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* CALVIN C. CIAWSO}

Ο Ταξιδευτής των μαθηματικών

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Page 1: Ο Ταξιδευτής των μαθηματικών

* CALVIN C. CIAWSO}

Page 2: Ο Ταξιδευτής των μαθηματικών

CALVIN C. CLAWSON

O TA=IAEYTHET O N M A O H M A T I K O N

H e[epe0vrlon rnq evtunootorrlgtotop(og turv opt0ptitv

K E A P O I

Page 3: Ο Ταξιδευτής των μαθηματικών

llotrtrd drogo Bo{oqoov yro vo o}ortrr1pooe( our6 to prpl(o. Euloptottit

r6ro(repo touq gltrouq pou oro epyootrlpro: Mop('Evrouopvtq, Mnpouq

Tdrlop, A(vto l6nopvr, Oiltq Adtrrnept rot Mnpdtov Xfppnept, Yto rrlv

unopovrl U€ rrlv ono(o 6rdBooov to Xetp6ypoqo Kot Yto rlq notrtr€q rt

e(otpetrr€q npotdoetq touq. Oo ri0etro en(or1q vo euXoptottloo tous Ko-

Orlyqtdg pou oro lloventot{pto tqg [to0ta, tdoo oro Tgqpo Ortrooo-

rpiog 6oo Kot oro TUnUo MoOrlpottrtirv, Ylo rrl gpovrOo Kqt rrlv oqo-

o(rrrorl pe u1v ono(o pou 66o(ov vo oyontir to go0rlporrxd'

Page 4: Ο Ταξιδευτής των μαθηματικών

EIEAIOTH

KEOAAAIO 1

KEOAAAIO 2

KEOAMIO 3

KEOAAAIO 4

KEOAAAIO 5

KEOAAAIO 6

KEOAAAIO 7

KEOAAAIO 8

KEOAAAIO 9

t l6qpre tpdge; . . . .

ApXrr€g popq6q oP(OPr1oqq.

H op(0gr1oq oe dtrtro e(6q .

Or opr0po( oqv oPXor6rlro .

Krv€(ror optOpo( xot optOpo( rou N6ou K6opou

flpoptrripoto orov floPdbetoo . .

Or opvqttro( oPt0go(.

Awrpeton(ovroq ro dnetPo

H toprj rou Ntdvrtrtvt: dpprlrot optOpo(

t3

19

37

63

8 1

1 1 9

143

177

195

229

Page 5: Ο Ταξιδευτής των μαθηματικών

KEOMAIO 10 H roropio tou n: unepponro( oprOpo( . . . . . .

KEOMAIO 11 Enerrefvovroq to BooiAero: gryo6rroi oprOpo(

KEOAAAIO 12 flpoyponrd geydtror:uTrrpn€n€poopdvor oprOgo(. . . . . .

KEOAAAIO 13 Or rbrorpue(g opr0popvf povss . 323

KEOMAIO 14 Tr oqpo(vouv 6 lo ou td ; . . . . . . . . . 343

KEOMAIO 15 Apr0go(: nopetrOdv, nopdv ror p€Itrov 365

l-luroodpro. 389

Btp t r roypog(o . . . . . . . . . 3sZ

Eupenjpro. 401

257

289

Page 6: Ο Ταξιδευτής των μαθηματικών

EroaYtlYrl

To prpl(o our6 eXet urg O€po roug optOpotig' Aev np6rttrot dptrtg yto

€vo ptBtr(o nou ooXotre(rot p6vo pte ro tuntrd po0qportrd' tlop' 6tro

nou e(vot otrqOera 6rt to po0qpottrd E(vot q petrdrrl tr'r-rv opt0gtiv' q

Xprlorl rurv opr0pdrv E(vot 16oo ouvurgoopdvq pa qv (6to rqv ov0ptlnt-

vq epnerp(o, tlore ro vo e[erd(et rove(g toug opr0potiq on' 6treq ttg

onrtrdg ytlv(eg efvot oov vo 6teptuvd notrtrd on6 ourd roOouto ro UU-

oqpto rqg ovOptirntvrlg q6orlq' Evtl Ero0ro ro BrBI(o €Xet urg rUpro 06-

po ro poOqportro, ovorp€perot eu(oqg orqv-ov0ptlnotroy(o' rr1 ptotro-

y(o, rqv rlruXoAoy(o, tqv ovorog(o' rqv torop(o rot rq rpttrooolll-^,^,,.

'EXouv onorotrfoet toug ovOpdrnoug ont0qrouq nou Xprlotporotouv

epyotre(or, (nt0rlKouq nou-Xprlotponorofv tq tpuxtd' Kot (o;lilofvrtq

nr0qroug,. oo qrov opt'q ;qioo' ourot6 ^ov

poq onoxotrofoov ontOq-

Kouq TIou rdvouv unoloytopoOg>' Ot opropo( *::.ot '":]:]l:i:::t:

nepvoriv rov noltrtop6 poq' IlpoonoOrlore' ov 06trtte' vo qovroor€tr€

rqv rotvov(o ptoq Xurp(g'opiOuo':q q unotroytopotjC :(Uoo:.:::::::

poo;-r€vot oroug opt0godg' tilore rouq Xpnotponoto0pe Yto vo P€rpopt

rd0e ntreupd rrlq (onq pog' Irlgertirvoupe nof pEvoupe pe optOpodq

oroug 6p6poug. Tqtrerptlvotipre o €vog orov dtrtro Xprlotponottivrog

opr0pofg. To Xpqpord ;rog Boo((ovrot orouq oprOporiq rot ro (6to oup-

fr

€ 3

Page 7: Ο Ταξιδευτής των μαθηματικών

poiv€r F€ ro rllEpol6yro Kor ro poidyro gos.'Orov {6nvtoo oip€poro npoc y'jp'oo Kor Ko[To{o ro n}€ftpovr(d po}6r Fou.

'Hrqv 6.45'.OuF4orl(o 6Il orip€po 4rov I l/l ludpo rou F4vo Kr our6 Uou un€v-oUlro€ 6! €ixo vo [^!pd,oo ]10 6o]dpro yro rn 660l rou quroxrv{rou

Uou. Ko06( alpoyo nporv6, dKouyo pq6rdq0rvo o l ouxv6rqro 97,1(or n^npoqop40nKo 6r ro XplFq(or4p(o €iX€ ov€p€r t4 Fovd6€(.'ErorouvExioqK€ rl Fapq Fou (Kor rl gipo 6^r,rv rrlv dtrtrov ovep{nrov). lrq-pr(oFoor. o,ou( opropou( yro ro l 'Uoupvouu' o t ouytrpovn {orvo-vio Fo(.

H o(Kovogio, I r€Xvoioy(o poq KoL I lnorigl Eqoprrivror on6 ul

xp4ot rurv opr0F6v, Kr ouq I E{dpqot qo(v€ror on6 ro n6oo vopi(

luorF€ ro norbrd Fo( orou( optogo!( avo on6 ro np6ro npdyporqnou rou( 6r6doKouFE €ivqr nlos vo U€rpodv oq ro 6:Ko. 001600, nop'6tro rou I Fflol a€Krvd 1600 vopi(, noltrd €Ko{oFurpto Au€p((ovo(ovorprxrd<ouv dro\r Kdnoro( ovoq€p€ror oro poorl$qr(kd - oq F€l€rrlrov oproudr!. (E(vor nolU ovropd", rqpo[ov(otivlor. <Eivor no] 6tjoKo^o K €Xouv ivo orip6 ofFpolq"- Kdnoto( 6lr,J( on6 ouroLj( oKplp6q rouq ov0p6nouq fftyoivouv oro Aoq Bdyxo( Kor n€pvorv dtrn Illvdl.ro roi(ovrq( dvo norx\r(6r np6oe€ol( oproFdlv, ro prtroK r(qK. -H

rqi(ouv avo opr0FnrK6 noxv(6r F€ (dpro, ro pnoppnoffl. lvlnop€(on^diq vo fdvouv oro onir rouq Kor vo no(<ouv Mov6no^n, fvo nortrva6r nou onorrEi on6 ouroLi( vq npooo{rouv tls Kourffi6€q 660 (oprolvKor F€rd vo unoloyi(ouv rdo€q oao€r( npdr€r vo UtTqKrvnoolv. A\rqrdoouv oro oKivr[o Kdro(ou d]iou, eo npZn€r vo urotroyioouv rorvoiKro. Av ro qx(vnIo 6€v lx€r '6'oKr4rt, unoloyi(ouv 11600 Xp€rd<o-vror yro vq ro oyopdoouv. O^6Kilpo ro norxvi6r oror€i€irol on6 npo-oodo€t(, oqorpdo€rs Kor unoloyroFoJq Fs qpropots. Or dveponor io-tp€,ouv rouq oproloff. lMeprK6( qoplq loxup((ovror 6tl rouq Eivoro6'd,popo ro run..o pooqporrro o^io. ornv npoyuor.(or4ra,. i r r npor, Ior yro rdno,o f lor\ar6r nou nor(ouv,. , yro Jro rnrLoyn noJ nrp.-ptvoJv ono 'nv rupfp,noq, r ,r v o . no,W.6r oro Xpnporor4p.o. oldv0ponol ̂orp€Ljouv rou( opreFoJ(.

M€ 6€6oFavt q FUorK4 €ulopioqon noLr Fo( nopixouv or opro

L

1 4

Page 8: Ο Ταξιδευτής των μαθηματικών

poi sivor notri nr0ov6 vo onotro o€r€ ou16 ro prpl(o Apxi(ou!€ trlv

n4rnir€rd Foq €{€Id(ovro( ro0q quorKoLJ( oproFoUq €(€ivous rouq

qprouodq Us rou( onoiou( UdooU€ npdrro vo F€rpdp€, 6llo6i ro r' to

z, ro l, to I xor or;ro rooe{iq AaYowo( 'ol,ro Koo€EtiS"' €vvooJl€ 6rL

FnopoluE vo ouv€XioouFE vo F€lpdF€ ql{iivLo Kor o{: vo Url qtdoo0

FE orov t!I€u1o(o oPr0U6.

Or quorKo( qpr{]uoi irov oL np6ror opro$o( nou ovoKd^LJQov or dv'

Oponoi. ro npOto npdYpo noLJ 'Kovov U€ ourotq ilov vo UErpofv'

Ito KerpdloLo I 0o 6drootru€ tov oproF6 rn( opioFrlollq Kor fLErd 0o

s{spEu!4oouf€ €x€ivo ro rFi$o tou €YK€qd}ou Uoq nou 8(vor €firqop!r

opivo lr! rq! opiouqon Kor6nLV 0o E{€rdoot'U€ n6oo no^rd U(opEl \ro

sivor rt opi€|t\ol (qr or oprouoi Ko( rdl( XploruonoriolKo! Yro np6rn

Qopd (Ksqdtroro 2). Iro Keqdloro 3 0o €{EtdoouU€ ov d^}o pLoloYr(d

e(61 $Topofv vq F€lp4oouv :10 K€{pdloto 4 0o 6o0U€ no( XprlorFo_

noroLjoov rouq oprepoLis or npdtor oYp6t€( (or, oq ouvix€ro' 0o 60[_

u€ r( npdrsq KotoYpoqfq opreF6v ond rou( (oroiKouS tlq M€oorlo

,oIioq *o, -ouS ArYffllou!. -EnEtto eo €('rdoouF€ Ilq opxrK€( ouv€l

oqop( rtdv Krvi(ov Kor r0rv rooY€v(iv AllprKov6v 6oov oaopd rou(

q;r0go( (K€qdloto 5) Kor6nLV eo gd0ot'U€ Yro ivo p€Ydlo Uuoripto

llou oxEri(stor Fs rouq opropodq, !o onoio ollvdwloov Yto nptilq qo

pd or opxo(or 'EIIlvEq (ar ro ono(o nop'UErvs dtruro Yto ndvo on6 660

Xrlrd64 Xp6vro - tvo gr.rot{pro noLl o6{YqoE 1€hKd orou( nopdl€vouq

dpprtrouq oproFod( (K€Qdtroro 6) :10 Ksqd^oro 7 0o nopouordoouf€

rouq opvnr(oi{ oprouo!( Kor ro oJYxpovo rv6oopopr(6 o0orlFo

oproFdv' KL h€rto 0o Eipqor€ dtortror vo €{ErdoouU€ rlv 'woro rou

on€ipou Kot ro ndrq ouq n r6ao ouYKpoLjorllK€ l€ ls n€noroiotr( pos

yro iouq opLogo0q (K€q{i}oro s) 'Exovlo( oxotrou0lio€r g'xpr €6dJ tlv

s{d}rh tov opte[6v [t€ op(€Ili ouv'nslo, eo $nopofu€ orrl ouvdx€ro

vo lfoougs ro Uuotipro rov dpprll0Jv oprofdv (K€qdtroro 9) 'En€rto

€o qn€l€u0€pdroou!€ rrl qovrooio yoq Kor 0o ovoKo^t'qroul€ rouq no'

' . r o u A . o . n u o P ' \ ' 4 - r p o ' o Y o r n o o o 0 ' o u q n o ' i B p l o l 1 E

' o . o t u . r u r o . o ' o r o o ' o o o | ' € o v o 4 o \ b ' � ' O v r

Page 9: Ο Ταξιδευτής των μαθηματικών

pdq€vouq ufl€pBolxodq aprOFoU!, dno( ro ,! _ oproFoLjq, 6nio6i, rou€ivot 1600 r6rduopqor ajorE 6€v FnopoLju€ Kov vo rou( xorqypo{roul]€(K€q{iiqro 10). Kor6nrv Oo sa€pEUv4oouFs Uro €vr€^{i( vto oprognlK4K^oorl rou( llyo6r(otS opropo69 or onoio( XplortronoroJvror €upt{rrqo1r{ quorKaq EnroqF€( Kor oq glxovoioyio (K€q{i}qro jt)_ :q ouvE,X€ro 0o ovoq€peoUfs mnv r€tEuro(o oproltr(K4 xidon, rou( ur€pn€n€pootlvou( opr0!od( (K€qdioro l2). :ro K€qd^oro t3 0q KdvouF€Cvq 6rdtr€rFtro yro vo ovogapouF€ Kdrorou( ond ouroU( rou oErupodlrsUEyoioqut€( mov roFio r(ov opreFrirv.:ro onf€io ourd eo €(Foor€trorF_or vo p€^Erioouu€ rrt qdooogrKi ri€upd rurv oproFdv (K.qdloro14) :ro r€tr€u{oao KEqd^oro eo oioKlrlp0Jo€i ro 1o{i6t Foq UE Uto ovo-oKdnnol rov bnloo[do€uJv Kot rujv uflooio€uJv rou €((ooroi or(ivo6oov oqopd rlv €{6^r{��(f�q r0rv pooluotlK(iv orov €u(oor6 rp6ro ou6vo(Klqdtro(o t5).

trEr6{ or guor(oi oproFo( €ivor 1600 or€vd ouvugoopdvor UE ro\rro^rlogd Uoi Koedjq rou( lEtr€rdF€ eo 6o6F! ornv rpoyFolxoqroivo Ul(p6KooFo rl( rmop(of qq ovepon6rlro(_ npd(€rro( yro Uro(oropio y€Udq €fitri{€rq, nou poq rpoKol€( 6ao! Kur oao^ouol.

nop' 6io nou o( n€ptoo6r€pot onrt €Fdq ovlpsr0Jn((ouF€ ldl^ovun€poflr((d rl onUooio rov oprog{6v, oqv [op€(o r.u\r Xpdvov xdrotoraxouv ovoyv(,rpio€r ro p6tro (dv oprofriv. fldvo ord r€rpoK6oro Xp6-vro nprv on6ftll ydwlot rou XproroU, avo(,Ei^tvos, o Ori6iooq ordlov Tdpovro,' €(r€ 6l Koo€ri nou [nopo6F€ vo yv{rpiooug€ dxst Kdrorov Apr0U6, yrori, Xopaq our6v lrov ApreF6], eivor o6ijvorov vo oui^dpouUE f€ ro !ou ri vo yvlrlpiooug€ 0116{110r€.,

O p€ydloq 'tllnvo( 9 6ooqoq, o n^drov, 66ooK€ 6l or opreFo(

6Xr F6vo dxouv ouoroor(Kd pdio orov x6opo Foq, oitrd fo( o6nyo6v oso0r4 '(o0our{ rnv ol{oEro. I,dvo 6(dloyo ovdtl€oq oro g€wopd rou,

^"",^ i-"_T l:1 .r",*" n \ Loa,o nu€o\oo.,o, e loooqo\ I bpoo|ou' J " , i :

; i " " ' " t r p i o ! ' ^ ano 'o p \a " ' " " t " " " . , " ' " " ' " " ' " " , " .

L Kathteen F.eeman, ,4,.i//, ro the prc-Soctat. philotuphe' (Katrnptra hatuardUnive6ity Pfes, 1966), o. /4.'t6

Page 10: Ο Ταξιδευτής των μαθηματικών

rov toxp&D xl dvo qi o, rov ftrod(ovo, o nldrov €pol€ rou( Aorlvol

ouq vq oxotrrdoouv rou! opreuoi(.

IoKpdr$: Alid pr'tv loYorL(i r€ (ol dpr8Pnn(il fl€Pl oproPo! trdoo

Ioxpdrts:Tdoro 6a ya qolv.rd dYoYd trpdsdliBsru!llo6kov: Yrspqu6s $v onv.'

no(o dnoqrn ̂om6v unodllpicr out6 1o prplio; Tr ea^ou!€ tEitKd vo

romldpouF€ yro FoS rou( 6rou( 6(opd(owoc alo prplio nou ovqgapttor

oouq qpr0go6$ Erorro ro prptrio b€tXr'EL 6Il ot opteUoi Kor €roFdvo( to

FoOnFor(K4 €ivoL ovon6o omo (oBgftr rrlg (610( qS Q6olq FoS, 6rt 6Ev

pnopoiF€ vo lErroupy4oouu€ o( dvopuJnol Xopi! rouq opl0uo6!.

H ovriefll dro{rl €ivor qnloiK4 Kol, 6uoruxdq, qv ulooqpi(ouv

flollo{ AFsprKovo(: EvdJ or n€ploo6repor on6 €!d( F€tpoUv, (or Konou

rdmu rpooefroov I1 oqorporv drov €ivot onoirroq onopofIr]ro, ro

FoetFor(Kd o€ y€vr#S ypouFiq €(vqr 6vo EYX€iplUq pulFtvov rl,\rK({'

Fdvov y€vElog6pov owpdJv, orotrovoUtv{rv otouq q wtoavrouq nop

yoo( rov nqv€rroqf(ov. A€v €Xouv pEYdtrrl onFoofq Yro Uoq (ol q

(ol1 pqg dflos 6X€r, oq norB€, q oYdnrl, I o((oYav€rd FoS i1 n Kopdpo

Fo(. EnoFdvr0!, €(vor n€proo6r€po $TIEI{iq nopd po4o€ro

Aur6 ro prpiio 0o ono6eilfl to ovrlo€to. oo 6€(€1 6rr ro FoelUm�KdppioKonor 0oFu!-vo oro Fd€oq q( i6lo( Irl( rout6ulrd( Foq E{vol Ulo6pqootpr6qrq nou $oq npoo6ropicr. Av ip0€r q UCpo nou 0o to6(6d'

qroutr€ o' dvo UqKprv6 lhoKd odmllo (ot 0o enroK€9ro6p€ (''io €uquli

6vrq, Fop€( vq Uoq poqoouv: "floror E(or€;, Oo onovtiloouF€ r0r€ U€ufisplqdvEro: (Eipom€ ouro( nou U€tpo v Yvopi(ouu€ rous qploFoU(,.

4. to(pdrlq: Kr 6Il rl oPLopltrKi Kor o, unoloYtolo( oxai(ovrdr FE rouq opr€-

Iorp6tt(: Ko1 goiv{o, vo obryo,jv ro vou npoq qv or''ioao;fio!tuv: NoL U' 6vov tp6tro eoludoro. (nolr€io, Z', 525 Ae'32)

lPtata,Ihe Dialoguet at Plato, f(. B. Jowetl (N6o Y6pKl, Random House, le3z)

II

I. t

1 7

Page 11: Ο Ταξιδευτής των μαθηματικών

Ilrig petpripe;

f6oo poo'd ppioK€ror oogp€vl orqv (6,0 rt qUorl rou ov0p6nou Iopl0ptot; Or oproFoi; Eivor I op(0!qon Fro 6€ut€p€uodoq( otUooiq(rKovdrtro ro0 onoKroipE aqdrou ovoftuxer( nlipuJq o €yKaeo^6q UoC;t{oprdror dpoy€ ono(l€rorL(d 0116 qv nohl'ofrK4 Fdotory

'H Fino( n

opio|lfot Eivor, r(ord (dno(o tp6no, (6tKruuiFtq> otov €yKtqoi6 Uoq;tivor I opiotrlol 16ijl) poorKri yro rouq ov0p6nou9 6018 vo ono!€ielrtlifo rtq y!v!r'Kri( KirlpovoFLds Foq M€ dilo l6yro, fl6q F€(pd!8;

A€qrldyro

np'v i.KLviooufE rrl fEydtrl Foq nsprnar€ro, o( U'^tloout'! iiyo y'o ro

k{Li{iyLo. KdoE qopd nou 0aioufE vo €nrKorvov{oouu3 U' alov n'oorpLpri 1p6no, !ivo' Xpiorf'o vo opi(ou!E vtouq dpou( i vo opi(ou!3€{ vrou KoL !€ nro ouyK€Kp(Favo rpdno tou( nonorolj( 6pou(. Aur6porlod rous ov6p6nou( vo ouvswooiwor rolfr8po, !itE 6ro0irouv x{inor€( ouyKEKprgdv€q fl6Lr€q yv6ofl5, €ir€ osxoioJwoL F! rnv rqvoio

lio i p! r! quorKi( €rLonitEs. 0(11600, dvo €{€L6rk€rJuavo l€a(l6ytoonoKl€isr ouvrioo( dropo nou io0rq Ev6roqipowor vo pdoouv gro v{o l9

Page 12: Ο Ταξιδευτής των μαθηματικών

r€Xr'onoyio 4 EntoqFrl_ Kdnoror €noyy€luo €q, dRuJ! or yrorpoi, ot qru,X(o{por r(or or 6rKly{ipot, dxouv r(ornyopleri dll Koroo(€ud(ouv Fro€vr€i6( vfo yi6ooo 60r€ vq (pororiv o€ ordorool rouq o6ldxprrou(rop€(oorrous. A€ eo ooxoll0o6F€, 6poq, U€ ro ftFo our6.

To UoorluonKd 6vro( n€p oppdvouv ndpo noilo6( €a€r6rK€uFa_\rou( optoFoU( y(o vo undpx€r F€yolUr€pn oKpip€ro oqv €nKorvuJvio.EF€iq 0o {o (pooE}./iooulE ovog€powoq Kdo€ v€o dpo 6rov XpErd(€ror, oi^d ndvro 0o ropouord<oofE rou( oproFoJq F€ qnld l6y(o Kr 6Xou)v.roirnioKt ri qv pr€p6€F:wt Uopqi rouq. noMa( qopdf 6€v Ef-vor 6tioxo^o vo Fd0ouF€ Flo Korvodpyro avvoro, ro gdvo 6LjoKolo €ivorro ropdq€vo vdo i€q 6yto. k6xoq Fo{ €(vor vo xorovorioouF€ rl€plo-o'jrrpo rou! opL0fo6( Kr 6Xr vo gd0ouF€ or^riq vf€S tri€€(q. KdoE qop{inou 0o €(odyouF :vq ga0lFor(K6 6po, eo 6ivouF€ ivq oJvroUo, or 6oproud. A€ 0o €LodyouF€ vr€q i€arL( drov 6€ Xp€rd(Ero(.

Or quotKoi o propoi

fro \ro Korovo{ooul.r€ o€ pdooq ro l o(prpolq €ivor ol opreuo( Oo !{€,rdoouF€ npdro rouq quot(oU! op(ouoJq. OuorKoi oproFoi €(vor rK€ivoror oproUoi Fr rouq oroioos €iFoor€ nrproo6r€po €qorK€rofdvor €ivor oropr0Foi F€ rous oroious F€rpdls: ro t, ro 2, .!o 3, ro 4... To onoororlr(K(i olFoivouv rir( yropo(U€ vo ouv€Xiooulr€ vo U€rpdl! 600 Oa-iouu€ 6Ev undpxs dp(o mnv opioFno( Fo( nou vo ro €npdi^ouv ori6ror or oproFo(. Yndpx€r, p€poir0t avo npqKnK6 6pro, yrori pnopo[lsvo F€rp4oouFE U6vov 600 go( {o €ntrplr€t o Xp6vo(. [4€p((a( qopaq,or quorxoi oproFol ovopd<ovror (oKdporot oproFo() 4 .O€fl(oi oKdpor-or,, Kor, nop'6^0 nou qurt( or rp€rq €Kqpdo€r( €tvor rooodvolsq, npotlpdf€ yEvrKd vo XprlorFororodtr€ qv a(qpoo! .quorKo( opropoi,_

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Page 13: Ο Ταξιδευτής των μαθηματικών

oA0p6( roJv griiov €ivot ro avr€Ko Kot ro odvolo €ivqr rl oulioyi rurv€vr8rq p4,\r,rv. KdoE ovlr€ilr€vo Ev6q ouvd^ou ovoud(Erol orory€lo.trcFdvot Kdo€ Uitro oror€tr€( dvo ororxEio ror, ouvd^ou {ov U|itr{rv.

nn0K6t opt0y6t: O opteF6q rou npoo6ropcr ndoo ovr((€iF€vo ufldpxouv o, avo orivo^o.

tlyoro: Mto oul^oy4 ovr((€rFavov.frorxrloi Kde€ ovlK€if€vo Evd( ouv6trou.

l\4nopEir€ vo 6€ir€ €UKoiq n6oo Xp(orun €ivor yro rou( ov0p6rou(l aworo rou fl^lorKoi oprouoU. Aiv€r ordvtlorl om Epor4uoror n6ooouroKivTo urdpxouv oro Xdpo ordeF€uo!!; ndoo 6oidpro oE XoprovoUroForq qo oro noproqdir pou; n6oo ftql6{d dxouv o td6€iq6(poo o fouoqopvr Kor I yuvoiKo rou I tvltrptq; Or onovqosrq oe 6ioourd to €p{r{goro €i\ror n}lo(Koi oplouoa

^4nopoLjp€ rripo vo opioouf€ tro 6€dr€p! Xp4ol yro rouq oproFod(.Aritrl]ouv rl ox€flni osrpd rov npoygdr{rv. ,Orov pno(vouu€ o, avo Uoyq,(i nou flouidsr Koqd yro vo oyopdoouF€ rov oyorqFavo fo( Kqgd Bol-(0q plinouFs 6r undpxsr Uro F€ydlrl oupd. ndoo npan€r vo r€prFavou_pe yn vo yog e{unqpmjoouv; noipvoup€ rov qproH6 4/ Kor BiarouFs 6lI rnvqKoo qrnid ̂:EL.Tojpo €Euqpsr€iror o oprop6q 15,. t6{t, o oproF6q a/ 6€ pqS ̂48 n6oo npdyForo urdpxouv o, avo orvolq o_Ltd rodppoK6foms o€ oxaon # Kdnorov d^^o. yrdpxouv 6{i6€Ko r€trdrE( flprvorb W!. Enofivr,rs, ro 4/ 6€iXV€t r! o€rpd Fo( oE Fto oKotrouoio oplo-puiv. Auro[ rou €(6ou( o qpr0p6( €ivorivo( oryr.rdyglyo, opreFd(.

XprlorFofloroLi!€ rotr^oUq ouvr€roypdvou( opLolors. O opref6( rou6pduou 6rou pdvoufe €(vor ouvrsroygavoq qpreF6(. A€ F€rpdEr oni1lo{ ov0puinouq i (dnoro oUvoio ororxEiov. As}"vEr noU ppior€ror ro or!r lqS o€ ox€ol F€ ro d^lo onitlo rou orKo6outKo6 po( rErpoydvouKorrn( no,\!C Fo(. KdeE qopd nou €voq opr0F6q 6€ilv€Lrt ox€lKri 0d-0! tt'rv npoyudrov, nag€ 6l .ivor ouvr€roypdvoq op(0f6q.

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Page 14: Ο Ταξιδευτής των μαθηματικών

H rpiq Xp4on r0rv oproUdrv €{vor ortr6( o fipoo6roptou6q q( rou_1611110(. Auroi Etvor ot optglloLt Ktt€C. AE U€rpo6v ro orox€io €v6(ouv6^ou orir€ 6€ixvouv Kdnoro ox€IlKri o€rpd. O opt0g6q rou qtr€96-vou Fo( €ivol avo( opr0F6(-€lK€ro. A€v npoo6ropc( r6oor opleFo(

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FoL yror( (d0€ quorKds opreF6( Koral€r Fio Ko( govo6rKi 060rl ortvoKoiooe(o rov qpl0g{6v. O qpt0tl6( 5 €ivqr nd!'ro UErd ro 4 (or rprvon6 ro o. O\ €r rourou, or op'epor J ivor 610,r royt l vo..

Or I€nropap€r€q rnq o p i0p nonq

Aro qv npdq ror6rq rltrrKio Udq 6r6doKouv vo XptorFonoro0FE rouqquou(ori! opr0goLJq. Or yov€i( Fo( Foq 6i6q{ov vo F€rpdtr€ oq ro 66Ko.Iul^r'd ouri yt\r6rov 6€alvowo( 6ro6oltKd ro 8dx.rutro: tvo, 6Uo, rpio...Koe6( Kqrov6Fo<ov Kdo€ opref6, or yov€i! poq dyyr(ov lvo 6ro9op€lK6 6dx10io. NoFi(qp€ ro( ourd €iX€ ro^I4 ntrdKo Kor ypiyopo FdeoF€vq rouS FrlooFoor€. tdvro$o gnopoUoof8 vo €rovo^ogpdvouF€ lqop(ouqrKaq ia{€r( g€ rrl oLror{ o€rpd: dvo, 6 o, rpicr raoo€po... Oord,oo, ro vo UdoouUE or}llq vo l:F€ Il( ouJoris laq€t( U€ rl o{rori o€rpd22

Page 15: Ο Ταξιδευτής των μαθηματικών

6w s(vor op0gtoq. H opieFnon onorr€( Kdl ropondvo: vo onovtou-

p€ dro €prfiqpo *n6oo Eivor). Or Yov€(q Uoq lo( po40qoqv Kor o€ ou_

16. H Flrtpo Foq;pd<€ pnpoord Foq rp€rq on6 rou( (LJpou( FoS 'n6

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vo rou( UsrprioouU€. Irlv qpx4 FnopEi on^drq vo rou( o(opni(oFE rpl_

T€hrd 6FUrq FdeopE. MnopoJooF€ vo 6€(XvouPE ro npdYFolo Kor

v0 sflqvqtroFpdvouFs ro ot0ord ov6Foro Koe6q U€rpoLJoqpE MdeoF€

& ro 6vouo nou tr€Yop€ ritov o .opL0f6(' rou io€l€ I Untpo Uo(

Kqlotropo(vogs, dpqYE, tr'lv aworo rou .n6oo €(vo('; Av 4uoot€ noiu

f(poi, Fdilov 6Xr. 001600, Kdnoro olYui nprv i€((vrioouF€ to oxo

tr€io, iord( qrdoof€ vo Koro^opoivouU€ to .nl{€oq otolxEirxv' €v6q ou_

v6trou. Er6rd 6rov q p€Ydtrl Uo( o6€^9i llEY€ 611 6€v n4p€ €vo otpo-

flrrrdKL n sF€iS +poF€ 6n atr€rn€ ivo To €iXoFE !8tp4o€r.E66 €q npEflEt vo orogofu€ Yra vo 6dooutr€ Uro ox6!o tr:h Yto

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pr6rlro. A€iXvoF€ 6ro6oxr(d to 6dtrrutr{i fo( Kor ̂ dYaU€ tr( o0-tor'S

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pqelforKoJ( qv aflaK'vto\''O.da'l lEtpdF€, EKXopolpE Fio oKpr

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oir€ trlf. 6io IaE€r( 6rov on'i(oLrp€ :vo 6dtrtutro np6KErtor YLo Fro

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il{q, U6vo lao 6Fo( }fqrl. o( gooluorrKoi ro oloFd<ouv o|Jr6 olrE\lo'

voo1tlovL\ a €K6vto4 tot ppioK€ror oro €niK€vrpo oKprpdr( rlq 6Lo6'

Koo(oq rtq opioptorlq.

4iq+nvoo4tnwn aftElK'vto,? To vo lrqt'Jpo0F€ oKprpdiq €vo Kot

U6vo dlq orowio Ev6! ouv6lou (nX rov opr0Hrt(6v lfqEov)

os KdOE ororxE(o €vd( 6€Ur.pou ouv6lou (n.X. rov 6oF0luiv)

T6r€, orlv flp6q norbrni Uq( lhKio, Kdnoro otrYFl ovoF€oo otrlv

lllKio 10rv 660 Kor r0rv navtE Erdv, (otos Kotopeuiootr€ vo Kdvouue 16r

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6loKotrE! {wol€q rorv Foell$olK6vi pdoof€ vo ow oFpo\r6lc1or€ ro 23

Page 16: Ο Ταξιδευτής των μαθηματικών

npdyloro oE ouMoy6( 4 o&olo Kor vo KorolopqlvouF€ 6l KrioE ouvo^o ax€r €vo r 40o( orotx€fov, ro orcio rpoo6top(Gror on6 rov r 10((6opr0F6 rou- :6pqp€ 6tl 6roqop€lKd oUvoio Fnopodv vo dxouv rov (610r t0rK6 opr0U6 4 6(oqop€lKoUq n\orKoUS oproiroUq. lM{iooF€ gro 610-bKooio opioFlonq KoL €noFtv0rq, ovo(d^uqil( rou o0roroJ nllorKodqproUoU Ev6q ouv6trou. Auri €tvor Fro EKrlrlKIlKi ou^Ioyr1 6€{ror4niv-

Tojpo €itoor€ arorgor vo e€ooutl€ ro poorK6 €pdrtFo: n6( F€rp6-lrE; nor( ouiioyrooKa( 6r€pyooi€( Xplo(gonoroUvrou noro gipo( ron€yK€qdlou €v€pyonor€iror; YndpXEr pro ouyK€(prFaq n€prox4 nou Xpq-orfonoL€lror yf ouriv r! 6pooqpr6rqro; np6K€rro( yro Uro 6€{r6qrorou axoup€ onon(o€r onoKi€ror(kd laoo q! Fdololc { yto Fro 6€Erdrnro rou (irlpovoForF€; E(vor onopoirqrt I ytrolooo yro rtv opiourloq, 686ouivou 6r Xpqo(gororoJFE tr€{fl( 6rov g€lpdus;

H n^iprl( rpd{n Itf op(eftot( pooi(€ror o€ rp€rq 6poortprdqEq.npdrrov,oaloup€ vo onow4oouu€ oro €pdirrllo: ndoo €(vot; 'EXouF€

npoobropio€( avo odvolo Kor 0o ea^oF€ vo {tpouF€ ro fi^4ooq r{rvoTo(X€iov ouroJ rou ouv6iou. Or €n6FEv€( 660 €vipy€r€( ouFpoivouvrour6xpovo. Ayyi(ouFE i 6€itr^r'ouF€ 6ro6oxrKd p€ ro 6dx1ui6 poq roKdo€ ororx€[o rou o0v6iou xor ̂ ipE rlv Kordl^llq oproFnlKri laql. BE-poi0J(, ro vo 68iiar'ouF€ Kor vo n6U€ rrt oooq MEn €ivq( t 6poortpr6qro qq on€r(6vrotq. Orov r€lEroioouU€, I r€tr€uroio opreftlK4 ^iqtnoo €iro!3 €ivol o r^l0rK6q oproFdq nou qidlvoups (or rl opi0Ftorl

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Page 17: Ο Ταξιδευτής των μαθηματικών

llo(. r. gEprK& n€prmdo€rq 6€ lqi€ 6uvord (( qproFrFrKdq i€6€rq oi-

Jrd Fdvo rq oK€qr6poor€.'txoupt dpoyE, flpoo6opio€r Kor l( flo llKp6q 6pooqpl6qt€( nou

€{vor qnopoftrF€( yto lrlv opourlory Av €(vor dtsr m fipdYporo, 16r€ 0o

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rcu oflorcdro( Uc[Epq (4, rou]dxrmov, 6Xr vopirEpq) on6 q y]6ooo.

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on6 q o0wpovn yp4yoprt oud(o xor 0q FropoUo€ vo Ury €Eoprdror 0116

em(vs( u( n€poxt( rou €yr€qdlou nou ouv6dowor F€ trl yi6ooo.

Io ptp io Apt9pqnKAe k€ete Kat apolntKd oippo)o: Mo fto)tttolt-

Kt @ropfu tuv apoliu rou Kopl N,4€vN'v(€p, Efvqr ivo KioorK{i apyo

rlou Foq porled€r vo KorovorioouFE rC npri|t€( npooldo€r€( opioFrlorlq

ml rou( npdrou! opreFoU(.'Iro €pyo rou, o l',4/vru"/r,Ep €nronporvtr drr

or npftor dvoponor dxr B6vo nr0ov6i o}trd oiyoupo XpqotuonotoUoovq n yloroorri opieFrlorl. A6!0 Kor Flxpr tov e(oor6 ordivo, xdno€S

npot6yovE(' qu^t( €ixov 6io F6vo optolnrrKd( MqEt(: ro €vo Kor 1o

6Uo. 'Olo ro dllo 4tov qntrdq "noMd". Om6oo, ro pd^l oudv tulv

quldrv xpElo(6rov ouxr'd vo nopqxoiouooiv tov opr0f6 rov orotxEiov

€v6S o0v6lou rpoyF{irov nou €txov oqv koroxi touc. or }aE€lq "ivo,

mr n6Jo, or 6( 6€v 4rov €nop(€is y(o vq U€rp4oouv tprdvto p66(o {t!r€(o oKUtrd 4 6€Kdqr Koldoro f€ orrlpd. O uroloyroU6q Kor rl ropo-

roioiorlorl rou oplo$od r(liv rpoyF6rov nou E[Xov mnv Koroxri touq,

€6rrd &ov o€ ourd ro fipdyForo ouFn€pdogpov6rov rl rpo94, to Ko-

roqljyro 4 I ooqdi€ro ord rou( q(€poUq rouq, 6lUro0pYo6o€ roxupd Ki-

r. ftd Me.ninger, Numbe. Wordt and Nunbet synboh. A Cultuat Hirtory of^/uobed (NaoYdpKn, Dover Publicatonr, re6e), o.ll.

2. 866 Xprlo't'otror6 q Mh .npdr6Yovo(' YLo vo n.p'Ypd'lr@ qv mr6otddnlou noilnopo( qs pul'l( K 6X' y,d vo r€p'ypd'Ud ro 9uo(d Xqponlp'onKd Yvo-p(d[dro rov p€l6v IllS. Oio, oL dvopono( nou louv c'r f6p€S Foq .{vor orYxpovo'

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Page 18: Ο Ταξιδευτής των μαθηματικών

vqrpo ororrS ov0p6nouq yro vo ropor(oiouoofv rov nlrler(6 opreFdIov ouv6trr0v. Av dvo( ovl\tFoq yEirovo( ouvq(Gl vo (l€p€{ l{ oy€ld6€! oou q vrx'ro Kr €oU ouvq(Grs vo F! yvoprcrr our6 ro yeyovri9 16r€ €oU Kor I ou(oyav€rii oou Kv6uv€dn€ vo l$oKrov{o€r€.

O lvldvrvyKEp €flroqFo(v€r 6tr xdnorcq npot6yoveg qutr6s, 6nuJ( I qu-l( Sawor oro vrloi q( Ipr AdvKo, 6€ F€rpodv olEl(ov(ovro( opro$nn,#( id6€rq o€ morx€io, oil{i on€rKov((ovrqs xio6(d 4 d},.lo ovfl(€iF€vomo ororx€io nou rp6K€rror vo pErpqoofv. EnoFavoq, yro vo p€rp{o€t6vo otvo^o U! KopU6€S, o dvept'no( q( eu}4{ Biwo Frop€i vo on€cKov(o€l 6vq Klq6f o€ xde€ Kopi6o. 'Orov fti€rulo€(, o oop6( pE ro Klo-6rd nou 0q Kpord€( oIa Xapro rou €(vor o .opr8$6q, rov Kop!60rv. A€vgrop€i vo foq n€r n6o€q Kopd6e( ax€r, yrqr{ 6E yvr,rp(Gr qv Kord}-I rtop'ouqlK4 }th ylo rov rlrl0rK6 opr0g6 rou oripod, oild UropEi voKpor4o€r ro o(ip6 !€ ro xlo6rd y(o vo Hoq .6E(q€r" our6 rov opr0U6.

:'€ro6ro to oqu€(o gnop€i vo ovrrdq€{€ ro €rnx€ipqFo 6r( o dv-opurno( qq eu^(( Bawo 6€ F€rpd€r rpoyponKd llq xop(6€q. 'loo( e€-^€r€ vo urlooqpi[Er€ dr olnorv4 opiouton €(vor q (orov6qot q(oxolouo(o( rov opreF6v (avo, 6Jo, rpio, raoo€po...), (oedq Kor ron6( FfloporF€ vo F€rpdUE p€ our(.'Ourrq, yro rov dvopLiro qS quliq Bdvro 6Ev ax€r olFooio ov o€opoUUE rt 6pomtprdrnrd rou oll0r-vi op(egrlot ( on^6q ;vo Ei6oq "opioFnon( $€ (lo6(d,. Aut6 nou €XEronfooio yro K€ivov Eivor 6rt qrdvEr orov 00)016 nlqe(K6 opreF6 yro rooLjvolo n v Kopu6ujv F€ rlv o[€tK6vrol K^q6r6v o€ Kop66eq Kor, €ro

Ffvoq, orovrd oro €p6rrlgo rou o11600 €ivou,. Aruwdvro( o€ our6ro Ep6rtpo pnop€i vo uroloy(oEr oS xop66€C, ro p66ro Ko( ro di).oundplovrd rou. Aur6lo (dv€r UE q po4o€ro Froq XerprorLK4( 68qt6q-ro( nou 6€v onou€( q Xp'1ol yhioooq.

nor€! oKprp6q rKav6qr€( €ivor onopoirnrEs yro rlv onu opieFtorl

}l€ Klo6rd; npdrov, o dvgpuJno( tllq qu^i( Ba!'ro eo nptrEr vo KorEXElt|tv oqtprtFavrl 6v\roro rou niieouq ororx€(ov €v6S ouv6lou. Anlo6i,

. . ou \ r l 1ou r 1 L ' o rouo o ' ( o /o ro ! . t , ooouo l ( tu p ro^ , \ ro I v to .o .o ' pL2 6 ^ o ' r , ' o r ^ ; o o o . ' o , p . " i . p , . " - , , v r ) . . \ b o ^ , { o 0 . . . ! r . )

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plftovms Uo ou)'lofi 6lqgopsnx6v o\flKEluevov (Kopu66v) ovt op'

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MErOn0

EIXONA 2. O dv€pd)trrvo! .y(.qoiol. Tdoo o npotr.@noio< qIo6S doo KoL n r!p'oxlrou ouv6p6fou k(porpov ouvofowoLocv6 F. rqv opioFfol.

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6. Mathenatical Disabilitiet, InLp Gdrard Deloche, /'avier Seron (Xilow'i ' NLou 3l

T<dp4 Lawrence E baum A$ocial€s, re34

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rohoFoii ovoqoprKd g€ rl 6€qld (or rlv oprm€p4 Kqr€60uvorl Kol €fvolov(Kovo( vo (dv8 onlo6( oproFqlKorj( unoloyloporj(.

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lloro €(vor ro r€tr1K6 oupntpooUo 6oov oqopd ro no npoyForonor-€iror n opigglol mov €yKtqqlo; A€v ufidpx€r Kdnoro oqoupn ordvn]ol. To papolo €ivor 6( 6roqopEnx€S fi€prox€q eFnldKovror ors 6€qr6III€( IIou Uo( €{vor Yv{rrot6s o( €niYvdorl rov opreF{iv, qpoUnorl KqlurloloyloUof. H 6€ard nl€upd rou syK€qdlou ovoyvopGr qv 6nop{nnol'lonl6v owrx€iF€vov mo dF€oo omrKd Uo( n€6io, fioptxovrd( BoSfto o(oelon rou r 40ou! rov mo{Eiurv. O fipopE(rrnlo(os 9lot6f fdq6iv€r rrl 6uvor6utro vq ox€6rd(ouU€ oproFqllK{( npda€r(. Iro nioo Fi-po( rou qprmEpou nFtogqrpfou ouoxsrf<ouu€ ro 6dxruld uq( pE rlvqpfolqol. (oeo( ouppolv.r orr16, rFnl(xoul]t ro rlnForo rn( opror€-

7. Paul D. lvacLean, The liune Etain in Evolutbn (N€q Ydpnt, Plenum Pre$,1990), o.549 Deloche Ko1seron, o. 140-

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tlE0pd( Iou €yx€gdlou fiou ouv66ovror U€ q y]6ooo yro vo Foqoro vou ol qptoFrltrKtf ]€E€t(. XprloqJonordvrqf our€( 1! 610-

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pofF€ vo Koodpoor€ Kor vo oKodF€ rou( yov€i( Uq(. A€ft€pov, dvo(0116 rous yov€(q Uoq xdo€ror mo rdttuuo gq<l Uo( Kor, rfldvovro! rodvo Xapr troc, i:Er: .Koiro, qurd €ivor ro 6dx,Iuld oou,. Kordflrv otli(€((d0€ 6dx.rutro Kor trr€r rlv KordMlll oproFnTlKi ^iErl: "'Evo, 660, rp(o.Kdv' ro rdpo K €or). Aur6 E(vor Kd

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'Orov FoooivoufE vo g€Ipdu€ - F€ro{d 6Jo Kor r€oodpurv €rdrv -

pitnoup€ 6r fip6(€rror yro Kdr rou onorr€i npoondosto, xr dxr yro xd-t( rlou goeo(vouu€ EoqvrKd. Xp€rd(€ror rpoonde€to, (upiuJq €na64 o€yKteoid( Fo( ouvExi(€r vq U€yqldv€( Kr €n€r64 or yov€i! Foq rouovrlouxoriv Friroq 6€v Eipoor€ Ko( 1600 6{unvor, oq4vouv Koprd gopdvo qqv€i o q6poq rouq. Moq nr;(ouv, €noFav{r(, ri( rq dKpo lurv r(o-vooir0rv po( (Kdr nou (dvouv oK6Fo Kor rolioi oropy(xo( lov€iq)_:qv opx4, q opieFtorl €(vor Fro ooq6q €prp6e€q npdft yro Fo(,oeoJ nooxi(ouF€ vo ouFrlooU$€ rrl oororil opr0ullK{ ^ih Kor vo ulvro6F€ 6€Wovro( ro 0r'1016 6dx1uio. Mro oogdjq €FrpdeEq 6poorrlprdqrq 0q gnopodo€ nol! (old vo €l€n€ror on6 rov npolr€ronto(oqlor6. 'Yor€po on6 opK€ri €{doKrlon, oroFvrlUov€doup€ tlc opreunfl,K€S ^dE€rq oq F'o {€X'llprm4 uro6rqip€on rou ̂ E{ oyiou tro! Kor 6tFroupyofF€ rq ouv€rpFrKa( FoKponpdo€og€q gv(F€q yro Kdo€ opr0F6.Aurdq o' ouv€'pf"Ki( Uv4U€S n€p(loFpdvouv nnflKiq, onlKdq Ko( KrvnrKd( Fv4fe( yrq K{io€ opr0g6. Enrnlaov, Kde€ oproFdr rEpr^oFBdv€r qouv€'pg'Kri Fvripn rou Er6u€vorr opreFod Trl( oKotrouoio( (n.X. 61ovtrdo (rpio,, o €n6f€vo( opreu6q .r€oo€po, €(vor ivq( our6goro( ou-!Erpp6q). ApXrKd, SI€q ouraq or FviF€( €ivqr avrov€q olld !€ rov Kor,p6 KupropxoUv o' txtflKd( pvri$E(. T6r€ aXoUU€ mn 6rdeEo4 Fo( l(oomdq rpoqop'Kaq op'ounllKaq iaa8rS, €vri Il( ypomdq oproFtlKtq ra-

E€r( Koollq (or ro opopr(d opreFtrKd lirlqio ro Foooivoup€ ouvfeosoqdrou qpx(ooup€ ro oxol€(o. Or KrvQrKaq UV4U€q nou ouv6aowot ou-\r8(pgrKd p€ rooq opreFod( npo€pxovror onri q 6pooqpr6qro ltov

34 Soytrlkiv Foq - 64tro6i on6 ro 6tl 6Etlr'ouF€. AK6Fo n 6rov ryrth(url

Page 27: Ο Ταξιδευτής των μαθηματικών

dor{ n odvoEon ropoU{v€r roxup4, yrqr{ ouxvd Kor Xop{q vo roroilr€ qlryi(oupE rouq ow{X€lp€( 4 ro dMo 6dxru^o Uqq

p8rpdF€ pE ro vou foq. 'Orov

4 op(oplol ytv€r r€ilKd ourduo_oud( or Foxponpdo€oF.! ouvErpprr€( Uvnpr( €vToni(ovrar or 610-

Kf( ftproxt( rou v€opiotorj.llop'610 nou Fo0oivouu€ n6( vo p€rp{iu€, p^6nouu€ 6Il ol €yK€qo,

larcupyi6( rlou Fo( €nlrpdnouv vo FdeouF€ vo !€rpdu€ oror€_l6po( r4( proloyrK4s Klnpovol'trdr Fof. EnutrEov, I rporopxrx4op{qrnoB, ro vo ovopoll6lom€ rldoo ow(K€iu€vq urnipxouv

omft6 Fo( n€6io, goiv€rql vo rinpovoF€irot Kor vo urdpx€t o€mu( qvopriJnous. Iuv€ndrf, ro poorK6 €pdqFo (11600 €(vou,,

omio I op0Frlol Kor or opreFoi orox€iiouv vo 6uioouv qldvqol,€vq Ep(,lql]o ooFgdvo oqv i6ro qv ov0p6nvt g$ot pqq.

35

Page 28: Ο Ταξιδευτής των μαθηματικών

n6g popg€g opa0Flo1g

oo €(Erdoouf€ rdpo E6d Kor 11600 Korp6 or dvop0Jnor F€rporiv. Av Iopiofrlot Eivor pro np6oqoro oroKqpdq 6€{rdqrq rov ov0pd(ov,00 lflopoiootrE vo ouFn€pdvougE 6r(, r€iLKd, 6EV €{vor Boord ouv6€6€-

Udvt F€ rrt qdon gcrq. An6 rnv di^rl, ov ppofp€ 61r np6(€rror yro Uro6€{r6qm nou undpx€r on6 qv opxor6qro, rl ono(o Xpovo}oy€(roron6 qv Enoxi rov rpoovopdlnrv{rv rpoy6vov Fqq 0q Fnoporoou€ vo(qroli{ouU€ oro ouFnipooFo rin onor€l€i pdpoq qS i6(oq qS qdolqrou ovopdnou, 6noq oKprpd( ouuBofv€( p€ In yldooo (or rqv r(oro-oK€ui Epyoi€iov, nou €(vor K€qoiord6ouq onfooio( yro ro €(6oC Foq.

H op,opnot. .vor nold nolor6rrp' t on6 r lv roroyrypouprvq roro.pio Kor 6€v lnopoi!€ vo floUU€ 6r mr! 2 loutriou 1l0l r.X., orov N€ilqo PoyroFdv o np€opUr€po( €q€6p€ rtv opioFtot Kor rt 6i6oE€ orouqouw{iprovoUq rou- I opioFrlol €ivor fioli fiq.lql6T€prl on6 ou16. En€(6{q€r Kpuppdv€( rL( p((€S rlq oqv npoioropfo, eo fip€n€r vo ovon]doou-

F ro €{€lrxru6 nop€106v FoC Kor tl€rd vo FowdLlouF€ ndr€ y€wiorlK€

n 6€Et6rqro qs op(guorlq. oo ro KdvouF€ our6 Kor yro fl( 6Jo Fopqa(op€frytq: yro rtv opieFrlot tr Klo6rd, I oroio ornpi+ror orrt XErplm'K'l r€Xv'ni q( Erp€ort( rou ntrnorKor oproFou €v6q ouv6iou, Kor y(o!|] oU\1povn op€gnon, n onoto XplorHonor€i opreFFK€q I€{€r!. 37

Page 29: Ο Ταξιδευτής των μαθηματικών

Or p0 Kprvo i pog npdyovor

Or yv{to.rq Fo( yrq rnv Lorop(o q( €{dhh( lxv ovopoirov €(oKo^oir-0o6v vo ovomrooowqt (or no^Io( ov0porol6yor 6xouv ovflKpou6u€-v€( q\rrdfqr€rc yro ro n6! rpoluipno€ ouri q roropio. or r otKor,orouq oroiou( ouFr€pdoFpdvovrqr or yopa€C oL X'irnor(46€q, o' oupo'ordyror Kor or yipp{,v,( , Lpoivrror nu\,ho. or ro\ | lvo,rpor ouyyrv€i( foq nou EloKolouoo6v vo €nrpuirvouv orov rtrov{q. ArId ourd roraoo€po Ei6n, o Xrpnor<{S e€op€iror ouvieo( o nro Kovriv6q ouyy€v4qpoq. O A^q\r fo0fioov xor rl Mopi KiEp KrvyK, on6 10 nov€nroqFrol\lrapKn€i 1l( Ko^rq6pvro, ouvtrprvov trq npoteiveq tov ovepdrnov Korrov runor<46r0v Kor pp4Kov dr 6(oeapouv F6vo Kord l%, F€ i6ro rour6ioLro 99% rov rp{rr€iv6v. O Xrpnor<(( €Uqovcr €rion( dvo p€yd-^o poeF6 y€v'Kri( euquioq, qvoyvop((ovro( rov €ourd rou orq {sxopr-016 Kor 6roqop€urd ov. :€ r€or rou EyLvov yro vo pp€o€( ndoo yp4yopo or niolKor Unopodv vo ovoyv{rpioouv ro E[6o^d rouq o'dvov Koopdett, o Xrunor4iq Eivor o nro ypiyopoq oKo^ouo€i o oupoKordyKoq,Evd oqv rpirq oiot ppioK€ror o qrox6q yopAoi rou onoruyxdv€r T€

Or dvopuino( onor€loav ro trdvo yvriord 6vro nou FnopoUv oirl0LVd vq lErpotjv, vo npoo6ropi(ouv 6tlo6{ rov orror6 n^t0rK6 opro

!6 €v6( o0v6iou Xptorforor{nvro( Eir€ ulv opi0llot F€ KIo6'd, €irsrn\r opieFrlot tr€ oproFtrKt( 16{€r(. Erottvr0s, rl opioFrlon 0o npa,n€r vo {€KivtoE F€rd rtv €roxi ro0 n €{€irnrKi ypop!4 6roxop(orn,(3 o€ ovepdlnou! Kor nroiKou(- An6 qv d^Irl, I opioFrlon €lvqL no,ior6rrpt ord q ypon( y^r6ooo, I oro(o E€Kivrto€ on6 q MtotAvoroli r€pinou navr€ Xdrd6€( Xp6vro flp'v. Enotavors, 10 6tdoqFoKotd ro onoio €Beov(orlxr 4 op(6gqoq optoosr€(ror F€ro{rj rn( €ro

Xnq flou I nop€io rou ov0pdnou oroxop(orlK€ on6 rouq rlloiKou( Kqtrou 3000 n.X. 'Onur( 0o 6o(p€, our6 ro XpovrKd 6rdorlpq €i\rot fioid

O Anov touiloov Kor o B(vo€vr ldprrr, €niorl( on6 10 Mn€pKl€i,aKovov ap€uv€q rdvo orq oFor6rlrEq xor r( 6roeopa( rov nporrlvdlv

-|

38

Page 30: Ο Ταξιδευτής των μαθηματικών

tuv ovopdflojv Kor rl,rv ftO4Kov, npofioe{irwqq vo npoo6rop(oo0v rld

do Kqp6 rprv ouv€pl ou{ I €q€hfiLKi 6roK^d&rol. To ouuntpoogd

rcul flou o{F€po ro onootxovro( [o]tro( ov0pono]6Yol, €(vor &l o

6ox0rpoud( rov nlo4Knv on6 rou( ovopdnouq ouvdpl nptv on6 na-

vrE oc ah EKorouBdplo Xp6vto. And €(€(vrl qv €noxi1, o EYxdQotrd( uo(qa rprrlooroor€i o. UaY€ooq ouYKplv6F€vos u€ tov €YK€qoio tou Xl

Fnor(1, €vdr o opr0F6S r0rv v€uplilvov ix€t 6rnlooroor€(. O( Ex rotjtou,

q pEo-\Ur€pn 6(oqopd €ivol to FdY€eo( rou €Yx€qdtrou FoS, tbroir€po

ro Fdy€eoq rou v€oqioro, Uq(, Kor, 6no( €inoF€ vorp(I€po, o v€oqiot-

6( oror€i€t ro xI€6( Ytq qv (orov6loq qS opi€Ulorls.

floo( o(otrouo€t ornv €E€hKrl( nop€io U€td rou( Kolvofq Uoq rlpo-

y6vou( U€ lou( rre4xous; To onoh06lroto <6uJv rou €Xouv ovoKoiu

Qr€(, rq oro(o pord(ouv xd[oJ( F€ n(g{Kouq olld Kqr FE ovep6noul El

vor 1600 rclld nou or drq 6€v UropodlrE vo €(€rdoouF€ rlv npo€i€u-

ol 6llllv. Euruxdls, or qvoprrfiotrdYor dxouv roi|oglio€r to ono^lodrpo-

rq o€ 6rdqop€q ogd6€9 Koe€P{o €K rdv oloiurv €X€l aExurprmd Xopo_nlplmr(d rlou 6€ixvouv ooqds q moo€pt1 €qdhh on6 qv €uquiq

tuv 4iuN oq oLiyxpovrl ovopdlnrvn €uquio. 'O^o ourd ro <ciro rou Yo(6

louv l€ nr04(ous xor ovep6[ouq ovogd(ovroL ovep0]no€t64

Or noior6tEpor Av0purni6€S €(vor or AuotpotroflOnKor qq AqprK4(,

ro onoiro6uoto r0JV onoidv XpovoloYofvTot ofi6 n€pilou r€oo€po

€Koroguipro d0rq €vdBlor €Korol.tfdpro Xp6vlo flprv. Aur6( o griopd

mt €nouav0rq, 614prco€ pEYdtro XpovlK6 6tdorlUo - 6u6ptol €KoroUFf

pro Xp6vro. Oo €iFoor€ oEldrorvor ov Unopfooup€ vo ouvoYrrvrmodF€

Me r 4roro<ov or Audrpolon(en(ou To KUpL6r€po Xoponnplotr(6llou rouq 6roxopiGl 0116 rous rno4Kou( l(o( trq $oigot6€( qS €noxiq

pos €ivor rl lKqvdqrd rcu( vo po6i(ouv €ueut€v€iq, o€ 6p0ro otdon,

(dI( [ou 4rov ro rp6to op6orlFo tqs rop€(os rlpoq tou( ovopdinouq'Htuv grKp6oogor op(nd Kdro on6 dvo Fdtpo - Kor (6Y((ov K{iro

r. Roger Lewin, Eonet of Contention: Contrcvesiet ih ke Search tot Hunan ,gOrgirr (Nao Ydprt, simon and schuster, 1987), o 103.

