Words and Pictures: New Light on Plimpton 322 - Mathematical

16 Pages·2002·0.94 MB·English

Words and Pictures:
New Light on Plimpton 322
Eleanor Robson
1. INTRODUCTION. InthispaperIshalldiscussPlimpton322,oneoftheworld’s
mostfamousancientmathematicalartefacts[Figure1].ButIalsowanttoexplorethe
waysinwhichstudyingancientmathematicsis,orshouldbe,differentfromresearch-
ing modernmathematics.One of the most citedanalysesof Plimpton322, published
some twenty years ago, was called “Sherlock Holmes in Babylon” [4]. This entic-
ing title gave out the message that deciphering historical documents was rather like
solving a fictional murder mystery: the amateur detective-historian need only pit his
razor-sharp intellect against the clues provided by the self-containedstory that is the
pieceofmathematicsheisstudying.Notonlywillhesolvethepuzzle,buthewillout-
wit the well-meaning but incompetent professional history-police every time. In real
life, the past isn’t like an old-fashionedwhodunnit:historicaldocumentscan only be
understoodintheirhistoricalcontext.
Figure1. Plimpton322(obverse).Drawingbytheauthor.
Let’s start with a small experiment: ask a friend or colleague to draw a triangle.
Thechancesarethatheorshewilldrawanequilateraltrianglewithahorizontalbase.
Thatisourculturallydeterminedconceptofanarchetypal,perfecttriangle.However,
February2002] WORDSANDPICTURES:NEWLIGHTONPLIMPTON322 105
if we look at trianglesdrawn on ancient cuneiformtablets like Plimpton322, we see
that they all point right and are much longerthan theyare tall: very like a cuneiform
wedgeinfact.AtypicalexampleisUM29-15-709,ascribalstudent’sexercise,from
ancientNippur,incalculatingtheareaofatriangle[Figure2].Thescaledrawingnext
toitshowshowelongatedthesketchis.
57;30
54
2552;30
Figure2. UM29-15-709(obverse).Drawingbytheauthor[26,p.29].
Wetendtothinkofmathematicsasrelativelyculture-free;i.e.,assomethingthatis
outthere,waitingtobediscovered,ratherthanasetofsociallyagreedconventions.If
asimpletrianglecanvarysomuchfromculturetoculture,though,whathopehavewe
inrelyingonourmodernmathematicalsensibilitiestointerpretmorecomplexancient
mathematics?UnlikeSherlockHolmeswecannotdependsolelyonourownintuitions
anddeductivepowers,andwecannotinterrogatetheancientauthorsorscrutinisetheir
otherwritingsforclues.Wethereforehavetoexploitallpossibleavailableresources:
language, history and archaeology, social context, as well as the network of mathe-
matical concepts within which the artefact was created. In the case of Plimpton 322,
for instance, there are three competing interpretations, all equally valid mathemati-
cally.AsIshallshow,itisthesecontextualisingtoolsthatenableustochoosebetween
them.
Plimpton 322 is just one of several thousand mathematical documents surviving
fromancientIraq(alsocalledMesopotamia).Inits currentstate,it comprisesafour-
column,fifteen-rowtableofPythagoreantriples,writtenincuneiform(wedge-shaped)
script on a clay tablet measuring about 13 by 9 by 2 cm [20, Text A, pp. 38–41].
ThehandwritingoftheheadingsistypicalofdocumentsfromsouthernIraqof4000–
3500 years ago. Its second and third columns list the smallest and largest member
of each triple—we can think of them as the shortest side s and the hypotenuse d of
a right-angled triangle—while the final column contains a line-count from 1 to 15.
Partofthetablethasbrokenawayatthebeginningofthefirstcolumnbut,depending
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on whether you believe the column has fully survived or not, it holds the square of
either thehypotenuseor theshortestsideofthetriangledividedbythesquareofthe
longer side l. Whether it lists d2/l2 or s2/l2, this column is in descending numerical
order. The numbers are written in the base 60, or sexagesimal, place value system. I
shalltransliteratethemwithasemicolonmarkingtheboundarybetweenintegersand
fractions,andspacesinbetweentheothersexagesimalplaces[Figure3].
