WPCs $5gF04k a]W][Vw*y˙>tƟۍa"k|._\D >vUD7Θ|k'* weçU)qVR?~K3md#1~Tc?N@iJβFwa `^o~E^u9{׿n!BBc/hy'bYɛ.M|fN#w|3fx?n&Ɲ`;(mx \9'UX)L)$Ou &6#K{_e 2z.In)r+f| EUnjrȠT*>@Ï$7qrz1y  V?o .&(`:tf_EePH'0}ڡn$10zvgx'{Ʌ{8- Zmom0#( UCN %7 0(=^ ewq4u me 0l 0Db 1u(b D+A 0 6lUNU> 09...f 0w 0> 0 0h 0 0 0 1(66 1^^^ 0N 3333333fM AO 0K7777 AQNa@Zo%@VDD?DDoFFaHHHH B 3H3HPH 0DI 0DZI 0DI?I D3J@XKKKKKKKK 72uKuKuKuKoKK?9O@ZGPo!!PP?q@Xrrrro\2s AyyM7z7zM;zN?z?z?AzMO{@XS{M{\{o }? 09[bX\ 09DYl D5~ B* D5ݕ B* D5< B7q B7 D5ߖ B D71 D/h%@zGZ*[i*.XH\^|@r`.qq·[42f!z;C3+^83d8 O8_KH fEPSON Stylus C80 Series0(9 Z6Times New Roman RegularX($͝s- g[- W<.3|x2U((3$ !      0  (#$  0  Level 1Level 2Level 3Level 4Level 53#37=CIQYag1.a.i.(1)(a)(i)1)a)Kutz  X"  _  *  **  ***  ****'dxd0.Strong xhH  Z6Times New Roman Regular\  `&Times New Roman(+T$ XXӀ2G3  0   XX (O;$0  2#  a  .3  0` (#(#(b$0  0` (#(#2#   .3  0 ` (#` (#(xir$0  0` (#(#0 ` (#` (#2#(  0  )3  0 (# (#($0  0` (#(#0 ` (#` (#0 (# (#2#(  a  )3  0h(#(#(F$0  0` (#(#0 ` (#` (#0 (# (#0h(#(#2#(   )3  0h(#h(#($0  0` (#(#0 ` (#` (#0 (# (#0h(#(#0h(#h(#2#  0  )3  0(#(#({$0  0` (#(#0 ` (#` (#0 (# (#0h(#(#0h(#h(#0(#(#2#  a  )3  0p(#(#"#$%&'(xGGaeimquy}Bullet ListBullets List(CEKQW]cioAutoList11)1)1)1)1)1)1)1)(O$WorksCited>Y= I&mage <=8C HKKKK( $ Figure  1  Level 1Level 2Level 3Level 4Level 5WXYZ\C<<CLevel 1Level 2Level 3Level 4Level 5Level 1Level 2Level 3Level 4Level 5TABLE ALevel 1Level 2Level 3Level 4Level 5JC:\MYFILES\PYTHAG~1\GRAPHICS\GNOMON~1.GIFWPC```` ~~~}}}|||{{{zzzyyyxxxwwwuuutttsssrrrqqqpppooonnnmmmkkkjjjiiihhhgggfffeeedddcccbbb```___^^^]]]\\\[[[ZZZYYYXXXVVVUUUTTTSSSRRRQQQPPPOOONNNLLLKKKJJJIIIHHHGGGFFFEEEDDDBBBAAA@@@???>>>===<<<;;;:::888777666555444333222111000///---,,,+++***)))((('''&&&%%%###"""!!!  =!}) o {  ]?ѹн2*.+.  22b 2M&2U    2       2    2  in32 Wϩ5;ж4ڞ'>j  ȹ$rد 2 ٽ` ¶\"ϺoӺk 溳ތ>մî2۷ğ-巰ú$ ¸9 ƴ8漹2 漽i  7ܺŲʻ߸ թ ݺ2 ޺ ٽ jζį Jٳ Lճ ɺ>2 佺Ġ Ľļ  Q XϹ b嵾? =Ṹ Aع!S⺫氹̽E 澶ڌ܂~`ඵỼ.L漺F(S~W\V@L#+a 7  !0$QbG4J4,%NA1P2 0&' !  B    (  E !   #46%(#-$B]`F7$\յļ6  BW  *% !" $ ,.  %c 7 $'k 3    S  $c!LPV$c  363IƬ;.z_-CP% <ۼ 7+׼պϼxUվ> ϲ& 潺4θ΄ּA   nشHķ ü溶㽂 Ǽ U  ݹ. ׺ -ⷺ "Զ湺 nijS ׽> 㺼 +պ ۺܻ` uϹ" Z傺 ົ  乳  渴  fӸSƸ - Ѻʟ йPպ! LƳ ִZ C ȯ  ðNߦ' (W“ ed4 $x[ ֚  aŶu  ,!       ;,    8'>BN  %BH"l7(Ld   $  !#'".2.;$1:5/49E849N6EH?FE<>B]OB@\Lf]odkuyz{xo~/_ĵǫ}zxcxzcpxZgpf|N:2g*.& C 53&"X H  (+ ," 7  5   %6h^Qn_F= ݴBMա' Գ&ʹ \wӼܷ»F 㴽̻V漸Gʸ úgʳ >ʼø  Iм1湻ȶ 4پ}溺Ǻ  ߾a Gк! ˬ‹ ں ƺ¾  ݹ 1 ܶː   ¾ ׼ʵ 溷 溵h ھ Z׹ȁ濾 iӸƧ˟Ź Ϭé   ĵ̹dQ潼Ǻ¿+ּs[̅׿w2  ˣ4z[(PoU{3VT   C  @'5 ,    O=!-( 1 /i43    ! $"+'&#*0>%3.(::E-8;ZT;L^hb+(܂'ҷѲ  :) O| O&OnOf- OI%%'K   .˥ "ſ%Vj 7?( C2 3  jƾZD" h=Eܳ庲ϪųQ ູ 湼 ̽7WGۼгtз źi L䲶Ƹ?ݷŷ13 ݽ0ڹxػ0 1ñ"4ٶ ޸X  ͵pɿs ,4 Rӹ8 (ڲ-湸? ཻ {Ϲ滭ȰH =޻5为 g޼* ðrɻ҈\ضFþb& ۺD-ŵb:s\=Wʽ:G5nפ  ;3NgTHݬ]"Ub? 4    2  </ OC: O.#O<@@O 8!O:!$ 2"F7?B@:-188@V;2<>\TRqThteuhmzλܪ5+ 2ȭų£r`wrkYii]]W_loF[UEPE?U<9A1?Ѱ*;6b:      U  $6CI ( ;P" >$6l  B 7  Sr&  ˸#L- Ȼ֘%бS  .  1, ?2  ⿼τ毺eɬ# eW @P cԹ Iܱ˹~ⷳ&湴HDZΟ  ð|ĵ"  fӰǥxNո淵ɪU DԷ0E `د  Ĺ* ^ַ˜p ϸĸ!  ҼR : Գy ͻ渾Mm ǺnVɂK H Կ¯‰ ߺǛ\潬ާI mѼP ߽] rͽȋ5 ǵ3 8rb串Ƽd~T渻J歺=Ӊ ! Z~üUڿ֚>YȾOE<6j<ն3ƙN SǴ /j 6KhM6Zd<69 dV6I]_ 6\~R 6+wW7 + 'P<RHZOC:\MYFILES\PYTHAG~1\GRAPHICS\1POINT.GIFH@TC:\MYFILES\PYTHAG~1\GRAPHICS\GNOMON~1.GIFC:\MyFiles\pythagoras\graphics\gnomon_even.gifLevel 1LevWPC``ndndnd`` @rrrqqqiiihhhgggKKK@@@===888777///...$$$nd}m/231 20 &*-' 10 )7::92#1/&8:.1/+:4# 1/,:5%1/'9:0! 1/ 1:96(10 ".35/( 20 "$! 22  4m%TABLE C dLevel 1Level 2Level 3Level 4Level 5(3$ !  (3$ !  ($$   1  K@PC:\MYFILES\PYTHAG~1\GRAPHICS\1POINT.GIFC:\MyFiles\pythagoras\graphics\1point.gifVc wLevel 1UUUU'dxdiC:\MYFILES\PYTHAG~1\GRAPHICS\2POINTS.GIFi)WPC``ndndnd`` rrrqqqlllhhhgggKKKJJJAAA???===999888333000...---$$$###?nd}m',  *',)!  -37/$) "0=CB:) 1JKKJ@*(0EKJ?#-IK:!(&65@HK77K4.JKA+(;K-#?KF0(*@KJ2  *9BD=0") %.83-  !),')   -m-]@RC:\MYFILES\PYTHAG~1\GRAPHICS\2POINTS.GIFC:\MyFiles\pythagoras\graphics\2points.gifVcvLevel 1mC:\MYFILES\PYTHAG~1\GRAPHICS\GNOMON~2.GIFWPC```` ~~~}}}|||{{{zzzyyyxxxwwwuuutttsssrrrqqqpppooonnnmmmkkkjjjiiihhhgggfffeeedddcccbbb```___^^^]]]\\\[[[ZZZYYYXXXVVVUUUTTTSSSRRRQQQPPPOOONNNLLLKKKJJJIIIHHHGGGFFFEEEDDDBBBAAA@@@???>>>===<<<;;;:::888777666555444333222111000///---,,,+++***)))((('''&&&%%%###"""!!!  }* L   $)" $ $% 52 00.(*8'35651*=J:D.=8+DTUBR/ ZނF̯þ«ӆ  !   P:+   ". +   -   u           q  Ƕr  op O; |ӝ ?ȿ #Ѳ4 ¿˻ .IJ Ƹ  r潮d  ƹ̣Ȼi ø  -㾸 oм.Q݃ ༴ʼ Bӫ c ڼ¸Є v̺Ⱥ Rٹÿ lӱ۹” Ϲi ɹ Bۻ½ ͸ ȻźT <ỵ  &潹̨ļ 派Ͳ[ųŸω̽\üa泲F֞oڪ<#Jؑm$I`1D %HC  ^ $#* "              ! H    +   ,," /27+#(,$(><:(9@@5QHXaQZRIWghX`gka{{ou}qvI   PɻȾؿǿűıˮlepok{}oy]uhkbi^RzcNi GP         p    e_uyu gur}^KE0AnICvRSWȳmpm ޝidܴwuվ~潻Ÿ2渻ļWVǵvÄ   潻έ0 Ҽʺ lֿH澽m ṷ  aлļp ϽýʼΫ  ۼ 滼F cػ \з˻ȸǏ ú ޺ Ҹi Yŵv %潨5 rӿQ ڽų˒̸a|滳˜  Ъ ̼Ϻ Fݹ  _濻Yo{溸 ݰù`ۗ|w̓-Z6tX:῞=fq : u~x0Cfugxuvu_u_D  '!%.8&78!;B/8blC橷З ۂĵƹF{*86@3=@9;(69&.-"#    ԍ\˃ukx\js  \u  b+b^{@Ɏ  x*7O$a{h4[J ˆٻ!lǽ֨,kwEֻ Ŵj{ڶtk‚\ T㼸t0淹{溽 z滴j㺺 ºǺYW Y۾¯ς] ػk ྾ Ƚ ެŸM ̿Ĵe $׿d⹺ ɫR ڼѾϓun潺}sҺ tN ɺty3}тʾ 潵W漽ܖoT%>,.!s}fsX z93YDF{wJ 3]8 ~v3 Cq{@  %  w| < (vw)&  m~e C I u' Wwq\,P2E^p}svqq~w~{å̲ż±;ryy2wogwu~[un[BcW@JZBFVF\CI;:TK>78H2CE03G& ֈ ŏ/+н8q4F n ɷȾ:ŲTH漽ת2#> ξh6 { 滸  溻:u湶-D gճ Ƚ ʲe# Ҳ̺h㺺A ºµ kǸ ðقR޻= ö˶& NԹʹ ղųŖ& ûi潃Us 悺Έ+幂# *伽nun⼺׌Cƺw]ȼ ںrw̴դGqHҍ"]۞eC=T-@  8# =/*> VD*5?R* ,WQ%   ,  -W-B׍* L*&+ *+ 0l@TC:\MYFILES\PYTHAG~1\GRAPHICS\GNOMON~2.GIFC:\MyFiles\pythagoras\graphics\gnomon_odd.gifVc{Level 1LetC:\MYFILES\PYTHAG~1\GRAPHICS\3POINTS.GIFWPC``ndndnd`` }}}zzzyyytttsssrrrooonnnlllkkkiiihhhgggfffaaa]]]ZZZXXXSSSOOOKKKJJJIIIFFFBBBAAA@@@>>>===<<<888333222000///...***'''%%%$$$###   nd}m150 %(!3. +FY]Q73. ,Wxyi@2.Gxyb02.%Yyk? 2.(]ynE2.!Oxyg8 2.6tywT&2/ :yulpxyG 3/ oyyd%" dy2TyN%"IyThy9$!*yo'uu'# hy2!9yh" LyP#NyT!/yk#dy>!hy/% ry- LyL'3yk/yh(Jy^ky-)_yHPyL+oy12yh,.yroy*.Dya TyI/XyM5yd 0ky5oy'2)yu$TyA4:ye  %(!:ya5 RyS%(! +FY]Vuy%5 fyZY]Q7 ,Wxy[4 6xyi@Gxyc04Gxyb0%Yyvh\Z[`qyk? (]ynE!OxyjC% "1Xxyg8 6iysT&46iysT& @bkmfU1 6 @bkmfU1  0?F9& 6 0?F9&   9  m&Draw Object <<=8C HKKKKp@RC:\MYFILES\PYTHAG~1\GRAPHICS\3POINTS.GIFC:\MyFiles\pythagoras\graphics\3points.gifVcvLevel 1C:\MYFILES\PYTHAG~1\GRAPHICS\4POINTS.GIF R `%  1  .  _Diogenes_Laertius_,LivesofEminentPhilosophers,Trans.R.D.Hicks(NewYork:G.P.  PutnamsSons,1931),Vol.II,BookVIII,line21,p.339.Henceforthabbreviatedas:D.L.,  Lives.WPC``ndndnd`` }}}yyyxxxvvvtttsssrrrooonnnmmmlllkkkiiihhhgggfffcccaaa___]]]ZZZXXXSSSRRROOONNNMMMKKKJJJIIIFFFBBBAAA@@@>>>===<<<:::888333222///...---+++***)))%%%$$$###  nd}m.8- -0&6+ 5XqthG#6+ 6nQ5+Y{<5+-qP5+0tW5+&fH 5+Fk.5, JY6, T_"6, kS'w Wz 5+ D} $uqa5* /$uC4)qR$k1,3)Iz $iK2($($iii2'rL$iK1'Iu$i,10&$$$iD/%uI$iaq.%Lu(i zV.$($1i$:-# zI1i =!,#Rq1i Zx +"/1`  u^+! zD 1^ A*!Rk 1^ 7%) / 1^ O}( }? 1^ kc(\k 1^I'2 1^//&? 7^I%`g=^ck$;=^}O$;=^%7#``=OA";=O^u !7=O xZ!g`=O= ?=O:$2=OVz k\=OoaD} AOD/EO1,kREOJ  -0&Jz EO ij -0& 5Xqtm-EE rqthG#  6nsEE FQ Y|<EEY{< -q~SooSU]pP 0tJ6>BbbBEKKW &ft1EE@H  FK EEAzk. #Q{lFbE JE Cr:#Q{l=  `%  16  .  _David_Fideler_, Introduction,InThePythagorean_Sourcebook_ԀandLibrary(GrandRapids,  Michigan:_Phanes_ԀPress1988)p.23. `%  17  .  _F.M._Cornford_, MysticismandScienceinthePythagoreanTradition,TheClassical {{ QuarterlyVol.XVII(1923):p.3. `%  18  .  __Archytas_,_Ancilla_,DK47,fr.1,p.78.  `%  19  .  _Iamblichus,TheTheologyofArithmetic:OntheMystical,MathematicalandCosmological  SymbolismoftheFirstTenNumbers,Trans.Robin_Waterfield_,(GrandRapids,Michigan:_Phanes_  Press,1988),p.35.Henceforthabbreviatedas:_Iam_.,Theology. `%  20  .  _Iamblichusquoting_Speusippus_Ԁ(Heath,1931,p.81).  `%  21  .  _Theseimagesappearin_Fideler_,1988,p.23.Theultimatesourceis:Aristotle,Physics.Trans.    PhilipH._Wicksteed_ԀandFrancisM._Conrford_.(Cambridge,Massachusetts:HarvardUniversity !! Press,1957)Line203a10. r `%  22  .  _Aristotle,Metaphysics,LineA.5.986a23. h `%  23  .  _Pythagoraswassaidtohavediscoveredthemusicalintervalsbystumblinguponablacksmith Z%Z% bangingonmetalwithdifferentsizedhammers.Thedifferenthammersproducedharmonious G&G& notes.Pythagorasthenmeasuredtheratiosoftheweightsofthehammersandappliedthesame 4'4' ratiostoweightswhichhehungonstringfromarod.(_Iam_.,VP,26.115)Thetaleisphysically !(!( impossible:itisthelengthofthestring,nottheamountofmass,thatcreatesawholenumber )) ratio.(_Burkert_,1972,p.376n24)Forahandsondemonstrationusingimagesandsoundsee:J. )) BoydBrent,Pythagoras:MusicandSpace, ** 4 2O  5  62O   7 Accessed21April,1999.http://www.aboutscotland.com/harmony/prop.html,  -- C  `%  24  .  _JohnBurnet,EarlyGreekPhilosophy,4thed.(London:AdamandCharlesBlack,1963),p.  107.  `%  25  .  _Thesesubjectsarelistedby_Archytas_,_Ancilla_,DK47,fr.1.;Introducedasthe Pythagorean RR QuadriviuminIvorThomas,GreekMathematicalWorks(Cambridge,Massachusetts:Harvard ?? UniversityPress,1967),VolI,p.5. `%  26  .  __Whibley_,Leonard,ACompaniontoGreekStudies(NewYork:CambridgeUniversityPress,   1931),p.8889. `%  27  .  __Zeller_,1881,p.354360;_Burkert_,1972,p.115120.  `%  28  .  _Forexample:_Cratinus_Ԁ(c.49020)TheShePythagorean,_Aristophon_Ԁ(4thcentury)The 88 _Pythagorist_.Freeman,ThePreSocratic_Philsophers_(1949),p.260,DK58E.  `%  29  .  _ Hewasformanyyearselectedgeneralof_Tarentum_,underthedemocraticconstitution  whichordinarilyallowedthisofficetobeheldforoneyearonly;andduringhistermofofficehe  wasneverdefeated._Archytas_,_Ancilla_,p.234,DK58A1. * `%  30  .  __Proclus_,CommentaryonEuclid,BookI,InGreekMathematicalWorks,ed._Friedlein_ ii (1967),Vol.II,p.149.Seealso:Heath,1931,p.37.  `%  31  .  _J.A.Philip, AristotlesMonographOnthePythagoreans,TransactionsoftheAmerican    Philological_Association_Ԁ94(1963):185198.  `%  32  .  _Aristotle,OnthePythagoreans,InTheWorksofAristotleTranslatedintoEnglish:Volume  XIISelectFragments,Trans.SirDavidRoss(Oxford:ClarendonPress,1952)VolXIIp.134. 1 `%  33  .  __Burkert_,1972,p.4.referstotheworkofErwin_Rohde_Ԁ( Die_Quellen_ԀdesIamblichusin_seiner_ OO _Biographie_ԀdesPythagoras_Rheinisches_ԀMuseum26[1871]554576,27[1872]2361).This << workarguesthatPorphyryandIamblichusused4thcenturyauthors_Aristoxenus_,_Dicaerchus_, )) _Heraclides_Ԁ_Ponticus_,and_Timaeus_Ԁassources. `%  34  .  _GillianClark, Introduction,InIamblichus:OnthePythagoreanLife,(Liverpool,England, !! LiverpoolUniversityPress,1989),p.xiv. `%  35  .  _Constantine,thefirstChristianEmperorofRome,ruledfrom312336C.E.Iamblichusdied m$m$ c.325C.E.  `%  36  .  _GregoryShaw,TheurgyandtheSoul:The_Neoplatonism_ԀofIamblichus(UniversityPark,PA: '' ThePennsylvaniaStateUniversity,1995),p.4. `%  37  .  _DominicJ._OMeara_,PythagorasRevived:MathematicsandPhilosophyinLateAntiquity )) (Oxford:ClarendonPress,1989),p.33.  S,S, C  `%  38  .  _Pierre_Grimal_,TheDictionaryofClassicalMythology,Trans.A.R.Maxwell_Hyslop_,  (Cambridge,Massachusetts:BlackwellPublishersInc.,1996),s.v. Heracles, Perseus,   Achilles. `%  39  .  _ThispoemalsoappearsinPorphyryVP,Section2.Thepoemisattributedto aSamian ?? poetbutisnottreatedcritically. ) `%  40  .  _Plato._Phaedrus_.Trans.C.J.Rowe.(AtlanticHighlands,N.J.,HumanitiesPress,1986),lines   246e248c;Seealso:_OMeara_,1989,p.37. `%  41  .  __Iam_.,VP,30;Porphyry,VP,Section28;D.L.,Lives,8.9.  `%  42  .  _Seeinfra,p.22;_Burkert_,1972listsacollectionofotherlegendsthatwereattributedtoeither 88 Pythagorasor_Pherecydes_Ԁatdifferenttimes.Hewrites: Thesemiraclesallbelongtotherealm %% ofprophecy.p.145. . `%  43  .  _OrphicHymn#87ToDeath,InAncientMysteries_Sourcebook_,ed.MarvinW.Meyer(New  York:_HarperCollins_ԀPublishers,1987),p.109.line3. `%  44  .  __Xenophanes_,_Ancilla_,DK21,fr.7. `%  45  .  __Burkert_,1972,p.120;D.L.,Lives,8.36. `%  46  .  _Homer,Iliad,trans._E.V._Ԁ_Rieu_Ԁ(Baltimore:PenguinBooks,1975)BooksXVI,XVII,p.315  317. Z `%  47  .  _D.L.,Lives,8.5saystheshieldwasinthetempletoApolloat_Branchidae_._Iam_.,VP,14.63 uu andPorphyry,VP,Section27locatetheshieldinatempletoArgiveHeraatMycenae `%  48  .  __Hendry_,M. PythagorassPreviousParents:Why_Euphorbus_?,Mnemosyne48:2(1995)p.  210. E `%  49  .  __Burkert_,1972,p.141pointsto:_Kernyi_,KarlPythagorasundOrpheus(Zurich:_Rhein_Ԅ_Verl_.,   1950)p.19.  `%  50  .  __Hesiod_,Works&Days,Trans.XLXXXHughG.EvelynWhite#XLXXXLۆ#XLXXXL(Cambridge,Massachusetts:Harvard [#[# UniversityPress,1950)#XLXXXL#Ԁlines274279,p.23. `%  51  .  __Burkert_,1971,p.155;_Iam_ԀVP143. `%  52  .  _EitherCroton,_Metapontum_,or_Tarentum_. U `%  53  .  _NotingthattheroosterfunctionsasareincarnationofbothPythagorasandApollo:_Miroslav_ g)g) _Marcovich_, PythagorasasCock,AmericanJournalofPhilology97(1976),p.331335.   , ,   `%  54  .  _ Pi:TheGoldenSpiral,PiTheMovie,ArtisanEntertainment  Accessed1May1999.  `%  55  .  _ ArtsontheT,MassachusettsBayTransitAuthorityWebSite,MassachusettsBayTransit RR AuthorityAccessed1May1999.  `%  56  .  _A.K._Dewdney_,AMathematicalMysteryTour:DiscoveringtheTruthandBeautyofthe   Cosmos(NewYork:JohnWiley&Sons,1999),p.v. !  _  8LXXdd8   `     h   wXX  "BetweenMathematicsandMythology:TheHeroicFigureofPythagoras#XLXw#"  wXXL  Abstract#XLXw# @    ThisthesislooksatthewaysinwhichPythagoraswasadmiredintheclassical F  world.WhileheisfrequentlythoughtofasapioneerinMathematicsand 3 Geometry,healsohasareputationofbeingamysticalfigurecapableof   performingmiracles.Asamysticalfigure,Pythagorassharesmanycharacteristic   withtheGreekMythologicalheroes.Somesharedtraitsare:beingtheoffspring  ofagod,havingtheabilitytotraveltotheunderworld.Thegoalofthisthesisis  torecognizedthatPythagorasisauniquefigureandtounderstandhow  Pythagorascouldbeadmiredforbeingbothamathematicianandamystical q _demi_Ԅgod. ^ TheresearchbeginswithPreSocraticfragments.Theseareclosestintimeto 8 Pythagorassassumeddateoforigin,550B.C.E.Thenexttestimonycomesfrom %u theAcademicandPeripateticSchoolswhodiscussthePythagoreansocietyand b PythagoreanPhilosophy.NeoPythagoreansofthe2ndand3rdcenturiesC.E., O especially_Iamblichus_,areanalyzed.SomereferencestoPythagorasin < contemporaryartandliteratureareincludedtofurtherthepointthatPythagoras ) isreveredasaherowhopromotesasystemofunderstandingthatisboth   scientificandreligious. ! .