Page 31: Ο Ταξιδευτής των μαθηματικών

on6 oopdvro K q m0ov6v vo Forpd<ovrov ulv rpoq{ roue Uloo ortvorKoyiv€lo Kor rrlv (ovlv|l nopio, vo €ixov q 6oUi q( nupnvrK4q ol,Koyav€(oq Kor {o0r( vo XplorFololo oqv lvo 6rqxoproFd €pyoouiv F€pdon m qflo. nop' 6^0 ourd, o €y(€goi6( rou( €{oKoiou0o6o€ voal€( ro i6ro n€pirou Uq/eoo( U€ rov €y(Eqotro rou mo4Kou _ Kord F:oo6po 450 cmr, o€ otl/Kpron F€ ro 400 cmr tou eyx:qdtrou tou yrgnor(i(or rcr 500 cm' rou Ey(€gdiou rou yopao. A€v undpxouv n€rmtKCqoro6E[E€rq 6u or Auorpo]onioqKor KorqoK€io<qv €pyotr€iq, 6r !rpo_yov Kp€o(, o1 Xplotuofioroloov q qofld, dll fqtroxr'ov porixo, dTldeopov rouq v€Kpo6S tou( { 6fl 6rlFroupyodoov oufpoirK{ rq(vn _ nou6,\0 onor€lodv XopoKqprmr(d rurv F€roy€vfor€prrv Avepu)ft66v., Toflro orluov1l(o XopoftrlproI|(6 rou l€(n€r on6 rou( Auorpqlonre4rous€ivol o Lr€ydios €y(€qq^oq. Aurd uno6liulv€r {wovo 6

6€v €(Xov rlvrKovdqro vo ovr oFpdvovrot ro €p{iqFo "rdoo €(vou, Kcr U€rd voqrdx.r'ouv Kdfioro r dvo yrq vo ppouv Fro trion. EropdvuJq, FropodF€vo Korotr4(ouF€ mo ouFripool.ro {iu 6€ F€rpoloov.

I]€pirlou 660 €(oroFFfpto Xp6vto nprv, Kor yro U(o6 €KoroffUproXpovto, €vo v€o €[60( €Uqovimrlrc orlv AvorotrlK4 AepKI, o Homohabilis 4 Enr6CEro{ AveArro( (homo olpoiv€r "6veporof, mo iolvrKd). Ovopd<Eror arol yro tvqv nol6 Kotr6 trdyo: Koroo(€fo<€ }io(vo €pyo^€io. Aur6 {rov pro onpovtlK4 rp6o6oq oE ox6on g€ rou( nolor6r€-pouq AuorpoionoiKou(. Emniaov, o €yK€qotr6( rou {rov lr€yoiir€posoa6 rov €yKfqqlo rov Auorpotrolto4Koq xord FCoo 6po 250 cmr o€ouyKptorl Uc ro 450 cm'. Aur6 qlor€^€i U€v Fro onpovllKri o({rlon, Ko-r{i rq d},}o, duoq, o Homo habilis €Foto<€ rotrJ !€ rou( [o^or6r€pou(Auorpolonro4(ouq. E{X€ dqios Kdro 0116 dvo p6rpo Kor pdpoq n€p{nouoopdwq x d Kord gtoo 6po. H €€{roon rorv 6ov(6v rou 6Eii^r'€r 6u€(vor nt0ov6v vo 6rpoy€ qpoUrq 6no( ol AuorpcdonielKol r(or, €rI!oni 611 6€v {rov noUqdyo( dnurq ro U€roy€vaor€po €i6n rou ydvooqAvopono!. E{orc}ouoofF€ vo pr1 ouvovrdF€ q Xp4on qq q@ndq,

40 '".1; *:;lr l;l"" th. tietd cuide to ta4\ Man tt\'a \op1 ta(\ on f"e

Page 32: Ο Ταξιδευτής των μαθηματικών

Ilrv polxov i qq r€Xwt(. flop'6io ourd, o F€Yoidt€po( €Y(dqotroq(or I KorqoK€u4 €pyo^€(0iv onor€ioUoov Y\jovfloio p4Foto npo( ro

Efnpdq.Yndpxouv, 6pqy€, oroLx€(o nou vo uno6nldlvouv dl o Homo habilis

Xprprfonoro o€ Kdno(o Uopq{ opiegrlolg H ondvqol €ivor 6Xt. o

Homo habili5, dnoq (or o( fipoKdloxoi tou, KorotKo o€ ornv AvqronrKi(or q N6(o AqprK4, 6nou ro oro0€p6 (Xgo €rdtp.rE rl ouYKoFr64

rpoQiq 610 ro xpdvo.''Erp(,y€ epofro Kor 6€ Xp€lo<6tov vo ouoo0)

p€U€r oroe€Uoro lpoqiFov yLo opY6rEpo. A€v undpxouv ono6€i{€r( drl

(oio€ o€ [6vrFou! r(oroutrrouoJq, Kdtr nou 0o ouv6€6rov p€ rtl ouooti'_

p€uoq lropx6wov. Xopiq urdpxovro Kol UE 6tooqohoutvrl q oro0€_

p{ nopoxi lpoq4(, €(vo( nroovdv vo Ul XpEro<6tov r|W opio$lorl.

nop' 6tro nou EiX€ pEyoldr€po €YKdqotro on6 touq Auorpolonre((ouq,

6sv undplouv onobEiq€rq dr €iX€ YpriYopn qovnrr(4 ogdio

nspiflou €vduror €xotouFdpro Xp6vro nprv, rt €qilLen nips Uro €v-

6ogdpouoo mpog( p€ lnv €Uqdvroq rou Homo erectus 4 'operou

Avop6nou, o onoio( €nd<rlo€ Fixpr n€piflou 100.000 Xp6vlo nprv. O

Homo erectus onorE^€( g(o q€roolF€(oq [pdo6o o€ oxaot] U€ ro

nponyodfEvo €i61. To nopoKdl0r XopoKrlprol(d Yv(dpioForo rov n€_

pLypdeouv:

. EiX€ nold u€Yoldt€po €YK€qo^o on6 tov Homo habilis

. EiX€ nro n€p([^oxo Epyotr€io ond rov Homo habilis.

. XptorFonoLofo€ tr'l Qt trd

. I4€rovdorEuo€ 6{t0 on6 rrlv Aqprnl, oqv Eupdlq Kor rrlv Aoio.

. EiX€ ur( pdol rou €nox(o(oJq lFrF6vlpou( Kotouirououq.

. Koroo(€ro(€ Koroq6yro.

. 'Hrov qLrroqdyo( Kor Kp€orogdyo( (noFgdYoq).

A116 ro! nopqndv{, Kordtrolo propodF€ vo 6oUgE Uto €vrunooroKi

np6o6o orov Homo erectus o€ oxaol 1600 Us touq Auolpq^onr04(ou(

600 Kor FE rov Homo habilis. H oJqnoq oro UdY€ooS Iou €YK€qdiou

41

Page 33: Ο Ταξιδευτής των μαθηματικών

Iou 4rov orlFowtK{. En€164 o Homo erectus diloq€ onfovllKd oro 1,2€(oroUufpro Xp6vro rl( (rlop{iq rou, our6q o rotr[lou6q 6loxopGroloTle TrEpt66ous Tou qpxl(of Homo erectus Kol Tou lr€roy€vdor€pouHomo ercctus, Fo U4/Eeo( rou EyK€gdtrou tou opxtKo6 Homo erectus4rov Kord Faoo 6po 900 cmr, o€ ofvKptol !E ro 250 cm, rou Homohabilis (Elx6vo l). O Erdpqioq rou !€roy€vlor€pou Homo erectus ou_E4elKE o€ 1.100 cm', nhotd(o\,.roq ro Fdoo 6po rov t_400 cmr rouorlyxpovou ovop{irnou.

To €pyoi€(o tou Homo erectus {rov p€trroufvo o€ ouyKpron U€ ro€pyol€io rou Homo habilis Kql 1600 n€pinlo(o dtor€ ot oriyxpovot ov-0pono^6yoL fipoorloo6wq( vo Fwl0o6v ( lKovdqrt( rou, Xp€dorn(ov F4v€q €qdoKnoqf yro vo Fnopfoouv vq ro Koroorcudoouv. yrdp,

Xouv olo6€i{er( 6Il o Homo erectus (orqoK€ro<s ontrd Kqrogdyro, gtorKovdqro rou 6i€lrl€ qrl6 rov Homo habilir. 001600, I Tr(o ollowtK4r€XvoioyrKi np6o6o( rou Homo erectus 4rov n €{orxEiuJon F€ rn qo-

El(oNA 3. Kpov(o rq( KpovroKi Ko 6qro rou n€tKovopdnou erecrL6, rcu trprircuiiomo ereclus ro!1 ovoKoiuqlta ro l3et on6 rov hom(v Nrpnoud (EuqeneDlboii) do vqolqs ldpo(. Io Kpov(o .(vqr 45o.ooo .riv (or I Kpovronl Ko 6qra€X€LXopnr'K6rt( e00 cm'. (rx€6ro on6 ro Stock Montagq r(dyo)42

Page 34: Ο Ταξιδευτής των μαθηματικών

(4 flou €ivor nroqvdv vo ouvipn oq 6rdpK€rq Bto( r€pr66ou FroodExqro|pupiou Xpdvov, on6 €vdUlol arr( ivo €KorogFipro Xp6v(o nprv.[vr6 o Homo erectus gdoorv€ vo XpnoruonotEi q qold, F€roKv{Olr€Fq(pd ofl6 qv Agprnl rpo( ro €ir(porq Ktrfuoro ]q Eupd]rqc (or rl(

A(6Uq Kor onlv €Ugdvrori rou tuoto<€ n€proo6r€po on, 6,r( ot npo-rdrolol rou F€ rous oU\lpovouq ovepdrouq, ytori o Homo erectus{rov rlrl,l6r€poq qn6 rov Homo habilis, rdvo on6 :vo Utrpo (ord fdoo 6po, F€ 110]6 nro €rir€6o np6ooro. To (€edir rou 4rov F€yo]dr€-po r(or i(o orpoyyul4 0ior6oo eEoKotrooeodo€ vo ax€r 6royK(rp6voomd ndvo on6 ro qpd6to lou r(o{ yupr6 p€roro. Av rov ptrlrot€ vru-F€vo FE KoorouFr vq n€prFfv€r ro rpavo, Fnop€i vo rou pil ror€ Ltlo6EfrEp[ €rilrovn gold, oMd r€l(Kd eo rov npoor€pvoroqr€. A\,,rl0€-ruS, ov pllrqr€ vo ordrEr mqv onopdopo ivo( Aumpo^onienKo( 4tvos Homo habilis, n{0ov6v vo KoloLjoor€ qv ErolpEfo npomoo{osZ6ov llyowqs 6r ivoq nionKoe ro €iX€ oKdo€r.

H F€rovdor€uol rou Homo erectus on6 qv Aqpr( oqv Eup0jnrlIqr rnv Ao[o Koraoo] 6uvor{ €n€l6i Xpnoruonorofo€ rn qol4 Koro-or€6oE€ Koroqdyrq Kor {rov ropgdyoq. Ito 6p(g€io Klilqro, or grrtldsKor ro rorqeUyto rov poionoov vo €r(pt6o€r o{ou( Kpdou( XErFUiv€q.Emflltov 6rqv m qpodro Kot ro loxovlxd €{ogovi(owov ro q0rv6n0r-po, propoUo€ vo Kuvnyd€r &o Kqr vo Kpord€r ro (p:o{ royogavoorous €(o1€prKoLi( Xdpou(. Xopi( rn g(,fld, rq (oroq6yro l(ot ro (pfqi Eivqr opqipotro qv 0o €X€ €nrprd,o€r FoKprd on6 rqv ooqo^4 <€oro-ord qS AgprK4(.

l\,4floporio€ vq gapdEr o Homo erectus; A€v undpxouv oro6€iq€.Sft o Homo erectus €lX€ yp|lyopn qu}vtnK( ytunooq. Erogtvot 6E U€-rpo6@ Xpnorgonordwqs lrq oiyxpov€S oproFrllKiq it{Er(. [,44no!0p0rq pErpofo€ Xpnotgofiouivro( dq6(d; Aurd €{vor tvo €p{irqgo nro6Uo(olo. Iiyoupo 6Ev €ivor oropo(qro vo yvop((ouF€ Kd(oto yirioooyrq vq axouF€ olv o(ootorl rov opr0Fdrv, yrqri qv dxouv 1600 ro (60doo Kor or dvoponor. O Nrdrp(vt l.:Fro, arc tp\o H larcpfu tuv pa-e4parx6t/, l1oitdkt: 43

Page 35: Ο Ταξιδευτής των μαθηματικών

Mrq nolU yvooni n€pim0ron €v6( Kdqdlo^ou oyoproU mu qndKqo€yviiro't Illv oprordv noparnp.6vro( rq 6d),.uld rou, nprv oKdto 6160},T€i vo !.rpde! 6€i}.,,€r dll n dworo rou oprofoi 6€ Xp€rdlsrqr vo qvop6v& qv ovdmu{rl qs npoqoprnls y}6ooqs, ( Eror €vo( nporoyovo!dvep0ro( Fnop€i vo ovrdrqo€( rc rpio, Xopis vo ptrop€i vq ovofdoarout op'eUoUS ndpo ond ro 6do.,

A€v undpx€t Kdnolo dpEoo molx€io fiou vo 6€iiv€l6Tl o Homo ere,ctus XprlotFono(orio€ 6vro( Irlv opioFlon F€ do6ld. Or opxqloidyot,6rl^064, 6Ev axouv pp€r onFd6ro opioFrloQ( oro €pyo^€rq rou il ox6tjrop€ narp€q nou vo napor#Fnouv o€ ap(0fo6( mo a6oqof r{rv Korquhog6v rou. noro fuF€oo molx€io uno6li6vouv rlv opioFnol F€ do6rdyro rov Homo erectus; npdrov, €rX€ U6yd_lo €yKlqolo rou, €r6tKd ytorov g€roy€v€or€po Homo erectus, nlloioc o€ !6yEOo! rov €yK€qqiorou oriylpovou ov0p6nou. A€6r€pov, Kor6p0oo€ vo KorooK€udoer n€-pinioKq €pyoi€io. fro vo KorooK€udo€t xdno(oq ivo EpyoiE(o, rpfr€rvo rpoox€6tdoa gto o€tpd Ev€py€tdrv nou Oo (oro^4{ouv o, avo €nr0uflrd r€irKd rlpoi6v. 8€poi{rs, our6 o(prp6( €ivor ro €(60( ul( €yK€qo_^rK4q krroupyios rcu Xp€rdcror yro Irlv on€rK6vro! rdo6r{iv, Kopu6rujv4 XoirKrdrv €r( €v6( ouvdiou ovllK€ruavov.

O Homo ercctus pEroKtvj1erl(€ lroKpr{i ofi6 ro O€pF6 (liFo rrl(Aeptnl( npoc ro €roxtKd p€ropqM6u€vo d(fo rQf Eupuinnq (or q(Aoios. Ynoxp€{rFavo( vo €fir(4o€r mov KpUo X€lpri,vq, o Homo eredusouood'p€uE oiyoupo orooaForq rpoqtpov. -to0rq ouro vo qrov q rp6-rrl n€prpotr^ovlK4 n(€on rou €vodppuv€ rtv opiogtor} fE Klo6!d.[,4nopei vo unipxqv mr d]]o npoBijltoro rou €vodppuvov ulv ovd-nruqn qs opoltol(. n6o€( Utp€s roqi6, 4rov Ffxpr rnv Kowrvdr€prtKo(v6qroj I€ 1160€( Fap€q €(X€ (ovoAnvo (K dpo eo gropodoov vo(uqy{oouv ul vdxrq); 11600r Kuvlyoi Entqoov F€rd qv Koro6io€nord I|q opKo66€9

Xopis vo ordpx€r Kdnoro dF€oq or66Erfi, eo uroo{oou}r€ dll o

44,* ; . : : : ' " * " t f ' th H^ to to a t Md 'heqd ' � i ' � \ tN jo rop ' r ' Dove Pub 'a r ion \ .

Page 36: Ο Ταξιδευτής των μαθηματικών

Homo ercctus €ivor nreov6v vo ov€muq€ rnv opoFrlorl |r€ Klo6ld. Au-

16 0o ouvapn o€ ouv6uooF6 g€ 10 F€Yo^d1€po €YKaqotr6 tou, ro Kotr6-

upq €pyol€io Iou, ul Xp4orl q( qurrld( Kor q P€rovom€uon o€ (puo

d(uoro. tivor on(oovo q opieBrpn U€ Klo6rd vo ovqmdx€l(€ rourd-

Xpovo o€ 6tr8( rC qutra( rou Homo erectus, to fileov61€po €ivol vo

ovomq(EK€ oKov6vrolo. 1oo( vo ovomfxeqK€ o€ 6rdqop€S n€proxdq,

tro vo €qoqovror€i opy6t€po (ql U€rd vo €Fgqvlor€i €K v€ou lrnop€ipro ptpo vo ldoouU€ out6v ro Ypf9o, qv (dnoto( rux€pd( opxolo^6Yoq

ovoKoir{€r Kdnoro opxqtoloyr(4 rofioo€oio tou Homo erectus nou eo

nEpd)(€i o6ldv€um€( Boprupi€q. l'46xpl r6rE 0o np€n€r !o KdvouF€ €r'

Kqoie( oXEuKd $€ our6 ro €pdqFo.n€pino0 rp|q(6or( X td6€( Xprivro nprv o Homo erectus dpxlo€ vo

eqoqqvilsrol Nopir€po, (o(,( Kor [€vroK6or€( Xdld6€q Xp6vro [plv, 6vq

vto €6oC Tou yavoof Avoponoq €ggovionlK€ oqv EupdrTln, qv AqplKl

r(dr qv Aoio."Hrov o Homo sapiens i 'Euqpov Avepurfio( o Homo

sapiens qfiora\€i Fro orlUowK4 €{ehin rou Homo erectus: ro UEY€0o(rou 6y(€qdlou ou€4e!(€ fipoo€Wi(ovroq ro orXlpovo FCY€ooq tu]v

1.400 cmr. 001600, o Homo sapiens 6€v 41ov o(Yxpovo( dv0ponoq, otr

ld oflor€troio€ dvo U€roponK6 ord6o ovdU€oo otov Homo erectus Kl

€ldq 6€v €(X€ ro onoldros mpoyyul€pavo kpov(o, to uqr 6 Udtono (ol

ro Eflir€6o npdo0rno nou Xopqnlpi(ouv mus o6\lpovou( ov0p6Tlouq.

Av unooaoou€ oulmd 61r o Homo ercctus XplolFororodo€ qv opio-

Hon F€ xlo6rd, 16r€ oi\/oupo q XplorFonorodo€ Kol o Homo sapiens.

oUro( 4 dllos, o U€yotrit€poq Kql nro nolJn^o(o( €YK€qoioS rou

Homo rapiens eo irqv TIro 6€nLK6S orlv ovdmirEn vorltrKdv 6€Eloq-

Ilnv 6no( I op(ounon U€ Klo6rd. Eq6oov onotrr0drporq rou Homo

sapiens ppt0rlKov 1600 ornv Eup6nn Kol rrlv Aoio 600 Kol orlv Aqpr_

nl, Xp€rdorrlKE Kol our6q vo €nlptdro€r o€ n(o Kprro Ki(Foto Xpnoruo_

noro|jo€ oillpovn ytr{irooo; O( on6$€r( 6roqapouv doov oqopd qut6

m oqro, qltrd or €r6rKoi ouFqovodv U€roqi rouq 611 6E Xprloruonorod_

s. Paul D. Macle.n, The Triune Etain in Evolution (Ndo Y6pKq, Plei'rm Pre$, 45

Page 37: Ο Ταξιδευτής των μαθηματικών

oc 1t 6rnl Uo( ypiyopn govnrKri oF (o. Yndpxouv, (,01600, ororx€io6rL 6vroq XpnoLUonoroJo€ (dnoLo €(6os nporofriio( nou poor(6mv o€

lro n€proproFtvt yKdFo qov'tlK6v ixov.'OrdvouF€, r6po, oro rEiEuroio ordbro qq !iai'li( to(: rtv €Uqd-

vron ro0 ldyxpovou Av0p6nou, noo ovoud(€ror Homo sapien5 sapiens.Or r€iLxaq oiloyaq nspddupovov ro\r 6rK6 Foq !€ydio €yKdqolo KoL{vov n}(p{rq ov€nruy!€vo ^dpu\ /o Kor ̂0r!6 yro q yp4yopt oF (o.

napo on6 rqv KorooK€u'i nEpinioK'rv €pyon€iov,odpouF€ rou( vEKpo'jq poq Ko' 6tF'oupyoLjUE rtlvn.

nrirE EgqovioqxE np6q gopd o Homo sapiens sapiens; YndpxouvKdnor( 6Loqovirq oXErKd p€ qur6 To oafq. OL oplorol6yoL dlouv ovo-rciJ{€r oord l]3 opxoLo^oyrKolc Xdpou( nou Xpovo^oyodvror rou}dxr-orov rprdvro { rprdvro nEvr€ } rd6€q Xp6vro nprv. oordoo, trLo or,Jp.ionp6oqor0rv orox€(ov uno6nldjvEr dr or orwpovor dvoponor efvor nonU noloL6rEpor. XptorFonoL{tvro( 6Jo vd€( olq €y6U€v€q r€XVrKa( yrort xpovo^6yfot ro\r opxoro^oyrx6v EpLoX6v r0rv €r6drv rou ylvouqAvoponog or oplorotrdyoL aloo! Koro^rl{€r 6r (dno€( n8proxiq opre-

foriv €(016 Xdrd6€S Xp6vro <o4q. Av outq o' XpovoloyiE( nx6ouv, r6rEo Homo sapiens sapiens urdpx€' Ebr0 xor eKor6 XLnrd63q xp6v(o Kr 6Xr

F6vo rprdlro nawE XdrdbEq Xp6vro.noo, ro rop6' , ro rohjLrpo nou pnupouf, vo noJp, .ho o,r o

Homo sapiens €e€^ix0rlKE oE oUyxpo\ro dvopono oro 6rdollFo on6 nEvro(6oL€q do( KoL €Kor6 XL^rd6€S Xp6vro rpLv. Or Enroqgov€( unooa,oJv o,. I ouylpovt Viuooo ovo,.uxorl . , o,o nporlyou/.vo,<oro r lhd6Es Xp6vro. nreov6v vo ovonrfxqKE F€ pdon ro noior6r€po rpoyotbro (Dv r.^Eroupyr{nv nou yivo\rrov yupur on6 u( eortq r0iv Ko-rou^roprirv Kor oror€ioroo\r avq p6oo y(o rlv €( v€ou 6r(ylon qqroropioq (or rou nohrofod qq qu^4q. Kord ncioo nr0ov6qro, I op(o

/qot f , opr0uq, ' \J, i , I , , ovon,uxOtt .ono,o o,. \pr l oqoroL, opI.ooF€ vo Xprto'FonoLo'jtt€ q odwpovq ottdia n€pinou nprv ond €KoT6

XllLdbr! xpdvro.

46 6 Rkhdd E. Leakey, Oalr@r (Niq YdfKf, [.p. Drtton, ]t//), o. 205

.L

Page 38: Ο Ταξιδευτής των μαθηματικών

Apiepnon p€ ro borruAo Kor iJ t ro ourpo

flpN npox0rp4oouu€ orrl ylooolK4 opiounon, eo npdr€r vo €q€rdoouu€

dvo €i6oS opofryrt( nou q€K(vto€ on6 qv op(0Urlorl U€ rio8rd i, nroq

vdv, os ouv6uqold U€ rlv opieulon U€ Klo6rd Awi vo XprlorFono€frlo6d, Xotr(no ( Kox6^lo, our6c nou g€rpooo€ XpnorFololotoE ro 6d_

trruld ro0 F€ rov i6ro nEp(TIou rp6no 6nos Kor ro olF€prvd nol6ld 'Evo

rl€ovdKrnpo, pepo(u4, €tvor drr to 66xrulo Efvql potrrKd. 'o$oq, q opi+

Hoq F€ rq 6dxrutro ax€r 6do onFovrtKd F€rov€Kr{poro: ro 6dx'tu}o €ivqr

nsprcploEvo oE opreF6 (oK6Fo Kt ov TlpooeioouF€ (ot 10 6:Ko 6dxlu^o

lurv tu6rdJv) Kor 6€v €(vor pohxd o( opx€(o. Av KpfldU€ ovuqroFavo

ond 6dxrutro dorspo 0116 Fro opoFlorl U€ to 6dxlu^o, €(vor dBo^o vo

n.pq€p6pqm€ fror 6lrl Udpq, Kpordrvtos ovuqroFfvo to 6dxtuld Uo!rr{ dpx.io qurod rou Lrstp(oq[€. E(vor nold (o^6r€po vo Fgtpqr€ U€ Xotrino Kor g€rd vo ro pd<outr€ ornv dKpl yto vo to {ovoKotni{ouF€ ofl6'

r€pq. (EqoKo^ou0otpE vo unooarouU€ €6dr 61 our6q TIou FErpd€t 6€!

q(€r onoKr{o€r rnv oqlplF:q dwoto rou oprouoi, (r &ol 6EV Unop€( vo(eupdIor, rov oprqr6 ondr Yro vo tov Eovoxplo$onor4oE( oro UAlov )

H opfopryl F€ ro 6dxlulo €(vor or(ouF€vlK{ o€ 6}0 to Fdpl rou Kd

oFou, or dveporTror Xplorliono(oioov oro nop€106v KdTIoro Fopq4 op(0

non( F€ ro 66xtUiq. AK6F0 K oq6rou Kqoop(oq(€ Kot dpxLo€ vo Xpl-oqolodror I olg€prv4 opropnt(i oKolouoict or (orvovfE( E(qKotrou_

eoUoov vq XploLlono(odv qv opiqrloq U€ rq 6dxlulo. Ol PoFoior ul

Xprprpo[orodoov yro vo UEtpo(v FEXpr ro 6aKq X rd6Eq A(6Fo Kol 04'

U€pq sqqKotrouoorjv vo Xpqoruonoroivtor €qEirypdv€( FopQE( opopqorl(

Kor unotroy(op6v Fe ro 6dxrutro, rup(d! on6 rous ontroU{ ovep6nouq

rcpd on6 qv E^ir r0Jv €nrotqp6vurv. Aqof 6€v €(vol i€nlK4, rl op(oFloq

F Iq 6{ixru}o gnop€( vo €(vor €(oou notrord F€ tlv optgUnol F€ (io-

6d. 'Op0rq I opiegloq U€ ro 6d}'rulo Uo( €v6loq€pE( n€po ond ro ou

oifioF ut( F€ rtv opiopqon $€ Ktro6L4 Yroli onor€lei Uro otlFovrni

flp6o6o o€ 6,r oqopd qv dworo tou oprouod. Opioqp€ rpolYouF6vos

6Jo €(6n opreU6v: rous n\orKodq kor touq ouvl€roYUfvou(. o n\0{K6Sqp(0u6( ovonoprord Iov opreF6 rrv ololxEi(rrv €v6q ouv6trou, EV(i) o ol} 47

Page 39: Ο Ταξιδευτής των μαθηματικών

wEroyt'ivoq 6€ii1€r rt o€rpd €v6q morx€iou U:oo o, dvo oUvoio. trt\ronli opi0ulon FE Klo6rd ouvovrdF pdvo rlnerKoLj( opreFo6(. O lsiL-Ko( oorpoq FE r( rtrpe( i ro Fdroo g€ ro Klo6rd ovonoprord rov n^lor-Kd opr0g6 rou FErpolp€vou ouv6trou, oi^d 6€ Foq 6iv€r Kdfloro o(oerlotTf( o€Lpd(. A€\r ufidpxouv ouvr€royFEvor qp(ogoi yrori rq xlo6rd Kor rq\o\r ' ro ro ovL,ptrunr(oupr (J( ,oJroo4po {or onl | 'oroo'do.Uo.

'Orov dgur( p€rpEr€ U€ ro 6dtr,ruld pol lrrroporig€ ov eEioqrE vq npo-oeaooup€ q 6dto€n oqv opieFtori foq. To 6dlrunq 6€v €ivor i6lq p€ro6jrouq, qouv 6roqop!T0((i ovEroro Kol, 6mv Xplo{ronoodwor mnv op€tqoD lErprlwor p€ Uro ouyKEKp{rdvl o€Lpd. fro nopd6Eryfc! os oMd tfptrou K6ofou or dveponor (€KNoriv vlv opioFrlo! on6 ro FrKp6 6dxtulo rouoprm€poD l8proa, F€rd npoxopofv mov nopdF€oo KoL Usrd mov Uaoo.AUId uno6n^drv€L 6rfto{D oqof o rcpdp€oo( apxrror gftd ro FrKpd 6dFu-io o.Ud npN on6 rov !aoo, 6rtuX oKprp6q o op'opdS 6io ipleror Und roivq rcr nprv ond ro rpio. Oq .K roUlou, rl opioptlof tr€ ro bdxruio Uo! 6av€r rlv .uKorpio vo €rooydyoug€ rouq ouw€royUdvouq oprogorj( rqv ow(if{ii loq yro ro,|q opr0FoU(. AeoU Fnopor!€ vo tr€rpdg€ [€ ro 6dxru^ooKolouoriwo( lro ouyKEKprlaq o€Lp4 or opLeUoinou Xplollronoro[F€ K]!-povoloLJv fro ouyKeKp$6!,rl oErp4 Kr aroLyEwfror I oproutrni oKo^ouo(o.

An6 rqv opiop'toq F€ ro 6dFulo o6tyl0iKofE oqv opiefrlot F€ro oriFo. E66, our6! nou U€rpdEr 6EiXV€r np6ro ro 6dtrru^o Kor U€Tdd^io Uiprt rou o6uoro(, Koe0j( ou{dv€( o opr6F6( qrlopieFrlotq. Ko-rdioLno Tnq nolor{iq op(eUqotq !E ro odrFo.{oKoioueodF€ vo ouvo-vrdp€ o€ UEpt q( Aoorpo^ioq Kor rfq Nioq fourvao( Mro qu^4 orovloi rnc Ndo( lourvaos ax€r ro roi(r^oKo oJoqFo op(ouorlS p€ rooduo nou pifnoof€ orov llivoKo l.'

H XptorF6qro rn! opioptoqc U€ ro ouigo €(vor oi-rio0.r( qov€pri ovoK€qroitrE 1r( r6ro(rEp€q qnorqo€r( ouvrovroFor UroS ofd6oq xu\rq,yrilv i noi€fLor{nv. H rKov6rrlro 6LoBiBool( 6roroydv otr^d Koi p€rd6o,on( nhpoqopriv on6 on6oroon, Xopi! I xpriot rixou, livor f€ydlonieovanfFo 6rov r^qord(Er( Kpuqd ro e'ipqfo i rous Exopof( oou.

48 , KarlMenningcr, N,Db€r rvo r(ts and NLnbet 5ynboh, o. 3s.

Page 40: Ο Ταξιδευτής των μαθηματικών

NAKAT L nopd6ELyFo €v6q oooriFoToq opi6Ulot!Ir ro oolfo orouq rooy€v€(q €v6( vloro( q( nonoro Nio( fourvioq

I = 6r{i p*pb odtaulo2 = 6.(Ld! nopdt.oo!I = 6EE d! *oo(4 = 6.(Ld! 6.kqq

8 = ar{Ldq 6fo9

l0 = ljlii I',ir

r5 = opLoEp6! oyKrirvo(l7 = op'or.p':iS Kopra!l3 = op'ocp6( owixapo!rs = opLor.poq 6.kif(

2l = op'oGpo( nopd!€oo(22 = opLorEpo Urp6 66xrulo

ftolrio ond (o rprrt4 ({< ,iiffr a Kat opl1trnlxa attllo)a, raa Kopt Mr\\xLp(Nio Y6pKr.' Dover pubkaronr, r96e), o. r5

H opi€flort trE ro 6d}.ruto KuL I opioFnol F€ ro o6Fo 4rov Kdnor€ 6ro6s6olavE! oE dlo rov (doFo (or Eivor oqoupo rotrd noloLaq. Toy8yovoq 6rL Fo( 6(vouv ouvr€royutvouq oild KoL nlto((oUq opL0uo6(0flor€^€L fro otuovlKrl tt€roK({ot npoq qv oqfpnFivn aworo rovou0rvdv opr0F6v. Av poliooufE avov oypdrn n6oo Filo ax€r oro (oidoLrou, pnopEi vo orovI|io€r 6Eixvovrd( loq avo fdroo do6rd. Kocr(ovrot oflldJ! ro fdroo, 6€v lnopoiF. vo notfs noroq sivor o rr^ter(6( opL0!d( rurv g4^ov- orlv KonLJr€pt r€p(m{!of, ixou!€ Fro oioolq qq roooqrd( rou(. OUUJ(, ov Foq 6€i{Er tlo ouyKsr(prpivo tfpo(rou orrl.lorot rou, 6nurq yiv€ror oIrlv opioFlol tr€ ro odrFc! 16r€ ourriI or^t X€rpovouio Fdq 6ivEr rov oKp(p{ ninerK6 opreu6. Mnop€i voUlv €Xoup€ avo dvouo yf ourd rov opr0[6, oMd {ipoutr€ oFaoo!noLoc Eivor. Xpforforrordwoq qv qpieFlo! tr€ ro o6Fo, o (6roq pp{KEoiid F!ra&ro. Kor o' €Udq rtv roKlKri Fi rou opr0uo6. Erouav{r(,yvopi(oul8 lt ox€IlKi oaorl rou oprogoJ o€ oxtot U€ dilooq opro,uou'. o, oxro.6. ioo^ pr ro o^,^o prp, rou ooporo\ nou ovonopcororv oprogoUs. 49

Page 41: Ο Ταξιδευτής των μαθηματικών

Apiou non p€ I6q€rq

H op(elrllort U€ oproUlKi( ^a{€lq onor€}€i ivov (l(6po po0F6 oqoipEolq yLo rou( qvepd)rou(. -Orov

Xprto(fonoLoLjg€ rnv opi0pnon F€ Klq6(d Kor ut! opioFrlot U€ ro or0Uo 6Ev axou!€ oltorvd oqfptFtvorJ(opropor(, yrori n 6Lo6LKooio ro0 oKoiouoo f€ €ivor plxovrxri Kor 6Evororr€i oqtpnUavn oKftlt_ An6 rtv dnlq, ro oqtpn!€vo ovllK€if€vo€ivor ovrKriF€vo rrlq oKirlilq (or or quor(o( oprouo( onor!troLJv rii.ronopo6.iyporo r€ror{rv oglprlUav{rv owor4ttov. Av 6U{r( ro Fdvo rour(dvouF. €rvor vo Xpnorpofloro[U€ xlo6Ld yro vo ro ouyKpivoulr€ !! Kdroro dlio quorKd o\r K€iF€vo €\16( ouvdtrou, r6rE 6s Xp€rd<€roL \rooK€qrouoor€ oqtpnUtvo. Aft6 rn otly!4 dpoq flou 0o €(qapouF€ avovlxo yro vo ovonopooriooup€ tvov opr0F6, 16r€ dx€r oplio€l I oqoip€ol. 'Onoq

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f6r€ rc^rKd oroqdoroov oL dvoponor vo npoo6ropioouv rou(oproUorC Ur la(€(( ovr( vo XpnorUororoUv K^o6rd, Kox(^ro 4 r6rp€q;nr0ov6\r 6Xr v{rpir€po ord €Kor6 Xdrd6!( Xp6vro rpr\r, oqo6 16r€ EF-qqviortKE np6q qopd n ypiyopn eovrFrKri yirirooo. 001600, oroproUrFK;q ^a{€rq 6EV €FqoviorlKov {oqvrKd pdnr( or dvoponor dp oov vo xpnorFonoro.v rl y^rlooo. H ovdmuln rov opropnl(6v traqEUrv rirov opy'i Kor npoxriptoE oro6roKd Ualpr ro otF€prv6 6EK06FKo o0orlFo. Iqv npoyfotlx6Tlro, 0o Fnopodootr. vo unoetoo0g€6r n €{€ir{t rurv opreFnllK(irv ii{€{rv Koouoraprlo€ €nEr64 o Homo5aprens sapi€ns (or€ix! 461 .{orp€lKd Koliq F€0d6ouq yro 1o\r npoo6roproFd rou rlterKoi opLoFor ro! ouvdtr0Jv, Udoro rt( op(0Ftoq(tE Klobrd i qq opioFlonq !E ro 6dFU^o. tLori ̂ orr6v vo Xp€ro(6rqvrq ia{€(q;

lro vo oxtgoTiooupE pro L6Co qq €{a^r{q( r{rv np6r{!v opreFrll50

Page 42: Ο Ταξιδευτής των μαθηματικών

ror \ [ .ov, oq ro ro[o'r ' o oror, l -oro opio-nol( /onorurv np0r oyovlrv qutrdv nou undpxouv uaxpr oriu€po Kor nou rL( alouv gll€r'ioE'

or qvop{rflo^6yor. Onoq axoutr€ ovo9ap€r, (drloL€( qulaq 6rtu( olApoo,) ' rq, Auo,ool io, . orov oJ\ oJvovrtoo\ nod,- l oopo o.oJllpovor tuponoior 6Ev pnopoLioov vo UErprioouv npoqoprKd nCpoon6 ro 6 o.'Eixov uro ̂ a{f yro ro .avo' Kor lro lih yro ro "6Jo,, or'6inor! dl^o 6[0's irov 0rr^6( .ron^d'. H qu^i Mr€pyKvrdpo qq N6noq Aqp'Kris E|XE Kor ouri !6vo 6Ljo opr0utlLKa( la{€rg ro .avo, Kol

ro "6i0,, Kor or6inor€ dlio npoo6ropL<drov oq (no^id'. A€v undp

Xouv oroLxlio yro ro ov ouraq oL qu^iq Uropo'joov i 6r vo f€rpoLiv f€(tro6Ld.0o npir€r vo rpooi{oufE vo U'tv unooioout'€ dl rl ouvo}LK({orov6not xor }prjol r,rv oproFdiv 0116 pLo qul4 nEpLopi(Ero' orovopr0fd r0rv opr€lqr(ov ^a(€uJV rrou undpxou! oq y^6ooo rouq. ofrpdxof OlryK, oro apyo ,4ptttl,toi: H to.opio Kat rc h\pd toue,.nlo"r

t(vor otUovrKd €6.i, vo Kpoloou!r rov (660!0 rou (!6Jvou. €v trpd-na vo onliooupr vo oufnrpdvouts, 6noq fxouv KdwL (dnoLoL qlopo.

noldyor, dr oL dl|]potror 6.v ovrLlot'FdvowoL rouq opr0UoLJ! napo!roLr opiou nou rous eirou! oL opLeplr'Kdq li€!'q nou XpttoLfonoLoliv.nolm ouxvd PranouF. d! o, ̂ ala! oulo6.LiovroL otr6 Xflpovo[i€!. npinrLroLro 5!o !o €p|'rlf0oi! ourord np'! Kordl'llou[E oE ylvLK.i ou

floltrd( dln€( rrpur6yo\r€( quirq, onrt rnv Aqp'K'i urq q N6lo AF€prKi Kor rt Ndo lourvfo, €ixov u6\ro 6do oproftttKi( ni{€Lq, ro tvo Ko'ro 5Jo, otrtrd rLs ouv6Jo<ov yLo vo usrprioouv !€yol6r€pec oFd6€! o€

our6 nou livor yvrrrord or 6uo6LKri opieFrlot. Iq 6uo6rni opiowo']enovqlq[pdvouF! onnd]q rq oprouttKi! ii{€rq nou ovroroL}oUv oro

3 Craham I eqg, Nunbe6 fhtough the ̂ qes ( ov6i!o, MacMl an Ed!.ations

e Crahah F eqq, Nun,be6: Ihen Htstory and Merrrg {Nio Ydprf, schockcn

-r.�

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€vo Kor ro 660, 6o€f qopdq Xp€r{icror. ,tvo nopd6€rygo qq 6uo6rx{f

opi€gtol( onor€l€i rl quiri tKoulroulyK{ii mtv Auorpotr{o.,o H }aqqrou XpqorFolorofoov yto ro €vo 4rov ro "oupqn6v" Kor I ldln yro ro6Uo 4rov ro .ou(oo{ip,. M€rporioov F€ rov ropoKdro rp6ro:

I = oupon6v Gdvot2 = ouxoodp ((6rot3 - ouxoodp-oupardv G6(o avota = ou|(ooop ouKoo6p ("6do 6do,)5 = ouKqodp ooxqodp oupor6v G6,jo,6Uo 6vo,5 = oukoodp-ouKqodp ouI(oodp ((6Ljo-660-6do,)

. _ To 6uo6rK6 odoqFq opiounorls nou Xprlotlonorofoov ourds or qu_

^e( grcpoDo€ vo npoxorp{o€t Kol o€ F€yolfr€pouq optoFof(. Iuv40oqolroc oEv ro €Kovov, ytori n o€lpd ii6€ov rlou Xp€ro<6rqv yro p€yotrJ-r€pous oproFouq oJwoFo yrvrirov un€ppokkd 66oKolrl yro vo UrIopoDv vo Irl ouForjvror. Aur6( nou Xplorlonot€i rrl 6uo6tK4 opioulonouvo6€i€r ould qv npoqoprKi opfetrlol U€ Frq opiogrlon |]€ ro 6d,X'ru,\o. EnoFivo(, ro ofoqUo qq 6uo6rKr1S opieFrlonc €(vor rlotri n€-ptoprouCvo Kr 6Xt r6rq{r€po Xpriorlq oqou Fnop€( €UKo}q vo qvrll(om_oro0€( F€ rnv on^4 opiofnol U€ ro 6dxr0io (or rorv 660 X€ptdlv. :rrlvnpoyFo(Kdrtrq, n 6uo6rKi opielrqol U€ tr6{€r! €tvor notrrj try6r€poonorE^€ogolKrl Kor Xp4orFt oftt rrlv opiogton p€ xio6rd. [/€ qvopiolnol !€{io6r4 j(dnoro( Frop€i dxr Fiivo vo F€rpd€r laxpr F€yq_^orEpou( oproFoljq, oMd tx€r Kor Fro rlgrF6vurn Korqypoei rns op(0_Fnonq_U€ ro Fdroo ruJv (^o6r0jv { F€ ro oop6 rov KoXUh6v. Iqv rpo-goprKi opio!4ot, o unotroyropd( Fordcr vo €{ov€Fi(Eror ox€66v !6^r(l€U€ rlv r€.I€uro{o opreFr)Iwi trEh.

Aoyo ruN notKaov qpxolo^oytKdrv ronoo€or{iv 6rou ovo(oluqrnrcI 6uo6rK( opieFtort 0116 qv Aqp(K4 oS rn N6ro AUEprKd Kor oC u)vAuorpoi{o, oKo}ofelo€ Fro ouqqorl yro ro ov rl 6uo6rn1 op(eFlo4ovomilqK€ {Exrprmd o€ (d0€ 4r€rpo 4 ov qvonr6x0lK€ o,dvo !{-

52 10- Flegg, Nunbe\ Ihrough the Ager, o. 9

Page 44: Ο Ταξιδευτής των μαθηματικών

(or fsrd 6ro66erlx€ o€ 6ll rnv uq4ho. odtos 4 d^ldf €(vo( ofYounolor4 nroqvdv 600 Kor I i6ro ! yl6ooo. 'Evo €n6F€vo o1d6ro

I E(aI{(n qq 6uodrK4q op(oFlolf, aro( 60r€ vq ouFn€pddp€r Iov

Kdnouv opr0F6v yro vo [poK6qrouv F€yoi|Jl€pol

(owqr !s rp€r( 6roqop€r#S ̂ d{€rq Kqr F€rd to .46, 0o €ivol .660

rpdpro,, rTo0 onuq(v€r 660 €n( rpio ow( 6Uo ouv tpio. Aut6 ovoUd(€

rdl vfo 6L,o6rK( (vEo6uq6rK4) opieUlol Kor tn ouvovrdF€ o€ 6rdQopo

Ffl rou K6oFou rro ouxvd on' 6,tl lv (oeopd 6uo6rK4 opiotqon,

ouv{oo( 6Fo! o€ n€proxaq rlou ppioKowor xowd o€ oura( drlou ouvo_

rd!€ rlv Koeopd 6uo6rK{ opteUton.To sn6!€vo ord6ro qq €El(qnq rov oproulttK{iv trfq€ov €ivol rl rI€

!r06rq op(efiorl, nou ur( r6to 0o npan€r vo fipoixuqrE on6 tov oplo-

Fd r(lrv 6oxrdluJv rou (d0€ X€proU. 'Ovtoq, o€ Kdno{€( rpu}t6Yov€q

ytrdoq€q flou Xprlo(goflorodoov qv n€vro6rn1 op€pnorl, or oproFrltl_

{f( r€{€r( nr po ond ro r loo.po oqv npoyForK6nl ro nrp,Ypdipouv rr \

lapovoFi€q nou XprlorFonorodoov yrq vo F€rpioouv F€ ro 6dFulo

rov onfororxo opreFd. Ardgop€S €(6oxd( q( n€vro6rqS opoUnolqqouv ovoKo^uql€( or( fi€proo6r€p€q nfi€(pouq. I' tvo orioqUo nouqwKtrLiqrrlKE oq N6r(o AU€p'K4, 6XouF€:

ravrE = (ol6Kntpo Xtpuaqr = "dvo oro dllo X€pr,6€Ko = .6do oi6Kupo Xapro,Kor avl€Ko = "avq oro fi661"

'Orqv qrdoouF€ oro €fKool dxouF€ ro .€voS dveporlo(o, rlou uro6l_

ldva rq 6dro 6{i}'ru}o ro! X€plllv Kor ro 6EKo 6dFU}o 10rv [o6r6v.

To oUoquo q( r€vro8rx{q optounortq EE€IiXqK€ o€ 660 6(o9op€

nKd €i6rl op(eUlot( F€ oprourKas li6€r(: qv n€!'ro6rK4-6€Ko6rK(

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006116 oforqFo Fnop€i orlv npoyFofl(6rlro vo 8[vol noiqrdrtpo. O

fto nopd6€rYgo, ro .6vo", ro .6Uo' Kor to (rp(o' fipoobrop(_

11- Fiegg, NunbeB: Iheh History and Meaninq,..24. 53

Page 45: Ο Ταξιδευτής των μαθηματικών

nINAKAI 2. f€v((€ufavo ox€6rdypoFuo r€vro6lniq 6€Ko6rK{(Kor n€wo6(K4q_€rK0006rK{( opi0glorl(

n€vro6l|('l 6EKo6rKi opouqol n8vro6rn1-€tKooo6rKi opi0f not

15 = "6aro Kor rtvr€"

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35 = .rpio blKdpro Kor rdwE,40' "Taoo.pq 6€Kdprq,

(Eixoor (oL 6Jo tr.wdpro"(nKoor '(o' rplo trEvrdpLo,

54

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Or opt0poi (l)q rdr6rnr€q

'Orov or oprofoi npoob(opiorrlKov yro nprirl gopd UE i€ftu(d ov6FoTo,6€v €ixov ro oqrlpqFtvo v6rlFo nou axouv oriu€po.Irnv opxl1 XprforFonotorivrov ruq E(io€rc! yto vo n€prypdqrouv ro quot(d o,vr((€iF€vonou F€rprdvTov. E{o(oiouooUF€ vo p}irlouF€ unoi€(Uuoro ouro6 ouloJwpovn yi0iooq. Iro oyy^rKd, ̂aFE , yo*e of oxen (t\o ku\6pt p6-6rq) Kor {apoup€ oFio{rrs dl €wooJu€ 660 B6610. d€ Oo ltyoF€ (ori a

l_

Page 46: Ο Ταξιδευτής των μαθηματικών

ofglover (avq Cuydpr ydv(o). To /ole (<€uydp0, oS opr0glr((6

pair ol people (tvo <cu\6t ovep6nov). H ooonl op'eFntlK{ Idqrl6Uo ovop{irou( Eivor ouv4oo( ro couple ((€uydp0.' OL Loqy€v€((

oulvd 6loeop€rKCq oprourlr((cs trC{€rs ylo 6(oqop€(Kd

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npoir0rv qv0p6ndv, tr6oo avrovo xup'opxoLloov rq npdyfqrq srousopr0uoUs. ^e(pouv enlols To votrK{i efn66rq flou anp€r€ vo E€n€pd-oer o np0rr6yovoq dvopono(, 6Xr fdvo yq vo or€leu€epdo€r rlv o(olou0lo rorv qpr0t6! otr6 qurd Tq rpoKoropn(kd srd6ro, qlld Kor yro

vo €i€KT€(!€r quras nq rpdr.q opxas. '

tryd oryd, or oproUnl(t( iC{€r( onCKrnoov y€vrK( rox (or Fnopo0-oov vo lplo(go[ornoodv yrq q ldrpnol 610 Kor n€proo6r€pov ou]lo-y6v owrK€rFtvov. Aur6 m+mqo€ 6uvoq qv ovrlr8rdnrol r0rv qpro

Fdv oC moopd oqqpnF€vurv owrK€rF€vov rou 6€ ouv6€ovrov g€ Kd-

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13. Menninger, o. l l .1a. Fle99, Nunbe6:Then Hittory and Meaning, o- 11.15. Menninger, o. 32. 55

Page 47: Ο Ταξιδευτής των μαθηματικών

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*

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pov yro qv ropoytoyi tpoqig €e(vqo€ otq fdvtUl HFroannvo rprvqIId n€p(nou avr€Ko X td6€S Xpdvro. Tr ouv6po(v€ Tlptv or'5 €K€ivq qv

sllol4; To nolorft€po dF€oo oro(X€(o nou txouF€ ox€rrKd F€ rn! op{o'

Frpl €avoL €va x6Ko^o trrKou tprdlIo X ld6urv Etdtv, 10 oTloio ovoKoir'

qrlrc oqv ToExoolopoK(o ro 1937 on6 rov 66Kropo Kopl Agnootrov ''

'Evo oKi{oo rou K6(oiou gofv€tot oqv Etx6vo 4- Eftdvo rou undpxouv

oKoholev€q nEvivro n€w€ €yKoTIdi oUo6onolnpdv€q ovd n€vr€ - plo

17.flegg, Nthbes lhtough the Aget o. 3/. 57

Page 49: Ο Ταξιδευτής των μαθηματικών

EIKoNA 4 IKftdo €vd( K6(oiou lr(ou rpLdwd XL^rd6bv aotv, ro otroio ppd€t€ oqvTodxooropoKio ro terz ond rov 6d(ropo Kopi Alnoolov. or nEviwo navE oUo6ono'!#ver .yKon( o.iraoov ft our6( trou ro Xdpo+ propoio€ !o H{pda. [To d(aroo ay'v. qnd qoroypoq(o, Ot otoptN< pGc tuv onidv Werydrrrjq rov ^ouKoqMnov! oaLn T(6ouvs, T(oK Mn(wtrw (Nio ydpq, Dover pubtkation, o.2).1

ooq{r( npoF€^€rlF.vn (oroypo9i opioFnon(. Aur6 ro (dKotro }dKouuro6nidv€' q Xpion €v6! r€wa6rKod ouorigoro( qpioFrlot(. troud_voq, or rpoibroprKol Kuvnyolrpoqooui^dKrE( nlpo ond ro n€wo6(K6o[orlFo opieulorl( Frop€i vo Eixov ovonrta€r Kor ro roiqrdrEpq ou_oriHoro 6uo6rniq Kor v€o6uo6rni( opieFlon(.

H Kooop4 6uo6rKi qp(eFtort XpqorgoroEi rlv opri qq rp6oo€ot(. O opropds rdoo€po €ivor ro.66o-6do,, ro oroto onFo(v€r p€poi,oq 6Uo ouv 6tjo. To aqr rivot ro .6Uo-6to 6rju,, Lo onoro roouror p€6Jo ouv 6Ljo ouv 6do. Ooiv€ror €nouivo( 6Il or Kuvlyoi rpogoou^id_Kr€S, nou (odoov rldvo qrl6 rprdvro X rd6€( Xp6v(o rprv, (orovoodoov rlv opx4 q( np6o0rolq FrKpdrv quorK6v opreFUjv, oqoi I 6006r_x4 opa0pnot unipx€ npr\r onrt qv ruvro6{ni opielton. MropoiF€€naonq vo uno06oougE 6Il, 010rl€(r,166( louldxrorov, Kqrovoorioov Korrov roi^ontroorooF6, €r€r6i n v€o6uo6(q op(oglorl, nou €FqovioqK€F€rd rnv Kooopd 6uo6u<i opioFton (or rprv 0116 qv n€vro6rK4 op(0-Ftoq, n€pddfpov€ rov flo}tron^oo(oog6 FrKp6v quoudrv qproF{nv. Au-16 ro pldrouF€ orov opr0p6 nou ovlorotxodo€ oro:{r, o onoao( qrovouv4o{r( o <6Jo-rprdpro,, rou oigo{v€ 6do €ri rpio Kor dxr 6Uo ouvrpio. AqoJ I qui{ nou Xdpo{e ro K6xoio rou ̂ dKou, ro onoio ppdotK€orlv ToExoo^opoKio, €(vor n0ov6v vo Xplo(gonoroio€ qv [€vro6rK(58

Page 50: Ο Ταξιδευτής των μαθηματικών

o p , g u o r . u n o o o u e v o u n o o o o ' J p o ' ' ' \ q r n v I n o ^ q \ c i f l o o r o v

Uir* , " " , , , " ' * " ' ro orob 'o r l ( "oo ' ioo ' " opio l rnol ' cu(o

; ; ; , ; , r , " ,"" ;r" ' l u, ' ̂ , ,uvn\o, poeooJ^^l ' , ' ' 'q\ \ '4 ' �oq\ ' �o '

au, pr ' lo o 'o ' 'g ' ,n fno\n' {noJ rhoov rpbor 'v o ^^rd6r\ u\

,n.", rO,U*, tpt* tp*l Koroldporvov rl! onli ilp6oe€on Kor rov

, oro o,o, u ovo' � ro inoq\ ' o ' �pou \orro'o v 'oouoo'o o' � rornuo

'" nrgrrorr | | 'oouo' vJ \onorlorroruuoov nv o{olp'oq 'o vo o l

," .J"" ' " ' "1,u""" o"u rMvo\ ronor ' ou' \r '

; . ' . " , " ' ' ' o o o r r u \ o o p o F ' v \ Q J ^ r \ o r ' l u J l r l o ' i v o I n r 6 d r" : ; " ' ; ; , " " o , . , , .

i , ' r o o ' o ' \ ' o o \ " o u q b u i o '

" . ' 'o ' o*o"o n o uftoo oo ooo' ov o'rqro' iu ' nuoo or lFov" 'o uo

; l ' ' " . " ' " ' o uoipooPo tq\ poLp- Yro o^4 o no^o^ po ov-

i" ' "" "" " t ^o 'o ' rov duTu "r"ru ' 0o n r r ' r 'o 'o o\ool ioo'" t .

"o^.1 , . t a " . " '1 ou/o/oro'Yro o on^ouorlpo rrnouo ' t oo

"rr-Ji," t,la<* ,n" *oro rqq 6roipEolq tv6q npdYporo! oto 600'

oio ri*epo *, *o tpio oo irov nopdloYo ov dv0purno' nou uno

;;";; t"*** *'ro.exvrxt! uil'q o€ oK6v.o' vq opYovdlvouv

l'".* "",* -r"'i '. *' "o €nrprtilvouv oro Kp[o rou ]€rgti!o' 6€v 't

*" *lii ". U",O*-\' Kdrr olo 6Jo Qor6oo' I Yv6ol q( bto6r-

looioq rts on^4s 6ro(p:orlq 6E oquoiv€r 6rr or KuulYo(-rporpooutrl'Kr'q

, " . ̂ " " , , ' n " , " . " t r u v ' / o o u o L r ! u \ o p 0 u ' d v n p o l P o r " 6 ' v

; ; . ; . " , : ; " o,o,^, ,o Lr ouo'*r o,u,p'o 'r \o noo\po'o o 6io

: ; ; ; ; " , : " o " p o u ' ' o u , ' o ' � o o l q o J ' o r r ' o n p o v r o ' o \ o 6 i o

i,"'"*".,utv" (or 6xr blfroLrpYio 6[0 lLrotJv Err€r6i 6tv unopxouv

;;;r;e:,t;," - *'re'.", e" npin€t npoq ro nopdv vo uno..oolr*€

;.r;;, ;o;.i", KUVTYo i r po{poou^tr6KrE( 6€v €ixo'r olontLjl'r rr\' t'voro

onoLou6inorE opreFoLj n'po\r ltDv Quol(rirv opr0$6v-

t o*1 o'o o ru | 0o H'rop'uoor ' vo ol 'u oo\ot 'u \ o I r

f , . q n I " . " " t . " ' ^ o " " " " ' r l ' " " ^ r l \ \ ' u p \ r o ' I v r o ' ' / r d b

,0U". ^lr-, a-O ndoo nr0ov6qro' or nEptoodrspor dvopono' o!o-

Page 51: Ο Ταξιδευτής των μαθηματικών

yv6pr<ov rou( fipdjrou( quot(od( opr€Fof(. Kdnoror trooi rouldxrorovgnopoloov oxt tr6vo vo gErpoUv noi6 Enr6fEto, XplorFonot6v{o( ron€wq6r(6 odornFo opieglorlg otr^d ioo{ Unopouoov vq X€rpi(ovror/rKpoa\ quorhou( oproporj( FF poorl Lr( npo{fl( r).). opr6pq Kn( Invnpdoo€ol, qv oqoip€ol, rov noitror oorooF6 Kol, (oru(, oK{igo (or ul6(o(pEorl. 001600, 6€v rptfi€t vo oufrEpdvouF€ 6rl or rruvlyo(_rpogo-ou,\iaKr€( Eixov r6ro(r€po gsydi€( r(ov6rrlr€( oro foonFoTlKd. no^ioirtuvrlyoirpoqoouiltfiE( nou ixouv €ntpr6o€l o( q oU\lpoq €nox4rEprop((owor ouv{oo( oro uroruflri6€S v€o6uo6r(6 odorrlFo opiotqolq. Yrdpx€r, 6UoC, oK6Fo I nroovdqro Kdfloro vao qpxorotroy(Kd €u-p{goro ri pro €K vlou €pFtv€(o rov noior(irv €uplUdrov va p(qouv n€_ploo6r€po qurq mrl UoonFollq qwAlqrn rov npoyrivov Fq( nprv on6qv €noxi qq y€{!py(oq.

