Birth, growth and computation of pi to ten trillion digits

59 Pages·2013·0.81 MB·English

Agarwaletal.AdvancesinDifferenceEquations2013,2013:100
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RESEARCH OpenAccess
Birth, growth and computation of pi to ten
trillion digits
RaviPAgarwal1*,HansAgarwal2andSyamalKSen3
*Correspondence:
[email protected] Abstract
1DepartmentofMathematics,Texas
Theuniversalrealconstantpi,theratioofthecircumferenceofanycircleandits
A&MUniversity-Kingsville,Kingsville,
TX,78363,USA diameter,hasnoexactnumericalrepresentationinafinitenumberofdigitsinany
Fulllistofauthorinformationis number/radixsystem.Ithasconjureduptremendousinterestinmathematiciansand
availableattheendofthearticle
non-mathematiciansalike,whospentcountlesshoursovermillenniatoexploreits
beautyandvariedapplicationsinscienceandengineering.Thearticleattemptsto
recordthepiexplorationovercenturiesincludingitssuccessivecomputationtoever
increasingnumberofdigitsanditsremarkableusages,thelistofwhichisnotyet
closed.
Keywords: circle;error-free;historyofpi;Matlab;randomsequence;stabilityofa
computer;trilliondigits
Allcircleshavethesameshape,andtraditionallyrepresenttheinfinite,immeasurableand
evenspiritualworld.Somecirclesmaybelargeandsomesmall,buttheir‘circleness’,their
perfectroundness,isimmediatelyevident.Mathematicianssaythatallcirclesaresimilar.
Beforedismissingthisasanutterlytrivialobservation,wenotebywayofcontrastthatnot
alltriangleshavethesameshape,norallrectangles,norallpeople.Wecaneasilyimag-
inetallnarrowrectanglesortallnarrowpeople,butatallnarrowcircleisnotacircleat
all.Behindthisunexcitingobservation,however,liesaprofoundfactofmathematics:that
theratioofcircumferencetodiameteristhesameforonecircleasforanother.Whether
the circle is gigantic, with large circumference and large diameter, or minute, with tiny
circumferenceandtinydiameter,therelativesizeofcircumferencetodiameterwillbeex-
actlythesame.Infact,theratioofthecircumferencetothediameterofacircleproduces,
the most famous/studied/unlimited praised/intriguing/ubiquitous/external/mysterious
mathematicalnumberknowntothehumanrace.Itiswrittenaspiorasπ [–],sym-
bolically,anddefinedas
distancearoundacircle C
pi= = =π.
distanceacrossandthroughthecenterofthecircle D
Sincetheexactdateofbirthof π isunknown, onecouldimagine that π existedbefore
theuniversecameintobeingandwillexistaftertheuniverseisgone.Itsappearancein
thedisksoftheMoonandtheSun,makesitasoneofthemostancientnumbersknown
tohumanity.Itkeepsonpoppingupinsideaswellasoutsidethescientificcommunity,
forexample,inmanyformulasingeometryandtrigonometry,physics,complexanalysis,
cosmology,numbertheory,generalrelativity,navigation,geneticengineering,statistics,
©2013Agarwaletal.;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons
AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproduction
inanymedium,providedtheoriginalworkisproperlycited.
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fractals,thermodynamics,mechanics,andelectromagnetism.Pihidesintherainbow,and
sitsinthepupiloftheeye,andwhenaraindropfallsintowaterπemergesinthespreading
rings.Picanbefoundinwavesandripplesandspectraofallkindsand,therefore,πoccurs
incolorsandmusic.ThedoublehelixofDNArevolvesaroundπ.Pihaslatelyturnedupin
super-strings,thehypotheticalloopsofenergyvibratinginsidesubatomicparticles.Pihas
beenusedasasymbolformathematicalsocietiesandmathematicsingeneral,andbuilt
intocalculatorsandprogramminglanguages.Piisrepresentedinthemosaicoutsidethe
mathematicsbuildingattheTechnischeUniversitätBerlin.Piisalsoengravedonamosaic
atDelftUniversity.Evenamoviehasbeennamedafterit.PiisthesecretcodeinAlfred
Hitchcock’s‘TornCurtain’andin‘TheNet’starringSandraBullock.Pidayiscelebrated
onMarch(whichwaschosenbecauseitresembles.).Theofficialcelebrationbe-
ginsat:p.m.,tomakeanappropriate.whencombinedwiththedate.In,
theUnitedStatesHouseofRepresentativessupportedthedesignationofPiDay.Albert
EinsteinwasbornonPiDay(March).
