The Great Year in Greek Persian and Hindu Astronomy--B L Van Der Waerden

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The GreatYear nGreek, ersianandHinduAstronomyB. L. VANDERWaERDEN

1. ntroductionPlato definesTimaios 9C-D) thePerfect ear s a period t theend ofwhich hefixed tars ndplanets eturno thepointfromwhich hey tartedtheirevolutions.ristotle *calls he ameperiodGreatestear,ndsays hatin thewinterf thisyear floodkataklysmos)akesplace, n the summerconflagrationekpyrosis).n laterGreek ndLatintexts his osmic eriod scalledthe GreatYear.In a seriesof papers2'3'4'5'6 and in a joint paper withE.S.KENNEDY7 Ihave investigated he history f the doctrineof the "Great Year", and the

application f thisdoctrinen Persianand Hindu Astronomy.My papers on this subject have been vigorously criticized byD. PlNGREE8'9'10.n hispapers,manyofmyargumentsre distortednd manyare left ut altogether.Moreover, ome ofmy papersare not easilyaccessible,and recently have found new connectionsbetweenHellenistic nd Persianastronomy. or thesereasons, t seemsdesirable o present newexposition fthe whole subject.All my argumentswill be explainedanew,and PlNGREE'scriticism illbe refuted.1Censorinus:De dienatali, hapter 8.2B.L.van derWaerden: Das GrosseJahrund die EwigeWiederkehr.ermes80 1952)p. 129-155.3.L. van derWaerden: Das GrosseJahr esOrpheus. ermes 1(1953)p. 481-484.4 B.L. van derWaerden: DiophantischeleichungenndPlanetenperiodenn derindischen stronomie.ierteljahrsschriftaturf. es. Zrich 00 1955)p. 153-170.5B.L.vanderWaerden: Ausgleichspunkt,ethodeder Perserund indischePlanetenrechnung.rchive ist.Exact ei.1 (1961)p.107-121.6B.L. van derWaerden: Das heliozentrischeystemn der griechischen,er-sischen nd ndischenstronomie.eujahrsblattaturf. es. Zrich 5 1970)7E.S.Kennedy & B.L. van der Waerden: The World- ear ofthe Persians. .Amer. riental oc.83 1963)p. 315-327.8D. Pingree:AstronomyndAstrologyn ndia nd ran. sis54 1963) .229-245.9D. Pingree:ThePersian Observation"f he olarApogeen Ca. A.D. 450.J. fNearEastern tudies 4 1965)p. 334-336.10D. Pingree:Review f6. Centaurus0 1976)p.258-260.



360 . L. VANDER WAERDENIn my expository aper6, have presentedomeargumentsn favour f thehypothesis hat the astronomy f the Persians and of RYABHATAwas in-

fluenced y theheliocentricystem f Aristarchos of Samos. In thepresentpaperI shall eave this ubject side. I hope to return o it on another ccasion.

2. The "Great Year" andtheDoctrine f EternalReturnnGreek SourcesFrom a fragmentf Eudemos quoted by SlMPLiKiOS11we know that thePythagoreans believed in the eternal returnof all things "according tonumbers".fone believes hePythagoreans, saysEudemos, I too shall stand

hereagain in future, olding his ittle tick n myhand and tellingyou myths,and youwillsitherebeforeme ust as youare sitting ow.Accordingto Nemesios12 and othersources,the Stoic philosophersheldthe same opinionas thePYTHAGOREANS.here willbe a conflagrationausedby theplanets, o they aught, nd afterwardsheworld will be recreated ndeverything ill be as before: herewill be a PLATO and a SOCRATES,nd anexactrepetitionf all motions n theskyand on earth.The same doctrine f annihilation ollowed ya new creation nd an exactrepetitionfeverythingn the world can be found n Hindu sources,notably ntheMahbhrata and the Laws ofManu13, as we shallsee presently.The idea of a Great Year with Flood and Ekpyrosis seems to be ofBabylonian rigin.Berossos, a priest f theBabyloniangod BEL,whocametoGreeceabout300 B.C. and founded schoolofastrology n the slandKos, tellsus that conflagration ill takeplacewhen ll planets ometogethernCancer,and a delugewhenthey ome togethern Capricorn14.n anotherfragment15,Berossos gives hefiguref120saroi as the umof theregnalyears fmythicalkingsbefore he Flood, 1 sar being 3600 years.Hence, a certainpart of theGreatYear was,according o thechronology fBEROSSOS,120x3600 = 432000 years.

Severalfigures or he durationof the Great Year are given n laterGreektexts.Of thesefigures shallmention nlythose that re composedof factors ,3 and 5. Accordingto Censorinus (De die natali, Ch. 18) the followingGreatestYearswereproposed:Heraclitus and Linus 10800years,Orpheus 120000years,Cass ANDRUS 3 600000years.11Simplikios: ommentaryoAristotle'shysics,.732 ed.Diels).12Nemesios:Anthropology8,p.309 ed.Matthaei).13G.Bhler: TheLawof Manu.SacredBooksoftheEast25 Oxford,larendonPress 886).14Seneca: Quaest.nat. ll 29.15P.Schnabel: BerossosLeipzig 923) . 261-263.



TheGreatYear 361On the otherhand,Aetios tells us: "Accordingto Herakleitos he GreatYear contains18000years, utaccording o Diogenes the Stoic 360 suchyears

as thatofHerakleitos".Aetios II 32; see 2,p.134).I shall leave aside theGreat Year ofHeraclitus and Linus, because itsconnectionswithBabylonian,Persianor Indian cosmicperiodsare not clear.More important orus is the "Great Year ofOrpheus".There are threeperiodsformed rom he factor12 multiplied y powersof10,viz:a) the "Chaldaean Dodekaeteris"of 12 years,mentioned y CENSORINUS,also called Dodekaeteris of Zeus (i.e. of the planet Jupiter) nd ascribed toOrpheus or to Zoroaster16,b) a Persianperiodof 12000years, onsisting f12millennia,ach governedbya zodiacal sign 7,c) theGreatYear ofOrpheus of120000years3.Periodsa) and c) wereboth ascribed o Orpheus; thisprobablymeans thattheywere mentioned n the "Books of Orpheus". Periods a) and b) areconnectedwith he12 zodiacal signs.The Great Year of Berossos was composedof thefactors 20 and 3600.Justso, the Great Year of Diogenes was composed of the factors360 and18000, nd the Persianperiodb) of 12 and 1000. The Great Years ofOrpheusand Cassandrus maybe obtainedfrom he same factors:120x1000= 1200003600x1000 = 3600000

3. The IndianYuga SystemThe Laws of Manu and the Mahbhrata both contain a passage onYugas,i.e.cosmicperiods.The twopassages clearly ome from common ourcewhich,accordingto BHLER13, xistedalready in the second centuryA.D.BHLERhas translated he twopassages and printed hem n parallelcolumns(13,p. lxxxiii). he Mahbhrata passage beginsthus:In the commencementxists he Brahmanwithout eginningr end, unborn,luminous,ree romecay,mmutable,ternal,nfathomableyreasoning,ot o befullyknown.Next the authorproceeds o an explanation f thedivisions ftime, tartingwithnimeshas twinklingsf the eye) and thenpassingto days,months ndyears.A year s called"a day and night fthegods". The next argerunit s the"yearofthegods",consisting f 360 ordinary ears.Our passagedescribes heduration fa "night nd dayofBrahman" nd thefour uccessiveYugas orWorld-Agesas follows:16J.Bidez& F.Cumont: LesmageshellnissParis1938, es belles ettres)., p.120-127,nd II, p. 138-187. ee also the article odekaeterisyF.Boll in Pauly-Wissowa's ealencyclopaedie.17E.S.Kennedy: Ramificationsf theWorldYearConceptn IslamicAstrology.Ithaca 6 VIII-2IX 1962Hermann,aris), .23^5, especially hapterV.



362 . L. VANDER WAERDENTheydeclare hat he Krita ge consists f four housand ears f thegods;thetwilightrecedingtconsistsfas many undreds,nd thetwilightollowingt of the

samenumber.This means thattheKritayuaga s a periodof400+ 4000+ 400= 4800yearsof thegods.In the other)hreeges,with heirwilightsrecedingndfollowing,he housandsandhundredsre diminishedyone ineach).

Thus,theTretyugahas300+ 3000+ 300= 3600

yearsof thegods,theDvparayuga200+ 2000+ 200= 2400and theKaliyuga100+1000+100=1200

yearsof thegods, .e.1200x360 = 432000 years.

The readerwill rememberhat n the Great Year ofBerossos thesumofthe regnal yearsof the kingsbeforethe Flood, was also 432000 vears.Thisperiod was a well-defined art of the "Great Year" of BEROSSOS, nd theKaliyuga s a well-definedartofthe ndianMahyuga.One might rguethat thisagreement f numericalvaluesmighthave beenaccidental.But now look how the numbers 32000 are obtained n bothcasesThe units n thechronological ystem f BEROSSOS resaros= 3 600years,neros 360years,sossos= 60years.