Avo K€q0Io i(l)on

'EXouU€ onoKoiJ{i€t avov opreFd plFdrov rq oroio, 600 Kotrrjr€polrnoporF€ vo unooaoouu€, o6tyoUv qn6 rnv rp{rr6yovt opieFrlon F€xtro6rd oIlS oUyxpovE( opreFtnKdq id{€r( r(o( mo ouor4goro opr0F6v.nqp' dlo nou oul,d i€(rouv or dp€o€( oro6€iE€rf, pnopofp€ vo Kd-vouf€ (dro€q unoodo€r( doov ogopd oro 6ro6oxrKd pigoro nou oSiyrloov oulv Korov6!ol lrv quorKrjjv opreF{irv. Or rp6yovo( Foq Xp€rdorlKqv dvo €Korot/FUpro Xp6vro yro vo o6tyleodv oqv oqrlptgtvnaworo rlxv opropdv. Enofavoq, q rop€fo npo( utv oqrlplF€vt eworqr0iv oproFoiv iro! pooovrotlKd opy{. 001600, our6 ro o€vdpto oqpa(€( €niorlS qv dnoQrt 6tl I opieFnon 6!v 4rov gro orwpovn €eah{rtrlou EqEupiolK€ on6 rou( A(ydmtouq Ep€(q, or oroio( €ixov urEppoir-Kd no^, Xp6vo orn 6rd0€oi rou(, o^id [ro 6€qr6qro nou onoKrielKEKord rl 6rdp(€ro no^irov XrtrEr(6v, pto 6€€16r|lt0 fiou ptop€i, g€ Kdro'o Fopq4? vo Eavor 1600 ro^ord 600 Kqr I XprlorFofloqot qq qo_r(d( on6 rou( ov0p6rou(.

tMflo(vouF€ orov n€{poop6 vo ooFn€pdvouls 6r €U€ig, ol dveporrlor60

Page 52: Ο Ταξιδευτής των μαθηματικών

€rKoolof oldjvo, Eiuoot€ oopo( Kol €qunvol, €vdJ ol TlpoYovo( Fo(

, Bn0ulnoivE(. f(qri Xp€rdolrl(ov 1600 no^6 Kolp6 Ylo vo qvonr6-

rv rouq oproporlq ror tqv opourlofi no1( onl6{ 6€v dpo(ov ro guo-

rouq vo 6oultrla r;ror: vo ow lqooiv qF o0J! louq oqrlpqu:vou('Evo rtroro ouprdpqoBo, 6pos, oYvoEi ro dl ol np6Yovo(

r6oirEpo Kod rrl 6ldp(€lo rov r€I€uto(ov €Kot6 X t66Liv €Iti)v'

ro (6(0 o[u6€p(€(q (or €uqu€(( F' €ldq 'ono( Kor oriu€po' 610'

16r€ Oo npEflEl vo un4pxov P€_Yd^€( 6(qqoptq o'nv €uquio rrrlv qv_

rllov. Kdlloror, ovouq(polo, itov iry6r€po !{unvot' dMor om6oo

\,0 4rov Lueudoloror ' ' lo €ntn€6o tq( Lnr6rh6lnro( ror rqtu_

oflrrduroq nou Xp€rdolnK Yro (dnoro ono to opxoio niprq Kor Uvn

urto. Onr,rq ro ftdouvxtfi( ror or nupouf6r(' brrtr!€r 6u or (d€r rnoxil

rou Homo sapien, 'apiens 0o np'nt( vo (ouoov L{unvor dvoponor':r

€fior( nou dlouv rcpdoer fipo notrIo[, ioo( dTlorol r6ro(t€po €uquEiq

nn€or,,no' *on,o(ou uo (orovorloouv Irlv lvvoro lur! oproUurv kor rl\

oploFlon( ouro 6. olPoiv.r , dyu(,6rr !Ft(( t ruoorr t (unvorLpor on6

rou(npoYdvou( Uo( €n€l6i Udeof€ rou( oprouotq 6mv {Uoo(€ not'

6d. A d( ono^oFpdvouu€ to oq'lrl qS xilpovopLdq outo0 nou Xp€t-

dorlrc nolla( Y€vt( Ylo vo ovomuxo€('

H nop€io [poq u]v avvolo tov oploFdJv q€K(vqo€ rlotrd qpYd Kor ro

npdrm FlUqm 6r4pK€oov €(oro\'(d6€S Xttrrd6€q Xp6!(o Koodrq rlpoxo

oo,loolre npoq .rlu ouan.uqtl €v6q ouo lfoto( op10F6v, to on$ovn(6'

-rpo p4poio yiuo*ou 610 Kor TIto YpiYopo Au{ I rdol ouwx((€ror

m( oqv €ndFevrl Qdol rou to!6rod FoS, nou ovoq6p€ror orrlv lmo-

p(o rurv ypqmdJv op{e!6v oqv quY( rou no}rrloFo6'

61

Page 53: Ο Ταξιδευτής των μαθηματικών

opi0F[o4 oe ril]ro ei64doo o{u6€p*{ G(voU

Tinolo 6€v o6ty€( nLo ypliyopo o€ ovrnopdo€ol on6 Uro ouclrnol YLoqv ruguio rov (drov ouroi nou oyonoUv tou( oKiiouq srflrl0€vrol o€

ouroLj( nou qyonofv rq ydIsq, or ldrp€rq rur! oi6Yov ro Kotrd<ouv u€0ou|looF6, sv6 or qovoflKo( rtuv Youpouvr6v (d0ovror ovonoulKo mr

lqlroyr\oiv Us 6io our6 to Fn:p6sfo. E{ioou 6X{i<Er Kor ro €plillfo ov

ro (ulq Fnopofv i 6€v lnopofv vo U€tpriooov. Kdnoror EnLoquo!3q Kor

oL n.proo6T€por r6roxrir€q KororKi6rov <6t0v (oxLrpi(ovtor 6n ro oYonl

Fvo rouc (uio 6Xr !6vo Fnopodv vo FEtpoJv olld pnopofv vo npoo0a-

rouv, vo oqorpoLjv Kor, o€ Er6Lxaq n€pLmdrosrt vo 6ropd(ouv q oKaq,l

ruiv dLoKrtrdJv touq. "ndrq li{€p€ o :ndpKL 6l niYorvo vo ndp0r ro }ou-

pi rou;" llEr o oqoor0Jttivos r6roK|iq( o(6^ou. .Oo Fnopotoo vo €ixo

olK,r0ei yro vo ndo ortv Kou(ivo. To {(€p€ 6utu(. n46to€ KL &p3{e oqv

lnpoorLvi ndpro. H fdvll s{iytol €(vor 611 6rdpoot rrl oKa(]Jn Fou '

To Ep6rtgd Foq €(vor nro o0yK.KprF€vo on6 In YevrK4 Euquio rov

(drrv ovoyvopi(ouFE 6r KdnoLo eoivovror F{li^ov o(u6EpKri, Evti d}^o

orol[ruJ( ]o4d. Aurd nou e:]outr€ vo UdoouUE Eivor ov Kdnolo <60

Fnopoiv vo !Erp{oouv Kor, flotl:voq, ov ovtdopp{ivovror (dno( rou(

oproFoLjq- O npotyolF€lo( optoF6! Uoq Yro tq ofwpoq opollol

rEprtrdppov€ rtv on€LK6vron opr0plrLKdv i€{€o! €ni ttrv otorXtiov 63

Page 54: Ο Ταξιδευτής των μαθηματικών

€v6( ouv6lou. Aqod ol dvepurnor €ivql ro U6vo <ovrovd €(6n nou Xprl-oluonorodv ul qovn K{, yp{yopo oU odg€vq y}6ooo, rGroiopoivoug€ 6tl o cv ldyo optoF6( orod€{El 610 ro <{iro 0116 ourod rou €i6ousqv opi0Ftol (l€ m0ov4 €{oip€oq ro 6€}qivrq Kqr nS qd}orv€S - (60nou owos qoiv€ror vo €ntKo(vovodv F€ ypiyopo0q 4xou(). 1oo( 6Fo(our6s o oproF6( vo €ivol un€ppolK{i n€ptoplol(6c_ A016 fiou npoyForrxd o{louf€ vo qdpouU€ €(vor ov Kdrlolo (60 Fropouv:

1. No ovoyvopi(ouv ro n^4eoq rov orolx€lov €v6C ouv6^ou.2. No €flreuFoUv (r1 vo Xp€rd<owor) vo )vopi(ouv qv noMonlftn,

ro €v6( ouv6lou,3. [r€ rl po{e€to rrlq qr€l(6v(olq, vo u[oloyi(ouv rov r n0lx6

qpr0U6 €v6( ouv6lou.

Or nopondvo rp€r( anomio€rs oul]govoriv U€ rov oproF6 q{ op6un-o{q lr3 K}q6d ro0 66ooF€, o oro(o! €Eo}€iq€r rnv ovdyq Xp{o$ qSyl6ooqq yro q 6{}oon rou n\olKoi opouof €v6( ouvd}ou. Ax6Fo KorU€ our6 rov mo oooq4 opoF6 qq opiopnonq, ot €6lKoi 6tqgovofv oX€-!K(i U€ ro r Fnopouv vo Kdvouv ro <0ro. O fKpdXoF Oleyx, ifnopoq poenlorx6v mo AvotKr6 nov€rnoolUlo xot {vo! on6 rou( 6purd(-UAri rou,m0ui( xor np6lv np6€6po( rlS Bp€rovrK{q Erqp€(o( yro qv torop(o tuvMo0nUor(6V oxoh{i<€t qvoq€p6Fvo( m|( loropi€C 4iJuJv flou FErpoiv:

A'JIdC Kor dlks ropduoq @opks 4ow oqy4o€r rdnoous ovopdnous@o lov0qo#vo oulltrtpoouo 6r undpxouv nldqrfto €ft6s on6 rcv dv-opono nou pnopodv 6wos vq pflpoJv... Irc spdq|ro (lnopofv Iq (da volnpo"w €(gqor8 unoxpn0Elor \,o onovolooup€ ll€ tro oro€€p4 dplrpn.r

001600, yrq ro (6to €p.arlFq txoug€ rlv ovrlo€rn yvrilFl rou Kqen-ylq X. Kd.luoug on6 ro nov€filoquoK6 Kol4^o rou Aov6ivou:

A€v 'rndpx€( nu ote66v xoF{o qrEpo}to 61 Kdnoro (6q 6tus or oK(-oupo' '1 or nqnqyd.ro', utropoUv vq €Knoro€urclv vq !erpoov... t{qvdq-r€( opioFrprlq a)(er ovoqrp0€i 6n undpxouv mrs e6r€q qou{ opou,

4 1. Ctahan tlegg, Nunbe6: Ihei Hittory and Mea1jng, o.7.

Page 55: Ο Ταξιδευτής των μαθηματικών

ror oro dvroFo nou p€roqapouv q YUprl. Kdnorq (6q $opoiv6rorp(vouv opropLjs oe Kord Io dllo trop6Fo'o om(d oritorq,

d o lnoporiv vo €Kno(6€uro'jv vq ovqyvorpl(ouv, qKdfq (or vq IIo-

o\orouo(r( oroooT.ko! orlpdrov. ̂ lyo <oo pnoporrv oKdpo vo

vo €nqvqlofpdvouv fe },!on'1foro rous qpr€foUr ruv

dp0trrFK6v {lqi(lJv ror yponrdlv oufp6lr,rv €ivor o I6yoS yro rovoorx€ldv ((ouffr60r) €vdq onrffoo ox€6iou... H onouolo npoQopKnv

i orc(o no^loi 6@6(ouv vo ono6sFoiv m <do or UootforKoUs.'

.lld,( ylwmr 6Jo €r6rKo( orov rogCo qS ouun€prqopds lurv 4ilov vooE oKprp6q ovrio€ro ouFr€pdoForo; noro €ivor I ord\rqot -

11 6€v $ropoJv ro (60 vo F€rprioouv; fto vo poteteo'ip8 o€

,Il oqopd qur6 ro €p6rrluq, o( €q€rdoouu€ xdrlol€q orld t|( rllo €v

vo npooof i f i Kor vo oqorp. i . O XovC F .nr6rh6rr l ro. onovlod.

n€prnroo€r(,

0rK€( n€prnribo€rq

Il€66v 610l dxouF€ oKorio€r rlv roropio rou 'E{urwou Xovq, €vri( otrdrcu pnopoUo€, g€ €v1o^4 rou €Krlor6€uri rou, 6X( p6vo vo g€rpd€l

oE ooord oro foolFqrtxd €por4Foro rou rou €rio€vro FUrdJWo( nqonld( rou oro X{irFo. Auoruxdrq, ol €p€uvlra( ovoKdluqrov riu ro dioyooqv flpoyuorK&qro qvll6poto€ o€ oouv€i6rlro Ulv6Uoro rou €fioc6rur1 rou' Kqr orouoro(o€ vo Xruld r(q onla( rou oro Kord^trrltro orl-p.{0. H ol{e€(o E{vqr &r 6€v p(opo6o€ orr€ vo g€rpioEr oft€ vo Kdv€r

2. H. (almus, .Animah as Mathemalicians", Natue 202 (20 louviou r964), o. rr56,3. To nopd{.vo ootv 'orop(o ou 'E{unvou Xov( (r6iq rou 19ou qL.) '1rov d( (6F

sl'( oo@( onoF(oas oK6uo KL6rov o.mq'6.Niq rou 6€ pp'oKdrov rdrcu KGft.lvos v.qpdq 'Iuxoidyoq pp'lc q llorl rou Fuo]lp(our rc dloyo €lr8 ovqm!{€Lqv rcv6qro vo (6ropdcb oouvoioolr€q ovlopdoaq uv o!9pdnuv rcu rou utr(pouov €p@do€q fiov f9lov€ ollt o@@r1 oftrqqn, n f@' qqrolouo€ ro )cllnt!0. Av 6tros ro'r €(ov€ Un €pdr'tol (dno'os rou 6ev '1(€pe rnv qrdlrrtol, o 'Eq$

nvq Xovt dxov€ rS FqolFol(dslou '(ovdqles. (r.r.E.) 65

Page 56: Ο Ταξιδευτής των μαθηματικών

oproFlrKar npd{€r(. Konpdv€ Xov(l Auq I nEpiml!\ol ao€o€ o€ €rtqu-

loK4 rou( €p€uvnrt( yto dltro n€prororrKd 6nou €Kror6Euta( 4 €nloul_

Uov€q or€trvouv oouvoioorllq Flvigoro, to onoiq ovr[olpdvovrol tq

lopoqpqrrKd (60 (l €ror n}6vouv" to rlpophlgqro nou rou( e€vlol

Mdhoro, ro trdel tov Ep€uvtlrlirv nou a€Klvodv on6 trora olloo{Fom

dxouv ovoFoor€(: oqd^Uoro tou -Equrwou Xovq. 'txowoq tov 'Equrwo

Xovq (ord vou, or €pEuvlraq Eivor ol(6uo nro npooEKrrKo( doov o9opri

o(rs ovtnop(Liq rp€uvr( 4 orou\ un.ppotrrKou( roIuproFotj(

H €fl6gEvrl n€p(nrurol €(vol iry6r€po Yvoonl, oltrd n€proo6t€po €v

6roqfpouoo. 'Oro! ov!9€p€ o o€p T<ov Ao[UrcK, qmpov6po( Kor Fo'

eluqrK6( rou 6€Kotou tvotou ol6vc! €vo( K[]Forioq €vox]otwov on6

fro €Kv€uprorx{ Koupo6vo nou aqrro}',/€ 6(opx6q rr} qu}h6 rnq Uioo

oro nopoutpqqpr6 tou.'Av ipnorve mov nfpYo Y(o vo 6ldlE€r rtlv

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troy(oS oro flov€nror4Fro tou opdl!fioupYx, 6Kqv€ n€lpduoro 6nou

4. Levi Leonaid Conant,.Counting,, oro The Warld ot Mathenatks,6V l,

66 -,r,. tut", n. r"*.un (Naa YdPq, simon and schusteL 1es6), o 4r3

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sfior6sUovrov vo ovoYvop(ouv 6loqop€TlKofq oproUo'J( KouK-

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r€( fvo trro0€p6 oproud oufdr{tv nou 0o XplorF€tjoou! o( tpoq[ Ko_

noro si6o( pd<€r niw€, dllo 6iKo xq( avo €i6o( ororFd<€r €(KooL ldooE-

po.To rapdforo i6€r{ov 61 6sv npoooltouv 06Foro Fi}pr nou vo Y!

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ooov. ndq eapouv 6Il €Xouv qrdo€r oto o{tm6 opr0u6; n6q q6pouv 6t'

npd(€uor oKprpdq Yro 6axq i Yro €iKooL t6oo€po;

5. Donald R. Cdfli., ,4nmal lhdllng (KaFtrprr( Harvard Univ€rity Pre$' r934)'

6. u iwolo of4 oloqap*qr o. n€Prntdo.Ls 6nou ovdYvopicro( ro nueo(

uv otr(a|ldvov, rop( vo (olovoudq.rol (t.r.M )7. O. Koehler, iThe Ability ol Bnd5 to "Count,'' oo fhP wo d ot Mathena c5

dU.r,€nLf.lam6 R. Newman (Nao v6pdl,simon and SchLrsier' r956), o 49r'67

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rdTopror, qlqiou, fior drots vo €ivoL mo €UKoll rl oflopvlFove0ol €nro Fo_

9 Cuy woodtuft kor David Premack, .Pdmate Mathemati'al Concepts r' the

Chimpanee: Poportionality and N!merosity,, Natorc 293 (15 Onoppiou r93l),

ro. Tll'9ovq ouvolLlio !€ tov Kive€ I N6PLS, ouwqFo0xo @olYllti 9uoFni!,o.opio! oro nov€n,oquro tdvro Ktou(rl! (aNq6pvLq, re No'Pppiou ree2

I

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F€yolOr€po poeF6 quq qv dF€ol ovranl|]n rcu rl4oous Eviot€, 6tov 69

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9{( ouil oL,!6aflq( l]€ ro po}oyu6 e6os Ious Ko, rs ov\Kss nftuox.''�

12. john Mclekh, /V Dbe.(Ndo YdpK!, Fawcett Columbine, r9er), o. /. 71

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Or dvoporol Kqt ro (uiq y€vvrorivro( F€ ouri qv €rliyvu]orl rov !r_xpriJv opt0F6v. Aur6 ro €(60S €riyvoorlq €ivot ofiopoirnro yro vq opxi-o€r q 6ro6rKoo(o q( qp(oglol( qLld, mr ndlL 6€v np6rcror ylq opioUlor)/ oDr€ r(ov yro (p0rr6yovl opi0uorl g€ dq6td. M€ rnv Cvvoro,iorrov, U€ rqv ofiofo aXouUE opio€r rlv opioFlon, FropoUg€ vo ou_Fr€povoug€ ort [po( ro rlqp6t 6Ev txouF€ oro(X€io ril| Kdroro E6oF€rpd€r. frori €ntglvouF€ 1600 nolU 6oov qqopd 6qv oplotqo!; Oopropoloop€ vq noUUE: "Kofto, o qrurx6q, Xo<6f XrFrorqq q€XOpfCr ovor (ou( 6F( Eivor n/vr€ n rr oo€pr(. A( rou 6oooupt pro I uKqlp(o oKd.go - yvuJpict rou( llKpo6( opleFodq nop, 6tro nou 6€v Frop€( vo rousovoFdo€r,. Aurd Fnop€( vo toxd€L ol_ld ovo(rlrdf€ (dfl fi€ptoodr€pooro Uto €niyvoorl rov ox€6(ov rou rl€p oFpdvouv on6 Uio lo( €nrd(ouK(6€(. Avoclrdg€ rlv tKov6qrq rov <riurv vo €rrEloJv Frq rlpd{rlon€r(6vrorl(. Avo4lr{iF€ pro €riyv(lon rov quolld]v opleFdrv ot ono(or6EV qouv dpro !€ pdon q loyrK{. Avo<trdF€ rtv r(ov6qro vo owr-lolpdvowot 6Xr Fdvo ro rpio oild Kol ro rprdvro Kor ro rpro(6orq.M€ dMo i6yro, rl op(eFlorl [pdr€r vq €ivor Flq ovorKrrl 6lo6rxoo(qnou 0q ouv€Xcro( Uaxpr vo tKovonoqeEt n ovdyq Fo( vo Kooopioou_F€ rov rll0t(6 oprold onorouoilor€ ouv6lou UdS €v6roetp€1. Orpoo6oprou6q FrKp{irv opreF6v 6€v rKovono€i r( onorrio€l( Fo! oE4r( oqopd rnv opiounorl.

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74

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lupfidq n6rE €60j Kol 116r€ €K€i (q( v, ovoiy€t( ro m6uq oou drov ouvo-wdq fvo (ond6r qropldrv; Aurd ro Koiooo(oio K4q Xp4orFololodv,dpoyE, rov oyKuj8l €yK69o^6 ro0S yro vo K6vouv Koloooroi€( oKE{€r(;

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o€ 6rdprcro fil Frori 6po (o( n€pdxouv on6 6vo €(oro$-

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or ourd ro Xrrprd, rpopl{Foro nou 6€v ovru€idrmoov noft ro {o64tr'gro rout or tuvrlyoirpoqooulla(t€q -E!o poorK6 al€ovaKrqFo 1{0voyporolirEpy€r6v €(vor 6r UnopoUv vo nopdyouv n^€6vooFo rpogdiv.To nl tovoopo orqpuv rn.rp,n,ro' ,vov nlqouoFd vo nopop,rvr o,rdno(ov rdro Kor vo KorooK€udoEr F6vrFo oniro, F6vov 6FLiq ov rooqpd Fnoporv vo npoqu^qlrodv Kqr q fioo6rqrd Iou( vo gErp!]oE[.

oAoFfvo orqpd lnopo(v vo Xprlo(go(orl0o(v yro rpoq( q q€rv(

XpovLd, olld dvo rU4Fo rouq np:rEr vo qu^o},r€( yra vo onop0E( rrlv€ndF€vt Xpovl4 Kr ivo oK6Fo {F4Fo rouq Urop€i ov Xp€rom€i, vo y('

vE( qw(rciF€vo ovrolloy{( F€ oyood rou U€roqlpovror 0116 di}o Xo'pLd oypoKoll[py€(6v. En([^tov, oulti Kqoouoi rl (oih€pyoiuEq yr] 83

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npir€r vo yo(poor€i ooo{d Kor vo rlpogu^oxr€(. M€ qv Koi^(apy€roTrl( yns €FqovGror Fro vao 6(dorool mrlv opieFto[ 6Xr U6!o yrorinpan€r vo F€rprofvror rq otrnpd, o^id yrori npdr€r vq Faprdror Kol Iyq. E6poL6elK€ €rou6vurs o ouoxEtloF6( ovdp€oo orou( quorxoiqopreFoi( (or ut parplorl.

ndpo on6 q orox€(prorlrov oypolv (o( rov oo6€uiv, or [€ploo6r€-po( dveponor €npEn€ vo opyovooolv orot€trEoFotlKd ylo vo qur€i-oov, vo (ollrpyoUv Kor vo o€pi(ouv q oo6€rd, Kor $Erd €np€n€ vo €(,vor onnropdvor ylo vo npoquldooouv rq oo6€rd Kol rl yl on6 rou(6rdqopou( €nr6popEi(. Aurtq or 6pooqpr6qr€q ororroUoov n€prood-r€po on6 qv onhl opieutorl. noltroi dvoponor, (oedq Kor or r6roKr(-r€S olo0rlK6v rpoqiFov Kor oypor€loxioU dpxr(ov vo nt(ouv rou(ypoltror(oUq Kor rou( r€p€(q vo npoootrouv, vo notrtro[troord<ouv, vooqorpoUv Kor vo ototpo6v. Mro dpxouoo rdq qvorcfxoqK€, qq oroioq or ypotrpottKoi unoi6yr<ov rou! q6pou( rlrv or6Fov Kor ltov vorKo

H Enox4lou Xp6vou Kor or Korprdq ou!e{(€( dpxroov vo onoxlodv€nrntrdov orlFoo(o ord ul olygi nou dpxro€ v'ovqnr6oo€1o( rlysop-yio. Or oyp6rE( Xp€ld(ovrov avo lF€po^6yro yrq vo npoo6rop((ouv qv'(ord^\il € oX( nou 0o qUt€uov Kqo€r(. A01d onorodo€ n€ptoo61s,p€S uno^oyrolr(tq 6€66qrE( yrq qv [opoKo^o 0lorl r0rv 06o€0rv rovdorpov (or ro ox€6roofd rov g€i^ovlKdlv o:o€{iv rou(. 'Evo rcpd-6€qFo rou rlfEpotroyrqKo, unoioyroFori nprv orld tl]v €e€Up€on rnqypoq4( (3s001100 n.X.) Eivor ro opxoio popul0rv(o(6 aroq, ro onoiodpxr(€ Fr rtv eqprv( roqF€pio (21t Mopraou). O npdro( trivoq 4rov opivoq rou Torpou. Aur6 uno6ttutv€r ril| ro lp€pol6yro Koo€p{ionKso€ fro €floxi flou o'H^ro( ppLoK6rqv olov Toripo, yUpo ol( 2t Mopri,ou. L4ro r€roLo rEpioSoq dpx(o8 n€p(nou ro 4700 r.X. Ot Iougdpror,dvoq oK6Fo nro opxqioq lo6q, Frop€( vo Eixov Kq0r€p6o€r ro 6rK6 rou(

lF€pol6yro ydpo oro 57oo n.X.''Ol€( quras or onopo(rrlr€q 6poorlp(dqcc - or 6rdrpopor unoloyr-

84 l. Davld Eugene 5mith, o. r7.

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rcu oxer(owqv lr€ rl( onoo{rcS tpoqtu0'v, q Yl, rou( qvopd

ror r( €noxi( - nopoKivloov rous {up€pvdJwE( Kor rouq ̂ oYror€q

sq€dpouv YpnYop6r€pou( roL €uKol6rEpouq rp6nouq X€lproFo6 rtrv

otn v. Avom{XolKov rp€(S 6roqop€1lK'{ }to€rq 6oov oQopd oqv

opreU|lllKolv opxEiov: or pdp6or PE eYxorl6('' ro oKolvlo u€ Ko

(or m n4trrvo Koun6vro i ftoPKE(.Or pdp8or F€ €Y(onaq {rov (6Ko}o { qdhv€( pdp6or nou €(Xov poe-

bnoEi 4 Xopoxt€i oK6nrFo, aror drore !o uro6ql6vouv oprouofc'

rpd!'to x rd6ov €r6v K6l(otro ^6Kou rlou ovoq:popE npoqYouU'voc

mls( flqp{i8€ryFo groq npot6Yovl( pdp6ou U€ €Y(ordq -Evo d^lo

i6€ryFo €ivot dvo KdKolo nou ovorcidq l(! o' 6vov qrop&ono

ltuvn -Evrouopw olo Zoip, ro onoio XpovoloYEitol U€to(8 9000

5500 n.X. Kor ro 6Jo ourd K6Kolo ixouv nol^or 'S €YKofl€( non

(ouv oproFqrrKd Uotipo 'oF(,(, or r€proo6t€p€S pdpoor U€

E(vor Eftrrv€q. XpnolFonolof\rov Eni X^Eri€S ono lou( on^ouq

Iro vq roroYpogo'rv to oYooo, rr( oulotrloYr\ (or ro oup

rou( (ErK6vo 9). ArlBoqftiq 4tov I 6rnl4 pdp6os Fs €YKond(

nou )plqrFonoroUvtov lLq tlv KoroYpoq4 tov Xp€6! Mro UoKpro' l€

n( pdp6o( Xopooodrov Ue €Y(ord( Yro vo 6€ixv€r to l,Vo( tou Xpd-

o(. Kor6rN rlv doxr<ov oro 6Jo o€ ol6Kllpo ox€66v to U4Koq trl(

[o onUor((ovrov 6 o pdp6or nou toipro(qv rl Uio F€ rlv d^lq H pd

p6os !s ro lqdlo d(po 4tov ro Xpdoq (or Irlv KporoJo€ o 6ov€rmiq'

ir1 pdp6o pe to urKp6 dKpo rlv anoLpv€ o oQ€d'q( Kdroro orryUi

0py6r€po, 6tov 10 6do rF4uoto q( pdpbou €vdrvowov ndiL 1600 o

6qv€o{q 600 Kor o oq€ddrrlc |(nopodoov !o Koto^dpouv ov €ixov no-

pomqo€i ot €YKona(, Yrori to 6Uo tF4goro {np€n€ vq rorp(d(ouv on6_

Or pdp6or F€ €YKoniq XpnoLuonorodvrov nowoo otlv uqq^tq oqv

AqAnl, oqv Eupdrnl, oto vlor{i rou Erplv(Kof OKEovo6, orqv Au€prn1

Kor oqv Kivq. H lt€rl tally npodpx€ror on6 q lor(v(K4 IEel raie4 rou

onpoiv€r .KoUU€vn pdp6o(, To Yponl6 odUpotro rov Kwd(ov Yro ro

85

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tl(oNA e. IXa6Lo urqq {pNtrdv6LK'i( pdB6ou U€ Ey(o;i< nou Kqraypdg, ndot 6ou,^.Ld ala yiva. To r(Ko( ritq e(vor 2s ch. EXa6m onj qoto\portio, ApatltnktE )i-taE KotdptqflaKd otiypo)q rou KopI Mtrvy(.p (N6o y6pq, Dover p!bticarons,196e), o.21,1.1

ouup6ioro ax€r rpio rp4Foro: oro ndv0r Utpo( undpxouv 6Uo xqpoKtipEg nou o ivoq ouFBo^Gr Fro pdpbo p€ EyKont! Kor o dllo( €vo poXoipl, Ev6 Kdro undpx€r avo( XopoKr{poq fiou orluoiv€r .FEyd]o,. Eno-p€vot ro ofFpo^o ovorop(md Uro (F€yd^n pdp6o F€ €yxort(,. Mro€vbroqapouoo roroprK{ n^tpoqopio yrq 1r( pdpoour F€ Ey(ofla( npotpXEror ond q Bp8roviq, 6nou ol €v i6yo pdp6ot XplorFonoroJwov lixptro 1828 yro rlv Koroypoqri rov qdpov. M€ydlor oopoi rdrorov pdp6urv €ilqv oroolx€urEi orq fi(pro rou Korvopouliou. To 1814, I pp€ro-v'Kri Kuplpvnorl onoedoroE vo l( KorompdLlj€r Koiyo,v,rd! r€C F€ orot€-^€sFo vo yivouv o1d}.q Kord ̂doo( (or rq Kr(pro rou KorvoBouiiou.

H 6€drsprl U60o6oq {pnot( opreFluKdrv op}€(r0v €ivor ro Kop66vri ro oxorvi F€ K6Fnou(. Kr ouoi I Ui0o6o! irqv 6ro6€6ouivn Kor ovoKotr6gqx€ o€ 6rdqopo ldpt ortv AeprK4, oq B6pEro Kor N6Ilo Ap€-prKrl (or orlv Kivo_ O Ado ro€, o Klv€<o( qd6ooqoq rou sou n.X. o(ri-vo, ouvaoqoE vo srllorpall]ouv ol dvopurnor mo 6:orFo K6Fftuv oE(opbdvrq, XptorUonouivroq ro urq Uro paeo8o ypoq{(. To rlto €q€hyF€vo Kop66vro p€ K6Fnouq KqrooKEudolnKov 0116 rou( .tvKo oro nrpo0. Ov6Fooov ro kop66vro rouq &rnou Kor ro Xprlouonororoov yro86

Page 76: Ο Ταξιδευτής των μαθηματικών

Il( €n(orlu€( ouvol^oy:q (ElKdvo l0). Tq rltrou €ixov

ooQop€rt(d si6l K6Un0rv (o( 6rdq€pov €n(olc oro Uri(o(, oto

(or oII] odoll IUJV ouu[]€YFdruJv rrxv (6unurv H ga0o6o( {Iov

notrialoKl, drorE rdv(, louq KoroYpdqowov oK6Uo Kot Frl oplo'

r 4poqoprl.H rp{rt ud0o6o( qplorl( opr0pnrrKtav opx€irxv €i!o( rl nro orluovl-

ro nil^lvq (oondvro |1 pdpKEq. flo noi}ds 6€Ko€t(€S, or €p€uvrlt'(pp(o(ovrov o€ o6\luol Elorr(oS lulv FL(pdlv mitrLv(,v t€

rov rou ppaelKov ottq ronoo.t(Lq t0rv v€ohelx{tv r6i€ov

fuJ( 3500 n.X.) Iq( f6vrFnq HFro€l{vou Aurd ro r€XvoupYiFoto

Ko0ndvro ovqKoliqrlKqv mLS notror61€p€( roroesoi€f oYpo(onl€p_

IIKONA 10. t(olv( ritod tdv'lvKd Or K6|no'e(vo( u q,nqio. O, opLouoi 6,qp6<owo' qn6

io ndv@ pfpo( rcu (opiovrou f,poq rc (dro

(oorcypdq{d on6 ro stock Montaqe, t('iYo) 87

Page 77: Ο Ταξιδευτής των μαθηματικών

88

y€r(ilv, off ro Todvou Nrdpo oro noKrordv oq ro Mr€trvrtBnr mrl vGro6unKi Toup(io KqL v6lq, uJC ro XoprodB mo lou6dv. To rq.lol6r€-pq Kounovto rpo€pxovrol 0116 ronoe€oi€f oro lpdv xot XpovoioyoLi-wqr oro 8000 r.X. To Koun6vro txouv ogorpr((i ox4fo, oX4Uq 6iorou,KurvrKO uro€r6ds, rpryovlK6, op0oyr6vlou ropoMlloypdFFou Koe6q Ktd).lo (op€Fq€p{, rnpdq€vo oxilorq. To Fri(o( rous (uFq(vEror ouv4-0o( on6 avq 6urq rdoo€po €(oromd (ErKdvq |).

Or opxotoidyor o{€qrnKov roLloUq rp6rou( U€ rous orcious 0qFropouoov vo €ixqv Xplorporoln€€( ourd ro Kourldvlq, 6rloi ylo rq_pd6€ryFo, oS nol6r(d lotxvibro, o( nt6vro oE oFo6l(d rurx1,(6ro, ofqoi^rKd odFpolo, o( yuvorK€(o €r66fuo, uJ( Kopqtd (or of pdlot.Omdoo, ov €(vot oooq I O€op(o fiou {€Kivlo€ 0116 qv Nr€vi< IFow_[,4n€o€pd mo nov€rtoriFro rou Tdlos, rdr€ KoFld oft qur€( l( uno-eio€rs 6€v Eivor ooor4. :tls 6€Ko€r(€( rou 1970 Kqt 19s0, n lFovr_[4n€o€pd €p€{vlo€ 6€{o6rK{i 6aKo Xrild6€( (o0r6vto, on6 €Kordv 6€r(odq(qpxorotroytKd( roloe€oi€S, ror qvoKdtruqr€ 6! ovorqplmofoov ouyrc_Kprg€vcl ovrrK€ig€vo xqr Xplolgonorodwov ylo rlv olpnon opx€iov.rLlo ptq b \\ nptv an6 q ypo94, rl llovr [,4n€oepd t€€r:

Io Koutr6vrq FopoUv vo ovo)doJv aq Nsolouil Enoxi, ydpo oro sooon.X. Eqs-A(X€trov qKolou06'1os rs qvdyK€s q( orc\a(oq, opxl|(d yo qvnqpoKorodoryq rurv npotiJvrov tuv oypoKqnrpy€rdv (oL oqv om,K,lEnoxt rlXp{ol rcus €r€ndqK€ onlv nopoKotoJeryn tuv npoi6wov nouKorooreud(owov glo spyoolllpo..

ILjpguJvo U€ qv un6e€ol ouni, ro Koun6vro XprlorForrotouvrov €u-p€os mouq toyoplooFo{( ofi6 qv oplri ul( dy6ol( Xdt€rioq (navr€X rd6€( Xpdvto [plv ond rqv €q€dp€orl q( ypogi(). To fipriro (ourl6-vro, nou XprlorFonor4onKov 0116 to 8000 aoc ro 4400 n.X., €txov or 6q01Fo (ot oxd6lo, Kor ro 6roqop€lK(i oxriford rouq ovqnoprorouoov

, 3. Denise schmandrBejserat, Befote Wtjtiog, Vot. 1: Ftoh Couhting to Cunei

rorm {'odr€, unive[iry ol Texas prers, ]992), o. /.

Page 78: Ο Ταξιδευτής των μαθηματικών

llKoNA 11. KounrjvLq dPGflorl9

on6 q 6u { Aoio (ooroYPoqio

t. qv o6dd rlq Ntvi( tfovt

Mn€Gpd (oL rou Mouo€rou lou

Page 79: Ο Ταξιδευτής των μαθηματικών

6rogop$r(d ovrK€iF€vq. flo nop66€ryFo, ro Kou(6vt U€ uro.r6lc oX4-

Lro ovonoproroJo€ dvo {(ofnt f€ }d6t €v6 q p(pr1 oeo(po 4rov UlqlE(oUpo orrlp0lv.' ,,i€rd ro 4400 n.X. XplolUololodvrov 11016 r€ptoo6-r€po oxdbro, Ev(ir ro 6ro(pnrKd orlud6lq rouq ayrvov fl€ploodr€po (olrl(o noldntroKo. Kord q 6rdpK€to rrls npdrq( n€pr66ou rov oypo(ol-l€py€rdv (8000-3100 r.X.), ro (oufi6vro XplorFoTrolodvrov f€ oxiorlavo npo! dvo o( rpo( ro ovt|KEif€vo nou U€rpodoov. Tpio Krodnro I,r€id6r F€rpr6vrov U€ rp(o Koufi6vro ulo€r6ofs ox(goro(, €v6 rdoo€ptf

F€<oLjp€( orrlpdrv g€rpr6wov !€ raoo€pts oqoip€q. Aurd 4rov tvo €q€-trryuavo €i6os op(oFnonq V€ Ktro6rd, 6[ou ro (ouft5vro 4rov tro( gTloy-

U6vo 6orE vo npoo6ropi(ouv ro ovr(K€(g€vo rou F€rpl6wov. To (oun6-vro 161€ 6€ XpriorForotofwov yto vo ovoroprorodv oqnpnFlvou(qpr0UodC.

Or loup€pror €q€upioKouv rn ypoei

:q Xo^(oherK4 EroX4 (4500-3000 n.X.) ouvaprl Uro €vrunooroK4 oiloy4 oq friv'Fn Hptodlqvo, I oroio 4rqv ox€66v ro i6ro ouwqpqKlK4

Fs ro E€Kivrluo rr'rv oypoKoMr€py€r6v. H o^lo}11 ou{ 4rqv 61 10 Fr-Kpd Xoprd oypor(oitrr€py€rulv qvomixon(ov oE olrlolviq rld^Et(. H ovd-mu{l Illj! n6l€ov ouvlr€o€ U€ r! Xp4ol nro rofuin}oKurv Kounovuirv.Or npdf€q ndtr€r( €gqov(orqKov oro Kordrr€po rpriuo qq n€6rd6oq q(1.4€oonoroF[o!, oE Fro n€ptoxil yvooq VE ro 6vogo :ooF€pic! nou r(i,po orot€i€i ro v6uo lpdK. Eni n€vr€ X €ri€(, onoF€rvdpro ourou rouo66ioyou fio^rrouoU nop4r€vov xpupuCvo Kdro on6 qv dUFo rn('poKrv{( €p(Uou (or, F€Xpr vo ovq(oiugtodv mo np(iro Uto6 mu 6t-Korou dvoror, ouivo, 6€v u oqJ|o<6lomov Kov qv (rlopE4 rou(. Orloouaplor lqrooov oro v6r(o lpd( yUpo oro 3500 fi.X. K( €qttoEov UtoouroKporopio rou 6({pK€o€ F€Xpr trlv KordKrno{ In( 0116 rouq Bopu-,\dlvrou(, n€pinou ro 2000 n.X. Mixpr ro 3000 n.X. urnipxov n€ploo6r€-

90

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0116 6668(0 [d^€(q orqv n€proxi qq loug€ploq, p€ U€yo]Ur€pnII6ll Oup. EK8f <o(oov fl€pinou €fKoor navr€ Xdrd6€( drogo, €\rdr

unlpxov lu( ror 610(60rr( Xdrd6r( droFo oro rpryupu Xop,d Fr*py€r€q. 'kn( ouq rnv €rox4 €ixov ovomuxe€i KL dli€( 116

oq f6vrUt HFrod^rlvo.'0r vf€q ndl€rq rnq 6uru{( Ao(qq 6rluo6pynoov 6vo fli€6voouo

wrpyof qv0pdrn(vou 8uvogrKod, ! €pyor(K1 6(vopl rou ono(ou Kor€u-orrol(3 rpoq Eq8r8rK€uu€vo €noyy€^Uoro Kor flpoq rtv nopoyoyri oyo-06v. fro vd oqrdoouv or r6lErq, €npEr€ vo ovrail{iooouv €trrop€fFo& rqr npdr€( dI€( F€ dil€( rdl€r(. fto nopd6€rypo, oqv n€prox46nou BpioKovrov or nrii€rq r{rv louu€pi(ov, 6€v umlpx€ rpdoFoorl o€Kol{ {utrsiq i orKo6ogrKors ^ieous i g&o^iq 6nurs o 10l(6S, ro oo{glKor o Xpuo6(. H ovdyq Ufioqopdq r{0v oyooriv mr €no^4e€uolc r(,v

{oFiov axovov oK6Uo U€yo^dr€p.C rC onorr4o€rq yro }oyrmeq. Hnpondosro Enauorl( ouliv rov npoplrlFdr0rv 0116 rou( IouF:prouqo6{noF o,o no}unloro rounov,o Ko,, ,!irKo, o,q ypoql.

'orov 6vo eoprio U€ Kprodpl <dro yro r( qypoKqitrtpy€€( 4 nopqy6

|l8vo oyood cnp6K€11o vo Lr€Ioq€pe€l o' ivo\r d^]o 16no, ro eoprfodnp,n, vo ouvo6'uL,o, ond,vo,rn/po,po, L,or oo, ' o oyopoo,n\ vo

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0'avo oorcJl, oqoU o Koodvoq FnopoUo€ vo Ki€qr€r oyoed (oL F€rd voda|l)er mr rov Kord^}rio opr0F6 (ounovrdv, aror rim€ ro qop(o vo eoivsror flI4p€s. EnoFdvoq, oL n6l€r( rtq ltvrFrl( HFro€Iivou €mv6toov

Uro fqurun rsxvft4 yro vo qpoupoiv ro Koun6vro: ro Koiumov F€ nrltr6(or F€rd aqrnvqv rov nl^6. A016 6lgroupyoro€ Uro odnpil Urldio on6qA6, ldoo orqv ono(o pp(oKovrov ro Koun6vro. 'Orov ro qopdo nopo-666mv mov oyopoor4, €K€ivoq aonoy€ rov nl^6 (or €ro^4e€u€ 6r ro

o. Vo linq (hdmber., Rd)mot C pw D.v d He lih), Tl-eodorp cdbb ro l. pr

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Page 81: Ο Ταξιδευτής των μαθηματικών

Kou(jvlo nou rl€p(€ix€ ovumo(XojoovEuqudql

oro oyo0d nou nopo66owov.

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-rlv n€p(ox{ rou OopoLj(oFnovr, rl€pinou €Kor6v n€vri\,rq

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f l 'o:ovrov proo o,o poK€^o orqv r{offprx4 ,ou ,mqov.,o np,v {r lvr LnoU/vru(,- onoro( ,r\f ornv {otoxl rou ,o 90Ks4o unoporjos

ofl,16q vo (6topdo€r> oro €{rdr€prK6 rou roro Koundvro n€pcx!. Apo,6mv nopt6r6€( o, dvo qoptdr4 tvo qoprao F€ ro qoK€16 rou, pnopoU-

:: vo 6,opooH orqv r to{rp,q rou J n,qovero no,o uno,rurror nu(

lrov r0 9oprio trov ) ,hKo npoop,opd. o oyopoolnq propouoE voovoi{€r ro qdK€tro Kor vo €fiolno€jo€r rdoo 6tl ra or]F{ibro o1o €{m€_prK6 ouugovoUoov Fs ro Koundvro oro €o{drsprKo, ooo Kor ou ro gop_

- To nopondvo oJorrlgo tr€roupyodo€ (o_tr4 ldxpr nou or i{unvor

Ioup,pro Korolopov or. ro rounovro noJ nrprFrlov or{dNtor irov nr-prro. rolovo nou ,rpoypo,r(o Xp o(o,ov.trov vo qr lof i q nnt,q cnlqov€lo trovuJ orlv onoio €ixov XqpoF€i ro Kouftdvto nou €ixov (oroF€lpte€r. ly,(ortMrqi!v . ao!,v(r! tou\ poKt^ou, Xdprv rov on^r,Jv nifuvruvn^onor{rv (nrvoKr6uJ\), rr nor yswllotKr I ypo,pd. Irlv opft, ro n{^rvo-r-1o(oro^orornpouoov ro poo|lo Xopor(I4pro xo {ur! Xopoyprvuv qoxr-^rrv Kd0r roun6vr rtoKo,^ouoouor vo ovonopru,u rvo orry^rrp{prvo

92 /. 5chmandr-8cserar, o. ll4

Page 82: Ο Ταξιδευτής των μαθηματικών

yrori or opreuo( 6€v €(Xov 6loxopor€( qr6 to U€lpouF€voAu16 ro €6oq opioFrtons ovotiltro\ ouyK€Kptqtvn apiap\'

0rT6 rqv Nts!( :Uovr-N4n€o€p{i Kor Xpnor$onorofwqv o€ qoKtlou(

nioKtoro Fflo{t l5O0 Ko( 3l0O n-X. Ta^oq Ydpo oro 3100 nX., ol

6roxdJproov ro oftotun6loro fiou ovonoplototoov tov oplo-

r(ljv ourK€lg€vov on6 tq i6lo 10 owLK€(F€vo Ot oqlprlFCvol qpLoFo(

v orov nrll6, €vti to n(Kr6YpoFFo lou F€tporpEvou ovtr_

Xopooo6rov F' avo €pyotr€(o Ypqq{! ndvo orov nri6 (ElK6vo

l3). Aur6q o 6oxurproF6s opleu6v Kor ovllK€rFdvov ttov noiU nlo qro-

Kor 6r€uK6truvE rrl Xpiot tov nlq0rK6v oprepdv nou 64

l(tlvdv qv rco6rn1o, oE ouv6uqoF6 U' avo ontr6 nlKt6YpoF$o, nou

npoo6dprc ro €i6o( mu qvrr€rFavou. An6 tq orYFri nou rq qvrtK€(p€_

w 6s ouox€r@vrov nlo U€ tou( oploUoLit ro nl(IoYpduForo TIou to

ovorTqplmoUoov Uropodoov vo Y€vr(€uloriv 6m€ !o ouFpotrihuv non-

ld( 6uqop€r(ac (6d€q Kr tror YEW4olrc n Ypoq{ H tUovr'['1n€o€pd ro

ouvo{ic( ur( €q{(:

Io KoDn6vro fds trpoogapouv fro pooUrEprl Korov6lol q( Alsls qs

ypoqlq. Ano6€rKliouv 6rr orrlv EYYJ( Avotohi q Ypqq{ npofKu{e otr6

ivo opreFF(6 raxvoopd (or 6Tr, orqv npoYFqrL(6q1o, rl Ypoq4 iTqv

dvo noponpoi6v qS qqqpnFavns opGfnons. Orov 6roxUrpi@'l(ov ol

awors r0rv oprefdv on,t 15 Cwoleq rov f€rpo6F€vov ovnKe'pavov, rq

nrnoypdwom 6s 6ilovqv nl6ov onoKlao(rd rov opreu6 tuv oYoe.0v

t€ fro ovT(ororxio dvq npoq 6vo ME qv €A€UpEo'l rorv opLeprlfttirv

{rt9iov, I rKoypoeio 6.v trsprop(drov triCo! orouq loYqp'doporjl,

o),id str€ndEKE Kor o. dllss ftproxds tuv ov0p6nrvov €n6r6[.ov H

€q€rlpeon Iov qqqpluflurv op'olrrlrrKuiv qrqql0rv rltov q apxn rov Uq

onporK{nv xqL €n(ons, I opxi uls Ypq94s'

[/ro on6 r( nro oEr6IoyE( 6nUroupy(€q rl( ovopdrftvl( qun{q, rl

ypqq4, !$vs fipqyForlK6tr]ro on6 rou( IouFtp(ous tr€ro{f 1500 (ql

ll00 n.X. nprv qn6 ouq lrlv €noxt1, I rohlloltrKil K^lpovoprd arp€n€

vo fflo6i6rror ol(q €n6F€v€( Y€vr€q F€ rqv rpoqoplK{ €nr(orvovio Kor

93

Page 83: Ο Ταξιδευτής των μαθηματικών

:': €n'6€'{€'(.'Oio ourd dtrio{ov Ir€ q ypqq4. or dvop{rnor UnopoJ,oov vo.psro6i6ouv Fllpoqopi€q 6Xr f6vo orouq dF€oous onoy6vou(

l:u^(: oild (o. or ioorjs Kor yrv,r, nou oafrtruv no^u ono rou( ibrou(.

:lrt r""":", ro nporo ypomd, oxnpori(oupr Fro I rKovo rod nurs (ori-

: :" : l " l tor- , kor no( i (ovov unoloyrosoul ornv oprordrnro. H

":::t1..::::Yo"el, r0Jv op,oF0jv rlrov ouri nou 16(J0€ opxrKi rrvoolot yro qv ovdntuqq rnf ypoqil(.

H opxot{ir€pl- ypoqi furv louF€pirnv irav on .o( I ororonoorl rovKounovtriiv oE go_lqKj nqtrd yJpo oro 3500 r.X. :q ouvEl€to, y6p0) oro1l0O flX., fitofr rqv ojorunurol Kourovrtrv, r,or uorr vo ouppolr<ouvoprupoD! |'or q \dpo{rl nrnoypofl,orov, , ror (,)orr vo ovonoprmodv

::TT""; * q 1opoft n<ivu or an^o xpnorpono,ouwov pro {d^rvn 4rore^rv! oqulqrq ypoqibo. )Kono( ourr.jq rrlr lopot4q 4rov vo xoro-ypo9ouv or lF€polnvtq Kq( vo 6rluloup\Ioojv rolo^oyor !€ ro owrK€!

f€vo._ fdpo oro 3000 n.X., or :ou*por ovantuqov rl oq4vorr6ri yporpr,l

{€ fdol m rpolyoff€vo €i6o{ ypo9rl(. M€ q oqnvo€rbi ypoq4, ro

^:-".1:ll1 rurv rounovuiiv pL ,qv op,0p rr4.aopoo"n*a ypo,p4

ow (orooloun^ov u! gdvlru(o orjppo^o r(or ovomul€q*r ruo qryrp,o,o,oqlpqpevo oliorlgo oplo!4o'l(. H ypoq,l opxrKo yrvo,ou or or ,,, onoot(o npo( To oprorrprj. Auro o,^,\otr pEro ro 1000 n.X. (or n ypoqri yrvo-rov o€ ypqrFdq on6 opror€pd npoq to 6Eq{{i. Euruxdls yro Fof, ro !!so:fqIn:

{mv llJnldvoq q^d( n eror 6roqp40t{ov o( rpoorouE€( rovroug€piov yo ndvo 0116 rtw€ X rd6€! Xpdvlo.

To poOn pol Kd rolv loup€piulv

A,yo tNor yvnoro yro ro Uoonlrollio Tov )ouf,prov, o,^lo ono ro fl,\o-

ij:'o-::u-:u:(or4o, rNo, eov.pd 6,r yvuJp(ov r,( ,,oorpr( poorr/( apo

<.r( Tl( op,opllKn(i r4v flpooeJon, r4v oqoiprol, rov aoMonioorq

:i:..l:l lt t:,1

:ll H no^un^oKoqro ,4( Ko,vovio. ,uv rourrpuxv

oflorrouoE .n6€Erdrlro oro X€rpro!6 nrrv (puou(irv oproprJv. An6 ronpuJrq xounovto ilou olrlnpoo0jn€uov op(oForjq Bi€rloulr€ dll ro 060rl_94

Page 84: Ο Ταξιδευτής των μαθηματικών

opioFrlol( r0Jv :ouy€p(ov ilrqv [otrrjnloxo ouyKpnrKd F€ to opxo'6-

6uo6rr6, fl€wo6l(6, oK6fo xor ro 6€Kq6rK6 o|joIrluo opieFlorlt

{rov dvo €EqKo,".ro6rK6 oUornuo F€ pdon 1600 ro €qivrq 6qo Kor

o6dKo. no^ld qrl6 ro rioKi6ro qo(vEror vo €ivor Epyo 10rv gq0r]r€u6$€

wv ypoqaov rlou €qqoKoiwov ytq ro Foo4Ford rouq. An6 ourd yvt pl(ouw 6r or:ou$Cpror Fnopodoov vq 6outr€6ouv F€ roii Feydlou! ol^d

ror Us flofuj FKpodq opreFodS, Xplo(gono(drfio( tdoo qKdpqrouq opLo'pod( 600 Kqr KidoForo. ho vo Xopda€r rov opleg6 .6vo, o ypoqao(

nii(s q ypqei6s ndvur mov n 6 un6 yov(o. 'Eror oxnpo1l<6rov ivo

llFKj(lo noo agolq<€ (dno( oov KEgolo(o .D" o(ouFnroFa\ro nldYro(Eftvo l2). To ofppolo yro ro "avo' €rovoloFpovdrov yro vo oxltrorF

orodv or FEyolir€por oprogo(. O ypqqaq( 6rqx{ipr+ Fro ouilofi opre-pll(rdv lllnqiLiv rou ouFp6h<ov dvov opr0f6 0116 qv undlorftt ypoqi

ronoe€rdrwqq ro Ffoo o' €vo opooydvro fiopo^^ni6ypqUpo. 'Orov

l$ovov oro 66(q, n ypqqi6o rrl6rov K60€ro Faoo orov nql6 Kl drol

onFor(&ov dvoq pr(p6q KdK^o( (o). To ofmquo 6Foq rov louFap(t0v

6€v €iX€ urq pdon ro 6tKo, q^ld ro €{4vrq. A116 ro avo loq to n€qvto

svvto XploruonoroUlrov 6vo( ouv6uoog6q povdourv (t) Kor 6€Kd6ov(o). Irq ro Eq4vro ({rypoqr<6rov €vo F€ydio odFpolo U€ ro oxiFo tou

D, ndh o(oognrogdvo r dyro. To er6F€vo p{Fo 4rov {o 600 { 60lq ro

orcio oulpol(6rov U' avo D, ro onoio Eq€ avo FrKp6 KfKio oro €ot 1€-pr6 rou (O). To 60.60 4, o^^r0ls, ro 3.600 <oypqq(drov F'€vo FEyd^o(riKio (O) xor ro 36.000 (10 60.60) ilov 6vos U€yqio( (6Kno( F' rvo Fr-Kp6 (iKtro oro ForxrprKo rou (@). Or 6rdqopt( popqd( ourov rov np{,

r0rv op(opdrv rurv touF€pirxv qo(vowor oqv E(6vo 12.To 6rx6 goq ofyxpovo oroqFo €{oprdror on6 q oaol Kd0E +rtqiou

mov opr0F6. EnoUavoq, ro rp(d\.ro €md €(vor 6roqop€(K6 on6 ro €p6c

F4!1o rpio yror( ro 6(0 qrrlgfo" 10 rp(o Kor ro €nd, fxouv 6roqop8rK6q

e6osr(. o( €r rodrou, ro 6lK6 Fo( €(vor {!o xoeop(6p€vo 0116 q olot

odoqpq oplopdrv. To rpdro opr0FrFr(6 oioqFo rulv loug€pi(,)v lirov

avo g[ Kq0opr(6u€vo 0116 q 06ot oioqFo, ogol, o ypogao( ord6(np6oe€r€ 1r( nUa( rov 6rdqop0Jv ouFp6i0rv yro vo erdo€r oto dopoF

oFo. H rF4 n v oupp6lov 6€v €{op!6rov 0116 q e€or1 toug 6noq mo 95

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npoirorop0Fo(rov AptepoirovIouF€piulv Bopulov(ov

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6r(6 FoS, odwpovo ouoquo. Iq g€rtn€rro oqlvo€r6i ypoq{, {o d(poq( ypoqi6q( €iXE riroro otr1iuo dor€ vo oqlv€r rpry{rv((d otFd6ro F€oup€q ndvo mov nni6.

nEpilou ro 2400 n.X. ptrdnouF€ 6l ol louFdprol aKopov €nlroyi(,uroldyr<ov q yn o€ oop, <dyr(ov o€ rdiovro (y/roup), F€rpo(oov uypd

t

Page 86: Ο Ταξιδευτής των μαθηματικών

m(or ufloldyr(ov r6Kou(. AKdpo nro o{tofip6o€Kro €ivor ro yEyovdslp4olJronomfoov ro (^douqro €vo 6€dr€po (t/2), ivo rp(ro (t/j)nfw€ fno (5/5). Auq €ivot q nolq(dr€prl yvoor{ ovq\r'dp{ol rou

6fl m rldoforo €ivor oprouof. To xdpro XopqKqprmKd rour(Dv :ouFEpi(,)v €ivor 61l tx€r ri( pdon ro €{i1vro, F€ 10 6!

v0 oror€ki a\ro €v6rdg€oo piFq, (oodiq Kor I €Fqdvrorl rurv KlooFdtrot ol louFlpro( BropoLjoov vo ypdgouv dxt U6vo F€yd^ou!, oi_

ffi F(rpoU( opr0po(S.Ooo Kr ov rov(oouF€ q orlgoo(o nou qouv rq Kldouqlo rrov :ouu€-

&v opKei. An6 ro EE(iurlHo q( ovoFmrK4q orcpoplorlr, 6rov ordpxloov vo yivowor oqlplpavol or F6vor 6loodorFor opteFoi

0rQool(o{ opr€lloi. nporoEfqov(mrlKE €noFdvrn( ivo vZo €i6os oproAv fious opxrKors opreFo(( 66enKov ov6Foro n€pinou E(or6 X d-

6t( Xpdv(o nprv, FE qv ovdnruq rt( ypriyopn( qovrlrK4( og ioq, 16r€

tlxoNA 13. nil|v'l nrvokt6o tuv loup.piov. ndv@ orov nfl6 txouv Xopoxa€i roop$pllKd 'ftqiq rcu olrqlorxoriv oto rr. rrov nllld 6Xr qoypoqor( ro nrKd_ypq+o €vd( Koun'o! g€ id6l Aurd To opxq(o op€pl(Kd LIryio nou ovnororxouvdo r €Xouv lorp6oEvo oX{Ud. (M3 rl! d6ao qS NEv(< rfok-Mn.ospd on6 ronown'or4u'o rou ftqoq oro 'oor€v KoL tou BooL^(o6 Mouo.(ou rou Owdpro orov

97

Page 87: Ο Ταξιδευτής των μαθηματικών

ro 10{6r onrt rou( quorKofq qprolo6( oro Ktrdofo{o 6({pK€o€ n€pirou3Kor6 Xdrri.(l To Ktrdoporo XpsrdmlKov rctrLj Korp6 ytq vo €pqovrmoUv.:qv npoygfirK6rlrq, eo 6oUU€ 6tl oUI€ or A(yrlmror odr€ or ,Alrlv€( o1lo_68i,rqKov rl! r6fo rov (^ooFftov lioo or6trum 600 or loupapror.

Or o6t6ioyot Bo puArivror

lJpur oro 2000 r.X., o iortq rov Apoprr{nv €roipoi€ oq yq r0Jv IouFE-pirrv KoL r(oraKqo€ l( n6i€rq rou(, Koroorpfqovroq qv Oup ro €roq2006 n.X. Auroi o( civeponor ayrvov yv(,roroi oS Boputr0lvror Kor 6luro'jpynoov oryd oryd Fro UEydtrn ouroKporopio nou nsp dFpov€ rEpro-XCq rou orlp€plvod lpd(, rl( top6ovios r(oL qq:upioq. -Exr(oov rolid(ro^31q, ouUr€pdoFpovoFivnq rnq r€p(qngn( Bopui6voq, (or I ouro-Kporopio rouq lirip(€oE Ftlpr ro 518 r.X., drov I Boputr.Jvo ln€os moxapro rou KUpou B'qq n€poioq.

