Mathematical Cuneiform Texts(1)

7/27/2019 Mathematical Cuneiform Texts(1)

1/155

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ORGEA.KENNEDY

IETY

UT

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RIENTALRESEARCH

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mehasbeenaided bygrants

lofLearnedSocietiesandthe

lAssociationofAmerica.

ety

ANCASTER,PENNSYLVANIA

OFAMERICA

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eHistoryofMathematics,

and

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he editionofhitherto

altextschieflyfromAmerican

thisnewgroupof docu-

viouslypublishedmaterial

ts,butisitself sufficiently

pressionof themaintypes

caltexts.Newfrontiers,

beennomore thansus-

edbyatablet(Plimpton

ean"NumberTheory.

rialwe haveenjoyedthe

tyofmanymuseums.For

clyextendourthanksto

collectionsofTheOriental

ofChicago,TheFree

ol(England),TheMetro-

ew YorkCity,ThePlimp-

iversityinNewYorkCity,

mofTheUniversityof

hia.TheDirectorofThe

ndlygrantedpermission

gtothe Morgancollection,

eUniversity.Thelate

edthephotographsof

enerouslyagreedto

preliminarycopiesoftwo

m.Dr. F.R.Steelewas

gusdescriptionsandcolla-

-textsinTheUniversity

ofPennsylvania.Several

me museumweremadefor

ecialmentionmustbe

anCollection,thesource

materialpresentedhere.

em-textshavenowbeen

ionthanfromanyother

.J. Stephens,thecura-

bakingandcleaningofthe

ubtfulreadings,andin

aphsofthetextsatour dis-

oustable-texts,mostof

edinthisbookweredis-

Wearealso deeplyin-

ving contributedtothis

hapteronthedialectsof

dianmathematicaltexts,

emeansatour disposalfor

)thetabletsof unknown

ntingwerebeingworked

calproblem-textsweredis-

dStephens.Thesedocu-

Eb,Ec,Ja,Pa,Sa,Sb,Ue)

thisvolume.Wecall

seofthe possibilitythat

thesetextsmayhavebeen

ffortsto takeaccountof

eAmericanOrientalSo-

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ationofAmericaforhaving

meof TheA.Buffum

ardtheprintingcosts.We

mericanCouncilofLearned

canSchoolsofOrientalRe-

e publicationofthisvol-

btedtoTheRockefeller

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e

mberSystem2

cription2

calCalculations6

StandardType11

Larger Extent13

andIrregularNumbers16

rocals17

9

les20

nTables24

aria

ts;CubeRoots33

39

s41

852

5) 44

2)45

tary46

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7)93

tofaVolume-

ntary102

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ntary117

26

Coefficients

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ON

romancientBabylonia

es:the"table-texts"and

abletextsareundoubt-

reinturncloselyrelated

epracticalneedssatis-

xtsareevident,sincethey

arrangementcertainmeas-

hs),theirmultiples,and

bletoreaddirectlyfrom

of acertainlargemeasure

berofunits ofasmaller.

tiontableimmediatelygives

ationofagivennumberby

tors.Allothertable-texts

gtothesameprinciple.

ometimesexhibitonlyvery

y,withpracticalquestions.

ntendedtoillustratethe

emswhichareproperly

fcourse,doesnotmean

also belonginthecur-

ols;indeed,manymulti-

ythosefromNippur,are

enasexercisesbypupils.1

eenproblem-textsand

act thattheordinary

muchlowerlevelofscribal

deuseoutsidetheschools,

tsareessentiallyamanifesta-

dwereundoubtedlywritten

eryrestrictedgroup of

t allsurprisingthat

anproblem-textshavebeen

eswhenweattemptto

oestablishits exactdate.

e table-textsareofknown

NippurandKi,'butitis

ontainsthesamemultiplicationtable

No. 99,3).

utlessthan 100problem-texts.

followingstatisticsfortable-textsof

b1,Assur1,Babylon1,Kis48,Larsa6,

9, Lagai2,Uruk1.Iti sworth

gletable-textis knowntohavecome

orethan15000tabletsfromthis site

wtimelimits forthese

problem-texts,onthe

-Babylonian(i.e.,theyare

esaround1700B.C.),but

nthedarkas totheir

einformationobtainedfrom

hmakeitpossibletodis-

nandsouthernOld-Baby-

ewtextsfromthelater

wofromUruk6andtwo

ermaynot evenbemathe-

evertheless,thattheextant

extsshouldbetracedback

ves.Thiscanbegathered

ons.(1)Manytabletsof

angedserially,asisex-

eringinthecolophons.8

dJ9 formaclearunit.

nofproblems,groupsof

Jinexactlythe sameorder

wooftheproblem-texts

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STEM.TRANSLATIONANDTRANSCRIPTION

berSystem

nthe mathematicaltexts

perties* (a)itis basedon

agesimal")and(b)ituses

esecondpropertyisespe-

reathistoricalimportance

t fromitoriginatedthe

onsystem.13

plyinthe followingis

elyaspossiblethe char-

otationusedinthe texts

alfortheunderstanding

eusedinBabylonianmathe-

mentofthesexagesimal

ould,tomentiononlyone

eadvantagesofdivisibility.

alculationwithfractions.

entsnotonly20 unitsbut

$.The corresponding

theinfinitefraction

ynotanadequaterepro-

Butalso 3wouldnot

ionbecausethiswouldmean

s"20"ina textcanrepre-

or 20timesanypowerof

etc.Analogously,1,20

0=80, butalsofor1$,

fo=-fc,etc.Onlywhen

ateintegersfromfractions

ol";"as amarkofsepa-

fori\. Theuseofthis

rpretationofthe textand

yin ourtranslationsand

etranscriptions.

onprevailsinthecaseof

wouldbewritten1,0,10

.Onlyinthelatest period,

notationwhichclearlycor-

leucidtexts,bothmathe-

,usefor"zero"thesign^

asamarkofseparation;

bythesymbol".".The

dinglybewritten1,.,10.

symbolisverydark;the

be certainisthatnosign

ld-Babylonianperiod,

kspacewasemployed

own,however,whenand

wasintroducedinthe

urtranslationsandcom-

hasfrequentlybeenin-

ethecorrectplace-valueof

mustalwaysbekeptin

espondingsigninthetext

allcaseswherethe text

wewrite".".

cetoemphasizethat the

ianwritingofnumbersis by

Justaswe multiply0.0325

or 325by732in exactly

malcomputationcanbe

eplace-value,whichcan

thesameway.aswe

decimal-point.Arecip-

ssibleordersofmagnitude:

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SCRIPTION

mesclear ifonetriesto

hereasahar isaddedtoa

ngto reproducesucha

otheground."

ltechnicalexpressionsjby

ntermsisnotthe onlypoint

abandonastrictlyliteral

fmanytablets,especially

anwritings,issocondensed

sclearonlyif thestructure

ntoaccount.Manysuch

atablebutcouldbest be

ticalformulas.17Inorder,

betweentheoriginaltext

aveattemptedtomakethe

ebyinsertingexplanatory

.Wecannotemphasizetoo

s,alwaysinsideparen-

moderncommentaryand

ofthewordingofthe text.

ationis raisedbySume-

compatiblewiththerules

nysuchformsaretobe

ylonianmathematicaltexts

ninAkkadianaswell as

callyAkkadianwritings

themeaning"subtract"

ed";dab"add"or"I

he suffixes-e,-ra,and

hese peculiarformscon-

tternimposedbythe texts

Akkadianandbythefuller,

whichsometimesoccur.

he firstpersonsingular

n thedescriptionofthe

dpersonsingular(imperative

perationscarriedout

roblem.

ollowtheoriginaltexts

treplaceSumerianwritings

nts.Suchcorrespond-

abularygiveninthelast

.below)Nos.15-18.

xtsofthis typeisalmosta"calcu-

f.forthis methodNeugebauer[3].

suchwritingsin mathematicaltexts,

dvantagesofashortenednotationare

personandothergrammaticalrefine-

ecauseof theratherrigorousliterary

thematicaltexts,aretheforerunners

hic"writingsinAkkadiantextsof the

ngofthe bareSumerianrootcarried

nandtense.

ke"thelengthexceededthewidth

formitywiththemodern

orAkkadianandromanfor

ntionedforthe benefit

ot Assyriologiststhatno

omthe useoftheSumerian

rplaceoforiginof thetexts;

mewhatsimilartotheuse

s.Noneofourtablets

bout1800B.C.

course followthesystem

ureau-Dangin.20Afew

hichare nottobefound

n thestandardlistsof

l,are:

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NDAREA

only1to5 beingpossible)

ymbolsareaccordinglyun-

ndr arenotcombined

atedtothe requiredmulti-

onslike"burbur bur"

r)and 2(ee).Tobecon-

Cbur'u),etc.,evenwhenonly

occurs.Thesign"iku"

whicharelargerthaniku

eas ifitwerea deter-

o indicatesuchthings

the text,seebelow

ationsbetweenthevari-

,volume,weight,etc.,is

rstandingofthemathe-

gofmetrologicalruleswas

ofmanyexamplesinour

efromthe mathematical

e masteryoftheratios

Inthefollowingpara-

tlineofmetrologicalrelations

exts treatedinthisvolume.

wever,thattheselists

fgeneralvalue regardless

nces;althoughtheyun-

portantstandardforOld-

o meansexcluded(indeed,

thatdifferentrelations

texts.22

resoflengtharethe

0inches)andtheGAR,

aTherelationbetweenthe

iveninthefollowinglist:

etationof thisandthe

clearfromthe following

ntallinemeansthat

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RICKS.WEIGHTS

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METROLOGICALCALCULATIONS

ereusedtodetermine^Vth

eral,ascanbe seenfrom

tc.,wasmeasuredby

ntiononlythe following

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TIONS

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TIONS

50SAR

50SAR

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TIONS

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TIONS

fractionofthesewritten

nisi se,incontrastto

67,68,and72).It is

notethatonlythe special

tionwithGAR,whereas

by4 ku (cf.Nos.9and

u,ginandSARare

nalsigns66,$and .All

xceptionin theOld-Baby-

lem-textsandmetrological

xampleisgiveninNCBT

ngcopy),whichreadsas

sforf GAR.