This system f counting s typically abylonian:The Sumerians lreadyhadspecialsignsfor he numbers 0,360 and 3600.Now the sum432000 is obtainedas 120saroi= 432000 years.Sincethesaroscontains10neroi, ne can also write hisequationas

1200 neroi 432000years.On the otherhand,theHindu "Year of theGods" is ust theBabyloniannerosof 360years, nd theduration f theKaliyuga s calculated s1200yearsof the Gods= 432000 years.



The GreatYear 363This is a strong rgumentn favourof a Babylonianoriginof the Indiancalculation.The description f the four ges in the ndian text s very imilar o that ofHesiodos. I shallquote theversion f Manu:In theKrita geDharma s four-footedndentire,nd so is)Truth; ordoesanygainaccrue o menby unrighteousness.n theotherthreeges), yreason f unjust)gains, harma sdepriveduccessivelyfonefoot,ndthroughthe revalencef) heft,falsehoodndfraud hemeritsdiminishedyone-fourthin each). Menare)free romdisease,ccomplishll theirims, nd ivefour undredearsn theKrita age),but ntheTreta nd the ucceedingages) heirife s lessened yonequarter.Next the sum total of the four ges is calculatedas 12000yearsof thegods,i.e. 4320000 years.Thisperiod s calledoneage ofthegods. n later stronomical

texts the same period is called Caturyuga = fourfoldYuga) or Mahyuga( = GreatYuga).Manu proceeds:Butknow thatthe sumof one thousandges of thegods makes) ne day ofBrahman,nd that isnight asthe ame ength.This "day of Brahman" s called in latertextskalpa:1 kalpa=1000 Mahyugas 4320000000 years.

The Mahbhrata passage adds:Whenmperishablerahmanwakes t the ndof his ight,e modifiesimselfndcreateshe lementcalled) heGreatOne and)from hat hemindwhich sdiscrete.The idea of a repeatedcreationof the world by the SupremeGod alsounderlies Hymnof theMagi summarized yDion CHRYSOSTOMOS18.The Manu passage ends thus:To whateverourse f ction heLord at firstppointedach kind fbeings),hatalone t hasspontaneouslydoptedn each uccessivereation.Whatevereassignedoeach at the first)reation,oxiousnessrharmlessness,entlenessrferocity,irtue rsin, ruth rfalsehood,hat lungafterwards)pontaneouslyo t.This is just the same doctrineof endlessrepetition s Eudemos ascribesto thePythagoreans, and Nemesios to the Stoics.All in all, we have found wholenetwork f interrelationsetweenBaby-lonian,Greek and Hindudoctrines.

Note onPingree'sCriticismIn his review10 fmy paper6,PlNGREEwrites:Anotherfthe uthor's laws s his failureo look athis sources ritically:ny extthat an be interpretedo fithishypothesess accepted,nd all othersgnored.hus18SeeBidez-Cumont16,,p. 91and I,p. 142-143.



364 . L. VANDER WAERDENSeneca's itation fBerossuss taken orepresenttruly abylonianourcepp.21-22)though oneofour hundredsf thousands f extant uneiformextswouldustifyanderWaerden'snterpretationf t.This statementfPlNGREE s wrong n severalrespects:1) Seneca was notmy onlysource. On page 141-143 ofmy paper2. havepresented everalarguments,ll pointing owardsa Babylonianoriginof thedoctrineof the Great Year. One of these argumentswas used by PlNGREEhimselfn page 238 of hispaper8.PlNGREEwrites:The lastyuga, hen, hekaliyuga,s 1/10mahyuga,r 432,000years. his s aBabylonianumber:exagesimallyt wouldbe written,0,0,0. t is the panof timegiven o theBabylonian ingdomefore he Flood in thehistoriesfBerossos ndAbydenos.tseemsikelyhat t should avebecome significantumbern ndia t atimewhen ther abyloniannfluencesere eing elt2) In my expository aper6,written or "Neujahrsblatt", had to restrictmyselfo a fewtestimoniesupportingmyhypotheses,ut in mymorefunda-mentalpapers2'3'4'5and in the oint papers7'22 have alwaysquoted all textsavailable to me. So PlNGREE's ssertion hat selected extsfavourable o myhypotheseswith theexclusionof others s not ustified.t would be nice ifhewould think t over and withdraw is unjustaccusation.

4. Al-Brnnd Abu Ma'sharIf, s Berossos says,theDeluge tookplace when all planetscame togetherin Capricorn,then it must be possible to date the Deluge by means ofastronomical alculationsof conjunctions fplanets n thepast. According oAl-Brn, severalastrologers ried o date theDeluge bythis method. n his"Chronology", RN19writes:Era of theDeluge. The next ollowingra s the ra ofthegreat eluge,nwhicheverythingerishedtthe ime fNoah.Here, oo, heres such differencefopinions,and such confusion,hatyouhaveno chance fdecidings to the orrectnessfthematter,ndyoudo not ven eelnclinedo nvestigatehoroughlytshistoricalruth.hereasons, n the irstnstance,hedifferenceegardinghe eriod etweenheEra Adamiand theDeluge,whichwe havementionedlready;ndsecondly,hat ifference,hichwe shall have to mention,egardingheperiodbetween heDeluge and the EraAlexandi.or theJews erive romheThora, nd thefollowingooks, or his atterperiod ,792years,whilst he Christianserive romheir horafor he sameperiod2,938 ears...The astrologersave tried o correct heseyears, eginningrom he firstftheconjunctionfSaturnndJupiter,orwhich he ages mong he nhabitantsfBabel,and theChaldaens ave onstructedstronomicalables,heDelugehaving riginatedntheirountry.orpeople ay, hatNoahbuilt he rk nKufa, ndthat t was there hat"the wellpoured orthtswaters"Sraxi.42;xxiii.27);hat heark rested ponthemountainfAljd,which s notvery arfrom hoseregions. ow this onjunction

occurred29years 08days eforeheDeluge. hisdate heytudiedarefully,nd tried19Albrn:Chromology,ranslatedyC.E.Sachau (London 879), .27-29.



TheGreatYear 365by hat ocorrecthe ubsequentimes.o they ounds the ntervaletweenheDelugeandthe eginningf he eignf he irst ebukadnezarNabonassar),,604 ears,nd asthe ntervaletween ebukadnezarnd Alexander36years, result hich omes rettynear o that ne,whichs derivedromheThora f heChristians.This was the era which Ab-Ma'shar lbalkhwanted, ponwhich o base hisstatementsegardinghemeanplacesofthe tars n his Canon.Now he supposedhattheDelugehad taken lace t the onjunctionfthe tarsn the astpart fPisces,ndthefirstart fAries,nd he tried ocomputeheir lacesfor hat ime. henhefound,that hey all ofthem stood n conjunctionn the pacebetween hetwenty-seventhdegree fPisces, nd the endofthe first egree fAries. urther,e supposed hatbetween hat ime nd theepochofthe Era Alexandi,heres an interval f2,790intercalatedears months nd 26 days.Thiscomputationomesnear o that ftheChristians,eing 49yearsnd 3 monthsess han he stimatef he stronomers.ow,when ethoughthathe had well stablishedhe omputationfthis um ccordingothemethod, hich e has explained,nd whenhe had arrived t theresult,hat heduration f thoseperiods, hich stronomersall "star-cycles",as360,000 ears,hebeginningfwhichwas toprecedehe ime f heDelugeby180,000 ears, e drew heinconsiderateonclusion,hat heDelugehadoccurrednce nevery 80,000 ears,ndthatt would gain ccurnfuturet similarntervals.Thisman,who s so proud fhis ngenuity,adcomputedhese tarcyclesnly romthemotionsf he tars,s they ad beenfixedythe bservationsf hePersiansWhat does BRN meanby"the Persians"? claim thathis"Persians" arethe authorsof the "Tables of the Shah", whichwere composed in SassanidPersia underKhosro Anshrvan about 560 and revisedunderYazdigerdIII about 640 (see20'21'22).This claim s supported yseveral estimonies:

1) BRN ays n hisChronology 19,p. 121)thatthe solaryearhas365; 15,32,30

days according o "the Persians". fone leavesout of account the 365daysandmultiplies he fractionalpart by 360, one obtains the so-called excess ofrevolutionf the fixed tars n a solaryear:93; 15 degrees.