Throughoutthehistoryofπ,whichaccordingtoBeckmann()‘isaquaintlittlemir-
rorofthehistoryofman’,andJamesGlaisher(-)‘hasengagedtheattentionof
many mathematicians and calculators from the time of Archimedes to the present day,
andhasbeencomputedfromsomanydifferentformula,thatacompleteaccountofits
calculationwouldalmostamounttoahistoryofmathematics’,oneoftheenduringchal-
lenges for mathematicians has been to understand the nature of the number π (ratio-
nal/irrational/transcendental),andtofinditsexact/approximatevalue.Thequest,infact,
startedduringthepre-historiceraandcontinuestothepresentdayofsupercomputers.
Theconstantsearchbymanyincludingthegreatestmathematicalthinkersthattheworld
produced, continues for new formulas/bounds based on geometry/algebra/analysis, re-
lationship among them, relationship with other numbers such as π =cos–(φ/), π (cid:2)
√
/ φ, where φ is the Golden section (ratio), and eiπ +=, which is due to Euler and
contains  of the most important mathematical constants, and their merit in terms of
computationofdigitsofπ.Rightfromthebeginninguntilmoderntimes,attemptswere
madetoexactlyfixthevalueofπ,butalwaysfailed,althoughhundredsconstructedcir-
clesquaresandclaimedthesuccess.Theseamateurmathematicianshavebeencalledthe
sufferersofmorbuscyclometricus,thecircle-squaringdisease.Storiesofthesecontribu-
torsareamusingandattimesalmostunbelievable.Manycameclose,somewenttotens,
hundreds,thousands,millions,billions,andnowuptotentrillion()decimalplaces,
butthereisnoexactsolution.TheAmericanphilosopherandpsychologistWilliamJames
(-)wrotein‘thethousandthdecimalofPisleepstherethoughnoonemay
evertrytocomputeit’.Thankstothetwentiethandtwenty-firstcentury,mathematicians
and computer scientists, it sleeps no more. In , Hermann Schubert (-), a
Hamburgmathematicsprofessor,said‘thereisnopracticalorscientificvalueinknowing
morethanthedecimalplacesusedintheforegoing,alreadysomewhatartificial,appli-
cation’,andaccordingtoArndtandHaenel(),justdecimalplaceswouldbeenough
tocomputethecircumferenceofacirclesurroundingtheknownuniversetowithinthe
radiusofahydrogenatom.Further,anexpansionof π toonlydecimalplaceswould
besufficientlyprecisetoinscribeacirclearoundthevisibleuniversethatdoesnotdeviate
fromperfectcircularitybymorethanthedistanceacrossasingleproton.Thequestion
hasbeenrepeatedlyaskedwhysomanydigits?Perhapstheprimarymotivationforthese
computations is the human desire to break records; the extensive calculations involved
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havebeenusedtotest supercomputersandhigh-precisionmultiplicationalgorithms(a
stresstestforacomputer,akindof‘digitalcardiogram’),thestatisticaldistributionofthe
digits,whichisexpectedtobeuniform,thatis,thefrequencywithwhichthedigits(to
)appearintheresultwilltendtothesamelimit(/)asthenumberofdecimalplaces
increases beyond all bounds, and in recent years these digits are being used in applied
problemsasarandomsequence.Itappearsexpertsinthefieldofπ arelookingforsur-
prisesinthedigitsofπ.Infact,theChudnovskybrothersoncesaid:‘Wearelookingforthe
appearanceofsomerulesthatwilldistinguishthedigitsofπ fromothernumbers.Ifyou
seeaRussiansentencethatextendsforawholepage,withhardlyacomma,itisdefinitely
Tolstoy.Ifsomeonegaveyouamilliondigitsfromsomewhereinπ,couldyoutellitwas
fromπ’?Someinterestingobservationsare:Thefirstdigitsofπ addupto(which
manyscholarssayis‘themarkoftheBeast’);Sincetherearedegreesinacircle,some
mathematiciansweredelightedtodiscoverthatthenumberisatthethdigitposi-
tionofπ.AmysteriouscropcircleinBritainshowsacodedimagerepresentingthe
firstdigitsofπ.TheWebsite‘ThePi-SearchPage’findsaperson’sbirthdayandother
well-knownnumbersinthedigitsofπ.Severalpeoplehaveendeavoredtomemorizethe
valueofπ withincreasingprecision,leadingtorecordsofover,digits.