[ta]-ki-il-tis.i-li-ip-tim
[sˇa1in]-na-as-sa`-˘hu-maSAGi-il-lu-u´ ´IB.SI8 SAG ´IB.SI8 s.i-li-ip-tim MU.BI.IM
[(1)59]0015 159 249 KI.1
[(1)5656]5814500615 5607 12025 KI.2
[(1)5507]41153345 11641 15049 KI.3
(1)531029325216 33149 50901 KI.4
(1)48540140 105 137 KI.[5]
(1)47064140 519 801 [KI.6]
(1)431156282640 3811 5901 KI.7
(1)41334514345 1319 2049 KI.8
(1)38333636 801 1249 KI.9
(1)3510022827242640 12241 21601 KI.10
(1)3345 45 115 KI.11
(1)292154215 2759 4849 KI.12
(1)27000345 241 449 KI.13
(1)25485135640 2931 5349 KI.14
(1)23134640 28 53 KI.15
Figure3. TransliterationofPlimpton322.
Therehavebeenthreemajorinterpretationsofthetablet’sfunctionsinceitwasfirst
published[Figure4]:1
1. Some have seen Plimpton 322 as a form of trigonometric table (e.g., [15]): if
ColumnsIIandIIIcontaintheshortsidesanddiagonalsofright-angledtriangles,
thenthevaluesinthefirstcolumnaretan2 or1/cos2—andthetableisarranged
sothattheacuteanglesofthetrianglesdecreasebyapproximately1◦ fromline
toline.
2. Neugebauer[19],andAaboefollowinghim,arguedthatthetablewasgenerated
likethis:
If pandqtakeonallwholevaluessubjectonlytotheconditions
(1) p>q >0,
(2) pandqhavenocommondivisor(save1),
(3) pandqarenotbothodd,
thentheexpressions
x = p2−q2 [ours],
1Incidentally,wecandismissimmediatelyanysuspicionthatPlimpton322mightbeconnectedwithobser-
vationalastronomy.AlthoughsomesimplerecordsofthemovementsofthemoonandVenusmayhavebeen
madefordivinationintheearlysecondmillenniumBCE,theaccurateanddetailedprogrammeofastronom-
icalobservationsforwhichMesopotamiaisrightlyfamousbeganathousandyearslater,atthecourtofthe
AssyriankingsintheeighthcenturyBCE[3].
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line α p q x 1/x
1 44.76◦ 12 5 224 25
2 44.25◦ 104 27 2221320 251845
3 43.79◦ 115 32 2203730 2536
4 43.27◦ 205 54 2185320 255512
5 42.08◦ 9 4 215 2640
6 41.54◦ 20 9 21320 27
7 40.32◦ 54 25 20936 274640
8 39.77◦ 32 15 208 280730
9 38.72◦ 25 12 205 2848
10 37.44◦ 121 40 20130 29374640
11 36.87◦ 2 1 2 30
12 34.98◦ 48 25 15512 3115
13 33.86◦ 15 8 15230 32
14 33.26◦ 50 27 1510640 3224
15 31.89◦ 9 5 148 3320
Figure 4. The proposed restorations at the beginning of the tablet according to the
trigonometric,generatingfunction,andreciprocalpairtheories.
y=2pq [ourl],
z= p2+q2 [ourd],
will produce all reduced Pythagorean number triples, and each triple only once [1,
pp.30–31].
Thequesthasthenbeentofindhow pandq werechosen.
3. Finally, the interpretation first put forward by Bruins [5], [6] and repeated in a
clusterofindependentpublicationsabouttwentyyearsago[4], [9], [30]isthat
the entries in the table are derived from reciprocal pairs x and 1/x, running in
descendingnumericalorderfrom2;24∼0;25to1;48∼0;3320(where∼marks
sexagesimal reciprocity). From these pairs the following “reduced triples” can
bederived:
s(cid:7) =s/l =(x −1/x)/2,
l(cid:7) =l/l =1,
d(cid:7) =d/l =(x +1/x)/2.
Thevaluesgivenonthetablet,accordingtothistheory,areallscaledupordown
bycommonfactors2,3,and5untilthecoprimevaluess andd arereached.