This same value is also attributed o "the Persians" by Al-Hshim (see20,p.147). On the otherhand, the Escoriai manuscript f the Mumtahan ZJascribes the value 93; 15 to the "Tables of the Shh" (see again20,p.147).Hence, fwe identifyhe"Persians" of BRN nd HSHIMwith heauthors fthe "Tables of the Shh", the statements f the threeauthors are in perfectaccordance.20E.S.Kennedy: A Surveyf slamicAstronomicalables.Trans.Amer. hilos.Soc.46,p. 129-130, o. 30.21E.S.Kennedy: The Sassanian Astronomical andbookZj-i Shh.J. Amer.Orientaloc.78,p.246-262.22J.J.Burckhardt& B. L.van der Waerden: Das astronomischeystem erPersischenafeln. Centaurus3 1968)p. 1-28.



366 . L. VANDER WAERDEN2) Brn states nhis treatise n Transits23 hatthe maximalequationsofthe sun and themoonaccording o the "Tables of theShah" agree exactlywith

the Hindu values 214' and 456' "because these values had passed from heHindus to the Persians". n the first alfofthis sentenceBRN peaksof the"Tables oftheShah" and in the econdhalf f the Persians".Obviously, RNis referringo thesame set of tables n bothpartsof his sentence.3) In the sametreatise23, RN ays 30:10) about theequationsofthe fiveplanets:The Hindus nd Persians ave a common pinion,o thezjesof theShah, ndAbuMa'shar,ndYa'qb bnTriq ontain othingn whichhey ifferxcept nly nething,hedifferencefwhich oesnot xceed ne minute.heyhaveforSaturn Jupiter Mars Venus Mercury837' 56' 1112' 213' 4In this tatement,RN peaksfirstf"the Persians" nd next,n the amecontext, ftheZj of theShah. Once more,the statementmakes senseonly f"the Persians" are regardedas the authors of that Zj.- The third authormentioned yBrn, namelyYa'qb ibn Triq, is the authorof a zj called"SindhindZj", which s based on Hindu astronomysee 20,p. 134).4) So forBRUN nd Al-HSHIM he situations clear.Next we shall showthat bn al Qifti also uses theexpression the Persians" n thesame sense. nhis commentaryo the tables of Al-Khwarizmi (see 24) Ibn al Qifti statesthat n these ablestheequationsof theplanetswerecalculated ccording o the"Method ofthe Persians".For an explanation f thismethod ee 5 or 22or 24.In thepaperofBuRCKHARDT& van der Waerden22 itwas shown that the"Method ofthe Persians"was actuallyused in the "Tables of theShah".We nowsee that n BRN's tatement bout the sourceofAbu Mashar'scycleof 360000 yearstheexpression cycleswhichhave been based upon theobservations fthe Persians"may safely e interpreteds "cyclesmentioned ntheTables of the Shah".In thenext ectionwe shallsee that Abu Ma'shar himself laimedthattheworld-year f 360000 years is due to "the Persians and some of the Baby-

lonians". As far as the Persians are concerned, is claim is in fullagreementwith the statement f Al-Brn. This agreementmpliesthat Abu Ma'sharand Al-Brn use the expression the Persians" in the same sense. Themeaning f "theBabylonians"will be discussed n a later ection.In his "Elements of Astrology"25,BRN gives a list of numbersofrevolution f the planets in 360000 years accordingto "the thousands" of23Al-Brn:On Transits,ranslatedyE.S.Kennedy. American niversityfBeirut riental eriesNo. 32, ection4: 7-11.24O.Neugebauer: The Astronomicalablesof al-Khwrizm.ist.-filos.krifterDanskeVid.-Selsk.,no.2 (1962). ee also the arlierommentaryfH.Suter in thesame krifter,theries (1914), specially. 33.25Al-Brn:The Book of nstructionn the Elementsf theArtofAstrology,transi,yR.R.Wright London, uzac1934), ection 04,p. 113-114.



TheGreatYear 367Abu Ma'shar26. He states that Abu Ma'shar has recorded these numbers"fromPersian sources". As we have seen,thesesources nclude the "Tables oftheShah". So the "Tables of theShah" werecertainlyne of the main sourcesofAbu Ma'shar's astronomical ystem. iruni's figuresre confirmedy al-Hshimand al-Tankhi (see Pingree26, p. 30).

Note onPingree'sCriticismIn hisreview10, ingree writes:Inordero endweightohisdoctrine buMa'shar, ollowinghe ead of numberof arlyAbbsid cholarsf ranian rigine.g.,bn NawbakhtndMiskawayh),oughtto ascribe tsdiscoveryo ante-Diluvianimes,ndeventuallyoGod Himself. o thisendhe inventedmythologicalistoryccordingo which hefirstfthreeHermes",who s at once he ranianAbanjhanHshank),heHebrew hanukhEnoch),nd theArabian drs received nowledgeromGod at AkhmmPanopolis) n Egypt. hisknowledgencluded he astronomicalSystem f thePersians",fwhich ahmrathburied descriptionn theSrawyatJayn sfahan efore heFlood,butnotbeforeBdhsafthe Bodhisattva) ad carried he systemo India.This manuscript asexcavated ot ongbefore buMa'shar's ime,nd he claims hat isdoctrinesdirectlyderivedromhis nte-Diluvianource. is theoriesnd themany ragmentsfhis twobookshavebeen horoughlyiscussedythepresenteviewern TheThousands f AbuMa'shar, ondon 968.A littlemore han milleniumfterAbu Ma'sharpublished is fiction rofessorvanderWaerden as resurrectedhismythology.owever, e has beenmislednto

thinkinghat y"the Persians"AbuMa'shar ntendedo refero theSasanians atherthan oHsank ndTahmrathf egendaryame. e thus onceives fAbuMa'shar'ssystemsrepresentingnearly asanian ij onp.48 heeven ries opush tback o theperiod etween 200and - 250 because l-Sijz, uotingAbuMa'shar,tates hat heBabylonianssed t. n AbuMa'shar's istoryheBabylonianso indeed erive heirimperfectnowledgefastronomyrom hesecond Hermes",who, n thetimeofNabrzBni whoever e is supposed o be),was thefirst o revive cience fter heFlood.To seethis ource uoted s serious istoryn 1970 sastonishingndeed.All this is completelywrong. nevertried to resurrectAbu Ma'shar'smythology. y mainsource was not Abu Ma'shar but Al-Brn, who saidquiteclearly hatAbu Ma'shar's "Great Year" and hisnumbers frevolutionwere derived fromPersian sources. The "Persians" of Al-Brn were de-finitely ot the legendaryHsank and Tamrath,but real Sassanid Persians.Also, never ried o pushback Abu Ma'shar's system o theperiodbetween- 200 and -250. These time imits re invented y Pingree. My ownopinionabout the time of invention f Abu Ma'shar's systemwill be explained insection 11 ofthepresent aper.

5. ThreeWorld- ears ComparedIn an excerptfromAbu Ma'shar's "Book of theThousands",written yAl-Suzi (about A.D. 1000),theWorld- ear of360000 years s comparedwith26D. Pingree:TheThousands fAbuMa'sharLondon,Warburgnstitute968).


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368 . L. VANDER WAERDENtwoother,ongerWorld- ears.The excerpts preservedn twoversions,whichwere translatedby E.S.Kennedy and discussed in our joint paper7. Bothversions re said to come from hetreatiseAl-Jmi l-Shhof Al-Suz. Thefirst ersionA begins hus:A world-year,ccordingo thegeneralityftheastrologers,s fromhe time farrival f he lanetst the irstfAries ntil he ime f heir eturnothe ndofPisces,without here eing differencen their mountsi.e., ongitudes).s for hose n aregionf ndia nd theirdherents,hey ay hat he even lanetsnd theirpogees ndnodesbegin hemotion romhefirstfAries,ndtheyonjoin t the ndofPisces n4,320,000,000ears. sfor hepartisansftheyear fArjabhazAryabhata),hey ifferfromhemnd make heworld-year,320,000ears. hepartisansf theyears ftheArkand aiddifferentlyrom his. he Persiansahl Fars)and someoftheBabylonianssaid that heworld-yearsre 36[0],000 olaryears, fwhich here re 365 days,15minutes,2] seconds,and)24 thirds, ithoutequiringheirpogees nd nodes tobeat Aries ). fwe divide heyears f theSindhindya thousandhere omeout theArjabhaz ears.fwedivide y welve housandhere omes ut heyear f hePersians.Version gives ssentiallyhe ameinformation,ut adds a fewwords boutAbu Ma'shar, the authorof the"Book of the Thousands":As for hose fone of he egionsf ndia, heylaim hat heyears f heworldre4,320,000,000,hey eing hepartisansashb) f he indhind.owever,he ther roupof hem,hey einghe artisansf heyears fArjabhaz,heylaim hat heyears f heworld re4,32[0],000.ut the uthorshib) f the Book oftheThousands" sed heyears f hePersiansor he yclesndtasyrt.owever,ome f hemoderns sed heworld-yearsccordingo theway hepartisansftheSindhindxplainedhem,utwenow,n this ook,willutilizewhat he uthor f he Book of heThousands" sed.In these exts, hreevariants f the"World- ear" are compared,namely:1) a World- ear of 4320000000 years,ascribedto "those in a regionofIndia".2) a World- ear of4320000 years, scribed o Arjabhaz,3) a World- ear of 360000years, scribed n versionA to "the Persians ndsomeof theBabylonians", nd in version to "the Persians".ThisWorld- earwas usedbyAbu Ma'shar in his"Book of theThousands".The period 1), of 4320 millionsof years, s the Kalpa used by Brahma-GUPTA nhisBrhmasphutasiddhnta27,ritten .D. 628. ^The period2) is theMahyugaor Catoryugaused by ARYABHATA(calledArjabhazbyArabauthors) nhis treatiseryabhatya28. ight t thebeginningof thistreatise,he author aysthat he was ust 23 yearsold in A.D. 499,when3600 yearsof theKaliyugahad passed. So the treatisewas writtenn A.D. 499or later.The period3) is the "Great Year" ofAbu Ma'shar al-Balkhi. Accordingto Al-Brn, thisGreatYear was derived rom thecyclesofthe Persians".Abu Ma'shar too ascribes the period 3) to "the Persians", ust as heascribes1) to "the Indians" and 2) to "Arjabhaz". The ascriptions f 1) and 2)27Brhmasphutasiddhnta,yBrahmagupta, ditedwith isown ommentaryyM.S.Dvivedi Benares, edicalHall Press 900), eprintedromandit, ol.24.28W.E.Clark: TheAryabhatyafAryabhataChicago 930).