Webelievethatthestudyanddiscoveriesofπ willneverend;therewillbebooks,re-
searcharticles,newrecord-settingcalculationsofthedigits,clubsandcomputerprograms
dedicated to π. In what follows, we shall discuss the growth and the computation of π
chronologically.Forourreadyreference,wealsogivesomedigitsofπ,
π =.
.
AboutBC.Themeaningofthewordsulvistomeasure,andgeometryinancient
Indiacametobeknownbythenamesulbaorsulva.Sulbasutrasmeans‘ruleofchords’,
whichisanothernameforgeometry.TheSulbasutrasarepartofthelargercorpusoftexts
calledtheShrautasutras,consideredtobeappendicestotheVedas,whichgiverulesfor
constructingaltars.Iftheritualsacrificewastobesuccessful,thenthealtarhadtoconform
to very precise measurements, so mathematical accuracy was seen to be of the utmost
importance.Thesulbascontainalargenumberofgeometricconstructionsforsquares,
rectangles,parallelogramsandtrapezia.Sulbasalsocontainremarkableapproximations
√
  
(cid:2)+ + – ,
 · ··
√
whichgives =....,and
(cid:2) (cid:3)
√  
π (cid:2)(– )= √ ,
+ 
whichgivesπ =.....
AboutBC.AryabhattawasborninBCinPatliputrainMagadha,modern
PatnainBihar(India).Hewasteachingastronomyandmathematicswhenhewasyears
ofageinBC.Hisastronomicalknowledgewassoadvancedthathecouldclaimthat
theEarthrotatedonitsownaxis,theEarthmovesroundtheSunandtheMoonrotates
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roundtheEarth;incrediblyhebelievedthattheorbitsoftheplanetsareellipses.Hetalks
aboutthepositionoftheplanetsinrelationtoitsmovementaroundtheSun.Herefersto
thelightoftheplanetsandtheMoonasreflectionfromtheSun.Heexplainstheeclipse
oftheMoonandtheSun,dayandnight,thelengthoftheyearexactlyasdays.He
calculatedthecircumferenceoftheEarthas,miles,whichisclosetomodernday
calculationof,miles.InhisAryabhattiyam,whichconsistsoftheversesand
 introductory verses, and is divided into four padas or chapters (written in the very
terse styletypicalofsutraliterature,inwhicheachlineisanaidtomemoryforacom-
plexsystem),Aryabhattaincludedversesgivingmathematicalrulesganitaonpure
mathematics.Hedescribedvariousoriginalwaystoperformdifferentmathematicalop-
erations,includingsquareandcuberootsandsolvingquadraticequations.Heprovided
elegant results for the summation of series of squares and cubes. He made use of dec-
imals, the zero (sunya) and the place value system. To find an approximate value of π,
Aryabhattagivesthefollowingprescription:Addto,multiplybyandaddto,.
Thisis‘approximately’thecircumferenceofacirclewhosediameteris,.Thismeans
π =,/,=..ItisimportanttonotethatAryabhattausedthewordasanna
(approaching),tomeanthatnotonlyisthisanapproximationofπ,butthatthevalueis
incommensurableorirrational,i.e.,itcannotbeexpressedasaratiooftwointegers.
AboutBC.GreatpyramidatGizehwasbuiltaroundBCinEgypt.Itisone
ofthemostmassivebuildingsevererected.Ithasatleasttwicethevolumeandthirtytimes
themass(theresistanceanobjectofferstoachangeinitsspeedordirectionofmotion)
oftheEmpireSateBuildinginNewYork,andbuiltfromindividualstonesweighingupto
tonseach.FromthedimensionsoftheGreatPyramid,itispossibletoderivethevalue
ofπ,namely,π =halftheperimeterofthebaseofthepyramid,dividedbyitsheight=+
/(cid:2).....
AboutBC.InatabletfoundininSusa(Iraq),Babyloniansusedthevalue
  
= + ,
π  ()
whichyieldsπ =/=..Theywerealsosatisfiedwithπ =.