'Orov or Bopui6vror r(oraKqoov rl yrl r0jv:ouFEpiov, uLooaqoov160l) rt oqtvo€r6{ ypoq|l 600 (or ro goetFotlKd rouq.,Exouv pp€0€iXri'd6€( poFu^urvroKd( nvo(i6sq g€ oqtvo€r6{ ypuqt, Kuproe ono ro2000 r-X. Kor ro 600 n.X. O( noiql6rEp€S (oroypoq;( FdS rp6og€pov6vo\r fliorro r tpoqopLdv yro rq poBu^ovroKd [oetUofl(d. nop, 6]0nou or Bopul{ivror xpdvtoov ro poorK6 odoqpo opiognon( rov :oup€piov, nou EiX€ or Fdon ro 6€Ko Kqr ro EFivro, €yKor€lEruov ro €(6r-Kd odlpolo ylo ro 60, ro 10 60, ro 60'�, ro 10.60: Kor ro 60r Kor Kpdrl-oov F6vo 6Jo oUtpono: ro rpiyovo f€ rtv K6e€q oupd, nou ovopo(6-rov oq4vo (T) Kor ouFp6hl€ ro 1, xor ro rpivrjvo p€ Ilq 6Uo oupi( oror dy(o (<), nou ovoFo(6rov dyKrorpo Kor suFp6trr<€ ro 10. (Tp€rS €{orpao€rq E66 oror€ioUv ro (^doporo ivo 6€dr€po, tvo rp(ro Kot 6Uo rpi-ro, nou ro Kooavo EiX8 ro 6rKd rou rbroir€po orFpoio.) Kordnrv np6,oeEoov €vo xopoKqprolKd nou ixov€ ro oljonlFo opiOUo(( rou( vo{E}rop(Gr ovdu€oo oro dltro rov trqdv rt! opxor6qro!. Xplouo(oin-oov avo Ko0opr<6F€vo on6 q oao! oUornFo, ro onoio fnopouo€ voovonoposrrlo€r tsydnou( oi^d Kqr FrKpotq opr0po6(_98

Page 88: Ο Ταξιδευτής των μαθηματικών

To 6lK6 Foq, oJ\lpovo 6€Ko6rK6 oioqUq €(vor :vo Ko0opr<6U€vo

iq oro,l "j"-'tu;.* p'q Enrtp€n€r vo ovonoproro6-:::::-l']-1'

i"" -", *t"ls "p,elt",Jq *or vo €rt€trotFE rrolftIoxouq unoloYr'

fr,o nopat.vr,", ." , ouupotr(€r Fdvo tov oprof i"l -"

ll iio opreFOq O*o, Ev6 to 1 00 €(!oL o opreF6( €Kot6 H o€on tou 1

ini[* -n" op,eun"^4 "rl4 nou np'n€r vo tou €rqopno€(' tK€gr€i

,oJ opoua *r. t; 6vo url Ko0opr(6u€!o 0116 q O;on oioqFo' ou'

i. "p',erroq 0. 'i.* t t , t I = 14 'opo(' oto 6t(d u* :o':f ::i*-oio *, = ̂ o. oo.3 i €moKdorq oopdllo rpio Av F€to

qin poq eilou$t vo ropom{oouuE Kidouolo' lono€€rodg€ uro

rln' oto ie(rd Kor ro €r6F€vo qnlqio ouFpotrt(ou! 6'Koto'

*in. i."u,".,, -o rt,t .ivo( ro s.lo + / + 3 (sr ) + 2 (t) =

+ 7 + A , J + A , 0 2 = 5 7 , 3 2

ro *oeop,ltU.to ono .,1 e6orl populorvrox6 o(otrlu" T:t"'p.11:".j

*-"" 4,".0U"", qlld €iX€ ur( pdor\ ro €Qivto (r 6Xr ro 6€Ko Enop'-

;oc o opLotas ozl (nou €6dl €X€r Ypoq€( U€ opoprKd Lliqqio) 6€v qrqv

rc s{oK6oro €iKoor €vo olld to:

6 502 + 2 60 +'l = 21 6a0 + 12a + 1 = 21 721

BsBo(oe 6rov €Ypoqov ro o6Upolo nou or€lK6vl(€ ro 621 6E Xprl_

druoflorodoov o orrci po( opoprro ouupo^o' otrIo rr\ 6t'r( ro'r( rpr

Vul*rr . oqn*.. lnop,""( ' o oprofd( nou ouppoh( ro2 /21 i rov0:

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i""J.rt", -" rs," u€ to 5/10 (n€vrE 6d(oto) EnoF !o(' f6vo p€ ro

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*oilop46u.uo ono .l olotr oioqud rouq' urlopotoov vo Ypdqouv no-

^, un"^*, *, *nl *-pou\ oprouou\ {or vo ro'r( \onoruonorotjv Yro

""t"tr"r" u. ,^ '*o",,t ," unoio', 'o"o" o' tn'o"ir 'o*c ouopt'"o' 99

Page 89: Ο Ταξιδευτής των μαθηματικών

wor yori or :ouFdpror €na€{qv o( pdol ro E{4vro, Fro Bdol nou

Xpqorfonorloov Kor or BoButrorvror. (Lrvor nreovov vo €npd(Erro VroFro €{dh(n rou nponyion(s rov :ouU€pir'J4 oro nlo{olo rou ouor4Fo-roq rov Kourovr(iv nou xpnorFono{ofvrov mn 6ulK( Ao(o nprv oftq }pion qq ypoe4!.) [4ro nreov4 €t{yqon Eivor 6tl ro €E4wq Uropslvo 6rorp€e€i rti€ro on{t notr}oi( F(Kp6r€pou( opr0Fo6(. Aqtro6ri, ro€E4wo 6rorp€iror rai€ro on6 ro 6rJo, ro rp(o, ro raooEpq, ro na\rr€, rod{L ro 6iKc! ro 666€Ko, ro 6€(onaw€ Kor ro rprdwo. Aur6 Kdva nroor 06( ron^ots poo(KodS ufloioyroFofq. To ouor(Foro rov touF€-pirrv Kor rov Eopu^ov(rov XplorForolodvrov €0pa{0( yro vo uroloyi.

<ouv pdpl, noo6qt€( Kor €Fpo6d €(rgov€(6v ync - 6^0 b€uKotrivo-lIov on6 q 6rorp€rdrlro rou €(i1vro.

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(q) 12 60'� + 2 60 + 30 = 41.150(p) 12.60' + 2 + 30.(*) = 2.5e2.002,5

(\) 12.60+ 2+ 10.(.n) = 722,5

(6) r2 + 2 (e) + l0 (&) '= 12,041666 (n€p(nou)

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El(oNA 14. Opooydvb rp(yovo do oroio qoodo{ Io ruOoy6pao 0.6flpo: A, + s, = d.

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1r, H.L. Resnikolf (d R.o. welh,oover Publications, r97l), o 76.

lr., Makenati6 ih civllunbn (N.o Y6pdl, lO3

Page 93: Ο Ταξιδευτής των μαθηματικών

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13. David Lugene Smith, o. 41. 105

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Kol qploFo( Kor oproFo(u N6ou Kriopou

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Xopon4p€q oro pvlu€io. npos F€ydil vrpornj rov tonov6v KoroKqrdv,KoroorpdqtKov dio EXr6q on6 rpio prpi(o (rou nipo pp(oKovror o€ €uponqi(:( 116^€r()." lvlro Fapo ro 1541, orrlv r6irl [,jdvr q( X€poov{oourou frou(ordv, o qpoyKroKov6( Uovox6( Nl6/Ko w€ Adwo txoqj€ To p!pi(o r0Jv lv4dyro, xqrompaqowo( yro ftivro ov€KriF,It( o{io( Ksrt4l|olvoq onou6oiou trooU. ,EX€r ovoqEpo€( dr( o Nr€ Advro Kordiop€ opydr€-po ro ayKlrlgd rou €vdvno o!ou( Mdyro r(or ntpoo€ to un6^ouo q( (oiq rou Fo<€rovroq oenEos(C yrq rov noir0oFd rou( 01(j ovopolmu(nov Mdyro flou <oUoqv oK6go. MoC Ougct dpoy€, ro (dqjrFo ourotv rurvp'Biiurv Kdror€q dtiEq ropd^oy€( rpda€r( (orootpoqic To 211 n.X., oKrva(o( quroKpdropq( Ir XoudvyK_r 6dro{€ vo KooJv rotid Kvt4xo pFpi(o Kor, ro 48 fl.X., o toUiroq Koioqpog €v6 6rdq€uy€ on6 qv Atr€{dv-6p€ro U€ ro orpord rou, rpo(d^€o€ fEyd^€( Koroorpoqtc orl orou6oioprploeiKrl q( nd\g uo prp^rooiq rou n€prEax€ tpyo ord q p6fl,rfv lv6ic! qv En^d6o Kor rlv Ai\?umo. To l9t F.^., o qovolKo! Xplolo,v6( norp(dpxt( O€6e oS (rov €rllK€qo^i( €v6q r ioou! flou aKoqr€ qIr.iX€ orog€iv.r ond q prpiroo,iKn qq Ais{6v6ps(o, €rEr64 o XprmrovdqoutoKpdtopos 0€066010( o [.4q/o( €narpsqj€ rqv Kqroorpoq4 6k0v rov€r60rtroiorplKriv vodv_ Asv ro Xr,Jp(i€r o vou( rou ovep.inou n0jq rdoo ̂ iyo droFo gnopodv vo rpo(o^aoouv qv Korompoqi 1600 U€ydloo rfi-Uoroq rtq rloy(6oF(o( KltpovoUrd( qq opxor6rrlroq.

To n6oo KorovooriF€ ro Ksiu€vo rov Mdyto o{p€r^aor, oq Eni ror^€iorov, o€ €Kor6v EiKoor opxo{otroyrKaq roroe€oi€( nou n€prfxouv n€-pinou oKroxdoro ilqF€(o Kor (rioporo !€ r€poylugrK4 ord ro onoioaxouv f€roqpoor€( f6vo ro n€woK6oro. A€v jlrov o{ MOyro ouroi rou€ftv6noov rl ypoqi oq M€ooo!€prni, oqod undpxouv ororx€io 6r(6voq fiporlyoiF€vo( lo6q, or O]gaxor, rurv onoiov o [oirlo!6q 6r{pK€-o€ on6 ro 1200 r.X. Flxpl rEpinou ro I U.X., lprlorpofloroJoov dvo (€-poYiuqrK6 onqdpnro. Iqv npoyFqrK6rlro, 6tr€( o( rporlyFav€( Korvovi€! rou l\,4€qrKoJ (or q( KEvrprKiq AF€prKi( Xptorfofloroooo\r ou.rd ro

132 .",i: ::,::i::":::::; ;::,::io6\' rc code\ rro co',es anuj oq Mq.ptr,l

Page 121: Ο Ταξιδευτής των μαθηματικών

r€poyiuqrxd olqdpnro Uo<{ p€ ro 6rTIi6 rlF€potr{tyro tov r€viwo 6do

€r6v, d.id or l4dyro 4rov ouroi nou ro dqtooov oro ono(opdqoFoq( E(d^r(i( rou(.

Mdxpr ol1!€pq 6€v urdpxouv €(F€vo r0rv Mdyrq rprv 0116 qv €r-

opohl tuv lonovdrv rlou vq ovogdpowor dF€oo oro goorluorrKd rou(.

tnopCvoq, 6,r mrolopo(voug€ yro rq UqoqFolKd rtov MdYro npoap-

X8ror KUp(o( qnd rou( oproFodq nou urldpxouv rdvo mo FvrlF€io,

[vllFEo nou Xprlorp'uov yro Inv KoToypoqi rurv rlptpoptvlrv or]Fo-

wr(13v yEov6rov. A€v axouu€ Kdroro mmypogi rldv ErlqKprp6v oKa

{sr0v rou( ox€r(Kd F€ ro UoolUqtlK4 €v{ir ourd rou yvt0p((ouu€ y(o ro

Uootlrol(d rous ogopo(v Kupio( oro rlFepol6yL6 rouq.

0r nqtrordrEpor oproFo( rov 1\4dyro El,rqovi(o\tor ndvo o€ FvrlF€(oyUp(lr oro 400 F.X. To oUotngo opteunoq( tuJv [,1dyro €X€ tllq pdon 10

r({oor, €np6{sr1o 6r1io6q yro,vo rrKooo6rx6 or iol l l t ,o r(r oxr Yro,vo6€Ko6rK6 odoqUq,6[uJ( ro 6rK{i Uoq. To o oqud rouq n€pddupov€

1600 {Uf( 0aons 600 (or ro Urt6av. or opr0go( rou( ypdgowov Kdo€'

ro, F€ ro qrlqio U€ rq UrKp6r€p€q r$d( oro Kdnr F:poq rcr ro U€Yold-rEpq oro €rdvo. Iq yl6ooo rt,)v lirdy(q umipxov 4Exoprm€( opreFn-

ftas trd(Elq yro rou( opr0vo6( on6 ro I €oq ro 10. ot opLoplllK€q l€-qa( yro rous op(gpo6q orld ro 1l iulq ro 19 Ford<ouv [o]{ U€ rr( oyyhKt( rpoqoprKds opreBll((€s tr€{€r( or }€{€rq yro tlq Fovd6€q ouv'

6ud<owor F€ q ii{n yro ro 10. E(vor o{roonU€itJto ro yEyov6f 6r }p€r-d(ovrqv p6vo rpio odFpolo yro vo ypdgouv rou( qpr0Fo6( rous: troro0xK[6o yto rov opr0p6 1, Fro Fr(pi ^t0p(6o yro ro 5 Kor dvo 0ro€r6a(o$1iro y(o ro 0. Or ypomoi oprofo( on6 ro I aorq 1o 19 oxrlunilovrov0116 ouv6uooBods KoUKK(6urv (yro ro 1) Kor irrpi6o! (yro 10 5). Or rd

vr€ rprir€q rFaq oaorlS rou 6rxori Uo( rv6oopoprKoi ouor|1BoroS, olo

onoio or rUL( hooopr(ovro, on6 rn eforl, €ivor quorKd or:

10.000 + r .000 + 100 + 10 + 1

or naw€ npdr€( rp6( otol( rov Mdyro 4rov or:

1 4 4 . 0 0 0 + 7 . 2 0 0 + 3 6 0 + 2 0 + l 133

Page 122: Ο Ταξιδευτής των μαθηματικών

Or lluaq ourtq €ivor ro orord^€oFo r{0v sqq noMor ooroopdlv:

20.20.20.18 + 20.20.18 + 20.18 + 20 + 1

trora xplorFofiorEiTor ro ]8 ovri vo sivor 6^0 20; To ̂ oy(K6 0o 4rqvvo XplotlororoJvro( oro oJoqpo noilofttrdoto rou 20 o€ KdO€ piFo.OFUiq, or Mdyro XprloruoTrorodoov ro oUoqgo opieFloiq rouq yrq voxoroypdqouv tF€poFlv(€q. fro vo gl{iqouv ro t[€poi6yrd rou(, Xo]pLoov ro:roc o€ 6€(ooxrdl Friv€q rov €iKool rlu€p(ljv. AUr6 o\rlmotxoJo€ pdvo o. rpro(6oG( €Fiwo lFipEq, KL €rot npdoo€oov dilE( ntvuludpE( yro vo ysFioouv ro (€v6. Auri( or rtvr€ lFapE( irov ol uayeb

or drux€q tpapr(. EroFivo(, ro 6€KooKlir Uiv€C €ri €iKoor lFap€q€qty€i ro 6€KooKr6 oro oUoqlo eton(-Tlpi( nou €ixov. Im €poylu-qrKd rouq, I lFri uno6ttov6rov 6X( F6vo 0116 r! eion qq, o^trd ou-vo6€u6rov K on6 tvo nopdE€vo rllKrdypogFo (EtK6vo t/).-Evo nopd8€(yFo rorJ ypqnrod opr6pod 45_783 rov Mdyro qq(vErqroulv trKdvo 18. '€Xowoq ypoq€i opr<dvlo, (ior€ vo 6ltrlirvovrq( or l-pa( oaoni axouur:

6.7.200 + 9. )60 + 17.20 + 3 = 46.783

i41.200 + 1.240 + 340 + 3 = 46./81

O 6EUrEpo( oprqr6q oqv ErK6vo l8 (4.126) 6€ilva q Xpqot rou U|l6s-voq ono rouq [.4dyro oq 6s6ftprl eforl. An6 qv ErKdvo 18 axoup€i

1 2 ( 1 8 2 0 ) + O 2 A + 6 = 4 . 3 2 6

Auro ro oLjorlFo op(eFloq(, nou EaX€ oq pdol ro €iKoo( f,dvo 6sKooK(l ollv rpiq odorl qofv€rol ovoilo noiUrloKo. A[v npoopl(6rqv6g10( y(o vo ro Xprlorforotodv ot or o( noifrEg olld €(XE ol€6roor€ion6 ro0! r€psiq rov [,.tdy(o yro ro r€pin]oKo nFEpoi6yrd rou(. Ko{d nd-oo nr0ov6qrq o gtoo( dvepornoq XploruoloroLjo€ l( €(Koor ol oUUE-

r

v. \ opropnr, \ ^r fur\ (o, , ooor Mdyro (o.oloporvov r l vpo, lr i , r reo| .'+ vdv vo Xo.orpono,ouoov y'ro onl4 frbolq r{rv ouppo^uv In( roLl.r,6os

Page 123: Ο Ταξιδευτής των μαθηματικών

$KoNA 17. N'lvlwio 10rv M6Yo To Fvq!€io s(vqr nsoF o mo rld" @ot6oo td

oo,eunr,ro,uno. ruv Movo Pl uq Kou6l61 to' r'( /'op'oI( oo(vo(or roeoPd

.r" pn.\ L, -*, o.p"r' ,outoYpoqo rdv o6rlquv MnPoou!' l(p\vY\)

mr q( lopi6oq Yro 6,rr Xp€ro(&ov vo Ypdpouv o$lv Koenpepoq rous

(ur4.1o nlip€q o6oqPo rpoopt(&qv Yto rou( L€p€i(-KUp€p!tF€('

To K(vnrpo Yro dvo 1&olo notr0nloKo o$ornFo opteg6v ttov n oprl-

oK€(o rov MdYro, 6rcu lorp€dowov op€ro( 0€o(, p[Ioq{ tuv oroitdv

tvo( 0€6S yro rov 'Htrto, Fto O€d Ylo q leqvn Kor, nL0ov6v, dvo( 0E6q i

Fro 0€d yro qv AQpo6(q, rl ono(o €qov ovrdlqe€( T(nq ftov 1600 o

Afloor€p(rN 600 Kor o ArrY€pN6( fto vo nopqKoloueodv rou( oupdvl-

ouq o€ods rouc Xp€rd<o\,rov €vo rci6 nU€potr6Yto Kor, oqv npoonde€d

rouq vo to Eqqogqtr(oouv, ofilv npoYForMrlro 69rlo6ov no^ld lFEpo-

l6yro. To ioiKd n$€potr6Ylo, nou ovoln(dtov oriln, {rov rc ol,oquo fiou

€hyodoE tlq €noxds xor €(K 6€xooKrri, Uliv€f rov €(Koor nu€p6v o Koe6-

voq. Aur6 ro nU€pol6Ylo npooopFo<drov rlpooe€Io!'toq ndvl€ r]Ffp€( Kd-

e€ Xp6vo, €vril Yivo'"Iov €nrlldov npooopUoY€( (Yto vo €BNoiv Kd€€

Xpovo qv npr;q 4p{po KdOE rprFnvio() nopotnpdwo( nlv Aqpo6(q'

Evo orouo no o(prp1( dlo\ unoloYloqir ono io'r<. rcpeiq "'rv vayro 135

Page 124: Ο Ταξιδευτής των μαθηματικών

6\7.200

9x360

17x20

l

= 46.783

12x360

.l' l l:l

' l l laaa

o"zo ]

, a l= 4.126

EtKoNA 18. To oproptrK(i Lllqio ro! MdyLo yLo rou( opLotoLjg 46./sl (oplorspd)(o' 4.r16 (6.{Ld).

oro Kondv, or oroiol runonoinoov ro &oq or(q 165,2420 qFapsq, npoo€y-y(owo( fror Xyo n€proo6r€po rov oiferv6 oprof6 rav ]i6s2422.ll']l€ptjrvon' 41 ro 6lX6 Uoq, tplyoprovd nu€po}6yto rov 165,2425 \r.p6v.'Evo di.\o tg€poi6yro poor<6rov o. €i(oor .rdqutv.( lfip.( 6E(o-rpniv KfK^ov t v 6roKooiov E{iwo tp€pdjv. Aurd 4rov ro Epd r4olft,ytFEpoidyro. To 6Jo tp€po^6yro ouv6ud<owov fior, r6m€ Fro ouyr€Kpqravn tf€pouvio onorrotu€ fro opreFlgtvq tuapo rou F{vo r(otrK(v ouv F(oopr0tqfawt llipo rou Urivo odFr. tro vo Kdvouv rvov r ipl K6Klq KorTo 6to tp.po^6yro Xp€rd(owov n€v4wo 6Ljo Xp6vro. H n€p(o6o( ouliovoFq<drov .rtpspoloyro(6! K6x^oq,. n{po ofti ro odln, ro r<o^Kiv Korrov rlF€potroyro(6 KjKtro Xplorponoroioov Kor r! .loKpoxp6vrq xoropa

-__ pnon". rvo olor| luo \povoroy,ov nou oox,/ . ur rr lv lp.po roio qt

| 50 oro,o $upouoov or ' ryrw r bqF,oupyro Auln I uoKpotoovro /o,op.rprf

Page 125: Ο Ταξιδευτής των μαθηματικών

q poo(drqv olo opr0Fqrrx6 oiotqpd tou! rcl Usrporo€ o€ n€pr66ou(

baktun 044.000 nU6p€{), katun (7.200 nu€p€q), tun (160 nU€p€C), uinal

(20lF€psq) KoL ratros, kin (l qFdpo). Or MdYlo nfol€uov 6r( ppioKowov

orov #Fmo Kljxtro qS Foxpoxp6voq Kolqpftflonl nou F€ 10 6rxd FoS

nFpoldyto q€KrvoljoE to 3133 r.X Oq 4rov oqdluo vo uTloo€oou|n '51ol

Mdyo dplloov vo Xplorpnorofi ro qU€pol6Yd rouq ond tdoo nolod

noqvdv vq Ko06poov qv npEponv(o ouni oqpl<dF€vol oE F60ouS Ylq

rous O€odq rou(. H 6lKri Uoq qFEpoFrP(o 5 lovouop(ou 501 g X. 0o o!'t(

morxo|Jo€ ouvotrrd o€ 3.113 + 501 = 1534 Xp6vro on6 rn 6nB(oupYio Au_

oo|Jtql F€ 1.326.410 rlUtpeq (ov owo4oouU€ lq €fi(n^6ov nF€pE( rov

6(o€fiorv €njv). t€ out6 rlpooeeroup€ rq 5 nUapE( tou lqvouopiou. H Fo-

Kpoxp6vro r(orouftpqot tlxv MdYto Y( quni qv qu€poFrlviq €ivor €vv'o

boktur! rdoo€po katun, too€pq tun, oKrdr uinal (ol 6EKoravt€ kin

EK!6q on6 q Uo(poxp6vro xfioFfrplon, I orloio 6r{pK€o€ ndvur

dn6 ndvr€ Xdrd6€! Xp6vlo { 6€Kqrpio baktun, ot MdYlq 6€Xowov Fto

0Kdfo U€yoldrEpn XpovtKl r€pio6o, rl ono(o oror.lofvrov on6 noild_

[trdoro rou baktun Kor ovouo<{itov alautun M(o t€roto n€p(o6os €ixe

6rdpKflo rdnou rqiv'o rpio €KorofPlpro XpovroIuvowovtqq or MdYro ovdmu(ov lvq €Kn^nKrlK6 ol,orqFo ople-

!{iv. To 6r(6 goq rv6oopqpr(6 oioqFo ![op€i vo €(X€ ovofiux€€( on6

ro 500 U.X., ol^d to rfleov6t€po €(vol vo €nlvo4el(€ orllv lv6io Yljpo

r(o 800 !.x.'' tnoutvoq, ol N,'ldYro €(Xov ivo (oeopl<d$€vo ofid rn edorl

sUoqFo nou n€p dupov€ ro Fn6dv rtoo€prq ot(iv€! nplv ovortux€Ei

rc 6116 Foq o{yxpovo oimlfo, Kot Uro o}6K}rlpl X €tio np| ro w6o'

opoplx6 orouluo urce€rle€( on6 qv Eupdrrq tqv tpoYFor$6qto, n

€gqdvlorl rou pq6€v6q lov MdY(o nreov6v vq Frl Xp€ldmqrc ndvoJ on6

triy€( €mrovrd6€( Xp6vro, €v6 mov nqtror6 K6oUo Il qrqdvroi mu Xpet-

dfirl€ opK€t€( Xil€rieq '" To rlF€pol6Y6 tous l1rqv mo notrrntro(o on6

ro €updnqiK6 nF€po^6Yro q( i6ro( n€pl66ou, €vdr I €nroI4url (ol rl

l l - Bunt (d. , o .226.1l.Ihonas .Jump, fhe Anthropology of Nunbes (NtoY6pK\, Cambrldge Uni

137veuity Pre55, 1990), o.46.

Page 126: Ο Ταξιδευτής των μαθηματικών

oorpovoFio rouq 0o fnoporioov Kdi^roro vo €qov q€n€pdo€r rou( owiororxou( roF€iq mtv E0p6nrl. fv6pr(ov du I Aqpo6(q Xp€rd<€ror 5sarlufpEq yrq vo rEprorpogEi yripo ond rov 'Hiro xot UropoUoov vo nopoKolou04oouv Ilq nvio€rq Ttq F€ oqdluo tlKpdrspo r0rv 6Uo opdjv o€rEvrqKdoru Xpdv(o.'

-Evo cn6 ro 6loo0Jeivro X€rp6ypoqd rouq, roCodex Dresdensis, n€pril€r oorpovourKoUq rfvoK€( psydltq oKpip€ro(tooo yro rnv Agpobirn 600 Kor y(o q lE^riv!. Mdvo vo ovop(ul6Fom€lropoLj!! n6ooL Eooupoi ruiv loenForu((irv r0rv Mdyro XdolKov y(qnowo ono ro rpoyrKd KdqIFo rr0v prp^iuJ! rouq on6 rov NIs Advro.

H ypoqi Kor ro tF€poloyroK6 oUoqFo Tlrv l4dyro uroo€riorlKovond F€roy€vaorspou! no^(1lloFoLjg oufr€p oppovoFavov fuJv rlo]r1-og(,lv rov ToiraKov, rorv MroraKov Kor rov AftiKrjv_ -OFoq, alJrq rlv!nox( (0v A<raKov, or loxpir€p€S qF€potoyrox!( n€p(060r €(xov €yKqroiELq0€i yro rov KfK^o r0Jv n€!{wo 6io €Tt0v Kol I ypoqi €iXE €Kqu-A'or€i. O' A(raKor 6€ Xprlorpororofoov rro q tr0rp(6o yLo vo ouFpo^(_<ouv ro navtE Kot sixov €yKoro^Eirli€t ro Fl6av.

Or ' l vKo

f p0r oro l4t0 9.X., or'lv(o, oL ono(or 4o[oov oulv Kot^d6o rou KoUoKo oro n€po6, dpxroov vo roipvouv ur6Iov ii€yxd rouq rouc yE(rova( rou!. Iuvaxroov vo €n€Kr€(vouv ro otivopd rouq Faxpr qv dqr€lrou OpovoioKo nLodpo ro 1512. Or tonovoi si6ov Fro ouroKporop(onou rKrErv6rov 0116 ro pop€ro6ulKd EKouo66p oC Irlv K€vrprxi XdiKor rEpddUpovE ruriporo rou [€po6, q( BotrLpioq (or lrl( ApyEvlv,iS,KoiUmowo( aror grov or6orool r€osdpov Xdrd6ov Xdrouirpov ondro poppd r0( ro v6ro. O nilououd( rov'tvKo unoioycror 6tl Kulorvdrov qn6 a(L aoq 6uj6€Ko €(oroupipro,'' KdI| nou ovroroLx€i orq F€yq

t5. la.quer Sounette, M€rt@ (Nad ydpKD Wond pub jhing Compant 196z),

138 16. F iedet , o . 15.

Page 127: Ο Ταξιδευτής των μαθηματικών

fuJr€pl ouro6UvoFrl ouroKporopio rou Niou K6ogou. fto vo ovonirl-privouv onoro64norE npooorplv4 p€iooq rov afioo€Fdnrv rpoqi(

4ouio( Ev6q noliuou 4 Fro( (oroorpoqiq q( oo6€rd9 or 'lvKo dqr(qIov Frydtr ' ( onoo4K€( o€ oi6{ i lp l Inv ouroiporopio Kor I . y iproo!

F rqr€+uyp6vEs nqrdrEg nou ovoFd<ovrov chuno- 'Evo orjorqgo 6n-

f6o|(l)v 6p6Fov, ouvol'Kod tr4Kou( Kdnou rprdwo X rd6ov X roud-rpurv, flou ovoFo(6rov rl "trEoqripoq rou i^tou', ouva6€€ rlv npurr€douoo oro KoUoKo F€ orouoKpuogivo rg4Foro qq outoxporopioq. frovo rrcuFovrdpouv our4 ulv r€pdono ouro(poropic! or 'lvKq 6ngtoip-

\qoov tvq otoull.ro 6poFCurv F€ pdp6l€f or onoiot ovogd(ovrovchasquis Kor g€169€pov n^npoqop(€q 0116 r( €rqpxiE( orlv npor€dou-oo. [(o vo 6(€u(o]6vovror or 6poF€i( 6lf(oupy{enK€ dvo orio lFoorooFdv (ord F4Koq rov 6p6!{rv, nou on€ixov n€pinou Eqrd X r6F€-rpo o fvo( 0116 rov di^o-

Or'lvKo 6€v €ixqv ypqnq yidsoo K tnp€n€ v'onoFvluovEuouv ro

Urlvfuoro Kor ro €rlfollo 6[/poqo. fto vo 6r€uKotrrvovrot o€ our4 rqvonoFvlg6v€uol ovanruqov fto notrfrioKo orioqBo K6un0rv ndvo o€oKo(vrd, nou ovoFd<ovrov r,/ro4 yro vq rfioypdqouv optoFodq Ko06qKor ro €(60( ttov tr€rpodF€vurv ovlK€(gav0rv. nollaq dM€( Korvrxvi€(

XpnorForoirloqv raroro r€XvdoForo yro rlv Koroypoqri rov qptoudv

Kdnorq orypri orlv roropfo rouq, 6roq oqv AqprKri ro oiKoro dvorooLutvq, olrlv opxo(o El^d6o, mnv Kivo Kot oq Xopdn. OrlvKo r(oro-offfooov n€pir€Xvo Kor rolfrioKo fltrou (p^irl€ E((6vo ]0 oro K€ed-lqro 4). nop' 6^0 nou axouF€ oq 6rd0€oi gos noLld k/,rou Kor FnopoJF€ €6Kono vo oroKpunroypoq4oouUE n6( Koroypdpowov or opro

FoL 0r i€moFapE€q qq Xpions rouq Xdoq(ov yro tros oqoU 6€v urqp-

X€ ypom4 yl6ooo.To /d?ou irov avo €pyol€(o Koroypoq4q r{iv opt0p6v nou Xpqotgo-

no(odlrov on6 rous €rno4Fouq, rou( KapaydK (\ KitouKayaydK), olomior {{ov ur€60uvor, oE 6rdqopo Erl[n€6o rl( r€popxioq FEoo oqvrupapvtorl yro rnv (oroypoqi ororou64nor€ oyoood €x€ €v6roqapovyro rou( Kup€pv4r€(. Or KqUqyl6K €firor6€dowov o€ €r6rKd oxotr€iorou irqv F6vo yro Dlv €iir lC (orvrrrv(o( rov 'lvKq. Or goorltfg tnperc 139

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Page 129: Ο Ταξιδευτής των μαθηματικών

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2.000 = ,B3.000 = , r

5.000 = ,E

7.AO0 = ,Z3.000 = ,H

2 0 0 = r300 =T

8 0 0 = o900 = a 147

Page 136: Ο Ταξιδευτής των μαθηματικών

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onou6oio 60ipo yro I|tv ovoK{itruqr! yEoF€rp((ujv otrnosrdv (o! ouv6uo-oF6 FE ro e€optr|K6 y€oF€rprKd oxiFord rou), oi^d or,E}"\rlv€( r{rrv!€rir€rro y€vE6v €(Xov €vrunooroorEi 1600 ord trl f:eo66 rou {.im€rrtv €qdpfo<ov ̂ov0ooFavo o€ 6io ro €i6t oino€r{,jr Kdn nou Ko00oraplos rrtv qvdnru{t qS €nro(U!(. ynip{oU.uruxLirS, rdroror'EiilvEq oroxqordq nou irov rpdeuFor vo r€rpoFolorodv Kor vo orq-prtrloJv ornv €lr€rpio. O Aprororailg rov rEropro ot6vo n.X., aKov€ovorotrKa( €{€rdo!r( yro vo oun^aE€r pLoioy(xt! rtrrlpoqopi€( Kqr o ApXrt'i6nq rou {€xivrlo€ rlv ip€uvd rou rov rpiro ouivo r.X., aKov€orrld n€rpdForq lrpooro06wo( vo ovo(o^UqrEr rouq v6Fouq qq quot-K4(. Oo n€pvoUoov 6U{r( no}io( or{iv€q ujonou oura( or n€rpoFor(Kd(npooEyyio€rq \ro €{Ehxoolv oq oLjyxpovrl €rllor(Frl.

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To oqvrK6 r{tpo iruv rrorpo yro dvov on6 {ou( onou6ordr€pouq Foon_Forxor( 6lr,rv rlrv Enoxrirv, mv nu0qy6po (ErKdvo 20). ArdqopE( rtfr€po-pnvi€q tiivovroL yro rl yawnorl Kqt ro edvord rou, olid Uro Koli npGotyyrot qq €notr1i( nou €<rloe 0q 4rov ovd!€oo oro 580 Kor ro 500 r.X.lEvv{oll(€ oq:duo, flou 6€v ofl€iXE ro^, ond q MAtro. Kdnoror uro,oqpi(ouv dr 6do(o^6q rou irov o i6roS o Oolig oqo6 or <uJdq rou( €vFap€r ouvarLnrov Xpovrxd, €v6 dl_trot ifvE 6l our6 €ivot anioovo ldyornq 6roqopdq nhKio( rou €ixov." fvopi(ouF€ dll o nu0oy6poq roat6€q,€nol, mo vrdrq rou Kor €nroKdqrrlK€ rlv Aiyunro o^trd Kor rrl BoBu^lljvo.EnoFdv{rlq, fnop€i vo €ixs i vo ptv €k nd€r oq Mitrfro drov <oro€ oOoI4q. Ev ndorl n€prnrdoEr, €ivor 6woq nroovd vo onod6ooe orfv trovrni :Xoii (or vo apoes EX€i llv onoy{rlyrKil fteo8o rou Oo}i_

Apydrspc! o 1lu0oy6po( oKoloLiolo€ ro nop66€Lytro rou Oo]i xori6puoE Fro oxo^4 orov Kp6{r0vo, orn lv4€ydtrq Ei^d6o (v6lo troniq). H

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Page 143: Ο Ταξιδευτής των μαθηματικών

ElKoNA20, nuooydpot n.pinou 530.500 nX.(Yl(6 on6 roug o6.lqoLj! Mnpooov, traphvy()

nu0oy6pE'oq:Xoli 4To! Kdrl n€proo6rrpo on6 Fro oxoi'i, 4rov ortvnpoyporLKdrlro !(o oplorcurLK{ Korv6rlro F. pf}l rprq(dotouq nEpiflou nlo[o'ouq Kor roXUpoUS vEopoLj( dvrp!( qr6Inv r€proxi rou Kp6lrrvo. Or opqoK€urKtq Fuoroyr,Jyi€( n€p$dUpovov rElstq yro rov !{oyv'of6 qq UUX'ir, oqo'j o nu0oy6poq nior€u€ 6rL rl (]Jufi FnopoLio€ !oEn'p'djoE' U.rd ro odvoro xor vo (ororxrio€r oE dtrtro o{nporo, oKdpoKor o! oujforo (0lurv. Aura( or truoroyoyiE( rEp dtpovqv Kor ororx€(o

Page 144: Ο Ταξιδευτής των μαθηματικών

FoerlporLKriv. OL Uoetrds rou irov X0rp(suivor o€ 660 oud6rq rovEor0rEpr(6 KLjKno i td64ga KonC Ga ou Kor rl 6rxi Fo( ia{rl .FoorluorLKdr, nou irov FurlUd\ror oIlf rLo Kpug:( o^40€re( rou 6oo(d^ourou(, Kor rov €{orrprKd KfK}o (t aKouop xoic gourci rou a{oUvt,or onoior 6r6doKovrov F6vo rou! Kov6v€( ouFflEprqopdq q( Korv6q-roq. Koodvo! ofl6 rcuS paeqLrauKoit tnp€n€ rp0iro vo n€pdo€( 10ordbLo fJqolq oC aKouapo x6e, nprv rou €rllrpor€i vo sroaio€r orov€orrrr€plKd KLjKtro. To pall rl( xorv6rlroq opxi(ovrov FuolrK6qrc, Kcirlnou efn66r<E rl 6rd6oot lrv r6€dtv rou!_ O (6roq o nuooydpo! 6€vdqrto€ ypanrd KEiF€\ro, Kr 6101 6io doo yvrrlpr<oufl yro ro €nLr€uyFororon €ivor on6 6Ear€po Xapr.

nop'6io rou or oKprp€is ouvoi(€q rou oovdrou rou nuooy6po oF-eLolitroavro', derooE 6woq o€ npoxoplfivl liu(io. Iq ouvax€ro,E{oLrioq ronrKdv notrrllKolv ovoropox{irv, I oxo^4 8rolUerlx€_ KdnoLoran6 rou\ poqnrylKoit EyKoloordolKov orov Tdpo\ao, Eniorl( orll\leydll EMd6o, dnou ouvixroov Ilq gooqFoTl(a( ono06i! rou(. nol\o(ott6rou\ oKouopa KoiC iyr\rov n€pro6E6ovr€! FUouKtmi(_

nopd rn 6rdtruon q( oxoiiq rou, or r6a€f rou nLreoy6po ouvfxtoovvo 6La6f6ovror orov EltrlvrK6 K6oUo. OL poorFa( rou i6p0oov v!€S oxoidq KoL 6i6o{ov vdo0q Uoetrac KoL or flLrooydp€(or ouvaxr(ov vo €qp€,d(ouv otpqvrLKd q oKaqrl 1lrv Elt'iv0rv ovd rou( or6vE(. O nlcirovnou rollo( rov o€oporiv ivov ond rouq lsyoirilrpouq qnoooqou( rouKootou (€vov roxuproF6 nou dtr^or oFqropnroLjv p€ rd6os), €iX€ €nt-p€oor€i L6Lo(r€po on6 rouq nu0oy6p€rous qq lnoxi( rou (ErK6vq 2t).O nldlov irov 1600 n€nEroFi\ro( yro o.t ottrooio 1t,rv UoetForu((iv,6orE orlv Eioo6o qq Axo6tFio( rou, BroS oXo^4S nou umipx€ Faxprro 529 [.X., 3iX€ Xopayurvo ro ptr6: M46:it d/s{rtdrprros doira, 6\iq6i vo gtv Un€r Kov€(q rou 6E yvop((Er y€ortr€rpio.,

Or $ goelpqflKd( n€noreio€r( rou nu0oy6po 4rov d\ro n€pispyo[€Ly[o npoytrol|K6rtro( (or .?ovlooio(. O 6rK6q rou K!:)6rKoS ooun€pr

156 "Jl::'ff:T::";: .tr::1i':"y:"^'^' val 1' fhe o sin' (N.dv6pK\,

Page 145: Ο Ταξιδευτής των μαθηματικών

ETKONA 21. ftdr0!. 427 riz n.X.(YlLtu dnd rou! o6!Iqo[( Mnpdouv, ItplLvya

qopd( aForo(€ UE lro nspi€pyn ouiloyi ov6rlro\r r(ovdv{rrv: Mfv rpdr.

9006^10, ovo6lLi€T€ ndwo rlq ordF€( nou o.p4v€r Fro qolld. 0()16(]0,di^8( rEnoro{o€r( rou 4rov rpoxruprlpav€q yra ropd6Erygo, ovoqap€ror dr unooripL<€ roq I lt livor oeorprK4. "

l 5

Page 146: Ο Ταξιδευτής των μαθηματικών

Or oprOpoi o t lpcpovo pre rov l lu0oy6po'Orov opli<oug€ vo s{€rd<ougE rlv iworo rov qproFrjv oJgqovcr !€rov nu0qy6po, pproKdFoor€ {oqvrKd F:oo o,lvov 1016 ord fr}€yutv€( oKa{€rq (or ovo}opnd( }qfrp4q ttrv€uorl(. Or .ti}tv€q livor olyoupo ol KoI€ixov l( BoorKa( urotroyror((t( rKov6rlrs! rlv €fl0l11 noudpxroov IlC €Kr€rouiv€( Ep[oprKtq ouvo]ioyaq !E rou( OoivrK€C, rov6y6oo oL(ivo r_X. OL 6€ar6rrF€q quri( ovofd<ovrov ,oforlri Kor ono-rEioUoov €pyoi€io y(o rouf €Fndpouq Kor rouq ypoqEi(. H loyror(KrinepricipBovr I|q npdqEr( rnr rp6o0!olq, qq ogoip€orlq, Tou floilonlooroopo[ (or rnq 6Lotp€otq. n€p dUpols.n(otr q Xpiol oKapol',rv opr0F6v oild Kor Klooldrov. Enrn^iov, oL -E^itv€S iluvov flpopiitoro nou oqopoooov q 6(ovoFri Kor rl ouyrdwprool oyq06v, rooroiq oip€po 0o o€opodooFE ororx€rdr6t ypoFp[q d^y€ppo. O^€!oura( or 6€{r6rtr€! rirov onopoirlrs( y(o ro0q Epn6pou( Kor rouq ypo-qEaS, or onoior irov Enrqoplogtvor U€ q 6roK(lnot rDv qyoeuiv Korro XErproU6 rurv KpolKrirv urooaoEr,rv. Erofav{r(, rl loyrolK4 {rov ou16 nou o{flpo ovot\<aul:" opt1ttnuK4. .OFoq, rl ioyrol|K4 0€urpo6_vrov 6€{rdqro i €nd\,./s^Fo Kr dxr Uro o^nervd rvEugolK,i ovo<4rtot.Ttv ovlF€rdnr<ov, Iorn6v, n€pinou dnuJ! ro Vdp€go ,i qv quioupyl|oi.

Ano Tlv dnlr} or 'Ei^n\r€q oro66o<ov ourd rou ovdloqav apapnl/oi rtv nvEUpol|Ki F€trarn rov oprot6v. H opr0grlrK{, U€ utv opxoroE^ritvrKi fworo, 6€! riro\r rl 6rni FoC opLoutlKri, olld our6 rou o,itrEpo ovoUd(ouir€ q€uptu tuv ap6\tiv. Olopodoov qv oproFtrLKrl avoyvr0olk6 ovlK€i!€vo ovrd{Lo ruJV ottervdv Enro ltdvrJv.

Aur6S o 6roxoprot6( q( UEIirf( rov opr0f6v o€ oproFnl&l (Ioyr-mrKri) KoL o€opio r{!v oproldv (oprofnnK,i) (rov oru}i(. :igoLv€ dlo6roqopoUoov yro ulv opLoFtrlK4 Kor ooxoioUwov F6vo !! q O€opior(,)v opLoudlv_ Oordoo, rLq npotyod#!€( X €rieq I rp6o6o( rtq qvoprrn6qrqq doov oqopd orouf oproUoLj( nopuKLvourov ono flpoKflKd npo_Btuitoro, Kr ourci ro rpoplitoro Fo( o6iynoov mouq qoorKouq op(o_

r!o for, ror oro , \ooporo. O. 14.\(( Iorrov of loqooroov vo ot\ ,oqooLrvlJ6 ro ̂ , propo ou ,uv ruv oprouuv ro \o.-r . . \ pdoo rv o. (o I o t^o.

Page 147: Ο Ταξιδευτής των μαθηματικών

l6rq(r€po or nueoydp€ro{ €(Xov oo},rlvEur€i 0116 q o€uJpio rurv opre-t(,]v, I onoio prop€i vo qEK{vno€ qrl6 Fio Kor p6q ovo|(diu|l)n noudKov€ o i6ro( o nu0oy6pqs. Au{6 rou ovoKdluqj€ irov I oxtoq F€roEUrou U4(ouq B(oq r€vroF€vn( Xop6{S (or rou r6vou rou nopdy€r drqvq Xrun{oouF€. npdrov, 6ror(moo€ 6r{, plKpoivovro( ro t{Koq q(oro 1106 ro0 opxrKod rnq F4kous, I Xop64 nopiy€ avov r6vo ul|)nl6r€-p0 rold Hio oKrdpo. T6vor noo 6roqdpouv (ord g(q oKrdpo ppioKovror"ff qpgovio" (q( 6lFroupyofv qv TIlo €uxdprorl oioelon m, ouldFoS. Irrl ouvfx€ro ovqKdtruQ€ €K€ivo ro FiKn Xop66v rlou nopdyouvIl( E r6F€v€( opFovi€(. 6nio64, Ulo Xop64 nou q grkpo(vouF€ orqTp(a rirqpro rou opxrKod q{ UrlKouS mpdy€r rnv rdrqpq, €v(i ov ellrrptvou!€ mo 6io rpiro nopdy€l rlv ftUftrl. O nueoy6po( qvqpo-n10lK€ r nopiy€ our€C Il( opgov(€( U€ro{|, njv 6rdqopov pnKdrv rrl(Iop6{(. H ordvqo4 rou 6E ox€l<drqv (o06trou U€ q quorKjl, F€ Il(6ovi1o€rq 6nio6r1 {ou qfpo (cr rlv oKoumrK{ KorooK€uri ro}v oufl(,ivpo(. Avrr06roS, Koralq€ oro ouunapoogq dll or opuov(€q irov €€Kd_Sopo qnorCtr€oFo rov tr6yov oploulvov opr0!6v {trog yto nopd6Ety-p0, rorv n6yov r{,rv oproFujv I fipoq 2, ] npo( 4, Kot 2 npoq l. ArId ou-r4 (dvovro( 6vo €fioyrrytK6" diFo. Kora^na€ oro dn 6^0 ro (pdyForo€{qptdwor on6 rou( oplego6! Kor ro0S l6you( rov oproF6v.

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{6rl Ktrdouoroon6 rouq Arylnlou( Kor rous Boputr6vrou(., noi6 ooord, Kql o ooq6(

rr. hoyoyi: Iuldy@yi ysvrc|J ouutr.pddpotoq(r.r.M.)

otro €n(p€pous 'o-u*'o 159

Page 148: Ο Ταξιδευτής των μαθηματικών

flu0oy6po( onl{n( oyv6rlo€ ourd nou Xp€rciorrtKov or6vlq yro vo €nr-IEUX€€I. frori iorn6v 0o npdn€r vo o€urpodFE onou6oio UqolfolKd ro\rnuooydpo, oqo qoivEror noq EyKora^€rqjE rq (^doForo yro vo ooxoll0!i p6vo U! rou( quorKois opr0Fo69 H roropio tr6h( opXGL.

MnopoUus r6po vo ovrLtrnqooUtlE rrt orv6€ot ovdtl€oo orov roluprofd rou n0eoy6po 6l ro ndvm €{opr6lroL on6 rou! oprogoJq _ Kor rlm€v'i oxaot t{rv Fin(vov f'€ rqv oroylJryrni Erllor4ln. Av 6^0 ro rpciyUo-m lqopr0lwor qn6 rouq oproFoLjt 16T€, yro vo gdeouFE dn€q lq oiie€rEq,ro p6\ro nou npdnEr vo KdvouF. €ivor vo ouvdyoutrE onoy0ryrKd Ttq o^4ft€q yro rooc oproFolq. EfloFivurc {€nvd!€ !€ ourqn66ELKr€S o^ioas(yro rouq opLoFoJq, Koroiriyoups U€ onoyoyf o€ 6rdqopo oupn€pdoForq,mr ro oounEpdoforo ourd 0o n€pfxouv oi{oErsq yro ro quol(6 K6ofoFfoo orov oroio <orjfs. Auri ! 6ro6xoo(o uno6n^6vEr 6rL n onoyrrjyrnl.nroqn b. xperd<€ror vo y€^€r4o€r q qljot. N4nopoJF€ \ro oyvorioouFson€( outEq rLq n€rpoFor(,( ovonoi€q. To Fdvo nou Xp€Ld<€ror €ivor vosp.uvrioouf€ rouq op(oFoLi( Kor ro €v6toqtpo\,ro XoporrlproIl(d rouq(rt osropio rov opre!0iv) Kor r6rs eo ovoKoiUQou|l€ 6^rl qv oi{e€ra.

npoonoedwo( vo yEqupdoouv ro XdoFo ovd!€oo ovl o€ropio rurvopLogolv Kor orov rpoypor(K6 K6oUo, or Iuooydp€ror ovoyKdoqKov vouroortpi(ouv (dnol.! €{orp€t|xd nopd{€v€q o€opi€(. :qv npoyuolK6qm, ovanruqov ivq €i6oq op(ogllKoLj guorLKrofod nou oK6go Kor oi,!€po €rvor o^oqdv€po( o lv opreFolovio. tLo nopd6€rygo, or n€prrro(opregoi irov opo€\^Ko( xor or dplor qpLouoi rirov ofiuKoi. To ratlro r€-rpoy{rvo, omdq ro raooEpo xor ro €wio, ouFpoi(qouv q 6rmroo6vt. ron,vr€ ouFpoicr ro y{iFo rrl ouvavool rou n€prrror (qposvrKd) ps rocipro (etluKd). To a{r irov o opr0pdq qq qrutrji( Kor ro €md ouupdh<€rlv ovralqrt xor qv uy€[o.'' To oupdvro orirtroro rnpEn€ vo BpioKoworo€ onoordo€rq nou oL FEro([ rouq ]dyoL vo s(vor (ooopro*vo( Kor arp€_n€ vo Eivq( opFovrKd 6rorrroygavo. 'qoup€, €noutvoq, qv opfovio rurvoupo\^ov oqorpdJv. Auri (ooouri n U\ rirq\r oqv ouoio qlqyptvtord opr0uoJq. H i6ro I 6lFroupyio rou odgnowo( npoK^ionxs on6

r60

Page 149: Ο Ταξιδευτής των μαθηματικών

. ApX(€ pE ro 1 (q lMovd6o, ro n€poprotrivo), ro ono(o F€rdoE 2 (q Aud6o) F€ Bdon rnv opx4 rcu Ar€pr6prorou.

to vo 6ofF€ ndrq €{€}ixonk€ qur6s o opr€FlllKds BuonKotdg Eivqlvo €q€tdoouF€ n6( ol flueoy6p€lol X€lpf<oyrov rou( "ople-

Or nueoydp€rol Crlq(ov F€ XotriKlo yu vo 6nFroupyo6v 6ldqo-oxlForq rq ono{o 0o onoKdlumov nEproodr€p€( ofu108€( yro

mu( opr0Fo6(. Aurori rou €i6ou( I CpEuvo nr0ov6q Trpoilo€ on6 qvopxoio npql(tu.rl rov unoloytog6v F€ XoliKo ndvo o€ unoioyrmlKdnlvdKo. noMa! laq€r( oX€IlKf( B€ rov uroloytou6 npofpxowql on6qv €lllvrKi iah yro rq XotrtKro, r( {r4qouf.'r Kdno€q oT(i Il( ovoKo-hj||laS nou dxovov onorflEoov ro e€Failq qS od\lpovl( oE0rp(o( rovqp0ldv. Ilo rqpd6€tyFo, ovo(diuqrov 6Il oproF6v€( outrtroy€( )(drr-Ktiv Fnopoioov vo ronoe€rloodv trol 60r€ vo oxnuor(ouv rpiyovq.Tph Xol{Krq 4rov q Fr(pfi€prl rdrolo ouitro}l1 Kor, €fiou€voq, ro ] €yl-v€ dvqs rpry0rvr(6( opr0u6s (EtK6vo 22). AMol qpleuoi rou Fropouoovvo qngor(oouv rp(yojvq irov ro 6, ro 10 Kol ro 15. Irl ouvaxElq IIo-pm{pnoov 6r ouroi ot rplyovrKoi opr0gof (ro 3, ro 6, ro 10 Kot ro t5){rqv ro 6ro6oxrKo oopo(oForo guorxurv qploFuv:

1 + 2 = 31 + 2 + a = 61 + 2 + 3 + 4 = 1 01 + 2 + 3 + 4 + 5 - 1 5

EfloFdvuls, glq on6 C rprilr€( ovoKo^dqr€rq rrlq o€op(og tov opro

Ftiv {rov 6Il ro doporoFo 6lo6oxlK6v opreF6v €ivot avo( rpryovrK6(op0Fdq. Av (orrd{oLru€ ro dopolouo 6to6oxK{iv n€pttr6v oproFriv, 0o6oUge 6rr no(pvouge rerpoyovlKods oprouods:

1 + 3 = 4 = 2 2l + 3 + 5 = 9 = 3 . 31 + 3 + 5 + 7 = 1 6 = 4 . 4

rr. Ittv oplo'6rll,d, I opI'Kt orlfdord rou t'1q(d lro! .op'opu, urolotrfuFr(ouqo( p(po ( lr€oJq {Viqor\)". Bl. (o, ucootpruorli (r r M ) l6l

Page 150: Ο Ταξιδευτής των μαθηματικών

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H rorlol rlrv XohKr6v i rov y€rrrp€rpr(6v Ko0KKi6urv f€ rous oproFofq i6oo€ rrl 6uvor6rlro orouq nuoqy6p€rouq vo roul(oouv ro quot_

- _- Ko olrlForo p€ ouyx€(prFtvouq oprofoJ(. Enogtvrds, ro ordoior€po162 6roOrdmoto opjpo, ro rpiyovo, rour(oqK€ g€ ro l, Kor ro oalojor€po

Page 151: Ο Ταξιδευτής των μαθηματικών

Ipro6Ldororo oX4Uo, ro t€rpd.6po, rouriorrlK. !s ro 4. ME ouro rovrpoflo,0r nu0oy6p€ror (orooK€dooov y€r0F€rpLKd or€p€d ro onoio,0q ou\raxEro, tflopodoov vo €qy'ioouv l|tv d^t. Iro Epyo rou Mf&jra quorn4 o Aprororailq oxo^ioo. ulv n€noierlon rov tu0oyopriurvdrL 0r opLogoi orn! npoyforLK6qro, irov ourd Ko€ould ro dropopLoq uhKiq npoyFolKdrnro!.

fn,l 6i rq fiv drlq rois opSpois iqdjvrro rit!.pao! oqoforoo8q,nooov, o b dplpoini ,ot ! l t ( qao.o9 trpdro, r , r raD!. tp, t l fov.ro.Xiio r(nv ovrov orolsid rdlrov unalodov si!o,, Koiro! diov o,tpovo!oplovqv .ivo, Koi (p 8p{iv.. Ooivovr( 6i (dt oaro, ro! op,tov vofiaon.SopXt!.i\1r Koid)S ijltv rois oi)o koi(ij! ndtq r! (oiift,s.rl

nopoKri(,r oro i6ro prplio, o Aprorora^tq tra€r:

Oi 6i nuSdldpf o b o 1 D opov n, )ilir rinv opr0fov na(!,t nTaiplovrd ivro( oioDtroig on4no,v, Livor iri.v qp Spoi,S inoitoov ,d ovroi ou Xoproro i '9 b i , a l l ' iE op,Bpdv ,n ovra 60 r i 6 i r 'Or

ro ndq rq rovop,lporv.\, iPtrolio iJrdp!fl (uiiv re rjapov(n (ris! roliois iillors.,5

.ll trtrroi ro n6vro €itdilowo n( quroJ( 0r! oqotooflvo trfo! rouq opr€.

[0u!, (o, !r.Lb'i ror'lpolvro oL opropoi o( a( oiddnpov 1f! qioLv nporopx(oi0no roy 9r^ooo'lor retirov, .E.toupdlowo 0( oro\iid, ro 6€ rLjrnov €v 10 ouvb,ro ro! o( oppovro {oL ofr€fdt... npoqovoq or nueoydpnoL np.ol},jouv drL oopLutro( rowdvn 't dph ror ldloro ! nlf rou Eivor, (q rpanov dror vo ouvdyouv$ rov {ouopLol'dv (oL rov olroE0jv rov oproUov rou( (ooopLoUodq KoL ro( oxior! rou Eivor. (LK6 ndrupo(, t9i5)

lArjtorlc, hI/)e 8rJ,. tlr'a t ot Atittatte, p{ J. Arnd Gr{p. Rnhad M.Keon)(Nio YdpK[ Raidom Hau\c, 19q). rh. Metaphysi.,, es6o, I ] KoL t5.ts, Oxtord

l5 oL fueoyap.ror duoc, !n.164 ipknov 6r €( ro oroelrd oduoro .up(oKoroLnolloi LbLdqr( rov opLou{rv, nop.D*oloov p€pdio( drL ro dvro livoL opLopoL, Xop( vo €louv duoq Xoprorlv ouounop(iov or opLofoi olrd drL€ivdL ouororrd oroLxEio rov dvruv drori 6€; En€L6i or r6ral|l1( rdv opL€$iv tropouor{ilowdLdXL fdvov lioo (( ro( opUovio!, otrid {or.iq rov oupov6v KoL #oo .( noltd dttronpriyPotu. (Lx6 nd Dpo(, 1975)

1) n , r0e00, 20-25 . j 16 l

Page 152: Ο Ταξιδευτής των μαθηματικών

'EXouF€ Iorn6v €6dr rou( nu0oy6p.rouq vo unoorqp((ouv Uro qropr

(4 0e{rpio, 6nou ro droFo o}rlpqti(ovror on6 orlf€io oro Xrirpo, rlou

Kor ro i6Lo €ivor opre!o(. A€v €ivo( v'o[opei KovE(! nou ouroi (oedS

Kor no^ioi ditror 'Etrtrlv€! nou €nnp€dotl(ov 0116 outo6q, niorELrov 6I(

rl U€iirl (0v op(opdv 0o oroKdlunr€ 6^0 ro puorrKd rou ofFnovto(.

Av Ko' ouq n nioln orouq oproFoUq-droFo Fnop€i vo Uo( qoiv€tor

onloixi, np6r.r vo euF6uool€ noro €[vot I un60€oq nou roxJ€r oig€po: I ill ororrtr.iror on6 drogo Kor ro dtoFo ourd r(vot ox€6dv €{

o^o(^(pou K€v6( X6poS! 'Orov

Fdhoro npoonooofFs vo npoo6rop(-

ooulE ro dl^o Uapt rou or6Fou, 6rl^o6i ro t^eKrpdvro, ro npurr6vro

<or ro v, rpov.o, oJ'o brotruovror ot Uro nheupo a({rvrprxv o!r( .rpr

vov nou Ford(ouv vo (olofiotodv 6tr€( Il( ovtrii{i€r( l.ios Yro ro X6poKo( ro xp6vo.

A\r or oproFoi Eivor oivoio KoriKK(60rv (or KoroloFpdvouv xuipo, eq

Fnoporoogs vo ovop0rrleodF€ n6oo onaxouv or KouKKi6€( FEtoqirous. Arl^06{, ov avo r€rpd€6po undpx€r oro X0lpo (Ko' €ivo' o op'o

Fo( a), ro n600 onr{oUv ro onu.o nou ovonoorotot jv rrr Ynvr ' rrou; H ondvrrlon rov nuooyopEi{rlv irov 6r or oxao€lq ovdu€oo oro

FiKn nou xopi(ouv ro orlF€(o anp€r€ vo eivar i6yor oKdpou0v opLo'

Fdlv. M€ d^lq i6yro, ro 6rdqopo F{q lurv ni€up{iv rrrrv yEt0u€tp(6v

opretdv npan€r vo ox€ri(ovro' F€ro{'j rou( f€ i6Yo0q oxaportiv opro-pdv. np6KError ooq6( yro ivo KpiorFo oqtr€io, avo olp€io nou or

npoiror nueoy6pEror eo npan€r vo o!{rpotjoov nporpovas. Mdvov opy6r€po, pErd ro ecivoro rou Uuooydpo, ouri rl r6do €n€otp€'l€ Yro vq

:€ ouri rnv Kpiolpt qdot rqs €qa({qq 10rv opr0u6v, 6XouF€ troouv€vr,ron rov qproFdv FE rl y€Lrp€rplo. fro rouq nu0oy6pErouq, or

oproFo( irov y€of€rprKoi, oKprpriq 6noq ot €ueEf€( Kq( or KfKtrot.

npdyFor, or ono8€G€rc rurv nuooyopEi0rv yro rq ovo(oioqr8r( rouq

nou ol€ri(ovror F€ q e€opio rurv opr0B6v litov y€of€rprKa( n 6Xr ol'y€pprKaq. Apy6rEpo 0o 60JF€ Kdno(o rqpo6€iygoro yl our6.