The GreatYear 369are correct,o I see no reasonto doubt theascription f3) to "the Persians".Moreover his scriptions confirmedyAl-BRN.

The period1) is 1000 times heperiod 2), and 2) is 12 times3),as VersionAcorrectlytates.The three stronomicalystemsmentioned nder1) 2) 3) are similar nmanydetails.Theyare all based upon theassumption f an approximate onjunctionofall planetsnear thepoint0 Aries n February, 102 B.C. In thesystems )and 2), thisconjunctionmarksthebeginning f theKaliyuga.In thesystem )adopted by Abu Ma'shar, theconjunction f3102 B.C. was connectedwithFlood, and it took place, according to the calculations of Kennedy &vanderWaerden7, in the night ust beforeThursday,February 17. Theduration fAbu Ma'shar's World- ear was 131493240days, nd itbeganon aTuesdayand ended with Friday, s Al-HSHIM tates see PlNGREE26, . 35).The mostnatural nterpretationfAl-HSHIM'swords eemsto be that hefirst ay of the World- ear was a Tuesday and the last day a Friday.Thisinterpretations in full ccordwith the calculationsof Kennedy & van derWaerden.In Pingree's opinion 26,p.35-36) the first ay of Abu Ma'shar's GreatYear would be a Wednesdayand the last day a Saturday,but this is incontradictionwith theverywordsof Al-HSHIMas well as with our calcu-lations.I shall now comparethe numbers f revolutions n 360000 years accordingto Abu Ma'shar and ryabhata. The latter omposed about A.D. 500) twoastronomical ystems.n thefirst,he"midnight ystem",which s knownfromVarha Mihira's account of the old Sryasiddhnta29nd also fromthetreatiseKhandakhdyakaof Brahmagupta30, the conjunctionwhich marksthebeginning aliyuga s assumedto takeplace at midnight etweenThursday,February17 and Friday, February18, 3102 B.C. In the "morning system",which s thesystem ftheryabhatya,heconjunction akesplace at sunriseon Friday, 8 February. he twosystems iffernly n details.The numbers orAbu Ma'shar are taken fromPingree's book26,thefigures orryabhatafrommypaper4. Numbers fRevolutionsn 360000 yearsPlanet Abu Ma'shar Midnight ystem Sunrise ystemSaturn 12214 12214- | 12214 iJupiter 30352 30352-^ 30352Mars 191402 191402 191402Sun 360000 360000 360000Venus 585199 585199 585199Mercury 1494751 1494750 1494751+Moon 4812778 4812778 481277829Varhamihira: Pancasiddhntik,d. Thibaut & Dvivedin (Benares1889, eprintedahore1930).NewtranslationyNeugebauer & Pingree,DanskeVidensk.elskab. ist.-Fil. krifter(1970-71).VBrahmagupta:Khandakhdyaka,ranslatedyP.Sengupta Calcutta 934).



370 . L. VANDER WAERDENIn my paper4 have shownthatthemean ongitudesomputed romryabhata's theoryrevery oodfor he ifetimefryabhata (A.D. 500),

but lessgood fordatesone or twocenturiesarlier r later.The "sunrisesystem"s even better han the"midnightystem".My explanations thatryabhata adjusted is numbers f revolutionso observations ade nhisowntime. n histheory,hemean ongitudesf all planetsresupposed o beexactlyeroat the ime fthe onjunctionf 3102B.C. Ifthis s assumed,heonly ossibleway f djustinghemean ongitudess, ochangehenumbersfrevolutionsn a Mahyuga.nryabhata's systemll numbersfrevolutionsmust e divisibley4. Ifoneofthenumberss increasedy4 units,hemeanlongitudesorA.D. 499are ncreasedy112' see4,p.157). nthisway, goodadaptationoobservationssalways ossible.In AbuMa'shar's system,he situation s different.ecause there s aconjunctionn themiddle fhis "GreatYear",his numbersf revolutionsorSaturn, upiter, arsand the moon mustbe even. fone of thenumberssincreasedy2,themean ongitudeorA.D. 799 s increasedy748', o thatgood adaptation o observationss not alwayspossible.The only waytoovercomehis ifficultystoadd constantorrectionerms o the ongitudes.tseems hatAbu Ma'shar did ustthat, orBiruni19saysthat ccordingoAbu Ma'shar's calculationor he time ftheDelugetheplanets stood nconjunctionnthe pacebetweenhetwenty-seventhegree fPisces nd theendof hefirstegree f Aries".Pingree (8,p.245)made he mportantemarkhat n thekalpa ystemfBrahmagupta the meanplanets re precisely etween he limits et byAbu Ma'shar at the beginning f Kaliyuga. t is quite possiblethatAbu Ma'shar used thiskalpa systemn order o obtain uitable orrectiontermsotheplanetaryongitudes.e maywellhave aken isperiod f360000yearsfrom ersian ourcesas BRN ays),but his correctionerms romBrahmagupta.The three stronomicalystems),2), 3) agree n so many oints hat heymust ave commonrigin. n this oint agreewith lNGREE,ut do notsharehisopinion hat his riginwas Indian. n order o statemy rgumentsclearly,must irst iscuss hehistoryfthe Tablesofthe hah".

6. The Tables oftheShahThe Arabicname of this ableset s,according o Kennedy20, Zj ash-Shh.It is a translationf a Pahlavi original,whichwas probably alledZlk-iShatro-ayr.The history f this Persiantable set is told by AL-HSHIM s follows(see22,p.4):KhusroAnshrvn,henhebeheld hedifferenceetweenhe Arkandnd whatPtolemysserted, e gatheredogetherhepeople earned n computationnd in(astrological)udgments,ndhe ooked ver hese wobooks.He foundheArkand obethemost ccurate yobservationndeyesight,nd udgmentsasedupon tsplanetsmore ccurate.o he workedut zjcalled TheShah" ndhe made t nkardajaswithfour ardajas.



The GreatYear 371Theywereusing t until hereign f Yazdigerdbn ShahriyarbnKhusro.Hebroughtuta zj andhe named tafterhe xample f theShah.He made t n three

kardajas nd called t "The Triple". ts explanatoryext nd apogees nd nodes ndmeanmotions)ndequations orrespondo those f heArkands tomidnightepoch).People tillworkwith t.The workArkand,whichoccurs three imes n thistext,s probablyderivedfromSanskritKhandakhdyaka.This is the name of an extanttreatiseofBrahmagupta30, in which he calculationofplanetary ositions ccording othe MidnightSystemof ryabhata s taught.Now KHUSRO ANSHRVNreignedn Persiafrom 31 to 579,and theKhandakhdyakawaswrittenn665,so the astronomers fKHUSRO could not use thistreatise, ut theycould usetheunderlying idnight ystem,whichwas composedabout A.D. 500. In fact,thecalculations f Kennedy21 and ofBurckhardt & vander Waerden22have shown that the Tables of theShah agreewith theKhandakhdyakaandhence with the Midnight System n many points, ncludingmean motions,apogees and maximalequations.Al-Hshim states that Khusro comparedthe Arkand with a book ofPtolemy. It follows hat a book of Ptolemy existed t the Sassanian court.This is confirmedya statementn the fourth ook of the Denkartto theeffectthatShapur added a copy or a translation) fAl-majisti.e.theAlmagest) othe books in his Royal Treasury31.However, he astronomers fKHUSROdidnot use the Almagest.Accordingto Al-Hshim, they preferredo use theArkand, .e. theMidnight ystem f RYABHATA.In the Tables of theShah,theyearwas supposedto contain in sexagesimalfractions)

