,”’ Revista del Museo Nacional 19 (1950).

(b)the Thecloth quipus thatare were contained ( in bag AS59—AS67. Both .: \ forms of top cords are used on AS66. This set of quipus has also been dis- O™, i\E cussed by C. J. Mackey (Knot Records). The quipu sets for which 4 e*— Cuzco

color construction is contrasted are .d\n AS147-AS149 and AS33 A-G. The quipus with top digit alignment are ChancayLb >|

AS146 and AS164. Fig. 4.1A is AS195 .

and fig. 4.1B is AS194. Figure 4.2A Lima Areal 4 | is AS173. Ancon m/| ureAS159. 4.2C isFigure AS174.4.2B Plateis4.6 showsFigChaquitanta AS33. Thearequipus suspended Puruchuco carved bars discussed in further from Pachacamac Chincaj detail in section 10 of chapter 5. One of Maranga 4 them is quipu example 5.7. The set of Lurin Tambo de Mora

large quipus are AS69-AS71. Two of them are discussed in further detail in section 5 of chapter 6 as quipu

example 6.4 and quipu example 6.6.oniche ea valleyazca :x Ocucaje

Ulluhaya e

Cayango

Chala

Ts Se Ce .

Quipu Locations 0 100 200 300 Km.

Arica te

Plate 4.4. Quipu locations.

68 CODE OF THE QUIPU

a|.

individual are a small and biased sample of his workmanship. The second problem is a consequence of the fact that all quipus have come from graves in the desert. (The seventeen locations are shown in plate 4.4.) It is in the desert that conditions are good enough for the preservation of cotton and wool. We do not know how many quipus are still below

‘

( eee nan val ih - B° 3 aed

- as os —* = ® en _ 2 -— ,~* ., »

-_res ASG : ~“f « // eee=s. ¥ : ~, ; = F >» eae ts ¢.' = «. he ~* :r >?

:yf ear. ; _ “ys ive He te 4j ; StCo a= .@ es, 2os = ' —e 4 . ve. *» x mata . = = &-

3. Sa , at 7 mrt ES ' ae|5> 3 :" aarre— —4 . . | r= < OT eae«| =~ ine ys Cie tt or? 7 . ' % 1S t 7 . > : : ry ¢ ¢ ry , Pa — © satiny: AS Cea a¥t-

RR cet , eeSPN diet Ne.’=te aa ep ate 23 ~ fe

}CeieieSSS a cals —— ;-Aen+Ge . 7Re :Foti Boe Rh :ere eteloe .*F -c ; - beeen :se * ce ec

Plate 4.5. A quipumaker’s bag. (/n the collection of Oscar Nunez del Prado, Cuzco, Peru.)

THE QUIPUMAKER 69

the ground, but it is unlikely that they will be discovered in mountainous, wet regions. This tells us that whatever the individual differences, they are limited to those among quipumakers who died far from rainy Cuzco, the center of Inca power. The third problem is the biggest block in the way of answering the question. After removal from graves, quipus were not only separated from persons and artifacts, as stated above, but they

were often separated from each other. Some of those that were not discarded were dispatched individually, or in mixed bundles, to museums spread across three continents. Once there, the museums, not knowing exactly where the quipus came from, added them to other quipus in their collections. It would be impossible to associate one set

| of quipus with one quipumaker if this had happened in every case. There are instances where we know, or strongly suspect, that a set of quipus was made by one person. We draw upon these to answer the question. Quipumakers differ from each other in the way that no two people write alike. Quipumakers also differ in the way that some people write more legibly than others. For example, no two quipumakers made knots exactly the same, and some took more care than others in the formation and placement of knot clusters. The differences point up individuality in workmanship and give a sense of person, but better expressions of it are found in the exercise of self-conscious choices. One example of this comes from a cloth bag (plate 4.5) which contained two bundles of prepared pendant cords and a set of eight quipus. In explaining how to make a quipu, two equally possible ways of forming top cords were given (see fig. 2.5). The quipumaker whose bag this was, chose to use both ways, and, uniquely, he chose to use them both on one quipu. The inclinations of different people are shown in the way they elect to combine colors. In explaining how to make a quipu, three ways were pre-

sented to construct new colors from already dyed yarn (chap. 2, section 6). One quipumaker used one kind of construction to produce four colors and a second kind to produce a fifth color. A fellow quipumaker, also interested in producing five colors, used the first construction for two colors, and the second to make three colors. And, finally, there is the case of the rather imaginative quipumaker who showed volition in the way he placed knot clusters when recording numbers. Normally, knot clusters for the units position are aligned from cord to cord. He chose, instead, to align the positions of highest value. This would be analogous to our lining up the leftmost digits in a column of figures rather than the rightmost digits; it is as if we were to write

1564 1564 93 rather than 93

427 427

70 CODE OF THE QUIPU

A set of quipus made by one person is often conceptually related. It is as if an individual at the time of his death was working on a single problem that necessitated the creation of several quipus. We extrapolate backward and suggest that a quipumaker often worked this way. Figure 4.1 is a schematic of two complete quipus. At first glance, they look identical. The cord layouts are the same and so are the values, accounting for the initial reaction. A closer look brings out the differences: figure 4.1A has an additional subsidiary, and it ends in a bulb, while figure 4.1B ends in a cord placed at a good distance from the pendant. The color signing differs: the main cord in 4.1A is a mix of two colors while the main cord in 4.1B is a mix of four colors; and in three places, where 4.1A has color Cl, 4.1B has color C4. Finally, the order of two

subsidiaries is reversed. The quipus are not identical, yet they are cer- Fig. 4.1

C1 & C4 C) End C1& C2& C3 & C4

broken broken

) C1

Blank

end cord

120 62Cl4Cl4 Cl 4 C4 120Cl8C2? C4 62Cl4 C4 4 Cl C2?Cl8C1

& C3 & C4 | & C3

THE QUIPUMAKER 71

AB

24 more 5 more

7 consecutive pendants pendants 8 consecutive pendants pendants

1 1 16&116 16 18 12 16 |

. 94&1 94&1

1 116&1 16 16 18 12 16

18&2 48& 1 46 18&2 48&1 46

pendant ( \

Se

single } 12 consecutive pendants, alternate values shown

C

94& 1

VO A8& 1 46\ a

118&2&1 25 23&1 16 14 10 6 4 4&1 3 3&1 4 0 Pendant sum= 94&1

Subsidiary sum= 18&2

Fig. 4.2 tainly close enough to be called two versions of one record. A different

| kind of conceptual linkage is seen in figure 4.2. Here three quipus are

involved. They distinctly differ from each other in layout. They refer, but only in part, to the same numerical data. Some numbers are repeated from quipu to quipu, but other numbers summarize or recombine those © on another quipu. Notice in figure 4.2 that related numbers on all three

quipus are consistantly placed on either pendant or subsidiary cords | and in either the higher or lower position of a cord with multiple num-

bers. Clearly, these quipus are related around one idea. In some in-

72 CODE OF THE QUIPU

Ph Ali ill oe smmocae Ya daar Bee coe Ohi q ae

ti - ro) Ay .s i Mo ainsi ray, MNS & | :

roncoun _ 4RtNioeway BP 4 Be Ars Net m in an a ae

ram itOVE" ae ABgi Ary Ledps.oP! 7 /wey {a ai\ En) ark foe Wee 4 Ded

ae ee 4 | ne do eee) Ae A A a ee Os Ce oo, | ln 4 DM pos yap ae. SAE ner. Fé “Wie

OAR a 4 il Be ee cdPF a wf . se “ LY] f vi \ *Wau : fy . ij fd, A an ha

Plate 4.6. Seven quipus tied together. (In the collection of the Peabody :

Museum, Cambridge, Massachusetts.)

stances, a quipumaker tied a series of related quipus to each other. This,

no doubt, made it easier for him to keep them together; it certainly _ makes our task easier. An example is shown in plate 4.6. There are actually seven quipus in the photograph; three were tied to form a circle,

and four more were hung from these three. This set of quipus can be | summarily described as a rolling series involving colors, construction features, and numbers that take on particular prominence.

Quipumakers who can be identified as persons may have worked at

any number of places, say at a building site, or with armies on the move. It seems likely that some quipumakers were in charge of, or responsible for, the work of their fellows. Given the wide range in sets of quipus, it is tempting to say that some people worked in the arena of the ayllu,

the smallest traditional organizational unit, whereas others were attached to the administration of larger units within the organization of the Inca state. Put another way, quipumakers were privileged persons, but some were more privileged than others. The temptation to draw

THE QUIPUMAKER 73

conclusions on which persons were which comes from a comparison of the paraphernalia associated with a set of quipus and from the size of

the quipus in different sets. The quipumaker who kept his work in a tattered cloth bag may have been less privileged than another whose formidable work was intricately suspended from carved wood bars that

were the mark of importance in the region where he died. (Compare plate 4.5 with plates 5.1, 5.2, 5.3, and 5.4.) In a similar vein, let us compare two people in terms of the sizes of the quipus buried with them.

The quipus of the first person have already been shown in figure 4.1. There are two in the set, each with one pendant cord. The second person was buried with four quipus. The number of pendant cords on each of his quipus are: 80, 102, 251, and over 2,000. Returning to the first person, note that the pendant on one of his quipus (fig. 4.1B) carries three sub-

sidiaries on the first level, and two on the second. By comparison, a

. single pendant of the smallest quipu made by the second person might Carry as many as Six levels of subsidiaries; that is, on one pendant, there are Subsidiaries, of subsidiaries, of subsidiaries, of subsidiaries, of subsidiaries, of subsidiaries. The size of quipus and the paraphernalia associated with them suggest differences in degrees of privilege, but we cannot say that for sure.

5 For analysis of oral literature, and 5 What did the quipumaker have to know? Recall that in his travels

the selections used here, we draw upon . P . . . :

Jan Vansina, Oral Tradition: A Study Cieza de Leon gathered information about quipus. Cieza does not anin Historical Methodology (Chicago: swer the question for us, but we return to what he wrote as a good place Aldine, 1965), source for the Burundi

narrative; and Albert B. Lord, The to sta rt to find an answer. : ;

Singer of Tales (New York: Atheneum, Cieza devoted one short chapter to quipus. More on the subject is 1974), source for the “Song of scattered throughout his chronicle. On two occasions, he tells us about retrieve fragments of Inca oral litera- censuses kept on quipus. Some categories in the census are given, but ture: all are too late; Christian and he settles for the general notion rather than a full explanation. In like For examples, see Inca selections in manner, Cieza writes about accounts kept on quipus O B00. s entering

Bagdad.’ There are several attempts to , Lo. .

Western influences are painfully clear. Cj t bout ts kept . f d ter;

Sebastian Salazar Bondy, Poesia Que- or leaving storehouses; again, full exposition is lacking, as it is in places chua (Mexico, D. F.:: Universidad . . . . Nacional Autonoma de Mexico, 1964): where he mentions quipu accounts of tribute in the form of goods for

and John H. Rowe’s analysis, Eleven the Inca. In three places, Cieza writes about songs, ballads, and lays Inca Prayers from the Zithuwa Ritual, concerned with the deeds of Sapa Incas. He associates quipus with these Papers nos. 8, 9, (Berkeley, 1953), State histories in song, but the connection is not clear. Nothing is gained

Kroeber Anthropological Society . i, oo _ .

pp. 82-99. — _ by dwelling on what Cieza could not, or did not, write. It is important

The characterization of statistics as . : . .

‘political arithmetic’’ originated in that he tells us that quipus served in many different contexts. seventeenth-century England. For more Chroniclers who followed Cieza in time add to the list of contexts in

on statistics and its connection with hich . tl d. Cristébal de Moli t bout states, see Harold Westergaard, Contri- Ww ic quipus were apparent y used. Crist6bal de Molina writes abou

butions to the History of Statistics quipus and calendrics, Garcilaso de Vega adds the context of law, and 1932). The connection continued into Bernabé Cobo links quipus with peace negotiations. These three and

(London: P. S. King and Son, Ltd., P : . . . 4s

modern times as is clear in many of the others, for example, José de Acosta and Polo de Ondegardo, echo Cieza’s

74 CODE OF THE QUIPU

statement connecting quipus and oral history. At least one thing can be articles found in John Koren, ed., The concluded from what Cieza and others wrote: given the varied contexts ned of wtOL8). (New York: Macin which quipus were used, they must have been the realization of a There are several general books on

eneral recordin svstem. writing. I. J. Gelb,of A Chicago Study of Writing very 8 . 8 SY . . . (Chicago: University Press, It is easier to guess how the quipumaker might record a census, trib- 1963), is comprehensive. The general

. . . . ess, .