66and72.

onhasshownusthat theexcep-

temporaryeconomicdocumentsare

cttofindthereasonsfor thesespecial

olematerial.

ed.

2f SAR6fgin

s

of asquare).

rea).

ormedtostandardunits)?

iku,2fSAR,(and)6fgin.

ndthemetrologicalunits

inesareuniquelydeter-

ivenin line5.The

40SAR

R6fgin.

emistobe foundonthe

bletwhichwaswritteni n

dperhapsinthe earlypart

estobeimpressedbythe

first halfofthesignBI.

Thetextreads:

entedinthe lowerright

eof thesquare.

in].

d1,46,40atthe topofthe

=1;46,40.Theremaining

ethat1;20wastreated inthe

wasfoundthrough 42-0;202.

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bymeansofwhichall

erecarriedout,playeda

elopmentofOld-Babylonian

onianastronomyinthe

beseparatedintotwo

group,mainlyfromthe

dmostfullyrepresented

icationtables,and(2)from

oup oftablesofgreat

ntforastronomicalcomputa-

olongerlinkedtogether

oupare self-explanatory,

sintheolder tabletsarenot

oldergroupfallsinto two

etscontainingasingletable

products;and(2)larger

ingagroupofconsecutive

ctionononetabletofthe

letables.Fromthecom-

mthecollectedmaterialof

tall thesetablesarear-

lowingwellplannedsystem:

owedby aseriesofabout

tables,afterwhichcome

e-roots.Theconnection

rocalsandthemultiplica-

hatthelatter concernthe

umberswhicharerecip-

forthemostpart contained

ciprocals.Thisshowsthat

ereusednotonlyfor the

simultaneouslyforgeneral

ocal5ofc.69

s,it ismostconvenient

gtothefollowinggroups:

dtype11

xtent13

andirregularnumbers.16

procals17

bles20

ontables24

ts;cuberoots 33

significance,seeMKTI pp.4ff.

ngenpp.18ff.

owever,thatbythis

shappensthattableswhich

esametabletare listed

procals,e.g.,whichcome

edtables,arelisted under

mthemultiplicationtables

eholdsforso-called

,e.g.,acombinedmulti-

obverse,butoneormore

ationtable(c)written by

nthereverse.Inall such

vebeengiven.

inthesexagesimalnota-

resentationoftherecip-

btainedbydividing1byn.

therthis divisionendsafter

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TABLES

sbetween1and 1,21."

mentionedin theremarks

wingtextsarepublished

ssotherwisenoted.'

;

T

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OFLARGEREXTENT

LargerExtent

gletablets,all ofun-

ptCBS29.13.21,whichis

e firsttabletisundeter-

lasttwo areOld-Baby-

to theSeleucid(ora

everseis destroyed.On

handcolumnispreserved;

wayofdeterminingthe

notabsolutelycertain;

The followingrestoration

splausible:

hat thistablet,con-

ongstotheearlierpartofthe

wasfoundatNippur.

riginaltabletispreserved;

elow.Thefrequentoccur-

ethat thetextdealswith

ewellknownpatternn

olines.Beforecomment-

holetext,wegiveatran-

portion.

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OFLARGEREXTENT

30]

8,45

2,30

elastpair seemstobegivenerroneouslyinthe textas

theright-handsideis correct;fortheerrorcf.p.13 note68.

orestore,withincertainlimits, thereverse.Thatwecannot

ntyisdue tothefactthatweare nowdealingwithfourdifferent

renot quitedetermined.Thefollowinggivesthemostplaus-

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OFLARGEREXTENT

enobtainablebycontinuous

0

opairsofnumbersgiven

eforedoingso,however,

ea simplenotationfor

umberswhichcanbede-

fpowersof2, 3and5

hallnow applythisnota-

stion:

s4,38,5,29,9,1,24,22,30.

rum73of1,20is 18,32,21,

avea verysimplestructure

25,0,1)

(23,0,1).

eobtainedbymultiplying

eis trueofthenumbersin

I),whichbeginswith

eisthereforean intro-

ction.Whyjustthese

notclear.

nasto themeaningof

haveseen,theoperation

givennumberbya high

reciprocal.Thisprocess,

weare byitsverysim-

nstodeterminethe specific

Weshallagainmeetthe

fferentcontext.74The

emstobethemultiplication

meriana-ra-kara,the

n-wordarakarum,76toa

s,sincea-raisthe technical

n,althoughnoneofthe

araseemstogivea clueto

ghere."Factor"or"coeffi-

ywellboththe literaland

gs.

essofcalculatinglargertablesof

nNeugebauer,Vorlesungenpp.9ff.

e(bothstartingwith2,5)are published

,top ofp.52.

);andtherefore30" =15,0=(2,2,2).

thertooccurredonlyonce:in a

oebelHGTNo.148,line20,which

ru-ut-um.

ereverseis destroyed.

wopairsofreciprocalseach.

,10]

22,30.42,40]

,26,24.[41,40]

,12]

53,20]

40[41,.,22,30]

,36]

30[. 40]

13,20]PublicDomain,

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ARANDIRREGULARNUMBERS

e doubtful.Inline4,

and[4]8doesnotagree

hichindicateratherclearly

ebeassumedtobean

consideredaserious

oration,which,however,

spacingofthe numbersin

Babyloniantextwas

the fragments.Itis

areinvolved.Thefollow-

ptionwhichhasbeenchecked

freciprocalsaregiven

39,48,28,38,31,6,40

3,4,25,23,10,56,47,24,26,40

2,9,37,9,15,32,10,54,17,25,48,

eachofthethreekey-

powersof 2,3and5 is

highpowersof3:

)

).

n transcriptionMKTI,

s.Bya differentcombina-

302,

hichwascalled"obverse(?)"is

togetherwiththelastsign ofthe

erencestoothersplitwritingsofthis

ne shouldhaveread:9,37,9,15,32,10,

f57(?).

,8as216 maybesignifi-

mbers7,30and2, since

andIrregularNumbers

thisOld-Babylonian

chmorethanone-thirdof

opunlesstheoriginal

g.Eachlinefollowsthe

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OCALS

ervedpartof thereverseis

procals

eYaleBabylonianCol-

d-Babylonian,arecon-

mmonpatternwhichis

ple:

columnarethe reciprocals

uesgiveninthe third

fthesecond,andthe con-

n thefourthcolumn.The

ght corneristhesumof

thirdcolumn.

owthesamestructure:

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OCALS

s,thepreviousexamples

onceptofreciprocalsfor

ely,pairsofnumbersnand

alsapowerof 60;thelast

dealwiththecaseofpairsof

whereais anumberwhich

orsofn notcontainedin60.

n,theremainderofthe

hepreviousscheme.The

case(YBC11125)40times

othercase(YBC7353)

n.Thesumof thenum-

givenbythenumber

Thefourthcolumnis

3016,41=8,20,30,respec-

thesetextscan accord-

gn andnorn andn'

n'=a=71113);

corcaor c1,40(wherec

10). Intheupperright

rsthe numberwhichisthe

n'.Ahintas tothepos-

ortinthe preparationor

msisofferedin theAppendix

edingtexts,butinvolving

rnumbers,isPTS 247,an

hwasdiscoveredbyDr.A.

til20

.76mAsasampleofthe

wegiveobv.710:

eciprocalsil of6,8,12

econdcolumnisn2,andthe

1,12,whichisrepeatedin

,45of thethirdcolumn

column.Thewordson

ousand,3hundred and

lequivalentofthisnumber

enumbercin thelast

tedas1,12,0.

pp. 287-289)con-

lessparallelexamples

um761);unfortunately,no

egiven.Theonlyattempt

eproblemsindetailwas

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n questioningthisinter-

efactthatweare nowina

certain thatthereisa

betweena,|8,7,5 onthe

er.

nthecase ofYBC11125

e sameterminologyas

fthetype

fferenceof0;0,1,40gin

textandthevaluecalcu-

ssmalldifferenceisamagni-

tionofa seconsideredasa

nit,andwasthereforedis-

ark astotheexact

oincidencestotheproblems

eemsobviousthatthereis

eenVAT7530and the

whosemultiplesarelisted

"principalnumber"of

eprincipalnumber;then

givenina tablewithprin-

texts,is40or 30,andyis

en inthethirdcolumn.

=40ork=30wouldy ie ld

aluewhichdiffersfrom

530by0;0,1,40gin,which

e otherexamplesin

r tablescanbesetup.

ollowingrelations:

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ABLES

hevalueofc2or acatch-line

followsinthecanonical

oterminology.The

"times".Themaintypes

etables,Cin combined

meintermediatetypes

sented:79

ommontypes,Abeing 50

bles

2asRCT1;overlookedinMKT.

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ABLES

byHilprechtinBE 20,1p.16

ormEABp.532);overlookedin

tanbul (cf.HilprechtBE

;probablyfromLarsa.19

ritten5wedgesover4.Colophon:

phon;thefirstlinecan berestored:

.-kam],"MonthVIII,day"

withthe tableoccupyingonly

gpartof theobverseempty.

yed;writtenonthereverseof

ableNo.160.

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ABLES

".

ocopiesonthetablet;on the

ultiplicationtableNo.168.

olophonatend.

