Mathematics and Measurement

66 Pages·1993·4.41 MB·English

by  Oswald Ashton Wentworth Dilke

READING THE PAST
MATHEMATICS
AND MEASUREMENT
QA.W. DILKE
Mathematics andMeasurement
Coverpicture:DetailfromiMcsimile<ittheI'eutinper1ahle,acopyofaRom.mroadmap.Rome
isinthecentre.1heoriginalisintheO&terreichischeNationalbibliotheic,Vienna(seefig.37).
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Romanbronzeset-square,dividers,proportionalcompasses,rulerandplumb-bob.
BritishMuseum.
Mathematics
and Measurement
O. A. W. Dilke
UniversityofCaliforniaPress/BritishMuseum
Th±» o
Acknowledgements
Iamindebtedformuchhelpto tin utre,ageographer,andourson, a mathematician;
tomycolleagueOrJ.R.Ravetz,ReaderintheHistdrv.inJPhilosophvofScience,University
ofLeeds; to IVott-ssorB. Wesenherp; totheMuseoHgizio, Iiinn;.indtoni.inv menihers
ofstaffoftheBritish Library andtheBritish Museum.TheseincludeparticularlyDrIan
Carradice,MrBrianCook,MrIanJenkins,MrThomasPattie,DrJeffreySpencer,Mr
OiristopherWalkerandDrHelenWalHs.1wishtothankthepublishersforenablingme
to reproduce illiistrnrinns from my book ihe Row.iu {..vniSurveyors (l*^~r, nowout
ofprintexceptinItaliantranslation;andThamesandHudsonLtdforfacilitatingtheuse
ofillustrations from my GreekamiRomanMaps (1985): figs4(fromJ.Ball, E^yptin
theClassicalGeographers^Cairo,1942),32,33,34(afterPauly,Reat-Encyclopadieder
ClassischenAltertumswissenschaft,supp.x,Stuttgart,!965),.^8,39,
Thanksarealsoduetoallinstitutions,acknowledgedinthecaptions,whohavekindly
suppliedphotographs,aswellastothefollowing: HBMC(England),Hg.22;Courtauld
InstituteofArt,London,tigs31,57and58.ArtworkbyChristineBarratt,SueBird,John
DixonandAnnSearight.
^
Volume2in ReadingThePastseries
LibraryofCongress
Cataioging-in-PublicationData
Diike,OswaldAshtonWentworth.
©Pub1l9isKh7edrhbiv-B1rriutsitschcsMuofstehuemBrIi'tuibslhicMautisDensumLtd MBiabtlhieomgaratpihcys:apn.dmeasurement.
46BloomsburyStreet,LondonWCIBiQQ Includesindex.
1.Mathematics,—Ancient.
Designedhv NrthurI.ockwooJ 2.Mensuration History.
Frontc<)\erdesignbyGrahanu'Dudley 1.Tide. 11.Scries.
QA22.D55 1988 510'.09'01 86-30803
SetinLinotype202Sabonandprintedat
TheBathPress,Avon ISBN0^520-06072r-5(pbk.:alk.paper)
Co(., y :odmaterial
Contents
Prefnee 6
1 TheBackground 2
2 NumbcrinfibyLetters L3
3 MnthematicnlFducntionintlieGreekWorld 12
4 MtMsiiremt-nt 23
5 MathematicsfortheSurveyorandArchitect 22
6 MappingandtheConceptofScale 15.
7 TellingtheTime 40
8 Calculati()nsforTradeandCommerce 4£
9 MathematirsinleisurePnr<iiiit<iandthe
Occnh 53
10 TheSegue! 52
Bibliography 62
IndpY 64
Preface
ThesophistProtagoras,wholecturedalloverGreeceinthefifthcenturyBc,proclaimed:
Themeasureofeverythingishuman'. Although hespokeinawidecontext,hishearers
maywellhavetaken him literally. IntheAshmolean Museum,Oxford,isareliefin the
shapeofa pediment,showingthehead,chest,outstretchedarms and foot imprintofa
man[helotv).Whateveritsexactmetrologicalinterpretation(andthisisdisputed),itshows
notonlyawarenessoftheimportanceofhumanmeasurements,butasenseofharmony
andsymmetry.SuchanapproachisvisibleinGreekandRomanartandarchitecture,and
intheesteemaccordedtotheteachingofmusicandgeometry.
Inourcomputerageofelectronicaccuracy,resultshyancientmethodsmayseemunreliable.
Yet in many respectswecan showthat theancientscouldoftencomeclosetoanexact
measurement,thoughattemptsatmodernconversionmaybeproblematic.