. oe . . . . ing writing, is treated in a bo

ute, and the like, than it is to imagine how he might record oral history. issue of human comanmcation, includ There is no contradiction in the notion of a record of oral history: if thet title: . is Colin Cherry, On Human

history was sung, as in a ballad, the transmission was oral even if a MIT Press 1966) see Mass. : record was made, just as the musical accompaniment for a ballad is not ‘The Sumerian tablet cited in full is the ballad itself. But what do numbers have to do with oral history? tablet 124 from James B. Nies, Ur

“4 Dynasty Tablets (Leipzig: Hinrichs’sche In oral history, they were probably used both ways. contents of tablets were compiled from

Remember that numbers can be used to express quantities or as labels. Buchhandlung, 1919). The general

Used as labels, numbers could have served to mark a series of the several sources, including Edward same or closely related phases. Repetitions of a phrase, later followed Telloh, Yokha and Dreham (Philadel-

_. Chiera, Selected Temple Accounts from

ee . . . . . ws and Tom B. Jones and John W. er, . . . . . versity of Minnesota Press, 1961). . : . South America. On this view see: lowed by three episodes (BCD). The starting phases of D repeat, with R. Larco Hoyle, “La Escritura Per-

by repetitions of a different phrase, or a return to an earlier phrase, are PB of enesyivane. aie distinctive of the literature of peoples who transmit their tradition orally. Sumerian Economic Texts from ihe. " Take, for example, a narrative from the native African state of Burundi. Third Ur Dynasty, (Minneapolis: UniThe narrative is about a king and his diviner who set out to overthrow There is an assertion that writing another king. There are ten episodes, A—J. The first episode (A) is fol- antedates the Inca state in western

slight variation, the start of C; and the start of C is a repeat, with slight uana Sobre Pallares,’’ Revista variation, of the starting phases of B. The story continues with a middle de Geografia Americana 20 (1943): 345— episode E; there is another set of repeats (FGH]) with the pattern shown Pallares,” ibid., 18 (1942): 93-103.

; ; . 54 and ‘*La Escritura Mochica Sobre

below, and the narrative concludes with episode J. That is, _ The oral world of the medieval scribe

is described in H. J. Chaytor, ‘““Reading and Writing,’’ in Explorations in Com-

ABCODEF GH ii J. munication, E. Carpenter and VN Ns Marshall McLuhaned.(Boston: Beacon

oe . . ; on the world of the scribe in fourRepetitions in oral literature are called formulas; they define the fun- teenth-century England is “The English

Press, 1960). A highly informative essay

damental structure a narrative will take. The quipumaker could easily Civil Services in the Fourteenth Cerrecord the formulatic expression of the Burundi story. A quipu, where Frederick Tout, vol. 3 (Manchester: numbers served as labels only, could be designed in several ways. Here Manchester University Press, 1934).

. . : . tury,” The Collected Papers of Thomas

. : Le: ‘ng: The consequences of the replacement

iS a very simple possibility of our own making: of the acoustic by the visual in modern

Fie. with particular to media, &. 4.3 *. istimes, the main concernregard in Marshall

McLuhan, The Gutenberg Galaxy (New York: Mentor, 1969). The point is developed somewhat differently in K. Heyman and E. Carpenter, They Became What They Beheld (New York: Ballantine Books, 1970).

1222345456 THE QUIPUMAKER 75

Another common feature of oral history is the incorporation of catalogs. There are four catalogs, for example, in the Slavic narrative ‘““Song

of Bagdad.’’ They appear in the part of the narrative devoted to the summoning of chieftains to a meeting. The four catalogs are: invitation catalog (al); ordering provisions (b1); arrival of provisions (b2); and arrival catalog (a2). Catalogs in oral history contain numbers that express quantities. It would be easy on a quipu to combine numbers that express quantities (such as those that occur in catalogs) with numbers that serve as labels (to record structures). One possibility is to place numbers at two levels: say, the lower ones for labels, the upper ones for quantities.

For other aspects of state oral history, the quipumaker could use a variety of colors and cord spacings. The paradigm for recording oral history fits peace negotiations, law, and calendrics. It also works for contexts mentioned nowhere in the Spanish chronicles in connection with quipus. For example, the quipumaker could make a recording of a completed building. He also could set down the plan for a building that was supposed to be erected and - never was built. But there seems little question that he was most often doing statistics. Statistics originally meant the collection and processing

of data in the interest of the state. In this sense, the one used here, Statistics is political arithmetic. Image the quipumaker at work toward the end of a day in a storehouse area and you have a picture of him engaged in political arithmetic. Among other things, his quipu might show sums that were subtotals of different kinds of goods that left the storehouse on that day. One group of pendant cords (we do not know how many) along the main cord (of a length unknown to us) could look like this, where the pendant values (perhaps recordings of quantities of different kinds of food) are summed on a top cord: Fig. 4.4

76 CODE OF THE QUIPU

Can we settle the question of what the quipumaker knew by simply saying that he could write? Writing is a subject with several parts; the quipumaker wrote or did not depending on which part is taken into

account. One part has to do with the notion of a general recording system. With every quipu, its maker realized a system that was general compared to special purpose systems, for example, records of chemical reactions. This should be clear from the preceding paragraphs. So, with regard to only this part of writing, the answer is that the quipumaker could write. Other parts of writing are its contents and setting. To examine these, we return to the Sumerian scribe. The marks that the Sumerian pressed into clay were drawn from a repertoire of 600 or more signs. Of these, about 100 to 150 stood for syllables in his language. For example, one sign was used for the sound

ba, another for dal. The Sumerians are credited as the first people to develop the use of speech sounds in a recording system. There are at least 100,000 known samples of this system, each one a separate clay tablet.

For the first thousand years of its existence, the system was used solely for political arithmetic. Scribes worked for the temple or for the

palace. As already noted, his clay dried fast, so the scribe recorded daily transactions and later summarized a week, or a month, or a year’s account on other tablets. These kinds of things were recorded: allowance for travel; account of a grain harvest; inventory of copper tools; rations of beer, bread, onions, oil, and spice. Here is the full content of a typical transaction: Shirra received from Abbashagga 56 sheep, 6 lambs, 159 ewes, 99 female lambs, 5 milking goats, and 30 female kids, 6th month,

Bur-Sin 2nd (and in the margin, a total is given: 355). Another unexcep-

tional tablet records the work done in eight palace workshops for a twelve month period. One side of still another tablet has already been shown (see plate 4.1). On this tablet, the scribe recorded payments, mostly in grain, to water pourers, guards, female singers, shepherds, and other workers in the temple of Dumuzi. The signs

| > and On

respectively. :

do not stand for syllables; they probably mean full time and half time, Like the Sumerian scribe, the scribes in our culture use speech sounds in their recording system. Our culture calls what they do writing. Therefore—it is generally concluded—the Sumerian scribe, who for a thousand years used a similar system to write solely the political arithmetic

THE QUIPUMAKER 77

described above, could write; others, who used systems that did not incorporate marks for speech, could not. Writing for the Sumerian scribe,

however, was much different than it is for us. It is more than a matter of what is written and under whose auspices it is done. As children, we learn to write by saying words, hearing what we say, and making appropriate associations with what we see we are doing with our hands. By the time we are young adults, sound has been eliminated and replaced by vision to the point where we ask to see something—say a letter read to us—in order to grasp its meaning. Not so for the adult Sumerian scribe: he existed in a setting where hearing and saying words were never disconnected from seeing them, as it is with us, who write and read in silence. Some say that this is to our detriment. In any case, the difference is profound enough to define two distinct settings. In

, Western culture, the medieval scribe in his noisy booth was probably the last of his profession to understand recording in terms similar to the Sumerian scribe.

} Now the quipumaker shared with the Sumerian scribe, the Egyptian, and indeed with all scribes in the formative periods of civilizations, these three things: (1) a general recording system; (2) contents, namely polit-

ical arithmetic (they worked for the state); and (3) a setting where recording involved saying/hearing as well as seeing/touching. It seems fair to say that the quipumaker knew how to write. Yet, he did not write as

we generally understand that term. He used cues from a shared informational model within Inca culture, and particularly those aspects of the model related to state affairs. A quipumaker told to make a recording of a village census brought to his task an understanding of what a census meant to the Incas. An Inca bureaucrat, sharing the same informational model and told that the quipu was the record of a village census, could read what the quipumaker recorded. We, for example, given something and told it is our calendar, would know to associate divisions of seven with days of the week in a certain order, and divisions into twelve with

. months in a certain order. And Americans, seeing a distinctive color or mark for the fourth day of the seventh month, would know that it means the memorial of the signing of the Declaration of Independence, while Mohammedans would recognize a special color for the twenty-seventh day of the ninth month as meaning the commemoration of Mohammed’s

receiving the Koran. However, we are not party to the informational model that would permit us to associate a particular quipu with a particular cultural meaning. For that reason, it is impossible to say that a

: certain quipu is a record of a certain census, or a song in praise of a Sapa Inca.

78 CODE OF THE QUIPU

Study of quipus, however, can yield an understanding of the general

recording system and some of the mathematical concepts, structural principles, and expressive techniques that underlie it. The following chapters convey our understanding of these. Your pleasure and understanding will be increased if you attempt to do the exercises we have provided for you.

THE QUIPUMAKER 79

~5 Format Category, and Summation

l Quipus are records. If many records, particularly those that contain numbers, were gathered together, they would differ from each other in form as well as in content. A monthly bank statement, a racing form, a ball game scorecard, a train schedule, a desk calendar, a stock market report, a telephone directory all look different but, as contrasted with most records, they share the special feature that each has a formal

layout. Each has a format designed for the display of the data particular | to it. A specific format has been arranged to convey the data concisely while displaying significant relationships and enabling significant com-

parisons. In fact, without the formality and consistency of the data arrangement, most of the information would not be accessible to the reader. Imagine, for a moment, a telephone directory in which either the given name or the surname preceded the other, in which sometimes the address came before the phone number and sometimes vice versa, in which no set spacing was used so there were no columns on the page, and in which names were not listed in alphabetical order. All the infor-

mation would still be in the directory, but it is almost impossible to imagine finding anything. Designing a format is an exercise in logic because its goal is an explicit

visual statement of the logical structure of the information. Repetition is implicit in the formatting of information because it is categories within the data, not individual items or facts, that are being made explicit. In the example of'a telephone directory, the format normally includes visual categorization into name, address, and phone number because these data

are repeated for each individual. The use of alphabetically arranged names is a way to order the repetitious data. Obviously, not all records contain formatted information. Also, if there

were only one sample of a particular type of format, it might easily go unrecognized. Similarly, some quipus are formatted and some are not. Moreover, the fact that there are only a limited number of quipus still around means that some formats will not be recognized as such. However, the quipus that exist lead us to conclude that formatting was of very great importance to the quipumakers. In this chapter, we explore in detail the logical structure of one of the most prevalent quipu formats. It is characterized by cross categorization and summation. We will first build a description of cross categorization and then give some specific quipu examples. The ways that summation are involved will then be discussed and specifically illustrated.

2 When dealing with seven pieces of information that consist of, for example, weights of three people and outdoor temperatures at four times

FORMAT, CATEGORY, AND SUMMATION 81

of the day, the data can be described in two categories. They can be written in many different arrangements but, once having recognized that there are two categories, it would be more convenient for future use to record them as a group of three items and another group of four items. They can be written for example, as: w1, Wo, Ws; t1, to, ts, tg. Were these data to be recorded on a quipu, the cords could be arranged as a group of three pendants separated by a space along the main cord from a group of four pendants. Or, there could be three pendants of one color followed by four pendants of another color. To summarize these written

or quipu arrangements, the mathematical notation w; (i = 1, 2, 3), ¢; (j = 1, 2, 3, 4) can be used. (This is read as: w subi where i = 1, 2, or 3, and ¢ subj where/ = 1, 2,3, or 4). It simply means that two categories have been conveyed, one with three elements placed in order and the other with four elements in order. Next, consider recording six pieces of information consisting of the scores of three games involving two teams. This data can be viewed as in two different categories: scores of Team 1 and scores of Team 2. Or, it can be viewed in three different categories because it describes three different games. Actually, both can be done simultaneously. The data can be organized into a chart:

Fig. 5.1 Team 1 Team 2

Gamet] |

Game2] |

Game 3 , pe Reading down the first column, the scores of Team 1 are associated, and reading down the second column, the scores of Team 2 are associated.