Colophon:im-gid-daU-bar-

5-kam.88

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ABLES

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ONTABLES

singlemultiplication

inMKT.Thecomplete

blescontains40individual

onofthetables for4888

blesarenowalsorepresented

nTables

clistof 30combined

addedto the44published

rieflythemostcomplete

p,thecylinder A7897

mple.

ancylinderfromthe

gtheremainsof12 columns

ation.Fortunately,only

mnsisnowcompletelygone.

eris about10.6cm(4}

pproximately22.5cm

1.3cm(| in.)diameter

ralongits axis.88Itmay

7897is thefirstknown

nonacylinder.89

TI pp.34f.andthefollowing

dinmostof thecombinedtables;

roup ofmorethan150sculp-

spurchasedforTheOriental Institute

aghdaddealers,Messayeand

n60of thetabletsareOld-Akkadian

-Babylonian(ca.1700 B.C.).The

theDiyalaregionasthe provenance

corroboratedbymanycoincidences

etweenthesetabletsandothersactually

eabythe IraqExpeditionofThe

accordinglynotthe slightestreason

nderinquestioncomesfromthesame

toDr.ThorkildJacobsenfor having

raphsatourdisposal andfordata

nthe dealer'spossession,someone

engthsto makethecylinderappear

ofpreservation.Thiswasdoneby

r fragmentsfromthesamecylinder

gtoothercylinders(A7895 andA7896,

wgenerallycalled"forerunnerstothe

Whenafragmentexceededthe

nstrumentwhichleft comb-like

heback andedgestoforma well-

shly,twoholesinsidethecylinder

of twoeconomictablets.Accord-

econcomitantdetachmentofthe

ccomplishedinthespringof1941,

hwritingvisibleoneverypart ofits

particularlystrong nativepastehad

ostanticlimactically,allthefrag-

paredandjoinedwith somuchcare

side-downpositions.Atpresent,

gepiecewhichformsthe bulkofthe

agmentswhichcannotbe directly

idualtablesover the

eschematicdrawinggiven

ningisthetableof recip-

above,p.12No.40.The

8and 45representthemost

altables;manycombined

8.Thefollowingtables

tandardarrangementwith

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ONTABLES

speculiarin twoways:the

e tablefor44,26,40isfol-

5againstthecanonicalorder.

xtsinthe followinglist

extensivetablebeginning

ecedesatableof smaller

wernumber.Theserial

sedinMKT.'2

nofthiscylinder,provenance,

orthetableof reciprocals

IIp. 37,MKTIIIp. 49.The

beaddedinMKTII Ip.49:120a

AP

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ONTABLES

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ONTABLES

seep. 12No.42.19written

seep. 12No.41.

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ONTABLES

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ONTABLES

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ONTABLES

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ONTABLES

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ONTABLES

eempty.Catch-line:1

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OOTS

4.

itten20-l-1.Smallfrag-

ed.

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OOTS.CUBEROOTS

thereverse,theremainder

etableof square

be noted:

earethesquaresof 1,1,1

ntcontainingfive

oots.Reverseempty.

he fragmentispreserved:

ishedbyVander

.Six-sidedprism,ofwhich

ol.VIaredestroyed.The

ostallof Col.Illcontain

measuresoflength.

l,allofCol. IV,and

tuteatableofsquare

onthroughCol.IV, which

.Varemissing,butcan

gcontainedthesquares

and 57inclusive.Thefew

nCol.V readasfollows:

en.

ad14,1-eetc.

tc.

,25-eetc.

ntofasingle table.

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gnfor10, butnosignat all

ndcase.Thesquareof

where"."againstands for

for10;butin thiscasethe

is nottoindicatea"zero",

om 6topreventa mistaken

ercasewherea unit

aten-groupofa higher

thistextshedsno new

euseofa signforzerobefore

oneouslygivenas43,12,15

ll probabilitybelong

od.

6by 5icm.)hasaline

rside andslightlyleftof

reenumberswrittenina

umbersissimple:

nscribed.

hesametypeis YBC

\by6cm.Theline onthe

ftofthecenter,butthis

side.Thenumbers,which

eexplainedbythe factthat

restingtonotethat,asis

xts,thepresenceofthe

tedin anywayinthe

ninscribed.

ofa similartype,cf.

esofexponentsa",where

and10,anda isoneofthe

notethatall oftheseare

WenowhaveanOld-

nswersthequestion:to

numberabe raisedin

ber?Thisproblemis

garithmtothe baseaofa

ugebauer[6],andNeugebauer,

.

stion(MLC2078)is

ttraces;alledgesarepre-

nd ontheleftmargin

u(?) ukPI(?)...ma(?)

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i8isusedin bothgroups

achedonthebasis ofthe

hatib-si8 (orba-si


47/155

XTS

ourknowledgewith

d placeoforiginofour

we cannotexpecttobe

alaccordingtosuch view-

merelytoseparatethe

niantablets(ca.1900to

ssyrian(ca.700B.C.)and

ucid(lastsixcenturiesB.C.)

yloniangroup,whichcon-

Ue,we havearrangedthe

ccordingtosubjectmatter.

aterdate,neitherallow

classification.

varygreatlyastotype

scontainonlyoneexample,

f thesolutionoftheprob-

gof thetext.Weeven

nonlyapartof theworking

blem(CandUc).At the

whichstatehundredsof

sedformbutgive noanswers

esetwoextremeslieall

s:textswithtwoor more

edoutindetailandwhich

ngtothedegreeof mathe-

tswithmanyexamplesof

rangedverycarelessly(this

exts BM85194andBM

KTI), andtextswhich

tionofcoordinatedproblems.

presentlargenumbersof

nswersbearcolophonsgiv-

ber.Thisgaveriseto the

nMKTfor thiswhole

t wise,however,to

sethenewmaterialmakes

rdersofthisgroup.It is

s noevidenceforcanonical

hetypeoftheastrological

rthe lexicographical

metabletnumberoccurs

etswithdifferentcontents;

ce,iscalled"Tablet10"

risgivento anothertablet.98

earlyrelatedtoU,butU is

Thus,it isfairlyclear

etextsimplies nothing

.385.

toftabletsofvarious groups

norder.

textswhichstateproblems

ersandtexts whichgivethe

celyillustratedbyproblem-

textGstates31problems

ewritingdownofthedetails

problemsrequiredthree

fwhicharepreservedinH

memight beassumed

eproblemsandcorre-

amples.

dbelowstand somewhat

ndtheproblem-texts.The

werstoa problemcon-

bers(orPythagoreantri-

eserveddocumentin

hesecondtext( 3)deals

rootinaspecialcase.

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CRIPTION

milarmathematicaltexts

terpretationof economic

gicaldata.

ypicalpartofBabylonian

blemsofapredominantly

rstthreetextsinthis group

varioustypes:asimple

o.2),linearequationsre-

tionoftheweightof a

onditions(R),andquadratic

rectangle(S).Thelast

eits geometricalelement

anthefactthat thearea

heproductxyofits sides.

roblemofdeterminingx

c txyandasumx+yora

sisthe normalformfor

hichthemoreinvolvedforms

tweightattributedtothe

tionsinBabylonianmathe-

ur material.Thisisalso

hichcontain247and177

lleadingtoquadraticequa-

ematiccompendiaofexer-

toaconsistentscheme.The

mustbeusedinorderto

similartoouruseof an

weareallowedtosub-

eletters a,b,cetc.which

athematical"textis

9).Thetabletscontain

re addedshortexplana-

ematicaltexts.Wefind

erringtobricks,workassign-

hoseparameterswhich

dealingwithvarioustypes

rthefirsttimewehave

acterofpagesfroma general

otamianhistoryarevery

urmaterial( 10).Only

eSeleucidperiodwerepub-

anothertextfromthe

mewhatdoubtfulLate-

W).Betweentheselate

ianmaterialtherestill

thousandyears,towhich

anbeassignedwith cer-

however,thatthereis

cterbetweenthelatest

ylonian.

]asag*-...-ii

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49/155

ENTARY

bletrepresentstheright-

Thepresenceofmodern

gof thetablet,ontheleft

hemissingpart musthave

ionofthe tablet.The

4| by3|in.(12.7by 8.8

ythatmuchmore than

xistingpartis missing.

abylonian,i.e.,itfalls

0and 1600B.C.Thesign

erimposedrowsofthree

osarenotindicatedby a

paceoccursinlines 3and

;ontheotherhand, lines

blankspacedoesnot neces-

ngtheprovenanceofthe

endelsohn,Catalogue,

hsafterthissectionwas

raphofthetabletbeforeit

chwepublishon Plate22

dtablet.Thecontentof

as "commercialaccount"

goreantriangles":right

ntegers.Let/ denotethe

farighttriangle,d its

areintegerswhichfulfill

asealmostlinearly(cf. fig.

suchPythagorean

secondandthirdpreserved

ght assumethatthe

correspondingvaluesof/.

givestheratios ofd2toP.

ndbvaryin averyirregular

fromlineto lineisvery

sisvirtuallyequivalentto

ostlinearly(averagedif-

,almost0;1),andweshall

properformulationofthe

problemwithrespectto

at westartoutwithalmost

value ofb:lwhichcorre-

e3]is 0;59,30)andgradually

n/andd stepbystep,the

31415

* 3f

exactly31.Itmust,how-

theactualsizeofthese

blyowingtothefactthat all

of(1) andnotapproxima-

in thelastcolumnhave

precedingnumbersbut

erofthesteps, liketheunits

ram.Theprecedingki

fordinal numbers.110

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50/155

ENTARY

wseemsprobablealthough

astwordofthe secondline.