About  BC. Ahmes (around - BC) (more accurately Ahmose) was an
Egyptianscribe.AsurvivingworkofAhmesispartoftheRhindMathematicalPapyrus,
 BC (named after the Scottish Egyptologist Alexander Henry Rhind who went to
Thebes for health reasons, became interested in excavating and purchased the papyrus
inEgyptin)locatedintheBritishMuseumsince.Whennew,thispapyruswas
aboutfeetlongandincheshigh.Ahmesstatesthathecopiedthepapyrusfromanow-
lostMiddleKingdomoriginal,datingaroundBC.Thiscuriousdocumententitled
directionsforknowingalldarkthings,decipheredbyEisenlohrin,isacollectionof
problemsingeometryandarithmetic,algebra,weightsandmeasures,businessandrecre-
ationaldiversions.Theproblemsarepresentedwithsolutions,butoftenwithnohintas
tohowthesolutionwasobtained.Inproblemno.,Ahmesstatesthatacircularfieldwith
adiameterofunitsinareaisthesameasasquarewithsidesofunits,i.e.,π(/)=,
andhencetheEgyptianvalueofπ is
(cid:2) (cid:3)
 
π =× =....,

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whichisonlyveryslightlyworsethantheBabyloniansvalue,andincontrasttothelatter,an
overestimation.Wehavenoideahowthisverysatisfactoryresultwasobtained(probably
empirically),althoughvariousjustificationsareavailable.Mayavalueofπ wasasgoodas
thatoftheEgyptians.
About BC. The earliest Chinese mathematicians, from the time of Chou-Kong
usedtheapproximationπ =.Someofthosewhousedthisapproximationweremathe-
maticiansofconsiderableattainmentsinotherrespects.AccordingtotheChinesemythol-
ogy,isusedbecauseitisthenumberoftheHeavensandthecircle.
About  BC. In the Old Testament (I Kings vii., and  Chronicles iv.), we find
thefollowingverse:‘Also,hemadeamoltenseaoftencubitsfrombrimtobrim,roundin
compass,andfivecubitstheheightthereof;andalineofthirtycubitsdidcompassitround
about’.Hencethebiblicalvalueofπis/=.TheJewishTalmud,whichisessentiallya
commentaryontheOldTestament,waspublishedaboutAD.ThisshowsthattheJews
did not pay much attention to geometry. However, debates have raged on for centuries
aboutthisverse.Accordingtosome,itwasjustasimpleapproximation,whileotherssay
that ‘...the diameter perhaps was measured from outside, while the circumference was
measuredfrominside’.
AboutBC.ShatapathaBrahmana(Priestmanualofpaths)isoneoftheprose
textsdescribingtheVedicritual.Itsurvivesintworecensions,MadhyandinaandKanva,
with the former having the eponymous  brahmanas in  books, and the latter 
brahmanasinbooks.Inthesebooks,π isapproximatedby/=.....
AboutBC.AnaxagorasofClazomanae(-BC)cametoAthensfromnear
Smyrna,wherehetaughttheresultsoftheIonianphilosophy.Heneglectedhispossessions
inordertodevotehimselftoscience,andinreplytothequestion,whatwastheobjectof
being born, he remarked: ‘The investigation of the Sun, Moon and heaven’. He was the
firsttoexplainthattheMoonshinesduetoreflectedlightfromtheSun,whichexplains
theMoon’sphases.HealsosaidthattheMoonhadmountainsandhebelievedthatitwas
inhabited.Anaxagorasgavesomescientificaccountsofeclipses,meteors,rainbows,and
theSun,whichheassertedwaslargerthanthePeloponnesus:thisopinion,andvarious
otherphysicalphenomena,whichhetriedtoexplainwhichweresupposedtohavebeen
directactionoftheGods,ledhimtoaprosecutionforimpiety.Whileinprisonhewrotea
treatiseonthequadratureofthecircle.(Thegeneralproblemofsquaringafigurecameto
beknownasthequadratureproblem.)Sincethattime,hundredsofmathematicianstried
tofindawaytodrawasquarewithequalareatoagivencircle;somemaintainedthatthey
havefoundmethodstosolvetheproblem,whileothersarguedthatitisimpossible.We
willseethattheproblemwasfinallylaidtorestinthenineteenthcentury.