<J  R  A'"%   wXXLAcknowledgments#XLXwO # @  *   FirstIwouldliketothankthestaffoftheClassicalStudiesDepartmentat F  _Brandeis_ԀUniversity.Thankyouforkeepingtheliberalartsalive.Leonard 3 _Muellner_Ԁwasanincrediblethesisadvisor.Hewasalwayssupportiveand   encouraging.Hisgentleguidancehelpedmetoproduceanacademicworkthat   ismeaningfultomyself.Ihopethatothersfinditmeaningfulaswell.Ann  _Koloski_Ԅ_Ostrow_,myundergraduateadvisor,wasaprincipalfactorinmydecision  tomajorinClassics.IwouldalsoliketothankPatriciaJohnstonandCheryl  Walkerforreadingmythesisandofferingcriticism._Stasia_Ԁ_Sutermeister_,the q departmentadministrator,wasanessentialresourceduringmythesisprocessand ^ mytermasUndergraduateRepresentative. K Icouldnothavefinishedthisprojectwithouttheloveandsupportfrommy %u amazinganddedicatedgirlfriend,RachelV._Flamenbaum_.Myparents,Archie b andPatricia_Kutz_,alsogavemelotsofpositivefeedback.Iwouldnotbewriting O thistodaywithoutthem.Lastly,IwouldliketothanktheReferenceLibrarians < ofthe_Brandeis_ԀUniversityLibrariesforprovidingresearchassistanceandflexible ) employmentwhileIcompletedthisthesis.   Josh_Kutz_,  November19,2002!   #!    6   T&!$ wXXL0  #XLXw# XXLTableofContents#XLX ,#wXXL> (#(# -,XX XX-  0  0(#(#Abstract (#(# 0  0(#(#Acknowledgments  (#(# 0  0(#(#Introduction.......................................................0 (#(#Page1  (# (# 0  0(#(#PythagorassConnectiontoMathematics...........0 (#(#6  (# (# 0  0(#(#TheMysticalPythagoras....................................0 (#(#21-  (# (# 0  0(#(#Conclusion.........................................................0 (#(#44B  (# (# 0  0(#(#WorksCitedW(#(# #XLXw#. X,XX X.wXXL  BetweenMathematicsandMythology:TheHeroicFigureofPythagoras    JoshR._Kutz_#XLXw#  wXXL  May3rd,1999#XLXw#    ! 8LXXdXXd8Introduction     Muchcanbelearnedaboutaculturebyexaminingitsheroes.Aheroissomeonewho  appearslargerthanlife,someonewhoembodiesthequalitiesandvirtueswhichhisorher b culturevalues.Aheroisanindividualwhodidwhatothersaroundthemcouldnot,orwouldnot, <  do.Theheroicactionsofthatindividualchangetheworldforever.Themostimportantheroes  f arethosewhoarecreditedwithmakingthegreatestimprovementstosociety.Inthisway,ahero  @  representsaperiodofchangeinthehistoryofhumanity;theworldwasmadeabetterplace   becauseoftheactionsofthatperson.     Throughouthistorymanydifferenttypesofpeoplehavebeenadmiredandtreated ~ heroically.Whenlookingbackwardsintime,thevaluesofaculturewilldeterminewhich X figuresfromthepastarerecalledasheroesandwhichcharacteristicsareparticularly 2 emphasized.OnefigurewhoisseenasaheroformanydifferentreasonsisPythagoras,aGreek  \ ofthesixthcenturybeforethecommonera.Althoughmanythingsarenotknownabout 6 Pythagoras,whereverheisadmireditisforhisintelligenceandhisteachings.Pythagorasis  rememberedasaherobecauseofhisintellect.WhenpeopletodayrefertoPythagoras,heis  mostcommonlythoughtofasagroundbreakingmathematician.Aftersomediggingintothe t  bodyofmaterialthatpraisesPythagoras,additionalreasonsforhisheroicstatusarise.Inthe N"  classicalworld,Pythagorashadthereputationofamiracleworker,amancapableof ($x" performingfeatsofmythicalproportion.Thesetwodifferentabilities,superiorintelligenceand &R!$ supernaturalpower,combinetoformauniqueheroicfigure. ',#&   TheintentofthispaperistodeterminehowPythagorascametobeassociatedwithboth )%( mathematicsandmythology,andtounderstandwhatthefusionoftheseconceptssaysaboutthe +&* waythattheGreekssawtheirworld.Pythagorasrepresentstheconceptthathumans,through  reasonandintelligence,canuncoverthetruenatureoftheuniverse.ToPythagoreansthis  uncoveringisbynomeansatransgressionagainstthedivine;itisthewayforhumanstobecome d moregodlike.Pythagoras,asthesupremehumanintelligence,isclosertothedivineand >  thereforedisplaysgodlikecharacteristics.  h   Beforeproceedingwiththebodyofthispaper,itisnecessarytostatesomeofthe  B  problemsfacingmodernPythagoreanresearch.Intheancientworld,Pythagoraswasaman   aboutwhommorewassaidthanwasknown.Hisperiodofinfluenceisthe6thcenturyB.C.E.,   whenthewrittenrecordwasinitsinfancy.Scholarlyattemptstocreateahistoricalpicturecan  provetheexistenceofthecultofPythagoreansinthe6thcentury,butdetailssurroundingtheman Z himselfarescarceandmostlatestoriesapocryphal.EventhenamePythagorascomesinto 4 question.Somelateauthorsexplainitasacombinationofthewords'Pythia'(_ +;_),the ^ oracleofApolloatDelphi,and_agoreuein_(_s# A _),fromtheGreekverb tospeak.  &  1       8   Pythagoraswassaidtohaveleftnowritings  &  2      ,andnoworkshavesurvivedwhichcanbe  successfullylinkedtohishand.  &  3      ׀Thetextsofhisthatarementionedarelostorspurious.One  sourceofconfusioncomesfromthenumerousOrphictextsthatdatetothe6thcentury. v  Diogenes_Laertius_ԀpointstoafragmentfromIonof_Chios_,amid5thcenturywriter,asproofof P"  Pythagorasswritings(D.L.,Lives,8.8).Ionsaysthat Pythagorasascribedsomewritingsto *$z" Orpheus.  &  4      ׀ThisfragmentshowsthatIonwastryingtofindtheauthorbehindsomeOrphic &T!$ poems,butitdoesnotprovethatheknewoftextsthatweredefinitelywrittenbyPythagoras.  &  5      ׀ '.#& ThePythagoreanandOrphiccultshadsimilaritieswhichwillbediscussedlater.Theywere )%( frequentlyconfusedbyGreeks.Herodotus,whoomitsmentionofPythagorasfrommostofhis +&* history,makesanattempttoclarifythisoverlap.  &  6      ׀    AnotherfragmentusedbyDiogenes_Laertius_Ԁcomesfrom_Heracleitus_Ԁwhosays  _ Pythagoras,sonof_Mnesarchus_,practicedresearchmostofallmen,andmakingextractsfrom d thesetreatises,hecompiledawisdomofhisown,anaccumulationoflearning,aharmfulcraft.  &  7      ׀ >  ThisfragmentmentionstreatisesthatPythagorasexamined,butitdoesnotconclusivelystate  h thattheywerewrittenbyPythagoras,onlythatPythagorasusedotherwritingstoassisthisown  B  efforts.     AttemptstoconnectsayingsandconceptstoPythagorasarefurthermuddledbythe   practiceofthecultmemberstoattributealldiscoveriestothemaster.  &  8      ׀Thiswastheirwayof  payingrespecttotheirfounderanditwasalsotheirwayofexplainingPythagorassamazing Z abilities.Thelatercultmemberswouldnothavebeenabletomakeinquiresintomathematics 4 andphilosophyifPythagorashadnotbeguntheprocess. ^   SincePythagorasleftnowrittenwordsbehind,everythingwhichissaidabouthimhas 8 thequalityofhearsay.TheearliestreferencestoPythagorascanbefoundinpreSocraticwriters  ofthelate6thcenturyand5thcentury.Thesemenareclosestintimetothehistoricalperson  Pythagoras,butfewfragmentssurvivewhichmentionPythagoras,andevenfewerdetailsare v  related._Heracleitus_Ԁof_Ephesus_,whowasinhisprimearound500_BCE_,mentionsPythagoras P"  twice.BothtimesheisexpressingcriticismofPythagorassknowledge._Heracleitus_Ԁdoes *$z" relatethedetailthatPythagoraswasthesonof_Mnesarchus_Ԁ(_Heracleitus_,_Ancilla_,DK22,fr. &T!$ 129)._Heracleitus_ԀandIonaretheonlypreSocraticstomentionPythagorasbyname. '.