asagtii

ousdifficulties.Themost

stobe:"Thetakiltumofthe

ubtractedsuchthatthe

a-k]i-il-tiis,however,

donot knowhowitisto

ementionofasubtraction

ondlinecouldindicatethat

b2.

ngfirst columnandifd

unknownquantitiestobe

(1).Weare,however,un-

nslationofthispassage

onofgreathistorical

thematiciansoftheOld-

tonlytosolvethe Pythag-

ersbut toadaptthesolu-

onthat theproportionr

bya numberdeviating

h?

estion,wemustfirst

,b,and dgiveninthe tablet.

owingtablewiththecorrec-

es4, 11,15,and17men-

he transcription:

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ENTARY

osereciprocalsarefinite

sandq'salso yieldsthe

wthenumbersof ourlist

solveequation(1)but

mitsgivenproportionsj

proportion

e reciprocalsofpandq,

ds,ourproblemrequires

mbersinorderto obtain

sexagesimalfractions.

single exceptionof

pandq ofourlistare con-

ularnumberswhichcon-

es."118Aswehavealready

etesystemof"multiplication

e"combinedmultiplication

of allproductsa-b(1= a

ofproductsahwhere6

arnumberincludedinthe

atourtabletwascalcu-

spQandqp fromcombined

that(3)hasa valueas

quiredvaluesofj; Pythag-

formedwiththesevalues

.

obemade.Theexcep-

veisnot tobeconsidered

thattheusualreciprocal

enlargedinthisverydirec-

dofusing(3),onecanalso

mbersbyusingoneparameter

rea=.122Buta com-

rlines

r,Vorlesungenpp.6and12ff.;and

MKTI pp.23f.(Nos.3and4).

p.163ff.

eitheranora canhavebeen

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ONALOFASQUARE

py:Plate22)

sl

di-nu-kum

um

ma

-im

ebroken;remainderun-

sin(EN-ZU)writtenabout

n,startingatabout half-

beroot.

3,22,30?

30,theydidnotgiveyou the

ecuberootofwhichtheydo

7,30,0?(Theansweris)

30,0,and(theresultis)

30,(andtheresultis)27.

27?(Theansweris)3.

ot, by30,theothercube

30.

0is1,30.

aninconclusivediscussionofthis

possiblysomethinglike"model

w,"in line5toindicatethe relative

ine6isapparentlyfurtherproof that

erewrittenandmeanttoberead

respecttothepositionconventionally

orevidencefromthefiguresinmathe-

49,note135dandNeugebauer,

76.

umberb=3,22,30is to

e problempresentedin

thease.The textproceeds

oducinganauxiliarynum-

hecuberoot^7,30,0=30

umablyinatable-text).

alculatedby

,it isnecessarythatthe

thefollowingthreecondi-

eofa rationalnumber;

er;(3)thatyj-canbe

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eside aofthesquare,and

(1) forV2~isverygood,

9,38,1,40.

'1;25 occurredin

erial(fromtheSeleucid

ue,however,nowalsooccurs

coefficients,published

din line10theentry

areroot.

eshowthevalue(1) for

wingmightgivethe

etofurnishdirectproof

followed.Twofactors

anation:first,thatthe

hevery numberfoundin

sameprocedureis attested

heapproximatevalueof

consistsin thealter-

abyarithmeticandhar-

yfoundapproximations.

onofVasuchthatai >Va.

proximationofVabutde-

n theoppositedirection

> Vathat0i


54/155

OBLEM-TEXTSBANDC

bverseofYBC7290126i

obtainedby

rapezoidwithoutinscribed

onthereverse.The

a trapezoidwithnumbers.

atisfiesthe followingrelation

dsides. Asfortheremain-

that

;303,0=31, 0

efficient3aswellasthe

ar.

reofa circlewhosecircum-

ehavec2 =9andfor the

7by 8cm.

isroughlycircular;diameter8cm.;

k

C1112012"

0;11,15.

C9852

75)

py:Plate1)

u ikuu-1-e5,10u-2-e4,50

-a-bi2(bur)ika

ma-naH-nazu-ii-uz1"ta-al-li

m.Thefigureappearson the

nscribed.

-a(?)is doubtfulbecauseofa

owpositionofthe lowerwedgeatthe

seofthe unevenarrangementofthe

t,e.g.,line20)andperhapsafourth

ereadingofthesignswhichfollow,

cleaningofthisline byDr.A.

e textwasprepared.

tu-u-usandtranslate"I

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TRANSLATION

kima-silu-ul-ku-un-ma

na1 (bitr)ikula-ni-i-im

-si u-lugud-dalu-ul-ku-

haga-me-ru-u-tim

ar-maba-a-[H-n]ate-he-pe-

a-ku[mt]a-pa-ta-ar-ma

sag-ki-tai-te-ru

a-al-H-ma2i-na-an-di-kum

n-natu-ul-ta-ak-ka-al-ma

b-bi4,49

49a-he-er-[t]um

-umi-il-li-a-kum

amlai-li-a-[kum]

r-maba-a-[H-nate-he-pe]-e-

ta-p[a-(a-ar-ma]

a-na-[al-H-m]a

-k[um]

a-na-al-li-m[a]

k[um]

2,4u-gid-d[a]

ra-af-ma

e-el-ma1(bur)ikusd-ni-iq

mqd-ab-li-a-am

ka-mar

10i-il-li-a-ak-kum

na1(bur)ikua-sa-t?ta-na-

-li-a-ak-kum

ta-na-al-H-ma

3 UStu-us-fa-ab-ma

ta-ha-ar-ra-as-ma

ta-ak-ka-al-ma

teofB(YBC4675)

elasttwol ines,thedis-

thetwotextsisthe same,

52)

py:Plate1)

al-lamqd-ab-li-a-am]

ta-ka-mar]

10[i-il-ii-a-ak-kum]

a1(bur)[ikua-a-tmta-na-

a-ak-kum]

a-na-al-[H-ma]

us-fa-ab-ma]

ha-a[r-ra-as-ma]

a1(bur)iknsd-ni-i[q]

nscribed.)

)

nscription.Dal-murub

ne.")

thesecond

wer width7,itsarea2bur;

,each(part)1 bur.How

-line?

e longerlengthandthe

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COMMENTARY

whichresultedforyou

the arakarum,and

4whichresultedforyou

dthe(resulting)2,4is the

he second2US\and

shorterlength.Youshall

cedure,andthe(re-

theoneside).

he middledividing-line

nd7,the lowerwidth,you

10 willresultforyou;

ocalof10,and youshall

rea,and

The3US whichresulted

,thearakarum,and

ushalladd6 to3US,and

longerlength.Youshall

e shorterlength.Youshall

plication,and

ers ontheotherside.

ntary

pezoidofgivendimen-

Atobe dividedintotwo

e figuregiveninthetran-

ngthd ofthebisecting

xpressedas1 bureach,

0,0GAR2;inthe caseof

s notgivenbythetext

auseh= 5,10isabout

,thetrapezoidis along

oneshouldread li=5;10

pondingly/2=4;50i sex-

would have

50=17=bu

ndicatethedegeneration

glelineoflengthbi (fig.9).

ethestretchedformofthe

umbers(1). Thisalso

hareexpresslyindicated

rminology.IMThescribe

er,thatthenumbersgiven

basicassumptionofhis

dareparallellines.From-

2,itwouldfollowthatone

es 61 62,hand/2(cf.

m(1) that

0+10=5,0

lityofsuchatriangle.In

l lengthsofhandh are

procedurefollowedby

b2anddare assumedtobe

esare

ensionsexcludeanarrange-

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COMMENTARY

areaAof thefigureis

approximativeformula

followarebased onthe

rthepartialareas

+M2

steps ofthecalculation,

rrelationused inthetext.

tlikethatindicatedinfig. 8,

bi

ityAi =Aiand(3), we

mi+ m)

gequationsbymultiplying

ndby d+bi,we findthat

d-h)

erivedunder theas-

ativeformulas(3)andof

dd.

odescribetheprocedure

s actuallybasedon(4),

dform.Insteadofcalcu-

nd(1),thetextemploysa

if onlybi biwereknown

suseofthe numericalvalue

procalof this5,0butfails

by1,0,0which,in view

nnotbeexpressed,would

mberswhicharewrittenin

licationbyA =1,0,0

bi2-h> )= i(V +b&

orderof magnitudefor

(4), wehave

13,

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tersectionofthe dividing

des.

ep'sby

eatthesameprocessforthe

nglesteps are

ningpartsofthelonger sides.

chisusedtotransform

ec-

Foradiscussionofthe

sspelledta-al-li(obv.3),

al-lam(obv.17,rev.7)

umerianwordfromwhich

addal,notri.184 Italso

otthe uniqueAkkadian

lwiththemeaningindi-

contexts.Thetranslation

tocoverthetechnical

kum,namely,theline

ea(e.g.,circle,trapezoid,

py:Plate2)

nu-umza-eki-ta[-zu-de]

a5,20i-H-ma1[6]

-LU-bu

^(?)-t>136s1

L-gar1,20

0a-a[i-H-ma4]

zi 2[0sag-an-ta]

tfortracesatend.

length,

and thelowerwidth?

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20and/' =30.1,6.In

0ofthetrapezoidis given.

oftheparallelsidesof

problemin question

,+ wi)and\(wuwi)

ferenceoftheseexpressions

xpressionis easilyfound.

oid,itfollowsthatwehave

= 16= i(w. + w i),

anation.

fig.13and thenfind,from

(m, wi).

ence,aright triangleisassumedin

llypossibleto assumeanisosceles

wouldthen alsohavetoassumethe

rtheareaofatrapezoid,whichis found,

ep.47, formula(2)).

al,calledthe"upperwidth"because

0 inclockwisedirection(cf.p.42,

r,Vorlesungen,pp.34and176).

write

ationusedbythe scribe.