365; 15,32,30days see20,p. 147).On the otherhand, n theMidnight ystem,heduration f theyear s only

365; 15,31,30days32.From thesefiguresne sees that 3600 years thetimefrom heconjunctionof 3102 B.C. to thelifetime fryabhata) ontainone moreday accordingtothe Tables of theShah than accordingto the Midnight ystem. n the lattersystem,heconjunction ook place betweenFebruary17 and 18,3102 B.C., atmidnightocal timeUjjain. Hence theTables ofthe Shah musthave placed theconjunction t about midnightust beforeFebruary17. This result greeswellwithBRN's tatement hattheZj-i Shah usedmidnightpoch in contrast othegeneralpractice fusingnoon.See Kennedy's Survey20, . 130.Let us recallthat heEpoch ofthe Flood according o BRN'sChronologi-cal Table (19,p. 133) is the same midnight eforeThursday, ebruary17,andthat thismidnightwas also the middleof the Great Year of Abu Ma'shar.Once more,Abu Ma'shar agreeswith the Persians".31See R.C.Zaehner: Zurvan, Zoroastrian ilemmaOxford955), . 139.J^Ihis duration as tound ydividinghenumberl days n a Mahayuga ythenumberfyears.



372 . L. VANDER WAERDENIn ryabhata's Midnightystem,he onjunctionf3102B.C. tookplaceatmidnightetweenhursdayndFriday.n his Sunriseystem,he ystemf

the treatiseryabhatya,he conjunctionookplace at sunrise n Friday,February8. So theThursdayateoftheTablesoftheShahwascertainlyotderivedromnyofthe wo ystemsf RYABHATA.On the otherhand, t seems hatryabhata knew bouttheThursdaytradition.n the reatise ryabhatyae writes:1.3.There re14Manus n a dayofBrahmana kalpa), nd72yugas onstituteheperiod f Manu.Since hebeginningfthiskalpaupto theThursdayftheBhratabattle Manus, 7yugas nd 3yugapadas ave lapsedtranslationfW.E.CLARK28,p. 12).This calls for omeexplanation.What ryabhata calls a yuga s themahyugaf4320000years.t is dividednto yugapadasrquarter-yugasf1080000years ach.Thekaliyuganwhichwe now ive s the astquarteryugaofthecurrentuga.Hencethebeginningf thekaliyugas themoment henthree ugapadas fthecurrentugahaveelapsed.ryabhata identifieshisbeginningf thekaliyugawith theThursdayftheBhrata attle".On thispoint,ryabhata is in accordancewith an earlier radition,or n theMahbhratat s saidthat hegreat attle fBhrata egan tthebeginningfthekaliyuga.fThursday asthefirstayofthebattle,hekaliyugamust avebegunn thenight eforehursday.his s ncontradictionithryabhata'sown ystem,nwhichhekaliyuga egan t sunrisenFridaymorning.How can weexplainhis ontradiction?he most aturalxplanationeemsto be thatryabhata found heThursdayate n an earlier radition.hisearlier radition as perhaps onnected ith he Persian octrinef a GreatYear consistingf two halvesof 180000years ach,separated y a GreatConjunctionn thenight efore hursday,ebruary 7,3102 B.C. Becauseryabhata's yearwas shorter,e was forced o shift hisconjunctionoFridaymorning,ut whenhe mentionedhe Bhratabattlehe keptthetraditionalhursdayate.Another actpoints ntothesamedirection.n theold Srya-Siddhnta,which s basedon theMidnightystem f ryabhata, a lunarperiodof180000years s used33.The numberfrevolutionsfthemoonduringhisperiod ccordingo theold Srya-Siddhntas exactly he same as in thePersian ystem,amely406389. t seems ossible hatryabhata took hisperiod f 180000years romnearlierradition,hichwassomehowonnectedwith hePersian reatYear.We thusare led to the conjecturehat the"PersianSystem"with tsThursdayonjunctionnditsGreatYear of 360000years xistedvenbeforethe ime fryabhata, i.e.before .D. 500.In thenext ectionwe shall eethat n earlier ersionftheTablesof theShah ctuallyxisted efore .D. 500.33Varhamihira Pancasiddhntikl4: Accordingo he aura = Sryasiddhnta)therere n 180000years 6389 ntercalaryonthsnd1045092 unar ays."



The GreatYear 3737. PersianpredecessorsftheTables of theShah

Al-BirOni informs s inhis Masudic Canon34 that Khusro Anshrvanconvoked,n theyearA.D. 556,an assemblage fastronomersor hepurposeofcorrectingheZj ash-Shahriyr.e. the Tables ofthe Shah.Fromthis estimony,Taqzdeh & Kennedy (20, p. 130) concluded that a Persian set of tablesexisted lready t thatdate.Anotherveryvaluable informations given by Ibn YNIS in the Leidenfragmentfhis HkimZj35. Ibn YNIS states thatthe Persiansobservedtheapogee of the sun at 7755' about A.D. 450 and again at 80 about A.D. 610(see7,p.325).Now whatdoes Ibn YOnis meanbytheexpression observing hesun'sapogee"? The solarapogee is not an objectone can see in thesky.One hasto determine t from hreeobservations f the sun's position,e.g. fromtwoequinoxesand one solstice methodof HlPPARCHOS nd PTOLEMY)or fromthreeunareclipses. bn YNISknewthis: hewas a competentstronomer, hocollectedmany ncient bservationsnd made observations imself. o wemaysafely nterpret is statement s follows:The Persianastronomers eterminedthe apogee of the sun by observationsmade about A.D. 450 and again byobservationsmade about A.D. 610. What was the situationof the Persianastronomers bout 610?Theyhad thechoice between everalvaluesofthesolarapogee. They mayhave knownPtolemy's value 6530'. The value 7755' wasknownalready bout 450,as we know from bn Ynis, and thevalue 80 wasknown from he "Arkand" system,.e. from he midnightystem f RYAB-HATA.The Persiansmade observations nd decided to adopt thevalue 80,forAl-Brn informs s that thevalue 80 was used in theTables of the Shah.BRNprobablyhad in mind theYazdigerdeditionoftheTables,whichwascomposedbetween 32 and 642. Hence thestatementfIbn YNISconcerningtheobservations ftheapogee about 610 is in accordance withwhat we knowfrom ther ources.I feelwe should nterprethe other tatement f bn YNIS nthe sameway.About450,a hundredyearsbeforeKhusro, the Persianastronomerswere ndoubt aboutthevalueofthesolarapogee.Theymadeobservations nd adoptedthe value 7755'. The same value was also used in the tables of Al-KhwR-izmi24.

Whydid thePersiansobservetheapogee? The mostnatural nswerto thisquestion s: because theywantedto compute table for hemotionofthe sun.So we may conjecturethat about 450 the Persians actually computedas-tronomical ables.This conjecture s confirmedy BRN'stestimonyuotedright t the beginning f the presentsection.Combiningthe statements fBRNand Ibn YNIS,we mayconclude that a setofastronomicalableswascomposed bout450 or a little ater, nd revised nderKhusroAnshrvnbout560.34Al-Brn:AlQnn l-Mas'd. smaniaOriental ublicationsureau, yde-rabat-Deccan1954-56), . 1423.35IbnYnis: Az-Zj l-Kabr l-Hkim.eidenCod.Or.143, . 124.



374 . L. VANDER WAERDENNote on Pingree'sCriticism

In hispaper9and againin10PlNGREE laimsto havedemonstratedhat hevalue 7755' ofthe ongitude ftheapogee of thesun,reported yIbn YNIS asobservedby the Persiansabout A.D. 450,was "in factcomputedbymeansofthe parameters of an Indian text, the Paitmahasiddhntaof theVisnudharmottarapurnaand computedfrom the beginning f the Kalpa "Now letus have a look at this demonstration".In theAppendix o hispaper9,PlNGREE ompares wo textswhichhe calls(B)and(P):(B), theBrhmasphutasiddhantafBrahmagupta,(P), thePaitmahasiddhnta,ncorporatedn theVisnudharmottarapurna.PlNGREE shows thatmanyparameters,ncluding he motion of the solarapogee from hebeginningfthekalpa,are ust the same inboth texts.He alsocalculatesthe ongitude f theapogee of the sunaccording o the textsB) and(P), and he finds 7; 54,7, 7 degrees,whichmaybe roundedoff o 7755'.So far agree,but thenext tep n theargumenteemsverydoubtful:"As the atterwork theVisnudharmottarapurna)as compiledbetween a.A.D. 450 and 650, the Paitmahasiddhntamustbe even older; and as it isreferred o by ryabhataunder the name Svayambh (ryabhatya,Go-ldhyya, 0), t mustbe dated at leastA.D. 450."To this onclusion have twoobjections. irst, t s not at all certain hat hework referredo by ryabhata under the name Svayambh (whichmeans