AboutBC.HippocratesofChioswasbornaboutBC,andbeganlifeasamer-
chant.AboutBChecametoAthensfromChiosandopenedaschoolofgeometry,
andbeganteaching,thusbecameoneofthefewindividualsevertoentertheteachingpro-
fessionforitsfinancialrewards.Heestablishedtheformulaπrfortheareaofacirclein
termsofitsradius.Itmeansthatacertainnumberπ exists,andisthesameforallcircles,
althoughhismethoddoes not give theactualnumericalvalue of π. In trying to square
thecircle(unsuccessfully),Hippocratesdiscoveredthattwomoon-shapedfigures(lunes,
boundedbypairofcirculararcs)couldbedrawnwhoseareasweretogetherequaltothat
ofaright-angledtriangle.Hippocratesgavethefirstexampleofconstructingarectilinear
areaequaltoanareaboundedbyoneormorecurves.
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About  BC. Antiphon of Rhamnos (around - BC) was a sophist who at-
tempted to find the area of a circle by considering it as the limit of an inscribed regu-
larpolygonwithaninfinitenumberofsides. Thus,heprovidedpreliminaryconceptof
infinitesimalcalculus.
About  BC. Bryson of Heraclea was born around  BC. He was a student of
Socrates.Brysonconsideredthecirclesquaringproblembycomparingthecircletopoly-
gonsinscribedwithinit.Hewronglyassumedthattheareaofacirclewasthearithmetical
meanbetweencircumscribedandinscribedpolygons.
About  BC. Hippias of Elis was born about  BC. He was a Greek Sophist, a
youngercontemporaryofSocrates.Heisdescribedasanexpertarithmetician,butheis
bestknowntousthroughhisinventionofacurvecalledthequadratrix(x=ycot(πy/)),
bymeansofwhichananglecanbetrisected,orindeeddividedinanygivenratio.Itisnot
knownwhetherHippiasrealizedthatbymeansofhiscurvethecirclecouldbesquared;
perhaps he realized but could not prove it. He lectured widely on mathematics and as
wellonpoetry,grammar,history,politics,archeologyandastronomy.Hippiaswasalsoa
prolificwriter,producingelegies,tragediesandtechnicaltreatisesinprose.Hisworkon
Homerwasconsideredexcellent.
BC.Aristophanes(-BC)inhisplayTheBirdsmakesfunofcirclesquarers.
AroundBC.PlatoofAthens(around-BC)wasoneofthegreatestGreek
philosophers, mathematicians, mechanician, a pupil of Socrates for eight years, and
teacher of Aristotle. He is famous for ‘Plato’s Academy’. ‘Let no man ignorant of math-
ematicsenterhere’issupposedtohavebeeninscribedoverthedoorsoftheAcademy.He
√ √
issupposedlyobtainedforhisdayafairlyaccuratevalueforπ= + =.....
AboutBC.EudoxusofCnidus(around-BC)wasthemostcelebratedmath-
ematician. He developed the theory of proportion, partly to place the doctrine of in-
commensurables(irrationals)uponathoroughlysoundbasis.Specially,heshowedthat
theareaofacircleisproportionaltoitsdiametersquared.Eudoxusestablishedfullythe
methodofexhaustionsofAntiphonbyconsideringboththeinscribedandcircumscribed
polygons.Healsoconsideredcertaincurvesotherthanthecircle.Heexplainedtheap-
parent motions of the planets as seen from the earth. Eudoxus also wrote a treatise on
practicalastronomy,inwhichhesupposedanumberofmovingspherestowhichtheSun,
Moonandstarswereattached,andwhichbytheirrotationproducedtheeffectsobserved.
Inall,herequiredspheres.
AboutBC.Dinostratus(around-BC)wasaGreekmathematician.Heused
Hippiasquadratrixtosquarethecircle.Forthis,heprovedDinostratus’theorem.Hippias
quadratrixlaterbecameknownastheDinostratusquadratrixalso.However,hisdemon-
strationwasnotacceptedbytheGreeksasitviolatedthefoundationalprinciplesoftheir
mathematics,namely,usingonlyrulerandcompass.
AboutBC.ArchimedesofSyracuse(-BC)rankswithNewtonandGauss
as one of the three greatest mathematicians who ever lived, and he is certainly the
greatest mathematician of antiquity. Galileo called him ‘divine Archimedes, superhu-
man Archimedes’; Sir William Rowan Hamilton (-) remarked ‘who would not
ratherhavethefameofArchimedesthanthatofhisconquerorMarcellus’?;AlfredNorth
Whitehead(-)commented‘noRomaneverdiedincontemplationoverageomet-
ricaldiagram’;GodfreyHaroldHardy(-)said‘Archimedeswillberemembered
whenAeschylusisforgotten,becauselanguagesdieandmathematicalideasdonot’;and
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Figure1 Circlewithdiameter1.