164

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ouv€roqop€q rov n ueoyop€icov

p.[ (droro( vo Un€r mov [Erpoou6 vo fl€r 6

or ov6r]ror nueoy6-4lov ontr6s og€tr€{S Uuo(Kror{!, or oro(or fiqp€Uld6loov ulv

r0Jv Uoenuor(Kdlv (l 6F orouboiol oroxoort( qq opxor6q-. 'OFoS, fipan€r vo (dvouF€ Cvo p4fo rlpos ro n(oo yro vo {Xoutr€Eupor€pl €r(6vo rrv €t[€uypdr0rv rouq. '[mvov noMt( Foquqll,ovoxold{€rq, UftqEU rov ofiofov n€pdoFpdvovrov Ulo U€itrn ytororo y€ou€rprKd oxllForo Cxouv ioo €Fpq6d, ro o€Udho qq 0€o-

rov ovqloyr6v, noiid o€{rp4Foro ylo rq rpiyovo, rous KJKtrouqmpdllnlEs €u€E(€! (or r( oqoip€s, Kqo{is Kol ot opItKC( ovoKotri-qf o€(,p(of rulv oproF{i}v-

0m6oo, ro mo orlFovlKd tpyo rouc irov oiyoupq ro op€oy{ivtorp{ynvo Kor ro g€6pnuo nou 6^0r ovoud(ouF€ nu0ay6peo gErlpnpo.To 0€.bprlpq our6 Efvor noid onl6 l orlFooio roo 6pos yto q p€roy€-vdor€pn oKdqrrl €(vor 6uoovdloyl F€ qv qni6qrd rou. To e€6pnFo it-€l 6rr ro deporoFo n|f rerpoy{iv(dv rov 6fo Kde€rov r}€upriv €v6qopeoyov{ou rpry{ivou roo0rql U€ ro r€rpdylvo rnS unote(vouooq (ptrt-n. tu(6vo 14 oro KEqdlqro 4).

tvqs qn6 rous oEBdtrrouc trioouq rov pqolporrKtirv €(vor rl dwororou o€0rpiporo(. Ivlnop€( vo yv(rrpi(ouu€ 6vo y€vlK6 (ov6vo yro vo Kd-vouF€ unoloyloFods, Xrxpis vq lXooFE Kould on66€(n &r ourds o Ko-v6vo( roxri€l [6wo. Orov €fuol 6uvoT6v vo ouFn€pdvouU€ qnoy{dylKdtvov xov6vd on6 dtrlq o€urp|luoro i oqt(i)Foro nou e€orpofu€ 6rl rcDd-oov, r6u ourd( o Kqvovo( YrvEror e.opqFo, yvrrpi(ouu€ bnio64 orl oKv6voq €q lox6€l [dwo. H aworo rou o€op4pfios npo{10€ on6 rou(IryvEs. IrS Bap€q Fo9 nollo( o€opqnKo( Uoenlorrci o€oporjv drlro 6rK6 rouf dpyo 6€v €(vor Iirloro n€ploo&€po i dnoro iryft€po 0116qv onoyoy4 o€oprlfdrorv 0116 oUvolo o{r0Jl6rov.

A116 notrol6r€po uh(6 yvopi(ouF€ 6It or nu0oy6pElol opLolo{ (61-tro6{ rp(o U{xq nou, ov onor€ldoouv r( Tri€upts €v6S rprydlvou, oxl-Uor(ouv 6vo opeoydvro rpiyovo) {rqv yvooro( Kor o€ no}ordr€p€qKowovt€S. Or Arydmror ronoypdqot (fiou p€rpodoov cvtrivowoq oxor- 165

Page 154: Ο Ταξιδευτής των μαθηματικών

vrd) yv{ipl(ov 6r ov XurpioouuE dvo or(orv( oro f(Kn rp(o, raoo€po (or

navr€, 16T€ Oo ndpouF€ dvo opooyuivro rpiyovo (€n€r64 3'+ 4'- 5'),olld Fdilo! 6E yvulpr(ov ro ylvrK6 0€6pnFo. or Bopuirivror Unop€i voyv{i)pr<o! ro y€v(Kd 0€6pnpo, oi^d gdtr^ov 6ev ri€€pov ndJq vo ro ono'6Ei6ouv. Ko' o' Krva<or yvripL<o! ro e€d,plgq np(! on6 rouq'Eilnv€9oild 6€v €ixov xriro'o onoyoy'Kri on66€rqq yf our6. EnoF€voq, nop'6io nou Kr dlior notrmoFoi yvripr<ov ro y€v(K6 Kov6vo rou rueoyop€(-ou oropqpolo\. otv pnopouoov vo 6' r iouv ol | ourd( o \ovovo( roIJ, r

H F€yoloquio rov EM{vov .ivor 61( yv6pr(ov ro y€vrK6 (ovrivo o-lid Kor Unopo oov vo rov ono68iqouv, npoo6(6ofioq arol orov Kqvdvqro Kjpoq €v6q o€op'itroroq. or nueoy6p€roL €ivor of\7oupo 6r €ixq\r grq

on66s(q rou nuooyop€(ou eE{rp4uoro(, €vd Kdnoror nrmEUouv 6r oi6Loq o nu0oy6po( a6'xo€ ouq qv onrt6€rq. Oo rEprypdQouFE rtvon66'r{q nou l6uro, o Fu(}Fr6n( oro r /o, !ra? rou, Xtdpi( vo r tv nopouo oooup, l , moFtpo( l rnrorfp, u( prtop, I vo ppr. (ovLi\ oLo ntproodrEpo prpl(o rorop(oq rov troorlForKiv.

" Yndpxouv 6do Kqiol l6yor y(o

vo nopouoldooutrE ouri qv on'i6€'qt: npdrov, ro 0€6plFo €ivor orf

Uovr((d yro rlv €a$rqn q( dvvoroq rou opr0Fo6 ortv Eild6o- Kor, 616-r€pov, q on66€rqq 6€qv€( rov rpdno p€ rov onoio or 'El\veq

lprlorFororofoov It( y€otr€rprKaq ono6€G€rq. H ErKdvo 2l 6€ixlreL avq op0oy6-vro rpiyrxvo trE r€rpdyovo <oypoqrop€vo o€ Kde€ ni€upd rou rpryrirvou,ro onoio rxouv ovoloorEi EnrqdvE'Eq 4 B KoL f. Of^ouu€ npoeovd'q vo5liiouF€ 6It ro doporotro rov €t'po66v ttov €n'qov€'ri,v A Kor I roo6rorg€ ro €p9o66v qq rftq{ivEroq r. Aur6 FoS 6iv€r 6r o' + p' = y', 6rl^06li6,r ^aa 10 o€!,iptuo. Aur6 o EUKI€iblq ro ord6nqE Xopd(ovro( !'oypouFli nou {€(lvorloc on6 qv op04 yurviq 6dox((€ rtv enqdverq rKo' rirov nopdli n or( nl€up€( q! f. Irl ouvaxELo ona6€([e 6r( ro

16. AUo 6Lqqop.r(Ki! otrrKi! onoo.GLt npo€pXovroL otr6 ro0( Stuart Ho ing..ldle, Make6 ofMathenaftt\ o!6ivo, Penguin Books, rese), o l9 (or Eri. TempleEel, Mathenatict: Queen .nd.tervrrl olt r'er.e (Nao Ydp(D Mccraw.Hill, r9sl),

1 6 6 . , *

Page 155: Ο Ταξιδευτής των μαθηματικών

ETKoNA 21. Mro on66€rh rou nooolop.(ou o€opiforos lo F (P6 op9o}6vD nd'poMtidypotfo aBr qq €n$dlads lls sfFo66v Y' rdouror Fs ro €ppd66v o', evdro.$q66! rou F.ydlou opeoYov(ou ropoUqlolpdppu Btzr oofiq{ |r. rc 3$o6dv P'. a|)16 6efia€r 6r or+ P'=Y'.

+po66v qS A roorJlot FE ro €Fpo66v rou oprm€pod opooY{Dviou flo_

po),lrltroypduFou rrl( €nrqdv€ro( f, €v(i) ro qlpo66v rl( B rooUrol LIE ro

qrpo66v rou 6€qro0 opeoYov(ou mpd'inloYpdUuou q( €mqdvEros f.

A016 oiFoLv€ 6r ro €Upq6d A + B = €Upoodv f.

Mro 6€iEpn, ouoqpd omrKl 01166€(r] npor€iv€rol oqv Elk6vo 24

Iqv optorEpi rl€upd rls ElKdvoq 24 6XouU€ 6vo r€lpdYovo Fcoo o'

dvo 6110 r€rpdy{ovo. To €Fpo66v rou 6€dr€pou, grxpdrepou rerpoyti- 167

Page 156: Ο Ταξιδευτής των μαθηματικών

r

B

frfifi ,1""."j,:#tijj[1fi ili':Jal;;, ;^.il::i"^r;.ll;ilfrii,.i:

n#'#ur*Jrnnil:*#"t*{:ffi

5ffi:,*'*;#*:tt-,"*n*::Iffi T;Jil j:l

fi:ul'''^11':""':-"**"'o rn( I U'�(^r io'�r

i#ftfltfli{''+}flftfiiffi,., ;i't,'llJ,l[*lt*#**ii*fr;#**il

B'

Page 157: Ο Ταξιδευτής των μαθηματικών

o#llln vo rcpoqoeo0v orov nspiqrluo Xpuo6 qdrvo tlrv Ei}ivov. Ol|jqolFonK€S Kot glloooglKa( r6t€S .rou ovdmulov ot nuooy6p€ror €nn,pdqoqv [oMoi( U€roy€vam€pous oroxomds, l6ro{r€po rov n}drovo,ro! nortoo rou lo€okouod.

T6po npdna vo €{€rdoooF€ Frov dlh ovo(dlu{,rl oxeflKd FE rou(qpreFodg fiou Kl ofi4 oq€O€Ior orous nu0oy6pElouq. Oordoo, ovri vom€ft€iv€t ro 6pro r0rv opleF6v, n €v l6y{r ovqKd}uQl qvd(o{E yro ol_dlv.( rlv [p6o6o rov goonFoItKi)v.

To oortr p€rpo

0r nu0oy6pElor nior€uov 6Tt ol l6yor rov oTroordo€{nv rov y€oF€tpl-xtnv qpl0u6v (olp€(ov) np{n€l vo E(vor trdyor oKCpdl('v qpreprirv. E6ri)€lvor nou or nu0oy6p€ror luTrtr€€qv kor, d0€id mu9 qvoKdiuqrov Ulo€1r€16S vdo ddon oproudv. Koftd€r€ qv Etk6vo 25. eo mootoouF€6u or n^€upd( o ror F dxouv n me€Uio F{rcq ioo U€ rl l'lovd6o. n6ooF{Kof fx€r rl unor€ivouoo. 6nlo64 n ntr€upd y; Ol nueoydp€tol ClEov6tl n llEupd y (ol pl fil€updq o Kol p lrplrl€l vo ouv6aowol p€ kd[oloi6yo 6fo o#pqlov opteF{iv. nolo( eo Flopodo€ vo €fuor ourds o }6-yos; Mnopo0u€ vo Xploluololiooulr€ ro fiueqydp€to 0€6pluq ylo vo

EI(ONA 25. 'Evq opeoyolvo rpi:yorc 6no'r or 6Uo Kdo.r€S n^eupfc fxouv FlKo( (oo 11€ r. Hutrocrvouoo q€, F{Kos it, rcono(o 6e pnop€( w ypde€{ o{l6yoq 6'io otdpo@v dp6F6vxory €noF.vdt €(vo( oo',FFs.pof€ to plKt r(llv 6!0 Kdoflov

15,

Page 158: Ο Ταξιδευτής των μαθηματικών

ndpouF€ ULo ouppo^roi ovonopdoroorl ouroJ rou qprouod (6!io64,rou pi(ou( q( nl€upd( y).

Y ' = o ' * P 'yr - t) + t, = 2, €noudvorq,y = , ! z

'Opoq, 11600 €(vor ro tr4Ko( ou{( q( ropd{€vnq r€rpoyovrKiq pi

<o( Io0 2; looc FrodF€ orov r€rpooud vo o(€9rorF€ no( our6 ro F4Koq, ro onoio ouv6aEror U' avo ouyK€KptFavo opooyrilvro rpiyovo, €ivolpoMov oorl"oflo yro vo po( rv6roqlp'r. npoKJ uor dpo\ y.o pro ouvn-oroFaq rpri, oqou Eivqr €nionq ro t4Koq rt( broyrrviou €v6q r€rpoydr-vou rou I f lLupd rou rvor roq pr ( l rKovo 2b).

O nu0oy6poq €ivor oivoupo rl(rlq r fi€u€ 6l gnopoUv vo ppEooUv 660oprouoi, o ̂ 6yoq rov onoi{rv vo roouro( pE ourl1 qv rLF4. Aumu}6a Kdrcro( nu0oy6pEroq, nou 6€ yv{rrpi(oup. o4F€po ro 6voFd rou, on!6€r{€

F€rd ro 0dvoro rou nuooydpo 6tl 6€v undpxouv rarorot quotKo( opt0Fo(lAt^o64, ro priKo( qq y €tvo\ oorJypnpo npo\ ro F4Ko( rns q { qq p.'Onoroq n qv €(vor o ̂ 6yoq y npoq o 'i y npo( B, 6Ev Unop€( nort vo €ivoroKprptirS (oo( tr'tvo )y4oro Ktrdopo. Or'EL\v€S ov6Foov ourd rq U4Krlri,ioyq'' nou ortFoiver "ovdKqpqmo', 4 ripp4rq'* 6 o64 "6ev orrroror

XoLjv or Kdnoro idyo,.' H dfloqrn our4 Ko*mpEqr€ tuq K6p(o XopoKqpr-mKd rlq nu0oy6p€roq (ooFoloy(o!. ndlq 0o FnopofoE or6l1not€ orlq6on vo €{oprdror on6 rouq quourcLj( oproUoUq (oL rouq ̂6youq rou(, ovundplouv raroro oorgF8rpo F4q; Aurd rq pikn, npogovdq, 6€ ouFqrovo'Jv H€ rov rp6no F€ rov oroiov o nu0oy6poS €iX€ op(o€r rouq opropou(.

Oo FsuioouF€ ul olFoo(o Tou y€yov6ro( ov ro6Fr ontr6( dr( novoK(iiu'lrl ouq npoKd^Eo€ oKdv6o^o. Yndpxouv noltrdq 6(o9op€r(Kd(roropie( yro ro r( ouvdp! Kor 6Ev Uropotv 6^e( vo €(vor oirlorvdq.

I7. ,\ioyo(v): ourd nou un€ppoiv{ t! IoyNi oKd+ll Kor !q dprd q(. (r.r.M.)r3. Applroq qp,0!dq: o opLopd! nou 6€v lnop.i vo ypqq.i o! (ldouo f. o(r-

ooLou( ooou( ( t .1M )170 'o r i i " " . . , ,

Page 159: Ο Ταξιδευτής των μαθηματικών

ErrcNA 26. tvd r4p6y@-!o, 6nou xdo€ d€upd qsrtitlKos r. H 6rqy6voc noutXA plioq 12, stvor oorFlr€rpl f€ ro p1q ruv

OploFavot. €K€lvq qv €[oX{, dn€6{doov rlv ovoKd u0n o€ (dnolov'lnllooo crl6 ro l\y'€ron6wro, o oflo(of €ho€ rov lltpmo or6vo Kol{rov !a}o( rlq KoN6rnros rurv n0oqyop€(0rv.,0 l\,4ro lorcp{o lt€r dTlolmooo( qno|(qlu{€ rlv dnopqn lov oodlgErpov Fnxdv o€ l./4 gdll

t{!v nuooyop€fov flopqprd<ovroq &ol rov xov6vo q6 puorrxdtrpogqS (orv6rrlrq(. M€prKo( ldvE 6Il o nueoydpoq Cnvl€€ rov ,lrmooo ytoqv lpo6oofo rou, qltrd our6 €tvol €{orps{Kd oFglpolq xoe69 oonou6o(oe 6doKoioq €(X€ Tloov6roro n€Odv€r €rciwl qv EnoX4. ,A.ll€(loropi€( i6v€ drr qur6( rlou ro ovoKdlu+€ E{6pyto€ 1600 [o]rj rou(o€odf, 160r€ npo(61€oov rov rMyg6 ro0 oq edisooo.,' Mro dMrl ex_6oXI }l€l 6rr ol nugoydp€rct rov n&o{ov 0116 ro filoio orn €diooooyrq vo rov nviqouv, 4 d( rov frwr{ov o€ lro trqvn. ,Orlolo Kt ov €{vol rlol4e€ro, or larop(€( oura( 6€qvouv 11600 (orqorp€rlflK4 rlrov our{ novoKdlu{,l yto qv rlu0oy6p€ro dnoqrn rl€pt opreFiu _ y,o _r1u n.no{-040r}.6 0611, d( ol opreuoi irav guolKd U€ylorl flou Erl€rr€Ivowov

\z

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Page 160: Ο Ταξιδευτής των μαθηματικών

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Yflipxov opxeroi ̂6yor yro vo uroo€rteoJv 10 vio oproFtlKd qrl-qio, oMd un{pxov €n(on( avrov.( nr{o€r! yro vo 6roqpqe€i ro no}oLdp(rrpoi(6 oJornpo. To BoorK6 rtr€ovafirlFo ro\r (v6oopoprK{nv opref|ltrKulv l]Jtgiov {rov 6tl gropodoov vo XpnorFororrlooUv yL, qurodq KooouroU( rouq unoloyroForj(. Anio6i, qvTi vo XprlorUonorEi Iov dpoKoi ro unonoyrotlK6 nrvdKro, o ypoqdqs fnopoio€ vo Koroypdq.L rou!opr6Foiq Kor F€rd vo X€rpcror KdO€ op(eptr(Kd qjtqio, droq qKprp6qKdvouu€ o4pEpo 6rov pd(ouF€ o€ or4^s( opreFoJ( yro vo rous npoooioouF€, vo rouq oqqrpaoouF€ ri vo rous nollortrootdoouUe. :ro no^q(or€po ooorlfo, o ypoqto( X€rpr<6rov oqorpi6ro oro unoioyLmr(6nrvdKro ri olov dpoKo (or, lirov r€iEiurv€, Kordypogs o€ Xopri i ftpyo-HVi U6vo rtv ordvqot. f(o rnv qn^( Koroypoqri Ev6q oproFod, ropr'rFoi{d oproFt0Kd Qtqiq 4rov ra}€ro.

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UEydlo 6Fo( p€rovdKrtFo rrlq Koroypoqi(ruJv npoplqudmv 4rov rl Koroldkoot xopfloJ, Kqr ro Xopri rilov oKpL-B6 o^trd Kor 6uor6psro. 'Evo 6EUr€po p6oo Ko{oypoqir oqv Eup6nrl4rov or nspyoplvd( rou qrLdtrvovrov ord 6ipfolo, KdrL nou Eriorl({rov qKprp6. M€ d^}o i6yro, or Eupur[oior rou [,4€oo(rjvq 6€v €ixov 16-vou( qqvou Xoptloo yro vo FouT<ouprtvouv. Xplorponordwo( dvounoioyLorKd n'vdK'o 4 avov dBoKo, o ypogdoq fKqv€ orKovoJr(o moXqpri Kor or(q rEpyollva! yro ro rfto ollovflKd rF|lfo rou ufloioyLoFoU, qv Koroypqq{ qq ondl1qot(_ Mnopodgs !o ovrdtqooUt. n6-o0 olFovnK{ Eivo( n 6100€0(g61110 Xoplod ov oK€qrorp€ 6rr rl rlpoj_q Fovdoo Xoprororioq oolv Eup6nn €FqqviorrlKE orrlv torovio roI154, €vri I Boy6dq i6n 6rieEr€ g(o on6 ro /94. 1

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H €rdu€vrl rpoondo€ro €(ooyrnyiq rov rv6oapog((0jv opregrltrK{nvrllqiov orrlv tup0lq eyrv€ on6 ro povo6rK6 6rarp€ml FoeluollK6 rouM€ooiolo, rov A€ovdpvro rl( niG (yvuro16 Eniolq oe Orpnov{irot,n€pirou 1t8O 1250).' To t2O2 o A€ovdpwo typoql ro Liber abaci (Bt-plio n€pi ro0 dpoKo), avo prBtr(o flou ovoq€pdlov mouq unotroyropoJ(Kor r€prEix€ ro vao oproFtn(d qrtqiq. To prpi(o rou Xplouono(orvrov€ni ordlv€( oq prptrio ovoqopdq yrq louq unoioyrouod(. tvrq 6€dr€prl6010l oro vio opteFrln(d qrlqio i6uJoE o lohannes de Sacrobosco, ooroio( 6f6oorc oro flov€nroqFro tou floprorori. To t24o iypoqr€ 6voKsiFevo yro q Xp{ot ruJv rv6oopoprK6v opr0Frlrudv ll)nq(ov oroururo^oytoFou(, ro oroio XplorFonoroJvro\r €uprroro Eni o({,jv€(.

Aur6q or npoondosr€( 6Fo( 6€v €(d^€ulov 6^€S tl( ovlrpp4o€{( oX€r(Kd fE ro vao o6FBolo. 'Ooo( npolpodoov rov dpo(o, unoonlpr<ovg€ fl€iofo lq nanots ondqJ€rr Kor ro nolord oprourFrKd Unq(o Ko(ovoFd<ovrov <opqKrori(,, €vd ourof nou ie€lov rlv €rooy{rry4 rov t!_6rK6v opr0FrIrr6v {nq(orv rirov or *o}yoproFrm6(,.'0 H UdXn ovdpEooorouq opoKrori( Kor rou( otryoptoFrortq Kpdrrlo€ €(orovrd6€q Xp6vro,FEXp( 10 6rKqro eKro o(rivo, qp(€rd p€rd qv opx4 qq Avoydwnonq.Taioc, opK€rd €yx€rp(6ro fiou ovopd<ovrov Rechenbucher (Ao\rcr|ioPrplio) Tun60rlxqv oq f€pFovio K( €yrvov s€ raro(o poeF6 oro6€xrdof Eyx€(p(6ro ovoqopdg dor€ To uroloytolK6 nlvdxro Kot o dBqKo(ovrKoroordorlKov on6 qv navq Kor ro Xopri. BEpoio(, ot Eup0rno(orropriyov r 6ov a€pLoo6r€po Xopri Kor n ti^€rqrn XoploU 6sv irqv nror0oo HEydlo npdp\Uo.

To yrori n tup6nn ovra6poo€ oro rv6oopopr(6 oJornFo 6€v €qlyEi-rqr n&p{rrs 0116 qv d^iErqrrl xoplori rour rputrouq or6v€(. Iqv npoy-UorKo lro, Eivor d{to oropio( yror( 6€v uroeaqo€ lvo odoqpo F€ 16oo {lKdeopo ntr€ovEKr4goro_ T600 ro u[o]oyror(K6 nvdK(o 600 Kot odpoxo( poor<ovTov orlv tlF4 oaon!, Kqr ro 6€Ko6r(6 opr0FllK6 oU-ortUo rirov €updo( 6(o6€6ofdvo mnv Eupdrrut. AK6Fo Kor or €uponol

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:rotnfo {onpDxvov rou( Fuponoiou( rpo(, ,vo ouy}povo orjortpo

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opr0!6v nou Kirlpov6floov ond rou('EtrltvEq xor nou rl€pr€q€ g6vo

0€I|Ko6( oproFoUq. To oUmluo ilov un€pponrKd r€proproudvo Yro lrS

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dno+rl 6fl 01 opvNrKoi opreFo( €ivor ono6€Kroi yro rou( unoloYr-

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oqv dnoqj|l 6r I I on groq Eqioooqq 0o Fnopo o€ vo €(vqr avo( qp

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o(o^iq rou( vo ono6€l'roUv oLirodq lou( nopda€vou( opr0Uo6(. O Mi

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Fo, iq). 'T€trrKO o OtroFov66( Uo0nUorK6( AlFnip Zrpdp (1590'1633), oro

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rrlrE(. AK6Fo Kor ro 6a(oro 6y6oo q(dvo, Kdroro tyxErpi6Lo 6€v nopou

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porouq oproFodq ro und^orro €ivor apyo rou ovep6nou).'IIlv npoy

r2. Hollingdal., o. r0e.1t.lane Mun, Ot Men and Nrmbe (N(o YdpKrI Dodd, Mead and compaiy,

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o Kp6v€x€p rl(or€u€ dIl 610l or opropo( €Kr6( 0116 rou! qu-rc6s eo npCn€r vo orloyop€uro|Jv ytor( rc vo pooqouu€ ro FoorlFo-

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nou ox€r(o\.ror U€ ro dn€rpo (ot Kot6nN pd^rooov rpoonoedrwo( vo rq

{€6rq^dvouv. Or n€ploo6t€por F€rqY€vim€por oloxqmd( uoofqoov touq

opropoUq Kor rq owft1{€l( rov Entlvov l(l lnEoov oro i6lo ra$oK

An€rpo p€y€eoq

'ooov oqopd ro 9uolK6 Ii uhK6 K6opo, oqv npoYuqtrK6 lto 6€ Yvopi(ouF€ r(ov ov undpx€r Kdnoro dn€lpo UiY€oo(. E(vot to ofFnov oTI€i

puq U€ydlo; A€v {apouF€. "'OUoq, n6S ptop€i to oounov vo €rvol rI€

n€pooldvo) iiEr o Ki€rorogoprK6(. <Oo npan€r vo oov€Xictor 6roprdq

y(ori ov rctd prdvo!€ oto t6^os, r{ eo untlpxE on6 qv dllrl ni€upd;,

O qrpoqoprK6s qrovrd: <To ofurlov FfiopEf vo ovo6rn}6v€rqr.

MnopFt vo rrvor n.ntpooptvo XUpr( vo rrvor n.proproFavo

o Kl€roroqopr(6(, 6Foq ouv€X(Gl: .Aurd €(vor o6(vqto. Kovavoq

6sv Fnop€i vo eqvtoor€( avo on€pL6proro, (€n€pooFdvo o{frov dpo

6Ev Unop€f vo ufidpxEr (drl ratoro,.

lv(opoi$€ vo 6ntroupytiooup€ rop6poro uno0€trxd enrlerp4goto 197

Page 185: Ο Ταξιδευτής των μαθηματικών

rou ovoqtpowor orlv or€po\rroofq rou Xpdvou. n.iJ( Unop€i ro oLjFnov vo orogcT|lo€r vo undpx€U npar€r vo ouv€X(Cror 6ropK6q yn4p-

X€ dpoyE ndvro o Xp6vo( ri urqp{€ K{iroro opfl, 6no( oq O€op(o rou

Kr aror ouv€X{Gror, rioo-Fnpo(, F€ rlv Kdo€ n^€upd vo Xplo{Fono€(€n EyFdvo roupordKro ^oylciq yro vo XopqFoiv oro vou goq ot &xE(q{ on6Q€r(.'Evo odlro nou €v{,jv€r r(( 6Uo r^€up€( Kor ouxr'd €rovqrqovGrol o€ ouroU rou €i6ouq ro €nrx€(p4Foro €ivor ro ov n Fio ,j n dMnfll€upd FropoJv vq (qovrooroJv, xdro(o awotq. Aur6 nou ouv|1eo( €v,voodFE o€ ourf( Ilq n€ptnr6o€rq €(var I txov6qro vo gowoorolF€ glor6do. Mrop€i Kov€(q vo golrocrr€i ro o(Frqv vo ouvqcrqt 6rop(69Mrop€i (ov€i( vo qovrom€i ro orurlov vo r€]€rdlv€(, Xopis 6FUJS voundpxouv 6prq O rpuFfavo( o0Moylo!6( €66 €ivol 6r yro vo ufidpxa(dr, o( dvopono( rpar€r vq Fnopofv vo ro qowqorofv (ot, oK6po r€proo6l€po, vo ro (6ouv, F€ ro .€oorEprK6 FdrD rou(. ndoo [€piEpyorpdyForl Aurd ro ouyr€KprFivo €nrx€ipllFo q€t ropdq v€( ouvtr€€S,yror{ urovo€( 6r nprv ono(o64ror€ <ajo yfv€{ op(nd oqu6€pKls dm€ vofnop€i vo gowqor€i tvo F€yq.trrrEpo ori!rcv, tvo rfroro oUpnov 6EvonipxE. Iq ouvax€ro, 6rov ot np6yovo( Fo( qjrvov opKErd €uqu€is dlm€vo Ko(rd<ouv To dorpo Kor vo ovqporodvrol Il ufidpx€r nioo rou, roorunov €pgov(oqK€ {ogvlKd {r( €K eo6poroq. Ev ndorl n€pwdosr, ro€nry€iprlFq 6€ orrKEr. To utrKd rpdyuoro 6€v oq€Aouv Dlv 6nqpq|i rouqmo y€yovoq 6tl or dvopornor Fnopodv vo ro qqwoorodv.

Mro d^^l ouvlnao IItq owalqJq( 6r n oKdqr! ou\,rnpEi rrlv npoylo-r|(Oulro Efvor n 6roph €v6q nqwoyv(irorrl vou nou qo\,rd<€rot 6rop(6(ro ofunov Kqt, €noF€voc, ro Kdv€r vo ouvEXGr vo undpx€r. Aur6 6Fo(6€ onpo(v€r dr( €UEiq, ro n€rl€pooFdvo ovop0jnrvq 6vrc! npoKolodF€ FEq o(a{l Uo( rlv drlapq rov ui(((irv rpoygdrov. EnoFfvoi ro ov Fro-poUU€ i 6€v FflopodFE vo gowqorojF€ KdIl on6 ro guorKri K6oFo 6€ox€tGro( g€ qv ,ropq{ rou. Iro Uot our6 rou urdpx€r, undpx€r, Kr or,-

tp8 a.,.';.i'n Bane i ! M'Ydtul'EKpfir: o€op(o rq 6lfLoupYrds rou olutrovroe.

Page 186: Ο Ταξιδευτής των μαθηματικών

noo 6€v urdptr€r, 6€v urldpx€r. oo npCn€l vo Tlpootxoug€ 60r€ v'

onoq.0yoo!€ ouri qv ono(ulon vq (pllnoug€, U€ ro €ot0t€pr(6 UoS

llftr qur6 rc oloio ou<ltdF€, Kot vq €iuom€ np60ugol vo oKEqtoJF€ dv'

vol€( fiou 6€v Fnopofp€ vo tr( govrqoto{U€ mo oJvoi6 rous

To ouuntpooFo 6^n( oud! q( q[oooqrK4( ou<4qols €ivor drr

6w Ffiopodg€ vo nodu€ ov urldpxsr { 6Xl (drloro uhx6 dn€lpo pdY€

0o(. Anldrq 6€ yvopi(ouF€. Au16 6€v €ivor oflo}6r0x KoKd. MoC oqivEl

q 6uvor6rlrq vo KdvouF€ unoelo€l(, kr oure( ot urloedo€rq Unop€f vo

loq snrptqiouv vo gowomodF€ rnv TlpqYFotr(6q1o U€ broqopErt_(oU9 nro €v6roq€povr€( rp6nou(.

or 'E),lrlv€( 4rov o( rp6tor nou 69loov r(oo tou( €vo qq{6}oYo

rN€uporud apyo yro qv awolo rou one(pou. 'Onur( 0q 6o6F€, to d[€lpo

krcupyodo€ outr /d u\ omoloylK6s nopdyowo( oulv €),irlvrKri rcogo^o_

r/iq. Mro ofl6 rrq rlp6t€q ovoqop6s oqv aworo tou on€ipou rpoEpx€tol

on6 ro KoFFdl €v6q X€rpoypdqou rou o€p€Kd6t on6 q l6po (dp6oFoc

i tno( q(rvo{-n.X.), o onoioq €(n€: <ZdS llv Kol Xp6vos iocv d€f. ,,

6$o6i: "O Aio( Kor o Xp6vo( un4pxov ndvror€...".'Aut6 6€q1€r dtt o

o€p€Kr6n( [(or€u€ noS q 6(doloor] rou Xp6vou 4rov dnnpn O Avoqi

Fqvbpoq o [4fi1oro( (tKro( o({irvo( n.X.) FLtro o€ Kl ourd( Yro dn€{pp

Xp6vo: "'Apxiv Kdi orolxeiov rd dn€rpov... Td piv U6pq H€ro8dtrl€rv, rd

6l Trdv o!€rd8llrov sivor', 61tro6{ n ouo(oortKri qdol], 4 411 K( ov €ivor

our6, rou Ml n€proprop€vou, €ivor ouivro Kol oYapooq.'

An6 louq opxoiou( 't-llnvEq o Aprmo{tirlq €[vo( our6( nou nop€{X€

qv rflo nEp€KrK4 ovdluol rou qr€ipou (Er(6vo 30). To prplio lll rov

ouor&iv ro0 €(vql otr6Ktrlpo ox€66v oq€po$€vo oulv dwoto rou on€C

pou. O Aprorord^n( doqo€ (plrlKi t6oo orov llidr0rvo 600 Kor orov

nueqydpo, En€r64 n{or€uov 6Il to 6[€lpo 4rov Fro oue6nopKu] ouoio

n 6r flo rbrdqro (dnorou ditrou npdYFolo(. O Aprmortlrlq 6EV qno

6€X6m! 6Il qurd Ko0our6 ro dTl€rpo 4tov ouolo Kor 6Xt 16l&110 Kd-

roroq d^\( urqpft4q ow6qlos. o Aproro*llf ovqYv6pr(€ Il( 610-

199

Page 187: Ο Ταξιδευτής των μαθηματικών

ItrKoNA 30. Aprorora\i rs4 322 tr.x.

(Y\rd on6 rou( q6€^qoLiq Mnp6ou!, rrapllvy()

qop€rxaq €qopFoytq €v6( on€ipou - da€rpo( Xpdvoi dr€rpt 6rorp€r6qTo - o^Id ordpprnr€ ro dr€rpo gE€0o(.

'AiI' ioos oilrt pdv aol (o86lou il4irnoq lrdllov, 0iav6rx€rorrb dnr-pov K.rl av rois $8npan(ois slvor(dl av rois votrois Koitq6av axouo, lr'y38o9 ip.is 6' .ToKonoDpsv r€pl rov qio0tlT6v (oi r.pt Alv trooa}reBqritv pa8o6ov, dp' dorN av oLjrois rl ou( ;o1 oopo dfterpov iri ril! orjtt-orv. ̂oy K6s pav oi)! o(orouFvo$ aK raDv robv6s 66€ssv d! orlk etvo,.r

a. AiI' (ob( rc <irrlto rorro €ivoL troi|J y.v(6v, (u ro firnFo .dv io dn€Lpoveivor 6uvor6! vd undpxrl (or !$ ro foolpon(d npdytoro KdL ers ( vorld Kdr €,1rkivq nou &v axouv Kov&o fayE€oq tlls( otrpdrdureo sdv a( ro ooold npdyFdro/ nou onor€^onv rc ow(d(|l.vov !t! l18lar!( uq!, undpm { dxr dnapov o6fo.o( npog rlv oUhoLv. M(o ̂ oy61 E+rooLq eo €qoiv{o vo ono6awq 61 6€!undpxr (EK6. ndnupoq, reA)

lAtisrorle, Ihe Basi. Works af Aristotte, ?hytics, Book t , 204b, 2-9.1200

Page 188: Ο Ταξιδευτής των μαθηματικών

l6a€q ovogoptKd p€ ro on€ipurq p€Yolo

'EXouFE ou<qdo€r rn guorKi €ndxrool €n' dn€rpov I ro onErpo or0

QUorK6 (6oFo. Tl ouFpo(v€r p€ ro oqrlpllravo ovrlK€itr€vo lqq o#qrnq

rov K6oFo t0Jv r6sdrv; lMflopo[v vo urdpxouv oqlpr\Uavo dn8(po; 16'

po PproK6UoorE o€ rlro or€p€o a8oqoq Kor UnopoiF€ vq onovltloouF€

xoroqor(d. Nor, dxouF€ tdro(€( avvorE( tou o[€[pou tqv npoYpoll_

K6rlro, ro (ol[t€po fiopd8€rYUq, Kr €K€ivo noo oooricl ^lY6r€po rouq

qKpoqoBrKodc, €ivor rl avvoro t(,! quorKdv opreulirv luv€Xl(ovror 6rop

K6q. 'OUru9 €60i oKprp{ir( od}ouu€ vo pp€ooJFE, Yrqti o or6xo( UoS €i

vor vo xqrovo{ooupE rouq oproFoi(, Kor or oprouoi €ivor dr€rpor o\r

rou( oK€qroaFs (,( ofvolo

Aq oroeo6p€ Fro otrYu{ Yro vo ou(rlr{ooup€ l( avvorE! lou ons!

po( peydlou, ttoL 6l10rq tLq ouvo!'rdF€ otouq quorKodq qproFo'q or

dvoponor qovrd(ovror {rs 6roqop€l(6 rp6no rouq o(aporouq opro

poiq.'Orov ritrouv nor6i, €ixo o(€qrE( Kdnol€ 6n 6tror ro[S "p^'nouv'

6nurq Kr €y6 6q1o64, o opr0F6q I Bpio€Iol oto #!rpo, to 2 o€ FrKpn

ondoroot] 6E{rd tou (ftor 60r€ vo !n ouYXtouFE ro 1 Kot to 2 FE rov

opleu6 l2), KoL UErd o opr0f6( I6s{ld on6lo 2 Kor otJro Koe€fiq or

quor(oi opreuo( €KrE(vovtov npo( ro 6€Etd gou, Uixpt nou €{oqovi(o

vlov ltoo o' avo €i6o( ouixtrl(. Kdnoro o1ryU4 pdrnoo q Unt'po Fou

n6( dplEr€ ou4 rous oproFodq. .:Envofv on6 rnv KopuQrl F€ lov

opr0u6 l', Fou €(i'iYllo€, (Kor, Koe6( npoxopoov npo( ro Kdrrd oxrluo-

ri(ouo( dvo €(60S orpopilou, ou{dvovror,.-Huouv xordr rlxroq Aii6,

p€poi0rs, €(vqr onoldto! (orovorlt6 10 YEYov6s 6T1 6rqqop€r(Koi 6v-

oponor, U€ 6roqop€1(Kri un6po0po (ol EKroi6suort qovtd<olrol 6to_

qop€rKd rouq opLgFor(.or [€proo61€por on6 Uo( fil0ov6v !o (qowo(duom€' rouq q'ror

KoUq opreuod{ "p€ ro vou poq, oqv vo q€Ktvofv P€ rov opr0p6 1 Kor vo

ouv€Xi(ovrqr 0'! ro 6 4 liYo nopondvo, Kor 'I€hKd vo €io'po!((ovtor

ond ro n€6io qs €otitEprnlq Fo( dpqol( AE 'planouur' ro dn€rpo

ntr4oo, ruv oprouuv nlx\ t fpouFr 6rr undpxsr onrrpo ' r lqeo( opro- -^-Fov: Er.r6l uropoupt va govroorouur o,rrov nou nro'rLjoupr ol t r 201

Page 189: Ο Ταξιδευτής των μαθηματικών

vor o U€yolfr€po( opr0f6q xo( tl€rd vo npoooaoouF! oE our6v ro ]Kor vo rdpouF€ avov oK6Fo fEyai0r€po opt0F6. Erop6vo(, o lsyoiLJ-r€poS oprop6( yto Fo( 6€v rlrov r€irKd o g€yotrdrEpoq. :uv€r6lrolotoi_pe, ,\orndv, dTr 6€v undpx€t f€yoidr€pos oproFdq o( oproFoi ouv€X(<o-woL .rl dn€rpov. Aur6 ro oUvrouo Efiry€(prlFq, nou 6]0r Uo( 10 dxouFeoKEqr€[, XplorForo€( r! ouvnorolavn loyrKi fd0o6o rou €(\ro( yv{rooio( 6gp€ot orob€rnrK( Vieo,o| \ reductjo ad abrurdum (€r( drorovoroyuryrlt, nou npolyouFEvoq €nrolFdvog€ dll onorE^oUoE rnv oyo-qF€vn Fa0o6o ruJv Ei^4vov.

-Exoupe norn6v rou( {tuor(oU( opropod( Kor Eivor dn€rpor. T( €vvoodll€ 6l{rl( idyovro( .dn€(poq Aur6 nou €vvooup€ €ivor du 6€v urdpX€r Kovd\roq quorKd( op(0!6q aou vo €ivor o fsyolJrEpo(, or oproFoi6 o6ri €ivor or€ptdproror. Aurd ro orlFsio sivor olpowrK6, yrori p€rrlv EWoLo 11€ qv onoio XprlotFororoUU€ ro dn€rpo €66, our6 Ko0o0r6ro dn€rpo 6Ev €ivor opr0u6s. €fvor, ovfloar(llq, €vo lopoK lprotK6 yvd-prolo rou ouv6lou 6^{rrv rorv quorK6v opref{irv.

:E d^€( tl( €noxa9 ond rrlv €nol4 T|v Eliivov Frxpr ro 6€Koro€voro ordvo ((or, odtquJvo !E xdnorou(, oK6po Kor Faxpr o,lF€po), nfwo(o rou or€ipou orou! optoFoU( rpop^nFdfl<€ rouq grtroo6qouq.A€v propoioqv vo .6ouv' rqur6xpovq d^ou! ouroUq rou( op(0FoUq,d\rruJeov trorn6v 6r dtrot €(€ivor oL optoFoi rou( ono(ouq 6sv lnopofoov vo qovrooroUv,0o npanEr vo 4rov Kdr{r( ̂ (y6r€po unopftoi ord€Ksfvouq rouq FrKporq opreFo(( nou Xploruonoro6F€ KoonF€prv{i. Au-q qv (poodl./rorl uroodqoov oL opxoior nu0oy6pEror. Er€r64 6€\rrouq dp€o€ ro dnlLpo n^ieo( ro\r guorK{nv opr0f6v,0€6pqoov rir( or6dKq npohor oprogo( onorEloUoov q pdon 6iov rorv 0notroirurv, (ordr ro l0 EiX€ f(q r6roir€pt r€p6rtro. O O 6^ooq on6 rov Tdpowo(nalnro( ordvo( fl.X.) 4rov dvo( nuooydp€roq nou g€opEirot dl EiX€€nlpEdo€r Tn eL\oooq(o rou nldrovo. To i6yro rou €Kgp6<ouv ulvovriinqrn rorv nuoqyop€(ov 6l ro t0 aXEr gro r6roirEpn odon oqv no,poyoyi 6iov r{rv opreF6v: np&sr vo F!tr€r(oouF€ rL( 6pdo€rq Kqr rnv

--- o^t0'v'1 eUot rou ApreFoJ o(Uqovo f€ qv roxj nou €vurdpx€r orf202 Aexd6o (/EKa5d6tlru). y(orf ouq (n A*d6a) Eivot orlpovuK4, oio

Page 190: Ο Ταξιδευτής των μαθηματικών

xltpoH€vn, rupoq (oropeoq, n opxrl q( eEiK4( Kor ovepd,mvnq (o

{q Kor o o6ny6( q( Xurpi( ouri1, 6}0 to npdYpolo €ivor on€pr6proro,

6uov6To Kor 6uo6rdKprro.'Au{ rl n€noonon uno6 tiJv€t 6rt ot oprouoi fiou €fvol F€Yoiit€pol

0fl6 m l0 FrFodwor rouq optoFodq on6 to I E{nq ro l0 Kol oq€Aouv

qv Jnop6ii rou( ornv €novdlnqrq ouri! lov 6tKo oproPttv ErloFdvog

0r F€yo^dr€por opropoi €gqov((ovrot 6tov €novoloFpdvovtor o( qpro_

Foi on6 ro I turs ro 10 €vov n€n€pqoF€vo aptopd gopdlv. A€v urldpxEr

dr€rpo r 4eo( oproUdv.O nkiruv Xpnouoroilo€ (t our6s qv owAllln 6( i€(oupYoUo€ xd-

nou qpFl 6€m6(6rnros yro vo n€popf(€t rou( drapous oplouoiq: "o0r€

ydp av rois dprqrois rd anlriv KdBoip€olv dT€rpov urdpx€r, il Ydp povds

aldx|orov, our€ trl riv oLiqnv ldxpr ydp 6€Kd6oS norsi rdv op$tdv,.'

Kor o Aplororttrrlq onapptm€ rlv 6nopq dlElpou ntr{oous qplo

1,r6v. *"EKoorov 6'dT€rpov €ivor d6rvorov od)Fo trlv Ydp aorr rd 1ro-

vrn dxov srdoroorv, dTapov 6l rd dTepdvros 6rsornxdq, 60r€ rd

dnopov o6fo rovroxn ioror 6r.ornk6s €is dTerpov,.T oor6oq 4rovop(€rd t€urwo( yro vo ouv€r6rlrono01o€t drl I on6^uq dpvnol 6lov

tuv on€ipriv 0o EKo!€ no^[ o6€tro q q oooqio rou. fto vo nopoKdf'

U€r ouq qv q6€qr6qro, €qldp€ tvo FtKp6 (6lno €ir€ 6r ro dnErpo

6€v 4rov .npoytrolK6,, oitrd g6vo .6uvrlnK6,. ApvilollK€ €rofdvo(

qv r 4pl inop{n rou on€ipou, €v6 rourdxpovo npooro0o6o€ vo(pooio€( 4urwovi Kd(oro dworo rou on€(pou.

6. ErS rouq opLofo,i( 6€v undpxs o( opxl our€ ro dnnpov Kor6 Feidd(v, 66rL

I Fovds .lvol alo sIdXL@o!, oit ro dnapov Koi oi{rprv, 6LdrL rl oF0Fn1'n1 rouoapd olqrord a! qv 64d6o. (E(6. ndruPos, re75)

I rislotle, Physi.s, Eook l , 206b,31 !2,)7. To vo €ivornd-\rv ro (dos rL odfq dn€pov slvo' o6uvorov 6,61, odlo €ivor

d,r, rXsL 6ocdo€,s (e(s&€tao trpoq drdq tds Ko€uoovoaq, d dr6 ro dMo F6pos6nryov dvor 4r dxa onapow.! 6,qordd€q doc 8dv un6pn ald dndpov odFo,

I onap6q! eAa mtoldp€r 6ioq Eq 6,odrdon(. (EK6. ndnupoc reTt

[o.n., 204b,6 8.] 203

Page 191: Ο Ταξιδευτής των μαθηματικών

'All' arri tro^lo}ds ro .ivor, djor€p t) 4pdpo aori (dt o dydv 16 d.lo^lo Koi dlio Yiv.o8o, oijro koird dnopo!... id! tj,oijros on€! rdv^6]ov iiors d.i 1 rn oorb nlp,tof8dvuv payr0os, 6rt{.ror, 6d ro ndvrd nrqqopdvov dlorprio8o or(.DoDv dpropive.!

O Aprororatrlq {eE^€ vo ropo6Exr€t 6rl or guorKoi oproUo( lropo -oov rdvro flpoofloaF€vor vo oxlpqrf(ouv oloavo (or F€yolUr€pouqoprgFoJq, oild our6, unooqpr<€, 6€ 04Forv€ dnapo n^ioo(.

A n € r p n 6 l q t p € r d T n r q

O''E^iqv.( Booovr(owov on6 ro spdtqpo ov o Xrirpof Fnopodo€ vo6rorp€(ror €n'dr€rpov, KdIl rou orl ouvtx€rq rou( d(ov€ vq Fr€p6€u-ro0v oE d/fl oqopd rlv (ivlol Kor q ouvax€(o rou lojpou. H Ei€oru{:Ioli, rou r6pLj0lK€ 0116 rov noppEv(6t 0116 rtv Elao (n€p(rou ro 475n.X.) €iX€ ovrio.r€( ondqjEr( 0116 ourl( rov nuooyopEirrv, or ono(or ri-oltuov o1 o Xdrpo( onor€iorjvrov ord Ur(pd €u0dypouFa ru(Fororou €ivor o6roip€ro_ Or Etr€drs( 6Xr F6vo C6fl{ov utv owiqqoq oulvonoio obrty€i I dno(]Jt rspi o6roip€r{lv rtrtFdrr,Jv, otid a6€rqov Ko( ronpopiiporo nou npoKdmouv ond qv dfloqjl 6l o X6poq oror€^siroL

H p€roquotK{ rou flopf8vi6n sivor ivo r€^.ro nopd6€ryFo qq .nlsupiqq rurv E^^(v{rv vo onoqpd{ouv tl( on}i( oroorio€r( Xdprv q( ^o-yrKri(. O nopu€va6l( unoor{pr(E dll, oqo, ro ol|6inor.6€v pnop€i voflpoKuLirEr on6 ro rinorc! rdrE 6,1r Eivol, or 6( s(vot, Kor ndwo 4rov.To ov Eivor €vo Kor ovolioirrrro. Aur6 o6iylo€ oro o{roong€ioro ougnapooFo dll I kivtol eo rpan€r vo €(vor lro Q€06oiooqorl oeod rt

3. all' rq' 6oov ro €ivoL .Klol1p6laor dq troMdq onudoio(, dnoq | lnqp4(1'l( npapo< @ rou dy0tvo( .ivor fia ouv€X{q qvovaooq aror quUpdrva (or pr rcdnL(pov . Av 6p@! ouqdq rov idyov (rou tdppovoutlou rtipqroo o!tu( rirorndwor vo lotpdn ro i6Lov Uay.eo<, oo ro 6Lalof (Oc ro .qovdiorl). (Er6. fdnu

['O.n., 206a, 26 206b, ]r.l204

Page 192: Ο Ταξιδευτής των μαθηματικών

ETKONA lr. Z{vov o El.drqq, 49sl490 430 n.x(oortoypoq(o on6 rou! nivoK€( rqs qul.\oy1( Culv{, Nad Y6pq)

npoy}rorni [nopEn 6€v Unop€( vo o]^d(€r. Ouurq, qut6( nou €na6po-o€ onpovrKd oro goonpqtUd 6€v (rov o nopF€vi6qs, F€ tlv oqrdloYnp oooq(o rou nEpi rou €vdq oMd o poolt4{ rou, Ztlv0rv o Ei€drns(n€p(nou 10 aso n.x.). O Z{vov (EL(6vq 3l) nopouo(oo€ 6rdqopo €nr'

XEeriuoro nou 66€r},/ov, ofugovo F€ rlv El€otrK4 ^oYrKi, drr o K6-

ogo( 6Ev FfiopEl vo €ivor Fto noilonl6tlro [poYudrurv our€ Kor vo

oltrdcr, o€ o\r(0€ol U€ rrlv n(otn Iov nueoYopEfov orouq opleuoLi((Fovd6E() Kor oro U€yC0q nou Kvo6wql mo Xdrpo To rlro Yvoor6 €nl

X€iplFd rou €vundpxEr oro Fieo Yro lov AX I€o Kot rov oYtirvo 6p6_

lou, drtu)( lov 6llyritor o Aplorotin(l 205

Page 193: Ο Ταξιδευτής των μαθηματικών

A3or.pos 6'o Kolo0u€vos AXli€|iS dol [,onros, 61 rb 6po66r€po!o0bdtror. Koro^nqBio€r& Bdov r:trd roo roxtorou iprpooBsv ydpdvdy(oiov diosi! rd 6r.n(ov, 6Bsv 6ppto0 .rb qsalov, dror, d.i1 trpoa-Xd! a'loy(oiov rb 6po6[r€pov. 'EoI6a kotoiros o oJros ]6los r4 6f

To €rxliprluo rou Z4vovo yro q 6xor6unon trlEr or 0t( tu rl K(v!-orl €ivor q6Uvorl, yrori dvo ovlK€(g€vo rlpafl€r (poro vq qroo€t oroFioo q( qn6orooq( ord rov r€il(6 rpoop{of6 rou. eor6oo, ftpolyoufdvoq, 0o rpdn€r vo grdo€t oro Udoo rou Uaoou qS ordorool(,(or o0ro (oo€{iq. Erouav{oq, 6Ev grop€( vo Kqo€( Koediou. 4016 nou€66 o€{,rp€i 6€6ou€vo o Z{vov €ivor riIl I orldorool on6 €x€a nou E€Kt-vd€r ro ovlK€(g€vo Ftxpr rK€i nou Koro^4y€r p[op€( vo 6rorp€e€i dnsr_p€( qop€(/ (or ol ro ovu(€iF€vo 6€v FnopE( vo fi€pdo€t 0116 dvovdn€rpo opr0F6 onu€iov oE nsn€pqogfvo Xp6vo. Auq I qwarlul r€pid[€rprl( 6(orp€rio]roq onor€]€i rrlv K€vrprK{ avvoro rou or€ipuJc F((po6. Mnopoug€ vo 6odg€ our{ qv dworo orlv rpd{n ov XpnorFo_fior4ooug€ ulv €uo€io rojv op.0p6v Kol 6^0 ro orlF€(o rou gnopo[psvo npoo6lopioouF€ €ndvo oulv €uo€io,

H trK6vo 12 6€(X.y€r Cvo rliFo rQq €uo€(o( rov oproFoiv ovdF€oooro 0 Kqr 10 ]� Oo oi^{iqouF€ triyo rn 6lo6rKoo{o rov uno6rqrpao€urvKor 0o npoxop4oouFE 0116 ro 0 a|llq ro ] F€ 6^0 ror nlo lrxpo qrlFo_ro.:EKlv6vro( qn6 ro 0, flpirl€r np{irro vo qrdoout€ oro ong€ro l/2.N4€rd, rrp;n€r vo r(oiri|l)ouFE trlv €vqnogtvouoo ordoroorl y(o vo qrd-oouFE oro olu€io 3/4. To €n6g€vo €v6rdF€oo onf€io €ivor ro //8. oyFK4 n 6ro6rxooio ounl Frop€( vo ouv€Xcror yro lvov {irErpo opre!6hpdrov. A€ eo {prdooupE no1d oro I F,lvqv rEr€pooFavo oproF6 pn-

e. o 6.Ljr€po9 (ouMoy@Hd!) .ivor o Kdloltsvo( Ax|Msi( o0wovo O€ p. ourdv, ro ppooLir.pov €rq ro rpdBpo ouoatror€ 0o qooq0i otrd rc roxrr.pov. 6,d(ouro trou Koro6dr.( npan.r ndvroa vo qodv! a< ro otU€(o! qBJ ro otroio! (€K(v'lo€ ro 9.uyov, .'s rpotrov (iroc .ivq, oloyKqiov ro ppoolrpo! vo 6lo{pl nd-vror€ fro! otroordo'v on6 to 6.Ljapov. EivoL o i6ro! ooiloyLo@l U€ s.(vov qq 6LXoroUioq... (rK6. ndnupoq, r9A)

lcr .n. ,2 leb, t4 . l3 . l206

Page 194: Ο Ταξιδευτής των μαθηματικών

o'ffi]'n*"'** ""',"ov dnapo op,eud o!!€iovy(o * **;:",."

fdrury }rou aovla Oo Fnoporjp vo npoxdplooupr ,lryo okopo. tro vogrooouur rpdyFor oro I rro onoro qipoul. dlr ouuporv€r oro puorKok00!o - 6lio6il ovro( KrvodFoor€ ono rov d!n€r vo r€pdooup€ 0..6 .vo" ;";,;'";;,; ;1""

'o"o otov d^lo), npi-

l: j:ou\ Edlvr: ro,., -,",,. ;;-";:p^d[:::l1l;:::Ti::;foultorr(orj( t\.4aopa vo qorvsror oolp""," r,U" no * ool"nnoou-l,r, oqou 610r Unopoup, vo (orTotouF, yup{, po( ror vo boupr d

ro:TIT':

***" vovov orov opy(oup, vo ovoporropoorl n@\::l'1, .r-1"*

* (,vouvTor ro npoyrorq. opx,<,, uo y,eror o4po.v ro_ro rnr ir ip lgo rot Zlvovo. Av ,o o4p,,o Oeu rt lou p,yreo5, iouuo pflopouoopr vo orprpo{outr. on po of,epO ot,r, iru or ooootno,€/r ip6 \opo 0, ̂ o, ip, . t ro vo r ,*rror ivo ovrrnrprvo, 0o nprnf l o, 610.9oprflrl( ollylr\ vo pprok.ror oro,(o0dvo tqoprglo ollo Kor or oiof : l ] " :*

* o4p€io. oorooo. *, ou,o o*o1,o 6eu opa,1yls u6 q,.-oourr Konou. yror ov ro o4pr,c orv ryouv piyroog rou ro vo n,pvo-pi ono rvov dnJ rpo opreuo o, l l r , , ru Otu ono,,r , , ,ono,o, , ,Oou, i ,ui-ol . nur4 oiplpru( I buoKorro X.rpropou 14(

lill'f" *""'* "";; ;;; 1;il::Jl': -;:111:Jl,,T "1:T'

..-. n nnY4 r l ( opnxovro( po( doov ogopd qv on pl 6rorpJ ror lro , i .vor.ro.ott qovro(opoorr rvov an.rpo oprepo or lurrrrv vo orprpdxvo-vlor olo ovdp€oo oro O xql ro t. IT4v rloo

.y":',*-) ,* ,* ,," ",* ,,."o';, ;;1ff;i::l::": :T,::r.rr( iopow; Ouglorrff ripo5 orr ro orlgtio btv tlouv ourr pl(oq our,u{ro(. ̂fv Ko)oloppovouv (onoro Xrpo, ,{ , ror orJrr ̂or to one,poq 207

Page 195: Ο Ταξιδευτής των μαθηματικών

nonld ot!€io xorologpdvouv ornv npoyuotlK6rqlo Kdnoro Xdpo oulv€uoEio ov(iF€oo oro 0 xor ro I� EIEi( 6foq.vr0jeouF€" no! rl euo€ioonor.i8iror onii onu€io, Kr our6 ro ysyovdq FdS ovoyKd(€r vo o(oppinrouF€ ro dr€rpo ni46os oqF€(ov. E66, pEBo(oS, ufldpxouv 6LJo €po-qForoi !rop€{ o quorK6( Xdrpo( vo 6rorp€0€i En, dn€rpov Kot Fnop€ivo 610rp.0!r , n ' or. .pov o . wo,o^olrKor lopo(,

nol,\o('Ei^tvE( qddooqor, ouFrepdotpovopavr0v rov nuooyop€(-uJq Frtroioov yro qn€pr6prorq orory€ao i opx6( rlou ovop€tyvJovro! F€Kdl d^io yro vo oxlForioouv ro quorK6 KdoUo. O ntrdrov, o norapo!rou r6€o^ropoi, s(vor yv!ror6! Kord (Upro ldyo yro rnv r6ao rou n€pirii€urrv, olErdBlqrov !'66v. 0(J1600, oKotro etoE Kr ourdr rnv nopd6oon rov nuooyop€iurv Xprlorponor6vroq rov or€ipuJ( 6rorp€16 Xdtpooll\r KooFoioyio rou. Iro apyo rou ftlorog, Fdd€r yro rlv npol^€uon

Tas o!.plorou Kol {icl Kqrd ronrq sxoUoqs ouolos (oi lts dLj n.pi rilodjfdrd l,I!o!a!|s t...lovdporov sl6ds r, (oidpopqov, nov6rxts, lsrdlotr6dvov6i c rop(;rord q ron votror-, Kot 6uool0)r6rdrov...ri

O llldrov Udd€t yro ro <nov6€Xiq", €e6oov o Xrirpo( ouv€vr0v€ro(U€ K(inoro truolipro rp6no p€ ro €i6n i l( r6a€q yro vo 6tFroupyne€io oLo0nr6q K6ogoq. H ouvdlrdol npafi€r vo €(vor Fuorlpr0i6lq, ogoLj onndrov uaooqpi(€r €raotq 6l ourd xooourd ro €i6rt 6Ev €roapxowornori mo quorK6 KdoFo. Or nu0oy6p€ror ouv:v{rrvov ro n€poptosttvo(opr0f6q) !E ro An€pbpDto yto ,ro 6t!(oupyioouv ro 6rKd rouq oJgflo\r. Or opreFoi aForo(ov F€ orourKd oouor(6ro (or rirov 06roip.ro(,wd ro A €pt'ptorc i\ov ro xdo( rou Xdrpou Kor rirov on€ipoq 6rorp€-

r0. KoLqv Koroo€Uoo. hlv noyK6olro 'lruxil ond.. rov qoLorpfto (oroFidp\ro [K6oto rov t6.dv] Kor .nio!( dnrj ourd nou.(vor 6ropad (or ox.ricroL p€ro u^(d o.irforo [..] noo .ivor o Xdpos, (o( €ivoL oLoiqoq Kor 6rv sntrpfnd qv (o.roorpoq! (o' onor^ri rov r6no oLotovri( dtov rov npoyrdrov nou lxouv 6tF!