Themostprominenttheoretical aspectsofmathematicswereastronomy,in Babylonia
andGreece,andgeometry,inGreece.Thesesometimesoverlapped,andgeometrycontributed
tothedevelopmentofalgebraandtrigonometry. Kratostheiies'calculationof theearth's
circumferenceshowsacombinationofthetheoreticalandthepractical.Hemayhavebeen
moreinterestedinassessingtheangleofthesunatmiddayindifferentplaces,buthealso
based his calculationson a rough approximation to actual land distance, and wenton
todrawaworldmapwhichinfluencedhissuccessors.
.Artefacts fromexcavationsin Egypt,theAthenianagora andelsewhereemphasisethe
practical importance, fortradeandcommerce,ofstandard weights, measures andtime-
keepers.IntheRomanworldthepracticalsideofmeasurementlargelypredominated.Road-
making,surveying,militaryorganisation,watersupplyandsanitation,alldependedonwell-
definedsystemsotmeasuring.
Thepresentworktriestoshowwhatawealthofartefactswehave,particularlyinmuseums
andlibraries,whichthrowlightonancientmathematicsandmeasurement,andtosetthese,
alongsidetechnical literature from the Rhind papyrus rightdown to Renaissance Latin,
intheirculturalbackground.
1 Greekmetrologicalrelief.Oxford,AshmoleanMuseum.
1 The Background
Weknowthatanumberofancientcivilisationsdevelopedtheirowntechniquesotmathema-
ticsandmeasurementseparately.Muchisnowknown,forexample,ofthehiston,'ofChinese
mathematicsandmeasurement;buttheydidnotinfluencetheknowledgeofthosesubjects
intheWest,andsoitisnotproposedtooutlinetheminthisbook.Ofthosecivilisations
whichdidinfluenceWesterndevelopments,theprincipaloneswereEgyptandMesopotamia,
eachofwhichhadbeenevolvingitsownsystemfromaveryearlyperiod.
InEgyptwefindearlyinterestinastronomyandcosmology,culminatinginthedivision
ofthe yearinto 12 months of30 dayseach,to which fivedayswereadded. Egyptian
interestinthestarswastypifiedbypicturesintombsshowingastronomicalfeatures,including
mapsofstarpositions,tocalculatethepassageofnighthours.Suchinformationmayhave
beenofpraaicalhelptoafarmingcommunitysuchasflourishedintheNilevalley.The
Egyptianswentonfromthistodeviseasystemof 12daytimehoursand 12nighthours,
whichformedtheoriginofthemodernsystemofdivisionoftheday.Since,however,the
amountofdaylightvariedaccordingtothemonth,itwillbeseen (chapter7)thatsome
carefulobservationsandcalibrationswereinvolved.
EgyptisthecountryoftheNileandthepyramids,andit isnotsurprisingthatgreat
accuracywasattemptedwithregardtobothofthese.Theannualautumnfloodingofthe
Nile, caused aswenow know by high summerrainfall in Ethiopia,helped the fertility
ofEgyptianfieldsbyspreadingquantitiesofsilt.Torecordlevelsupstream,aNilometer
wassetuponPhilaeIsland,whichisnowsubmergedbythelakeoftheAswandam.Strnho
describes it as a well, in which the level ofthe Nilecould be read from markson the
side.Fromsuchreadings,scribescouldregisterthemaximum,minimumandmeanlevels
ofthewaterin cubits. Asis known from ancientwriters,annual compilationsofsuch
readings werecarefully preserved, butthesehavenot themselvessurvived. A Nilometer
atMemphis(Cairo) ismentionedby Hcliodorus,andsimilargaugeshavebeen foundat
Edfu,Luxorandelsewhere,withcubitmarkingsmostlybetween52.Sand53..^cmabove
theriver'slowlevel.
Whenthefloodsreceded,manylandowners'boundarymarkshadinevitablybeenwashed
away,so itwas importantthatsuneyingshould becarried outimmediately.Thereare
Egyptianrepresentationsofsurveyorsemployingknottedropes(theknotsindicatingsub-
divisionsoflinearmeasurement),themerkhet(asplitcentre-ribofapalm-leaf,used for
2 Reproductionof.imerkhcl,anF.f;yptiansurveyinginstrument.Themerkhetwasalinncdonan
objectbylookingthroughthesplitcentre,heldupwards,itwasusedwithashortplumb-lineand
plummet.London,Science.Museum.
1
sighting),and measuringrods.Thepriests inauguratedthis rapid re-surveyoftheland,
whichhadtobereadyforwintercultivation.Thereisnoevidencethatlandsurveymaps
wereusedindynasticEgyptforthisoperation.Butbymeansofexactareameasurements
and verbal descriptions, the status quo was re-established. Graeco-Roman writers from
Herodotus(c.484-^.420bc)rightdowntoCassiodorus{c.ad490-c:.583)attributetheori-
ginsofgeometr)',literally'measuringoftheearth',tothispractice.
Egyptian numerals werebasedon thedecimal system,thehighestvaluesbeingplaced
ontheleft. Uptonineofthesamesymbolcouldbeused,arranged inoneortwolines.