Reading across the first row describes Game 1 while the second and third rows describe Games 2 and 3 respectively. Alternately, the chart used could be: Game 1 Game 2 Game 3 Fig. 5.2

Team 1

data. } .

Here columns associate a game’s data while rows associate a team’s

A quipu record of these data could use space and/or color to display the categories. One possibility is:

82 CODE OF THE QUIPU

Fig. 5.3

Cl C2 C3 Cl C2 C3 where pendants in the first group contain Team | scores while those in the second group contain Team 2 scores, and colors 1, 2, 3 are associated with Games 1, 2, 3 respectively. Another possibility is: Fig. 5.4

Cl Cl Cl Cl Cl Cl where all cords are the same color. The first group contains scores of Game 1, the second of Game 2, and the third of Game 3. Within each group, the first position is associated with Team 1 and the second position with Team 2.

In subscript notation, both charts and both quipus are summarized

by pj; @ = 1, 2, 3; j = 1, 2). (Read as: p sub ij where i = 1, 2, or 3 and j = 1 or 2.) A value of i is associated with each game and a value of j is associated with each team. Thus pj, refers to the score in Game 1 of Team 1 and pg refers to the score in Game 3 of Team 2. To further clarify the meaning of the subscripts, figure 5.5 combines the chart forms and the quipu forms with the subscript form. Fig. 5.5.

Game | Game 2 Game 3

alfalfa qa} ala

Chart A Cl C2 C3 C1 C2 C3 Layout A

Team 1 Team 2

Game Game 3

Chart B Cl Cl C1 Cl C1 Cl Layout B

FORMAT, CATEGORY, AND SUMMATION 83

Before proceeding to a generalization, another step of cross categorization is needed. Consider a set of twenty-four pieces of information which are the prices of three products in four stores on two different days. To write them, we can form one table for each day as shown in figure 5.6. In subscript notation, the data is summarized as p;;, (i = 1, 2; 7 = 1, 2, 3, 4;k = 1, 2, 3) where i is associated with day; j is associated with store; and k with product. The twenty-four individual p’s have been placed in the tables in figure 5.6 in order to show how they correspond to the tabular form. Fig. 5.6

Day 1 Day 2

Prod. 1 Prod. 2 Prod. 3 Prod. 1 Prod. 2 Prod. 3

There are several other ways the data could be arranged in tabular form. The two tables of figure 5.6 could, instead, have rows for products and columns for stores; there could be three tables of eight entries each, where each table is associated with a product (and either rows or columns with days and stores); and there could be four tables of six entries

each, where each table is associated with a store (and either rows or columns with days and products). No matter which of these tabular formats is used, the same summary subscript notation applies because each format is a way of demonstrating the same cross categorization. Thus, the subscript notation is identifying the logical structure being exhibited by them regardless of which of the ways it is exhibited. It is for this reason that the subscript notation has been introduced and is preferable. When describing the logical structure of actual quipus, the subscript notation enables a concise exhibition of the structure without having to include the particular details of the quipu construction. Moreover, a statement of the logical structure provides the framework for examining the data within it. Without an understanding of the logical

, organization of a telephone directory, looking for someone’s number ) would be as hopeless a task as looking in a directory that had no logical organization. For the same data as in figure 5.6, many different quipu arrangements could be used. A few of them are shown in figure 5.7. Each has twentyfour pendants and conveys by spacing and color that four categories are

84 CODE OF THE QUIPU

being crossed with three categories which are being crossed with two categories. All are examples of the summarizing subscript notation p ;; G@ = 1,2; 7 = 1, 2, 3, 4; & = 1, 2, 3). Fig. 5.7.

C1C1C1ICICICICIC!1 C1C1CICICICICIC1 C1C1CICICICICIC1

groups (by large spaces) correspond to A = 1, 2,3 (product) subgroups (by smaller spaces) correspond to j= 1,2,3,4 (store)

position (in subgroups) correspond to i= 1,2 (day)

C1C1C2C2C3C3 = C1C1C2C2C3C3) = =©C1C1C2C2C3C3) —- C1C1 C2C2C3C3

groups (by large spaces) correspond to j = 1,2, 3,4 (store) subgroups (united by same color) correspond to k = 1, 2,3 (product)

position (in subgroups) correspond to i= 1,2 (day)

C1C2 C3C1 C2 C3-C1 C2 C3 C1 C2 C3 C1 C2 C3 C1 C2 C3 C1 C2 C3 C1 C2 C3

groups (by large spaces) correspond to i= 1,2 (day) subgroups (united by color pattern) correspond to j = 1, 2, 3, 4 (store) position (in subgroups) correspond to k= 1,2,3 (product)

FORMAT, CATEGORY, AND SUMMATION ~=_. 85

3 The quipus in figure 5.8 are: 3 Many quipus are formatted so that they display cross categorizations.

A ASIA onsen. fOas3)) Primarily, it is spacing and color coding that convey categories. Some7 (AS175): 8 (AS121). times, for emphasis or as an extension of these devices, additional visual markers are used. These markers include very long cords, very short cords, cord puffs tied on the main cord, or specially colored cords. A

| sample of the larger quipus of this type are summarized in figure 5.8. Some important ideas to keep in mind when looking at them: 1. The logical complexity of cross categorization increases as the number of subscripts increase. Our example used first two subscripts and then three subscripts. Two subscripts are analogous to a table of data and three subscripts to a set of several tables of the same size. An analogy to four subscripts could be several pages each containing several tables.

2. The highest value for each subscript is the number of logical distinctions being made. Our examples had highest values of 2 and 3 and of 2, 3, and 4. These values are analogous to the number of rows or columns in a table and then to the number of tables and the number of pages of tables.

3. The product of the highest value for all subscripts gives the number of items in the entire set of data. Our examples had 2 x 3 = 6 items and 2 X 3 X 4 = 24 items. Fig. 5.8. Some quipu configurations.

l.py i=l,...,5 3: j=l, ...,17 2.pi; i=l, ...,8 ;j=1,...,24 3. py i=l, ...,153j=1,..., 25

4. Dije i=l, 2 -j=1,...,10;k=1, ..., 16 5. pine i=l, ...,4 3 j=1,...,6 k=l, ..., 24 6. Dijk 1=1, ...,6 3 j=l, ...,8 ;k=1,...,9 7. Pijkm i=1, 2 ; J=1, 2, 3 ;k=1,...,5 3; m=1,...,7 8. Dijkm t=1, 2 ; J=1, 2, 3 ;k=1, ...,7 3;m=1,..., 10 4 To introduce summation, we return to the example of two teams competing against each other in three games. Now the record (shown in fig. 5.5) is also to include the sums of points scored per game and the sums of points scored per team.

86 CODE OF THE QUIPU

Fig. 5.9. Chart B of figure 5.5 extended.

Team 1 Team 2 Game Sums Game | Game 2 Game 3 Team Sums pistPavt ss

Summation of the scores in the first row of chart B results in the total points in Game 1 and summation of the second and third rows result in the total points in Games 2 and 3 respectively. These are recorded in a column that has been appended to the chart in the row associated with the game being described (see fig. 5.9). Similarly, the summation of the first column results in the total scored by Team 1, and summation of the second column gives the total for Team 2. In order to associate each of these totals with the appropriate team, they are placed in the team’s column in an appended row. Whatever format was originally designed to record the scores can be enlarged to include the sums in keeping with their associated categories. If the data had been recorded on a quipu, the placement of the sums would also depend on how the categories had been designated. Where the quipu had two spatial groups to distinguish the scores by team (see fig. 5.5, layout A), the sum for each team would remain with the group. It could be on a top cord associated with the group or on an additional Fig. 5.10. Layout A of figure 5.5

i. S| Ai. a ee OR+g + g q < x a} aloa a. at 9s q+ Q A;+ 8Q a toa ne SY a a, nn cord in the group (fig. 5.10). extended in two possible ways. Team } 4 pag Team 2

pu + Pa TPs sum pio + P22 P sum

— x o9 a A BS = nN 59 n a A e N

Cl C2 C3 Cl C2 C3 Ci C2 C3 C4 C1 C2 C3 C4

Team 1 scores Team 2 scores Team 1 Team 1 Team 2 Team 2

Scores sum scores sum

FORMAT, CATEGORY, AND SUMMATION 87

| sSNqCY33en Since, within each group, the scores for different games were distinguished by color, the sums for each game would be in another group still identified by their color (fig. 5.11).

Fig. 5.11. Layout A of figure 5.5 extended.

a +Q+}q + n $9

nN q39— a xQ$9a aa a Q a= —

Cl C2 C3 Cl C2 C3 Cl C2 C3 Team 1 scores Team 2 scores Game sums

Another possible quipu layout was shown in figure 5.5 (layout B) for the same data. On it, game scores were distinguished by spatial groups while

| team scores were distinguished by pendant position within a group. Therefore, in this layout, sums of scores would be placed and identified in kind. Figure 5.12 shows layout B extended to include sums. Although

Fig. 5.12. Layout B of figure 5.5 shown separately, both game and team sums could be included on the

extended. same quipu.

Cl C2 C3

A.Oe A.ES feoS S _P

(Game 1 sum) (Game 2 sum) (Game 3 sum)

= —N — N — N = N _ N —_ N OR

_ Ee EH FA i EH RIle| a el R| aA RL eI Nn

a pn ee Ne Nn LY Nett nnn

Cl Cl C2 C2 C3 C3 C1 Cl Cl C2 C2 C2 C3 C3 C3 Game 1 Game 2 Game 3 Game 1 Game 2 Game 3

= ON vod ON — ON = ON

- EH os os EH I EE EH Cl Cl Cl Cl Cl C1 Cl Cl

Game 1 Game 2 Game 3 Team sums |

88 CODE OF THE QUIPU

If the three total game scores were summed, the result would be the same as if the two total team scores were summed since each would actually be the grand total of the six individual scores. This value is, therefore, associated with the entire quipu regardless of its particular layout. If present, it could be on a first or last pendant set off from the

rest, on a dangle end cord, or even knotted into the main cord itself. : 5 Cross categorization with categorical summation is present on about 5 On fourteen of the eighteen quipus

25 percent of the quipus. Depending on the format used for categori- tat have tthe aseociated groupe zation, the sums are on top cords associated with groups, within asso- Two more probably would also but the ciated groups, or on pendants in groups that sum other groups. Grand anottng o numbers onto com Seems totals, which unite all categories, are separately and prominently placed. ie that on twelve of these eighteen Several quipus have more elaborate layouts involving summation. quipus, the pendant groups are of size Before proceeding to them, two observations fundamental to all sum- six. In the general sample, of those mation need to be made explicit. In our earlier discussion of numbers, only about 25 percent have groups it was noted that not all numbers are magnitudes. Numbers that are or woe six are mote grotinent groups labels, such as apartment numbers in a multifloor building, are not (67 percent) among pendant groups summable. The numbers in exercise 2.1 were the opening and closing wroupe in general O5 pone Lae ; hours of a service station so their sum would have no meaning. Even appear within their associated groups

’ quipus with distinct pendant groups,

when numbers are magnitudes, they might not be additive. If six pieces on Ass: Ast? and “s ae Sums are of data were the heights and weights of three individuals, recording the groups on ALS, / ALT, AS8, AS49, “ sum of the two values associated with each individual would be highly AS100, AS120, AS143, AS147, AS176, unlikely. The sum of the three items in the weight category could be of AS100, AS122, AS127, AS130, and

. ws . and AS178. Grand totals are on L49,

interest. For example, contemporary passenger elevators usually carry AS178. a statement of maximum total weight permitted. It is difficult to think of a reason for summing the three items in the height category. Therefore, the presence of summation is a significant statement about the rest of the data on the quipu. When a quipu contains categorical sums, we can conclude that the data being summed are magnitudes and that the

magnitudes are such that their sum is meaningful. | The second observation relates to the number zero and color coding. On some quipus, cord positions within a group are reinforced by color coding. That is, when, for example, colors 1, 2, 3, 4 appear in the groups in positions 1, 2, 3, 4 respectively. This redundancy enables a distinction

to be made between a value of zero and the absence of a category. Consider a set of data that is an inventory of the number of pairs of

sandals owned by a community of people. Say that the numbers are | recorded on pendants categorized into families by spatial groups and, within each group, categorized into men, women, and children as represented by cord colors 1, 2, 3. A cord of color 3 with no knots would have a value of zero and mean that the children of a family had no sandals. The absence of cord color 3 in the cord group would mean that

FORMAT, CATEGORY, AND SUMMATION 89

the family had no children. These are different in information content but either contributes nothing to a categorical sum. The fact that the

distinction is made is important because it corroborates that a blank . cord, in fact, represents what is symbolized by our zero. It also provides the key to many quipu formats. In cases where there are spatial groups with different numbers of cords but all are subsets of one color pattern, the logical structure can be identified from this overall color pattern.