Howarewetochoosebetweenthethreetheories?Internalmathematicalevidence
aloneclearlyisn’tenough.Weneedtodevelopsomecriteriaforassessingtheirhistor-
icalmerit.Ingeneral,wecansaythatthesuccessfultheoryshouldnotonlybemathe-
maticallyvalidbuthistorically,archaeologically,andlinguisticallysensitivetoo.
A great deal of emphasis has been laid on the uniqueness of Plimpton 322; how
nothing remotely like it has been found in the corpus of Mesopotamian mathemat-
ics.Indeed,thishasbeenanimplicitargumentfortreatingPlimpton322inhistorical
isolation. Admittedly we know of no other ancient table of Pythagorean triples, but
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Pythagoreantriangleswereacommonsubjectforschoolmathematicsproblemsinan-
cientMesopotamia.Thispointhasbeenmadebefore(e.g.,byFriberg[9])buthasn’t
yetprovedparticularlyhelpfulindecidingbetweenthethreeinterpretationsofPlimp-
ton 322. What we shall do instead is to make some new comparisons. None of the
comparativematerialitselfisnewthough:allbutoneofthedocumentsIhavechosen
werepublishedatthesametimeasPlimpton322ordecadesearlier.
First, Plimpton 322 is a table. Hundreds of other tables, both mathematical and
nonmathematical,havebeenexcavatedfromMesopotamianarchaeologicalsites.What
canwelearnfromthem?
Second,ifPlimpton322isatrigonometrytable,thenthereshouldbeotherevidence
ofmeasuredanglefromMesopotamia.Weshallgoinsearchofthis.
Third,Plimpton322containswordsaswellasnumbers:theheadingsatthetopof
eachcolumnshouldtelluswhatthetabletisabout.Someofthemoredifficultwords
alsoappearonothermathematicaldocumentsfromMesopotamia.Cantheyhelpusto
understandtheirfunctiononPlimpton322?
Finally,Plimpton322waswrittenbyanindividual.What,ifanything,canwesay
abouthimorher,andwhythetabletwasmade?
2. TURNINGTHETABLESONGENERATINGFUNCTIONS. Let’sstartwith
someverygeneralcontextualisation.Wecanlearnalotaboutanytabletsimplyfrom
itssize,shape,andhandwriting.
Plimpton322isnamedafteritsfirstWesternowner,theNewYorkpublisherGeorge
A. Plimpton (see Donoghue [8]). He bequeathed his whole collection of historical
mathematicalbooksandartefactstoColumbiaUniversityinthemid-1930salongwith
a large number of personal effects. Surviving correspondence shows that he bought
thetabletfor$10fromawell-knowndealercalledEdgarJ.Banksinabout1922[2].
BankstoldhimitcamefromanarchaeologicalsitecalledSenkerehinsouthernIraq,
whoseancientnamewasLarsa[Figure5].
Figure5. Mapofthearchaeologicalsitesmentionedinthetext.Drawingbytheauthor.
February2002] WORDSANDPICTURES:NEWLIGHTONPLIMPTON322 109
VastnumbersofcuneiformtabletswerebeingillicitlyexcavatedfromLarsaatthat
time.Severalbigmuseums,suchastheLouvreinParis,Oxford’sAshmoleanMuseum,
andtheYaleBabylonianCollectionboughtthousandsofthem.AlthoughPlimpton322
doesn’tlookmuchlikeothermathematicaltabletsfromLarsa,itsformatisstrikingly
similar to administrative tables from the area, first attested from the late 1820s BCE.
The tablet YBC 4721 [12, no. 103], for example, is an account of grain destined for
variouscitieswithinthekingdomofLarsa[Figure6].ItwaswritteninthecityofUr,
thenunderLarsa’spoliticalcontrol,in1822BCEandisnowhousedatYale.
Like Plimpton322it is writtenon a “landscape”format tablet (thatis, thewriting
runs along the longer axis) with a heading at the top of each column. Entries in the

Graindebit ForUr ForMar-... For...... Total Itsname
301<gur> 301<gur> 301<gur> Lipit-Suen
301<gur> 214gur,285sila 86gur15sila 301<gur> Nur-Dagan
296<gur> 180gur [60gur] 56<gur> 296gur Ili-eriba
277gur200sila [ ] 277<gur>200sila 277<gur>200sila Sˇamasˇ-kima-ilisˇu
From23gur40silaof[Sˇamasˇ-...-aplu’s]troops/workers.