IPlab, Ihe Diatogus of ptata, |[r. E. Jowert (Niq y6prn, Random House,208

Page 196: Ο Ταξιδευτής των μαθηματικών

16. M€ Kdrorov urEpquorK6 rp6ro, rc An€pbptorc 6roxripc rou(0progodS on6 rnv npdq govd6o (ro ?yo) yro vo brlproupyqeoiv rquilKd npdyuoro- O n^drov, €rrlp€oogdvoq 0116 rouq nu0oy6pErou(,ouvfvoo€ rq 166€S U€ rrlv uro6ox4 1rlq dnop{rl( (ro X6po) yro vo 6nfloupfio€r rov K6oFo. T600 ro An€pdptorc rlljv nuooyop€iov doo KorI u o6otr( rou flIdrovo €ixov opqlKo6( ouoxEnoFodg. Kql yro rouqnu0qy6p8rou! Kor y(o rov nMrovo, ro Koid a6p€u€ ofl( ovrio€I€( op-

XtS rouq: mo nEp@ptoltvo mr mr( 166€S. fto rov nldrovo, or r6i€((rov 1.^€r€q Kor oiol(inpoF:v€( (o( 6€v un6K€rwo o€ 6{o(p€ol, €v6 rluno6oxrl 4rqv or€(poc 6rorpEr4. Evrir o nld{ov Xp€ro(6rov qv opxrlqC 6rorp€r6rrlro( oqv Koouoloyio rou, 6€v rirov n oyoqudvn rou6do Ko( ovoqtporlr€ o€ our{ U6vo arov Tqrarc. Evti oro6axrqK€ !€qnpoeuulo qv dn€rpl 6lo(p€r6qro rqt ro dfi€rpo tdyEooq lou Xp6vou, 6€v Fn6p€o€ vo qldo€r oro onU€(o v' oro6Ex,r€( ro dn€rpo F€Y€eos 4 rnv €rdKroon €n' dr€rpov mo X6po.

fto rov Ap(mortil {rov rlro 6dsKo}o vo orqnpoyFor€ur€i q 6to(-p8ol En' dlErpov napd ro d(€tpo Fry€eo(.

'OrIo( r(or mnv r€pinrorsrlrou dr€(pou F€ydeoui unoor{prE€ 6

I olotp€oq €n'dn€lpov irovavo 6uvrlr(6 xl 6Xl dvo npoyFolK6 €rlir€uyFo. 'Onorq €ntolFo(v€r oPodwr Pd{€p, np6qv Koenytqq FoelFotlK6v oro fuvdrK€io Kotrfy(oPdlronq MtrKov, oro €(oip€ro plp^io rou To dt Etpo Kdt o vouc,1l rt6\6-(prol rou Kdv€r o Aprororalqq F€ro{U 6uvnl(od Kor rpqygollKodqn€(pou €ivor .npoFinpoflKrl, oqod, orrlv ouo(o, €qltp€ p(o 6€ur€-p€uouoo npoYFolKdrnro y(o !o or€ydo€( rq 6r(d rou dr€rpo. tlov KdoFo ro)v r6€drv, or guolrc( opreFoi uno6ll6vouv on6 q gdorl rouq€vov drErpo opr0u6 t6€0iv. To vo onoKoio6F€ ourd rov dn€rpo oplopd6€d! "6uvnrx6" K( 6Xr .rpoyuolK6' 6€ Fo( Folod€r Ko06^ou vo 1SKorovorloouF€ n€proo{ir€po.

11. Rody au.Net, tnfinny and the Mird (N(o Ydpq, Banhm Books, I9s2), o. r. 2O9

Page 197: Ο Ταξιδευτής των μαθηματικών

To qn€ipoq peyd,\o yiveror oneiptoq ptrp6

Auri I 6roloKl(6qro rrrv opxo({,v Etrtrtlvurv vo orlo6Exloriv ro dn€rpo

€ivor Korovolq, oqoU €ivor 6do(otro vo '6ofUE' 6lou( too( quorKoUq

oproFoiq U€ ro lrdItq rou vou poq Koodr( npoonoooiF€ vo qowq'

olotF€ oloavo Kor nEproo6lEpou( opLofo6(, ouroi ouv€Xi(ouv vo npo

Xopofv Kor vo Xdvovtor mo 6rdornuo (npoq ro 6€4rd, Yro F vo)' Kol

doo npoo€xlKd K( ov Kolldqouu€, ouloi ouvEx(louv vo npoxopodv vo

onoFoKpivo\rrot lltpo o(d rov A(o KoL tov lllofrovo Kor ntpo on6 ro

nho(6 oforqUo. ho vo Frlop€oouuE vq qowqorodg€ ro dr€rpo nli-

Ooq rov quorK6v oprOF6!, 0o XplorFoflot4ooup€ rlv Eue€io rrxv op(0_

Udv Fo(( !E l(yt npopohK{ YEoF€Ip(o." Ito (dro U'po( rq( Er(6!qq

lS axouF€ rqv oproFrlrlK4 EUOE(o F€ louq guoLxo!( opreUofq ono to 1

d0Jq ro lo, xoe6q Kor to 0 ndvo o(prp6( qn6 ro 0 undpxEr €vo( KU_

Kloq. tx€6rd<ovro( €u0€(€q on6 10 0 Kor rous quorKof! optgFoli( 6oq

ro oqu€io nou ouFpol(Gr to p6pELo 11610 rou K(xlou, FnopoU}r8 vo

npopd^o0$. 6^ous touq QuorKodq qproFoJq ndvo mov Kd(lo :uv6'

ouu€ Kde€ quorx6 oprep6 F€ ro orlp€io tou KfKtrou 6nou rl €ueEio Kor o

K6K^o( *uvovror. An6 rou( npdrouq npopoll6u€vou( opLoFoUq p}i-

nouf€ 6r or QUor(oi oprogoi n^nord<o0v 6io KoL n€ploodrEpo npoq ro

ndvo tdpo( too KiKlou npo( ro p6p€ro r6io

or opr0Fo( 6U{xq 6€ qrdvouv nold oro BdpELo n6}o, 600 6€8d n ov

npoxlrprioouf€ otqv €uo€io 10rv oploUriv ox€6ld<o\'ro( EU0€i€q npo(

rov (6KIo. rro vo qrdo€r ivoq quor(d( op(0U6( oKprpllq oto 96p€ro

1610, rl €ue€io rou 0o npan€r va €(vor nopd^ilh UE rq! €uo€io rLrv

opreFdJv, Kr oqo6 6^oL or quor(oi opteuo( ppioKovlor Endvo ornv EU-

e€io rdv opr0$6v, out6 6sv Fnop€i vo ouFp€( Enoldvoq, or quorKo(

oplopoi (popdltrovrol ndvo otov KfKlo Kor nllord(ouv o^ofvo Kor rI€_

proodrEpo ro oqu:io oulv Kopuqi, l0rp(q nord vo ro grdvouv npdY-

1r. npopolL4 Y.oUap(o €(vor o dd6os 1dv lqolforKri'v trou doxo^aldL F'

*€iv4 r! rodrlt! rd! Y€ol]{prdN olrlFdrdv nou 6N oll6<ouv' 6rdv ra onfq

210 ro rpopnlio*or or KdnoLo ordqopa(6 XdPo

Page 198: Ο Ταξιδευτής των μαθηματικών

0 r 2 t 4 5 6 7 I 9 t 0

s(oNA33. npop6Mouqs rcuqdnapou( quoLrols op€foUq trdvur o,flov KjKlo.

lfir, [nopofF€ vq o€opioouF€ 6{1 6}ot ol guotxo( opr€Uo( [popd]]o-wor ndvo orov Kido, g€ orordl€ofo vq CXouF€ dr€rpo onp€(o mpr-

frryFtvo dio xo( nro Kovrd Ko06q fipoxopoiF€ rpo( ro ndvo FIpo(rou KdKlou. MnopoiF€ vo roxuploroUv€ 6rL 6XouU€ 6vo F€y€0uvrl(6qoK6 (or (ord<ouF€ ro rldvo ndvo fdpo( rou KdK^ou. Oo 6odg€ 6r(610 Kor n€proo6r€po op(opr)IlKd olF€io nlrp(d(ouv rpo( ro onp€(orou p6p€rou ndtrou. To Kooavo Xopicrol on6 ro y€rrovrKd rou on6 oio-dvo Kor FrKp6r€pq ruiBoro rou KiKlou. Aur6 ro orivoio orlg€(ov €ivormp6uoro U€ ro dn€rpo nl4oos onF€iov rou rlolpvouF€ F€ pdon rooul^oyroF6 rou Z4vurvq €vdwlo ornv Kivqol.

'OporS, our6 rou dxoup€ Kdv€r €(vor ivq €(60! roxu6oKrutroupyr(oUr6inou, Fro FrKpri qndq. A1611 6€ 6qUtoupyrlooF€ ivov rp6[o yto vo(6oJF., ro dr€lpo [^40o( orw€irrv- Aur6 €ivor a8Livorov. Aur6 noudvotr€ €ivor vq npopdlouF€ qv on€(pou F4Kou( €uo€io rov opreFUjvnd!r0 o' tvo n€l€pooFavo rd4o, €ntrpdlovroq mq orlp€io fiou ouypo-li(ouv rouq oproFoUq vo npoo€\.y(ouv ro ivo ro di^o ndvo o€ Uro R€-n€poou€q KdFrUlrl. T6po, 6FoC, FnopoipE rouidxLmov vo 6€(ouF€Tov Kr(io Kor vq (odg€ 61 ro dr€rpo fll{eoc lrov quorKdv opr0F6vo0Fpo^ikrqr 0116 ro odvolo T|v olu€(ov fiou urnipxouv UCoo o,lvovn€fl€poogavo }6po. Ev ouvropio, qvro dqoF€ ro U€yd^o dr€rpo rn(€uo€io( rrv op(eFd;v F€ ro UtKp6 6n€rpo q( 6rqrp€r6rrlmq.

211

Page 199: Ο Ταξιδευτής των μαθηματικών

KoIIn pdvor o€ pdlro

Or opxoior -Ei^tvsq 6€v Fnoporoov vo {€n€pdoouv U€ pdot q loyrnirouq 10 6rdqopo rpop^iuoro nou rou( 6qutoupyoUoE rl tvvoto rouon€ipou. 001600, ! fv\roro rirov, Kor €(vor, ouorooruni yro 10 foolForK{i KoL q oEop(o tov oprogrirv. O Aprorortirl( 40€o€ l( pdoa( yroro XrLprop6 Tou on€(pou: 6Ev unqpxov npoyFotlKo unopKrd dnflpq,on^d pdvo 6uvtrLKd dn€rpo. Or lEroyEvaor€por Uo0tFo{rKoi Kor e 6-ooqor npoordorloov vq X€rprorofv ro dr€lpo tr€ ouli qv rlopd{€vl.6rJutr(K4' ovriinqrrl. H Fovo6rK4 sEoip€on 4rov o 0€6( rn( oprloKEiosrov AurK6v. O OE6q npdn€r vo €ivot dnsrpoq - our6 6guJq 6€v oror€loijor np6Fnrlpo, ogoU o 0vN6q dvoporroq 6Ev F(op.i vo n€prFav€rvo Koro\roio€r rov OE6-

To npopl(Foro rou ox€r((owor !E ro d[€tpo, oro oroio n€pL^oF-po'ov(o, o on,rpo ptvsorl , q on,rpt bro,prro.qro, - Krvrto. ror I ou-vax8ro, poodvr(ov rorr( oroxomr( q( AUol( flxpr ro 6a(oro avoro ordjvo, y(o lro r€pio6o EiKoor 6Jo ordvuJv F€rd rnv €rox{ rou Aprorort

\. Kord rn 6LdpK€ro ouri! rfq r€pr66ou roltro( rolovroUxor Kor a{u-nvor dvrp€C KoL yuvoiK€( npoonderloov vo Korovo4oouv ro d[€tpo, o^-id our6 nopaF.rv€ dvo oivryFo. To ropoKdro onoordoporo 6€il"r'ouv61r ro KoirIlpq Uuoid our6v r0rv 6Uo Xd€l6v €ixov pqlldioEr.

O T6Fo( XoUnq (1588-1679), 6voq on6 rou( npdrous Bp€rovo6(EFn.rprorag ontpp((]Jr rtv iwoLo rou orEipou €n€16(, 6noq Eins, 6€!

lnopolo€ vo €XEr drErp€q oKdqr€rq:

o,r ( ov qovroorouE civot nEnqaoptvo_ Ena&uut, 6€v urdpxsr Kd-noLo iwoLo 'j ovri !|l)ll ouro! nou onoKoloi$ ritr€ipo. (ov6vos dv-epotro! 6€v llnop.i vo dxa oro lou rcu fro a(6vq €vdS drapoo p.yr-oouS olr€ vo qond(€ror Iltv dn.rpt roxiqro, rov dn€rpo Xpdvo, vlvdn.Lpf orvol''t 'i qv d{.rpt roxtJ. '

zr z "r, ]i;Ji"Jll t"'bes' Leviathan: ?aft' t an'1 tt (Nro YdPKf' BobbsMedr compa

Page 200: Ο Ταξιδευτής των μαθηματικών

E6ri xplotponor€iror ro €r(x€iprlFo nou n€ptyp{i{roF€ rlpoqyougavr0t 6r( ov 6€v UflopodF€ vo gowomodFE (dl, 16r€ our6 6€v lfiop€(v0 urdpxa. I€_ our{ rnv n€pift.ool, ro €rllx€(plpo orlpi<€rot orQVo€o ou Frq oKdQl yro ro d[€rpo 0o fipan€t Kot I i6lo vo €ivor d[€rpl.0 Xopr( 66al€ E[(ol( n6oo eo^aq itov or iwor€q rrl( roxuIrlro( (orq( ouvdx€ro( €{o(r(o( rrrrv nopo66€uJv rou Zivovo.

H.Ktvrlart .Not p@ ouE^lii tyKati)€t,q pa, Otott, Kat Kardnq)t p@,)rt L..l Atd owtx4c €yrardrr{,4 yrorl Kovavo odFcr, ooo6itror€ trrp6, 6.! lrTop€( oloKhpotKd xor oFooq vo giy€( ond qv nponyod-

t'.v't^ 06on mu m( vo ndp€r |l(on d t, trpdy[orL, ol]d Kdnoro rtiForou 0o €ivoLqrifo fros odorls mu ovi(er Kor qr( 6Uq 6 q6i (or oI$on nou 'y\ov^..urr rq, o.r Orot nou ro.,^op. .

.O fdiloq p€vd Nr€K{ipT (Kopr6oroq, 1596_1550) irov dvos 0116 rou(

rQltr€pouq qdoodgou( Kor lo0!!qr{Ko|Jq. Efvor oxE6dv rloyKdoFrowdor6( y,o rnv ntprqtt t qpoor) rou "t(rmopor, opo unopxrr. . H0.-^r vo orrooouio| !,o loyr{q lrrorripl rou vo ornpr(rror o ( np{rrr!opxaS, (dr nou aForo<€ rold FE To gaongol(o !ovra,{o qC y€rr-l€rpio( fiou ovimul€ o E0K]Ei6l(. :rov op0oloyrog6 rou, o Nr€Kdprqo{v€ror 6n €ivor oKdgo noyr6EuFavoq oq p€Oo6o qq onoyoyrK{f€nrot l lFq( ruv Ltr) ivov. ) ' d, l oqopo ro onlrpu, {pqorponourrvo ro_^u €voro9ipov,nr l , ipn o yro r tv unoph rou Orou. O Nrrropr nr.m€u€ 6Il oqod irov [€r€poopdvo ov, pnopotio€ vo nopdy€r g6vo rE-ftpooFav€q t6l€(. AqoU }orflriv €iX€ rlv dworo rou qn€ipou, go flp!ftEr

: Io^u Inv {,X, 6oos, Kdnoro on,.po ov | flofr vur(. o OFo(. {r( Aa.rpo

uv, U0 nparJrvo t , I r pdi l , r lv,wo,o rou or-rrpou oro puold {ouNr€Kdpr.

EnoU6v0r( sivor otrol{r0r( orapo(rlro vo oug€pdvoufE, or/ 6^0 600ryo nE' trpotyouf6vor drL o O.dq undpx€r: yrori tolov6r I (woro

, ra Thomd Hobbes, (Set&rions frcm the De Corporo, orc rh osophe' Speaktot Ihensetve!

,Fron D6ana ro ro.*e Gmf. rv. smith Kdr Matone crefe) (Ikdyq Un vsrity of Chicago pre$, 195/), o. 144. 213

Page 201: Ο Ταξιδευτής των μαθηματικών

rt! ouqids undpx€' oro fuol6 lrou €{o(r(q( ouloU, lou 61r (l .yd oi6'o{ €(fqL p'q ouo(q, 6€ 0o arp€re dpur! vo fxur rlv avvoro qs dn.lprt! ouo(o9 Kogdoov e(for avo nen€pootavo ov, €(rot KL ov I'o'r endotono (dnoro ouoiq npoyfqr(d dn€Lp']. -

O Nr€(dpr 6€v (rov o p6vo( rlou npoonoeolo€ vo q€6ro^dv€L rodn€rpo. O Kopi Opivtprx l(douq (1717-1855) e€op€(ror on6 rous r€-proo6r€pouq FoetForu(odS oq tvoq on6 rouq rp€r( onou6or6r€pouq

FoetForKoUq rou ehoov rori. 001600, oK6go Kor o lKdouq 6Ev Fropouo. vo on'Lu0,p0re,rono Fro npoKordltqrq or o,u t l t r vo Kdvr. t / tro dn€'po. I' fvo ypdplo rou rpos Kdnoto ouvd6€iqo, iypoq€:

ooov oqopd orrtv otr66a['i oou, 0o trp6na vd 6'o$propl0r,] oqo6pdy , o , o v . p o l o , r , , o v o r o . o r p t o ' p o l o | . ! o o n , . p o . o o r v o v o ' r o r ltrou tnopEi vo yiv.L trpayfolK6r'tn, .qdoo! dord troTi a€v €n'rpan.roL oro pdelfor(d. To dmpo 6.! €ivo' nopd dvo ox4tq i6you... '

EnoFtvoq, ro Ko^fr€po FUoId E{oKo}ou0o6oov v' onoq€dyouv ro

dn€'po y'o tEydio rFriFo rou btKorou avorou ordrvo. AK6go xor o

Koupr lKavr€^, tvo( qnd rou( Koioooor( ruJV poorlporLKdv rou €r(o-

oroJ o16vo, 6€v pnoporo€ vo nd$€r vo vrdre€L of{Xovo nou lvo dn€r'

po nueos orlfEi0Jv gnoporloov vo ciuyxpotrloouv ro x6po: ":6uq{rvo

ttE ouniv q 6'o'o0tlK4 ivvorc! ov ouyr€vrp{noouF€ 6}0 ro orlu€io,

€{oKo^ouoodl! vo lqv a}ooge ur( onor6}Eopo qv €u0.io ro orlU€io,

ovrodroq, oxr'lFor((ouv avo €i6oq oKo^uroLd( ndvo olqv suo€(ou. '

E{dvr,\non r<or 6pro

nop'd^o nou or-E^itv€!, Kor oL n€proo6r€por FoqForLKoi afl€Lro ondourofq, oniqEuyov ro dn€Lpo, op'oFrvo npopliForo orqrroUoqv q

ls. Rend Des.arter, (Meditations on Fnnlat lh.nselves: Fron, Desertes to La.ke, o.7a.

r6. Hol ln9da c/ o. r59.214

Philo\uphy oro Philu\ophe6 Speak

Page 202: Ο Ταξιδευτής των μαθηματικών

Xp4ol rou oTrEipou. fro rqpd6€ryFo, 4rov ox€lKd €oKotro yro rouq'Ei.lqv€(, Kol yto rou( rohrioBorq nprv qrl6 ouroiq, vo unoioyi(ouv roqrpoodv €vds r€rpoy6vou uqr(ivovroq oro r€rp(iyrrvo ro !{Ko( rn(FraS ri€updS. To op0oy6v(o nopolln^6ypoFFo rlrov €d(olo, K( o(6gc(ol ro rp{yovo o(o^ouooioov or odq ufiotroytorrKor( Kov6v€S. Aur6oufpo(v€r €r€164 ro €v l6yo y€{d!€rprKd oxigoro ixouv 6tro r}€uplqnoo €(vor €uoEf€q ypoFFi(. T( ouFpo(v€r 6UoS F€ rov dKlo; n6( UrIo-poi$€ vo uloloyioouF€ ro €gpo66v €vds KfKiou, oqod o KUKtro( €(vq(novroLi Kopnf^oq;

flo €n|(pdv€l€( rou optoo€roriwq( qn6 '(oFndi€S, ot 'EMlv€S ovdnfl]Eov q pa0o6o qq €qd!'rinotq, rl ono(o dvor{€ ro 6p6Fo yro rov od\Tpevo orl€rpomr6 loylog6. Xorrd€r€ rnv Eudvo j4o, q onoio 6€fxv€r or 6qfvq r€rpdyovo nou €X€r ox€6rom€( Fdoo o, avov Kixlo. Mnopodll€ vo6oUF dr ro €Bpo66v rou KUK^ou €ivot l,rEyo^dr€po qrl6 ro €Fpo66v rour8lpoyr0vou. nop' 6tro qur4 pnopo{|r€ vo XprlorFonouloouF€ ro €Upo_6dv rou r€rpoyrirvou oS lvo €i6o( rlp6q( rpoot\1,tol( rou €UBo6od rouKjxlou. AS uno0aooup€ 6l I 6ldp€rpoq rou Kjdou €ivor ptq govd6o(dvq g€rpo, avo Exoroor6, Kor orirr! (o0€E4(). T6po ro €pdrulgo {oerur

El(oNA 34. H Fd0o6o! q9 €{dwlloil( yrd Illv .lpsoil rou qrpo6o, EvdS KlKIou.Iro (o) ro rerpdyovo onor€^€i uro ovstrdp&l K(uryn rou €spo6oU rou rlKtou.Iro (p) o {rp6yovo ouv ro r€oo€po rpiyovd qnocio(v |llo nolLj rcitupn s((f,roq rcu srrgqooi. 215

Page 203: Ο Ταξιδευτής των μαθηματικών

v'r( €{iq: ndoo Ko}i apooa\ /'ort d}oup€, ov Xptoruonor{oout€ ro €ppo66v rou r€rpoy6vou oq Eppo66v ToLr Kr1(lou. To €Fpo66v rou KjKlou €avor nl, 6tlo6i n lni ro r€Tpdy!,r\ro q( oKrivoq rou KrKiou. :€ ouri qvnEpimurorl, I oKrivo roouror U€ 0,5, oqo6 rl tirdpErpo( rooLjrqt !€ ]� Eno-piv{r(, i}oup€ n€pinou (1,1416) (0,5)'= 0,7854 yto ro €ppo66v rou K6dou. Aqoi I 6rdFErpoq rou KUKtrou €ivor,t 6roydJvroq rou r€rpoyrijvou,

Unopo'iu€ vo XptorFono'iooupe ro nu0oydp€ro 0EdlptFo y'o vo unolGyioouuE ro tt'iKo( o rt! Fro( nl€updS Kor, on6 our6, ro €ppo66\r rou r€-rpoy6vou. To ru0oy6p€ro osdJpluo la€r: o' + o' = t' ri o' = 0,5, nou eivorro lgpobdv rou r€rpoy0ivou. EnoFavurq, ro €!8q66! rou r€rpoy6vouroolror pe 0,5 nou onrx€r oprftrd 0116 rov opr0u6 0,/854.

MnoporjU€ 6foq vo Kdvouu€ Kdr Kolu13po. tlpoof(rE ortv ErK6volao dr raooEpo rwiForo rou KdKlou 6€v n€prloFpdvowoL oro €ppob6v rou r.rpoyrtvou. Mnopolu€ vo ouprEp'IdpouFE ro f€yolir€porlritro rou €ppo6oLj oLJrdv rov rFtudrov rono0Er.lwclq oro €oor€prKdrou! rpiyolo Kor npoo0frowo! oq ouvax€ro ro €gpo66v rov rpLydlv0Jv oro 3upo66v rou r€rpoy{nvou. Aurd To (dvou!€ orrlv ErK6vo l.4pKor unoioy(<oufE ro EFpo66v Kdoe rpry6voLJ o( €q{q:

Eppo66v rpryrvou = ](2r o) ( lo) jo(r , jo)

A v o , , o r o r o l l o n l o o . o o o u p r t r , i l o u / , :

Egpo66v rov r€oodpr'rv rpryrirvrd! = o(2r o) {

E F p o 6 6 v ( r / 1 t ) ( r 1 l N 2 ) = o , 2 o r j

Aur6 rdpo ro npoo0arouHs oro 0,5 Gfpo6dv rou rfrpoydrvorJ) (or !vio EKripnot rooJror Fr 0,/021. O opreFd( our6( ni'tord(€r noli nEpLoodr€po oro 0,7854 on'd,fl ro 0,5, rou irqv I nporq poq s(Iiurlol. Kortd<owo( qv ErK6vo l4p, pirnoups drL LJndpxouv ti)po oKr{t prKpd rUrj

Boro nou 6E oupn:pddpops ortv sKritrloi Foq. MropoLi!€ vo uro^oyi.ooutr€ ourd ro ltFo6d xor, npooearowdq ro, vo ntrtordooups o(6Fo n.-proodrEpo oq\r olto(vi uUi rou €Upo6oi rou KdKloo. 'OUoC ov€edprf-

2!6 ro on6 to n6oeq n€nepootrav€r qopiq €o Kdvouy€ our{v rq 6ro6rxooio,

Page 204: Ο Ταξιδευτής των μαθηματικών

6€ 0q (dpoulE no16 ro or(prp€q €Fpo66v rou KUK^ou, oqoU ndvro 0oundpxouv gr(p6r€po rF4Foro nou 6€v ixouv ouFnEp lq0€i Or'B'inv€(ovdmuEov oor{v q F:eo6o, rcu qv ovogd<oul!. U&o6o qe €€aw)n-ont yro vo ppouv ro egpq6d i rou( 6yKoU( r(oFnUiov oxtud(dv. Mrooumlp4 rpqypdr€uorl u[dpx€r oro trory€r'o rou EuKi€a6r], o].ld €(vorWoor6 6rr q Uaeo6o our4 rnv €k ovomfE€r v.upft€po o EU6o{o!.

O EuKtr€t6qq nopouoioo€ Fro rp6roon nou €Koar€r q FZ0o6o qq€{dvllloqS:

"Eoro 6uo fcyC8t 6! oo, rar AB, r 6v ps(ov rb AB ̂ ryo, 61, dd! dtrdiot AB dqorp3ot KiEo! rl rd ipoo kirlroi (oroldroFavou ps(ov ri rdilFou, Koi ro0ro osl ylyv'tror, r.rqSio€rol 1 ffyssos b ioror iiqooov

A016 ftou if€r r] nopordvo fipdrooq Efvol 6rL o! ro oqd\ro ovdp€,oo otn 6rni Fo( €rriFlorl Kqr oro llpoyl.rotlK6 €Fpo66v F€uov€rot rou,ldxrmov oro Fro6 qq ruilS rou Kdo€ qopd rou oxrluor((ouu€ lvo vaoodvoio rpry6vov, rdr€ r€l((d ro oqdlFo Urop€i vo yiv€( 600 FrKp6 e{-loupe. Auril q (6Co €(vor nop6Foro F€ ro fiopd6oao mu Z{vovo, aould€r 6r ndvro ro{r6€douF€ ro Frod qq qn6oroorlq on6 ro ordxo, orovom(o rorl 6€ qrdvoupE g' avov n€n€pooEvo opr0F6 prlgdro)v otr^dmv oflo[o, omdoo, ploporF. vo nllor{ioouF€ rioo Kovrd of^oup€. H1660 q( npoodyyron( Uroq rF{S 600 r€proo6r€po €n(€uFoif€ ppioKe-tqr olo €nfK€vlpo l( o'jwpovl( o€orpiq( rov op(ov, I onoio UE qoapd qS onor€I€( q pdot rou oTr€rpoor(ko( ^oyroFo6. Enrnldov, Ie€opio rov opiov roGr p6io KlEt6( orov npoo6roprotr6 vaov €t60ivqproltiv ornv €uo€io rurv oprouuiv.

ho vo Korovorioouu€ ro 6pro Xp€ro<6lom€ (iirorouq qn^otq opF

r3. A! qft onoo6'1noc Eweoq dqop€9ei €vo +iuq dxr t(pd€po on6 ro f!d6 rou, ( qv qft rc utrdlortro oqopaooupE n6h 6X, rry6Epo qn,j ro lrod roq (ov du{ n 0'q6roo(o qq ogo(rot! ou!:X'ori, ciL(d eq qnou.iv€r dvo lay€eoqF,rp6roo or6 ono'o6'1ro'. ,poro9op.oFlvo,rfy.oo( o ou ,ou L'6ous.

ltu4a. hFnpnt. &p^o||| {Nio IopKrI Do\e.Prbic o-,. c56). Tp4-o 4 217

Page 205: Ο Ταξιδευτής των μαθηματικών

oFodq. np{trot ei}ouus vo op(oouU€ Uro dro,ioud,tr opr6p6v oq dvorbro(1€po Ei6o( ouvdlou opr0p6v.

OP|I|VOI: 'Evo oJvo^o opropojv: A,, Aj, Ar,... A"... oxlforGr F(oo(olouoio oproldv ov or oprogo( €ivor r(oid 6rorErqyEvor,6tlo6i ov or 6€iKrE( axouv rn 6rdroh lt0v quorK{iv qproF{irv.

fro nopci6Eryfo, Unopri vo €Xoup€ rlv oKolouoio 2, 3, 6, 8, I oroioEX.L evov r€n€pooFavo op'0f6 dp{rv. .OFoq, or rpoytro{rKd €v6ro9€-po0o€q oKoloue(!( axouv dn€rpo opre!6 6purv.

oP|IMOL Mro oKo^ouoio opre!6v €ivor dnErpl, ov Krio€ dpoqrq( oKoiou0aqs dXEr tvov Erdu€vo ripo.

H r(o €Jxoln dnErpl oKo^ouo(o yro vo q qo!,romodfs €ivor I oKoioo,0(o rorv quoLK6v opr0f6v i ro t, 2, l, 4, 5, 4... Oftjj( Kor rportyouF6vur(,uno6nldvouu€ 6l our( rl oKoiouoio €ivor dr€rpl ]rE rrc rp€rS r€.lEi€( mo6!{rd qq.'EXoutlE avov on}6 Kov6vo yro vq onuori(ouFE rov {dO€ 61060Xr(6 dpo o lv oKoioueio rov quoL(dv opreF{rjv 6nio6i on}uj( npooeirou!€ rov opL0Fd j. Av ovopoqooiu€ r6so F€ydior Fnoporjv vo yivouvor 6pot {tpougs 6r( 6sv urdp}€( r(ovivq 6pro mo nroovd fq,€O6q rouq.'Ofo!, o' dpor oE KdroEq dn€rp€S oKotroue(E( nord 6€ U€yold\rouvnrpo ono lro ouy(€KprFav! lFi. tro nqpd8€rypo, o Kdo€ 6po( rtqo(olouoio(: i i, i, ii,... nori 6e yivr,u, F€yo^u1€po( ono ro I� Aur6|llropodl8 vo ro 6odF. qgdo{r( ov oK€qroUF rov rJno yro ro vroor66po, o oroio( eivor o (2 - t )/2. O rlroq 6EiXV€r 6rL o oproutqs eo €avor rdvro lrKp6r€po( ond rov ropovouoori_ EnoFalorq, I oKo^ouoiorota 6€ eo nop{iyEL a\rov 6po oKprpdJq ioo pe ro I ri Kdrorov 6po FEyq-iLjr€po on6 ro I� En€(64 oL 6por ornv oKotroueio FoS 6E eo y(vouv notdcin€rpor o€ !€yEooq, iiF€ 6! n oKo^ouoio sivor qpoytdq 6tto6i, dl0r opor eLVor rEproprop€voL doov orpopci oro ndoo f.yd^or Uftoporjv voyivouv- Iqv npoyuoflxdrrlro, yro q ouyKEKp(ir{q oKo^ouoio gnopoLj_UE vo noUU€ 6Il o oproud( t, Kor 6tror or oproFo( rou sivor g€yolUrEpor

__^ ond ro , orou^ou\ upoypo o r . , o.o^ouoro\. tovrvo\ opo! b, 0oI ! ! - ' r . , n o l p , r o ^ u r t p o i o n o o u . o r o g p o t F o r o .

Page 206: Ο Ταξιδευτής των μαθηματικών

Eipqor€ arorFor r6po vo e€oouF€ 6vo on6 ro 0€F€tr1661 €porigoro&Jv lroeqFoflK(iv. Av oK€grodF€ 6io ro gpdyforo rov 6pov qq rlopfidvo oKoiouo{oi uldpx€r dpoy€ xdlolo d\dxloro (llKp6r€po)9priygq Yndpx€r dvo ppdyFo f(p6r€po on, 6to rq dMo, U€yotrjr€pooFo( ord ofiorov6ilor€ ouyK€KptF€vo dpo qq omiouoaos; Ilo rnvoKolouoiq go( +, ,1, +, #,-.. n ondwlorl €ivql vor, yror( ro I €{vot ro!(por€po qpdyUo. Ovopd<ouF€ ro I dpro qq oKotroue(o(. Av €nrl{-{ouF€ onorov64(or€ opr0p6 FrKpdr€po ond ro I (oitrd Uqotrljr€po ordro +), Fnopofp€ rdre vq ppoiF€ avov 6po ul( oKoiouoioq p€yo^dr€poqn6 our6 rov opreF6, o^_ld FrKp6r€po qn6 ro t. Aur{ I 6wo(o rou op!ou onord\€( ro e€ldtr(o iioo rov ov.(Jr€pov lqonuolK0jv. nporo6 olqoqrae rov g€lK(irv €norrlp6v Kor rrrv [o0nuqll(6v prlopaoouv voolou6doouv ovrir€po FoongorKd, Trpfn€r vo 6r6dFo(v on€rpomrx6ioyrou6. Kr our6 nou Fqoo(vouv np{irro ap6ro orov on€tpoolK6 loyroFo €(vor ro ndwo yro ro 6pro.

f(o vo ovr nqeodp€ n^{po( qv lwolo rou op{ou, rp€n€t vo Korq-ldpouF€ Il €vvoorjU€ 6rov l6F€ 6Il ro 6pto €(vor ro prKp6r€po qpdyuo.To vo rorjp€ 6fl lvqs oprofds, o L €ivql6pro yto gro oKolouo(o oluo(_v€r 0l rl oKoiouo(o 0o rdlold<€r 600 fi€proo6rEpo yiv€rot tov L Xo]p(qrcrd vo tov erdv€r. npdypotl, ov 6told{oup€ oueoipEro ivov [olU FtKp6 oprofd, rov €, 16r€ fropofu€ vo Kolrd{ouF€ flpoo€rnrK(i rnv oKo_iouelo goq g{Xpr vo ppo !€ ivov 6po fiou ppioK€ror KovrJr€po oro L,qllfxowos ord our6v (or{i Uto tlF4 fiou €ivor UtKp6rEprl on6 rov €. E!pom€ drotpor rdpq vo 66ooug€ rov €niolFo optoU6 rou opiou.

OPIIMOI: H orrctrouoio 6pov: A,, A, A,... Av.._ dx€t 6vo 6po L ovyro Kdo€ oElKi lui E undpx€r ivo( oAoUdq N rlroo( 6a(€,yrq 0 o ro v > N, t qr6^uq r$i L_ d < €. Alto6ri lL _dl < €.

O roporldvro oplog6( qoiv€ror Kdros 6doKqFmoq Kqr lmi€yFdvoiov 6€v €ivor Kov€(( ouvnoroFivoq orouq UoelFollKod( oprouod(.Omdoo, €(vor €riKolo vo (orolripouu€ q po06r€pn avvoto nou dx€r, Krouq I €worq r0rv opirrv pp(oK€ro( mo €r(rcwpo rq( odyxpovlq FoolluoflK4( qvdluol(. f(o vo rfiqvo6roru[droouFe pe onioLiorepo ]riyro 2'19

Page 207: Ο Ταξιδευτής των μαθηματικών

rov nopondvo op(ou6, itu. dll or lluf( rov op{OFdv A,, Ar, Ar, Kor oU_ro Koo€Fig rltord(ouv 6^0 Kor r€proo6r€po orrtv lt{ L. f(o nopd-6€Lygo, o\r €ndt{out€ Uro noi6 frKp4 TrUi, o( notuE rnv €, 16r€ propoLjp€ vo ppofu€ tvov 6po rtq oKoioueios, o( ro6F€ ro vroord, Eror{iror€ Kdo€ dpos U€yo}Ljr€po! on6 ro vroord vo ppior€rqr fto Kovr6orov L on' 6/rL ro L !. Aurd qoiv€ror nopoorol(d oqv ErK6vo t5.E66 axoup! tv oKoioue(o po( +, +, +, +,... dnou o KdO€ dpo( q!n^lord(€r d^o Ko( n.ploo6r€po oro t. Av €n(liEooFs avo UrKp6 €, o(noip€ ro t/10.000, rdr€ €ivor nroovdv vo ppo6p€ avov 6po rt( oKolouoioq rarorov (iors Kdos €r6g€voq 6po( vo sivor nro xovrd oro I on,6,r ro (t - t/10.000). noroq Oo €(vor ourdS o 6po!; noipvowoq 611160-XK€l r!€q rou 2'B^tnouUE dr ro 2,, = 16.184. tnoFiv(o!, o 6tKoro!raropro( 6po( rnS oKoiouoiqq. ivqt o 16.381/16.184. Aur6 ro Ktrdouoppiox€ror nolLi Kovrd oro t, orlv rpoyuotlKdrtro 6ttro6{ Eivor Ko\rrUr€po on6 10 1 t/to.ooo (ro L €), rou €ivor ro K^oogo e.999l10.000.Eniolq, xor Kdo€ €r6F€voq dpos r(po 0116 ro 6dKoro rirqpro dpo O.rrivor nro Kovrd oro j ofld qv rLU4 rou f((po6 oqdilor6q Foq, rou €.Blanoup€ rdlpo 6r, ovEldp lTo on6 ro n6oo u(poq €rvol o E rrou €rll-^€youp€, gnoporu€ vq 6rol6{ouU€ fvqv dpo oqv oKolouoiq rou 0oppio(€roL mo Kovrd orrd rov E oro 6pL6 FoS, ro t. Aur6 rKovonoL€i rlvEnroufio UoS vo qrdooug€ 600 nro Kovrd €itrouFE oro L M. rlv no-pondvo oKoiou€io nltordqouur dto KoL nro Kovrd orov oprofd l

2 ' ' 1 2"-11

c = l /10.000ETKONA 35. Bpiokovro! ro olfE(o rou opiou. A(6Ho (L ov .nrldqoup. rov € vo livor

. ' r n d o . . p " o o ' ' o o o u p p r o . o " , o , o , o o . , , - . o . o o , e , o . - , . p . . o , p oz.v rov a ltp'oKovo' nro rovro

Page 208: Ο Ταξιδευτής των μαθηματικών

(drl(lrq mo rop{i6o{o rou Z{v0rvo), Xopi( rlot vo rov qtdvouu€.

npdyw(, rovdvo( 6por ouq( q( oKoioueioq 6E 0o qtdo€t i 6€ 0o

{m€pdo€r [ort ro L EnoFavuJg I oKoiouoio laltt 61\ ouy fu€toro1.

OPIIMOI: Mro oKolouoio nou ax€r dvo 6p(o €ivol ouyKiivouoo,6rq9op€uKd €(vor oroKlivouoo.

O l6yoq yro rov onoio XptorFonoroiU€ rnv on6iuq llFri tou (L - €)

oro! oprolio poc yro ro dpro , rvor yro vo Fnoplooupt vo npoot\1,ioou

F ro 6pd FoS Ko( ond Il( 6do Kor€uo0vo€rq. tro nopd6€rygo, n qxo

louoio i i, i, J;,... (^nord<€r ro I 0116 Kdlto (6n1o64, or npa(

6tr0rv rurv 6pov €ivor Fr(p6r€p€q on6 ro t). H oxoiou0io ], i ?, .r,...qord(€r ro 1 ord ndvo. E(vor nreov6v vo €Xoug€ Urq qrolouoio rou

dl€r 6plo, olid 6Ev rKqvoror€i rov oproF6 Fo(. fro nopd6€rypo, q oKorou0(o i i i, +, +, +, +.... ax€r 6io 6pro (F€prKaq eopa( layovror.otf€io ouoo6p€uonst, oqo( or 6por q( oKolouoio( orrYd(vouv€vqlld{ oro O Kor mo I� 001600, orov 6lK6 Uqq ontr6 opto$6 tou op(-

oq 0o qroKi€ioouF€ ourd( rS o(olooe(€(.:uppo^i(ouFe ro dpro L Urc( o(olouofoq U€ rov flopo(dIur rp6ro:

LtJ0 (A") = L

E6dr ro lim ovtororx€i oro limit (dpro) Kor Kdro on6 ro oU$-polo rou opiou €Xouge to v+-, rlou orlFqiv€r litr o oproFdq t(l)v

6pov, o v, ouqdv€r Xop(q 6pro. To A €(vor 6vos o6vrolro( rp6no( ypo-

El( rnq q(o^ouoio(. EnoF€voi ro 6pto 1r)( o(o^ouoiqt Koodi( o opto-

F6( lov 6pov nnyoiv€r oro drErpo, €ivor o L. Av n o(oiouoiq orordi-

v€r, our6 {o 6€[XVouU€ o( €ai!

l"rJl (A') = -

ro oloio ld€r 6r or 6por ouv€Xti,( ouEdvovrol Xurp(q 6pro.-Evo oK6Uo rqpd6€ryuq Froq rmoprqq o(olouotoS 0o $oq pot04-

o€r vo fi€p(ypdq]oup€ dvo Xoponlpror(d yvttpropo rov Foorl!filx(ir4ro onoio yorlt€r€r rooq gooquolKoU( Kor rouq dMouq €n(or{gov€s €66

Kor or6v€(. ouiqe€(r€ 6n ovoqdpoUe rpottyougavoq tov leovdpwo 221

Page 209: Ο Ταξιδευτής των μαθηματικών

rnS nqo, o oroio! ro 1202, pE ro Bp)b n€pi rou dBaKa, po4}lae eo€rooxeodv ro rv6oopopr(d oproFnllKd Vtq(o oqv Eup6nl. OvoFq(6-rov^€rior)( OlFnovdror, Kqt ovqxdluqj€ f(o €{orp€r((d €v6rogapouoqopr0Flri(4 {(oiouoio, nou n{p€ ro dvoud rou. np6K€trar od.0c yroqv oKo,\ouoio j , j , 2, 3, S, 8,13, 21,. . . (deE 6poq qq oKoiouo(oq Oc1,"":".: . vo ro dopo,ouo rov bdo nporlyo,.,!,v(0v opov. Au,t 4orolo,roio onoKhv!r prpordc, Kr ourd ro brrf,\roupr ,{rouvovros ro6pro qq oKolouoioq F€ ro {ir€rpo:

t i m ( 1 , 1 , 2 , l , s , 8 , 1 1 , 2 1 , . . . ) = *

^ Mnopoupt 6pux. Xplorponorovro( 6robo\rKou( opou( L4( oKoiou-

uro. orpnovdror, vo oxluo oouFr l ro o^in oKonouo,o y,o vo onpr-oupy4ooup€ Ki(ioForo, ulv o(oioueio +, +, i, +, +,... noporrlp4or6r Kdo€ K^dofo 6ngroupy€iror on6 6Uo 6ro6ox((o{q 6poug trlf oro,\ouo(oq OrFrovdrol f€ 10 Fsyotrfr€po 6po oq opropnri. Auq I oKo

ltj! (+. i ; i...) = (y't. rr2Tdpo,.ro 6pto nou undpxEr oro 6€qtd rou ioov Unop€i vo Fl oq-

FolvEr roMd ylo oqq, rKr6! xt ov €im€ avoq opxoioc ,tiilvcrq Uoo4uol(o( I oplrr€novo(. Auq t 1F4, ro (15+ t)/2 {rov yv{rrori orouq op-

Xoio^uq.'EMtv€( r,i( Xpuo6e Mtoo( 4 Xpuo6S 6yoC, xor 4e ro 6rx6 t4qofppotrq ro Q. Or -E-Iilv€(

e€{rrpoJoov ou14 rt\r lui €vo F€oo i avo }6yo ovdg€oo oE 6ijo UjlK[ yrori 6€v €txov xidogoro. troFavor(, o(lqrGvtov ou rl 'rlv l|Un on4(,( o! ro ioyo krv F4xuv ( r s . t) npo\ 2. H ELKovo 36 6!iXVEL avo opeoy6vro rp(yurvq or KdeLLlg r €up€q rcu onoiou€Xouv tr4q I Kor 2. To y€yovdq our6, pE pdon ro nuooy6p€ro o€dpnfo,(dv€t rrlv uror€ivouoo fo4 ge /s. tnopivrr4, ov 6rqtpEoouF€ ro UrlKo(rt_( uror€ivooooq ouv ro iriKoq rrlq Frprls Kdosqs ni€upd( (t) g€ rot'4Ko( rq( F€yd\S Kde€qq ni€upd( (2), noipvouu€ r" tr,q_ tjlz ,t _ouXpuo6.Mioo. Or-E&v€( l r'drpr<ov 6Il ro opooyrivro ropoLln^6ypoFpo,rl fEydit n}€upd rcu oro(ou irov iorl gr {i+ I Kor I fr(pi niEupdItov rorl g€ 2, {rov €uxdproro vo ro pidnEr Kov€(S, Kot €vooFtooov ou

222

Page 210: Ο Ταξιδευτής των μαθηματικών

!KoNA 16. O Xpuod( Mfoo!( Xpud6( l6yos €(vo' o rdYosrou r + l! npo< to z, tor

I6v ro ̂ 6yo o€ LrEdlo F€poS ul( opxfi€novrK4( rous, 6rmq (o( olov r€_

p(QlFo nopo€v6vo. M€roY€vfor€por mil|Idtrr€s, ouurl€prlo$povoFvou

ro0 Aeovdpwo vto Biwot, XplolgoTloinoov rov Xpuo6 Mdoo dto ipYo

rous. or nueoy6p€ror Xpqoruonotqoov tu n€vrdYpoFUo" o( odppoli

rouS, xor ro n€mdypoFfo oxlFolicl fio]trd nopo6eiYforo rou Xpuood

Mdoou (ptrdn€ E(6vo 37).M€ our6 ro nopd6eryFo UrlopoiF€ vo ovrdqqeotu€ rt €(vor our6

nou yoqr€6€r ouxvd lou( FooqUorrKoiq - n oltrnlooUv6€oll rov !ooq-

liollidrv. H KqronlrlKt(xri oxolouo(q orFnovdror €X€l o( 6pt6 rqs ro

Xpuo6 M€oo, Kor o Xpuodq l,ldooq €Uqov(Grol oE Kde€ E[6ouq onp6'

oF€vsf y€ou€rprKf( Kqroo(€u6(.H r6€q €v6q opfou €ivor npdYuol Fto €Ktr€nruofdvn Kor 6uoppl €v-

volq, yrori fnopoiFE vq n{ipouU€ dvo dn€Po TI}4eo( 6pov Kol vq rouq

ouq€r(oouu€ oro 6pto UE Kdtr n€Tl€pooFCvo. XprlorBonotovro( ouIrl

rlv r€Xvr(4, FIopoUP€ vo €p€uv4oouF€ o(6po pooll€po ro $uoriproIo0 ouor4uolo( I(l)v opl0l6v FoS, Ylo vo ofioKqioqioup€ €vrunooto

Kod( Kol ropdl€vou( vaous oproFoaqOq oovotro opreUdrv F€p(Ktq qop€S rl oKoloue{o €ivot dpotro vo €!-

oorForoo€t o' dvo goolFolK6 16T10, oqof rl o(olouofo, oC oul^oY4,

6€v €qrodrv€ror U€ Fto B6vo lF4, P€ rl ouvr\0lou6vn opleur]tlq avvoro

]e. Pentaqam: ovoud(€tor strioN n.vrdigq mr .(vor 6vo n.wdKr(vo dd'pr

trou npoKLjma dnd ro ou!6uoof6 nfw€ €golotov A, ro ono(o o' nu0qY6paoL 223oxe6{o<ov K Fd fovoKowul'd. (Lr'M )

Page 211: Ο Ταξιδευτής των μαθηματικών

511!*,1 t"^l*.*f"u*, 'ou ro xpnoLuono@loqv o, nuooydpaoL o! o,jupo^d

]iirr;l'fep;" -M"u< xpuoorrr M.oouq. q = AB/Bi = .oter = )1,) = zttiu _

'Evo( €dKoioc rp6ro( yro vo €firrdxouu€ our4 qv €vorllgdTu]oq rivotvo Xplorpororlooug€ fro dlil laorlpor{nj tvvo(o orn 0!on qq o(o_Aouoio( rQ o€rpd.

Irlv oxolouoio goq 11 i *, r!,... nl4ord(oupe Olo Kqr rEproo6r€poro I rpoooarowo( ivo Fr(p6 oprofd oE (de€ 6po, yro \ro 6nFro0py{-oo0F€ tov ErOF€vo 6po. Mroporitr€ vo {ovoypciqrouF€ our4 rlv oro-ioueio o( €{ i ( : + , : . + , : . + . ; j r + . * . , r , . . . rnoutvoq, npooeftouF€ to ] trlq tryrjq rou Kdo€ npolyodF€vou 6pou yrovo ndpoug€ rov €n6F€vo 6po. KdO€ dpoe ouqs rnq oKoiouo(o{ €(vorFro o€rpo rpoooio€rrv, { on}{i( gro o€rpd.

OPIIMOI: I8rpd €(vot ro ooiloyrKd 60poroFo €v6q ouv6^ou opro,Folv.

E(€i nou I orolouero €rvor uro ou,t)224 il ; ;;;,;ffi;;":,l:::*"Jll*fiilr%:TJ:i,i

Page 212: Ο Ταξιδευτής των μαθηματικών

0(olo00(o, n o€rpd pnop€[ vo ax€r fl€n€poopdvo 4 dnErpo opr0g6dp0v.

oPlIl\,lOI: H o€tpd A, + A, + Ar +...,\... €{vol dn€Prl ov undp-

Xouv dn€lpor 6por,

H nopqKdTo o€rpd €fvor l€n€pooFtn: 3 + 9 + 27, d€v €(vor orlo-pq(rFo vo mofor4oouF€ orou( rp€rs !6vov 6pous, oMd gnoporrp€vo Eoog€ dn€rpo opr0g6 6p{dvl

3 + 9 + 2 / + 8 1 + . . .

H IlFi rrlq nopordvur o€rpd( ou€dv€l ndpq 0116 611010 dpro r( ovnpoollooioouFE vo €nlpdlouu€. H o€rpd ooq6( orlo(liv€I.

oPlIMol Av rc dopouFo Flo( dn€plq o€pdq €i\ol n€nEpooFavo,16r€ rl o€rpd €ivor ouyKlivouoo, 6loqop€r(d €ivqr onoriivouoo.

Aq xord{ouF€ r6po q o6rpd nou rpor€(v€r ro nopd6oqo rou ztiv0)vo, dnou €(vql olopoiqro vq p€roKvofFoor€ xorri ro + q( on6oto

olS, (ol U€rd rcrd ro i rls o[dorooqf nou onopdv€I, rcr o6ruj {o0Eq4(. :€nv6vro( on6 ro I 4 oerpd 0o e(vor q eflC i + i + * + * +...

0r Foongo(Ko( 6lUlorjpvloov avq oiwoUo rpdro ouFpohopo0 I(,)voap6v. Xprloruoloro0v ro KEqo^o(o €MnvrKd ypdUuo ofyUo (:), qKo-

ioueof!€vo 0116 dvov r6no nou 6€(Xv€l fi6( vo grdxvoup€ rrl oErpd. Hnqpondvo o€rpd 0o ypog6rov o(:

:,oD)

6nou rc v no(pvEr 6ro6oxrKd rS (UCS l, 2, 3,,.. To v = I Kdro on6 rooiyfo uno6lldlv€r 6n q€xNdF€ rq o€rpd !€ ro v = l, xor ro oflpoiorou or€(pou ndvo ond ro oiyFa onFo(v€r 6tr ouv.Xi(ouF€ F€ 6louqrou( quorxori( oplouo'iq. To oiyFo uno6nl6v€r dr rpdK€rrol vo rpo-oofoouF€ 6lou( tou( 6pou(, xdl rou 0o Foq o6tyrio€r o' dvo F6voop(qr6. Evior€, rnv nopondvo o€lpd rrl ypdqouu€ Fro odwopq oddr(r,r( !(l/2') i, ysvrK6r€po, orq tA". 225

Page 213: Ο Ταξιδευτής των μαθηματικών

oPlI[.4OI: H o€rpd A] + A2 + Ar + ... A! ... Frop€( vo ypog€i uX

"4, (d)

6rou 4 €i\ror o vloor6( dpo(.

Or oEtpaq 6noc Kor ot oKoiouoi€q, plopodv vo dxouv 6pro. Av qo-vroorotF€ 6Il Fnopofu€ vo npooeioouB€ yp{yopo rov drErpo opreF66p0rv rnc o€rpd( t (t/2"), roro lFi eo mipvo!€; Oo noipvol.r€ ro t.Atio6{, ro I €ivor ro 6pro rl( o€rpdS. .O!oq, our6( o rp6noq oldiqs6Ev {xovono{€i rou( goorlUollKoi(, y(ori 6€v rou( qpaoEr vo trdv€ *npo_oearowos yp4yopo €vov dr€rpo qpreF6 6purv,. Aur6 rqpo€tvor ooo-gdq yro (€ivou(. Kord ouvtr€ro, rlpoo€yy(<ouF€ rov oprou6 rou op(oufro( o€tpdq X{rpiS vo XpnorFonorodF€ rov 6po.dfl€rpo). A:F€ 6r Frqo€'p4 I >A", tx€r avo 6pro 6rov:A! = L.

optrMor: H o€rpd:A! lx€r tvo 6pro, 6rov urdpxEr fvof oprOU6CL tdtoro( olor€ :A, = L

fto vq ow lqeoUl]€ dI| ro L €(vor dvo 6pro, Oo npirEl vo Korqvo{_oouu€ 01 ro L €{vor n oKprp4( IlF{ i o o(pfis apr0!6S nou roorjro{F€ ro doporofo 6iov rov 6pov rls dn€rpn( o€rpd( Uq(.

:uxv{i UnopofF€ vo ypdQouF€ Uto oErpd aror (iror€ o vroor6( 6poqvq 6€ilv€t n6( 6lproupy€(ror I s€lpd. Ilo [opd6EtyFo, ylo rn oElpdi + + + +... FropoUU€ vq ypdqroul.r€ t (t/2'). Aur6( rivor avo( roidBotrrKd( rp6no( ougpohopoU qq o€rpd( Kor 6Eixv€r orov ovoyv(i)oqn6( 6rluloupy€iror I o€(pd. fvop(<ouF€ noitra( ouyKlivouo€q d[€rp€(o€rpdq Kor Kdno|€( oflrj oort! dxouv o{lonp6o€Kro 6prq.

j - +. +r i. +... =t (rhp6roq opr0F6() =-

1 11' + 112' + 1 131 + 1141 +.. . = t (1lv ' ) = n'16

r - + + + - i + + - . . . = n / aH np6q o€(pd €ivql onlui( ro dopotolo rorv ovriorpoqorv 6furv

226 rr,rv nprntov opr0g{tv. Aur4 I o€rpd, 6rrrt( ror n o€rpd ruJV ovrimpo,

Page 214: Ο Ταξιδευτής των μαθηματικών

Qov 6trurv rov quorKdv qpl0u6, ono(liv€r. H 6€0r€pl o€rpd €ivor rodoporogo rr,rv owimpoqurv r0rv r€rpoy6v{w 6ro6oxlK6v opleFdv Korro 6pd tqc ovoKqtrUqrlK€ on6 ro orcu6oio po0nporrK6 A€ovopvr OllEp (lZolJ781). nopqrlp{or€ 6

ro n €ivol dvos on6 rou( 6pou( rouop(ou. nd( prlop€{ U(o 1600 on}rl o€tpd vo ox€ricror F€ ro tr6yo rls6oF6rpou €v6q (dKlou npo( qv fi€plqdp€ld lou; H rpirrl o€lpd ovopd-kro\ €va)^dooouoo oztpd, 0906 €vo^Idooowor rq np6onFo llrv6p0N rnq. Kor our4 dx€l tvo dplo nou n€prlouBdv€t ro r. Iqv npoy,

fqrrK6rrFo, ro 7r €FqqvGrol o€ 610 ro onpdoF€vo Fapl. AUr6, yro dI

\ Fro qopd, 6€W€r.rnv unapoxq ouvox4 rov goqForK6v.M€ pdol tl( dvvol€q rov o€tpdiv Kol T0Jv oKolouetriv, (oora( Kot rov

opirdv rou(, UrIopoUF€ vo ouh{ooug€ ^oyrxd rs 1616 lr€S rqs ouv.-

XoUq p€ropolijq. :1o ovujrrpo UqoluollKd, ro 6pro Fq( €(rrpanouv vo

FEl€r{ooug€ rl ouv€X{ 6roqopd, K 6Xr on}.6! tl( €}dx(or€S 6roqop:{nou uno6l^rjvouv or guo(Ko{ opteFoi EnoF€vur(, Unopodg€ vo onoKoldqroug€ ro Fo€lFolKd puor((d rov Kouru}0lv, r0rv €nrroxdvo€rdv Korrurv Kv4o€ov. To odllpovo FoolFoTlKd Kot ot guox{q €rnm4U€( 6E80 $nopoioov [or! vo ovomuxoofv ov 6€v €{XoU€ ovoKotrrqJ€t rq6po. H Korov6qol rov o€rp(iv Ko( rov oKolouer64 Koouls €n(ol( turruv opforv rouq, Uo( rpo€rolud<€r yto lv €r6F€vl onou6o(o 6r€6puvon r00 oum4Foro( opr0p6v - rou( dpprFou( oprego(q.