6 To further elaborate the cross categorization and summation, we again return to the score chart of the two teams competing in three games. The complexity of the chart is increased by including the fact that each team has four players and their individual scores are also to be included. Figure 5.13 is the extended chart but still includes all the data and sums from figure 5.9. The p;;, @ = 1, 2,3; 7 = 1,2;k = 1, 2, 3, 4) representing the scores of each player k on team / in game /, have

also been noted on the chart. It is important to realize that the team scores for each game are now sums of player scores. Therefore, each team total is the sum of three game scores which in turn are each sums of four player scores. Similarly, the three game totals are each sums of

Fig. 5.13. Figure 5.9 extended. pairs of team scores which in turn are sums of player scores.

Player 1 Player 2 Player 3 Player 4 Team 1 Game 2 Game 3 Team Sum

Game Sums

Player | Player 2 Player 3 Player 4 Team 2 Game |

Game2 | pn | pow | ps | ae | re || Pan trtat

P22+P31tP32 Grand Total

Game 3 Team Sum

— 90 CODE OF THE QUIPU

This type of cross categorization and summation can be viewed in terms of other familiar contemporary situations. An example is an accounting scheme for a company with several departments. A final report

might include total expenses categorized by department. For each department there is a ledger elaborating the different types of expenses; say sales expense, equipment expense, expenses of utilities and maintenance, and salaries. Then, for each department for each of these expense categories, there is a subledger further detailing the expenses. Another example is a company that makes several products and organizes its sales into geographic regions. The final report would be the total number of each kind of product sold, the charts would be products sold categorized into sales regions, and each sales region might have a subchart further detailing its distribution of each product. In these examples, as contrasted with the team scores, the subledgers (or subcharts) do not necessarily have the same number of categories within them and some ledgers might not even have associated subledgers. To summarize the examples in terminology that will be used for the quipu descriptions, we have described: 1. Team scores per game—these form a chart of p;; @ = 1, 2, 3; j = 1, 2)—each p;; is the sum of four items in an associated subchart. 2. Player scores p;jx G@ = 1, 2, 3; 7 = 1, 2; k = 1, 2, 3, 4) constituting two subcharts, one for each value of /.

3. Categorical sums: team sums (j = 1, 2); game sums (i = 1, 2, 3).

4. Grand total. 7 On several elaborate quipus, some of these arrangements are found. 7 Quipu example 5.1 is AS156. Quipu

Here are the details of four quipus involving subcharts. a ate 2. 5. Ouipu examele 3 is AS149. Quipu example 5.4 is AS175.

Quipu Example 5.1

1. Chart with elements p;; (i = 1, 2, 3, 4; 7 = 1, 2)—each p;; is the sum of seven items in an associated subchart.

2. Elements p;j, @ = 1, 2, 3,4; 7 = 1, 2; A =1,...., 7) constituting two subcharts, one for each value of /. 3. This quipu, therefore, has sixty-four values. There are sixty-

four pendant cords suspended from the main cord. By a repeated color pattern and spacing, these are separated into sixteen groups of four pendants each. By spacing and a marker, the sixteen groups are

separated into two, seven, seven. The actual data on the pendant cords is shown in tabular form (fig. 5.14). In the table, each group of

FORMAT, CATEGORY, AND SUMMATION 91

four values is ina row. Consecutive groups are placed in rows beneath

each other rather than strung out horizontally as they are on the quipu. Fig. 5.14. Quipu example 5.1.

i=] i=2 i=3 i=4 ]Chart

J-: “-

“~ :Subchart

:-°

:° : “! *‘-*

:-, Subchart “-°

‘-°

:-' C

92 CODE OF THE QUIPU

Quipu Example 5.2

1. Chart with elements p;;(i = 1,2; 7 = 1,2,..., 18)—each p;; for which i = 1 is the sum of three items in an associated subchart.

2. Elements p1;, (Uj = 1,2,..., 18;4 = 1, 2, 3) constituting one subchart for i = 1.

3. Categorical sums (j = 1, 2,..., 18). Quipu Example 5.3

1. Chart with elements p;; @ = 1, 2; 7 = 1, 2, 3, 4, 5)—each p;; is the sum of seven items in an associated subchart.

2. Elements pij, @ = 1, 2; j = 1, 2, 3, 4, 5;k = 1, 2, ..., 7) constituting two subcharts, one for each value of i. The five elements

for which i = | and k = 1 are each the sums of nine elements in an associated ‘sub-subchart.

3. Elements pyjim UV = 1,...,5; m= 1,2,..., 9) constituting one sub-subchart fori = 1,A = 1. Quipu Example 5.4 1. Chart A with elements p;; @ = 1, 2, 3; 7 = 1, 2, 3, 4, 5)—each Piz 1s the sum of seven elements in an associated subchart.

2. Elements pj, @ = 1,2,3; j= 1,...,5;k =1,..., 7) constituting three subcharts, one for each value of i. 3. The quipu contains another set of subcharts (B) with elements

dije @ = 1, 2,33; 7= 1,...,5;k4 =1,..., 7). The main cord of the quipu is evidently broken. Because of the layout of the quipu, broken end—subcharts B—chart A—subcharts A

we believe that it contained another portion that would have been chart B with elements qg;; (i = 1, 2, 3; j = 1, 2, 3, 4, 5). We have no way of knowing where the data on these quipus came from. But we can recognize formats, and indeed, the specifics of the format. This, in itself, tells us something very important about the Incas, namely the formal specific and rather complex ways in which they ordered their universe.

§ Before concluding the discussion of cross categorization and summation, the word sum needs some further examination. When we see the values 18, 31, and 49, we can describe 49 as the sum of 18 and 31. However, this carries no implication that we have seen the operation of

FORMAT, CATEGORY, AND SUMMATION 93

addition carried out. A quipu is not a calculating device. We do not know how or where the arithmetic was done. In fact, we do not even know what arithmetic was done. Just a few examples of how these numbers could be operationally related are: 18 results when 31 is subtracted from 49; 18 added to 31 results in 49; 49 can be subdivided into 18 and 31.

If there were only these three values, the arrangement might be fortuitous and we would even hesitate to presume there was some arithmetic relationship. Because the logical structure of the data was identified

, first, and because, within that structure, there is repetition of a patterned relationship, we can proceed to state the pattern. This still leaves as an open question the human actions that placed the data into this relationship.

9 Quipu example 5.3—continued: 9 Charts and subcharts of ball game scores might be a prediction for To clarify the effect that a large range the next season rather than the results of games played. A budget which the following situation. The prices of allocates money that is available to be spent next year usually has the $1.92 and $1.94 for an item are not the same classificatory scheme as a report of expenditures for last year. And $0.06, $1.92, $1.94, and $84.72, the two a record of number of products by sales regions could represent a plan

of values has on comparisons, consider ; ;

same. But when faced with prices of .

Having only integers also effects com- . .

would appear much more similar. or goal rather than a report of past sales. The important difference is parisons. If fractions were available, we that, in these latter cases, the summary numbers are determined first would not say that 1, 3, 5, 6 are each and then broken down into parts.

rere. 7, 10, because half of 7 is The data within the charts and subcharts of some of the quipus have With no 12 fractions available,

regarding 3 or 4 as half of 7 becomes a regularity that is more suggestive of a plan than of a result. Or, if it is appropriate. These notions are dis- a result, it is the result of something that was very carefully planned. Regularity of data is hard to define because it can show up in many different ways: values that are exactly the same (200, 200); values that are multiples of a common unit (155, 85, 315, 500); values that are a fixed proportion of each other (41, 82; 71, 142; 193, 386); values that are symmetrically placed (48, 11, 91, 11, 48). When, as in one of the quipu examples below, ten values show regularity but each is the sum of seven other values and some of these are, in turn, the sum of nine other values, it is more plausible that the ten values were determined first and then broken down into the associated 105 numbers than that the ten values were built up from them. In order to find regularities, the actual numbers on a quipu have to be examined. We first looked at the logical structure of the quipus and then at the arithmetic relationships within them. Summations, charts, and subcharts are common to both a record of results and a plan. Keeping within this common framework, but delving into the actual numbers

cussed further in chapter 7. ° ; . 7

94 CODE OF THE QUIPU

within the relationships, we can attempt to distinguish between them. Our question is whether the sums were formed from the values in the subcharts or whether the sums were preconceived values that were then decomposed.

Two of the four quipus that were just used as examples (quipu examples 5.1 and 5.3), exhibit distinctive data regularities. We, therefore, return to look at each of them. Quipu Example 5.1—Continued

All the values on the quipu were shown in Figure 5.14. 1. The eight chart values are each the sum of seven subchart values. But when the chart’s categorical sums are formed (by adding together the four values for which j = 1 and by adding together the four values for which j = 2), each turns out to equal 1,000.

2. Within each of the subcharts there are twenty-eight values. However, each has only four different numbers used over and over again. When the twenty-eight values are combined into fourteen pairs

(for each k, values fori = 1 andi = 2 are combined; values fori = 3 and i = 4 are combined), the regularities within each subchart and between the subcharts become more apparent. Each subchart is rewritten as a table of pairs in figure 5.15. Fig. 5.15. Quipu example 5.1—continued

j = 1 Subchart j = 2 Subchart

70 70 72 71 40070 6070 60 72 72 71 71

200 { Goo 00) ( 72 72

70" LL 400JO 70 60 70 72 70 60 70 72 ee1,000 ——500 wae500, ae 1,000

a) Each pair table (subchart) has only c) In each, the rows just above and below three different numbers. One has only the center row are the same. multiples of 10 and the other has con- d) In the / = 1 subchart, there is symmesecutive numbers. In each case, there try of the sum of all values above and are 8, 4, and 2 of each of its three below the center row: 400, 200, 400.

numbers. e) In the j = 2 subchart, there is symme- |

b) In each, the center row has a pair of try of the sum across a center line (ver-

maximum values. tically): 500, 500.

FORMAT, CATEGORY, AND SUMMATION 95

Quipu Example 5.3—Continued 1. The chart of ten values is related to subcharts and sub-subcharts

containing, in all, about a hundred values. However, when the five values in each of the two categories of the chart are compared, they have a remarkable numerical regularity. They are found to represent the same proportions of their own category. Because the numbers in the chart range from 660 to 21,243 and the proportions are not simple fractions, this might not be visible by looking at the ten numbers. To make the similarity apparent, each value has been divided by the sum

of its i category so that only the proportions themselves are written in a table in figure 5.16. Fig. 5.16. Quipu example 5.3.

j=l j=2 j =3 j=4 j=5

As usual, when dealing with real data, we do not expect the numerical exactness that is found in made-up textbook problems. Also, since all the numbers on the quipu are integers, with no fractions or decimal

fractions, accuracy of individual numbers is limited to the closest whole integer. The numbers in the table in figure 5.16 are, therefore,

rounded to three places and we accept as “the same’’ values that differ by at most 1 percent. 2. In addition to the proportions being the same, one of the pro-

portions is the sum of two others. That is, the proportion of the

for j = 1. ,

wholes for 7 = 2 and j = 4 combined, is the same as the proportion

3. Another regularity on this quipu is in the data in subcharts and sub-subcharts. Many of the values are multiples of the value 110. In fact, in one of the subcharts, 38 percent of the entries are multiples 10 Quipu example 5.5 is AS168. Quipu of this single value. example 5.6 is AS141. Quipu

cxample Othe. jurpus suspended fom 10 There is another type of quipu, similar in concept to the kind being wooden bars are in the same museum. discussed. They show cross categorization and have internal regularities They are AS106, AS112, and AS124. but do not have on them categorical sums. This lack is significant. Bevon Peru (Berlin: Impropylaen-Ver- cause they lack this evidence of summation, it cannot be presumed that lagzu, 1929), contains photos (p. 545) the values are magnitudes which are additive. It is only when we form anc 7 5 companion ASIA. ‘Oninu exann. the sums and find that they are patterned, that we believe the interpre-

Max Schmidt, Kunst und Kultur . . . .

ple 5.8 is AS109. tation is valid. These then are charts or subcharts. What makes them

96 CODE OF THE QUIPU

identifiable as such is their logical structure combined with the regularity of categorical sums formed by us. Two small examples of these follow. Quipu Example 5.5

1. Chart with elements p;; @ = 1,2; j= 1,..., 9). 2. Equal categorical sums fori = 1 andi = 2. (Nine values go into each sum.) Quipu Example 5.6

| 1. Chart with elements p;; @ = 1, 2;j7 = 1,..., 10). 2. Equal categorical sums fori = 1 andi = 2. (Ten values go into

each sum.) 3. The twenty values in the chart are only four different numbers repeated over and over.