Brhmaor Paitmaha) s the same as text B). There were several stronomicaltreatises oingunderthename Brahmasiddhntar Paitmahasiddhnta; lN-GREE (36, p. 178-179) mentionsfour of them. ryabhata mentions"theuniversallyrue scienceof astronomy,whichwas formerlyevealedby Svay-ambh",but I do not know what revelationhe refers o, and I think thatPlNGREEdoes not know t either.He only guesses, nd he presents isguessasif twerea certain ruth.My secondobjection s: Even if one admitsthat ryabhata refers o thetextB), then t would follow nlythat his ext xisted yA.D. 499,but notthatitexistedA.D. 450,as PlNGREE laims.So whynotacceptIbn YNIS definitetatementhatthePersiansobservedthe apogee about 450 and again about 610? The authorof the astronomicalsystemxposed n the textsB) and (P) maywellhave known Persiantable setinwhich heapogeewas locatedat 1155'.In Pingree's opinionAbu Ma'shar gotthe dea ofa GreatYear connectedwiththeconjunction f 3102 B.C. from ndia. He roundedoff henumbers frevolutionsn a Mahayuga so as to make them divisibleby 12, and thus heobtainedhisperiodof360000years.As far s I can see,PlNGREEhas no goodargumentsnfavour fhisthesis:he ust speaksof "Abu Ma'shar'srounding ffof theSindhind arameters" s if t werea fact 26,p.33).In myopinion, t is muchmoreprobablethattheHindu astronomersearntabout theconjunction f 3102 B.C. from he Persians. suppose ryabhata36D. PlNGREE The Later Paulisasiddhnta, entaurus14,p. 172-241.



The GreatYear 375and Brahmagupta replaced,forgood reasons,the Persian"Great Year" of360000 years by the Mahyuga and Kalpa theyknewfrom arliertraditions.The larger eriods nabledthe Hindu astronomerso construct etter nd moreflexible stronomical ystems. his hypothesis eemsreasonablefrom he as-tronomical ointofview, nd it agreeswiththe testimonies f Ibn YNIS andAl-Brn, for the latter ssertsthat Abu Ma'shar's systemwas exclusivelybased on "the motions fthestars, s theyhad beenfixed ythe observations fthe Persians".It is true that n Al-Brn's India a differentpinionon Abu Ma'shar'ssource eemsto be expressed. RNwrites37:"The contextf hese assagesmakestclear hat his estructionf heworld akesplace t the ndof kalpa, ndhence sderivedhe heoryfAb-Ma'sharhat delugetakes lace t the onjunctionf he lanets,ecause,nfact,heytand nconjunctiontthe ndof achcaturyugand at thebeginningf achkaliyuga."In thispassage,Brn seemsto say that Abu Ma'shar's theory freturn-ing delugeswas derived romHindu sources.However,his statementn "India"is rather ague, nd the reasonhe givesfor t s notvery onvincing, hereashisstatementoncerningAbu Ma'shar's Persian sources n the"Chronology" sclear and definite.When writing he "Chronology",BRN had the Persiantablesand the textofAbu Ma'shar beforehis eyes, nd he could comparethePersiancycleswith hoseofAbu Ma'shar.One moreargumentn favour fmy hypothesiswas presentedn my ointpaperwithKennedy7 on p.323:In classicalGreek nd Hellenisticiterature,hedoctrinefthe GreatYear"wasalreadyonnected ith hemythsf heDeluge ndEkpyrosis.hese atastropheseresupposed o returneriodicallyhen heplanets ametogetherncertainigns f thezodiac. n thePersianystem,e still ind heDeluge onnected ith conjuctionf heplanets.ryabhataoes not mentionheDeluge:he onlyalludesto the "battle fBhrata" nThursday,ebruary7,-3101. The ideaofa Deluge n themiddle f acycle f360,000 ears annot e derived romndia; t must avecome o Persia romtheWest.I admit hat heopinions xpressedn this Note on Pingree's riticism" renot certain, ut in any case theyrest on a betterfoundation han Pingree'sfirmssertions.

8. PersianAstronomyeforeKhusroAccording to the middle- ersian treatiseDenkart, the Sassanian kingSHPR (240-270) collectedwritings rom ndia, theByzantineEmpire, ndother ands,and whichtreated fmedicine, stronomy,movement, ime, pace,substance, reation, ecoming, assing way,etc. see31,p. 8). Amongthebookscollectedwas Ptolemy's Almagestsee 31,p. 139).The hexameters fthefirst-

century strologerDorotheos of Sidon were translated nto Pahlavi under37Al-Brn,ndia ,translatedyE.C.Sachau, p. 325.



376 .L. van derWaerdenShPR (see Pingree8, p.241). Also translatedduringthe Sassanid Periodwere the "Anthologies"of Vettius Valens (see again8, p.241-242). Fromthesefacts,Neugebauer39 rightlyoncludedthat "serious astronomical c-tivitywas takingplace in Persia n the third entury". ateron,underKhusroAnSHrvn, the "Paranatellonta" of Teukros the Babylonianwere alsotranslatednto Pahlavi38. The contents f the Paranatellontawill be discussedin Section11.Without stronomical ables it is impossible o cast horoscopes.Thereforewe may safely ssume that the Persianshad astronomical ablesfrom heverybeginning nder Shpr I. The observation f the apogee of the sun aboutA.D. 450 was probablymade in orderto correct xisting ables.In Kennedy's paper17,Persianmethodsfor omputingmean conjunctionsofJupiternd Saturn re discussed.These methods re found n severalPersiansources,but not in Hindu treatises. oth Pingree and Kennedy assume theconjunction stronomynd itsastrological pplications o have been inventedin Sassanid Persia 17,p.41).The lapse of timebetween wosuccessive onjunctions fJupiternd Saturnis nearly 20 years. The motion of Saturn during this time is given byAbu Ma'shar as 24225'17".

Otherauthorsgiveslightly ifferentalues 17,p. 31). Abu Ma'shar's valueis infull ccordancewithhiscycleof 360000years,n whichperiodthe numbersof revolutions f Jupiter nd Saturn are 30352 and 12214. The differencebetween hesenumberss 18138; hence there re 18138 conjunctionsn 360000years, nd themotionofSaturnfrom ne meanconjunction o the next s^360 = 24225'17"...

Startingwith the assumed conjunctionof 3102 B.C., the times of thesuccessivemean conjunctions an be calculated.If the mean motionsof thePersiansystem ased upon the cycleof 360000 yearswerecorrect, he differ-encesof themean ongitudes fJupiternd Saturnought obe zero at thetimesof thecalculatedmeanconjunctions.Actually, hey re not zero. According othecalculations fKennedy & van der Waerden 7theerrors approximately5 in the3rd entury,4 in the4th entury,2 in the5th entury,Io in the6th entury.

38F.Boll: Sphaera. eubner,eipzig 903. eealsoC.A.Nallino: Tracce ioperegreche iuntegli rabiper rafilaehlevica,accolta i scritti (1948), .285-303.39O.Neugebauer: Review fL'Indeclassique yL. Renouand J. illiozat,Ar-chivesnternationales'Histoirees Sciences ,p. 172.



The GreatYear 377From theseerrorsKennedy & van der Waerden7 concludedthat thePersian ystemnthe formnwhichwe know t,with18138 meanconjunctions

in 360000 years, annot have originated efore he 5th entury. ow we knowthat n 556 Khusro's astronomers ecided to replacethe old "Tables of theShah" by better tables based on the "Midnight System" of ryabhata.Therefore he mostprobabletimeof nvention fthe"PersianSystem"with tsnumbers frevolutions 0352 and 12214 forJupiternd Saturn eemsto be thefifthenturyA.D.This conclusion is well in accordance with the testimony f Ibn YNISconcerning he observationof the solar apogee about A.D. 450. So we mayconjecture hat about this time the Persianscomposedtables of thesame kindas the ater Tables oftheShah",butbased upon thecycleof 360000years.9. The HellenisticOriginofthe "Persian System"