227

Page 215: Ο Ταξιδευτής των μαθηματικών

t;1oupe e{etdoer p6Xpr t6po to 6rdrpopo e6q opr0trrti,v nou onor6treoov

e[dtrr(q rtrrv rpuotrtiv oproptilv: ro rtrdopoto, rouq opvltlKofq optogo0q

ror ro pr16€v. 'Otror outo( or oprOgo( po(( ovopd(ov:tar pryto( ap0po(.

Auroti tou e(6oug or opr0po( e(vot otre(ot oe 6trouq p€oo on6 rtq ouvq-

0ropCveg, roOqpeptv€q 6pootrlprdtrlteq, ond tov unotroytogd rou uno-

rov prp)\rop(rrrv xoto0€oeov ulq ro vo no((et rove(q pd\or r(ox.

e(poote Crotpot vo pd0outrre ere(vo ro e(6q optOptitv TIou ol TI€-

dv0ptrrnot e(te to oyvoodv, e(te od\tiq to ouvdvtqoov ypd-

rprv mo pdOrlpo qg dtryeBpoq oto yupvdoro. Qot6oo, uuto( ot v6-

opr0po( e(vqt on(oteurot Kot pnoporiv vo €vrurlooldoouv rot vo

rdoo rouq poOqporrroriq 6oo rot rouq Frl er6txo0g'

Iqpr(6pevor mq olyepprrCq e(rotiroers pnopo6pe vo 6o0pe ytor( on6

mo;g'i rurv oplo(rrrv Drlrtvtrrv, 6ev xotdqepvov or poOrlpotrxo( vo ppouv

trq l0oerq trrrv npoptrrlpdttrrv nou €vtutOov 6n 0o enpme vo undp;puv.

opqtrxo( oprOpo( mfluoov ontr€q e[rotiroetq 6ntlq: x + 5 = 0. Me tqv

rtov opvqtmtiv oprOFdrv, q ondvqorl e(vor ontrdq: -5. Tr ouppoG

6F,lS Fe ro: x' - 2 = 0; Me dllo tr6yrq noroq opr0p69 6tov noMontro-

pe rov eoutd toq tootitot pe 2; An6 qv enop! tov llu0oyope(trtv,

yvtirpr(ov dn q terpoyovrni p(o tou 2 6ev pnopotioe vo 229

Page 216: Ο Ταξιδευτής των μαθηματικών

K E O A A A I O 9

oupBohme( pe rdnoro ridopo ror, rord ouv€nerq 6ev rjrov prltdq oprg_p6q' Enopdvrrrg, ro o0votro drov r'v pqttirv oprOpdrv 6ev enoprorio€ yro voeppqvefoer dleq trq (lto0pevrq }foerq rov otryeBprrrirv efrotiroeurv. ymjp_xov dpoye dtrIor, dppqror opr0po( nou pnoporloov vo itioouv quto6 rou e(_6ouq lq e{ro6oerq; Aq orerproUpe ndtrr rqv euOe(o rulv oprOptilv. H Errdvo38 6e(Xvet ro rgrjpo r4q ovdpeoo mo 0 ror ro 2. ,EXouU€

ndper qv unoreivouoo ev6q opo0ytrrv(ou tprytirvou, 4 ndoe rdoerq nleupd rou onofou rierprlroq 1, rr 6xoupe ronoOer(oer ourr] rqv unore(vouoo ndvur ornv eu0e(orrrrv opr0pdrv. To oprorep6 drpo rou euOOypogpou rpr]goroq Bp(oreror nd_vul oro 0. Ie noro oqpe(o qq euOe(og owrmorXe( ro dMo drpo; And ro nu_Ooydpero Oedrpqpo (6poupr drr to yeogerprrd prjroq ourorj tou eu0rjypolr_pou tprjgoroq e(vor tE Av orprjooupe vo ndoer ndvo onrv eu0elo rurv opro-1.uirv ivo euO0ypoptrro rptipo, rl opXrj tou ono(ou pp(orerot mo rilog rougrjrouq y'2, ro orlpe(o mo ono(o 0o rprjoer qv euOe(o tttrv opr0pdrv 6evowrmorXef oe p4r6 opr0p6. Aurd ro orlpe(o 0etroupe vo ouoXer(ooup€ U€ rotlT, Onwg orprptirq ouoXer(oope dvo dllo oqpe(o 1.re tov opt0pd 2. Oo ovo_pdooupe uur6v ro v€o opr0p6 dppqro ap0p6.Tr e6ouq oprOpdq e(vor qu_rdg nou ouoXa[(etor pe ourd ro orlpe(o ndvur oqv euOe(o rov opr0pdrv;

An6 tqv opXordqrq to y'7roul(6rov pr, €vov on6 ourorlq rouq v€_ouq dpprlroug opt0porlg. yndpyouv dtrtro nopo6eryporo r€rourv dpprlrurvoptOpdrv; Arporj dXoupe 46q ono6e(er 6u to fe(vor tippqroq, pnoporipeeuKolq vq oTro0eqou1ie 6u 0r up€q {t \ft G no06g nor or rerpoyovr_riq p((rq 6trov r,v rpuorrtiv oprOgtirv ilou or 6ro1 6ev e(vor rerpdytrrvooprOpdrq e(vor en(or1q dpprlror opr0po(. Enop€vog, ot l[i, ,[i, VIZ e(vorpqroi oprOgoi otrld dleg or p(eq rurv opr0ptirv nou 6ev e(vor retpdyovo

,[,

EIKONA 38. To o4pe(o rqq euOe(oq rurv opr0pu.0v nou ov{xer oro rCtroq rou pulr,l[ 6ev e(vor €vo o4;..re(o nou ouoXet((etor pe rdnoro pqrj opr0p6 tfrq onp€b;234 lo6r], nou vo e(vor tr6yoq 6rio orCporov oprOptiv)

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opr0pdrv e(vor dppqror optOpo(. Aur6 orlgo(vet 6tt undpXet dnetpo ntr{-

0og dppqrov optOpdrv. Mnopotlpe vo oXqpor(ooupe notrtro0q dtrtrouq dp-

pqroug optOgo0g. fto nopd6€tYUo, o oprOpdg rou oflo(ou 11 rtpq too0rot

Fe (2 + {t np€ner en(o4g vo e(vot oppnroq' Mnopo0pe vo to ono6e(-

[ouge out6 unoO€tovrog 6rt o (2 + fz) e(vot pqr6g opr0g6g rot 6efiyo-

wog 6rr ouui 11 un60eor1 o6qye( oe ovt(tpoorl' Av o (2 + f4 e(vot prp6q'

dre undpXouv 60o rpuotro( opt0po(, o rot B, t€totot tita:.: 2 * '[i = ol}'

Trirpo, orpotpofpe ontrdrg ro 2 Kot on6 ttg 60o ntrtupfq rqg t[(otloqg rot

no(pvoupe {i -- ol9 - 2. 'Opr.ug, ro 6e(( or€troq ourqq rrlg e((ouroqg e(vot

npotpovdq dvoq pqt6g opr0p6g, otpo0 t6oo ro o/p 6oo rot ro 2 e(vot p4-

ro(opt0;ro(. Aut6 orlpo(ver dtt to tEe(vot pr1t6g, otrId r16q yvop(ouge

611 6ev e(vot. Autri q ovt(<pooq ono6errv0et 6rr o (2 + f4 npdner vo e(-

vot dppqrog. Mnopo0pe, pepo(urg, vo oXrlpor(ooupe notr0 neptoodtepoug

ouv6uool-totig pqrtiv rot dpprlrurv opt0ptirv Kclt' F€ rrlv (6to loyrrri' vo

ono6e(oupe 6rt ouro( nou npor0nrouv e(vot dppqrot opr0po(' rdrt nou

beflver [erdOopo 6rt ot dnetpot ouv6uootrro( prlttiv rot dppqturv optOptitv

o6qyo0v oe or6po ntptoodrepouq dppqroug opt0po0g'

Mdypt rtirpo €Xoupe ooXotrq0e( g6vo pe ltq TsrpclYurvrr€g p((eg rov

quorr6v oprOgdrv, otrId undpXouv notrd neptoodtepeq 6uvot6rqreg oXq-

portopo0 dpprpurv opt0trr6v. kerpre(te, yto nopd6etypo, rrlv e[(otlorl:

x' - 5 = 0. Av l0oouus oq npoq x dxoupe * = {f np6rerrot 6r1tro6ri yro

dvov opt0trr6 o onofog, dtov notrtrontrootoors( 66o <pop6g pe rov eoutd

rou, too0tot pe ndwe' Aur6 t(vot pto ruprrrl p((o ror e(vot en(or1g dpprl-

toq opt0p6g. Kotd tov (6to rp6no, pnopo0pe vo op(oou;re rt dtrtreg vto-

mdg p((eq, 6nou to v e(vot 6vog <puotr6g oprOgdq, Kot fiou oXe66v 6treg

0o e(vor dppqtor oprOpo(. Iqv npoyportr6rrpo, gnopodpe vc nopo0d-

ooupe 6vo ptrp6 0etirprlgcl, to ono(o pog tr€er noreq p(eg ordpott'tv

opr0ptirv pnopoOpe vo neptgdvoupe 6rt 0o e(vor dpprpot optOpo('

OEQPHMA: Av ot v ror o e(vot rpuotro( oprOpo(, t6te 11 vtoorrl p(-

(o tou o e(vor prpdg opt0prdq, edv rot p6vo edv ro o e(vot q

vtoorrj 66vopq ev6q dtrtrou <puotrof optOporj' 6to<popertrd' e(-

vot dpprltog oPt0P6g. 231

i

:

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I (EOAAAIO 9

l-ro nopd6erypq etuor 11 rapoyurvtni p(o tou 9 dppqroq { pqtdq opr0_pdq; Arpo0 ro 9 €ivor ro tetpdyurvo rou 3, ro ono(o efuor quotrdq oprOpd,g11 tapoyovxr] p(o tou 9 efuor pqr6q opr0;r6g. kegte(te qv ruprnl pr(otou 9. E(vor pqtdq oudq o opr0pdq; :6poupe 6u r1 ruBrnl p(o tou 9 6evefuor guomdq opr0p6g, yrot( ro 2 mov x6po rooriror ge g, evtir ro 3 movnlpo roorlror pe 22. EnopCvr4, 11 ruBrni p(o tou 9 np€ner vo e(vor 6vogopr0g6q ovdpeoo oro 2 Kol ro 3. Kord ouvdnerq dnoroq opr0p6q n ov e[_vor, 6€v e(var guorr6q opr0trrdq. Ago0 6ev e(vor guomdq oprOpdq, r6q pepdorl ro nopondvo Oerirpqgo, 0o npdner vo e(vor dppqrog opr0p6q. flopo-pofug, q rupm{ pr(o rou 8 e(vor prpdq oprOprdq, mer6rj ro 8 rooriror pe qvruprni p(o tou 2 (nou efvor quorr6q opr0p6q), oMd q crpoyrrrvrni p(otou 8 efuor dppqroq opr0p69 ogori 6ev e(vor rdnoroq guorx6q opr0p6q.

Ie xd0e nep[rnorrl nou dloupe Cvov dppr;ro opr0pd, pnoporige volqgroupyliooupe €vo yeurpetprrd pdyeOoq ilou vo owrororye( oe ourdtov oprOgd Kol vq ro tonoOeuiooupe ndvur mqv eu0e(o rurv opr0pdrv,pe to dvo dxpo tou oto trrq6€v. Ere( nou oroupnder ro dllo dxpo Clou_pe €vo oqpe(o orqv euOe(o FoS, ro ono(o 6ev ovnotorye( oe pqrd opr0_96. Enog€vurq, 4 eu0e(o turv oprOpdrv np€ner vo e(vor yepdrq ond rC-roro (dpprFou oqpe(o.

Aworiq roug vCouq oprOgo0q ro0q ovotrrd(o vpe dpp4toug apilpofuq, oA-trd qur6 efuor ontrdq gto ovopoo(o. lrq vo romvo{ooupe rohlrepo touqdpprpouq oprOporig npdner vo e{rrdooupe 6rjo loponqprourd rurv prp6voprOprirv. f-f ptinov, to dtr or pqto( oprOpo( e(vat ar)d 6nctoylttvot.

OPIIMOL Or opr0po( ev6q ouvdtrou e(vor ontrd 6rotetoypCvor ovro;giouv 60o npo0no06oerq:(1) fro onorouo6{nore 6Uo oprOgorlq, A Kol B, rou ouvdlou

ro;trfer 6vo pdvo on6 to nopordrur: A > B rj B > A { A = B(2) Av ol A, B Kol I e(vor opr0po( tou ouv6}ou, Kl ov A > B Kor

B > f , r 6 r e A > f . '

.t,,i l. 'Evo Kotrd 6rotetoyprdvo orjvolo e(vqt onld 6rotetoyycvq dtor drme rdOe uno-zrz o0votro vo neptdyer 6vo npdrto otorye(o. Bl. ro of(rrrpo qq end\oyriq tou Zcppetro.

Page 219: Ο Ταξιδευτής των μαθηματικών

Ot nopondvtrt 60o npoiino0Coerq go(vowor outon66errt€S ylq rousopr0porig, ener6{ fiouge ouvr1O(oet t6oo notrf vo uS XprlotponoroUpe 6rovoqptrodpome ;re owo0q. Oo(vetor nor6rdmrxo vo no0pe 6tr onoror6{nor 60o oxdporor oprO;.ro( npfner vo e(vor (oor rl 6n o 6voq np6ner vo e(vorgeyotrdrepoq ond rov dMov. 0o16oo, ouro0 rou e6ouq q 6rdro(r1 6ev

{mv {erdOopo 6tr umlpXe oe outo0q rouq nopd(evoug dpprlrouq opr&podq. 'Orov rorrd(oupe rouq pqro0q opr0po0q, <ptr€noupe> q 6rdro[rjtoug l6ytrr tou ouonlporoq oprOpdlv Trou Xprlo+lonororipe. Ar1tro6ri, roud-(ovrdq rouq npooertrxd, Btr€noupe 6"u o 27.991 e(vor peyolfrepoq ond tov3.990 Kol 6rr o 0,199987 e(vor prrpdrepog ond rov 0,1999998. Enop€vrrrg, oouppohop6q rou opr0po0 pdq 6(ver q 6rdro{q tou oprOpo0. loX0er 6pulqOn o (/-D.(/-t = ({-O; Av 61, noroq e(vor peyol0tepog; An6 our6 ro no-pd6erypo owrtroppovdpqore 6u ro roX0ov o0orqgo oprOpuirv 6ev qnoro-

trrimer ndwo q 6rdro[r1 rov rippqnrrv opr0prirv ror pnope( vo ovopouyOofpe e(vor dpoye ontrd 6roctoyp€vor, 6nurq eivor or pqro( oprOpo(;

To 6e0upo eptitrlpo qer(rror trre qv rleror6rqro. Tqv rlamdqto qvop(oo1re nporlyoup6vtrrq. Av 6vo orjvotro e(vor xtramd yro pro oprOpqtmjnpd$1, r6te rd0e gopri nou Xprlorgonororipre ouui tqv npdft mo ororye(orou ouv6lou, to onodtreopo nou no(pvouge ovrirer rr our6 mo otivotro. fronopd6erypq ro orivotro mrv quoxtirv oprOgtirv efuqr Ktrerord rdoo yro rqvnpdoe€on 6oo rqr ylq rov noMonlooroogd. Aqlo6{, rd0e gopd nou npo-o0froupe 66o guotro6q opr0po09 to onor€leopo e(vor n<ivro €voq rpuor-169 optOpdq. En(oqq, 6rov noMonloord(oupe onorouo6{nore 60o rpuor-xor( oprOporig ro onodleopo efuor rdnotoq dMoq quorr6q opr0p6q.

llop'6tro ourd, or ovr(orpogeq npd{erq rqq oqo(peorlq ror qq 6rofpeoqg 6ev e(vor rlerm€g yro rouq guorroriq opr0po6g. fro nopd6ery;ro,7 - 12 = -5. EOdt oqorp€oope 6rio guorrorig oprOprodq ror nripope €vovopvqnr6 opr0trr6. 'Orov 6rorpo0pe 60o rpuorroriq oprOporig, o1e6dv nd-wo no(pvou1.re €vo yvrloro xldopo rr 6;3 6vov ox€poro optOpd. And qvdtrtrq, 6lor or pqto( opr0pof (guorro(, opvquroi d\doporo Kur ro Frl-6dv) e(vor rlermo( Kor ylo ug t€ooeprq 0epetrui6erq npd(erg, pe povo6r-ni efo(peor1 q 6ro(peo4 pe ro pq6€v. Kord ouv€nero, 0o pnoporioolrevo noripe 6tr or prltoi opr0go( e(vor ouorqmrrd rleroto(. MnopoUpe vo 233

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K E O A A A I O 9

roroldpoupe yror( q rleror6rqro e(vot 1.rto oqpovrtr( t6r6rqto. rqpo!v€t ort, 6tov Xprlorponororipe rrg rdooeprg npd(erg rqg opt0prltrrr]g,yvtrrp(ouge rt 0o e(vqr to onor€treopo - €vog dtrtrog pqt6q opr0p6g.

Tr ougpo(ver dtov ouvOud(oupe dppqroug ror pqrodg opr0potiq; E{o_rotrou0e( vo roxfer q rtrerordrnro; Mnopo6pe vo 6odpe opdourq 6' or(6ror or dpp4tot opt0po( 6ev s(vor rtreroto(, yror( dXoupe Ofz).([4 = z.Kord ouvdnero, €Xouge pto nep(nttrrorl 6nou ro yrv6pevo 66o dppqrovoptOptiv e(vor pqr6g oprOgdq. To yevrr6repo eptirr4po e(vor ov Xdvouprqv rleror6rrlro drov dXoupe po(( dpp4roug ror prpodg opr0po6q. AvopX(ooupe vo Xprlorl.lonororipe ug tdooeprg npd(rrg trlq optOprptr{q oedppqroug oprOpodq, 0o opX(ooupe dpoye vo no(pvoupe nopd(eveq, trrr1oprOpqrtrdg ovr6rqreq og onovrrloerg;

H yevtrrl ovrlouX(o nou 6vurr0ov ot poOqporrro( oXerrrd pe rouq dp_prlrouq opt0podg npoXtrrpoUoe nipo ond to eporr]poro rrlq rtrerotdr4_roq Kor rrlg 6rdro(qq. H oUyypovq poOqpourrl ovdtruoq poo((etot orovonerpoorrr6 troyropd, Kor o or€rpoorrr6g troyropdg dxer ro oep€hd rouorrl 0eop(o rov op(trrv. AIld 11 opXrrr] 0etrrp(o rov op(trrv Xplorgoror_ouoe 6vvoreq on6 q yeoperp(o, dnoq, yro nopd6erypo, 6u pro orotrou_0(o o4pe(urv nlrlord(er o'6vo opror6 orlpe(o. Mnopotroope 6purg vo 0e-urp4ooupe 61 6lo ourd ro oprord orlpe(o oupBdlt(ov 0eptroUq opt0_porlg; Mrjnurq 0o rordppee o}6rtrqpoq o ndpyog on6 rponouldXopronou ouyKporoUo€ ro ovtirepo poOqpotrrd, ener6q q Oerrrp(o turv dpp4_rov oprOptirv 6e oqpr(6rov yepd orrlv €vvoro rov prlrtiv optOpdrv;

Or A6yor rou E6do(ou

flpotorj npoXrrrprloou;re mq o0yXpovrl npoypdreuorl rov dpp4turv oprO_ptirv, 0o npdner vo mo0o0pe pro mryprl yro vo e(erdooupe n6q npoond0rl_oclv vcr rous X€rptmo0v ot opXo(ot 'Ellqveg. Ae(ope npo4youp6voq 6tr 11ovordtrurfrrl rou tr?ovomdrooe rouq .Filrlveg go04go.ro69, yror( Bprirov€vo euOriypoprpo rp{po nou 6ev pnoporioe vo clvorclpooroOe( ulg l6yog

g3€ rpuorr6v opt0grirv (rurv p6vtrrv opr0pdrv nou ono66xovrov). Aur6 npordtrroe

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I T O M H T O Y N T E N T E I ( I N T

to broXurpropd rqq yeopetp(oq on6 qv dlyeppq ytot( 6tro ro owtre(pevoqg yerrrperp(oq 6cv pnopo0oov vo ovonopomooo6v olyepptrd. 'Otrttlg, no-

pd ro 6roXurprog6 out6, q yetrrgetp(o e{orotrouOotiot vo ovltp€ront(et npoptrriporo. H yecuperp(o tou Eurle6q Xprlotponotofoe onode(erg nou oqpf

(ovrov orqv owiAqrpq 6rt ro yeogerptrd geydOrl (g{q, eppo6( 6yrot)

rjrov ouyrp(orpo geto(6 toug. Oor6oq Kdnoto on6 ourd ro pteyd0rl {rovooripperpo prirrl. Mnopotioov ourd ro g{rq vo ouyrpr0o6v ptero[6 roug;

0r'E}trr1veg yeurpftpeq Xpetd(owov pto 0eurp(o yro ro oo0gpetpo p{rq.

O E66o[oq o Kv(6rog (nep(nou 400-350 n.X.) 4tov po0qrrlq tou l-ltrd-

Tovo Kor e(etr(X0qre o'€vov on6 roug onou6otdrepoug'Etrtrqveg goOq-

prolro0q. 'Htov en(oqg yrotpdq ror ootpov6poq, Kclt tou ono6(6etot 11

Oeop(o rov op6revtptrtv o<potptitv, q ono(o e(r1ye( ttq rpotvogevtrdg rt-

vr]oerg rtrrv oupdvulv oopdrov. O E06o(oq np6tetve pro Oeop(o rtlv

ovotroyuirv yro vo e(r1yr1oet ndrq ro oorippetpo pey€0q pnopoUoov vo

Xprlorpronorrl0oriv ou1 yeopetp(o. H 0strtp(o tou 6tooti0rlK€ Yto poq oro

ndpnro prpXo rurv Lroqtfuv rou Eurle(6q. H 0eop(o, 6ntlg 6rotunovd-

rov, {rov 66oxotrr1 rot roOdtrou lerdOopq. Or go0qpottro( rr1 pe}eto6-

oov 6re{o6rrd ptdXpr nou nopoyrov(otnK€ ro 6droro ivoto ottitvo. O

oprop6q nou €6rve yro rtq ovotroy(tg, 6noq undpXet oro €pyo tou Eu-

rtre(6q, onoretre( Xoporrrlptottr6 nopd6stYpo rou nveugottrof p6X0ou

rov Etrtrrjvurv nporertrrdvou vo ovrene{6}0ouv oe out6 ro np6ptrqpo.

'Ev rQ o0r(l tr6yrp pey€Sr1 triyeror elvor npdtov npdg 6e0repov r<ol tplrov npdg r6roprov 6rqv'rd roD np6rou roi tplrou iodrrg nollontrd-oro rdlv To0 6eur6pou roi re'rdptou iodrrg notrtrontrooiorv roB' 6noro-voOv notrlqntroorcopdv ir<drepov 6ror6pou q CIpo fnepdlq ii 6po ioo t1opa elleinq lqrpgdvro rorollqtro.2

2. To ptey€fi tr€ge 6rr dXouv tov (6ro tr6yo, to nptitto pe to 6e0tepo rot to tp(to

pe ro tfropto, dtov, ov ndpouge onoto6rjnote toonotrtrontrdoro on6 to npdrto ror

ro tp(ro, ror onoro6{note toonotrtrontrdotq on6 to 6efrepo rot ro tdtopto, to nptit-

to roonotrtrontrdoto o;lo(rr-rq uneppo(vouv, ogo(tlg tooUvrot { ogo(trtg unotre(novtot

rurv 6efrepov toonotrtrontroo(rrlv ov to ndpoup€ p€ Trlv ovtiototXq oetpd.

[Sir Thomas Heath, ,4 History of Greek Mathematics, r6tt. | (O[<p6p6n, The

Clarendon Press, 1960), o. 385.1 235

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K E O A A A I O 9

Tr 0o pnopofoe vo orlpo(vet pto 16oo orvtyportxrl 6r]trooq; O E66o-

(oq (6nurq rpo(vetor g€oo rou Eurtre(6r1) 0o npdnet vo ro06tov ot€treto-

req vriXteg npoono0tivrog vo ypdrpet rdrt nou rov€vog 6e 0o pnopotiot

vo rorotrdBet. fro vo xorovo{oou1.te ouuiv rq 6ritrrrroq 0o np€net vo 0u-

gqOofpre 6fo npdypoto Yto touq'Etrtrr1veg poOr1golro0g. [lpdrtov, o E6-

6o[o9 6e trltro0oe yro oprOpodg otr]d yto peydOq. Aurd to 60o 6ev rltov

(6ro rot 6ev gnopofoov vo ouox€rtororjv ro €vo ge to d}tro. Aeftepov,

or 'Eltrqvtg 6ev etXov rtrdoptoto, pttrotioov trorn6v yro trdyoug oprOptitv

ror l6youq geyeOtirv. Enog€vu-rg, to 6116 pog xldopo ? rttov yt' ouroUg

o ldyog 2:3. Irq yeurperp(o roug Xpero(drov vo pilo0v yro trdyouq, dXt

oprOpdrv otrtrd geye0rirv. fro nupd6erypo, yvtipr(ov 6tt o tr6yoq rov ep-

Bobtirrv 60o r0rtrov rooriror pe ro tr6yo rov rerpoydrvov tov brop€rptlv

rov r0rtrov. Aut6 trrnopo0Fc vo to 6e(oupe oq effg:

(eppo66v tou r0rtrou A):(eppo66v rou xtixlou B) =

(oxr(vo rou rrjrtrou A)':(orr(vo rou rUrtrou B)'

Ot'EtrIqveg inpene vo e(vot ofyoupor drt, 6tov or trdyot our6v ttrtv

preye06v oupneprtrdpBovov oofppetpo gr1xr1, (oXuov ot oX€oetq 6rotd-

(eurg. Mt dtrtro }6yro, 0o (oXuov dpoye or yeurperprr€g ono6e[(etg toug,

6tov ourdg ot ono6e(etg oupnepildprBovov l6youg ootilipetptov pq-

rtirv; O oprop6g nou ov€ntu{e o Eri6o[oq qrov pro npoondOeto vo 6tq-

orpotrioer 6rt 0o ioXuov. To pey€Or1 otouq tr6youq npoo6top(ovrot oq

e[r1g: npdrro:6e0repo = rp(ro:r€topro.

O E66o(og s(ne 6tt o tr6yoq rou npdrou Tlpoq ro 6efrepo g€yeOoq

roo0tot pe ro ldyo tou rp(tou npos ro t€ropto gdyeOoq edv, 6rov no}-

tronlootdooupe ro nptiro rot ro rpiro pe ro 6to g€ytOog, rot notrtrq-

ntrootdooupe rot to 6efrepo Kot ro rftopto pe xdnoto dtrtro p6ye0og,

t6re 6noro 6tdrolrl ndpouge preto{ri tou np6tou Kot rou 6e0repou 0o

6tou1pq0e( preto(tl rou rp(tou Kol rou r6toprou.

Auui q €qnYrlorl, ov xot rpo(veror ontrri, 6ev e(vot (erdOopr1.'Evo no-

pd6erypo 0o fe6ro]rlver to O€po. Oerrrpo0pe 6rt to r€ooepo pey60r1

€Xouv ro nopordro prirrl: 3:6 = 7:14. An6 our6 no(pvoupre ttg nopord-

236 to ovro6rrpeg:3 < 6 Kor Z < 14. O E66o{og tr6er 6tt, ov notrlontroord-

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I T O M H T O Y X T E N T E T I N T

ooupr ro 3 Kqt ro 7 pe xdnoto prdye0oq A, Kqt ro 6 xot to 14 pe rdnoto

gdyeOoq B, t6te 11 oX€oq 6rritofr1q ovdpeoo orq rpononotqp€vo 3 rot 6

0q e(vqt 6ro pe tr1 oXCorl 6rdto(4q ovdpeoo oro rponorotqpCvo 7 xot

14.'Eorur 6tr A = 5 Kot B = 2. Ilotrtronlootd(ovtog, i;1oupe:

4.3:8.6 = A.7:B'14 11 tS:tZ --35. '28

Iorptirq trorn6v '15 > 12 rot 35 > 28. Kotd ouvdTltto, o notrtrontrooto-

opdg pe ro 5 Kot to 2 Orotqpqoe rr1 6rdto[r1 tutv 60o tr6yu-rv. O optop6q

tou E06o{ou }€et 6tt, yto vo e(vot (oot 60o )\dyor, d}eg or ttpdq rov A

ror B 0o 6rotrlpo0v rr1 6rdto{r1 ovdpeoo mo ovt(ototXo pey60r1' Aur6

d6uoe mrlv etrtrr1vrr4 yeurperp(o rov oprop6 ttlv trreyt0tirv rtlv tr6ytrlv

nou lpero(6tov yto rtq 6rdgopeq ono6e(erq nou otrlp[(ovrov ortq ovo-

loy(eq. To trrey€Oq 6purq 6ev e(vot opt0po(, rot 11 ono(tqoq 6treg or up€q

rov A ror B vo trovonotodv tov oprop6 elorlyoye on6 u1v n(oo: n6pto

rqv Cworo tou one(pou. llopdtr to €pyo tou E06o(ou lKovoiloto0oe rtq

ovdyxeq rov ytu,p€rptiv, e(orotrouOorjoe vo prlv enopre( Yto rrlv Kdlu-

gq turv ovoyrtiv qq oprOprltrrqg ror rq otripr(r1 rurv dppqrov opt0-

ptiv oe pro yepq Oeopqttxrl Pdor1.flprv 6drooupe qv tetrrrr] ondvrqoq Yto rouq dpprltouq opt0trtotig,

go npdner vo ovoyvop(ooupe ere(vouq rouq poOrlpourofq nou rltov

npd0utrror vo ono6e;6rodv 6tt ot dpprltor opr0po( ovt]xouv otqv otroyC'

v€ro rurv oprOgdrv. Or rv6outorCq yevtrd, rot et6txdrepo o BpoXpoyxoti-

nro, 0ropo0oov trq dpprlttg p((eq oq 0eprt€g }0oerg e(totitoetlv. Kord

ouv€n€ro,6ev e(Xov ev6oroopofq vo ono6eXtoriv to {7ri to VTurq tr6-

oerq npoptrrlpdrov. 'Evog dtrtrog noo(yvooro( poOrlportxdq, yvtrtot6g

oq A6o11 urq norrluiq, qtov o Opdp Koyrdp (nep(nou 1050-1123). Evtit

or 'Etrtrqveg, yro vo enil0ouv e[totioetq, Xprlorponoiqoov yeogetptxdg

trre0d6oug nou ovttpettitnr(ov ttg trriotrq oq prlrq, our6q 0errrpo6oe 6tt

or }doerq rjrov oprOgo(. AoXotrri0qre en(oqq pe ro npdplqpo rou opr-

opo0 rrrrv dppqtov opt0pttitv.ho pelotrUtepo 6rdorrlgo tou Meoo(trlvo, ot Eupono(ot poOrlpottxo(

ono66Xovtov p6vo touq rpuotrorig opt0poriq rot ro rtrdoporo oq trrloerq

e(rotioeov.'OUoq, q ougBotrrrri <pt1oq qq dtryeppoq e[etrfy0qre ror, tetrt- 237

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rd, 4 nolord pqroptxrj p€0o6oq eyrorotre(rpOr1re. M6trrg or tr€(erg ovuKo_tootdOrlrov ye odpBotro, ot opv4uro( ror or dppqtor opr0go( (nou oup_Botr(,(owov F€ r€rpoyurvrr€S pt(eq oriparov opr0grirv) dpXroov vo €Fqo_v[(owor 6edd urq trUoerq. To vo no0pe, pqroprrd, du emd Xr]veq pe(ov6wero Xrlveg tooriror pe pe(ov rdooeprg Xrjveq (opr(er rr1 rpovtoo(o, yror(l 0o gnopofoe vo orlpo(ver ro ope(ov tdooeprq Xr]veqr; 0o16oo, pe tr1Xprion rnq oupBolrr{q ypoq{q ro}fnretor our{ q 6uorol(o, yrot( 6Xou_pe: 7 - 11 = 4. Me ou{ rrlv oqorp€urrj 6uvordtrlro tqg titryeppoq, orpo0rlpouro( dpXroov vo Xpnorponorotiv roug opv4rtro6q ror rouq dppq_toug oprOpo0q Xurp(S rdnorov onoOerrd opropd yro ro u orprBtiq e(vor.

Or rperq po0rlpouro( nou Oeopo0wor ouvqOoq or onou6otdrepor e(-vor o ApXrgrj6lq, o osp Ar(ox Nro0rov (loodr Ne0turv) ror o Kopl Op(_vrptX lrdouq. O Koptr Op(vrprX fr<iouq (1t77-1855), o ono(oq yevvrl0r6eo€ l.lro epyolr{ orroy€vero ror 0eopo6vrov nor6(_0o0po, rjtov outdqnou npo60r1o€ rlv ono6oXrj rtrrv dppqttrrv opt0prirv (Errdvo 39). Oqtpotr4qq dyporpe yro 14 6t6orroprrrj 6rorprBri tou 6vo ond to optoroup_yrlpotti rou, To Disquisitiones Arithmeticae. Ootdoo, ro Xe rpdyporpo ro_Ouotdpqoe vo er6o0e( rp(o Xpdvro. ho peto[rl €yporpe pro dllq ptrprjOrotprprj yro uq onorrjoerq tou muX(ou rou, q ono(o er660qre ro 17gg,6tov ere(vog rlrov e(roor 66o Xpovrirv. Ie ourrjv 14 6eriteprl €Tnorrlpovr_r{ 6qgoo(euo4, o fxdouq ond6er{e our6 nou dytve <To Oegilui6eq Oeril_prlpo rnq Alyeppoq". floro e(vor ourd ro Oetirp4go rou 0o pnoporioe voo{i(er rdtoro Ordrproq; To Oetip4go tr6er 6. rd0e notru,vuprrq e{(ouroq,p'6vov dyvooro €Xer p(o toutrdXrotov fujorl rj p(o. ,Evo n6propo ouroUrou Oeulprjporoq }€er du undpXouv r6oeq }doerg, doeg 6e(Xva o peyol0_repoq er06rqq rou oyvtitotou. Enop€vr.rrg, r; notrurlrvuprr4 e((ouroq:

Aoxu + A,xu-t +... + A"rx3 * Au-rX, + Au_rX + A" = 0

3. t1otrut'rvupu<{ e(ourorl e(vor q e(oooq dnou to oprmep6 or€}oq onote}e( or on6dvov q neproo6repoug oyvtrroroug (ouvrj0og rous x, y, r.trn.), or ono(or e(vor urrrou€vor o€6uvdperq ror noMontroord(owor pe oprOgoriq nou treyowor ouwetreor€q. To 6e(( ordloq

rr 6 qq e(ooorlg e(vor ouvriOurg pr166v. fro nopd6erygo, ro 5x' + 3x - 4 : 0 e(vor €vo noAutir-/'t6 vupo p'dvov dyvrrloro (tov x) ror ouweleordq rou( 5, 3 Kcu -4.

1ji

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EIKONA 39. Koptr Op(vtptX frdou9,1777'185s'(Ytrtrd on6 toug o6etr<porjg Mnpdouv, trt€ptrtvyr)

dXer orprB6g v tr0ottg. Ie rdnoreg neptnrtioetg €Xoupe rourdorlpeg trrjoetg,

6nog orqv e((ourorl x'- 10x + 25 = 0. Av nopoyovtonot4ooupe ourq r4v

e((oooq, €Xou;re (x - s)(x - 5) = 0, 6qlo6r1 x = 5 rot yro rrg 66o trfoerg. To

oqpovrrr6 orlpe(o e6r.ir e(vot 6rt to 0etirprlpo l€tt nrrrg prto e((otlorl 6ntlg 11

x' - 2 = 0 €Xtr 60o tr6oe rg. Aur€q ot trtjoetg e(vot, mrlv npoYuortr6rrlro, ot

ltl rat -r[i, nou Kor or 66o e(vot dppqror oprOpo(. Enop€vtrtg, pe rqv on6-

6er(r1 ouro0 rou 0epetrttir6ouq 0eurprl;'rotog, o frdoug ovoyrdorqre vo

ono6eXre( roug dppqroug opr0go0g urg trdoerg eftodrotu-tv. 'Onurg 0o 6o0-

ge opy6tepc, o frdouq npoetofgooe m(or1g ro 66orpog Yto rrlv ono6oXrl

Kor proq dtrtrr1q, wtetrdrg vdoq rtrdoqg optOgtilv, rov ptyo6trtirv opt0ptitv.

Me rov frdoug vo nt€(et yt'our6 to O€po orrg opXdg rou D€rotou

fvorou ouitvo, (er(vr1oe 11 ovo(rlrqor1 ev6q rorovoqro6 optopoU rov

dpprpu.rv opropdrv. e39

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I ( E O A A A I O 9

H undpoxn schni t t4 rou Nrdvr€Ktvr

Eipoore r6ro(tepo rulepoi drov grdvoup€ oro otiy;1povo opropd rovdppqrov opr0pdv. 'Olor or p4to( oprOpof pog pp(orowor Ooppdvortdoo poOrd oto nopel06v, 6ore 6ev pnoporipe vo nodpe ndre, yronp6r4 gopd, r<inotoq pftpqoe ro 6dyrutrd rou Kot e(ne: .AXti! Or gu_orro( opr0po(!u O0te pnopo0pe vo 6ropdoou;le trq ordrfrerq rurv poOr;_potrr6v nou ovordlurfov to rkiopoto ri roug opvquro0g oprOpodq.AIId q loyrxrj Oepetr(oor1 rov {ppqrov opr0ptirv e(vor olertrd npd_oqqrq Kor pnopo0pe vo oyopdooupr Kor vo Oropdooupe ro 6116 poqovr(ypogo wdq un€poXou rerp€vou. O oprotrrdg rurv dppqrurv oprOprirvnepiloppdveror o'Cvo lentd oe nriXoq lerpdypqgo rou p(1opvr NrC_vr€Klvr (1831-r916), evdq feptrrovod poo4po'ro0 tou bdrorou cvqrouorrivo, pe t(ttro IuvtXcn rat dpp4rot apfipo(.'O roOdvoq nou 0o oyo_pdoer ourd ro dpyo purope( vo rdroer rdro ror vo ro 6ropdoer otrdrlq_po trr€oo oe trryo lernd. E(vor }orovrr6, qltrd oe urfrqtrd enroqpovrrdenine6o.

O P(1opvr Nr€werrvr yevv{0r1re oro Mnpdouvopoip otq fepgov(o.Iro e(roor €vo rou Xp6vro, ro 1852, nrjpe ro 6r6orroprr6 rou on6 rollovenror{gro rou frftrvyrev, nopotroppdvowoq ro mu;1(o rou on6 rov{6q yvrrrord Koptr t-xdoug.

flop'dtro nou o NtCvrerrvr ovordtrurpe q Oerrrp(o rou ro 1g58,6€vq 6r;poo(euoe pr6Xpr r.o 1872. h4v oppj rou €pyou tou IuvtXen xatdppryrot ap0po( priq trCer 6tr 6ev {rov rrovonor4pivoq on6 qv €v_voto rrls ouvCleroq rou Xpnotponororjvrov orov on€rpoourd loyropd,r; ono(o orqpr(drov oe Cworeq boverop€veg ond rq yerrrperp(o. Agoripeldqoe to npdBtrrlpo, gp{re q 6rrrj rou }dor1 or.S 24 NoepBpiou1 858.

To nptirro npriypo nou €rove o Nrdvrerrvr {rov vo op(oer ro orivoro

4, Schnitt (yepU.): roprj. (r.t.u.)5. H oyyhrri perdgpoorl outoO rou dpyou undpXer oro: Richard Dedekind,

Essays on the Theory of Numbers (Ao Iotr, Open Court publishing Company, 1 948).

r lj

i

240

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dtrtrtv ttrrv pqtti,rv oprOgdtv wg oAor4po Q. k4 ouvdXeto, e(ne drr yro

touq oprOpodq rou Q roX0ouv ot nopordro v6por:

l. Av q > p ror B > y, tdte o > y. Kd0e <popd nou or o, y e(vor 6rio6roqopetrro( (rl dvtoot) opr0po(, rot o p e(vor preyol0repoq ond tov dvqror prrp6repoq on6 tov dtrlo, outd 0o to errppd(oupe odvtopro, Xop(qvo 6rmd(oupe ener6{ unovooriwor yerrrpetprr€q €woreg, }€yovtog: to Bpp(oretor ovdpeoo otouq 60o opr0po0g o, y.

tt. Av or o, y e(vor 6rio Otorpopetrro( opr0po(, undpXouv dnetpor 6ro-qoperrxo( opt0po( nou pp(orovtor gero{ri rov o, y.

lll. Av o e(vot onoroo6qnote ouyrerprp6voq opr0prdq, tdte 6}or oropt0yo( tou ouorripotoq Q 6rorpoUvtot oe 6Uo rtrdoerq, uS Ar Kor 42, rlroOeg(o on6 trq ono(eg neprloprpdver dnerpouq oprOporig. rl nptirq xld-ol, I Ar, qnotetre(tor on6 dtrouq touq oprOpotiq o, nou e(vor <o, 4 6e6-repq rldorl, 11 Ar, onotetre(tot on6 dtrouq touq oprOpo0q o, nou e(vor >o.o (6roq o opr0p6g o ;.rnope( vo equrpqOe( ele0Oepo e(te oqv nprirtq e(teotq OeUteprl xtrdor1, otpo0 e(vor ovuotofXurq o preyotr0tepoq opt0pdq qqnptirqq rtrdoqq ror o prrp6tepoq qq 6e0repr1q. Ie rdOe nep(ntoo4, oDroXtoprop6q rou ouotrlpotoq Q oe 66o xtrdoerq, Ar Kor 42, e(vot rdtoroq6ore rd0e opr0p6q tqq np6qq rldoqq R, vq e(vor purpdtepoq ond rdgeopr0p6 rqg 6e0tepqg rtrdoqg Ar.u

Oo {rov 66oxolo vo errppdooupe pr peyo}rirepq oorprjvero qv r6riotou Nrdwexrvt. Tov npdrto vdpo rov €Xouge rj6r1 ouvovrr]oer trier 6rr orpqro( opr0po( e(vor 6roreroypdvor. O 6eUtepoq vdpoq trder 6u undp-

Xouv dnerpor oprOpo( ovdpeoo oe 60o onorouo6qnote p4todq oprO-po6q. Aur{ e(vor q opXo(o e}}qvrrrj rivvoro rrlq rinerp4q Orotperdtqroq.O rp(roq vdpoq onoretre( qv npoygourrl ouo(o qg r6€oq rou. A6er drrrdOe pqt6q oprOpdq 6torpe( dtrouq rouq pqroriq oprOtrroriq oe 60o xtrd-oetg, 6nou 6tror or oprOpo( qq npdrqq rtrdoqq e(vor prrpdrepor on6toug opr0potiq rr;q 6eftepqq rldoqg. Merd 6rlltirver 6u o pr1t6q opr0-pdq nou Xprlorgonouj0lK€ yro vo y(ver our6g o 6roXoprotrr6q rov opr0-prirv pnope( vo oupnepilrypOe( e(te or4 pfo e(re oulv dtrtrq rtrdoq. Mno-

6. 'O.n., o. 6. 241

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q

II ( E O A A A I O 9

pe( vo e(vor o peyotr0tepoq oprOpdg qq rldorlg A, ri o grrpdrepogopr0p6q qq rtrdorlq R,.

M€Xpr rtirpo ov€<pepo ourd nou e(ne o Nr€vterrvt pdvo yro rous prl_roUq opr0pofq. 'Orov r]rov €rorpoq vo op(oer toug dpp4touq oprOporig,onltiq nop€tr€tq,E rrlv ono(tqo4 rou rpftou vdgou rou, to vo y(vovror611lo6{ 6tror or 6roXtrrpropo( rov pqttirv opr0ptirv on6 rjvo pqrd opr0p6.EfXe 6(no nou ro €rove outd; O Nr€vterrvt 6r]truroe 6rr:

Av lorn6v Oeu.rprjoouye 6e6opdvo onorov6{note Oto;trlptopd tou ouon]go_roq Q o€ 66o rldoetg nq Al Kor Ar, o ono(oq dXer y6vo ouu]v q XopqKrqpForrr{ t6r6trlrq 6tL 6ntro6{, rdOe oprOpdg or triS Ar e(vor ;urp6tepoq ondrdOe opr0pd oz rl( Az t6re yro tr6youg ouwop(og pnopoUge vo ovogdoou_pe outdv to 6roXtrlpropd ropaf [Schnitt] Kor vo rn ouppotr(ouge u4 (R,, nr).'

Mnope( vo ovopr.uq0oUpe yror( o Nr6werrw dvuloe nu4 rytov onopo(q_ro vo op(oer €vo vrio 6pq qv topr], ow( vo Xprloqloilor{oer qv Cworo rouoqpe(ou nou Xrrrpr(er dtrouq toug pqrodq opr0poriq qq euOe(oq rov opr0ptivoe 6rio rldoerq. oo np€ner vo 0uprlOofpe 6l r]0ele vo 6roxtrrp(oer qv iv-voro ru,v dppqrurv oprOptirv on6 ug yeurperprr€q owrtrrjrferq, dnoq nopo6eQ_poroq xdprv to orlpe(o ror nq euoe(eq. Aurd ro rordpoo0e orov nopondvuroptop6 rnq roptiq, yror( 6proe qv totruj pe pdor; 60o rkioerq opr0pdrv, dnourdOe opt0g6q qS Trpdrrnq *trdolq eivor prrpdtepoq on6 rd0e opr0p6 r4q6edtep4g rtrdoqq. An€rpuye €[unvo vo ovorpdper euOe(eq ror orlpe(o.

'Oproe qv toprj rou Xurp(g vo ner ndrq pnope( vo ilpoyporonorqOe(our{ q togrj. fvtrrpi(oupe rj6q 6tr pnopofpe vo 6qproupyriooupe ptot€roro roy{ Xpqorponordrwoq onorovOrinore p4rd opr0trrd. To eptitrlpo,trorn6v, e(vor ov ;rnopei vo y(ver n ropn Xtrrp(q rdnoro prltd oprOpd. Irourd o Nrfurerrvr ondwrloe urg e(r1q:

Atrld e(vqr e0rolo vo 6e(outrre 6rt undpXouv dnerpeq topdq nou 6e 6rlpr_oupyorlvtor ond prltotig opr0po0g. To nopordtur nopdbetygo prou dpXe_ror ypriyopo oro vou.

,ifi

242 7. 'O.n . , o . 12

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T O M H T O Y N T E N T E K I N T

'Eoto 6tr A €(vot 6voq 0ettr6g ordpotog otrtrd 6Xt to tetpdytlvo tvdg

qrepo(ou, t6te undpXet 6vog 0ettr6q or€potoq tr t€totog tiote:

l ' < A < ( t r + t ) ' �

Av orrl 6e0tepr1 rtrdor1 n, erXurp(ooupe rd0e 0etrr6 prgt6 opr0p6 o,

nou to tetpdyutvd tou e(vot >4 Kol oulv npdrtq rtrdor1 At 6louq touq dtr-

troug prlto0q optOgo0q o', outdg o 6toXr.r-rptop6g oXrlpot(et pto toprri (At'

Ar),6r1tro6r1 rd0e opr0gdg o, e(vor gtrpdtepoq on6 rdOe opt0p6 or"' Auul

11 topri 6e 6qgroupye(tor on6 rdnoto pr1t6 opt0pr6''

Tr oupBo(ver e6tir; flptirrov, sntorlpo(vst 6tt yto rdOe or6poto 4 nou o

6ro9 6ev e(vot retpdyutvo ev6g rpuotro0 oprOpo0, out6g o A Bp(oK€rot Us-

ro(0 66o d),Iov retpoytitvtrtv, ot rerpoytlvrr€g p(eq rurv ono(utv e(vot or€-

poror opt0po( nou 6rorpdpouv gero(0 toug xord g(o povd6o' fro nopd6ery-

U0, To 5 pp(oxeror peto(U rou 4 rot tou 9, Kot ro 4 s(vot ro 2', Evrh ro 9 t(-

vot ro 3'. Metd enilsyet qv togrl rou €rot 6mt to A, vo e(vot ro o0votro

dtrov rov opr0ptirv tuv ono(uv ro rctpdyuva e(vat prya)trcpo ano ro 5'

Me ourrl rrlv notrri €(unv4 npooqlytoq, o Nt€wertw €Xer rdvet pto rop{

nou op(er ro {1 nou e(vot dvog dpprltog opr0p6g, otrtrd Xp4otgonote( p6-

vo pqrodg oprOporjg yto vo op(oer ourrl rqv ropri' 'Etor, onorpedyet vo Xpn-

orpronor{oet dppqtoug opt0po0g yto vo op(oet roug dppqtoug optOpotig'

Og 6eftepo nopd6etypo pnoporlpe vo op(oouge pto ropq Yto lov

dpprpo oprOpd {7 pe rov nopordro rp6no: og unoO€ooupe dtt ro A,

neprdXer 6troug roug pqrodg opr0pofq TIou ro terpdytrtvd roug e(vot gt-

yotrdtepo on6 ro 2. 'Otror ot dtrtror pqro( opr0po( 0o pp(orovrot oro A1'

Aut6 Xpetd(etor trfur1 oxdrprl, otrtrd ypriyopo (ero0op((et oro vou Uoq'

dnurg orptpdrg 0o np€net vo ouvdprl Kot orov Ntdvrertvt, r6re, ortg 24

Noeppp(ou 1 858. Korrd[re rrlv Etr6vo 40, q ono(o 6e(Xvet q 0€or1 16oo

rcu ,[i 6oo ror rou 2 orrlv euoe(o rov oprogdrv. Oovroore(re ttitpo

onorov6{nore pr1t6 opr0p6 peyotrdrtpo on6 ro y'7 (o ono(og 0o pp(-

orerot 6e[td tou fz;. l-vurp((oupe 6rt to rerpdytrtvo ourof rou pqroU

oprOpoti 0o np€net vo e(vot peyohjtepo on6 ro 2'

8 . 'O.n . , o . 13 . e43

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K E O A A A ! O 9

p4ro( oprOpo( A, E prlto( oprOtrro( A,

EIKONA 40. Mro rogrj rou Nt€vterrvr. 'Otrot or pqro( opr0po( nou ro rerpdyrrrvdtouq e(vor prrp6tepo on6 ro 2 pp(orovror ndvo orqv eu0e(o rt'rv oprOpdrv oprore-p<i ond ro y') ror ov4rouv oro A,. 'Otror ot pqro( oprOpo( nou to retpdytrrvd rouqe(vor peyolftepo on6 ro z pp(orowor ndvto orqv euOe(o rov optOprdrv 6e[r<i on6ro y'7 ror ovrjrouv oro Ar. Aurd to 60o orjvotro pqrrirv opr0prdv, ro Ar Kor Ar, opl(ouv rov opprlro opr0pd rD.

Me our6 ro unipoXo riXvoopo rnq Xp(onq p6vo pqrtirv opr0truiv yrorq 6qgroupy(o roptirv nou 6toXurp((ouv rouq prpodg opr0;rorig, o Nr6-vr€Krvr 6ptoe rouq dppqrouq opr0po0q.

Onote6(note, lorn6v, dXoupre pro toprj (A,, nr), q ono(o 6€v TrpodKuqr€on6 p4to6q opr0poUq, 6rlproupyo0pe 6vo v€o, tvov dpprlto opr0p6, tovo, o ono(oq 0eopo0pe 6tr op((etot nl{pog on6 outq qv totruj, u1v (R,,Ar)' 0o nofpe 6tr o opr0g6q o ovuotorXe( oe outrl rqv toprj, ri 6u 6r1pr-oupye( outq qv ro1,r{. An6 6ur Kor oro e(r]q, enopr€vurg, oe rdOe ouyre-rprp6vrl roprl ovuotorXe( €vog ouyrerprpfvoq plr6q 4 dppqroq opr0gdq,ror 0eurpofpe 60o opt0porig \oEopeurolg 11 dvtooug ndwote ror pr6-vov 6tov ovrrotorXoriv oe ouotoortrd 6rorpopettrdq rop6g.e

To odvolo rcrrv oprOtruirv nou e(vor o ouv6uoogdg dlurv rov prltdrvror dppqturv oprOpdrv ovopd(eror o0oqpo ruv rpayLrartrdv oprOptirv.Op((ovrog 6loug rouq npoygourodq opr0po0q pe togdg, 6qtro6rj pe od-volo dnerpulv pqttirv opr0prirv, o Nr€werrvr 6proe ouorootrrd rougopt0po0q pe onerpoorivotro. Enop6vog, to o0orqpo trrtv opr0ptirv poqe(oprdror ond r4v dvvoro rou one(pou yto Fro oorpolrl Oepelitrroq. OropXo(or'Etrtrr1veq 6e 0o pnopoUoov nord vo ono6eyro6v 6rr our4 11 iv-voro onoretre( 14 Bdorl rcov po0qporrr6v.

O Nt€vrerrw 6Xt p6vov opi(er toug dppqrouq oprOpofq, otrtrd npoXrrr-

2M 9. 'O.Tr. ,

o. 15.

Page 231: Ο Ταξιδευτής των μαθηματικών

T O M H T O Y N T E N T E K I N T

pd mov optop6 tqg npd(qq rqq np6o0eoqg yo 6troug roug npoyportro0g

oprOpo0g Xpqotgonordrvroq rtq rog€g. Arpo0 entreuX€e( out6, e(vot efrolo

vo op(ooupe lg dtrtreq tperq npd[erq u1g opr0prptrrlq yto toug TrpoYgott-

rodg opr0pto0q. Or opropo( rou rou enrrp€nouv vo unoorrlp(et 6rt, 6tov

Xpqorponoro0g€ rouq npoYgonro0g oprOpo0q (prpo0g q dppqroug) ortg

rfooeptg npd(etg rqg opt0gqurrig, ro onor€treo;ro 0o e(vot ndvro €vog dtr-

trog npoyporrrdg opt0p6g (trre e(o(peoq rr1 6ro(peoq trre to gq6iv)' Aut6

rou nop€Xet rqv rtretor6rrlro rou oumrlporoq rov npoYgotlKdrv optOptitv

ror rou enrtp€n€t vo unoofflp(et rrlv ontrri ro6qto: ''l z'"1 I = tro' ttitpo

nou o Ntdvrertvt tonoO€rqoe oe yeprl pdorl roug dppqtoug optOpotlg,

pnopo0pe vo ovoporrlOo0ge: Tt oxprBdrg e(vot ot dpprlror oprOpo(;

Arooredd(ovroq p€ ro der<odrr<6 o0ornpo

H erpedpeoq rou 6ero6trod ouordprorog, ro ono(o Xpqorponots( rrlv uTto-

bromotr{ yto vo btoltrtp(oer ro or€poto rgrlpo ev6q oprOpori on6 to rtro-

oportr6 tou rg{pro, ur<pdtrqoe r6roftepo roug unotroytopo0q' To rtrdo;toto

{rov gne}otr(6tKo orrlv npdoOeorl Kot rlv orpo(peor1, rot rdnoto xtrdopo-

ro ouyrp(vovrov 6dorotro petofd toug. fto nopd6etygo, noto e(vot geyo-

tr0repo, ao * i to H; E(vot 60orotro vo nofpt. 'Ogog, 6rov rorrdlouge

ro nptiro tdooepo t|tr1q(o rqg 6ero6rrqq ovonopooroorig toug gnopo0pe

vo ro 6odpe op€otrtg, ytot( to # = o,zall evti ro # = o,zloe ' An6 toug

6ero6tro0q opr0po0g ptrdnoupe ogiourq 6rt to il e(vot peyotrftepo'

To 6ero6txd xtrdogoto eprpov(orqrov yro npdrr4 rpopd rord rov

np6ro ottitvo rrlq Avoyivvrlorlg. To 1492, o Opovtodoro fl€trog

(Francesco Pellos, rltov nopoyoyrr6q geto(0 1450 Kot 1500) 6r1goo(eu-

oe ro €pyo :rov Compendio de lo abaco, ro ono(o ne prtrdgBove rq Xpn-

or1 ev6g r6ppotog yto vo uno6rltrtivet €vo rtrdotrro nou e(Xe oq ropovo-

poor{ rdnoto 60vogr1 tou 6€ro.'o'oJt,lq, ot opXo(ot BoButrtirvtor e(Xov

Xpqotponor{oer ro e(rlrovro6rrd xtrdogoro, flou efiov rrrg pdorl ro e(q-

10. Boyer, o. 307 245

Page 232: Ο Ταξιδευτής των μαθηματικών

vrq. H nporrrr{ rrlq Xprlons e{qrovto6rr6v rtroopdtrrrv ro0repdr0qreorrlv ctorpovop(o rou Aurrrof r6oprou Kor, Kord rr1 6rdprero rou Me-oo(rdvo, Xprlorponororivrclv orouq ouvr]0erg unotroytogorig. fro rdnoro6tdorqpo prerd rov [16trog, ro efqrovro6rrd rtrdoporo 6rorrlpr]0qrov.Tetrrrd, o fdtrtroq po0qpour6g Opovooud Brdr (1540-1603) unoo{pt(ero 6ero6rrd ofoqpo KCtr, orrl ouv€Xero, o T(ov Ndnrep (issO-tOtz) ro6td6u.roe erodyovroq rr;v uno6roorolr] yro ro 6roXulprop6 rou or€porourp{poroq rou opr0pori on6 to rtrooporrrd rprlgo."

To orjyXpovo 6ero6rr6 oforqpo e(vor dvoq o0wopoq tp6noq ougpo-trropoti rov rtroopdrurv (r1 tutv rtrooprourtiv rgqpdrurv rov opr0pdrv)og d0porolro rtroopdrtrrv rou or nopovogoordg rouq e(vor ou{ov6peveq6uvdpetg rou 10. Enop€vtrrg, yro ro B34,522 6Xoupre:

834,572 = g.10'+ 3. ' |0r + 4. j00 + 5/10' + l l1o, + 2110.

'i

834,572= 800 + 30 + 4 + * *.'rz * .oh

H Xpqorpdrqro rou 6erobrrod ouorr]porog rpo(vsror op6orrrq p6hqouvel6rponoujooupe n6oo prrp6g Xtirpoq Xperd(erdr yro vo ypd{rouge6vov opr0p6, roOtirg ror t4v eurotr(o' ps ulv ono(o Xerpt(6poou rovop1096 oug opr0prprr€g npd{erg.

YndpXe; 6pulg, 1.rro r6rogoprp(o pe 14 6ero6rrq ovonopdoroorl. fro voovonopoorqooupe 6vo rtrdogo urg 6erobrr6 opr0p6, 6rorporige ouvrl0urg(Xp4orgonordrvroq otetrrj 6ro(peo4") rov optOpqrrl rou rldoporog (ro nd-vo rprlpo) F€ rov nopovogoorrl (ro rdrul rUnUo). fro nopd6erypo, rortrdopo f e(vor ro 2 6rd rou 5, ro ono(o toofrot pe 0,4. Aurdg o 6ero6tn6gopr0p6g ovogd(eror nmtpoolttvog 5era5rc6g opt0p6E, yrot( ro qryp(o 6e-(td r4g uno6roorotrqg e(vor nenepoop€vo - 6r1tro6q, moporodv p6vo toug.