4. In the chart, there is symmetry of the sum of all values above and below a center line and across a center line (fig. 5.17). Fig. 5.17. Quipu example 5.6.

i= 1 i=2

Two larger examples are of considerable interest: one because of its patterning and the other because of an unusual construction feature. Quipu Example 5.7 1. This quipu is distinctive in its construction because it is strung from a carved wooden bar (see plates 5.1 and 5.2). Unfortunately, the carvings do not help us to identify what is being recorded. It consists

of two subcharts with elements p;;, @ = 1,2; 7= 1,...,9;A = 1, ..., 10). Because of the way the quipu is attached to the bar, the subcharts are physically back-to-back. It is evidently related to another quipu believed to come from the same set. The related quipu is also strung from a carved wooden bar and uses the same style of color coding (see plates 5.3 and 5.4). It too, has a logical structure

that can be described as q;jj;, (A = 1, 2; j= 1, ..., 9 k= 1, ..., 10). 2. The subcharts of example 5.7 and their regularities are diagrammed in figure 5.18.

FORMAT, CATEGORY, AND SUMMATION 97

hime Ce A PR ee ' ) in ) Moe\Ff 3 2) 298 4 aye

| iBinta. PAWHALT ; i f nebin i: DD

‘| time | ;WY it out| Le Wy fis i\Ny 1 in tin . Be \

me TRS i ADE P. fi 1 if i] Hi i | (AYE tm) ; va) i} TALES | Hi f Rrwviis 1 we aR. i iy SU Pat

iM if HY : Hi ) | Mg 4 ‘a !] |

; 4 be ig el f :{Tr :}443 Nea | : hy ' i ' ti i|/ 7.RD Tainili i , 1]

fj / if A

| /; |/.mar} (}): |nf ‘74 iy | b

uf fs: qY: ps}| ayo{ =f j fs,

f i N jh] Ip if ; . AMP.Be 4) VATED FriboiS tale A i]

Plate 5.1. Quipu example 5.7. (Courtesy of the Museum fiir Volkerkunde, Berlin, West Germany.)

98 CODE OF THE QUIPU

Plate 5.2. Carving detail from plate 5.1. (Courtesy of the Museum fiir

S

y Volkerkunde, Berlin, West Germany.)

f|, Ph

EA pi ee oF: 7 . ; 7 y AN

al ‘~ AW

; iy “oh. > ae aa’ : _ a) a . TE Dy AY) oy our oe Sere) in itof j ih h!. My oi Ue VA. -: ijJa ii] A\E f/ ie | j : ti iJ lf PARAL LL DL a h/t) pf | } ; y | : . + 7 EN : ; ey |: HANNE -. )

JANN i Aa “i + \\ Ht. eit it a | tea i/ | .\ Bru "hh. & | ‘wat Ah, . f t, 4 ‘ ‘| ) 1h & ; y 7 : \ i. i

AA \Y Hil\ \||eB MN! WANA) y. {iV) |AY |st.YY}1B) Fal} ye) AD Re, } h iy ; \ b Hie Be Y : bi f

) "Fl TE ae Hy fe Re , h| 1i ay /| OGG A } i if i ; ; : ' aDAree / i oe. f { in ; 1 . . ie OPO ARTS WER Te

»- \}: Ik Wore iv ) a Wty Vy omeae atadj | i wil “ffi MAA BL. PTT STPESyee abe

4

Plate 5.3. Companion to quipu example 5.7. (Courtesy of the

m Museum fiir Volkerkunde, Berlin, West Germany.)

f. \ ;

g ™ . Rs

”Ate ae ” we ha Ley ft

Plate 5.4. Carving detail from plate 5.3. (Courtesy of the Museum fiir Volkerkunde, Berlin, West Germany.)

FORMAT, CATEGORY, AND SUMMATION 99

Fig. 5.18. Quipu example 5.7.

i = | Subchart i = 2 Subchart ~~ Maximum values —————_}

Ss ee ee CCT) 8s Co: x A A ee

ke P23 4567 8 9 jr te se 56789

ee Ee eee 2; Ge PT Pipi id t.Pipi 2 Eege Geyey Ee

ee Oe 28 Ge Ge Be eee BE Ee Ee

AG Geeeee ee 38 Ge eeBe 2 ee2 ee:

20 Ge ee eee Be ee ee Sums equal

a) Category k = 10 contains all zeroes in both subcharts. b) The categorical sum for k = 6 (excluding the value for j = 1) is the same value in both subcharts. c) In each subchart, the categorical sum for k = 4 equals the categorical sum for k = 5. d) The categorical sum for 7 = 4 and j = 5 combined, is the same in both subcharts. e) In both subcharts, for each k, the values associated with j = 1 are greater than those associated with 7 = 2, ... , 9.

f) In addition to the relationships in the diagram: the categorical sum for j = 3 on i = 1 subchart equals the categorical sum for k = 4 oni = 2 subchart; and the categorical sum for j = 2 and j = 3 oni = 1 subchart equals the categorical sum for Jj = 6 and j = 7 combined oni = 2 subchart.

Quipu Example 5.8

1. Seven subcharts labeled i = 1, 2,...,7. They are of different sizes. By color coding, the smaller subcharts are related to parts of the larger ones. Notice in the description of the seventh subchart that there are only the second, third, and fourth categories of those identified with /.

a)Fori=1,3,4 pix (W=1, ...,5;k =1,..., 10);

b) Fori = 2 Pijk G=l, ...,5;k =1,...,1); c) Fori = $ Pike (J =1, 2,334 =1,..., 10);

d) Fori = 6 Piik (j = 1, 595k =1,..., 8); e) Fori =7 Piik (j = 2, 3,4;k =1,..., 10). 2. We calculated the categorical sums which form the chart p;;

@=1,...,7;j = 1,...,5). Each p;; is the sum of eight, ten or eleven values depending on how many categories were identified with k. To display the regularity of these sums, they are written in tabular form in figure 5.19.

100 CODE OF THE QUIPU

Fig. 5.19. Quipu example 5.8 sum table.

j=1 j =2 j =3 j=4 j=5

:S

i =6 533} -| — 1,200 1,052 1,120 3. Within each of the subcharts there is repetition of groups of values. The repeated values differ from subchart to subchart. a) In subchart i = 1, one grouping of five values (associated with

j= 1, 2,..., 5) is repeated twice (kK = 2, 3) and another is

repeated three times (k = 6, 7, 9).

b) In subchart i = 2, one grouping of five values (associated with

j= 1,..., 5) is repeated three times (k = 3, 6, 9). c) In subchart i = 3, one grouping of five values (associated with

j= 1,...., 5) is repeated three times (kK = 2, 3, 5). d) In subcharti = 6, two groupings of four values (associated with jJ = 1, 2, 3, 5) are each repeated twice (k = 2, 4; k = 6, 7). e) In subchart i = 7, one grouping of three values (associated with j = 2, 3, 4) is repeated seven times (A = 1, 2, 3, 4, 5, 6, 7). For this quipu, particularly from the sum table (fig. 5.19), we invite the reader to find as many regularities as he can. Different readers may find

different ones. But, in any case, we think they demonstrate that the logical structure of the quipu has been correctly identified from the cord spacing and color coding; that the data is additive and so the process of summation justified; and that the data is reflecting something planned carefully and with attention to detail.

l l The next chapter will continue with other logical structures. A later chapter will deal with other evidence of numerical and arithmetic relationships. But the particular quipus described earlier—those with cross categorization and summation—compactly characterize Inca insistence. There is formality of structure, both physical and logical. Cord placement, color coding, and number representation are the basic construction features repeated and recombined to define a format and con-

FORMAT, CATEGORY, AND SUMMATION 101

vey a logical structure. The structure itself is based on repetition and recombination of smaller units into larger units. There are categories grouped into subcharts and charts with sums and sums of sums. The data within the structure shows repetition and recombination as values are repeated in groups as well as summing to repeated or symmetrical values. These quipus certainly were constructed after careful planning but many also are part of a planning process.

Exercises Quipu example 5.9 in EXERCISES exercise 5.3 is AS199. For purposes of

the exercise we have modified one .

digit by 1 and another digit by 2. Origi- Exercise 5.1

nal errors in counting, errors in Three sheds are being built. They are different sizes but each has walls

arithmetic, errors in knotting, and our .

errors in counting knots or in transcrip- made of cinder block and a floor and roof made of wooden boards. The tion all are possible. Therefore, we do materials used for the first shed are: 284 cinder blocks, 100 pounds of not consider a few errors of 1 or 2 in mortar, 28 boards, and 200 pounds of nails. For the second and third

individual knot positions to be very 2 ; ; significant. sheds respectively the materials are: 244 cinder blocks, 85 pounds of mortar, 24 boards, 170 pounds of nails; and 364 cinder blocks, 150

pounds of mortar, 51 boards, and 400 pounds of nails. Design a quipu and record on it the amount of each material used for

each shed and their sums. Show the quipu on a schematic including relative cord placement, cord color, Knot types, and relative knot placement.

Fig. 5.20 An Answer to Exercise 5.1

2sxX Is 2s 2s Is 3sx Is 4s 8st 3sf Is* 7s 8s 2s Ask 8sK 2sxXK 7s 6sX 5s 5s 9s*¥ 3s 7s

4L SL 4L*% SL* 4L 4L IE 2LK SLK 3L

C1 C2 C3 C4 Cl C2 C3 C4 Ci C2 C3 C4 Cl C2 C3 C4

102 CODE OF THE QUIPU

C1, C2, C3, C4 are associated with cinder blocks, mortar, boards, and nails respectively. Cord groups 1, 2, 3 are associated with sheds 1, 2, 3 respectively. Cord group 4 is associated with the material totals. (Note that there are no sums associated with the individual sheds as the amounts of four different materials are not additive.) Exercise 5.2

A small community consists of four families. The families have the following members: Family 1: man, woman, man’s mother, man’s father, two children Family 2: man, woman, woman’s older sister, four children Family 3: man, woman Family 4: man, woman, three children One hundred potatoes and 242 ears of maize are to be distributed among

the families. Each person in the community is to get the same share. However, no potatoes or ears of maize are to be chopped in pieces. The distribution among families is to be carried out with whole items. Any excess is to be divided between the two largest families who will distribute it to the oldest members of their household.

1. Calculate the number of potatoes and the number of ears of maize to be distributed to each family. Write the results in a table. 2. For each family, calculate the number of potatoes to be distributed to adults and the number to be distributed to children. Do the same for the ears of maize. Write the results in a pair of tables.

3. Design a quipu and record on it the distribution plan as described by the answers to parts 1 and 2. Give the answer in the form of a schematic.