#&   Thereareotherimportantfindingsfromthisperiodthatscholarsbothancientand )%( modernrelatetoPythagoras,butinthosecasesnopropernamesurviveswithinthefragment,so +&* theconnectiontoPythagorasmustbeextrapolated.  &  9      ׀Nevertheless,becauseofhisinfluenceon  laterGreekthinkers,thePreSocraticfragments,andtheexistenceofthecultinhisname,  Pythagorasmustbethoughtofasanhistoricalpersonofthe6thcentury.  &  10      ׀ d    h PythagorassconnectiontoMathematics     PythagoraswillforeverholdaplaceinthehistoryofGreekthinkers.Themostfamous  discoveryattributedtohimisthetheoremequatingthesidesofarighttriangle.Knownasthe b PythagoreanTheorem,itstates: Inrightangledtrianglesthesquareonthesidesubtendingthe <  rightangleisequaltothesquaresonthesidescontainingtheright_angle.  &  11      _ԀSincethethirdor  f possiblyfourthcenturyB.C.E.,thistheoremhasbornthenameof_Pythagoras.  &  12      _ԀModern  @  scholarshiphasuncoveredevidencethatthistheoremmayhaveoriginatedinEgyptorBabylon,   atdatesmuchearlierthanPythagorassperiodofactivity(Heath,1931,p.967).Historical   accuracyaside,Pythagorassnameremainsattachedtoatheoremwhichistaughtinmost ~ secondaryschoolsandhasapplicationsinmanybranchesofmodernscience,fromgeometryto X EinsteinstheoryofGeneral_Relativity.  &  13      _Ԁ 2   IfPythagorasdidnotdiscoverthistheoremhimself,hemusthavebeenfamiliarwithit  \ andincludeditinhisteachingsingeometryandarithmetic.Thesesubjectsareaspectsofthe 6 philosophythatPythagorasisassociatedwith.OurrecordsofPythagorasshowthathewasnota  mathematicianinthemodernsense,butwasinterestedinusingmathematicsandnumbertheory  toexplainhumanlifeandothergeneralphenomena.Pythagoraswasinterestedinapplying t  mathematicstoabroadrangeofsubjectsthatcannotbeeasilyreplicated.Heismostfamous N"  forexpoundingaphilosophyinwhichnumberswerethebuildingblocksoftheuniverse. ($x"   Greekphilosophyisconcernedwithlearningthetruenatureofreality,theundeniable &R!$ truthsthatliebeyondtherangeofmeresenseperception.Plato,wholivedafterPythagoras, ',#& solvedthispuzzlebypositingtheexistenceofForms:pureandunchangeableprincipleswhich )%( existfarawayfromhumansenses.Theworldthathumansinhabitconsistsofglimmersand +&* reflectionsoftheseformsthatcontroltheuniverse.Onlythroughintensestudyandphilosophy  dohumansperceivethesePlatonicForms,andthenthetruenatureofrealitycanbeperceived.  GreekscommentingonPlatoindicatethathewasheavilyinfluencedbyPythagoreanphilosophy d andmayhaveborrowedsomeimportant_elements.  &  14      _ԀThemajordistinctionbetweenthetwo >  philosophiesisthat,toaPythagorean,numbersarethecontrollingforceoftheuniverse,not  h forms.  B    ThefirstdescriptionofPythagoreanphilosophycomefromthefragmentsof_Philolaus_,a   manwhowasactiveinthelatefifthcentury.Hesays: Actually,everythingthatcanbeknown   hasaNumber;foritisimpossibletograspanythingwiththemindortorecognizeitwithoutthis  _(Number).  &  15      _ԀThePythagoreanwayofunderstandingtheworldwasthatitwasmadeoutof Z numbers,andthosenumbersgovernedhowtheuniversefunctioned. 4   ThefirsttenetofPythagoreanphilosophyisthattheuniversebeginswithunity.Unity ^ thenbreaksdownintoLimited(_peras_,_!7#'_)andUnLimited(_apeiron_,_t! #_) 8 components.Thesecomponentsopposeoneanother.TheLimitedcomponentsymbolizesthe  orderintheuniverse:itintroducesadefiniteboundarywherepreviouslytherewasnothing.The  UnLimitedcomponentissymbolicofthechaosintheuniverse;itcreatesplurality.TheUn v  Limitedcomponentisinfiniteinanegativesense:itcanbedividedaninfinitenumberof P"  _times.  &  16      _ԀTheLimitedandUnLimitedcomponentsrecombinetocreatenumbers.In *$z" Pythagoreanphilosophy,everythingintheworldconsistsofnumber.F.M._Cornford_,inhis &T!$ essayonPythagoreanphilosophy,offersalucidexplanationoftheabstractionbehindthis '.#& cosmogony: Thereis(1)anundifferentiatedunity.(2)Fromthisunitytwooppositepowersare )%( separatedouttoformtheworldorder.(3)Thetwooppositesuniteagaintogenerate_life.  &  17      _ԀTo +&* Pythagoreans,theMonadistheoriginal,undifferentiatedunity.Itunitestheentireworld  becausetherestoftheuniverseisformedfromit.ToPythagoreans,numbers,whichcreateall  things,areformedfromtheMonad. d   TheMonadreferstothenumberone,butitalsoreferstothenumbersfromonetoten. >  Thesetennumbers,collectivelyknownasthe_Decad_,constitutetheunifiedcontinuum  h (_Fideler_,1988,p.21),aspectrumofdifferentattributes(numbers)thatistheessenceofall  B  things.Theideathattheentireuniverseissomehowconnectedtoasingle,unifiedoriginis   necessaryforasuccessfulsystemofthought.Ifeverythingintheuniverseisconnected,thenit   ispossibletounderstandeverythingintheuniverse.Bystudyingaparticularaspectofthe  universe,itispossibletolearnabouttheuniverseasawhole._Archytas_Ԁof_Tarentum_,oneofthe Z earliestGreekstobecalledaPythagorean,offersthisanalysisoftheconnectionbetweenthe 4 universalandtheparticular: ^ 0  Mathematiciansseemtometohaveexcellentdiscernment,anditisinnowaystrange 8 thattheyshouldthinkcorrectlyconcerningthenatureofparticular_existences_.Forsince % theyhavepassedanexcellentjudgementonthenatureoftheWhole,theywereboundto  haveanexcellentviewofseparate_things.  &  18      _(#(# WBycorrectlyunderstandingthewaythattheuniversefunctionsasawhole,itiseasierto  understandeachsmallerpart. c!   ThetenPythagoreannumbersofthe_Decad_Ԁwerethoughttobedifferentfromthe =#! numbersthatappearaspartofeverydaylife.Theyshareattributeswiththenumbersusedto %g # measurequantitiesandconducttrade,butarefundamentallydifferent.InPythagorean &A"% philosophy,thePythagoreannumbersofthe_Decad_Ԁarethemysticalsourceofthenumbers ($' observedthroughouttheuniverse. *%)   TheMonadwasdescribedasbeingbothevenandodd,alsoknownaseven_Ԅodd.  &  19      _ԀThese  arethetwooppositepowerspresentinunitywhichseparateandrecombinetoformtherestof  theworld.TheUnLimitedcomponentispresentinevennumbers;theLimitedcomponentis d presentinoddnumbers.TheMonadisthefirstPythagoreannumber;itrepresentstheunityof >  thecosmosandthatiswhyitisbothLimitedandUnLimited.Thecombinationofthesetwo  h powersresultsinthesubsequentsymbolsaftertheMonad.  B    TheIndefiniteDyadisthenextPythagoreannumberanditisUnLimited.ThisfirstUn   limitednumbersignifiesastepawayfromunity,intothechaoticworldofduality,ofcauseand   effect,subjectandobject.    AftertheDyadcomestheTriad,thesumofthenumbersoneandtwo.BecausetheTriad Z canberepresentedasthecombinationofLimitandtheUnlimited,itsignifiesharmonyand 4 relation.Thenumberthreeisthefirstnumberwhichcanbebrokendownintotwoother ^ numbers,soitsymbolizesthebeginningofmathematicsandtheabilitytoequateonethingto 8 another.ItisthePythagoreanoriginofharmony(_{#;_)(_Fideler_,1988,p.22).    