Voltaireremarked‘therewasmoreimaginationintheheadofArchimedesthaninthatof
Homer’.Hismathematicalworkissomoderninspiritandtechniquethatitisbarelydis-
tinguishablefromthatofaseventeenth-centurymathematician.Amonghismathemati-
calachievements,Archimedesdevelopedageneralmethodofexhaustionforfindingar-
easboundedbyparabolasandspirals,andvolumesofcylinders,parabolas,segmentsof
spheres,andspeciallytoapproximateπ,whichhecalledastheparametertodiameter.His
approachtoapproximateπisbasedonthefollowingfact:thecircumferenceofacirclelies
betweentheperimetersoftheinscribedandcircumscribedregularpolygons(equilateral
andequiangular)ofnsides,andasnincreases,thedeviationofthecircumferencefrom
thetwoperimetersbecomessmaller.Becauseofthisfact,manymathematiciansclaimthat
itismorecorrecttosaythatacirclehasaninfinitenumberofcornersthantoviewacircle
asbeingcornerless.Ifa andb denotetheperimetersoftheinscribedandcircumscribed
n n
regularpolygonsofnsides,andCthecircumferenceofthecircle,thenitisclearthat{a }
n
isanincreasingsequenceboundedabovebyC,and{b }isadecreasingsequencebounded
n
belowbyC.BothofthesesequencesconvergetothesamelimitC.Tosimplifymatters,
supposewechooseacirclewiththediameter,thenfromFigureitimmediatelyfollows
that
π π
a =nsin and b =ntan . ()
n n
n n
Itisclearthatlimn→∞an=π =limn→∞bn.Further,bn istheharmonicmeanofan and
b ,anda isthegeometricmeanofa andb ,i.e.,
n n n n
(cid:4)
a b
n n
b = and a = a b . ()
n n n n
a +b
n n
√
From () for the hexagon, i.e., n= it follows that a =, b = . Then Archimedes
 
successivelytookpolygonsofsides,,and,usedtherecursiverelations(),and
theinequality
√
 ,
< < ,
 
whichheprobablyfoundbywhatisnowcalledHeron’smethod,toobtainthebounds
 
....= <π < =..... ()
 
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It is interesting to note that during Archimedes time algebraic and trigonometric no-
tations, and our present decimal system were not available, and hence he had to de-
riverecurrencerelations()geometrically,andcertainlyforhimthecomputationofa

and b must have been a formidable task. The approximation / is often called the

Archimedeanvalueofπ,anditisgoodformostpurposes.Ifwetaketheaverageofthe
bounds given in (), we obtain π =..... The above method of computing π by
usingregularinscribedandcircumscribedpolygonsisknownastheclassicalmethodof
computingπ.Itfollowsthataninscribedregularpolygonofn sidestakesupmorethan
–/n–oftheareaofacircle.HeronofAlexandria(aboutAD)inhisMetrica,which
hadbeenlostforcenturiesuntilafragmentwasdiscoveredin,followedbyacomplete
copyin,referstoanArchimedeswork,wherehegivesthebounds
, ,
....= <π < =.....
, ,
Clearly,intheaboverightinequality,thereisamistakeasitisworsethantheupperbound
/ found by Archimedes earlier. Heron adds ‘Since these numbers are inconvenient
for measurements, they are reduced to the ratio of the smaller numbers, namely, /’.
Archimedes’polygonalmethodremainedunsurpassedforcenturies.Archimedesalso
showedthatacurvediscoveredbyCononofSamos(around-BC)could,likeHip-
pias’quadratrix,beusedtosquarethecircle.ThecurveistodaycalledtheArchimedean
Spiral.
About  BC. Daivajna Varahamihira (working  BC) was an astronomer, math-
ematician and astrologer. His picture may be found in the Indian Parliament along
with Aryabhata. He was one of the nine jewels (Navaratnas) of the court of legendary
king Vikramaditya I (- BC). In  BC, Varahamihira wrote Pancha-Siddhanta
(The Five Astronomical Canons), in which he codified the five existing Siddhantas,
namely,PaulisaSiddhanta,RomakaSiddhanta,VasishthaSiddhanta,SuryaSiddhantaand
PaitamahaSiddhanta. He also made some important mathematical discoveries such as
givingcertaintrigonometricformulae;developingnewinterpolationmethodstoproduce
sinetables;constructingatableforthebinomialcoefficients;andexaminingthepandiag-
√
onalmagicsquareoforderfour.Inhiswork,heapproximatedπ as .