An Answer to Exercise 5.2 1. There are twenty people in the community. Since there are 100

potatoes and each person is to get the same share, each person receives 5 potatoes. Family 1 has six people so gets 30 potatoes, family 2 with seven people gets 35 potatoes, family 3 with two people gets 10 potatoes, and family 4 with five people gets 25 potatoes. There are 242 ears of maize to be divided equally among twenty

people. Each person’s share is 12. However, this leaves 2 ears of maize. The two largest families are families 1 and 2 so each of them

gets 1 extra ear of maize. Family 1 with six people gets 6 x 12+ 1 ears of maize, family 2 with seven people gets 7 x 12 + 1 ears of maize, family 3 gets 2 x 12 ears, and family 4 gets 5 x 12 ears.

FORMAT, CATEGORY, AND SUMMATION 103

Fig. 5.21 Fam. Fam. Fam. Fam.

1234

Maize 73 85 24 0 Fig. 5.22

12341234

2. Fam. Fam. Fam. Fam. Fam. Fam. Fam. Fam.

Potatoes Maize Here are some observations on the data in the answers: a) Note that each of the eight values in part 1 is the sum of two values in part 2. b) Because the 100 potatoes and 242 ears of maize are distributed according to the same scheme, the numbers of each per family are the same proportions of their respective wholes. The distribution scheme excludes the possibility of chopping up potatoes or maize. As a result, all the values are integers. The proportions are, therefore, “the same’’ to within that limitation. To demonstrate these proportions, Our answer for part 1 is rewritten below with the number of potatoes per family divided by 100 and the number of ears of maize per family divided by 242.

; Fam.5.23 Fam. Fam. Fig. , 5 3Fam. 4

Potatoes | 0.300) 0.350/ 0.100 | 0.250

Maize 0.302 | 0.351 | 0.099 | 0.248 , , c)- Because each person’s share is basically 5 potatoes and 12 ears of maize, and the family groups and age subgroups are sets of individuals, most of the data in the tables are multiples of 5 or 12. d) Because, within each family, the potatoes and maize are dis-

tributed the same way, the ratios of items for adults to items for children is the same in both tables in part 2. This consistency can also be seen in the ratios of number of ears of maize to number of potatoes for each subgroup.

104 CODE OF THE QUIPU

3. Here is one quipu schematic.

2sK 1s¥ 1s® 1s® 4sX 3s¥ 2s¥ 25 IsX 2sXK 1sX 2sXK 4sX 3s 3s 3sK isk 2s# Ts 8s 2sX 6s

SL OLX 7LX 4L¥4 4L SL¥ 4LX 8LX 6L SL SLA 3L*K S5L¥ 4L C1 C2 C3 C4 Cl C2 C3 C4 C1 C2 C4 Cl C2 C4 C1 C2 C3 C4 Cl C2 C3 C4 Colors C1, C2, C3, C4 are associated with families 1, 2, 3, 4 respec- Fig. 5.24 tively. Cord group 1 is associated with items distributed to adults (potatoes followed by maize); cord group 2 is associated with items distributed to children; cord group 3 is associated with items distributed to whole families. (Check your quipu design to be sure that it conveys that there are no children in family 3. Everyone was given food—none received a share of zero).

Another observation: the data in the tables or on the quipu can be described as: a chart of elements p;; @@ = 1, 2; 7 = 1, 2, 3, 4) where each element is the sum of two elements in associated subcharts; subcharts of elements pjjx Gi = 1, 2; 7 = 1, 2, 3, 4; k = 1, 2); 71s associated with food type, 7 with family, and k with age group. Exercise 5.3

The first seventy-seven cords (quipu example 5.9) are separated into eleven groups. Each group consists of six pendant cords tied together with a top cord. From the color coding and the way the top cords are attached, we thought that relationships between values might become more apparent if some cord groups were read left to right and others right to left. The values are written in figure 5.25 in accordance with our thought. Notice that there is a pattern to the reversals so that the eleven

groups form three sets: ( Subsidiaries group 22| } A Lower subsidiaries of group 2 ~——> Pendants of group 3 } B A Pendants group 3 Pendants of group Subsidiariesof of group 3 ———————»> Subsidiaries of group 44 } A

124 CODE OF THE QUIPU

The belief that this is not simply a numerical record is corroborated

by the nonstandard use of knots. Only single knots are used in all knot

clusters. Therefore, all or some may be number labels. The values of | the knot clusters are included in the chapter notes (p. 123) in case the reader wishes to continue the search for patterns on these quipus. It would be particularly interesting if rules of correspondence could be found that would transform the knot values on quipu I into those on

quipu II. | , Quipu Example 6.9 |

This is a most intriguing quipu. A long cord suspended from a loop

attached to the main cord can be seen on plate 6.2. This “‘tail”’ is clearly | | neither a pendant cord nor a dangle end cord. Small pieces of cord are

ee ee +. | ee 0 2 -— \ » | NE ge Oe a gE ee _OO0O0OX—_ a. | : . oe

=eco —F=ERS a eeerrr ORES Ee ie : a, oe ae igttm, =: es |Se| :me | Ae gg a:See | Hae 8 a oe Je - \ : i ‘ ee eee ae ke oe oo e : a8 ie - OO ee ee ae 2 oe oe Ee : a see LN . oe oo ee ee Ee ees ee

eee FAM fr) te Me =f

a. —-—_ {_ ee 1. - =) ~=~——hChC)rUlrlUrULUULCU

LP 146 lr rc rrr

Se . Sd fae ff . . Ff =. |

@# ; ° 8 8}»}§=»60x®_L—_zZ-e—r™ $f FF yg : - . fo...

rrr rrr ee errLrrrrrrrr—r lr

errr CL Fe"”?rrrrlh—lc rr rr -_-e__, rr r—r.-—re—i“=#RECOCC

lr rer cs er CCl CC CL Plate 6.2. Quipu example 6.9. (In the collection of The University Museum,

University of Pennsylvania, Philadelphia.) | ,

HIERARCHY AND PATTERN 125 |

attached to it. Because of the way they are attached, and because they are much too short to have knots on them, these are not subsidiaries but are ‘“‘flags’’ of some sort. This construction component, a tail with single knots interspersed with colored flags, was found on only two other quipus. Another striking aspect of the quipu is that each pendant has either one figure eight knot or no Knots at all. Rather than representing the numbers 1 and 0, these could represent any dichotomy such

as yes/no, present/absent, or in/out. The quipu has, in all, pendants colored C1, C2, C3, C4 or CS, each with a knot (K) or no Knot (0), in the following order: C1(K), C2(K), C1(K), C1(K), C1(K), C1(K), C2(K), C2(K), C2(K), C3(0), C40), C5(0); and a tail with flags of color C1 or C2 separating the single knots (s) into groups of one, two, or three in the

following order: 3s, C1, Is, C2, Cl, 2s, Cl, 1s, C2, C2, Cl, 2s, C2, C2. Comparison of the body of the quipu with its tail shows a basic theme

restated in different ways. There are nine knotted pendants and there are nine flags. They can be placed in a one-to-one correspondence with each other because they are in exactly the same color order. There are also nine single knots on the tail. Just as the flags separate the knots into groups, the knots separate the flags into groups. The numbers of knots in the knot groups are 3, 1, 2, 1, 2. The sizes of the flag groups are 1, 2, 1, 3, 2. The sizes of the groups are the same, and the orders in which they occur correspond to each other according to the rule: size of (5-i)” flag group = size of i” knot group i= 1, 2, 3, 4, 5; arithmetic is mod 5. To visualize the rule above, view the tail as being tied in a circle and list the knot group sizes clockwise:

21 12

Fig. 6.16 | First 3

rotate the circle two positions clockwise:

126 CODE OF THE QUIPU

Fig. 6.17

22 13 | First

and now read off the sizes counterclockwise: 1, 2, 1, 3, 2. Thus, there are essentially nine elements. They are associated with the color order C1, C2, C1, Cl, C1, Cl, C2, C2, C2 as expressed on the flags and reiterated on the knotted pendants. The nine elements are divided into groupings of 1, 2, 1, 3, 2 as expressed on the flags and reiterated by the knots. The relative order of the groupings remain the same whether they continue around or are listed in either direction. It is, therefore, the relative, rather than absolute order, that is being emphasized by the reiteration.

10 The information on any quipu is in both the data and the organization of the data. Planning, design, and construction, which embedded information into a quipu, had to be complemented by retrieval of the information. Recognizing a quipu as being of a prevalent format provides the conceptual framework for retrieving information from it. Interpreting and understanding quipus whose structures are more individualized also has to draw on a set of shared ideas. Prevalent formats and individualized quipus that are highly ordered compositions result from the same underlying system of concepts and rules for the expression of the concept.

EXERCISES Exercises (a) The afghan of exercise 6.2 is B~894 of Creative Afghans, Coats and Clark’s Book no. 223 (New

Exercise 6.1 York, 1972). The assembly diagram 1. Draw a tree diagram to represent the following game. A box is on page 7. .

. . (b) The quipu in exercise 6.3 is

contains three cards which are colored red, yellow, and green. The AS98. The second pendant in groups 14 player draws a card and receives $1.00 if red was drawn, $2.00 if and 15 each have a subsidiary of yellow was drawn, and nothing if green was drawn. The card is re- ted from the exercise.

. . : color C6. For simplicity, this was omit-

turned to the box and the player draws a card again. He receives $4.00 more if he draws the same color as he did the first time. He

HIERARCHY AND PATTERN 127

, receives nothing and must stop playing if he draws a different color. The card is again returned to the box. On the third and last draw, the player gets $10.00 more if he draws the same color and nothing for

a different color. |

2. Design a quipu to represent the same situation.

An Answer to Exercise 6.1

Fig. 6.18 1. Tree diagram

R 1Y 2G 0 R /Y|G R/|O YG 4Y 0 /G 0 0R 4; 0 0| 4

R Y| Y |G0 R Y|10 G 10 0 0G0R10 0 0; Fig. 6.19 2. Quipu schematic

RYG

1E | 2L Is

Is 1s 4L 4L 4L GYRGYRGYR GYRGYRGYR

128 CODE OF THE QUIPU

Exercise 6.2 The making of an afghan includes knitting squares of four different solid

colors and embroidering some of them with a leaf motif. The scheme for assembling the finished squares is diagramed in the instructions. Fig. 6.20. D=embroidered motif.

!

1. Design a quipu for displaying the assembly scheme. Give the answer in the form of a schematic. 2. Describe any patterns you can find in the planned afghan. An Answer to Exercise 6.2

1. The quipu could have four colors of pendants C1, C2, C3, C4 corresponding to the four colors of knitted squares. The dichotomy, embroidered/not embroidered, can be represented by a knot or no Knot. Each group of pendant cords corresponds to one row on the diagram.

HIERARCHY AND PATTERN 129

C1C3C2 C43 C2C4C3C1C2. C3C1C4C2C3) C1C2C3C4C1 C2C4C1C2C1 C1C3C2C4C3 C2C4C3C1 C2

Fig. 6.21 2. Embroidered motifs alternate within each row. However, in alternate rows they begin at the edge or one square over. Therefore, rows 1, 3, 5, 7 have DODOD and the alternative rows 2, 4, 6 have ODODO. Also, the first two rows and last two rows are the same colors: the color order in row 1 corresponds to the color order in row 6, and that of row 2 corresponds to row 7. Exercise 6.3 A quipu currently housed in Berlin consists of fifteen groups of pendant cords. Groups 1, 4, 7, 10 and 13 have six pendants each. Groups 2, 3, 5,6, 8,9, 11, 12, 14, and 15 each have five pendants with two subsidiaries on the first. The colors of cords in groups 1, 4, and 7 are Cl, C2, C2, C2, C2, C2 and those in groups 10 and 13 are all color C5. The colors of cords in groups 2, 3, 5, 6, 8, 9 are C2 (with subsidiaries of C1 and C2), C3, C2, C4, C2 and those in groups 11, 12, 14, and 15 are C5 (with subsidiaries of C5, C5), C3, C5, C4, CS.

1. Find a rule that transforms the colors of group 2 into those of group 11, and which also transforms the colors of group 1 into those of group 10.

2. Let X stand for the cord arrangement and color pattern of group | and X' stand for the same cord arrangement but transformed color pattern in group 10. Similarly, let Y stand for the arrangement and color pattern of group 2 and Y' stand for the same arrangement but transformed color pattern in group 11. Using only the symbols X, X', Y,Y', write a description of the fifteen cord groups on the quipu. 3. Using your own terminology, describe the overall pattern that results from 2.