= 1, 0+ 20= 1, 20

owprovidethefinalanswer

16+ 4= 20= w u

6 4=12=wu

notherproblemare

py:Plate3)

H-nazu-ii-uz1"]

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LATION

28],7,30zi-ma

gi-]5-gal-biid-ki-tadu8-

ma[8],26,15in-si

-el-kau-ka-al-lugaz-ma

na8,26,15dab-ma

p-pa-al-(a-ar

u-ul-ku-unla52,30dal-bi

15in-si15 a-na1lai-na

15u id-an-na

ta139al-ku-nuta-na-al-H-

u-umsag-an-nausag-

-nae-tab28,7,30in-si

n-si4a-na28,7,30nim-ma

052,30dal-bita-b[a-al\

ua-naH-nae-tab1,52,3(0

40 a-na1,52,30nim-ma

g-an-nata-ba-al

-]ma-ri-i-ka

ba-ma-at[1],15


61/155

ENTARY

uarekeepinginyour

d 4,41;15to8,26;15,and

of13,7;30isnotobtain-

,7;30whichwillgiveme

rocalof0;4willgiveyou

by1,whichIput

188andthe (resulting)

erstrip.

ythe5whichI put(aside)

engthofthelower

etheupper width

tiplythearea bytwo,

5,andyouwill get0;4;

d

[eaway]52;30,itsdivid-

d)

upperwidth.Multiply

d)[you willget]

30,andyouwillget

1,52,30,and

way1,0,the upperwidth,

lowerwidth.Inorder

e resultis)1,15;halve

7;[30];

0,and[youwillget] 56,15,

dtheupper

hebeginning.)

m of]

he width?[]

nd[ ]

thewidth["*]

olding."

?) thewidth;

sulting)

thewidth.7,12,0isthe

[and]thewidth

ength,[11,22],30thearea;

h.

ebrother('sshare)ex-

dIdidnotknow.

erexceedtheother?

operations),multiplythe

t is)22,45,0.

0isnot obtainable.

whichwillgiveme

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ENTARY

extiinds theunknown

following:

5=Ai+ Ai=A

torbywhichthe numbers

order toobtainthe

Thefirstgoalis thedeter-

ebydeterminingthecorre-

Tothisend,it calculates

2=18,45

appliedto thepartialareas.

0=-rj1=-(h+d)

=-(6X+d)- J(h+bi)

ve

^-- -^L-i.

onitfollowsthat

30

ence

rtial lengths:

=/ , .

ndbiby

00;4

+d)-d=h

00;0,40=1,15

-h-6, .

etelysolved.

tcalculates

30= 56,15

eofA1 +A2.

destroyedwiththeex-

o lines,whicharesufficient

haracteroftheproblem.

width"impliesa trapezoid.

"diagonal"iscorrect,we

xampleof theuseoffiliptum

ofatrapezoid)"sincethe

ngis"diagonal(ofarec-

right triangle)."

maged,butthenumbers

atethatbothdealwitha

width b=1,12,andarea

siblerestorationof No.3

, 12andb=%lwerethe

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CRIPTIONANDTRANSLATION

6 brothersbyequidistant

fthetriangle.Theques-

cernsthedifferencebetween

ers.

omputingthebasebiof

A

chgives

ialfield.Thenthecorre-

ividing biinto6parts

tep bystep148

b2]

b3]

= bt]

= bt]

bt].

ne tostatethat

heindividualfields,but

wthiscould bereconciled

py:Plate4)

[a-l]a-an2,20sag-kia-a

red.

somanytracesof erasurethatit

ourtextwaswrittenoverthebe-

]...[...]...

sag-du....[....]

},e-pS-ma1

ag sag-dula-ni-im

4i-H-ma1,20

ag-kisa[g-du]

0a-na1,20u[ M-nim]

H-na[fye-pe-ma]

-H-ma]

he-pe-ma]

S0[i-H-ma]

m]

-ki-gu4]

[mi-l]i[-ma]

]di-kum-ma

ki-nu-um1"1

g-dui-na-di-ku

g-ki-gu4H-li-ip-tim

la-nu-um

kisag-d[u]

scription.)

ofthetwolengths,2,20

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64/155

ENTARY

10by1,2[0,the second

hearea ofthe[second(?)]

hetriangle,[and]

30by1,20,the second161

heareaofthethird

]

thetrapezoidofthe

hediagonal,[and]

go]nal.

gth,andthe(resulting)

th.

uwill]ge[t]1,0,the

ofthetrapezoidofthe

e(resulting)1,40isthe



nedinthistext arevery

nbythe identityofcorre-

efirstexampleconcernsa

dedintothreetrianglesas

enat thebeginningofthe

smentiona"trapezoid"

eethatthis referstoan

icatesthatthedeter-

eproblemtobesolved.

catedinfig.16,theparam-

ndB =la+b=2, 20. A s

xtderivesthe values

20fromtheseparam-

reobtained,theareasin

bv.8torev.4) asfollows:

Ai

sprecedingthecomputa-

st discusstheshapeofthe

ealreadyseenthat

tertriangles,

thereforeright triangles

commonbaseandadjacent

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65/155

SCRIPTION

ghthe firststepofthe

stsincomputing

+ 1,0.

rioussituationwouldbe

lueb =20istaken from

ontainsallthenumbers

ellasthoser equired.An-

sistindecomposingthe

ts suchthattheouter

ngles;butitis hardtosee

ave beensolved.Weare

ethattheproblemis in-

sentform.

putationofc,the final

terminationoftheareas

called "secondlength"in

oftheoutsidecontour.

multiplyinga magnitude

by4,whichgives

pis simpleifonelooksat

above.Fromthe fact

ari ghttriangleandfrom

ollow sthatthethirdside

+42 =52.Themakfarum

ctor162bywhich3, 4and5

togivea,c andd.

esconfirmthepreceding

m.Herethetwooutertri-

ely;ineachcase,a, candd

mula(1),c beingalsocalled

iousdifficultyliesin the

gofNo.3: "makfarumof

al".Aswe haveseen

efactor20neededto en-

nglewithsidesmeasuring

mensions1,0,1,20and 1,40.

makfarumforthe diagonal

sofmakfarumoffersnodifficulties:

mtypefromtheroot kfr"tobind,tie."

rsoutsidemathematicaltextswiththe

"(cf.,e.g.,Ungnad[1]p.25b,note 2),

arel)(restorationof DeimelL597,

alsomakfaru"ofthemouthof a

nexplanationofnapsamu(forthe

unap.112,note1 againstDelitzsch,

ctransparencyof maksarumdoes

theelucidationofitsmathematical

ewaswritten,maksarumhasturned

2), obv.1,q.v.;themeaningisalso

milartriangleswitha com-

nnerindicatedinfig.18.

trianglewithsidesmeasur-

anglecorrespondsto the

betweenthetwodiagonals

called"thetrapezoidof the

asized,however,that

stanceofthissort of

abylonianmathematics.

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66/155

SCRIPTION

elength,

thevolume).

length,

(is thevolume).

) 22evolume.

uare],

n (and)22|e volume].

multiplyinglength,width

tly

gin7$e

in15e

0SAR= 2fSARgin 22|

osimplein themselvesto

urposeofthe figureasa

onmayhavebeencon-

etelydestroyedlinewhich

eedge,althoughitis not

twasinreality thefirst

ereconjecture,wepropose

evolumesasindicated in

ghtbe interpretedas

d apillar;thiswould

arterminologyofthedepths.

ctangularprisms with

py:Plate21)

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-de

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67/155

SCRIPTIONANDTRANSLATION

a4ta-mar4 a-na2,4[8,45]]

gar-ra

ii-H-ma

30fye-p6lu-tam-fyir-ma

ma]11,29,3,45

-thirds[ ]

GARIdi-

area

and thewidth?

erations),

40,multiply(it)by1,52;30,

2,48;45.

f0;151Mo,thefixedcoefficient,

y4by2,4[8;45],

(down)halfof

40,multiplyby5,of the

30,square(the result),

stroyed.)

nd

lypartoftheproblem

bjectisasegmentofa

iven as1iku12 SAR

ayedbythe 5GAR

clear;themeaningof

enmoreobscure.

alculationswhichare

msobvious.Usingthe

ns,atthe pointwherethe

simalorderhereisarbitrary.

hatthissentenceis outofplace;

illsee 7;30"inline9.

to

hiswasfollowedby

oftheunknownquantities:

xmustaccordinglyhave

terpretthisinthe geomet-

m.

py:Plate21)

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68/155

ENTARY

yby4,48,thefixedcoeifi-

2silais thethick-

falog.

]ceofthelog?

mely,)

eresultis)

f0;0,25is

erenceofthelog.

sinthelight whichit

thertoobscuremetrological

seofthe unitsila,other-

ureof capacity,162'to

flogshasbeenavailable

xamplethetechnique

altransformationfromthe

nunitsof GARtothe

nitsofsila.

renceisgivenas c=1ku

ess"tis thencalculated

oefficient"(i.e.,constant),

mainderofthe textsolves

hecircumferenceiscalcu-

lsomentionedinline58

whichisactuallyanOld-

ts,as"4,48ofthethickness

eand useofaare thus

lysisofthevariouselements

evalue4,48nottomen-

se ofsilaunitsin connec-

oteasy.Thehypothesis,

eem tobethemostplau-

acitymeasuresonp.6.

onianlettertreatedbyUngnad

Assyrianletter(approximately7th

yHarperABLVINo.566,12-15and

anRCAEIpp.400f.;see alsothe

blishedbyClayNBLENo.200,29f.