BC.MarcusVitruviusPollio(about-BC),aRomanwriter,architectandengineer,
inhismulti-volumeworkDeArchitectura(OnArchitecture)usedthevalueπ =/=
.,whichisthesameasBabylonianshadused,yearsearlier.Hewasthefirstto
describedirectmeasurementofdistancesbytherevolutionofawheel.
AboutBC.LiuXin(LiuHsin)(aboutBC-AD)wasanastronomer,historianand
editorduringtheXinDynasty(-AD).Liucreatedanewastronomicalsystem,called
TripleConcordance.Hewasthefirsttogiveamoreaccuratecalculationofπ as.,
theexactmethodheusedtoreachthisfigureisunknown.Thiswasfirstmentionedinthe
Suishu(-).Healsofoundtheapproximations.,.and..
AroundAD.LiuXin(BC-AD)wasaChineseastronomer,historianandeditor
duringtheXinDynasty(-AD).HewasthesonofConfucianscholarLiuXiang(-
BC).Liucreatedacatalogof,stars,whereheusedthescaleofmagnitudes.Hewas
thefirstinChinatogiveamoreaccuratecalculationofπ as..Themethodheused
toreachthisfigureisunknown.
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 AD. Brahmagupta (born  BC) wrote two treatises on mathematics and astron-
omy:theBrahmasphutasiddhanta(TheCorrectlyEstablishedDoctrineofBrahma)but
oftentranslatedas(TheOpeningoftheUniverse),andtheKhandakhadyaka(EdibleBite)
which mostly expands the work of Aryabhata. As a mathematician he is considered as
thefatherofarithmetic,algebra,andnumericalanalysis.Mostimportantly,inBrahmas-
phutasiddhantahetreatedzeroasanumberinitsownright,statedrulesforarithmetic
onnegativenumbersandzero,andattemptedtodefinedivisionbyzero,particularlyhe
wronglybelievedthat/wasequalto.Heusedageometricconstructionforsquaring
√
thecircle,whichamountstoπ = .
. Zhang Heng (- AD) was an astronomer, mathematician, inventor, geogra-
pher,cartographer,artist,poet,statesmanandliteraryscholar.Heproposedatheoryof
theuniversethatcomparedittoanegg.‘Theskyislikeahen’seggandisasroundasa
crossbowpellet.TheEarthisliketheyolkoftheegg,lyingaloneatthecenter.Theskyis
largeandtheEarthissmall’.Accordingtohimtheuniverseoriginatedfromchaos.Hesaid
thattheSun,Moonandplanetswereontheinsideofthesphereandmovedatdifferent
rates.HedemonstratedthattheMoondidnothaveindependentlight,butthatitmerely
reflectedthelightfromthesun.HeismostfamousintheWestforhisrotatingcelestial
globe,andinventinginthefirstseismographformeasuringearthquakes.Heproposed
√
(about.)forπ.Healsocomparedthecelestialcircletothewidth(i.e.,diameter)
oftheearthintheproportionofto,whichgivesπ as..
. Claudius Ptolemaeus (around - AD) known in English as Ptolemy, was a
mathematician,geographer,astrologer,poetofasingleepigramintheGreekAnthology,
andmostimportantlyastronomer.Hemadeamapoftheancientworldinwhichheem-
ployedacoordinatesystemverysimilartothelatitudeandlongitudeoftoday.Oneofhis
mostimportantachievementswashisgeometriccalculationsofsemichords.Ptolemyin
hisfamousSyntaxismathematica(morepopularlyknownbyitsArabiantitleoftheAl-
magest),thegreatestancientGreekworkonastronomy,obtained,usingchordsofacircle
andaninscribed-gon,anapproximatevalueofπ insexagesimalnotation,as(cid:6)(cid:6)(cid:6),
whichisthesameas/=.....EutociusofAscalon(about-)refers
toabookQuickdeliverybyApolloniusofPerga(around-BC),whoearnedthe
title‘TheGreatGeometer’,inwhichApolloniusobtainedanapproximationforπ,which
wasbetterthanknowntoArchimedes,perhapsthesameas/.
.WangFan(-)wasamathematicianandastronomer.Hecalculatedthedis-
tancefromtheSuntotheEarth,buthisgeometricmodelwasnotcorrect.Hehasbeen
creditedwiththerationalapproximation/forπ,yieldingπ =..