130 CODE OF THE QUIPU

Answer to Exercise 6.3

1. Every place that group 2 has colors Cl or C2, group 11 has color C5. Similarly, every place that group 1 has Cl or C2, group 10 has color C5. The rule of transformation is, therefore, replace all C1 and C2 by C3. 2. XYYXYYXYYX'Y'Y'X'Y'Y'

HEIRARCHY AND PATTERN 131

~~ / | Arithmetical Ideas

and Recurrent Numbers

1 Numbers with their sums have already been seen on many quipus. But there are other numerical relationships on quipus that show other arithmetical ideas. In order-to understand these ideas, we need to discuss more about numbers and their representations. 2 How positive integers are symbolized in our base 10 positional system and how integers are represented by knots on quipus has already been explained. But we write more than just positive integers. If, for example, one wants to represent one-half, it is written as the fraction 1/2 or as the decimal fraction 0.5. These symbols are in different notational systems yet they represent the same value. A fraction is a single entity but contains two distinct integers (for example, 2/3, 7/4, 1/324). The value of a fraction can be more or less than 1. A decimal fraction (0.372, 0.25, 0.414141. . .), on the other hand, is part of the base 10 positional notation. In base 10, a decimal point is used to separate integers from decimal fractions. Starting at the decimal point, as you move to the left, the value of each symbol in the collection is multiplied by one higher power of 10; starting at the decimal point, as you move to the right, the value of each symbol is divided by one higher power of 10. For example, 437.243 has the value

(4 x 100) + G x 10) + (7 X 1) + 2/10 + 4/100 + 3/1000. Every fraction can be translated into base 10 positional notation but not every decimal number can be translated into a fraction. Most people have internalized the translation from one notation to the other for a

limited set of values, such as 0.5 = 1/2, 0.75 = 3/4, 0.333... = 1/3. (The means of translation from one notation to the other is elaborated in the notes that accompany the chapter. Which decimal numbers cannot be translated and what we do with them, is also included there.) Since decimal numbers can represent any value that a fraction can, we could just use them but we persist in using both. This is probably because they stress different ideas and so are useful in different ways. Fractions give a direct statement of the result of dividing some number into several equal parts. For example, when 6 is divided into 8 equal parts, each resultant part is simply 6/8. Also, viewing a fraction as a ratio, one can find from it other numbers with the same ratio. For example,

6/8 = 2(3)/2(4) = 3/4 or 6/8 = 3(6)/3(8) = 18/24. Decimal fractions, on the other hand, greatly simplify arithmetic. Each fractional value has the same general representation: the parts of

ARITHMETICAL IDEAS AND RECURRENT NUMBERS 133

2 Every fraction can be translated into at the representation, if it is base positional notation. The translation translatable. procedure is long division. That is, to If the decimal fraction has a finite find the decimal representation of 3/4, number of positions, it can be written

divide 4 into 3: as a fraction. Examples of decimal 0.75 fractions with 1,their 2,translations and 4 positions and are:

28 20 05-5191. 10 2(5) 2’

4[ 3.0

20 7 5 70+5 75 325) 3.

However, there are some values that 0.75 = 10’ 100 100. 100 4(25) 4” can be represented in base 10 positional

- notation that cannot be translated into 0.6216 = 6 n 2 in I + 6 _ 6,000 + 200+ 10+ 6 6,216 = 8777) 777 .

fractions. One can tell, just by looking 10 100 1,000 10,000 10,000 10,000 8(1,250) 1,250 A decimal fraction that has an infinite To translate an infinitely repeating This leaves decimal numbers whose number of positions but in which, from decimal fraction into a fraction, the fractional parts have an infinite number some position on, the same digits are number is multiplied by powers of 10 of positions but which do not cycle. repeated in a cycle, can also be written such that when the multiples are sub- There is no way of translating them

as a fraction. For example, tracted from each other only an integer into fractions. This is not because we

remains. Integer arithmetic is then do not know the translation rule; it 0.3333 ....= 1/3; used. is because no fraction can have an 0.2143143143 . . . = 2,141/9,990. equivalent value. These numbers, incidentally, are inappropriately, but perhaps understandably, called irrational numbers. When such a number

Example: x = .2143143 ... 1. Let x stand for the value. . commonly occurs because it corresponds to a value that arises in a

10x = 2.143143... 2. Multiply by the power of 10 that common situation, it is often repreplaces the first cycle at the decimal sented by a special sign which is not in

point. either of the two notational systems. The most familiar is 1. The value of the

10,000x = 2,143.143143 ... 3. Multiply by the power of 10 that ratio of the circumference of a circle

places a single cycle to the left of to the diameter of that circle is an

the decimal point. irrational number. There is no integer that when divided by another integer is

9,990x = 2,141 4. Subtract the result of step 2 from equivalent to it, so it cannot be repre-

the result of step 3. sented as a fraction. If one were to

. ; write it as a decimal, the representation

x = 2,141 5. Divide and reduce if possible. is unending and, hence, most inconve-

9,990 nient. Any decimal representation

that uses only a finite number of digits,

Example: x = .666... Step 1. such as 3.1416, is only an approximaThere is no need for Step 2. tion to the value. Also, any fraction 10x = 6.666 ... Step 3. representation such as 22/7, is only an

Ox = 6Step Step x = 2/3 5. 4. approximate value.

134 CODE OF THE QUIPU

. . . ematica otations, VO. >; reprin

numbers that are used are confined to powers of 10. Thus, of all the 3 @) Fonian Palo, BL 1008 ie

fractions that are equivalent to say, 6/8, one uses 75/100 or 0.75. With ed., LaSalle, Ill.: Open Court Pablishthis systematized way of representing all fractional values, arithmetic ing Co., 1974), contains examples of

be d th h thej t ti ‘nulati What j th expressions of fractions in words by the

can | e one rough their sys ema ic manipulation. at is more, the Greeks. For example, Eratosthenes’

manipulations are the same as for integers. description of 11/83 of a unit arc of the

earth’s meridian is ‘“‘amounts to eleven

3 Clearly, oo,the; idea parts ofthingswhich meridian eightyof dividing into fractionalthe parts and the idea three’ (abouthad 200 B.c.E.). Also, he of common ratios can exist with or without a scheme for the represen- notes that unit fractions were fre; ; ; ; ; quently used by Geminus (first century tation of fractions. And fractional parts can be combined and modified B.C.E.) and later by Diophantus (third

. . . (fifth century C.E.), and that ‘“‘the

in systematic ways that do not involve manipulations of symbolic rep- century C.E.) and Eutocius and Proclus resentations. On the simplest level, even a child who has no knowledge fraction 1/2 had a mark of its own, of how to write fractions can divide up a set of items. But complex namely W or he » but this oe enation mathematics, too, can be built without fractional representation. The the Greeks than were the other notaGreeks of the fourth century B.c.E. and throughout what is often re- tions of fractions.’ See his pages 26-27. ferred to as the golden age of Greek mathematics (first century C.E.) are (0) To give some idea of how ratios a good illustration of this. They had no general way of representing we quote from Carl B. Boyer, A Hisfractional values yet they, nevertheless, devoted considerable attention tory of Mathematics (New York: John to number theory and to ratios. Their approach, however, was theoretical Eudoxus “Apparently the Greeks had and geometric rather than computational. The Greeks not only. had no made use of the idea that four quantati f of fracti | val thev had definite belief thatthat ththere titiestheare inratios proportion, a:bhave = c-d, representation fractional values, theya had a definite belief two a-b and c:d the if

. . . . . was no more adopted generally among

. : ‘ j were dealt with without fractions, b . . . Wiley and Sons, 1968). Prior to

should be none. | same mutual subtraction. That is, On quipus we find no evidence of the representation of fractional the smaller in each ratio can be laid off

oe on the larger the same integral number

values. We do find evidence for division into parts and the use of com- of times, and the remainder in each mon ratios. Just as in the case of summation, where numbers and their case can be laid off on the smaller the . same integral number of times, and the sums appeared together, we have no way of knowing how or where the new remainder can be laid off on the

calculations were done or even what calculations were done. former remainder the same integral

number of times, and so on”’ (pp. 98-

4 Several — OS 99). Then, Euclid’s Book which quipus show the arithmetic idea offrom even division into parts. deals with the generalV theory of propor-

. . . . tions, he states Eudoxus’ definition.

The simplest example is a quipu where the total of all the values is 200. “Magnitudes are said to be in the same The quipu consists of two cord groups, each of which has a total value ratio, the first to the second and the of 1/2 of 200 (100). Then, in the cord group made up of two pendants, multiples whatever be taken of the first each has 1/2 the value of the total for the group (50). The other group and the third, and any equimultiples has six pendants. The values are the result of dividing 100 evenly into whatever of the second and fourth, the

. third to the fourth, when, if any equi-

; former S1X parts. |equimultiples are alike equal to,alike or are exceed, alike less

° ae ° . . . COrrespo orade . .

Consider, for a moment, the results of dividing integers with the re- than, the latter ee taken in

striction that only integers can occur. If 12 is divided evenly into three (c) The Greek sy tat 2. ore should parts, the result is 4, 4, and 4. If 13 is divided evenly into three parts, be no fractional representation is . clearly set forth in Plato’s The Republic the result cannot be 4 1/3, 4 1/3, 4 1/3. It must be 4, 4, 5. This manner (Book VII, 525, as translated by Benjaof division is frequently used by us in situations where we restrict our- min Jowett). ‘‘. . . steadily the masters selves to integral results even though fractions are available. People are of the art repel and ridicule anyone viewed as units with integrity. We view many other things as funda- when he is calculating. . . .”

j ‘ . ; ; who attempts to divide absolute unity

ARITHMETICAL IDEAS AND RECURRENT NUMBERS 135

4 (a) The first quipu in this section is mental or indivisible units. For example, 9 pennies evenly divided into

Ah) Ouipts exemple 71 nae P13>

Sum of level 1 second branches and their branches = 19 x 141 Sum of level 1 second branches and their

branches = 19 x 19 x 5

branches on top tree | ARITHMETICAL IDEAS AND RECURRENT NUMBERS 141

The number nineteen is of calendric concern because it is related to the alignment of cycles based on the sun with cycles based on the moon. In a calendar based on the appearances of the moon, months are 29 or 30 days so that the same phases of the moon occur on more or less the same date each month. In a calendar based on the earth’s revolution about the sun, a year is 365 or 366 days so that the equinoxes and solstices occur on more or less the same date each year. If both relationships are

considered, the time it takes to return to the same association of moon,

, earth, and sun is nineteen solar years or 235 lunar months. To retain some synchronism between a lunar calendar and the seasons, lunar months are combined into lunar years. But the lunar years have to vary,

with some having 12 months and others 13. If there are basically 12 lunar months in a year, then within the nineteen-year cycle, 7 more months have to be inserted. Similarly, the nineteen-year cycle would be

| important to those with a solar calendar if they wanted to calculate the dates of new moons. A quipu emphasizing the number nineteen, containing multiples of nineteen and categorical sums which are multiples of nineteen, raises the possibility of a calendric association. The seven final pendants summing to nineteen are suggestive of a scheme that divides the nineteenyear cycle into seven unequal parts in order to place the seven additional months. As intriguing as the idea sounds, there would have to be a considerable amount of associated evidence before any single quipu could be so specifically interpreted. Nevertheless, it is certainly plausible that some quipus are recordings of astronomical values or other fixed values related to the natural environment.