ThefourthPythagoreannumber,theTetrad,hasspecialsignificancebecauseofits  relationtotherestofthe_Decad_.ThemostfamousPythagoreanfigureisthe_Tetraktys_,atriangle v  withfourlevels. P"    Figure1:The_Tetraktys_ *$z" dzBD40r~ `@E[%tt%z z (#(#        +&*   dThistrianglecanbeseenasacombinationofthenumbers1,2,3,and4whichaddedtogether  give10,thePythagorean_Decad_.Thataperfecttriangleisnaturallyformedbytennumberswas v evidencetothePythagoreansthatthe_Decad_Ԁisasignificantquantity.The_tetraktys_Ԁwas P  regardedasasacredfigure.ItwassaidthatPythagoreanswouldnotviolatethenamesofthe * z GodsorofPythagorasinanoathandinsteadsay:  T    Iswearbythediscovererofthe_Tetraktys_, .    Whichisthespringofallourwisdom,     TheperennialrootofNaturesfount.(_Iam_.,VP,29.162)[i   Thefirstfournumbersalsohavesignificancewhenappliedtogeometry.Thenumberone   signifiesapointinspace.Thenumbertwoisaline,whichisdrawnbetweentwopoints. l Numberthreeisatwodimensionalfigure,suchasadrawingofatriangle.Withthenumber F four,therealmofphysicalbodieshasbeenreached:athreedimensionalfigure(apyramid)can  p beconstructedfromfour_points.  &  20      _Ԁ J   Figure2:ProgressionfromOnePointtoaThreeDimensionalPyramid   m/m*UQddd Xdd Xdd X(#(#,dd ,dd ,dd ,dd +    j<K6&"`r!K `_E\66j  a j<]6&"`r!K `_E \66 j  a j<p6&"`r!K `_EP\66Pj  a j<~6&"`r!K `_E\66j a  dXXL   Figure2a  %  Onepoint#XLXdp#. ^&!!   qqdXXL   Figure2b  % " Twopointsformaline.#XLXdq# ^&!#  qqdXXL   Figure2c  % $ Threepointsforma ^&!% triangle.#XLXdr# #'s"&  rrdXXL  Figure2d  % ' Fourpointsform ^&!( athreedimensional #'s") pyramid.#XLXds#"'8#*   "ss  ForthePythagoreans,theUnlimitedcomponentofnumberswassubservienttothe *&, Limited.TheUnlimited,evennumbersarenotinsignificant,buttheyarelessperfectthanthe  Limited,oddnumbers.Anexampleofthisimperfectioncanbeseenwhenthemethodof  placing carpenterssquares,gnomon(_E3_)inGreek,aroundnumbersisreplicated.A d gnomonisarightangleLshapemarkedwithequalunits.Usingdotstosignifynumbers,a >  singledotrepresentsaone,andtwodotsnexttoeachotherrepresentsthenumbertwo.The  h singledothasthesameheightandwidth,andwhenagnomonisplacedadjacenttothissquare,it  B  formsalargersquare.Theaddedgnomonisthreeunitslong.Whenanothergnomonisadded,it   willbefiveunitslongandalargersquarewillbeformed.Eachsuccessivesquarehasthesame   heightandwidthandisanoddnumberofunitslong.     Figure3:TheGnomon _ Method   &  21      _ F CyVy*;UEdddd dd dd dd UQ(#(#,<dd ,<dd +  ,"66 ,r<l\<,(hr1V `EDr 9/!" 9r<H\<,(hr1l `E(`Dr >4&"   >dXXL Figure3a  r Carpenterssquaresplacedaroundanoddnumber. 7 Eachnewgnomonformsasquarewiththesameheight L andwidthratio. 9/! " 9 Figure3b  r Carpenterssquaresplacedaroundanevennumber#XLXd|#. 7 dXXLEachnewgnomonformsarectanglewitha $ t different#XLXdV~#dXXLԀheightandwidthratio.#XLXd~#1'% 9"   1Inthecaseofevennumbers,arectangleisformedbytwodotsplacedsidebyside.Thisobject #  hasaheightofoneandalengthoftwo.Whenagnomonisplacedaroundthisrectangle,alarger % " rectangleisformedwithheighttwoandlengththree.Thenewgnomonisfourunitslong.Each l'"$ additionalgnomonwillbeanevennumber.Theimperfectnatureofevennumbersappears F)$& whencomparingeachnewrectangle.ThefirstrectangleformedfromtheDyadhasaheightto  +p&( widthrationof1:2.Thenextrectanglehasaratioof2:3.Addinganothergnomonmakesa  rectanglewithproportions3:4andsoon.Eachnewrectanglehasadifferentshapeand  proportion,versusasquarewhichmaintainsthesameproportionasitgrows. d   ThecontrastbetweenevenandoddcarriedintomanysubjectsinPythagorean >  philosophy.ThebestexampleofthedifferencesappearsinAristotlesMetaphysicswherehe  h includesthePythagoreantableof_opposites  &  22      :_  B   /(83"T" XX  0  T ݀2  1  )3  0` (#(#  Limited0` (#` (#UnlimitedT v݌ (#(# Ќ XX  "T" XX  0  T`݀2  2  )3  0` (#(#  Odd0 ` (#` (#0 (# (#EvenT`݌ (#(# Ќ XX  "T" XX  0  T݀2  3  )3  0` (#(#  One0 ` (#` (#0 (# (#PluralityT݌ (#(# Ќ XX  "T" XX  0  T݀2  4  )3  0` (#(#  Right0 ` (#` (#0 (# (#LeftTR݌ (#(# Ќ XX  "T" XX  0  TH݀2  5  )3  0` (#(#  Male0 ` (#` (#0 (# (#FemaleTH݌(#(# Ќ XX  "T" XX  0  T݀2  6  )3  0` (#(#  Atrest0 ` (#` (#0 (# (#InmotionT݌m(#(# Ќ XX  "T" XX  0  T݀2  7  )3  0` (#(#  Straight0` (#` (#CrookedTA݌Z(#(# Ќ XX  "T" XX  0  T)݀2  8  )3  0` (#(#  Light0 ` (#` (#0 (# (#DarknessT)݌G(#(# Ќ XX  "T" XX  0  Tz݀2  9  )3  0` (#(#  Good0 ` (#` (#0 (# (#EvilTzя݌4(#(# Ќ XX  "T " XX  0  TƐ݀2  10  )3  0` (#(#  Square0 ` (#` (#0 (# (#OblongXTƐ݌!q(#(# Ќ XX  Thischartconsistsoftensetsofopposites,whichisareferencetothe_Decad_Ԁastheultimate K sourceofeverything.ThistabledemonstrateshowtheprincipleoftheLimitedandUnlimited % canapplytomanysubjects.ThePythagoreansbelievedthatnumbersdictatedwhatwasgoodor  evil,straightorcrooked.Pythagoreanphilosophyappliednumerologytomanydifferentaspects  oftheworld. c!   Becausetheuniverseisbuiltoutofnumbers,eachnumberhasaspecialpropertyand =#! attributethatitdisplayswhenitispartofsomething.Asdescribedabove,theMonadrepresents %g # unity,theDyadrepresentsduality,andtheTriadrepresentsharmonyandrelation.TheTetrad,in &A"% itscapacityasthe_Tetraktys_,standsforfoundationandstability.As_Iamblichus_Ԁsays,itisthe ($' fount,thesourceofallnumbers.ItisthereforethefoundationforallPythagoreannumerology *%) (_Iam_.,Theology,23).    ThePentadwascalledmarriage,sinceitconsistsofthefirstevennumber,two,  combinedwiththefirstfullyoddnumber,three.Fromthetableofopposites,malebelongsto d theoddcolumnandfemalebelongstotheevencolumn.ThePentadcanbeformedbycombing >  maleandfemalenumbers.TheHexadwasalsocalledmarriagebecauseitcanbeformedbythe  h multiplicationofthenumberstwoandthree.TheHeptadisaprimenumber;ithasnofactors  B  otherthanoneanditself.Becauseofthisitwaslabeledavirgin,notbornofanymotheror   father.ItwasassociatedwithAthena,agoddesswhowasbornfromZeusshead,notthe   productofsexualintercourse(Ibid,71).TheOctadisdescribedasthefirstactualcube,because  itistheproductof2x2x2.TheEnneadwaslabeledthehorizonbecauseitisthelastnumber Z withinthe_Decad_.The_Decad_,whichcontainsallthePythagoreannumbers,isknownas 4 wholenessandastheuniverse(Ibid,180).Thisisanabridgedlistoftheattributesofthe ^ Pythagoreannumbers.AsPythagoreanphilosophydevelopedovertime,numerousassociations 8 andattributesweregiventothesetennumbers.    Pythagoreanphilosophyisbasedonthebeliefthatnumbersandmathematicsare  fundamentalaspectsoftheuniverse.Apowerfulexampleoftheimportanceofnumbersisthe v  discovery,attributedto_Pythagoras,  &  23      _Ԁoftheuseofwholenumbersinthemusicalintervals. P"  AncientGreeksknewthattherelationshipsbetweenpleasantsoundingnotesonthemusical *$z" scalecouldbeexpressednumerically.Onastringedinstrument,ifonestringistwiceaslongas &T!$ anotherstring,thenotesplayedwhenthetwostringsaresimultaneouslypluckedwillbeexactly '.#& oneoctaveapart.Theshorterstringwillproducethehighernote.Ifthetwostringshavethe )%( ratioof2:3,whenplayedtogetherthenoteswillbetheperfectfifth,themostpowerfulmusical +&* relationship.Ifthestringlengthshavetheratio3:4,thecorrespondingnoteswillbetheperfect  fourth.Theseratios,alongwithmorecomplicatedwholenumberratios,canbeusedto  constructtheentiremusicalscale(_Fideler_,1988,p.25).Themusicalscaleisanaturallyexisting d phenomenon;certainfrequenciesofsound,bytheirnature,arepleasingtothehumanear.The >  factthattherelationshipbetweenpleasingnotescanbeexpressedbyratiosofonewholenumber  h toanotherofferseasilyreplicatedevidencethatnumbersareasignificantpartoftheworld.As  B  onescholarputsit: Ifmusicalsoundscanbereducedtonumbers,whynoteverything_else?  &  24      _     Pythagorashimselfmusthaveusedmathematicsinhisteachings.Arithmetic,geometry,   musictheory,andastronomyarefeaturedheavilyassubjectsthatmembersofhiscult_studied.  &  25      _Ԁ  ThecultthatarosearoundthefigureofPythagoraswasinstrumentalinpreservinghisteachings Z anddiscoveries.Italsomakesthepicturecomplicatedbecausetherearenowrittenrecordsfor 4 theearliestyearsofthecult.Anydiscoveriesmadebycultmemberswerekeptanonymousor ^ attributedtoPythagoras, themaster. 8   ThePythagoreancultscenterofactivitywasCroton,aGreekcolonyinItalytowhich  PythagorasisthoughttohavemigratedfromhisnativeislandofSamos.Thedateofhis  migrationcoincideswiththedominationof_Polycrates_ԀwhowastyrantofSamosandmasterof v  theseaintheyear533B.C._E.  &  26      _ԀThestoryofPythagorassarrivalinCrotonappearsintheneo P"  Pythagoreanbiographies.Inthisstory,Pythagorasconvincesthetownofhissuccessasa *$z" philosopherandgiveslecturesonmoralconducttodifferentsocialgroups:youthsinthe &T!$ gymnasium,youngmen,thetownwomen,andtheadultsinthesenate(_Iam_.,VP,3557).The '.#& convertsfromtheselecturesjoinPythagorasandbeginacommunalwayoflifethatincludes )%( studyingphilosophy. +&*   Thespecificsofthisstorymaybeexaggerated,buttheunderlyingthemeisthoughttobe  correct.APythagoreansocietydidexistinMagna_Graecia_,thelandofSouthernItalyandSicily  thatwascolonizedbyGreeks.ThesocietyhadpoliticaltiestothetownsofCrotonand d _Metapontum.  &  27      _ԀOthertownsthatappearwithreferencetocultactivitiesare:_Tarentum_,_Sybaris_, >  _Caulonia_,and_Locri_.AfterthedeathofPythagorasthecultbecamethevehicleforthe  h transmissionofhisteachingsandphilosophy.Manyofthecultpracticeswerekeptsecretor  B  transmittedasencodedsymbols,andthepenaltyforrevealingthemwasexcommunicationand   divineretribution(_Iam_.,VP,88).Fortheearlyyearsoftheculttherearenowrittenrecords;the   teachingwasdoneorally.NearthetimeofPythagorassdeath,therewaspoliticalupheavalin  Italy,anditappearsthatthePythagoreansocietywasdispersed.Twodifferentversionsofthis Z endingarerecordedbywritersofthePeripateticschool._Aristoxenus_ԀsaysthatPythagoras 4 withdrewto_Metapontum_Ԁtoavoidtheupheavalanddiedpeacefully._Diceaearchus_Ԁsays ^ PythagoraswasinCrotonatthetimeofthepoliticaltrouble.Thesediscrepanciesarisewhen 8 yearsoforaltraditionprecedewrittendocumentation(_Burkert_,1972,p.117).Thecontinuing  existenceofthiscultaffectsPythagoreanresearch;itisimpossibletoseparatethehistorical  Pythagorasfromthelegendstoldbymembersofhiscult. v    MostdepictionsofthePythagoreancultincludegradationsinthelevelofinitiation. P"  _Iamblichus_ԀandotherneoPythagoreansdescribetwomajordivisions:the_acusmatici_ *$z" (_s+%);_,thehearers)andthe_mathematici_(_);_,thelearners).Sometimes &T!$ apoliticaldivisionisalsomentioned.Accordingto_Iamblichus_,the_acusmatici_werethe '.#& exotericdiscipleswholistenedtolecturesthatPythagorasgaveoutloudfrombehindaveil.The )%( _acusmatici_werenotallowedtoseePythagorasandtheywerenottaughttheinnersecretsofthe +&* cult.Insteadtheyweretaughtlawsofbehaviorandmoralityintheformofcryptic,briefsayings  thathadhiddenmeanings.Thesemaximsareknownas_acusmata_andalso_symbola_.Theywere  firsttransmittedorallyandcanbedatedbacktoabout400B.C.E.(_Burkert_,1972,p.166). d _Iamblichus_Ԁincludessayingssuchas: Donothelptounloadaburden(becauseitiswrongto >  encouragelackofeffort)buthelptoloaditup.; Pouralibationtothegodsoverthehandleof  h thecup,asanomen,andsothatnoonedrinksfromthesameplace.Othersayingsarenot  B  accompaniedbysuchexplicitinterpretations.Forexample: Onemustputtherightshoeon   first.; Donotspeakwithoutalight.; Oneshouldmakesacrifice,andgotoholyplaces,   barefoot(_Iam_.,VP,18.8283).Byfollowingtheinstructionsinthesemaxims,the_acusmatici_  wouldreplicatetheasceticlifestylethatPythagorasintroducedandtheoreticallywouldimprove Z thequalityoftheirlives. 4   The_mathematici_weretheesotericmembersofthecultwhostudiedtheteachingsof ^ Pythagoras.Theywentthroughrigorousinitiationsandahighlystructurededucationalprocess 8 includingyearsofstudyunderavowofsilence.Ifastudentwasnotabletomaintainhis/her  disciplineovertheyears,he/shewasrejectedandregardedasdead(Ibid,l17).Asthename  implies,theywereconcernedwiththemathematicsandnumerologyofPythagorassteachings. v  TheystudiedthePythagoreanquadriviumoffourmathematicalsubjects:arithmetic,geometry, P"  astronomy,andmusic. *$z"   Itisuncertainwhetherthesedivisionswereformalsectsandnottheinventionoflater &T!$ authors.Whatisclearisthatinthe5thand4thcenturysharpcontrastswereseenamongdifferent '.#& groupsofpeoplecallingthemselvesPythagoreans.Onetheonehand,theterm_Pythagorists_ )%( wasusedforwanderingasceticsandunintelligiblemysticsinplaysofOldandMiddle +&* _Comedy.  &  28      _ԀThesecharactersareportrayedaspoor,dirty,unshodvegetarianswhoexpounded  hypocriticalbeliefs.Dramaisbynomeanshistoricalevidence,butinorderforthestereotypes  tohaveahumourouseffectPythagoreancultmembersmusthaveexistedwhocouldbethought d ofthisway(_Burkert_,1972,p.200). >    Ontheotherhand,therewereintellectualswhocalledthemselvesPythagoreansand  h pursuedmathematicsandphilosophy._Archytas_Ԁwasawellrespectedpoliticianin_Tarentum  &  29      _  B  _duringtheearlyfourthcenturywhoengagedinPythagoreanstudies.Aristoxenuswasascholar   inAristotlesPeripateticschoolandhepursuedmusicaltheoryasaPythagorean(Ibid,p.198).   TheseareonlyafewofthemanyGreekthinkerswhosawPythagorasastheirintellectual  predecessor. Z   Somewhereinitsdevelopment,Pythagoreanismgrewintotwodifferentstyles:the 4 mysticalandthemathematical.EvidenceofthissplithasbeensuggestedbythescholarF.M. ^ Cornford.ByexaminingcontemporarywritershededucesthatPythagoreannumerologywas 8 understoodtwodifferentways.Initiallythephilosophyofnumbersmeantthateverythingwas  madebynumbers,anddifferentnumbershaddifferentattributes.LaterGreekthinkersthen  modifiedthisintothetheorythatnumbersmeasurethequantityofamaterialthing.Aristotle v  noticesthesedifferencesinPythagoreannumbertheoryandasksthequestioninhis P"  Metaphysics: *$z" 0  Ithasyettobeexplained[bythePythagoreans]hownumbersarethecausesof &T!$ substancesandofbeing:whether(1)asboundaries,aspointsareofspatialmagnitudes, &A"% asEurytusdeterminedthenumberofeachlivingthing(e.g.manorhorse)bycounting '.#& thenumberofpebblesheusedintracingitsoutline;...or(2)becauseharmony,man, ($' andeverythingelseisaratioofnumbers.(Aristotle,Metaphysics,BookN,V1092b8.))