Mynexthypothesiss that he"PersianSystem"with tsconjunction f3102B.C. in the middle of a cycle of 360000 yearswas ultimately erivedfromHellenistic ources.Let me first ote that the conjunction f 3102 B.C. was not observed,butcalculated. n thatyear,no matterwhetherhecalculation s made forFebruary17 or 18 or anyotherdate in thatneighbourhood,o conjunction ftheplanetstookplace. On February17 the mean longitudes fJupiternd Saturndifferedby more than 40. The main reason forthis deviation s thatthe real Saturnmovesmuch faster han t was supposedto move in thePersiansystem40. hedifferencemounts o 26" a year, r 26 in the 3600 yearsfrom 102B.C. to thetimeofryabhata.So the conjunctionof 3102 B.C. was foundby a backward calculation,perhapsbyone ofthoseastrologerswho,according o BRN, ried o date theflood by computingconjunctionsof Jupiter nd Saturn.Now for such anextrapolation ne needsmethods o calculatemean longitudes fplanets.TheBabyloniansof the Seleucid era had no such methods: the notion "meanlongitude"does not occur in their heories.The Greeks of thehellenistic gewerethe first o calculateplanetary ongitudes y first omputingmean longi-tudes and next ddingcorrectionerms. his is themainreasonwhy supposethattheconjunction f 3102B.C. was a Greek nvention.Another eason sa passagefromAbu Ma'SHAR's"Book ofConjunctions"41.The firstpart of this passage was quoted in Kennedy's paper17, p.25.Thisquotationendswith hewords:

40See B. L. van derWaerden Vergleichermittlerenewegungenn dergriechi-schen nd ndischenstronomie.entaurus1 1965)p.16.41Kitb l-qirnat,scoriaiMS arabe937,fol. r.See alsofootnote42.42AccordingoKennedy7, ote 32],anotheropy f he BookofConjunctions",Kitb l-Qirnt,s nthe ibraryf heNearEastSchool fTheology,eirut, S PB20.A Latin translation,Albumasarde magnisconiunctionibusnnorum evolutionibusceorum rofectionibus,as published nVenice 1515.



378 . L. VANDER WAERDENAndtheres betweenhefirstayoftheyear ftheFloodand thefirstayof theyear nwhichwastheconjunctionndicatinghe Arabpeople, hreehousand nd six

hundredndseventyoneears,761.According o Kennedy, the "conjunctionof the Arabs" was A.D. 571. Ifone countsback from hisyear3671 solar years,one arrives t theyear3101B.C.,theyear ust after heconjunction f 3102 B.C.The continuationfthispassage in theEscoriaiMS Arabe937,fol.5rreads:Andverily NYNSand othersmentionedhat etweenhebeginningfthebirthofAdam, heprayersfGoduponhim,nd theFriday ightn whichwastheDelugewastwothousandnd twohundredndtwenty-sixears nd onemonthndtwenty-three ays ndfour ours..43.The name ANYNS mentioned ere s almostcertainly Greekname; itmaybe read as Anianos. If some dots are changed, hename can be read asABTNUS. As suggested y A.Sachs44, it is possiblethatthe Greek historianAbydenos is meant,who wrote book on Babylonianhistorynd chronology,partly ased on the"Babyloniaka" of BEROSSOS. ince Berossos wrote n theDeluge and on theGreat Year, thename Abydenos would fit xtremely ell.However,since the identificationf ANYNS with Abydenos is farfromcertain, shall restrictmyselfo stating hat Abu Ma'shar connectedhis dateof theDeluge with the name of a Greek author who had computedthe timefrom he creation o theDeluge.A similarstatement bout the date of the deluge accordingto a Greekauthor,probablydrawnfrom he same source, s found n BRN's "Chro-nology". quote from achau's translation,age 25:

Accordingo one ofthehistorians,nianus,he ntervaletween he reationndthenightftheFridaywhen hedeluge ommenced,s2226years 3days nd4 hours.This tatementyAnianussreportedy bn-Albzyrn his Kitb-alkirntBookoftheConjunctions)..In thistranslation he name Anianus is based on a conjecture ue to thetranslator achau. On page 374 Sachau notes thatin the manuscriptshename s written s 'THNYWS, hichmightmean ATHENAIOS.n hisopinion, heauthor Athenaios is out of the question; therefore e prefers he readingAnianus. This Anianus was an Egyptianmonk,a contemporaryf Pan-ODORUS,who was knownas a chronographer.he tworeadingsAnynS andAthnyws differnly n the diacritical ots.Note that a historianAthenaios, who wroteabout Assyrianhistory,smentioned yDlODOROS II 20. I shall quote the translation fOldwater inLoeb's Classical Library:Such, hen,s the ccount hatCtesias f Cnidushasgiven boutSemiramis;utAthenaeusnd certainther istoriansaythat hewasa comelyourtesannd becauseofherbeauty as oved ytheking f heAssyrians.owat firsthewas accordednlya moderatecceptancentheplalace, ut ater,when hehadbeenproclaimedlawful43This ranslationasmadebyDr.Anwar Sinnuandkindlyommunicatedo mebyE. S.Kennedy.44See8,p.243, ootnote14.



TheGreatYear 379wife,hepersuadedheking oyield heroyal rerogativeso herfor period f fivedays.And emiramis,ponreceivinghe ceptrendthe egal arb, n thefirstayheldhigh estivalndgave magnificentanquet,twhich hepersuadedhe ommandersfthemilitaryorcesnd all thegreatest ignitarieso co-operate ithher;and on thesecond ay,while hepeople nd themost otable itizens ere aying er heirespectsas queen,he rrestederhusbandndputhim nprison;nd since hewasbynaturewoman fgreat esignsnd bold swell,he eized he hronendremainingueenuntiloldage accomplished any reat hings.uch, hen,re the onflictingccountswhichmay e foundnthehistoriansegardinghe areer fSemiramis.

10. The "Chaldaeans" and the"Babylonians"We have seen that Al-Brn, in his account of the attemptsof theastrologerso date theDeluge bymeansofconjunctions fSaturn nd Jupiter,mentions the inhabitants f Babel and the Chaldaeans".We also have seenthat Abu Ma'shar ascribesthecycleof360000yearsto"the Persians nd someoftheBabylonians".Who were the "Babylonians" mentionedby BRN as well as byAbu Ma'shar?In ancient stronomical nd astrological extsone findsmanyreferencesocollectivitiesuch as"theChaldaeans","theEgyptians","theBabylonians","the Persians","the Indians".In a recent aper45 havecollectedmany uotationsfrom he Chaldaeans"and the "Egyptians", nd I have shownthatthese can all be understood squotationsfromspecific reatiseswritten y or ascribed to Chaldaean andEgyptian uthors.Concerning he originof thesetreatises have reachedthefollowingonclusions:1) All testimonieseferringo the"Chaldaeans" come from ne treatiseorgroupofrelated reatises)writtenn Greek between 30 and 170B.C., inwhichBabylonian astronomy nd astrologywere summarized.This treatise,fromwhichwe haveextensive xcerpts e.g. n the"Isagoge" ofGeminos), was basedon cuneiformexts.2) All testimonieseferringo the"Egyptians"come from ne astrologicaltreatisewrittenn Greek nHellenistic gyptbefore 50 B.C. The authors fthistreatise re sometimes uoted as "Nechepso and Petosiris", sometimes s"the Egyptians round Petosiris", and veryoftenust as "the Egyptians"or"theancientEgyptians".I shall illustrate he consequencesof theseconclusionsby an example. nLydus, DeMensibus II 12we read:45B.L.vanderWaerden: Die "Aegypter" nd die "Chalder".Sitzungsber.Heidelbergerkad.Math.-nat.)972,Abhandlung.



380 . L. VANDER WAERDENThe Egyptiansnd Chaldaeans all theseventh ay of the week and also thecorrespondingtar hainon.Ifthis tatements interpreteds a testimonybout the nhabitantsfEgyptand southern abylonia, t is just nonsense, or heEgyptians alled theplanetSaturn Horus theBull",and theBabylonians alled it kaimanu. owever, s astatementbout books writtenn Greek t s perfectlyeasonable. n both booksthe Greek wordPhainonwas usedto denoteSaturday s well as Saturn.This isa meaningfulnd interestingtatement.t impliesthat theplanetaryweek, nwhich each weekday s put under the supremacy f a planet,was used in thebook of the "Chaldaeans" as well as in thatofthe"Egyptians".Regarding the Persians", he situation s similar, s we have seen n section4. The "Persians"are authors f treatiseswrittenn Middle Persian Pahlavi),ormorespecificallyhe authors f"Tables ofthe Shah".The same thingholds for "the Indians". When BRN speaks of "theHindusand Persians" see section4), he is comparing stronomical arametersusedbyPersian nd Hindu authors.When Abu Ma'shar speaksof "those n aregionof ndia" who use thekalpa of 4320 millions fyears see section5),hejustmeans Brahmagupta.Now consider he"Babylonians".Theyare mentioned wice n the "Antho-logies"ofVettius Valens46.1) In VI 3, on page 249 ofKroll's edition,VETTIUS peaksofSaturn ndsays: "The Babylonianscall theplanet Phainon". This statementmakes sense

only if Vettius had beforehis eyes an expositionof the doctrinesof theBabylonianswrittennGreek.2) In IX 11, on page 353, Vettius makes a distinctions etween Chal-daeans" and "Babylonians".He ascribes o the Chaldaeans a certain uration ftheyear, nd to theBabylonians nother, amely

365+| + T4daysIfthe word "Chaldaeans" is taken n itsusual senseas Babylonian stronomers,a distinction etween Chaldaeans and Babyloniansmakes no sense. It doesmake sense fwe assumethathis figures eretakenfrom wodifferentources.In thefirst ource certain octrineswere ascribed o theChaldaeans,and in theother ource doctrineswereascribedto theBabylonians.The durationof theyearwas differentccording o histwo sources.