NBUpp.158f.

leltoline 35ofUe(p. 137).

merelyreferstothe diameter

doutbythefactthat the

areofthe circumference.

stakablytowardanarea

sthatanareais involved,

snotverylikelybecauseit

mysterioussila unit.

factthatall indications

tionofavolumerelation,we

veexplanationfora=4,48.

veninunits ofsila,itis

henowwell-attestedOld-

eenthecapacity-silaandthe

12volume-SAR.

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69/155

CRIPTION

inconvenientfractionsof

nthecirc umferenceisan

deliberatesubstitution

ouldbemotivatedbythe

n betweencircumferences

andcorresponding"thick-

berofsila.162k

caltextwhichinvolves

tillobscuredespiteall

culationsmadebythe

py:Plate5)

mib-si82\ku bur-bi

-hauku-babbaren-nam

\e2\bur-biib-si83\GAR 3

3| GAR3ku-ta-am7\

\ku {ku}

GARsagl\ ku bur-bi

saharerimku

GARsag3$ku bur-bi10

en3GAR

GAR4ku u Z\ku

GARsag

value-k =>3;7,30(decimally,3.125)

4volume-SARandthusdirectlyto

on,however,seemstous

approximationofjrisnot attested

mathematics.

,obverseI.Cf. MKTIII,p.61,

oden[1],p. 200,andvonSoden[2],

butionto thereadingofthistext

eading4ba-si2 ba-sifortheline given

thetranscriptioninMKT.

GAR4 kuSu 2|GAR

en-nam3$ku bur-bi

$kuS bur-bi10gine-

AR4 ku u sagen-nam

-bi

ku bur-bi10gine-kar

4 ku dirig

R4 ku u-bi2\GAR

Rsagi GARbur-bi10gin

rerimku-babbaren-

r-bi 10gine-kar6e

GARu \\GARsag-bi

ur-bi10 gine-kar6sea-bi

u sagen-nam5GAR

168sag10gin e-kar6e

ARu buren-nam

1"3sag10gin kar

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LATION

GARsagigi-3u

bi[en-nam]

1 GARsagigi-3-galu

rdab u buren-

ARsagigi-4-gal u bur

bu bur[en-nam]

sagigi-5-galu bur2\u

bur]

agigi-6-galu bur2GAR

agigi-11-galu bur\

am

ARsagigi-3-galu bur

ziu buren[-nam]

GARu ig[i-3-gal]sagbur

a bur[-bi]

[u-ri-a]sagbur3ku

m

3i[gi-6-]glsagbur \GAR

absagburen-nam

ig]i-4-galsagbur1ku

sag\bur

n-taib-si84 ku-ta-am

AR5ginsahar-bi

n4ku

b-si8\GAR an-ta

\GAR-

ki-ta4ku

n\ GAR-ta-aman-taib-si8

ur-bi

ginib-si8

r\ GAR4ku

iGARib-si8

n\ GARbarib-si8an-na

>ta\GAR

ta

gin} GARbur\ib-si8

en-nam\GAR ib-si8

n\ GARburu-ri-aib-si8

taen-nam\GARan-na

n\ GARburigi-3-galib-si8

en-nam\ GARan-na

gin\GARigi-6-gal

1Men-nam\ GARan-na

ku iseachsquare-

in(volume)the

Whatarethearea,the

orkers,andthePublicDomain,

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71/155

LATION

ilver ofaki-laare 5|gin

eft thelength,

olume)theassign-

Whatisits width?

silver ofaki-laare 5gfn

ku thelength,

olume)theassign-

ahiredman.Whatis

th.

silver ofaki-laare 5gin

th,10gin (volume)

r)the wages;

thewidth,and(theresult

atarethe length

nd)4ku isthe

.

silver ofaki-laare 5?gin

th,10gin(volume)

r)the wages;the

byi GAR(and)

itswidth?3GAR

GARis itswidth.

ngth,\\ GARthe

gin(volume)the

ewages.What

he(numberof)

xpensesin)silver?

n)silver ofaki-laare9 gin,

ume)theassign-

ges;Iaddedthe

the resultis)

gth andthewidth?

Ris itswidth.

sin)silverof aki-laare9 gin,

ume)theassign-

ges;

ewidthby2>\GAR.

he width?5

R)isitswidth.

n)silverofa ki-laare9gin,

n (volume)the

ewages;I added

(andtheresultis)

gthandthedepth?


n (volume)the

ewages;thelength

GAR.What are


volume)theassign-

ges;Iaddedthe

he resultis)2

166andthe depth?


volume)theassign-

ges;thewidthex-

R.Whatarethe

s45[SA]R,\ GARits

this thewidth

to)thewidth.

hewidth?

is45SAR,\ GARthe

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72/155

LATION

GAR(from) the

thand itsdepth?

>is 45SAR,1|GAR

engthis thedepth

AR (from)the

68]2ku [(from)

gth andthe

A[Ris thedepth].

s45SAR,1i GARthe

ngthisthe depth

AR (from)the

683 ku (from)the

thand thedepth?

s4[5SAR],1* (GAR)the

this thedepth

GARfrom) the

hand thedepth?

AR)isthe depth].

s45SAR,\\ (GAR)the

gthis thedepth

AR (from)the

hand itsdepth?

s45SAR,\\ (GAR)the

lengthisthedepth

to)thelength.

sdepth?

s45SAR,\\ GARthe

this thedepth

AR (from)the

kit (to)thedepth.

he depth?

s4[5SAR],5 GARthe

idthis thedepth.

edepth?\\

)is[its]depth.

s45SAR,5 (GAR)the

dthisthe depth

(to)thedepth.

e depth?

s45SAR,5 (GAR)the

idthisthe depth

(to)

(from)thedepth.

e depth?

s45SAR,5 (GAR)[the

widthisthe depth

u (from)the

and)the depth?

R)isthe depth.

square-sideis\GAR,

us\\GAR the

1SAR(and)5 gin

is1SAR(and) 5gin,


73/155

ENTARY

is1SAR(and) 5gin,

-sixthoftheupper

helower(square-

re-sideandthe lower

pper (square-

uare-side).

ntary

ematicallyarrangedlist

hthe volumesofprisms

ndatruncatedsquare

estion,however,cannot

minologyalonebutonlyby

ofthenumericalrelations

iesandtheindicatedsolu-

msappearstosuppose

all problemsofthegroup.

yV,wehave

9,22,30SAR

6,40SAR

45SAR

2)z= 1;5SAR

ythewidth(sag),z the

wer(ib-si8ki-ta),xuthe

n-ta)ofthetruncated

umes,however,onlyoccur

esthroughaspecificvolume

'soutputofwork(e-kar)

agesper manw(a)and

borE (ku="silver")are

relation

sameinallexamples,

er,whilethegivenoutput

0gin(Nos.1to 3)and10gin

hreeexamples(Nos.1,4

workers(erim-ha)isgiven

swered.Theanswerwould

erbecausethenumberof

ingmorethanthe number

k.

earrangementofthe

uesofthe magnitudesin-

asicrelations(1)and(2)

frequentlycallforquad-

9; 11to34;39 to44).

AR

0gin.

ewouldbe

R.

escribestheproblems:

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74/155

ENTARY

with arectangularprism

newsetof values:

10SAR

;2gin

.

irst fourexamplesinthe

nothingmorethanthe

ation(1).Thenext six

typesof quadraticequa-

fx, yandz,as thefollowing

softhis groupinvolvinga

arrelationbetweentwoof

eral typeax+by= c(and

z):

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ENTARY

bythesamestrange-looking

gouscontextsinStrassburg

.If.,13f.,rev.8f.Thus

terally,"(one)square-side

de;Iadded10to thelarge(r)

e small(er)square-side",

gdab,andzi inaliteral

esis inallprobabilityour

ntativelyofferthefollow-

on,whichatleasthasthe

henormalliteralmeanings

et"length","width"and

altextasunpreciseterms

c.,andifwethen denote

alwidthbyy, andthe

engthandwidthbyXi and

getinsteadofequations(4)

eadof(6),

=y i+5.

hattheseequationsare

(6). Inthisway,no

meaningsofdahand ziis

convenience,ourtrans-

thetype(4), (5)and(6)

).

:

loweranduppersquare-

hevolumecalculated

ativeformulaindicated.

otmentionedinthis group.

esame typealsocontains

uncatedpyramid,thevol-

accordingtothesame

here.11Theonlydiffer-

9ff.).

ngementoftheseexamplesfollows

e.

BC4708assumesthe

ormofa pileofbricksand

rsquare-sidegreaterthanthe

enttext,the uppersquare-

ndicatesthata holeinthe

nofabuilding orthelike,

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76/155

CRIPTION

-wordkalakkum,"cellar",

kala-k>kalakkum.1*

GlNcontainingAL,BUR,

alaandcorrespondingto

mel,L498,2;511,23; and

forwantofa better

-karthroughoutthisbook

al,e-karseemstorefer

r workturnedoveror

specifictask.In ourtexts,

the workwhichoneworker

one day.1Mb

4663,YBC4662

7)

y:Plate6)

RsagiGARbur-b i10

i[l-h]un-g[a]

]ku-bab[bar]

bbar

sagiGAR

un-ga

R| u

iGARbur-bi 10gin

g

u -b i1 iGARsag10gin

n-nam[ GAR]bur-bi

1* G[ARs]ag[i GAR

0ginsafoar]e-kar

Ru 1 iGARsagiG]AR

m6e]a-bil-hun-g

,which turnedupaftertheabove

ki-laand kalakkum;unfortunately,

hisconstitutesbrilliant proofor

proposedrelationbetweenki-laand

MB,p.XVIII .