Thehieroglyphicsignsare:
1 10,000
10 100,000
100 1,000,000
1000
Examplesare:
465 ttn1111
4323 nm?r.
Thesign for 1,000,000disappeared inearlytimes,andafteritsdisappearanceasystem
ofmultiplication,byplacingonenumberoveranother,wassometimesused;forexample,
1,100,000=100,000X II= ^
Themost famous mathematical work from dynastic Egypt isthe Rhindmathematical
papyrus,copiedbythescribeAhmcsorAhmosefromapapyrusof1849-1801bc.Itwas
boughtbyA.H.RhindatI.uxorin1858andisnowintheBritishMuseum.Initsgeometry
wecan seesome understandingofthepropertiesofright-angledtriangles, includingthe
simplestones, which havesidesthatareintegralnumbersofunitsm length (3,4,5;5,
12, 13).Alsoincludedisanapproximationfortt,theratioofthecircumferenceofacircle
toitsdiameter;thevaluegivenis(v)'=3.1604938,loolargebyabout0.019,
1-3
3 PartoftheRhindmathematicalpapyrus,dealingwiththecalculationof.ircas.StvonJIntermediate
Period,c.1575BC.BritishMuseum.
9
TheRhindpapyrushasseveralarithmeticalproblems,suchasthefollowing:*Aquantity
whosehalfisaddedtoitbecomes16*;inodierwords,'findfof16*.Theworkingsare
asfollows:
1(a)2 •I(bS)
^1
2 6
4 12
13 2
» 1 1
Table(a)showsthatilicdivisorshouldhe^.Thepurposeoftahlefh)istofindone-third
of 16. Column2multiphescolumn I by three. Theasteriskeditemsareaddedtogether
becausethecorrespondingnumbersintheright-handcolumnaddupto16.Thisestablishes
that1+4+ thatis,5i^i$one-thirdof16.Since|of16istobefound,theremaining
workingsare:
(c)
1 5i
2 10^
Geometrywasequallyrequiredtortheconstructionofpyramids.Inthefirstplace,care
wasoftentakentoachieveorientationnorth,south,eastandwest.North-southorientation
couldeasilybeobtainedbyfindingthedirectionofthenooiulav sun.Averticalpolewas
setupin thesandasagnomon.Thenthepathtraversedbythetipofitsshadow(.oiild
heobserved. Thepoints A and B. where ihis intersected asuitablecircledrawn around
thegnomon, wouldbejoined. Then the hue Aii was bisected toestabhsh thedirection
ofthesunatmid-day.Apossiblealternativemethod,outlinedbyI.E.S.EdwardsinThe
PyramidsofEgypt^wouldhavebeentobisecttheangleformedbytherisingandsetting
positionsofastar.
Egyptianpyramids(withtheexceptionotthemostancient,theSaqqarasteppyramid)
weresquareinground-planandfullypyramidalinshape.Thespecificationforthegradient,
knownasthe'batter*,ofsuchpyramidswasdenotedbythehieroglyphicwordikd^meaning
ratio.Itwasexpressedinterm^ofthenumberofpa'ms. inhalfdielengthofaside,per
cubitotvertical height (I cubit="palms).The Rhiiul papyrusisparticularlyconcerned
withthegeometryofpyramids,measurementsbemgreckonedinroyalcubits(seechapter
4).Anexampleoffindingthebatteris:'Apyramidwhoseverticalheightis93-, jcubitsj.
Letmeknowitsbatter, 140 [cubits] beingthelengthofitsside.*Halfthelengthofa
sideis7X70palms;thebatteristherefore:
70 210
since1palm=4ringer's-breadths,thisisexpressedas5palms,1tinger's-brcadth.
ThedimensionsandorientationoftheGreatPyramidwereverycarefullyfixed.Like
a numberofdynastic Egyptian buildings, its sidesfacethefourpointsofthecompass.
nMaerarsouwrleym—enmtosstalryebientwwheoelne52n.i3michmersanodf5r2oy.a5lcmc,ubiwtist,hoaftnenavreoruangdeeiolfo5f2t,.3w6hcimc.hTvhaerymoenalny
ofthefourbaselengths,whichareverynearlyequal,is440cubits.Theoriginalheight
fromplatformtoapexwas280cubits,givingabatter,asdefinedabove,of5^palms.
Thelengthofthedescendingpassage is75cubits.Thefloorlengthofthegrandgallery
is 88 cubits, one-fifth ofthe base length. The king'schamberis 20cubits long and 10
cubitsbroad;itsheightis 11 cubits,againafactorofthebaselength,at52.5cmtothe
cubit.
ContrastingsomewhatwiththisprecisionistheclumsinessoftheEgyptianmethodof
expressingfractions.Apartfrom},only'simple'fractionswereused,thatis,thosehaving
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