.LiuHui(around-)wrotetwoworks.Thefirstonewasanextremelyim-
portantcommentaryontheJiuzhangsuanshu,morecommonlycalledNineChapterson
theMathematicalArt,whichcameintobeingintheEasternHanDynasty,andbelievedto
havebeenoriginallywrittenaroundBC.(Itshouldbenotedthatverylittleisknown
aboutthemathematicsofancientChina.InBC,theemperorShiHuangoftheChin
dynastyhadallofthemanuscriptofthekingdomburned.)Theotherwasamuchshorter
workcalledHaidaosuanjing orSeaIslandMathematicalManual.InJiuzhangsuanshu,
LiuHuiusedavariationoftheArchimedeaninscribedregularpolygonwithsidesto
approximateπ as.andsuggested/=.asapracticalapproximation.
About.PappusofAlexandria(around-)wasborninAlexandria,Egypt,and
eitherhewasaGreekoraHellenizedEgyptian.Thewrittenrecordssuggestthat,Pappus
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livedinAlexandriaduringthereignofDiocletian(-).HismajorworkisSynagoge
ortheMathematicalCollection,whichisacompendiumofmathematicsofwhicheight
volumeshavesurvived.Pappus’BookIVcontainsvarioustheoremsoncircles,studyof
variouscurves,andanaccountofthethreeclassicalproblemsofantiquity(thesquaring
ofthecircle,theduplicationofacube,andthetrisectionofanangle).Forsquaringthe
circle,heusedDinostratusquadratrixandhisproofisareductioadabsurdum.Pappusis
rememberedforPappus’scentroidtheorem,Pappus’schain,Pappus’sharmonictheorem,
Pappus’shexagontheorem,Pappus’strisectionmethod,andforthefocusanddirectrixof
anellipse.
. He Chengtian (-) gave the approximate value of π as ,/,=
.....
. Tsu Ch’ung-chih (Zu Chongzhi) (-) created various formulas that have
beenusedthroughouthistory.WithhissonheusedavariationofArchimedesmethod
tofind.<π <..Healsoobtainedaremarkablerationalapproximation
/,whichyieldsπ correcttosixdecimaldigits.InChinesethisfractionisknownas
Milü.Tocomputethisaccuracyforπ,hemusthavetakenaninscribedregular×-
gonandperformedlengthycalculations.Notethatπ =/canbeobtainedfromthe
valuesofPtolemyandArchimedes:
 –
= .
 –
Hedeclaredthat/isaninaccuratevaluewhereas/istheaccuratevalueofπ.
Wealsonotethatπ=/canbeobtainedfromthevaluesofLiuHuiandArchimedes.
Infact,byusingthemethodofaveraging,wehave
+(×) 
= .
+(×) 
.BhaskaraIIorBhaskaracharya(working)wroteSiddhantaSiromani(crown
oftreatises),whichconsistsoffourparts,namely,LeelavatiBijaganitam,Grahaganitam
andGoladhyaya.The firsttwoexclusivelydealwithmathematicsandthelasttwowith
astronomy.HispopulartextLeelavatiwaswritteninADinthenameofhisdaugh-
ter.Hiscontributionstomathematicsinclude:aproofofthePythagoreantheorem,solu-
tionsofquadratic,cubic,andquarticindeterminateequations,solutionsofindeterminate
quadratic equations, integer solutions of linear and quadratic indeterminate equations,
a cyclic Chakravala method for solving indeterminate equations, solutions of the Pell’s
equationandsolutionsofDiophantineequationsofthesecondorder.Hesolvedquadratic
equationswithmorethanoneunknown,andfoundnegativeandirrationalsolutions,pro-
videdpreliminaryconceptofinfinitesimalcalculus,alongwithnotablecontributionsto-
wardintegralcalculus,conceiveddifferentialcalculus,afterdiscoveringthederivativeand
differentialcoefficient,statedRolle’stheorem,calculatedthederivativesoftrigonometric
functionsandformulaeanddevelopedsphericaltrigonometry.Heconceivedthemodern
mathematical convention that when a finite number is divided by zero, the result is in-
finity. He speculatedthe nature of the number / by stating that it is ‘like the Infinite,
Invariable Godwhosuffersnochangewhen oldworldsaredestroyedornew onescre-
ated,wheninnumerablespeciesofcreaturesarebornorasmanyperish’.Hegaveseveral