1176gur20sila 481<gur> 274<gur>485sila 420gur95sila 1176gur20sila
FromthegrainofLu-am-...
Andfrom23gur40silaofSˇamasˇ-...-aplu’stroops/workers.
Month1,day7,
YearthatRim-Sinbecameking(1822BCE).
Figure6. YBC4721,afterGrice[12,pl.XL].1gur=300sila≈300litres.
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firstcolumnaresortedintodescendingnumericalorder.Calculationsrunfromleftto
rightacrossthetable,whilethefinalcolumnliststhenamesoftheofficialsresponsible
foreachtransaction.AlthoughthescribesofLarsamostlyusedthecuneiformscriptto
write a Semitic languagecalled Akkadian, they often used monosyllabicwords from
a much older language, Sumerian, as a kind of shorthand. Like the final column of
Plimpton322,thelastheadingonYBC 4721carriestheSumerianwriting MU.BI.IM
forAkkadiansˇumsˇu(“itsname”).UnlikePlimpton322though,thetextisdatedinthe
finalline.ThereareabouthalfadozenpublishedtablesfromtheLarsaareawiththese
samecharacteristics:allofthemaredatedtotheshortperiod1822–1784BCE andso,
therefore,isPlimpton322.
SowecanalreadysaythatPlimpton322waswrittenbysomeonefamiliarwiththe
temple administration in the city of Larsa in around 1800 BCE, at least twenty years
beforeitsconquestbyBabylonin1762.SherlockHolmes,ifheevermadeittoBaby-
lon, would have been over 100 miles away from the action: no ancient mathematics
haseverbeenfoundthere.
AndthefactthatPlimpton322followsthesameformattingrulesasallothertables
from ancient Larsa leads us to dismiss Neugebauer’s theory of generating functions.
If the missing columns at the left of the tablet had listed p and q, they would not
havebeenindescendingnumericalorderandwouldthushaveviolatedthoseformat-
ting rules. Nor, underthis theory,has anyone satisfactorilyexplainedthe presence of
ColumnIinthetable.Thereareothergoodreasonstoeliminatethegeneratingfunc-
tion theory;I deal with themin [29]. The trigonometrytableand the reciprocal pairs
remain.
3. CIRCLINGROUNDTRIGONOMETRY. Wesawatthebeginningofthisarti-
clehowdifferentlyfromusthepeopleofancientMesopotamiathoughtabouttriangles;
thatcontrastwithmodernconceptsrunsrightthroughtheirplanegeometry.
For instance, YBC 7302 [20, p. 44] is roughly contemporary with Plimpton 322
[Figure 7]. From its circular shape and size (about 8 cm across) we know the tablet
was used by a trainee scribe for school rough work. It shows a picture of a circle
with three numbers inscribed in and around it in cuneiform writing: 3 on the top of
the circle, 9 to the right of it, and 45 in the centre. Now 9 is clearly the square of 3,
but what is the relationship of these numbers to 45? The answer lies in the spatial
arrangementofthediagram.Lookingclosely,wecanseethatthe3liesdirectlyonthe
circumference of the circle, while the 45 is contained within it. The 9, on the other
hand, has no physical connection to the rest of the picture. If on this basis we guess
that 3 represents the length of the circumference and 45 the area of the circle, we
shouldbelookingfortherelationship A =c2/4π.Wehavethec2—that’sthe9—and
if we use the usual Mesopotamian school approximation π ≈ 3, we get A = 9/12.
Thistranslatesinbase60to45/60,namely,the45writteninthecircle.