%((#(# Cornfordoffershisexplanation: +&* 0  IbelievethatthissecondviewistheoriginalPythagoreandoctrine,accordingtowhich  thingsembodyorrepresentnumbers,notarenumbers;andthesoul,astheessential  reality,isaratioorharmony,notamerecollectionofmonads.Theotheristhecrude  materialisticviewofNumberatomismthatthingsarenumbers,andnumbersconsistof w monads.d(#(# q0   (#(# 0  Itwasatthispoint,Ibelieve,thatthetwoschoolsofPythagoreans!theoriginalsixth >  centurymysticsandthefifthcenturymathematicians!partedcompany.Theytookvery + { differentviewsofthenatureoftheMonad,andconsequentlyofthegenerationof  h numbersandthings.(Cornford,1923,p.11,5) U (#(# CornfordsanalysisshedslightonthedivergentsectsofPythagoreanism.Theoriginal /  philosophyofnumberswasbothmysticalandmathematical.Itintroducedtheideathatnumbers   arefundamentalprinciplesintheuniverse.Arithmeticproofs,geometricalfigures,andmusic   theorywereallexamplesoftheimportantpositionthatnumbershaveinunderstandingtheway m theworldworks. G   Overtime,thenatureofGreekmathematicalinquirychanged,asGreeksfromIoniaput !q forthmorescientificviewsoftheworldbasedonexperimentandobservation.Atomism,the K beliefthattheuniverseismadeupofaninfinitenumberofunbreakableparticles,begantotake % hold(Burnet,1963,p.10,26,336).ButGreekscontinuedtolookbackonPythagorasasa  pioneerinmathematicsandgeometry.Proclus,oneofthelastGreekphilosophicalwriters,  wrotehisCommentaryonEuclidinthefifthcenturyC.E.asareviewofthedevelopmentof c! Greekmathematics.Pythagorasappearsearlyinhissequenceofimportantfigures.Hewrote: =#!  Pythagorastransformedthestudyofgeometryintotheformofaliberaleducation,examining %g # theprinciplesofthesciencefromthebeginning.  &  30      ׀Pythagorasisseeninaheroiclightas &A"% someonewhospreadinterestinnumbersandmathematicstotheGreeksofhistime.Heis ($' rememberedasanimportantinfluenceonthedevelopmentofGreekscience.  *%)  TheMysticalPythagoras     ThemathematicsandtheoriesofnumberthatPythagorasisassociatedwithhave  implicationsthatreachfarbeyondtherealmsofscienceandlogic.ThePythagoreanphilosophy b thatcanbetracedtoPythagorasstimepromotedamysticalknowledgewherebythetruly <  initiatedwouldbeincompleteharmonywiththeinnerworkingsoftheuniverse.Intheyears  f afterPythagorassdeath,therewerelegendsincirculationthatdescribedhimasamiracle  @  worker.TheselegendsdepictPythagoraswithsupernaturalabilities.Theydescribea   Pythagoraswhoismorethanmortal.Intheselegends,heiselevatedtothestatusofademigod,   abeingwhoisparthuman,partdivine. ~   Thesestoriesmayhaveoriginatedwithinthecultandbeenpasseddownorally. X Eventuallytheybecamepubliclyknown.Aristotlecollectedthesestoriesandpublishedthemin 2 thefirstoftwomonographsconcerningthePythagoreans.  &  31      ׀Thetwomonographswerelater  \ combinedintoone.Thetitlesoftheoriginaltwomonographscannotbespecificallystated,but 6 collectivelytheworkwasreferredtoasOnthePythagoreans(Ibid,p.186).Thiscollectionof  legends,gatheredsome200yearsafterthedeathofPythagoras,playedamajorroleinshaping  thesubsequenttradition.AristotlesworkisfrequentlyreferredtoinmostPythagorean t  biographies.Themonographisnolongerextant,butthelegendssurvivebecausetheywereso N"  frequentlyquoted.ThefollowingisalistofsomeofthelegendarydetailsfromAristotleslost ($x" monographwhichbecameimportanttolaterNeoPythagoreanauthors: &R!$ FragmentI  &  32      : ',#& 0  Pythagoraspredictedthatanapproachingshipwouldcarryadeadbody.(Apollonius. )%( HistoriaMirabilium6.)*%)(#(# 0  HepredictedthatashebearwouldappearinCaulonia.(Apoll.6.)+&*(#(# 0  InTuscanyPythagorasbitaserpenttodeath.(Apoll.6.)(#(# 0  HeforetoldofpoliticalstrifeagainstthePythagoreans;thenhesecretlywentto  Metapontum.(Apoll.6.)(#(# 0  HeaddressedtheriverCosasanditrepliedHailPythagoras(Apoll.6.)(Aelian.Varia w Historia,2.26.)d(#(# 0  HeappearedinbothCrotonandMetapontumonthesamedayinthesamehour(Apoll. Q  6.)> (#(# 0  Hedisplayedhisgoldenthighwhilesittinginthetheater(Apoll.6.)(Aelian.2.26.)+ {(#(# 0  HewascalledtheHyperboreanApollobypeopleinCroton(Aelian.2.26.) h(#(# FragmentII(Ibid,p.136):  B  0  ThefollowingdivisionwaspreservedbythePythagoreansasoneoftheirgreatestsecrets   !thattherearethreekindsofrationallivingcreatures!gods,men,andbeingslike   Pythagoras.(Iam.,VP,6.31.) (#(#   ThismaterialbecomesveryimportantwhenPythagoreanandPlatonicphilosophyexperiencea  riseinpopularityaftertheyhadgoneoutofstyleforhundredsofyears. Y   Theearliesteffortstoreintroducethemysticalaspectsofthesephilosophiesarethe 3 productofaRomangovernornamedNigidiusFiguluswholivedfrom98to45B.C.E.Inthe  ] firstcenturyC.E.ApolloniusofTyana,whowroteabiographyofPythagoras,claimedtobea 7 reincarnationofPythagorasandlivedasanasceticmystic.NicomachusofGerasa,a  mathematicianactive140150C.E.,wroteaboutthemysticalPythagoreannumbersinhisbook  TheologyofArithmeticandaboutPythagorasinhisLifeofPythagoras.Themajorityofthe u  informationaboutPythagorasthathassurvivedtothisdayandinfluencedsubsequent_people O"  comesfromthistime,whichisknownastheNeoPythagoreanand_Neoplatonic_Ԁperiod.The )$y" largestamountofmaterialisfoundintheworkofPorphyry(c.230c.305C.E.)andhisstudent &S!$ _Iamblichus_Ԁ(c.240c.325C.E.)(_Fideler_,1987,p.4042.). '-#&   ThematerialthatisfoundintheNeoPythagoreanauthorshascomeunderscholarly )%( scrutiny.Thetrendthatthismaterialfollowsisindeedstrangetoobserve:asthelengthoftime +&* increasessincePythagoraswasactive,authorshavemoretosayabouthisphilosophyandmore  detailsofhislifetorelate._Eduard_Ԁ_Zeller_,writinginthelate19thcentury,feelsthatthe   miraculoustalesandimprobablecombinations(_Zeller_,1881,p.310)foundinlate d Pythagoreanhistoryarenothistorical.Hesaysthatthedetailsthatarenotsupportedbyother >  testimonyhavebeeninsertedbyNeoPythagoreansandarebasedon dogmaticpresuppositions,  h partyinterests,uncertainlegends,arbitraryinventions,orfalsifiedwritings(Ibid)._Zeller_s  B  solutionistodisregardanysuspiciouslatesource.Butmorerecentscholarshiphasarguedfor   thevalidityoftheselaterworks.Walter_Burkert_ԀbeginshisextensivebookLoreandSciencein   Ancient_Pythagoreanism_withanexplanationofhowPythagoreanscholarshiphaschangedsince  _Zeller_stime.HepointstoscholarlyevidencewhichsuggeststhatNeoPythagoreansofthe3rd Z and4thcenturyC.E.used4thcenturyB.C.E.sourcesintheir_biographies.  &  33      _ԀHealsoseesvaluein 4 whatlaterauthorshavetosayaboutPythagoras.Hewrites: ^ 0  Thoughmanysourcesmaybelateandnotveryreliable,moremustliebehindthemall 8 thanasimplezero. _Pythagoreanism_ԀwithoutPythagoras,withoutchronological % positionoraplaceinthehistoryofthought,isnotonlyunsatisfyingtothescholar,but  impossibleinitself.A_minimalism_Ԁthateliminateseveryaspectoftraditionwhichseems  _inanyrespectquestionablecannothelpgivingafalsepicture.(Ibid,p.10)(#(# WiththeknowledgethatlaterNeoPythagoreansusedfourthcenturysources,Burkertisableto v  uselatertestimonytoshedlightontheearlierpicture.Thereisstilltheproblemofmistaking P"  newinterpretationforanauthenticoldersource: *$z" 0  Justasacitywhichwascontinuouslyinhabitedoveraperiodoftime,bychanging &T!