0 gine-kar6]ea-bi

6,30u ]sagen-nam

ur-bi10gin]e-kar6 se

u sagen-nam5GAR

ag iGARbur-b]i

m7iSARgagar45

iGARsag]i GARbur-bi

GA]Ru iGARbur-bi

bi

]GARu 1iGARsag

GARbur-biu saggar-

-nam

ARbur-bi u ugusag

1iGARsag

gar-gar-ma52,301iGAR

am5GAR u

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77/155

LATION

agi bur-bi10gin

amal-dib

agi bur-bi10gin

u u i-dib-be-e

sag GARbur-bi10gin

en-nami-til-e

Rbur-bi10 gine-kar

-e

bi

bur-bi10 gine-kar30

ag-bien-nam1

sag10 gine-kar30

-e

-bi

Rsag} GARbur-bi30

il-e

gine-kar

4-9-kami-til-e i164GAR

aggar-gar-ma

m5GARu \\GAR

4-9-kami-til-e i164GAR

usag3?GAR dirig

1|GARsag-bi

$ku bur-bi10gine-kar

[n-ga]

aueen-nam6SAR igi-

m-ha4(gur)5(ban)

)

gth,1i] GARthe

ginvolumethe

ewagesof ahire[d

olume,[the(number

expensesin)silver?

AR)isthe volume;

i.

orkers];9 ginisthe

ilver ofaki-la[are 9

AR,

ment,6e (silver)

AR{GAR}isthe

ilver ofaki-l[aare

GARits depth,

ment,6e (silver)

ARisthewidth.

ilver ofaki-la[are 9gin,

hewidth,10 gin

fahiredman.Whatis

th.

silver ofaki-laare 9gin,

hewidth,[|GAR]

ewagesofahiredPublicDomain,

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78/155

LATION

is45[SAR,5] GARthe

hatisitswidth?

s45[SAR,5] GARthe

hat isitsdepth?

s45[SAR],} GARits

(and)thewidth,and

atarethelength

GARis thelength.1"

is45SAR,i GARits

dthewidthby3\

d)the width?5GAR

ewidth.

rea(and)thevolume,

GARisthe width,

ength? 5

area(and)thevolume,

GARis thelength,

GARis[thewidth].

earea(and)thevolume],

>;|GARisits

and thewidth,

R].

d)the width]?

earea(and)thevolume],

GARis itsdepth;

dthby3 [GAR].

d)the widjth?

ea,45 (SAR)the

and thewidth,and

width,(and) itsdepth?

ea,45 (SAR)the

dedthewidthby

width,and itsdepth?

Ris thewidth;

ationofNos.19,20and 21,the

ally,"in".The syntacticfunction

espondingNos.(withtheexception

scapesus.

rea , 45(SAR )the

atbywhichthe

isits depth.

he width?5GARis

width.

gth,l\ GARthe

0ginvolumethe

didonemantake?

length,1GAR the

0ginvolumethe

did30workers

d) \ku length.

gth,\\ GARthe

ginvolumethe

did30workers

ay.

width,\ GARitsdepth,

ment;30workers

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79/155

CRIPTION

olume,the(numberof)

pensesin)barley?

)is thearea;

2,5isthe(numberof)

nisthe(total ex-

63)

py:Plate7)

* GARbur-bi

ku-babbaren-namza-e

a-ad-di-ik-ku

na-ad-di-ik-ku

-kua-na45 i-H4,30

i-kuki-a-amni-p[e-l]u

GAR* GAR

e[a-b]i

a-zusagHbur-bi

karpu-(H-ur

-ik-ku-um

ad-di-ku-um

-na-ad-di-ik-ku33,20a-na

Ru-biki-a-amnS-pe-luw

R u iGARbur-bi10

-da-zu-de

0i-na-ad-di-ku

H3i-na-ad-di-ku

kuigi6du8 a-na9ki-H

ARsagki-a-amni-pe-lu

1* GARsag10sahar

da-zu-de

a-di-kuigie

ku45a-na i-dii-H

u-(H-ur40i-na-di-ku

i-na-di-kuGARbur-bi

AR u 1* GARsag* GAR

eni-pu-iu.

-da-zu-deu Hsag

i-ku7,30a-nabur-bii-H

nai-dii-H

u-babbarpu-(H-ur

40a-ra1,30i-H17

e-kar

* GARsagfGAR bur-bi10

a-zu-deu Hsag

-ku7,30a-nabur-bit-lt

rdu8a-na45t-Jft

4,30du8

3,20a-na9ku-babbart-Jft

ea-bi-am6

}168

ar-ma6,30\GAR[bur-bi]

sag-bien-nam

pu-(H-ur

30 i-na-di-ku-um

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80/155

LATION

)

th,1 GAR


81/155

RIPTION

llget10;33,45;

,45,(and)

itssquare root,(and)

otheone,subtractit

and)thewidth.5(GAR)

width

pensesin)silverof aki-la

dthby3;30 (GAR)

n(volume)theassign

ges,

d)the width?Whenyou

ewages,multiplyby 9,

er,(and)you will

gnment,(and)youwill

GAR(,thedepth),mul-

get7;30;

lengthexceededthe

45;

willget3;3,45;

)youwill get10;33,45;

d)youwill get3;15;

:add1;45to170the one,

heother,(and)youwill

th.

GARisthewidth.Such

2)

py:Plate8)

ar45SARsabar-h[a]

0u sagu[bur-bien-nam]

gar[du8-]m[a]

6 bur-bii-na-d[i-ku]

ku10a-na45sahar-ha

lsaglagar-gar[-ru

5a-ra3,15U[R-UR-a]

30i-nali-ib[-bi10,33,45]

k[uib-si8-swle-qi]

[1,4]5dahi-n[a1 1,45

m5GAR[u 1^sag]

SARsahar-hau ugusag

bur-bien-namza-e

a-di-ku8a-ra45sahar-ha

bur-bidu810i-na-di-ku

,30 i-na-ad-d[i-k]u

u3,30fye-p[e]

a-ra1,45UR-UR[-a]

a-ra170li-ib-b[i3,3,45dab]

0,33,45ib-si8-Jfole-qe

a2lu-pu-ut-ma

a-na1 fiu-ru-uf

ARu 1sag

agar45 SARsahar-ha

iu sagubur-bien-nam

SARgagardu8a-na 45

mlala-ap-tu-mahi-pe

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82/155

ATION

bur-bi10gine-kar

-til-e

a-zu-de

-na-di-kuigie-kar

kuigi54du8 1,6,40i-na-

a-di-ku4,30

di-ku5GARu

bur-bi10 gine-kar30

ag-bien-nam

ur-biUR-UR-ta

du86 i-na-di-ku

di-kuigi>3du8-a20

-ku4,30a-na20to-na-

sag

ARsa]g10 gine-kar

]da-zu-de

0i-n]a-di-kuigie-kardu8

1,20i-na-di-ku

R-U[R-a4,3]0i-na-di-ku

na-di-ku|GARbur-bi]

Rsag]\ GARbur-bie-karen-nam

agUR-UR-a

ur-bii-si45 ta-mar

UR-UR-a4,30ta-mar

r13,20a-na[45 i-ft]

6[-kar]

earea,45 SARthe

ewidth,and(the

tare] thelength,

operations),takethe

multiply by45SAR,

lget6 (ku),its

depth,(and)youwillget

,thevolume,(and)

thelength andthe

er,(and)

[together]3;15times

takeaway7;30fr[om

et3;3,45;[takeits

4]5to theone,[subtract

)]

ndthewidth.5GAR

ewidth.]

)thearea,45SAR

xceeded>the

arethe length,

Whenyouperform

30,thearea, (and)you

mes 45,thevolume,

depth.Taketherecip-

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COMMENTARY

andthewidth.5GARis

width].

ength,1i GARthe

[gin(volume)the

nemantake?When]

ns),

idthanditsdepth,(and)

of9,(and)] youwillget

mestheassignment,

0].0;1,6,40(GAR)

ength,1 GARthe

gin (volume)the

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ns)],

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0 gin(volume)

0 workers]finish?

perations),multiply

ewidth,(and)

ly7;30by itsdepth,

assignment,(and)you

,(and)

ereciprocal of30

0;2;

ouwillget 9.

he9th day.

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ment;

he9th day.

youperform(the

dthanditsdepth,(and)

ciprocalofthe

lget6];

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will get0;1,6,40.

and9,(and)youwill

0;1,6,40,(and)you

sthelength.

ngth, GARitsdepth,

ment;30workers

Whatis itswidth?

operations),multiply

depth,(and)

thereciprocalofthe

get6;

youwill get3,0>;

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d9

d)youwillget4,30;

d

GARisthewidth.

ngth,1 GAR]the

assignment;.

[you]perform(the

ngthandthewidth,

kethe reciprocal

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OMMENTARY

a(gagar)ofthe base

pth(bur)

sabar).

lowsthattheki-lais con-

ortheworkon theki-la,

saremade:

o beexpected

ar,

permanperday

workindays(u4)

s(erim-ha)

days(erim-ha)

silver(ku-babbar).

grelationsmusthold:

eexamples(No.22),

kedfor; thisisdefinedasthe

uldbeexpectedtocomplete

ermsofthelengthwhichhe

thwhichhetook").This

rmula

tesmXj,i.e.,thesumofthe

ssumesa ki-lawitha

ndadepthh =3$ku.

maneachday,X, isagain

examples.Thewages,how-

barley:1 bn=10sila.179

are g=a2,V,m andE.

answers:g =6JSAR,

n, E=20,50sila=4

57)does notaskforh,but J

mputesitsvalue.

reequivalentinall theexamplesof

herelation that6sesilver corre-

ey,or1seof silvertofsila ofbarley,

sooccursinproblem-textK(cf.p.79),

period oftheThirdDynastyofUr

nof thefirst30problems

ationsontherightmargin

emsare treatedinH(YBC

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oneofthesemagnitudesis

whatit actuallyis;inthis

strictshimselftothe bare

veexamplesneednospecial

eystrictlyfollowtheformulas

8leadto quadraticequa-

n whichIb andlbare

forNos.9 to18,but

solvedin exactlythesame

.15and16 leadtolinear

17,18 toquadraticequa-

nsare againpreserved,

nd20solvethesame

s.13and14.

sistsinfinding/,b andh

45SARandh=\ (l-b).

nd7=3;30.