Whenweteachgeometryinschoolwehaveourstudentsusetherelationship A =
πr2; none of us, I would guess, would use A = c2/4π as a formula for the area of
a circle. In modern mathematics the circle is conceptualised as the area generated
by a rotating line, the radius. In ancient Mesopotamia, by contrast, a circle was the
shapecontainedwithinanequidistantcircumference:notethatthereisnoradiusdrawn
on YBC 7302. There are many more examples of circle calculations from the early
secondmillennium,andnoneoftheminvolvesaradius.Evenwhenthediameterofa
circlewasknown,itsareawascalculatedbymeansofthecircumference.Wealsosee
this conceptualisation in the language used: the word kippatum, literally “thing that
curves,”meansboththetwo-dimensionaldiscandtheone-dimensionalcircumference
that defines it. The conceptual and linguistic identification of a plane figure and one
February2002] WORDSANDPICTURES:NEWLIGHTONPLIMPTON322 111
3
45 9

Figure7. YBC7302(obverse).Drawingbytheauthor.
of its external lines is a key feature of Mesopotamian mathematics. For instance, the
wordmithartum(“thingthatisequalandoppositetoitself”)meansboth“square”and
˘
“sideofsquare.”We runintobiginterpretationalproblemsifweignorethesecrucial
terminologicaldifferencesbetweenancientMesopotamianandourownmathematics.
What does this tell us about Plimpton 322? That if plane figures were conceptu-
alised, named, and defined from the inside out, then the centre of the circle and the
idea of the rotating radius could not have played an important part in Mesopotamian
mathematics.Andiftherotatingradiusdidnotfeatureinthemathematicalideaofthe
circle, then there was no conceptual framework for measured angle or trigonometry.
Inshort,Plimpton322cannothavebeenatrigonometrictable.
Thisshouldhavebeenourintuitiononlaterhistoricalgroundsanyway.Nearlytwo
millenniaafterPlimpton322waswritten,Ptolemyconceptualisedthecircleasadiam-
eterrotatingaboutitscentreinordertosimplifyhiscalculationsofchordsofarc—but
thosechordswerefunctionsofarc,notofangle(Toomer[31,p.47]).(Ptolemy,work-
ing in Roman Egypt in the second century CE, was heavilyreliant on Mesopotamian
traditions: he used astronomical data from first millennium Assyria and Babylonia,
adaptedMesopotamianmathematicalmethods,andevencalculatedinbase60.)Over
the following millennium several generations of Indian and Iraqi scholars compiled
tables of half-chords, but the conceptual transition from arc to angle was slow and
halting.
ReturningtotheearlysecondmillenniumBCE,Ishouldemphasisetwopoints.First,
IdonotmeanthattheancientMesopotamiansdidnotknowthatcirclescouldbegener-
atedbyrotatingradii.Thereisagreatdealofvisualevidencetoshowthattheydid.For
example, BM 15285, a compilation of plane geometry problems from Larsa, depicts
severalcircleswhosedeeplyimpressedcentresrevealthattheyweredrawnbymeans
of rotating compasses [Figure 8]. But Mesopotamian mathematical concepts were as
112 (cid:1)c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly109
Figure8. BM15285(detail).Drawingbytheauthor[25,p.214].
sociallyboundedasoursare:althoughweoftendrawcirclesfree-handwithoutradii,
even in mathematics classes, it would rarely cross our minds to teach our students
A =c2/4π.Equally,radiiwereknownandusedinancientMesopotamia,butplayed
little part in the dominant outside-in conceptualisation of plane geometry. Second,
neither do I mean that there was no concept of angle at all in ancient Mesopotamia.
Gradientswereusedtomeasuretheexternalslopeofwallsandrampsinformulations
like“forevery1cubitdepththeslope(ofthecanal)is1/2cubit”(YBC4666,rev.25;
[20,textK]).Therewasalsoaroughdistinctionmadebetweenrightanglesandwhat
wemightcall“wrongangles,”namely,thoseconfigurationsforwhichthePythagorean
ruleheldtrueornot,withprobablya10◦ to15◦ leeway(seeRobson[24]).
To sum up so far: the theory of generating functions is organisationally implausi-
ble,whilethetrigonometrictheoryisconceptuallyanachronistic.Weareleftwiththe
theoryofreciprocalpairs—howdoesitmeasureuptoourhistoricalexpectations?