$ populations,presentstothearchaeologicalinvestigatorfarmorecomplicatedproblems &A"% thanasitedestroyedbyasinglecatastropheandthenabandoned,thespecialdifficultyin '.#& thestudyofPythaogreansimcomesfromthefactthatitwasneversodeadas,for ($' example,thesystemofAnaxagorasoreventhatofParmenides.Whentheirsystemshad )%( beensupersededandlostallbuttheirphilologicalandhistoricalinterest,therestill *%) seemedtobeinthespellofPythagorasnameaninvitationtofurtheradaption, +&* reinterpretation,andextension.Andatthesourceofthiscontinuouslychangingstream  laynotabook,anauthoritativetextwhichmightbereconstructedandinterpreted,nor  authenticatedactsofahistoricalpersonwhichmightbeputdownashistoricalfacts.  Thereisless,andthereismore:a name,whichsomehowrespondstothepersistent w humanlongingforsomethingwhichwillservetocombinethehypnoticspellofthe d religiouswiththecertaintyofexactknowledge!anidealwhichappeals,ineverchanging Q  forms,toeachsuccessivegeneration.(Ibid)> (#(# vBecausePythagorasissuchanimportantsymboloftheconnectionbetweenscienceandreligion,  h itisstillvaluabletolookatwhatotherpeoplehavetosayabouthim,eveniftheyinclude  B  materialthatcannotbelinkedtooriginalsources.The newmaterialshouldnotbetakenas   historicalfact,butitshouldbeanalyzedtoexplainhowPythagoraswasunderstoodby   subsequentadmirers.    Iamblichus,inthe3rdand4thcenturyC.E.,wasapproachingPythagoreanismfromavery Z differentpositionthan5thand4thcenturyB.C.E.followers.Hebelievedintheconceptof 4 theurgy(theiaergaortheonerga), divineworks.  &  34      ׀Bydoingdivineworksuchasprayingand ^ performingrituals,mortalscouldgainassistancefromthegods.Iamblichusbelievedthat 8 philosophywasatoolforspiritualenlightenmentsentfromheaven.HisbeliefintheGreekgods  standsinoppositiontoChristiandoctrine,whichwasbecomingthedominantreligionduringhis  lifetime.  &  35      ׀HealsostandsinoppositiontotrendsinPlatonicphilosophywhichsoughtto v  minimizetheimportanceofthegods. P"  0  IamblichusbelievedthattheworlddescribedbyPlatointheTimaeuswasbeingtorn *$z" apartbyanewkindofPlatonismthatdeniedthesanctityoftheworldandelevatedthe %g # humanmindbeyonditsnaturallimits.AccordingtoIamblichussuchrationalistichubris &T!$ threatenedtoseparatemanfromtheactivityofthegods.  &  36      &A"%(#(# ForIamblichus,Pythagoraswasanexampleoftheperfectlife.Herewasafigurefromapast ($' ageforwhomtherelationshipbetweengodsandmortalswasbetterthanitwasinIamblichuss *%) time.IamblichusportraysPythagorasasamanofincrediblewisdom.Partoftheproofthat  IamblichusgivesofPythagorasstalentsarehisreverenceforthegodsandhisdesiretoteach  otherstoworshipthemproperly.InPythagorasIamblichushadtheperfectexampleofhow d wisdomshouldbeused:tostrengthentherelationshipbetweenhumansandgods,nottodissolve >  it.  h   IamblichusssurvivingtexthasthetitleOnthePythagoreanLife(  #)  B   +#;+)andwasthefirstpartofhisoveralldescriptionofPythagorean   philosophy.Theexacttitleisuncertain,butscholarsrefertotheentireworkas On   PythagoreanismbecauseitcoveredthebiographyofPythagorasaswellasthemathematicsand  philosophyofthecult.  &  37      ׀OnthePythagoreanLifeismeantasthefirststeppingstoneonthe Z pathtocomprehensionofPythagoreanphilosophy.Iamblichusbegins: 4 0  Allrightmindedpeople,embarkingonanystudyofphilosophy,invokeagod.Thisis ^ especiallyfittingforthephilosophywhichtakesitsnamefromthedivinePythagoras(a K titlewelldeserved)sinceitwasoriginallyhandeddownfromthegodsandcanbe 8 understoodonlywiththegodshelp....Andafterthegodsweshalltakeasourguidethe % founderandfatherofthedivinephilosophy.(Iam.,VP,1.12)(#(#  AsIamblichussaysinthefirstline,itiscustomarytoinvokeagod.Then,aftermentioningthe  gods,IamblichuspraisesPythagorasasthedivinefounderofthisamazingsystemofthought.In v  Iamblichussmind,Pythagoreanphilosophycametohumansfromtherealmofthegodsvia P"  Pythagoras.IamblichuscastsPythagorasinaspecialpositionintherelationshipbetweengod *$z" andmanbecauseofhispietyandthephilosophythatheintroducedtotheGreekcivilization. &T!$ Becauseofthisspecialposition,Iamblichusfeelsitnecessaryto invokePythagorasatthevery '.#& outset.Iamblichusdoessoonlyafterinvokingthegods,reinforcingthestandardhierarchy )%( betweendivinities:demigodsareworshipedafterthegods.Thishierarchywasimportantto_the +&* Pythagoreans.Laterinthetext_Iamblichus_ԀincludesaquotationfromAristotlesmonograph:   thePythagoreansmakeadistinctionasfollows,guardingitamongtheirmostsecretteachings:  amongrationalbeingstherearegods,andhumans,andbeingslikePythagoras.(Ibid,6.31)This d fragmentfromthe4thcenturydisplaysthereasoningbehindtheheroworshipofPythagoras.He >  wasacceptedasbeingmoreintelligentthatanyotherperson,sohisintelligencemustbedueto  h powersbeyondthehumanscope.Pythagorasisseenasbelongingtoacategoryofbeings  B  superiortoregularhumansbutinferiortotheeternalgods.Thiscategoryisknownas_demi_Ԅgod,   a halfgod.     InthetraditionalGreekmythologyof_Hesiod_ԀandHomertherearestoriesaboutmenwho  displayedextraordinarytalentsandperformedlegendaryfeats.Throughtheirstrengthand Z bravery,thesemenperformedtaskswhichgenerallyimprovedthequalityoflifeforthecommon 4 peopleinvolved.Mostoftheseheroeshaddivineorigins;theyweretheproductofaunion ^ betweenagodandamortal.Onceamanhadprovenhisdivineoriginandachievedfameby 8 completingtasks,hebecameahero,anobjectofworshipforthepeoplewhoseliveshehas  affected.Thereisastrongoppositionbetweentheworshippaidtoagodandthatpaidtoahero.  Thegodsresideinthesky,inOlympus,andareworshipedintemples.Heroesresideunderthe v  earth(_chthonioi_)andareworshipedatgravesites(_Burkert_,1985,p.199).Oftenheroeswere P"  closelyassociatedwiththeregionofGreeceinwhichtheyperformedtheirfeats,butother *$z" heroeswereworshipedinmanyGreekcitystates.Therearetalesofmanyheroesthroughout &T!$ AncientGreece,butthefigureofHeraclesstandsoutasbeingthemostuniversallyworshiped. '.#& HeistheprototypicalGreekhero.Heracles(HerculesinLatin)wasrenownedforkilling )%( monsters,travelingtotheunderworld,andperforminglegendaryfeatsthattookplaceinmany +&* differentpartsoftheancientworld.    InthebiographiesofPythagorasslife,therearemanydetailswhichcomparetothe  worshipofthemythologicalheroes.Commontoallheroesisthenotionofdivineorigin.Often d itisZeus,therulerofthegods,whodisguiseshimselfinordertomatewithamortalwoman. >  Thechildisthenraisedbyitsmortalparentorparents.HeracleswasconceivedwhenZeus,in  h theformofthemortal_Amphitryon_,sleptwith_Amphitryon_swife,_Alcmene_.Perseusisthe  B  offspringofthewoman_Dana_ԀwhowasimpregnatedbyZeuswhenhetooktheformofashower   ofgoldandcametoherinprison.Achilles,theepicherooftheGreeksintheIliad,wasthe   childofamortalmanandthegoddessThetis.  &  38      ׀    SomeoftheancientGreeksunderstoodPythagorasasbeingsimilartothesemythical Z _demi_Ԅgods.OnesimilarityisthenotionthatPythagoraswastheoffspringofagod._Iamblichus_ 4 beginsPythagorassgenealogybysayingthathisparentswere_Mnesarchos_Ԁand_Pythais_,two ^ mortalswhowererelatedto_Ankaios_,thefounderofSamos.Hethendescribesth