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ES
RMTEXT.
Publ
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ORGEA.KENNEDY
IETY
UT
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RM
ACHS
IETY
RIENTALRESEARCH
CUT
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mehasbeenaided bygrants
lofLearnedSocietiesandthe
lAssociationofAmerica.
ety
ANCASTER,PENNSYLVANIA
OFAMERICA
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Archibald,
eHistoryofMathematics,
and
nt
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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-
s workinitsSeries,and to
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
16
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.
ningwithPublicDomain,
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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
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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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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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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
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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
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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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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
e(resulting)1,20isthe
e(resulting)1,0isthe
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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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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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)
Mb
-de
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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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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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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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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)silver ofaki-laare9 gin,
n (volume)the
ewages;thelength
GAR.What are
n)silver ofaki-laare9 gin,
volume)theassign-
ges;Iaddedthe
he resultis)2
166andthe depth?
n)silver ofaki-laare9 gin,
volume)theassign-
ges;thewidthex-
R.Whatarethe
s45[SA]R,\ GARits
this thewidth
to)thewidth.
hewidth?
is45SAR,\ GARthe
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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,
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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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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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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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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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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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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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LATION
)
th,1 GAR
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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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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
0workerstake?]When
ns)],
ngth,1i GARthe
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.
dth,| GARitsdepth,
ment;
he9th day.
youperform(the
dthanditsdepth,(and)
ciprocalofthe
lget6];
u willget54;takethe
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>;
0,(and) youwill
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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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-
]usab[ar]en-namigi-
4Higi-5-gal u4dusu
1ku dagal1ku
ffi-au-la-mi-i(sahar-bi
1ku dagal1ku bur-bi
gin196e-karlu-1-eu e[n-
1ku dagal1ku bur-bi
in196e-kar10erim-ha
]
USu 1kugdagal1 k[u
0u-U-mi-i\\\gine-kar]
-tilu4-75-kam[in-til]
agal 1ku bur-bideA-
la-mi-i($gin6- kar>
ffi-ia.
u-bien-nam5US
-bi2 kugdagal1kug
a-am-mi-i(
sahar-bien-nam1iku
u->bi 2ku dagal1ku
u-la-am-mi-i(1ku a-n0
1-eu en-namal-dib
-bi3 ku dagal1ku
u-la-am-mi-i(1|ku
-bien-nam[PublicDomain,
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LATION
uchlengthdid]
)6u-si.
elength,3kugthe
rthefirst) 1ku]
meisthelllutum;
h,208avolumeof204
uchlengthdid
k8GAR length.
elength,3ku the
thefirst) 1ku
umeisthelUutum;
h,203avolumeof204
manydaysdid
shed(after)1
he width,3ku its
depth,203is the
r)1month(and)75
gth?5 USisits
helength,3ku the
rthefirst) 1ku
olume)istheMutum;
th,203avolumeof204
ast) Ifku
5197ginis thedusu.
emantake? Hetook
ength.
e length,3kugthe
thefirst)1ku
olume)isthelllutum;
depth,203(avolumeof)
actionofa day
m?Whatfraction
u? Andwhatis
dayhe extracted
)4ginvolume;
fofa dayhedug
gin volume.
Sis thelength,1kug
ofitstarahhum.What
5gin.199
Sis thelength,1ku thewidth,1ku itsdepth;itsincrease
hum203;$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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