0is superfluousbecauselb

1;45

specificlengthXjcorre-

gnmentfora singleday.

llowstheformula(2c)

estroyed,isto berestored

as thelengthassignedto

s.24-28)arebased on

urof theremainingfive

astis easilycomputed.

py:Plate21)

m

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CRIPTION

.*

th.Put (down)0;30,the

um

",whichhe(or:they)men-

he(or:they)mentioned

gnment.

whichhe(or:they)men-

rs,

nment,by9,the(numberof)

g)

h,by0;30,thewidth,(and

rea.

45,and(theresultis) 1;20.

thevolume,(andthe

0thewidth,

by 0;30,thewidth,(and

a,by3,thedepth,(and the

e.

5,theassignment,and(the

ume,(andthe resultis)9.

umberof)workers.

rstandingthistextlies

uiteseethe roleplayed

lines 2-3:"inonekalakkum

7** Evenifthesugges-

stheAkkadianequivalentof

heconsequencesarenot

eproblem(lines14) is

GARandthewidth

ndthedepthh.In lines

stobeused intheactual

thetranslation,"ninekalakkuaccord-

9kalakkuofequalsize.He compares

"xcubitsaccordingtoone(standard)

sthe pluralofkalakkum.

ndwearetold thatthe

obeunderstoodin thesense

andthedailyassignment

s0;15 SAR.Thecalcula-

ssmoothly,andthedepthh

pparentlyservesas acheck

hefirstpartof thetablet.

oundby

ern)

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LATION

dagal1kit bur-bi$gin

-til-e u4-10-kamin-til-e

kit bur-bi$ginsahare-kar

n-til-e u en-n[am]5US

ku bur-bi\gi[nsab]ar

4G[ARu] al-dib

ku bur-bi\[gi]nsahar

8GARu al-dib

1kuSdagali kulbur-bi\

-12i-kami-til

1 k[u]bur-bi\gine-kar

)]5[si]la

t bitr-bile-a-amen-

kl-bur-bi$ gine-kar1

u bur-bile-a-amen-

dagal-an-ta2ks'dagal-ki-

n-gagagarsabar-[h]a

am

12,30erim-ha$[m]a-na

5 gin3kit dagal-an-[t]a

-bi

alu-bun-g[a]u en-nam

gin5USu 2ksdagal-ki-

bun-gad[agal]-an-ta

am

dagal-a[n-ta]2ku dagal-

n-namal-dib4fk 4

dagal-an-ta2ku dagal-ki-ta2ku bur-bi

u en-namal-dib12

dagal-an-ta2kit dagal-

u4 en-nami-til-e u4-

an-ta2ku dagal-ki-ta2

u4-2S-kami-til-e u

kit dagal-an-ta2k!

bi*ren-namkiii-ku\ ku

k dagal-ki-ta2kit

kui-kudagal-an-t[a

dagal3ku bur-bi10

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88/155

LATION

sof ahiredman.What

pth.

slength,2ku the

gin(volume)the

ke? Hetook2GAR

helength,2ku the

n(volume)the

erstake?Theytook

helength,2ku the

n(volume)the

5 workersfinish?

.

he width,1kus"its

signment;

10thday.Whatis the

he width,1ku its

ssignment.

e? Hetook4GAR

isthewidth,iku its

signment.

ke? Hetook8GAR

ARisthelength, 1ku

$gin(volume)the

ne manfinish?He

thewidth,1ku its

ssignment;1180

n](and)5sila

houldIgive you?

he width,1ku its

ssignment;1hired

)5sila

s

houldIgive you?

helength,3kulthe

werwidth,2kuSits

nment,6se (silver)the

tare thearea,

f)workers,andthe

2],5 (SAR)isthe

berof)workers;

otal expensesin)

n)silver ofalittlecanal are

theupperwidth,

its depth,

ment,6se (silver)the

tis thelength?

n)silver ofalittlecanal

USthe length,

itsdepth,

gnment,6e (silver)the

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ENTARY

,2ku

depth,[10]gin

ku depthisku.

sthelength,3 kulthe

g[in(volume)the

of ahiredman.

ntary

inthistext exhibita

the ki-laproblemsof

bjecttobeworkedon is

ecanal(paj-sig)ofacertain

larcross-section.188The

expenses,etc.areagain

toformalarge numberof

ntheseeightexamples

of thecanal

th(dagal)ofthe canal

ecanal.

erefore

oved

1ubu.

earthexcavatedbyone

mberofmenrequiredto

e day,wthewagesofone

the totalwagesexpended,

smusthold:

enin thetextis$ gin,

nfor $;butsincem= 2,30

ealsogivenin VAT7528(MKTI

U;wehavetranslateduniformly

be$SAR= 20gin,not

y188continuesthroughthe

thecanaldugbyone

btain

dugby15 meninoneday,

holework,atimeintervalof

hermoreletotiodenote

dto finishtheworkin10

otal expensesarehere

heformeramountsto

m =2,30menare

medtobe 6esilver.The

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ENTARY

ectionaregivenas follows:

R

enin thetext:/i=4 GAR

10.

inationofthe numberti

n todigacanal withthe

ditionalmagnitudeswhich

=1ku

1(ban)5sila

orwhich2bn seemstobe

thcases.Therelationship

emagnitudesisnotclear to

respecttothevaluesof

sectionofthecanalis no

nowatrapezoidwiththe

thanthel owerwidth(bi).

perwidth(dagal-an-ta)

erwidth(dagal-ki-ta)

e(sabar)isgivenby

askedfor(as inNo.1in

swerisunfortunatelynot

ubtasto whichareais

ermanper dayis

medin thepreviousex-

wexpressedinsilverin-

sforone man'sdailywork

respondingtothe number

e

gin.

es,wedenotethelength

gin onedayby/,the

finish thewholeworkin

meintervalforthecomple-

nbyho-The firstseven

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CRIPTION

eismeasuredbythe

toaverticalheight of1ku

ketheoneassumedhere,

tionof difbu,bi andhare

.No.22 assumesd,bi

orthevalueofbu. Both

vealso/andX.

unsymmetricalarrange-

ninorderto givethesimplest

eparameterdwhichchar-

uld,however,assumethat

in widthforoneku depth

correspondstoonly\don

urse,makefor atech-

fora canal,butthefact

xamplesassumesrec-

cal(!)sides,showshow

stimatingthevalidityof

thistype.

softwolines whichmerely

oblem.Thatthese two

catch-line"givingthefirst

assupposedtofollowcan

efactthat thefirstexample

,beginswiththe identical

blem-textLis thedirect

extK.

thistext wassupposed

,althoughthereareonly23.

moftheexpressionforthe

iouslypublishedtexts,

rminology:1'1

a-gal

i-ku(-e)

u-ul

men-nami-ku-ul

ar8 ii-ku-la19* i-ku-ul

erse,23)i-na 1ku bur

ers|ku i-ku;inreverse,

ope isgivenasi-na1 ku

ey termsisquiteclear,

enounku allmean"fodder,

dakalu mean"eat,con-

gofthepassagesin our

ku ofthedepthhowmuch

al194)eat,"theanswers

n1ku ofitsdepthit ate

meaningin allcasesis

tofheight(ordepth),the

stionrecedesbyacertain

py:Plate10)

gal3ku bur-bi[10gin

ahar erim-haku-babb[ar

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ATION

dusu30 erim-hau e[n-

al->dib

agal3ku bur-bi1ku

-lu-tum

ndusu30 erim-hau4en-

til

ugbur-bi1 ku Sm196

m2ku lm196

-tilu en-nam5U$

dagal4Jku bur-bi1ku

m

usu1 ku lw1968197

5f198kug3,20u-si u

dagal3ku bur-bi1ku

um

ulu-1-eigi-TE-enu4H-lu-

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4Higi-5-gal u4dusu

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in196e-kar10erim-ha

]

USu 1kugdagal1 k[u

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LATION

uchlengthdid]

)6u-si.

elength,3kugthe

rthefirst) 1ku]

meisthelllutum;

h,208avolumeof204

uchlengthdid

k8GAR length.

elength,3ku the

thefirst) 1ku

umeisthelUutum;

h,203avolumeof204

manydaysdid

shed(after)1

he width,3ku its

depth,203is the

r)1month(and)75

gth?5 USisits

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rthefirst) 1ku

olume)istheMutum;

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ength.

e length,3kugthe

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actionofa day

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u? Andwhatis

dayhe extracted

)4ginvolume;

fofa dayhedug

gin volume.

Sis thelength,1kug

ofitstarahhum.What

5gin.199

Sis thelength,1ku thewidth,1ku itsdepth;itsincrease

hum203;$gin196(volume)

chlength[did]

gth.

USis thelength,1kuS

;[itsincrease]

humi03;3gin196(volume)

uch]length[didj

h.

].5 USisthelength,1 kit5

];

ofitstarahhum203;[$gin

t].

0 workersfinish?

ays.20*

u isthewidth,1 ku

off>\ku ofits

heassignment>.

days.What isits

.

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7/27/2019 Mathematical Cuneif

Mathematical Cuneiform Texts(1)

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