6 Notations for fractions are discussed 6 Before discussing numerical relationships that demonstrate more in-

aon i Si oan mated tricate arithmetical ideas, a final point about number representation needs decimal fractions are discussed on to be made. We have distinguished between the idea of fractional values pp. 314-35. The idea of decimal frac- and a symbolic representation of them. We distinguish further between discussed in George Sarton, ‘‘The First a variety of representations and a standardized representation that is Explanation of Decimal Fractions and generally used.

tions and their notations are also . . . . . tory of the Decgnal Hien and 2 a HS: The use of spaced knot clusters with particular types of knots to Facsimile (no. XVII) of Stevin’s represent integers on quipus is sufficiently standardized to be recognized in particular pt Lil (sess: 0100). by us. Moreover, they occur on some quipus where the interpretation

pp. 167-86 of the article. can be corroborated. If different quipumakers had the same basic idea but represented it more individualistically, we possibly would not be able to recognize it. The history of Western notation for fractions and decimal fractions is an interesting example of varied and individualistic expressions for the same basic idea. It serves as a reminder that unifor-

142 CODE OF THE QUIPU

mity of number representation is more common to u: in our age of standardization than it was in other times or places. As late as the sixteenth century, individuals used special signs for special fractions, such as ~ or = (in England) and © or & (in Spain) for one-half, and ¢ for onefourth and ~ for three-fourths (in England). A horizontal line separating the numerator and denominator (i.e., ) became general usage only in

the sixteenth century, but in the seventeenth century there were still those who omitted it. Later, starting about 1850, because fractions had to be compressed into a single line of print, many people began to use the solidus instead (1.e., 3/4). Now, as common practice, we use both forms. Similarly, the idea and clear statement of decimal fractions is attributed to Simon Stevins (Belgium, 1585) but our notation for them did not become standardized for hundreds of years. Stevins wrote 23759

notations are: | for the value we symbolize as 2.3759. Examples of other individuals’ 123.459.872 = 123.459872 (Beyer, Frankfurt, 1603)

693 @ = 6.93 (von Kalcheim, Bremen, 1629) 16 | 7249 = 16.7249 (Jager, London, 1651)

3:04 = 3.04 (Rawlyns, England, 1656) 31,008 — 31.008 (Foster, London, 1659)

22=3 = 22.3 (Caramuel, Bohemia, 1670) 732,|569 = 732.569 (Raphson, London, 1728)

4.25= 4.2005 (Chelucci, Rome, 1738) And, of course, standard usage today differs from France (4,5) to England (4¢5) to the United States (4.5). 7 So far, in the quipu examples, when referring to fractional parts, we have used fractions rather than decimal fractions. Since we will now begin to use decimal fractions, we pause to first explain why and to

explain the difference in the associated statements about accuracy. | We have used statements of the form: ‘‘to within +1, X = , Y.”’ In each case, we focused on an integer (X), and associated with it another integer (Y) and a fraction that related them. The accuracy statement ‘“‘to within +1’? was connected to the integer X. In the examples that

follow, we wish to compare ratios of numbers. For example, we want ;

ARITHMETICAL IDEAS AND RECURRENT NUMBERS 143 :

to be able to say that a/b = c/d. Translating a/b and c/d into decimal fractions takes the attention off the specific integers involved and places

| it on the result of combining them. Similarly, our accuracy statement

integer. |

must be about the values a/b and c/d rather than about any specific _ Consider several fractions that are near 50/100 or 100/200: 49/99,

50/101, 51/101, 99/200, 101/200, 99/199, 100/199, 100/201, 101/201. Both

the numerators and the denominators vary, but they vary together. The numerators from 49 to 51 are combined with denominators from 99 to 101 and those from 99 to 101 are combined with denominators from 199 to 201. It is easier to see how much, if at all, the variation effects the results when decimal fractions are used instead. If each of these fractions is translated into a decimal fraction that is rounded to three positions, they are all near 0.500. All the values are between 0.495 and 0.505 and so the maximum deviation from 0.500 is —0.005 or +0.005. Since 0.005 is 1 percent of 0.500, we say that the fractions all have the value 0.500 to within 1 percent. Fractions with quite different numerators and denominators could be included in this generalization. For example,

76/151, 106/213, and 1107/2197 all have the value 0.500 to within 1 percent.

8 Quipu example 7.5 is AS120. Quipu 8 Quipu Example 7.5

example 7.6 is AS143. This quipu consists of a chart of twenty-four values p;; (i = 1, 2, 3; j = 1, ..., 8) and the categorical sums for j = 1, ..., 8. Also, for each i, for 7 = 3, there is a subsidiary value. Their categorical sum is on a subsidiary attached to the j = 3 categorical sum. To refer more | easily to the categorical sums, we call them ps; (j = 1, 2, 3, 3 subsidiary, 4,...., 8); the value of each ps; is p1; + Pej + p3j- No two values in

| the chart are the same and the range of values in the chart is considerable; the smallest is 102 and the largest is more than 400 times as much, namely 43,372. But, the ratios of the values in the chart to their categorical sums are remarkably consistent. The numerators of the nine

ratios p1;/ps; Uj = 1, 2, 3, 3 subsidiary, 4,... , 8) range from 102 to over 140 times as much (14,743), and the denominators range from 300 to 43,372. But, for all but 7 = 3, each ratio is 0.340 to within 0.6 percent.

(The exception, j = 3, deviates from 0.340 by 1.4 percent.) Similarly, for the nine ratios p2;/ps;, with the exception of j = 1, each is 0.425 to within 0.7 percent. And to within 0.9 percent (again with the exception of j = 1), each p3;/ps; 1s 0.235. Comparing the chart values to each other, there is a simple fraction relating each of the nine p;,; to the corresponding pg; and p3;. To within 1 percent, each P1j Is 17/33 of P23 + p3j- UV = 3 is still an exception; it deviates by 2.2 percent.)

144 CODE OF THE QUIPU

Quipu Example 7.6 ~ Consistency of ratios is also found on quipu example 7.6. It too has a

large range of values. In fact, it contains the largest number we have | seen on any quipu, 97,357. Even more interesting is that the ratios themselves are similar to those in the previous example.

Part of the quipu is a chart of sixteen values p;; @ = 1, ..., 4; j= 1,...,4) and the categorical sums for 7 = 1,...,4. The values in the chart range from 2,026 to 22,469 with no. two being the same. Again, calling the categorical sums p,; (J = 1, 2, 3, 4) and finding the

ratio of each p;; to its categorical sum: p,;/ps; = 0.110; poj/psj = 0.228; p3;/Ps;j = 0.437; and p4;/ps; = 0.225. Each of these is within 1.6 percent with the exception of j = 4 fori = 3, 4. (Curiously p44/ p64 deviates by exactly 10 percent.) Comparing the chart values to each other, the same simple fraction as appeared in the last quipu relates the P41; and po; to the corresponding p3; and p4;. To within 1.2 percent (but 2.7 percent for j = 3), each sum py; + Pog; is 17/33 of p3j + Pa4j.

Quipu Example 5.3— Continued

This is one more quipu that is similar in that it has consistent ratios. It has already been described in detail as an example of a chart which contains categorical sums of values in subcharts. And five values in the subcharts are themselves categorical sums of values in sub-subcharts. The quipu was further elaborated upon when discussing regularities on charts since the values in the chart, when compared to the sums of their respective categories, have consistent ratios. Again, the values in the chart are quite dissimilar as they range from 660 to 21,243. Figure 5.16 shows the ratios of the values to the sums of their respective categories. Now comparing them, we can say that, within 0.8 percent, the ratios

(rj;7 = 1,...,5) are ry = 0.122; ro = 0.017; rg = 0.534; rg = 0.105; rs= 0.222 for both values of i. In all three of these quipus, the values have a very large range. Yet, they are the same fractional parts of their own categories and not simple fractional parts of those categories. To have, as in quipu example 7.5, nine values that each are 0.425 of such diverse numbers as 1,068, 5,485, and 42,760, requires a concept of proportions and a scheme to somehow calculate them. If there were evidence of decimal fractions on the quipu, it could just be said that somewhere 0.425 was multipled by each of these numbers. In the absence of evidence of the representation of decimal fractions and of fractions, the conclusion has to be that the Incas clearly could work with fractions without having any representation for them.

ARITHMETICAL IDEAS AND RECURRENT NUMBERS 145

9 (a) The Greek study of proportions 9 We now pursue further the values that have appeared as consistent started in Pythagoras’ time (ca. 500 _ ratios. This is done in order to see if there is anything these values have geometric, and harmonic means. The in common that might give some clues as to the type of calculations they seventh proportion is said to have been arose from or to their use. We note, first of all, that there is insufficient and also discussed by Nicomachus data available about the three quipus to either confirm or deny that they (ca. 100 c.£.). For a fuller discussion, are related to one quipumaker. They do, however, have the same general / see Thomas Heath, A History of Greek oe ag ee Mathematics, vol. I (London: Oxford provenance and there are similarities in color.

B.C.E.) with the study of the arithmetic, ; ; ; j added by Myonides and Euphranor . . .

University 1921), pp. 85-86 Since there are three values characteristic of quipu example or M.PressGhyka, The Geometry of Art and . .7.5,. a_

Life (New York: Dover Publications, classification scheme devised by the Greeks can be used. They estab

1977). lished that for three values, a, b, c such thata < b) are accurate to within satisfies the relationship to within 1.6 percent and the latter to within 0.4 0.4 percent, 1—X? is to within percent.) Therefore, the three sets of values, when viewed as having | percent and 1-X- x(? =) to within three values each, do indeed have something in common.

1.7 percent. Fig. 7.3. Consistent ratios. (c) Figure 7.8 is quite similar to a

geometric form thought to be important . , . and persistent in the cosmology of Quipu Example 7.5 Quipu Example 7.6 Quipu Example 5.3 western South America. See W. H.

Isbell, ““Cosmological Order Expressed 0.340 0.110 0.222 in Prehistoric Ceremonial Centers,” b 0.338 Actes du XLII° Congres International des Américanistes 4 (1976): 269-97. 0.228

, 0.105

Cc 0.425 0.437 0.534 >? 0.656 0.017

a 0.235 0.225 0.122 146 CODE OF THE QUIPU

We pursue their commonality further in an attempt to better visualize it and to relate to it the individual values that were combined together. We present a spatial visualization that also serves as a basis for a simple scheme for generating the values. Consider, first of all, any three values a < b < c. Define c as a unit

and a and b as parts of that unit. Associate c with the area of a unit square and a with the area of a rectangle formed by cutting a strip off a unit square. Algebraically, the values a, b, c are being normalized to alc, b/c, and 1 and then a/c is being relabeled 1—X.

~H ;

Fig. 7.4 1 1-X X

| Area = om I-X 1 | Area = £ = 1 !

c | —+——— Strip that was cut off _

We next seek to interpret b/c as the area of a figure related to the unit square and the rectangle. Because of the specific relationship that exists between a, b, and c, b/c = 1—X?. That corresponds to cutting from a unit square a smaller square whose length and width are each the width of the strip that was cut off before. Figure 7.5, therefore, represents the commonalities of the three quipu ratios. The ratios of the areas of parts of this figure to the area of the whole figure are the characteristic ratios of the quipus. For each quipu there is a different value of X involved.

]y1

Fig. 7.5 | I 1-x x]

1—X 1 1—X

For quipu example 7.6, the value designated as b was the sum of two values. Splitting b into its two original parts, we find that the parts neatly fit into our picture. The portion of the figure associated with b simply

needs to be separated along the line that was used to define the width of the small square.

Fig. 7.6 I-X_ A

| pb ARITHMETICAL IDEAS AND RECURRENT NUMBERS 147

For quipu example 5.3, the value designated as c was the sum of three values so, in the picture, it is the unit square that must be partitioned. Again we find that the partitioning depends on X in a straightforward way. The three values separate the unit square into a rectangle of width X,acircle of diameter 1—X, and a rectangle of width 1—X from which the circle has been subtracted.

Fig. 7 y 1x 1g. 7./

| CO The three sets of ratios, with their three, four, or five values, are, there-

fore, still summarized by the original figure. Now, however, the figure has more subdivisions within it (see fig. 7.8). Fig. 7.8

1-X X Xx 1—-X 1—-X | :

|

x The model presented in figure 7.8 can be restated as a numerical scheme for generating the ratios in question. To generate a set of three values:

1. Take some whole and divide it into three equal parts: P, P, P. 2. Leave one part as is: P 3. Withhold some fraction of the second part: P(1—X) 4. Withhold the square of that fraction of the third part: P(1—X7). Let us, as an example, generate some numbers according to this scheme. Starting with 3,000, divide it into three equal parts of 1,000 each. Leave one part as is (1,000); delete, say 1/3, of the second part (1,000 — 333 = 667); delete (1/3)? = (1/9) of the third part (1,000 — 111 = 889). The resulting numbers are 1,000, 667, and 889 and their sum is 2,556. The ratios of the numbers to their sum are 0.391, 0.261, and 0.348. Starting with a different whole, say 4,230, and using the same fraction 1/3, the

resulting numbers are 1,410, 940, and 1,253. Their sum is 3,603 and

148 CODE OF THE QUIPU

again the ratios of the numbers to their sum are 0.391, 0.261, and 0.348.

Calling the ratios a, b, c such that a