4. WORDS COUNT TOO: RECIPROCAL PAIRS. We can start by recognising
thatreciprocalpairs—unlikegeneratingfunctionsortrigonometry—playedakeyrole
inancientMesopotamianmathematics.Ourbestevidenceisfromthescribalschoolsof
nineteenthandeighteenthcenturyLarsa,Ur,andNippur,wherethousandsofsurviving
practicecopiesshowthatscribalstudentshadtolearntheirsexagesimalmultiplication
tablesinthecorrectorderandbyheart.Thefirstpartoftheserieswasthesetofthirty
standard reciprocal pairs encompassing all the sexagesimally regular integers from 2
to 81 (thereby including the squares of the integers 1 to 9) [Figure 9]. The trainees
also learned how to calculate the reciprocals of regular numbers that were not in the
standard list and practised division by means of finding reciprocals, as this was how
allMesopotamiandivisionswerecarriedout(seeRobson[26,pp.19–23]).
Looking at the reciprocals proposed as the starting point for Plimpton 322 [Fig-
ure 4], it turns out that although only five pairs occur in the standard list, the other
tenarewidelyinevidenceelsewhereinMesopotamianmathematics(asconstants,for
instance),orcouldbecalculatedtriviallyusingmethodsknowntohavebeentaughtin
scribalschools.Noneofthemismorethanfoursexagesimalplaceslong,andtheyare
listed in decreasing numerical order, thereby fulfilling our tabular expectations. But
wehaven’tyetexplainedthepurposeofthefirstsurvivingcolumn:whatareallthose
s2/l2 (ord2/l2)doingthere?
Theheadingsatthetopofthetableoughttotellus:that,afterall,istheirfunction.
Wehavealreadyseenthatthelastcolumn,whichcontainsonlyaline-count,isheaded
liketheothertablesfromLarsawiththesignsMU.BI.IMmeaningsˇumsˇu(“itsname”).
February2002] WORDSANDPICTURES:NEWLIGHTONPLIMPTON322 113
Twothirdsof1is0;40. Thereciprocalof24is0;0230.
Itshalfis0;30. Thereciprocalof25is0;0224.
Thereciprocalof2is0;30. Thereciprocalof27is0;021320.
Thereciprocalof3is0;20. Thereciprocalof30is0;02.
Thereciprocalof4is0;15. Thereciprocalof32is0;015230.
Thereciprocalof5is0;12. Thereciprocalof36is0;0140.
Thereciprocalof6is0;10. Thereciprocalof40is0;0130.
Thereciprocalof8is0;0730. Thereciprocalof45is0;0120.
Thereciprocalof9is0;0640. Thereciprocalof48is0;0115.
Thereciprocalof10is0;06. Thereciprocalof50is0;0112.
Thereciprocalof12is0;05. Thereciprocalof54is0;010640.
Thereciprocalof15is0;04. Thereciprocalof100is0;01.
Thereciprocalof16is0;0345. Thereciprocalof104is0;005615.
Thereciprocalof18is0;0320. Thereciprocalof121is0;00442640.
Thereciprocalof20is0;03. <Itshalf>
Figure9. MLC1670,afterClay[7,no.37].
Thetwocolumnsimmediatelyprecedingthat,weremember,containwhatwecan
convenientlythinkofastheshortestsidesanddiagonalsofright-angledtriangles.They
are headed´IB.SI8 SAG and ´IB.SI8 s.i-li-ip-tim for mit˘harti pu¯tim and mit˘harti s.iliptim
(“squareoftheshortside”and“squareofthediagonal,”respectively).Thiscontradic-
tiondisappearswhenwerecallthatMesopotamianplanefiguresaredefinedandnamed
for their key external lines. We can thus adjust our translationsto read “square-side”
oftheshortsideanddiagonal,respectively.(Iamusingthetranslation“diagonal”here
ratherthan“hypotenuse”toindicatethatthisisageneralwordforthetransversalofa
figure,notrestrictedtotriangles.)
Lastandbyfarthemostdifficult,theheadingofthefirstsurvivingcolumnreads
[ta]-ki-il-tis.i-li-ip-tim
[sˇa1in]-na-as-sa`-hu-u´-maSAGi-il-lu-u´
˘
(forAkkadiantakiltis.iliptimsˇaisˇte¯ninnassahuˆmapu¯tumilluˆ),
˘
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