3) Threechapters f an anonymous uthor,knownas "The Astrologer f379",were opiedbyPALCHOS47.At the endof thefirsthapter listof authorsis givenwho have writtenbout the phenomenaof the fixed tars.The first46VettiusValens: Anthologiae, d. W.Kroll (Berlin 1908). See alsoO.Neugebauer: TheChronologyfVettius alens,Harvard heological eview 7

(1954), . 65-67.47The text fthe Astrologerf379" waspublishedyF.Cumont nCataloguscodicumstrologorumraecorumI, p. 194-211.



The GreatYear 381authors isted re "The Babylonians nd theChaldaeans". Just s in the textofVettius Valens, a clear distinctions madebetween he two sources.

The subjectmatter n thischapter ftheAnonymouss just the same as intheParanatellonta f"Teukros theBabylonian"38.Hence we may conjecturethatthe"Babylonian"sourceof theAnonymouswas ustTeukros.4) Alexander of Aphrodisias48, peaking of the number seven of thePleiads,firstmentions heChaldaeans,next theGreeks, gain theChaldaeansand finallyheBabylonians.Bythe"Babylonians"hemaywell meanTEUKROS,whomentions hePleiadsseveral imes n his Paranatellontasee Boll38, p. 122and 280).Let us now pass to Islamic sources.Abu Ma'shar and Al-Battani andAl-Brn ascribe certain stronomical heories nd numerical alues to "the

Babylonians"or to "thepeople ofBabylon". shallmention our ases.According o Al-Battn49 the "Egyptians"and the "Babylonians" as-sumedthe duration f theyearto be365; 15,30days.

Varaha Mihira ascribes hesame value to thePaulisa-Siddhnta.Nobodydoubts that this Siddhntawas based on a Greek treatise rom he Hellenisticperiod.5) A very nteresting estimony oncerning the people of Babylon" (ahlBabil) comes fromAl-Brn, who in his book Ris'il IV describes everal

methodsforcalculating he rising imesof the twelvezodiacal signs50.Mostmethods make use of trigonometry,ut the last three methods are purelyarithmetical. shallcall thesemethodsA, B, and C:A: Ris'il IV, 137: 2-12: Ris'il IV, 137: 13-138:2C: Ris'il IV, 138: 3-13.As Lesley has shown50,methodsA and C are based on the same arithmeti-cal methods s were used in cuneiformextsbelonging o the ancientBaby-lonianSystemA. However, hesemethodswere daptedto differentatitudes yusingmultiples f theequinoctial hadow ength. his adaptation s not made incuneiformexts;hence Lesley concludesthat theadaptationto otherclimatathan thatofBabylonwas made in theHellenistic ge.BRNascribes Method to "some of the books of the Persians",andMethod C to "thepeople ofBabylon". t seems thathe foundbothmethods none and the same Persian source,forafterhavingdescribed Method B, hewrites48Alexander Aphrodisiensis: In AristotelisMetaphysica ommentaria,d.Hayduk Berlin 891) .832.See alsoBoll38, p.366.49Al-Battn: Opus stronomicum,edited yC.A.Nallino (Milano1903), .40.50M.Lesley: Brn n Rising imes ndDaylight engths.entaurus (1957)p.121-141.Morematerials in Al-Brn's Exhaustive reatise n Shadows", rans-lation nd Commentaryy E.S.Kennedy (InstituteorHistory f ArabicScience,UniversityfAleppo 976, hapter2).



382 .L. van derWaerdenThe (Persian) uthor f theoperationaid."As for hepeopleofBabylon, heymultipliedhe quatorialhadow y twenty-fivend divided heresult y eighteen,nd

subtracted hat esultsromhirty.here emainedhe scensionorAries. hen heysubtractedwice he scension orAries romixty,nd divided heremainderyve.There esultedhebaseof ncrease or achsign, ndthey idit to the scension orAries o obtain aurus,nd to the scensionor aurus o obtainGemini,nd so ontoobtainVirgo"51.The last sentencemplies hattheascensiontimesof thesignsfromAriestoVirgoform n arithmeticalequence, ust as in theAnaphorikos fHypsiklesand in theAnthologies f Vettius Valens52.The conclusion s the same as before:The "Babylonians"were Hellenisticauthors.6) Another estimonyf Al-BRN has alreadybeenquoted in Section4:The astrologersave tried o correct heseyears, eginningrom he first f theconjunctionsfSaturn ndJupiter,orwhich he ages mong he nhabitantsfBabel,and theChaldaeans ave onstructedstronomicalables,heDelugehaving riginatedin theirountry.Once more,as in the testimonies f Vettius Valens and Palchos, theChaldaeans are distinguishedrom heBabylonians.This is correct, or ccord-ingto ourresults he "Chaldaeans" wereearlyHellenistic uthors,whereas he"Babylonians"were late Hellenistic uthors. guess the "Chaldaean" bookswerewrittenn the third entury .C., and the"Babylonian"books in the first

century .D.7) In Section 5 I have already quoted the testimony f Abu Ma'shar(VersionA):ThePersiansnd some f heBabyloniansaidthat heworld-yearsre360000 olaryears, fwhich here re 365 days,15 minutes,2 seconds,nd 24 thirds, ithoutrequiringhe pogees nd nodestobe atAries ).I supposethequotation The Persians nd someof theBabylonians"meansthat Abu Ma'shar found the description f the World- ear in a Persiansource, nwhich Babylonian ourcewas quoted,as in testimony).Can we make a reasonableguessabout this ource? thinkwe can.11. Teukros heBabylonian

In my opinion it is extremely robable that the commonsource of allquotationsfrom the Babylonians"was "Teukros the Babylonian".Severalargumentsn favour f this dentificationerepresentedn my paper53 n the5 This ranslationaskindlyommunicatedomebyE.S. Kennedy.52O. Neugebauer: On SomeAstronomicalapyri ndRelatedProblemsfAn-cientGeography.ransactionsf heAmericanhilos. oc. 32 1942), .251-263.53.L.van derWaerden: The"Babylonians"ndthe Persians". rismata,est-schrifturWillyHartnerSteiner, iesbaden977), .431-440.



TheGreatYear 383Festschriftor Willy H artner. As we have seen, "the Babylonians"werequoted byVettius Valens as well as by Abu Ma'shar and Al-Brn, andwe also know that these three authors have known and used the work ofTeukros. This seemsto be a strong rgumentnfavour fthe dentification.About the ifetimefTeukros notmuch s known.Boll showed 38.p. 8)that he must have lived before irca A.D. 100.On the otherhand,GUNDEL54concludesfrom he ongitudesn the"Paranatellonta"ofTEUKROS hathe lived"not long afterHipparchos". We may conclude that Teukros lived,mostprobably, etweenHipparchos (130 B.C.) and Vettius Valens (A.D. 140).In Islamic sources TEUKROS was known as TNQALSor TNQARSorTankalOsha. His "Paranatellonta" were translated nto Pahlavi underKhu-SRO Anshirvn, and this translationwas used by Abu Ma'shar in his"Great Introduction". n fact,book 6, Chapter 1 of thisworkcontains threevariants f the istof"Paranatellonta"or simultaneously ising or culminatingor setting)tars55.One of thevariants s ascribedto the"Persians",one to theIndians and one to Ptolemy. The "Persian" listwas derivedfrom hePahlaviversion ftheworkofTeukros. So the road oftransmittance as

Teukros

Pahlavi Translation

Abu Ma'sharI supposethatAbu Ma'shar's knowledge bout the Great Year of360000years nd theConjunction f 3102B.C. was transmittedythesame road.I don'tbelieve hatTeukros theBabylonian nvented he"PersianSystem".According o BOLL38,TEUKROS was not an original stronomer,ut rathercompilator. he man who discovered he conjunction f 3102 B.C. must have

been an astronomerwho calculated conjunctions n the past and found anapproximate onjunction fall planetsnear0 Aries ntheyear3102.Onlyafterthisdiscoverywas it possibleto setup an astronomical heory ased upon theassumption hatthisconjunction ookplace in the middleof a "Great Year" of360000years. MathematicalnstituteUniversityfZrich(Received anuary0,1978)54W.Gundel & H.G. Gundel: AstrologumenaF. Steiner, iesbaden966), .112

55A Germanranslationfthis art ftheGreat ntroductionyK.Dyroff wasprintedogetherith heArab ext,nBoll's Sphaera 8, .490-539.

Documents

The Great Year in Greek Persian and Hindu Astronomy--B L Van Der Waerden