Ag npoono0rlooupe rc0po vo prerorpdrfoupre ro rldopro 1 os 6Ero6r-

1 1 . ' O . n . , o . 3 4 8 .tz. H 6ro(peoq orqv ono(o ro undtrorno r(vor dpprltog q 6ero6rrdg nepro6tr6g

opr0prig. ( l .r . l , t .)3€$

Page 233: Ο Ταξιδευτής των μαθηματικών

H T O M H T O Y N T E N T E K I N T

xd opr0p6. Arorpo0ge ro I Us ro 3 oq e{rjq: to 1 6e 6rorpe(ror oneu0efoq Ur ro 3, €ror ypdrpoupe ro 0 p€ prto uno6toototrrj ror rono0eto0pe6vo 0 6(ntro oro 1 yro vo ro rdvoull€ .|0. T6po 6rorpodpe to 10 pe ro 3,fiou FoS 6(ver 3 Kor oq un6lorno ro 1. fpdqoup€ ro 3 6e(rd qq uno_Droorotrrjq rr iror dXoupe 0,3, eforotrou0o0pe 6prrrq vo iXoupe ro I oqundtrorno. A(ntro on6 outd ro un6trorno, to 1, rono0rrodgr dllo 6vo Oror ro rdvoull€ 10, Kl enovolopBdvoup€ rrl 6to6rroo(o on6 qv opXrj.Oordoo, rdOe rpopd nou 6rorporip€ pr ro 3, no(pvoupe trrq ondvrqorlto 3 ur un6}orno ro 1, Kor q 6ro6rroo(o ourr] 6ev retrertirvrr nor€. Eno-pCvrrrg, yro vo ypdrloutrre qv orprp{ 6ero6rrcrj ttprj tou }, 0o npdnet voypdrpoupe gro dnerpq oerpd tpropuirv. lto vq ro onorpriyoupe our6, ro_no0erodpe tperq tele(eg 6e{rd on6 ro retreuto(o 3, yro vo 6e(oupe drrro rprdpro 6e otopotoriv nor6. Enoprivtoq, ] = o,:11... ror rorolopo(_vouye 6rr €vog nenepoop€vog opr0p6g rpropuirv onoretrE( ndvto pronpoodyyrorl tou ]. Ie rdnoro re(pevo, outrl q enovdlqrlr4 <po(veror pepro rouxx(6o ndvo ond ro tetreuro(o 3, 6r;Io6rj pe to 0,33i _ our6 oq_po(ver 6rr enovotrogpdvrror ro rfqrp(o rdrur on6 qv rourr(6o. Aurodrou e(6ouq o 6ero6rrdg optOpdq ovopd(etor dnetpoE ntptoStx6E 6*a6r169 opr0p6q. An6 notrtrd rtrdoporo npor0ntouv dnerpor nepro6rro( 6e_ro6tro( optOpo( ror nollo( ond ouroUq 6Xouv neproodrepo ond 6vonepro6rrd tpnq(o. kerpre(re to rtrdopo fi, ro ono(o pog 6(vet ro 6ero6r_16 opr0g6 0,27272i. Iuv40urq enovologBdvoup€ rov r0xlo tou 27 rpEv'gopdq, otrtrd o opr0p6q rov enovotrrjgeurv e{oprdror qnd epdg. Oopnoporioope onltirq vo ypdrlroupe 0,27.

lro to rldoporo pnopo6pe vo no0pe: rd0e r]dopo pnope( vo ctvo_nopootoOe( e(re oq nenepoop6voq e(te urg dnerpoq nepro6rr6q 6ero6r_169 opr0g6g. Avuotpd<poq, rd0e nenepootrr6vog, ro069 ror rdOe dner_poq nepro6rrdq 6ero6rr6q opr0pdq, gnope( vo ovonopooroOe( urg rtrd_ogo. Aev e(vor 60orotro vo [ovoyup(ooupe ond ro 6ero6rr6 opr0p6 otortrdopo. Oo ooXolq0oripe npdrro p€ rous nenepoop€voug 6ero6trodgoprOpodq. Aq unoO€ooupe 6tt o 6ero6rr6q poq opr0p6g e(vor o 1,028.Merpdpe tov optOpd rurv qrrlq(ov 6e{rd ond rrlv uno6tooro}r1. To qr1_rp(o efvor tp(o. fpdrpougr rtipo rov 6ero6rxd og oprOprlrri u€ nopovo- 247

Page 234: Ο Ταξιδευτής των μαθηματικών

poorq ro 1. l-lotrtrontrqord(ou;re t6oo rov opr0pqrrl doo ror rov Tropo-vopoorr] pe ro 10r. O ex0€rqq rou 10 e(vot ro 3, yror( perprloope rpioQqrp(o 6eltd qg uno6roorotrqq. Enopdvoq, iXoupe:

1,028 = 1,02811= (1,028) '(1000)i1.1000 = rLff i8'EXoupe, trotn6v, €vo rldogo ror prnoporjge vo BpoUpe to ovdyoy6

rou ontronordrvroq rouq rotvo0q nopdyovreg ond rov oprOprlrq Kor rovnopovopooul, 64tro6r1 l*63 - #. Tdrpo [dpoupe drt o 6ero6rr6g 1,028toodror ge ro rtrdogo ffi.

H petorpon{ rcrrv dnerpurv neprodrrtiv 6ero6rr6v oprOpdrv oe rtrd-oporo e(vor tr(yo nro 66oro111, otrtrd e(vor e((oou oltdnroq. Aq ro Trpo-ono0rjooupe pe rov nepro6rx6 6erq6rr6 opr0p6 0,3333i. Merpdpe nprir-ro rov opr0p6 rov nepro6rrtiv rlrrlrp(urv, flou orrlv nep(nrooq our{ efvorp6vo 6vo. flotrtrontroord(ouge rtirpo ro 6ero6rr6 pog optOpd pe ro 10'.Av unqpXov 6do nepro6rrd Qqrp(o, 0o Xpqorponororiooge ro 2 oq rr0€-rq, 0o nollontroord(ope 6qlo6rj pre ro 't0'. llotrtrontrootd(ovroq ro0,3333i pe ro 10i no(pvoupre 3,3333i. lloporqpqore 6rr rpordpe rov(6ro optOpd rproprdrv 6e(rd rrlg uno6roorotrrlg. Aur6 gnopoUpe vo roxovoupe, yror( €Xouge ndvro odnerpo ntrrjOogo rpropuirv npog ro 6e(rd.Av orporpdooulls rov opXtrd 6ero6rr6 poq opt0p6 on6 our6v ro F€yo-trUrepo 6ero6116 opr0pr6, npoo6[re rr oupBo(ver:

3 ,33333u r l J ) J J

3,00000

'Otrq to nepro6rrd rlrrlrp(o <periyouv ror prEvoup€ F€ rov nenepoop€vo6ero6rr6 opr0p6 3,0. Tripo ro 3,0 rooriror ge 6€ro ond rouq opXrrorjg6exo6tro0g oprOpofg pre(ov €vov on6 outo6q. Enop€vtrrg, 66ro rpopdq o6ero6rrdg opr0p6g pe(ov p(o <popd o 6ero6rr6g opr0p6g = 3,0. Aur6oqgo(vst 6n o opXrrc6g 6ero6rrdg optOpdg - + ri 5, 6r1lo6rj our6 urpr-ptirg nou neprg€voge.

Aur4 qv npd{4 gnopofpe vo rrlv xdvoupe ge onorov6rlnore dnet_24& po nrpro6rrd 6ero6rr6 opr0p6, yro vo rov petorp€rlroupr o€ rldogo.

Page 235: Ο Ταξιδευτής των μαθηματικών

I T O M H T O Y N T E N T E K I N T

Bepo(rrrq, pnopei vo 6topoptuprl0e(te 6rt rdtt nopd{evo ougpo(vet

drov unooqp(oupe noq, ov notrtrontrootdoouge Fs ro 10 KdTlotov

dnerpo nepto6rx6 6ero6tr6 opr0p6, out6 getortve( 6tro to rprlrp(o gto

O€oq npog to optorepd (nou e(vot oov vo tr€ge 6u €rot perortve(rot 11

uno6rootolrj pro 0dor1 npos To 6e{td). fltiq [€poupe 6rt out6 o6r1ye(

oto enr0uprpd onor6treogo; Aur6 ro epdrrqgo o6qye( oe yto nopd(evrl

r6ropoprp(o rou 6ero6trof ouor{potoq, nou rpo(vetot vo Koronord rrlv

rorvrj loyrrri. Mnoporlpt vo 0iooupe to efiq eptirrrlpo: l oxptptiq e(-

vor o opr0p6q 0,9999i; E(vot o opr0g6g ourdq onldrg €vo onetpotldyt-

oro noo6 prrpdrepo ond rov opt0g6 1; Irqv npoygortr6tqro, ro

0,ggggi roo0ror oxptBtiq pe 1! Av orotrouOt]ooupe p6Xpt r€troug rq 6ro-

6rroo(o rrlq p€torponrjg rou 0,99999 oe rtrdopto, pnoporipe vq ro 6oti-

pe vo ouprBo(ver. ilotrtrontrootd(oupe ro 6ero6tr6 opt0p6 Lt€ ro 10 Kor

ogorpodpe tov opXrr6 6ero6116 opt0p6.

9,99999o,gs999

9,00000

'EXoupe lorn6v 6€xo <popdg ro 6ero6tr6 opr0g6 pe(ov g(o rpopd ro

6ero6rr6 opr0p6 = 9 rl, otr^t6q, o opXrr6q 6ero6rr6g optOpdq too6rot ge

! = t. Oot6oo, €Xoupe orrlptpe( oqv t6€o 6tt pnopofpe vo noltrontro-

ordooupe €vov dnerpo 6ero6txd opt0pt6 Kot vo F€roKtvrioouge ro Un-rp(o npoq ro oprorepd. Mnoporipe vo 6odpe drt out6 e(vot 6vo 0eptr6prlpo ov npoorp0youpe oro dpto. Ag ypd{toupe to 6ero6tr6 opt0p6

0,99999 urg dnerpq oerpd, 6qtro6r1:

O,ggggg = * * *a * r*ao * ..'

Tdrpq ov rdnoroq unooqpt(e 6\ o o,ggggg e(vor prrp6repog ond rov

opr0p6 1, r6te 0o np€ner vo e(vot prp6repog rord €vo ouyrerpq€vo nood,

og nofpe to e. Altr4 onoto6qnote ttpr] rr ov en€leye Yto ro €, gnopo0ge vo

npoo0€oouge oprero0g 6poug oro 6ero6tr6 ovdmuygo tov 0,99999 €rot

tiorc vo rprdooupe nro rowd oto 1 on6 to e. Enog€vtrtg, ro I np€rut vo efuot

to 6pto rou 6ero6moU ovorn0ygorog tou 0,99999. 249

Page 236: Ο Ταξιδευτής των μαθηματικών

MdXpr rtirpo, pe 6tro ro nopondvur, €Xoupe nopoBl€rper rde(og rougdpp4rouq oprOporjg. floro, 0o pnoporioope vo ovoporq0o6pe, e(vot ro6ero6rr6 ovdnruypo ev6g dpprltou oprOgori; E(vor dvoq dnrrpoq, pr1nepto6rr6g opr0;r69. Enopdvoq, €vog dppqrog oprOpdq 6nog o {i tyrlevo dnetpo, pq nrpro6rr6 6ero6rr6 ovdnruypro. t"ldrg 0o pnoporioopenor€ vq ovonopoor{ooup€ F€ orp(peto €vuv r€toro oprOpd, drov 6Evpnopoupe vo Xprlorporou]ooupe o0re rtrdoporo odte 6ero6rro6g opr0-podq; Irqv npoyporrr6rqro, 6ev pnopotjpe. E(orr(og ourorj orptBdrgrou XopoKrnprourof rouq, rdnoror dv0ponor 0eulporiv svoXtr4rtro0grouq dpprltouq oprOpodq. E(vot pnep6eg€vol, 6ev e(vot rorronorrlg€voroUre olortr4po;r€vor. 0o16oo, on6 r4v dM4, e(vor ouvopnoolrd 6to-rpoperrro( pe ro .6ero6rrd ovonrriypord roug vo orolou0o6v 6or6qtr6_6erg 6ro6popdg, XtrLp(q vo rorotrrjyouv nor€ o, 6vo ovropd, nepro6rr6por(po. Mdhoro, rdnorot poOrlporrro( ooyotroUvror egp6vurg pe ro 6e_ro6rrd ovonr0yporo rov ouv40top€vov dpp4rurv oprOptirv.

O tr6yog rrlg nepup6perog evdg rdrtrou npog rr1 6rdperpd rou e(vot 4 lpqrou n. En(or1g, qno6errvfetor 6rt ro n r(vor dppqrog opr0p6g. To nptirro 10.1{qrp(o rou 6ero6rroU ovonr6ypordg rou 6(vovror nopordto:

3,1 41 5926s3 s897 9323846264338327 9 5028841 97 1 6939937 51 o582097 49 4 4592307 81 640628620B99862803 48253 421 1 7 067 9...

llop' 6tro rou or poOrlgorrro( yvulp(ouv 6u ro por(Bo rou n 6e 0oy(ver nor€ dnerpo ror nepro6rr6, 6cv pnoporJv vo plv ovoporrodvrorprlnog undpXouv rdnolo dltro e(6r1 por(purv rcpugp6vo oro pdOog touovomUyporog rou. fro nopd6erypo, o dppqroq 6ero6rrdg oprOp6q nouovopd(eror oopt0p6g rutv opr0prdrvo (number number) ror oXqprorf(eroton6 6to6oXtro0q <puorrodg opr0poUg e(vqr o 0,j2f456789101j1213...Arpou our6q o opr0p6g e(vor dppqrog, 6sv ovonrfooer rov6vo nepro6r_16 porfpo otrld, oo16oo, 11 p€0o6og endrtoorlg rou ovom6yporog e(vorolorpdvsprl: roroypdrfre ontr6g rov endpevo rpuorr6 opr0;r6 rr our6gonoretre( ro en6pevo odvolo {rr1rp(ov. Tr ouppo(ver 6pog pr ro n; Mno-po6pe vo BpoUge rdnoro por(Bo nou 0o (ertrer6river ro guorrrd rou;

€S* MdXpr ro rp0rv6ntr;po rou 1989, brdrpopeq o;rd6eg poO4pourdrv etXov

Page 237: Ο Ταξιδευτής των μαθηματικών

p4orgonoujoer unepunotroyrotdg yto tov unoloytopd neproo6repovond 6vo 6roeroropp0pro rfqrp(ov rou rE, orrlv npoondOerd rouq voppouv rdtr onp6opevo, ror pdXpr r6re etXov onorfXer.'' Evrodtorq 6evro nopdtqoov rr o oytivoq 6pdpou ouveX(eror pdXpr orlpepo, yto vo

6ouv noro opd6o pnops( vo nopiryer prlXovrrd ro peyolfrepo opr0p6t|l11q(trtv.

kerpr6pevor rouq dnerpoug, p4 nepro6rxo0g 6ero6rro6g oprOpodq,

0o pnoporioops vo ovoporrl0oripe n6oo ouyvd ep<pov((ovrot 6to<pope-rud g4rp(o oro 6ero6rrd ovorudypord rouq. Me dtrtro tr6yto, undpXouvdpoye opr0go( nou, ov ;rnopotiooge vo unotroy(oougs rq ovotroy(eqrov y46evrr6v, rov povd6r.,-rv, rcuv 6uopuiv, ror o0ro roOeffq, 0o pp(-

oroge roOivo on6 ro 66ro oprOprprd rfqrp(o vo egrpovt(etot o€ nooo-

016 10o/o tov neprnttiroeu-rv; H ondvr4orl e(vot vor, rot ot oprOyo( ouro(ovoprd(ovrot ravovrro( ap0po(. 'Evog opr0p6q nou ro 6ero6rr6 ovd-

muypd rou nepr€Xer or (oeg ovotroy(eg dtro ro Qq<p(o rdOe Bdorlq ovo-pd(erot ano)iruq ravowr6E ap0p6; flop' 6tro rou ot po0qgortro(yvop((ouv drr 11 peydlrl ntrerorlrlrp(o 6trov rov optOptitv e(vot onoldrogrovovrro( opr0;roi 6ev pnopodv vo etr€yfouv ouyrerptg€voug optOpotjgyro vo 6ouv ov e(vot rovovrro(. Enop€vu-rg, 6ev [€poupe ov to n e(votrovovrr6g crpr0pr6g. To nopd(evo E(vor drt ere(vot ot opt0po( TIou ouvo-vrdpe orrlv ro0qpeprv( FoS (on t(vot oXs66v ndwo pqto( opt0po( rotoye66v rov€voq ond ourodg 6ev e(vot rovovtrdq, ev6 q ouvrptnrtrqnkrorfrrlrp(o 6trrrrv rov npoyprottrurv opt0ptitv e(vot rovovtro( optOpo(.'Evo nopd6erypo ev6g rovovrrof opt0po0, nou e(vor ror prpdg, e(vot oncpro61169 6erobtr6q oprop6g o,o1 234567 8901 234567 8901 2345..., orovono(o enovqloppdveror ro rd0e opt0prlrtr6 {r1rp(o pto <popd oe rd0errirlo.

fkig orptptirg unotroy((oupe rouq dpprlroug oprOptofg; Irqv npoy;.to-rtrdtrpo, 6Xoupe rj6r1 npoerortrroore( yr' out6 drov po0o(vope yto rtqouyrtr(vouoeg oetpdg. fro vo unotroy(ooupe ro 6ero6trd ovonrUyporo

13. Richard Preston, "Profi les: The Mountains of Pio,1992), o.36.

The New Yorker (2 Mopt(ou gg,E

Page 238: Ο Ταξιδευτής των μαθηματικών

rov dpp4rov opr0ptirv, ontrriq unoloyr(oup€ touq 6ro6oXrro0q dpouqgtoq dnerp4q oerpdq. fro nop<i6erypo, yro vo unoloy(ooupr ro y'7pno-podgr vo Xprlorponot{ooups rrlv ordtrouOrl oerpd:

{ t = , , 1 1 3 3 . 5 3 . 5 . 72 2.4 2.4.6 2.4.6.8 2.4.6.8. ' tO

Aur{ q orrpd rpo(veror pnep6ep6vq, oordoo e(vqr edxotro vo ulvunoloy(ooupe, yror( pnopodpe vo {ovoypdrfoups rrl oerpd itor tiorerdOe 6ro6oXrr6q dpoq vq e(vor onl6q o nporlyoUpevo( 6poq nollontro_orootrrEvog trr'6vo yvrioro rtrdopo.

^[7 = t * + - (+)(+) * (*x;) - (*x;) * (;*x#) _...Mnopodpe, Iorn6v, vo unotroy(ooupc ro 6ero6rr6 ovdmuypo rou

VTror vo 6oUtrre ntirg rdOe dpog pdq rpdpvet nro Kovrd oq ourorrj uprjrov €(t O€oeov rou 1,41421. O Il(voroq 5 6erXver rouq np6toug 6rron€_

l

fIINAKAI 5. tlpooeyy(ovroq ro y'7

'opos Trsri 6pou ot;;o ooou,

I

2

3

4

5

6

7

8a

1 0

1 1

1 2

1 3

1 4

1 5

+1,00000

+0,50000-o,1 2500+0,06250-o,03906

+0,02734-o,02051

+0,01 6l 1-0,01 309+0,01 091

4,00927+0,0080i-o,00701

+0,00620-o,00554

r,00000

1,50000

1,37s00

1 ,437 501,39844

1,42578

1,40527

1,42138

1,40829

1,41920

1,40993

1,41794

1,41093

1,41713

1,41159252

Page 239: Ο Ταξιδευτής των μαθηματικών

H T O M H T O Y N T E N T E K I N T

i

II

we 6pouq rou ovont0vporos' 'Enetto on6 6exon

]tiillll,:itpW € O p O U S L U U u v u f r l u l F s l v ) i - . ' - . r ^

fiet rprdoer oe orp(pero rpttiv odoeutv: *of€"T: -n-l.f:l]::::'.^::

rffi;il i."i'*"'t' BeBo(trrs' €vos.unotroYtmT 111fll:i::-.nri*,n,ii's unotrovtosouq ro 6eurep:ntl-o: *'

I lf:1]::1i:"';;ffi,'il;;il ;' ", o "vxp *n .'x' "roYll- : :" ::iilll, lil?Tlilitt t;J, r,o uo' o'v *'trtuouu vpqvo p6re po

:l "'T::".1"1.':::ffi,-.fi;;; ;n; ";tilortr6trpo nro notrontroreq ouvoptrloerq on6

irs onl( oetp€g nou nopouotd(oupe e6tit'

Ilopouot6oope npoqyoug€voq Uto oetpd nou ;rnope( vq Xpnotgo-

notrl0e( yto qv npoo€yytorl lou 7r'

n = 4 ' ( 1 - i * i - i * i - " ' )

ot npdrtot 6exon€vre 6por ourr]g qq oerpdq roroypdrpovtot olov

fl(voro 6' Me rov (6ro opr0g6 6purv 6nrrrq Kot rl nporlYo0pevq oetfl-d'

owi 11 oerpd pdg 6(vet g6vo ro npdrto $ryp(o tou n' ouyxtr(ver 611lo611

flINAKAI6' IlPooeYY((ovtclq ro 7r

Opoq TtPrl 6Pou AOPotoPo oetPog

1

3

4(

6

7

8

9

1 0'11

1 2

1 3

1 4

1 5

+4,0000-1 ,3333

+0,8000

4,5714+0,4444-o,3636

+0,3077

4,2667+0,2353-0,21 05+0,1 905

4,1719+0,1 600-o,1 481+0,1379

4,0000

2,6667

1,4667

2,8953

3,3397

2,9761

3,2838

3,O171a ) \ ? 4

3,0419

3,2324

3,0585

3,218s

3,0704

3,2083 253

Page 240: Ο Ταξιδευτής των μαθηματικών

K E O A A A I O 9

notr6 nro opyd on6 u1 oerpd nou Xprlorporou]0r1re yro ulv npoo€yytorlrcu'r[7.

Ard;.ro ror or p4tof oprOpo( e(vor oprorig rrpdg yro uq oerp€g, yrot(o ApXrp46rlq ovordtrurpe r4v nopordtur oerpd yto ro {:

i = * * # * * * * + . . .'Evo ouveX€q rldopo e(vor pro er6rrrl dnerprl oerpd nou onordeftor

on6 €vov ordpoto ouv dvo rtrdopo, Kor o [opovopooulg rou rtrdogo_rog e(vot €vog dltroq oriporog ouv €vo rtrdopo rou (6tou e(6ouq. Av torldopo ouveX((eror Xtrrp(q vo otoporder, ovoprd(eror pq nntpaopivoouveyiE x)dopa.'Evo opo(o pn n€n€poop€vo ouveXdq rtrdogo e(vor ronooordtctr:

irt + rfo\

| | J , '

_ = l f

I

1 + _ _ -1

1 * 1 * -

BeBo(urg, 11 rrprj rou oprorepori oritroug rrlg e{(otrto4g e(vor o un€po-Xog Xpuo6g M6oog poq. E(vor yvoor6 6u rd0e dppqroq opr0p6g pno_pe( vo ovonopooro0e( u.rg rivo pn Tr€n€pooFEvo ouveXdq rldopro. Eno_pevog, yro rov unotroyropd Tlq upr]g rd0e dppqrou oprOpod undpXaoq 6td0eorl pog €vo ouveX€g r<trdopo.

MdXpr rdrpo dXouge roldper nolld 06poro orqv npoond0etd pog vororovor]oou1_r€ rous dpprpoug opr0porjq. ,EXoutrre rov Kopq,d oprotrr6rou PtXopvr Nrdvrerrvr, nou ronoOere( rouq dppqrouq oprOpouq oe groyep{ 0eurp4trrrl Bdoq. :ipoupe 6tt e(vqr 6roreroypr€vor, 6ncuq orprpdrgror or pqrof opr0por, rr 6u ro o0votro ruJV rpoyporrrdrv oprOpc0v efvorrtreror6 yro 6}eg rtg npci(erg, err6g ond 14 6ro(peor1 pe ro pr166v. Mno_po6pe vo XerproroUpe roug opprlroug opr0podg 6noq ror toug prpofg.

' Ontog onoDrrrvderor, dXouge sppqverioer en(oqg 6lo to oqpe(o FrqS su_954 Oe(og, yror( or rpoygorrKo( optOpo( ovrrororxouv oprprpovoo4povro or

Page 241: Ο Ταξιδευτής των μαθηματικών

H T O M H T O Y N T E N T E K I N T

ro orlpe(o qg eu0e(og rr-uv opr0ptitv. Tr neproo6tepo 0o gnopofou-

vo 06troupe vo gdOoupe;'Evo epdtrlpo ngu onorp0yope e(vor ro n6oor dppqror oprOpo(

. fvrrrpi(oupre 6rr undpXouv dnetpot, olld undpXouv (n€pto-

odtepor,' ri utrry6tepor" dpp4ror opr0po( on6 pqrofq; 'EXet v6qpo vo 1it-trdpe yro neproodtepouq rj hydtepouq, dtov t6oo ot dpprlror doo rot ot

:p4to( oprOgo( e(vor dnerpor; Ttitpo e(poore 6tot1.tot vo 6tepeuvrjooupe:uurd ro epdrrqpo.

255

Page 242: Ο Ταξιδευτής των μαθηματικών

Mdlpr rtipo fXouge e{erdoer trq noluovuprr€g efrotioers rnq popqtj5

os{u * o,xut * ... * ou-rX + ou = 0, 6nou to oo 6tr)q o" e(vqt ouwetreot€qttrrv 6uvdgcrrlv rou x. Ovopd(oupe out€q ttq e(totioetq xavowrd noAutit-vupo.'Oyug, n€po on6 rqv notrutrtvuptr{ poprp{, undpXouv rt d}r}ot

rpdnor oXqpotropo0 e(rotioeov ylo rrlv enfiuor1 npoptrlpdrov. flto ou-yrerprp€vo, OCtrouge vo petrer{ooupe dltro rp(o e(6q e(totioeov: rt( EK-

0errdq, uq troyoprOprr€q ror lS rpryovolletprrfq e{rotioetq. Autotj tou

e(6oug or e{rodroerq e(vor Boorr€g ylo rrlv enitruorl tov npoB}qpdrcuvqq enoXrjq poq, on6 q 6r0hoq rou netpetro(ou piXpt qv ert6[euoq6rooqprrtirv treoqope(ov.

Or er0errr6q e(rotioerq nepilopprivouv oyvtiorouq orous erO€teg tutv

oprOprirv. Mro tftoro ef(otrroq 0o gnopo0or vo e(vor q efiS: 12^ = 144.

E6tir yperd(etor vo yvop(ooul.r€ o€ noro 60votrrr1 np€net vo utlttiloou;re to12 yro vo ndpoupe ro 144. Efrolo pp(oroupe du q on<ivrrloq e(vot to 2.'Etot, 122 = 144. 'Oprrrq, oe dMeg er0eur€q e(rotiroerq 11 ondvqoq 6ev e(-vor t6oo onl{. l-ro nopd6erypo, 1,0994'="17,32. Tr e(vor ro x; E(vot to xpqrdq n dpprpog oprOpdq;

Oo Oupdote nr0ov6v rouq loyopiOpouq on6 qv dlyeppo oro oxo-tre(o. E(vor or erO€req rtrrv opr0ptitv. fro nopd6etyJro, o loydpr0poq rou 257

Page 243: Ο Ταξιδευτής των μαθηματικών

100 pe pdoq ro 10 roorjror pe 2. Aurd or1trro(ver du ro 10 np€ner vo d;1erer06tr1 ro 2 yro vo poq 6riroer ro 100. Me o0ppotro, out6 ypdqetor rrrqfog,ol00 = 2. Or loydpr0por pe pdorl ro 10 ovopd(ovru 6ero6Koi troyd-prOpor ror ouXvd nopotre(noupe vo yptiqoupe ro 10. 'EXouFe, enop6-voq, log100 = 2.'Evo opo(o loporrqprotrxd trrrv troyop(O;.ulv e(vor dn

trrnoporitrre vo notrtronloordooupe 6rio oprOtrro0q pero[ri roug npoo06ro-vros rous troyopiOpoug touq ldyrrr rrlq rourdrqroq AY.AU = 416. fro no-pdDerypq dXoupe 3t.3t - 3'* = 3i. Kord ouvinrro, 6rov €1outrre €vo 6ri-orotro tj pnep6etrrCvo npdplrlpo notrtrontroorootrrori, trrnoporipe oqlvd voro perotpirlroup€ o' dvo edrotro np6Blrlpo npdo0eorlq troyop€ptrtv.Mnopodpr en(or1g vo enilfooutrre oprop€vo npoptrrlpoto nou exrppri(o-vrot F€ loyopr0purrj popgri. fto nopd6erypo, 0o ;.rnopo0oo;.re vo 6Xou-pe logx = 2 Kor vo 0€troupe vo pdOoupe noro e(vor ro x. Ie ouu] r4v ne-p(rnr.oor1, pp(oxoupe erjrolo 6rr e(vor ro 100. Or troydpr0por e(vor €voq

Bohxdq rpdnoE npoo€yyroqq tov npoBtrqpdrtrrv pe erO€req.Or rpryulvopetprx€q e{rotiroerq otqp(ovtot onq oX€oerq ovdpeoo mo

pfxq tov ntreupdrv ev6q op0oyurv(ou rprytirvou Kol ro pey€Oq rov yu,-vuirv rou. H Err6vo 41 6€(XV€r 6vo op0oytirvro tpiyrrrvo, q p(o yurv(o rouono(ou e(vor q 0 xor or tperq ntreupCq rou XopoKrqp(,(owor og unote(-vouoo, npoore(pevq rdOeroq xor ondvovn rd0mog. I1otrtr6q rptyovoll€-rprx€q ouvopr{oerq opi(ovror on6 trq oXioerq ovdpeoo oul 0 ror trqtperq nleup€q rou tpryrirvou, otrtrd e6tir 0o petreu]ooupe p6vo tq ouvdp-rrlorl rou rlprt6vou. To r1p(rovo proq yov(oq op(etor oq o tr6yoq {6on€vovu ntreupdq Trpoq rrlv unore(vouoo. Aurd ypdqeror oq: 4p0 =

ondvovu rd0etoq/unott(vouoo. Enop€vo6, to qgO, 6noq orprBtirq Kor olog100, e(vor dvoq opr0p6q. To qp30' = 0,5, ro ono(o trier dtr 6tqv ro 0€(vor 30o, 11 unore(vouoo €Xer 6rntrdoro prlroq on6 qv on€vovl rd0e-ro, Kor o tr6yoq touq e(vor: on€vovtr rdOetoq/unore(vouoo = 0,5.

Tt e(6ouq oprOpoOq noipvoupe drov Bd(oupe 6rorpopetrrCq ug€qorouq erO€teq, orous troyop(0pouq Kor ouq ouvopr{oerq rou qprtdvou;'Onoq €Xoupe 6er nopondvo, or Kdilor€g neprnrtiroerq no(pvoupe rpuor-rofg oprOgorjq { rtrdoporo - 6qlo6ri, pqro0g opr0po0q. Mnopodge no-t6 vo ndpoulle dppqrouE opr0poUq; Iqv npoyporix6rrlro, or neproo6-2s8

Page 244: Ο Ταξιδευτής των μαθηματικών

H I f T O P I A T O Y f i

onfvowr rtiOerog

npoore(yev4 rdOerog

EIKONA 41. Or tpryovogetprrCg o1€oerg neprtro;.rpdvouv trq 6rdgopeq olfoerq ovd-

F€oo o€ trro yov(o, tq 0, ev6g op0oyov(ou tprytirvou ror ro prirq rov tpuiv ntreu-prirv. H oXdoq rou rlprt6vou op(eror urq: r1p0 = on€vovtr rdOetoq/uuoteivouoo.

upeq lrjoerq oe ourorj rou e(6ouq rrq e(rotiroerg e(vor dppqtor oprOgo(.fto nopd6erypo, ov ro 0 e(vor ivoq prp6q opr0trr6g peto{0 0 ror 90o,rdu ro 4p0 e(vor 6voq dpprlroq oprOgdq, err6q edv 0 = 30'.

l- lpoo6ropi(ovrog ro o n poodr6proro e

H oXCoq ovdpeoo or4 6rdgerpo ev6q rdrtrou Kot rrlv neprqipeui tou e(-vor yvoor{ ond tqv opXor6tqro ror ovonop(ororot F€ to odpBotro n.O EtrBer6q goOqpourdg Adovopvr 'Orlep (1 707-1783) {rov our6q noudpXroe vo Xpqorponore( ro eltrr1vrr6 ypdppo nr yr' ourlv rq oXdorl (Er-r6vo 42).

'Hrov en(oqg €voq ond rous firo nopoyuryrroriq poOqptotrro0q noud(r1oov noti' ro €pyo tou (prpl(o Kot €nrorqpovrriq 6rltrroore0oerq) {e-nepvotiv to nevror6oro, rvrit nopqyoye oprerd poOrlporrrd yro vo y€-pioouv evevrjvto rdpouq.' O'Ortrep 6proe en(oqg q Xprlorl rou ypdpgo-roq e, Trou oupBotrr(er pro dllq 0egetruir64 oX6oq rtrrv poOrlpourdrv:

l. Hol l ingdale, o. 275 259

Page 245: Ο Ταξιδευτής των μαθηματικών

K E O A A A I O I O

EIKONA 42. A€ovopvr 'OtiEp, 1707-1783.(Yhr6 on6 rouq o6etrrpo6q Mnpdouv, tr€ptrrvyx)

. = l ' g ( r .+ ) "

Enopdvr,rg, to e e(vor ro 6pro nou no(pvoupre 6rov orprlvoupe ro vvo y(vetor 6tro rot peyotrOrepo. Me dtrtro }dyro, pnopoUge vo oXrlpor(-ooup€ Uro orolou0(o opr0gtirv pe pdorl rov r0no (1 + 1/v)", dnou ro vopxftr und ro 1 ror ou{dveror no(pvovroq rqv rrprj rdOr 6ro6oXrro0rpuorrof opt0po6. Aurry 11 orotrou0(o oprOpdrv 0o ntrqord(er dlo ror ne-260

Page 246: Ο Ταξιδευτής των μαθηματικών

H I f T O P I A T O Y f i

puodrepo oqv olrlOrvri uprj rou e. To e, enopdvoq, e(vor ro 6pto r4qorol\ou0(oq poq Kor ourd ro 6e(Xvoupre oq e(rjq:

. = l ' 111 r * f ) "'Onog orprptirs Kor mrlv nep(nrooq rou 7r, pnopo0pe vo fipoory-

yiooupe ro e {errvtivroq Fs ro v = 1 xor trrivovroq perd to (t + i/v)" rrlgrrpos To peyotr0repo v (ll(voroq Z).

Av ouveX(ooupe ouujv rq 6ro6rroo(q 0o rprtioouye oqv or6trou04 u-pti pe npoo€yyroq 6iro 6erq6rr6v 0ioetov: e = 2,7i82818284... M€ our6rov opr0p6 ouoXet(etor pro opo(o oerpd, 11 ono(o ovqrotrdgrqre on6 rovNeUrovo (1642-1727) ro 't665, drov {tov g6vo e(roor rpuiv er6v. Oo 0u-gdme dl 11 oerpd e(vor 6vo d0poropo 6pov rr dXr gro orotrou0(o 6prrrv.flprv nopouordoouge ouui rrlv unipoX4 oerpri, 0o np€ner vo op(ooupe prorbrofuepq origpoorl fiou Xprlorgonore(tor ouXvd mo goOqpolxd. Av rotrtro-ntroordooupe rouq oprOpods I Kot 2 eXoupe i.2 = 2. Av noltronloouioouperous rperq nptirtouq opr0po0q no(pvouge 1.2.3 = 6, ror o noMonlooroopdqruv uoodprrtv nptirr.uv opr0ptirv pdg 6(ver 1.2.3.4 = 24. 'Evog oriwopoqtpdnoq ypoqriq ourtiv tov notrIontroorool.uirv e(vor ro o0pBotro rou nopo-yow6o0,611lo6{ ro ..!u. Enop€vurq, 1.2 = 2! = 2 ror 1.2.3 = jl = 6, evrh1.2'3.4 = 4l -- 24. Mnope(re vo 6e(re 6tr or 6rdgop€S Tropoyovnr€g ug€qou{dvowor yptiyopo. fro nopri6erygo, ro i0! = 3.628.800. Iro go0qponrd

nINAKAI Z. llpooeyy((ovroq ro e

v (1 + 1/v)" flpoo6yytoq tou e

I

4

1r * i) '1r * i)'1r * 1)'1r * *)o

1r * i)'

2,000

2,250

2,370

2,441

2,488261

Page 247: Ο Ταξιδευτής των μαθηματικών

ouvowdpe t6oo ou;1v<i uS nopqyovurdq ug€q, ticne e(vor lprJorpo vosxoup€ ouniv u1 ouvropoypoqmi odgpoorl yr' ou*q.

H oerpd nou qvordlurfe o Ne0turv yro ro e e(vor 11 or6lou0q:

e = 1 * l * ] * a * 1 _ . . .0 ! 1 ! 2 1 3 ! 4 l

To o0prpotro 0! roorjtor e[opropod pe t. H nopondvur oerpd pnope(vo ypogeios: e = 1 + 1 + ** ** **. . . *o, pnope(vo oupBotrrote( 'qe = L(1/v!), dnou to v no(pver trg updq ordpourrv oprOptirv fexrvtirwoqon6 ro 0. Oo Ouprime 6u ro etrtrr1vrr6 rerpotro(o ypdppo orypo (!) oq-po(ver 6u npoo0Ctoupe 6louq touq dpouq rqq oerpdq pog.

O trdyoq ylo rov ono(o or poOrlpotrro( ror or enroujpoveg ev6rorp€po-vrol yro ro e e(vor 6rt eprpovi(etor {oqvrrd oe nol)\d 6rogopeurri oq-pe(o, 6tov npoonoOo0pe vo enil0ooupe npoptrrjpoto. Aut6 o;eil,(etor peqv rrrpo(o r6rdqro 6tr, yro prrpiq up€q tou x, fioupe e" = I + x. E6drdyoupe lpqorponouioer ro er6rrd odppolo tqq npoo6yylorls, ro -, yrovo 6e(ouge 6u or 6uo nleup€q rqg e{ioorqq €Xouv trp€q nep(nou (oegol\Id 6Xr onopofuqto (oeq. Mnopofpe vo 6o0trre dtr q nopondvur oXdorlrq16er ov ennp€rpouge oro x vo e(vor (oo pe to l/v, 6nou to v e(vor pe-ydlo (e'/'= 1+"tlv). YQdvouF€ rt0po rd0e ordtrog oq vrom{ 6dvoprlxor no(pvoup€ (e"")" = (l + l/v)" { e = (1 + l/v)", to onolo e(vot orpr-prirg o oprop6g tou opiou, 6tov ro v ou{ov6pevo rqyoiver oro dnerpo.

Tr orqv euX{ €Xer vo rdver ro e [r€ u1v Cpeuvd Foq yto roug oprO-po0q; 'Orov or trro0qpotrro( dpXroov vo ovopu,uodvrur u ei6ouq opr0-po( orprptirg r]tov ro e Kor ro n, ovordlurlov Fro evtetr6q v6o rldorlopr0prirv.

E(erd(ovroq ro ?[ Kcn ro e

Av floupe pro noluuxu;rml e{tuooq npdtou Bo0po( Oqlobri qv ox + B = 06nou to o ror p e(vor quorro( opr0poi r6re or hioerq qq oq npoq x e(vorndwo prpo( oprOpoi orporJ x = -9/o. Iqv npoypotrr6ulto, 6tror or prpo(

1,l

262

Page 248: Ο Ταξιδευτής των μαθηματικών

l l l fTOPlA TOY n

,optOpo( e(vot p(@q rdnotoq nptrnopd0prog notruovugrniq e[(orrrorlq. 'E1ou-

p 6er dtr to nolutitvupo ovtirepou poOpo0 pnoporiv vo €Xouv tdoo pqt6q6oo ror dpp4teq }doerq. Trg trdoerq ruv noluov0Fov fiou or ouvretrem€grouq e(vor or€poror opr0po( trg opi(oupe wc, aAy$pworlq apr0go06. l-ro voe(poore oxpQe(q, pnopofpe vo nodpe 6n 6treq or lfoerq rov notrurrrvripurvpe pqto0q ouvrelem€q e(vor olyepprroi oprOgol Autd ougpo(ver 6rdrt yrord0e notrutovutruni e((otrroq pe ouwetremdq rtrdogoto (pqro0q opr0trroriq)pnopodpe vc pporipe €vov or€poro pre tov ono(o vo noMontroordoouper0r ro 600 orelq qq e(oooqq, perotp€nowdq qv fror oe notrurirvutrro peordporouq ouweleordq. fro nopd6erypq to ]x + + = 0 pnope( vq p€ro-tponef oe notrurirvullo ge ordporouq ouvrelem€q noMonloord(owoq Kolu 611o or€trr1 u1g e(oooqq F€ ro 6, 611lo6rl: ro 5.(*x + !1= e.o y(vetor5'(|x) + 6-(+) -0 rj 3x + 4 : o.

Kor or 6do e(rotioerq, n *x + i = o ror rl 3x + 4 = 0, €Xouv tqv 6ro

16o11, qv x = -$. Enop€voq, ;.rnopo0pe vo ndpouge onoro6{note no-lutrtvuprxrj e((otooq pe prlto6q ouwetreot€q (rtrdopoto) Kor vo rrl U€ro:tpdgoupe o€ pro notruurvuprr{ e((ooor1 ge ordporouq ouvtetreotdq, roror lOoerq nopopdvouv (6req. Aurd oqpoiver 6tr or rpuorroi oprO;ro(, torldoporo ror or p[(eg ({v) e(vor 6)\o otryeBprxo( oprOpo(, oqo0 onore-tro0v l\doetq notrutrtv0ptrrv ouro0 rou e(6ouq.

OPIIMOI: AlyeBprrdq opr0pdq e(vor o opr0pdq nou qnotele( 16-orl Froq noluovuprr{q e{ioooqq, 6tror or ouvteleot6q qqono(oq e(vor orCporor oprOpo(.

O 'Oilep {rov o npdroq nou ovoprrrtri0rlxe, to '1748, ov ro e Kor ro

n e(vqr olyepprro( oprOpo(. Ar1lo6rj, pnopodv vo e(vor fuioerq notruovrl-pu,v F€ ordporouq ouvtetremCq; Aur6 to ep6tr1trro €pr trlo r6ro(repq oq-poo(o yro to n. An6 tqv enoX{ ttrrv opXo(trlv Etrl\rivrrrv, 6vo ovondvrqtoep6tqpo orpopofoe to ov e(vor 6uvot6v vo rqtooreudooupe €vo te-tpdyrrtvo pe eppo66v (oo pe out6 ev6q ouyrerprp6vou x0rlou Xprlolpo-nordrwoq p6vo €vo Xdporo n fvo 6roprjtq. To eppo66v tou nixtrou 6i-vetqr on6 tov t6no nf , 6nou r e(vor q qKr(vo tou rrjrtrou. Av [envrl-ooupe g'€vov rdrtro nou q orr(vo rou e(vor (oq pe 1, t6te to eppo6dv 263

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rou rooritor pe ru. lro vo rorooreudooupe 6vo retpdyovo nou ro eppo_6dv rou vo e(vor (oo ge n, npiner vo rorooreu<iooup€ pro eu0e(o nouto p{roq rns vo eivor (oo Ve flr, dtor t6ore 6rov urpo0r( oro rerpdyo_vo vo ntipouge ro eppo6dv nou enrOupoOpe.

Aur6 ro npdplqpo ovopd(eror rcrpayuvtolt6g rcu rldou rr €XouvononerpoOe( vo to l0oouv ovopi0gqtor epoorr€Xveq ror enoyyelpor(eqpoOqpo'ro( ro teleuto(o 60o xilrd6eq xpdvro. Av neproprorori;.re oulXp{o4 ev6q Xtiporo ror evdq 6roBrjt4, t6te or 6rdgopor rp6nor p€ rousono(ouq pnopori;.re vo Xerprororil.re to prjq nou oXeti(owor 1r, ivov rri_rtro neprop(ovror p6vo otov notrtronloorootrr6, orr;v npdoOeorl, orrlvorpo(peo4 Kor orrl 6ro(peoq. Aur€q e(vor or r€ooeprq npd{erq nou Xprlor_ponorouvror oro rovovrrd norurilvupd poq. Enopdvur(, ov ro n onorere(ItJo4 xdnorou ruvovrrod noluovrjpou, rdte 0o np€ner vo pnopodpe vorerpoyrrrvfooupe tov rtir]o, Kor ro ,[ 0o e(vor ivoq otryepprrdq opr0p6g.Me dllo l6yro, ov to n e(vor otryepprrdq opr0p6q, r6re o rUxloq puo_pei vo terpoyovroref. Arogopetrrd 6ev pnope(.

Av to e Kol ro ,r 6ev e(vor olyepprro( oprOpoi r6re tt oprOpo( 0opnoporioov vo e(vor; Or pooqpotrro( iXouv opX(oer vo qnorqlodv toucpr1 olyrBptrorlg oprOporig untpparwoag opr0poOq. oordoo, drov o ,or_trep i0eoe npdrroq ro epdltqpo, 6ev rirov yvurord ofre rov ro ov un{p_Xov ouro( ot oprO;ro(. llop, dtro nou ro ev ldyrrr eprirtqgo t60qre nprirrqgopd to t/48,6ev onowr]0qre p€Xpr to 1844, notr0 perd to 0dvororou 'Oilep. Ere(vr1 14 Xpovrd, o t-dlloq po0qpotrrdq Zo(€rp ArouBii(1809-1S82) 6rlprodpyqoe tov np6ro opr0p6 nou ono6e6eryp€vo ritovunepBotrrdq. O opr0ydq rou {rov o:

L = l * 1 * a * 1 *l o 10' lou , lo2o " '

re 6sro6rxrl popqri ixouge du L = 0,1 1000r00000000000000000100...O Zo(trp Atoupitr pn6proe vo ono6e((er 6rr out6q o oprOpdq 6ev pno_

264 podoe vo onotetre( tr6oq onoroo6{nore notruovugrr{q e{iorrro4q pe or€_

t = l - * i = * 1 . - * 1 * , i10 , , 10 , , 10 , , 10r ,

. . . ,1

Page 250: Ο Ταξιδευτής των μαθηματικών

porouq ouvretreor€q rot, enog€vtrtg, r]rov 6voq on6 touq onpoo6tdpt-

moug uneppourofq opt0po0g. Iq ouvdXeto €6et{e ntirg gnopo6pe vo

6qproupyriooupe dnetpo ntrqOoq unepBottrtiv opr0pdrv pe rrlv nopor6-

rur popgrl:

ol cl2 ol

1 d ' * l d r - ; i r " '

dnou to 6ro<poperrrd o €(vot oripotot opr0po( ond ro 0 €oq ro 9' Au-

ro0 rou e(6oug ot optggo( ovopd(ovtor apfipo( AtouprT. Aur6 ondwrl-

oe oprotrrd oto epdrrrlgo ov npdyporr undpXouv uneppottro( aptO-

po(. Oo16oo, rove(q or6po 6e yvtirpt(e rr t6t6rqteq efiov opr0go( 6nog

To tr Kot to e. 'Hrov unepBortro( ri otryeBprro( opr0go(; Tetrtrd, to

1873, o loptr Epg(t (1822-1901) on66er(e 6rt ro e e(vot unepportr6g

opr0p6g. To 1882, o Odpvttvovr <pov A(vrepov (1852-1939) ond6erle

dtr ro n e(vot unepBottr6q opr0g6g. Autd ondvrrloe optortrd oro np6-

ptrr1po rou r€rpoyovtopod rou r0rtrou, ono6etrv0ovtoq 6tt e(vot o66-

vorov vo y(vet.

Ae6op€vou 6tr ro n Kot ro e, ro0tirg rot or oprOpo( AroupCi, e(vot

unepporrxo( opr0po(, undpXouv dpoye rr dltror unepBottro( oprOpo(; Ot

erOerrriq xor loyopt0gtrdg eftotioetg €louv ouvqOurg trUoetq nou t(vot

unepBotrro( opr0po(. 'O1loq, ro vo npooilo0odge vo npoo6rop(ooupe

ov Uro ouyK€rprg€vr1 l0or1 e(vor unepBotrr6g 4 otryeBptr6q opt0p6g

pnope( vo poq onoOoppdvet. Kdnotot dltrot ouv6uoopo( gdg e(vot o16-

pr1 dyvrrrotor. Ilop' 6tro nou 6Xet ono6etyte( 6tt to e" e(vot unepBortr6g

opr0trr6g, 6ev e(vqt yvtrlotd ov ro ee ri ro n" q to n" e(vor unepportro( 11

olyepprro( opr0go(. Entntr€ov, t(orotrou0odv vo e(vqt dyvooteg ot tbr6-

tqreq ontr6v nopoordoeov, dnog ro e + fi Kot ro e'fi.

Ir]pepo yvurp(ouge 6tr undpXer gro otr6rlr1pr1 rtrdor1 er0ertr6v

oprOptirv nou e(vot unepportro(. To 1934, o Ptirooq go0qpottx6q Atre(€t

Oo(noBrtq frdl<povt (1906-68) on€6er{e 6tr 6}or or oprOgo( rnq popq{q

op e(vor unepportroi ov ro o e(vor olyepprrdg opr0g6q rot 6ev e(vut ro

0 d ro 1, rot ro p e(vor dppqroq olyepprr6g opr0g6g. Aut6 ntr6ov ovo-

pd(eror 1edprlpa rcu lx€Aqovt (ri, peptr€g qopdq, 9aip4pa ruv [x€A- 265

Page 251: Ο Ταξιδευτής των μαθηματικών

govt-f.vdurcp, ocpo0 o T€ovtop Ivdtwep on€6er(e ove{dpqto to 0-e6-

prlpo ro 1935), xat oqpo(ver 6tr oprOpo( 6noq ro 3V' Kqt ro (y'e)n'ei

vor unepBonro( oprOpoi, ogori ot pdoerq (r ror G) e(vor otryeBprxo(

oprOpoi ror 6ev €(vor ro 0 { ro 1, evtir ot er0iteq ({7 ror {s) eivor ap-

pqtor otryepptro( oprOpo(. AuotuX6q, our6 6e poq Borled€t vo onoqo-

o(ooupe tr r6rdtqreq €Xouv opr0po( dnoq ro e", ?f," { n", oqoU e6ti t6oo

or Bdoerq doo ror ot erO€req ri6r1 e(vor unepponro( opr0po( rot 6ev tro-

voTtorofv trq npoUno0€ottq rou l-r€trqovt.

And ro 0etirpqpo tou lxdtrqovt pnopo0pe en(or1q vo oupnepdvoupe

6u €vog opr0g6q 6noq o log2 np€ner vo e(vot uneppottr6E opt0p6q. O

log2 e(vor 11 60vopr1 oulv ono(o np€net vo ut[tl0e( to 10 Yto vo €loupe

og onordleopo ro 2.'EXoupe, enop€voq, 10ros2 = 2. Mnoporipe vo oTto-

6e[[ouge dl o log2 e(vor dpprpoq opr0p6q. Aq unoOfooupe 6tt o lo92

e(vor pr1t6g opr0trr6g. Aut6 0o oriporve 6rr undpXouv 6do or€potor, ot p

xor q, r6rotot tirote log2 = P/q. An6 tov oprop6 trrrv loyop(0gov no(p-

voup€ lo'oe'= top/q - 2. Yrfdrvoupe ttipo rot to 60o oritrr1 rrlq e{(oooqq

oq 60vopr1 q ror no(pvoutrre 10p = 2q. Tdpo, ro l0 €(vot to ytv6pevo

rou 2 en( 5, 6qtro6ri (2.5)o = 2c |l2p.5P = 2e.'EXoupr ttirpo 60o 6uvordqteq. Av o p e(vor geyotrritepoS on6 tov q,

rdte pnopo0ge vo onlonotqooupre to 2q Kot on6 to 60o or€trr1 rqq €q(-

ooons Kor vo ndpoupe 2p{'5p : 1, to ono(o e(vot nporpovdrg lov0oogC-

vo. Tr ouppo(va ov o q e(vor peyotrritepoq ond tov p; Ie ouui rqv nep[-

rnoorl od\orototipe to 2p xot on6 to 66o ox€trr1 tqq e{(otrroqq rot no(p-

voupe 5p = 2q+. Aur6 np€net en(oqE vo e(vot trov0oop€vo, orpori 6trot ot

o6vOetor opr0po( ovotrfowot p' €vo p6vo tp6no. Me dMo tr6ytq 6ev

undpXer rov€voq oprOpdq fiou vo pnope( vo yporpe( og 5 ut[ttrtp€vo oe

pro ordporq 6rjvopq rot oq 2 urptrrpfvo oe pto or€porq 6rivopq. Aut6 to

ovrilopBov6poore op€oulq, ov Oupr10o0trre 6tt to 2 utptrtp€vo o€ onotov-

Drinore or€poro opr0pd 0o e(vot dptroq oprOpdq rot to 5 utpopCvo o€

onorov6{note ordpoto opr0gd 0o e(vot neprrr66 opt0p6q (0o eivot nd-

vro €voq opt0p6q fiou ro r€I€uro(o $qrp(o tou 0o e(vot to 5). Enop6vtog,

or p Kor q 6ev undpXouv Kol o log2 e(vor dppqtoq opt0p6q.

Ttirpo, ov o log2 ritov dpprltog otrtrd rot otryepptr6q opt0p6q, t6tt

"c

266

Page 252: Ο Ταξιδευτής των μαθηματικών

H I ITOPIA TOY f i

.ond ro 0e6pqpo rou lr6trgovr 0o yvtrrpi(ope 6u to 2 npdner vo efvoruneppotrr6q oprOgdq, ener6{ l0"n' = 2.To 2 6prrrq e(vor prpdq opr0g6q,fror o log2 np€ner vo e(vor dppqrog olld ror unepponr6q oprOgdq. To6ro enrXe(pr1po roXrier yro rd0e logx, dnou ro x e(vor pqtdq opr0p6q roro logx eivor dppqtog opr0p6q. Aw€q or npoiinoOdoerq roplouv yro q

ouwpunrnl nlerorpqg(o tov trp6v tou x. Kotd ouv€nero, 6tror oXe66vor loydprOpor e(vor uneppotrxo( opr0pro(. And qv dtrtrr1, nop' 6Io nouor rovovrr€q rpryovop€rplr€q ouvoptr]oerg e(vor ouv(0oq dppqtoroprOpoi, e(vqr enrnl€ov olyepprro( xr 6;g unepBorrroi oprOpoi.

f l6oor unepporKof oprO;ro i undpxouv;

l-vtrrpil(ou;re 6u undpXouv notrtro( uneppotrro( oprOpot o(oupo undpXerdnerpo ntr{0oq ond 6oritoug. E(vor neproo6tepor or uneppouro( opr0go(ond toug otryepprrodq oprOpofq; 'Eyeq dpoye, v6r1po vo tr€pe 6u €voonerpoorivotro oprOpti,rv e(vor peyotrdtepo { prrpdrepo on6 Cvo ritr}o; HnpoondOero vo onowrlOo0v outd to epotrilloto dvor{e €vov ewetrtirqvCo opi(owo orrl po0qpour{ o16g11 oe d,tr orpopd ro dnerpo Kor lroqnopefie Jlo €vrurru,orord poOrd owflrl{q rqq q6oqq rou one(pou.

To npdro onerpoorivotro pE ro ono(o rjtrOov oe enog{ or dv0ponor

{rov to ofvolo tov quorr6v oprOgtirv. fro vo onorrfooupe dtreyXo rouone(pou, e(vor rotrftepo vo rorovorlooupe nptiro rouq rpuorxo0q opr0-poUq. 'Orov 6rio nenepoop€vo o0votro €;1ouv tov (6ro ntrr10rr6 opr0g6,Cloupe ner 6tr to dvo o0votro pnope( vo orr€rKovrotc( oprprpovooripovroen( tou dtrlou ouv6trou. Av 6ev neprooeder xov6vo ororye(o oe rdnoroon6 ro 6rjo orivotro, t6re €Xouv rov 6ro ntrr10rr6 opr0p6. Oo Xpqorpo-nou]ooupe u1v i6ro opXri 6rov 0o prtrdpe yro ro orrapoorivotro, ror 0oro ouyxp(voup€ Xpqorponordrvrog rqv onerx6vroq.

OPltMol Av to A ror B e(vor 6rio onerpoorivotro, rr ov dlo rqororXe(o rou A on€rKovi(ovtor dvo npoq fvo en( dtrrrrv tov otor-

1e[ov tou B, tdre ro A Kor B €Xouv rov (6ro n]r10rrd opr0p6. 267

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Agod to 66o odvotro A Kor B e(vor onerpoo0votro, o nl40rrdg touqopt0pdq 6ev pnope( vo e(vor rdnoroq nenepootrrdvog oprOgdq. luvendrq,0o e(vor onopo(rr1to reltrd vo op(ooupe rdnoro enrntrdov odppolo nou0o errppd(ouv rov ntr40trd opr0p6 rov on€tpoouvdtrurv.

Kd0e onerpoorjvotro nou pnope( vo reOe( o€ opgtgovoorjpovrq oner_rdvroq pre rous quotKodq opr0poriq ovopd(etor apilpqotpo orjvolo.'On'q ono6erxvUeror, to opr0pr]orpo otivotrq €Xouv rdnoreq o{roorlpeGurreq r6tdqreq. O l-rolildo frohl6r (lolrtro(og, 1564_1642) e(vor dvogon6 toug nto (orouoro0q enror{poveq ror poOqpottxotig. Be}riuroe torqleor6nro, ovordtru{e touq 6opugdpous rou A[o ror ug qhorig x4l(.-6eg, unoot{pr{e ro qtrrorevrprr6 o6orqgo. Avoxdtrurlre ro vdpo rou €K_rpepofg ror ond6er{e rr1 oroOeprj enrt<iXuvoq tov oopdrov ro0drg nd_qrouv orrf y4. Iro €pyo rou A6yot yApu an6 56o v€q enrcu1peg, rconoio er6604K€ ro 1636, o lolilo(oq enrorjpove €vo notr6 nep(epyo Xo_poxrrlptour6 rov on€rpoouvdtrulv.''E6er[e dtr e(vor ouvordv vo oftaKo_v(ooupe opqtpovoor]povro rous guorrofq opr0porjq en( rov terpoydr_vov rov guorr6v opr0pcirv. H onerrdvroq ouu] e(vor n e6riS:

i1 2 3 4 5 6 7t t t t f " +Y v v v + l +' 1

4 9 1 6 2 5 3 6 4 9

8 9 1 0 l tt t t lY v V +

64 81 100 121

Kd0e <puorrdq oprOpdq onerrovi(eror o, €vo p6vo terpoycuvrrd opr0_pd ror xd0e retpoyr.rrvrr6q opr0p6q onerrov((erqr o, €vo p6vo rpuorx6opr0trr6. Enop€voq, €Xoupe Fro opqrpovoorjpovr4 onerr6vro4, or4vono(o nrpiloppdvetor rd0e rpuorx6q opt0pdg ror rdOe retpoytuvrr6qopr0p6g. And ourd rorotrr]youpe oro €vrunulorord ougndpoopo 6l orretpoyovrro( opt0poi e(vor rdoor door ror or rpuorro( oprOpo(! Or guot_roi oprOpo( ovoprd(ovror oprOpr]orprot rot rd0e dMo odvotro nou pnope(vo teOe( o€ olrgrpovoorjpovrq onerxdvrorl pe curodq ovopd(eror rr ou_rd opt0prjorpo.

'Orov outd ro B}€noupe yro npdrtq gopd, pno(voug€ orov terpoopd

268 2. Boyer, a.36i.

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H I I T O P I A T O Y n

oqrtirooupe rprl}d to 16pro ror vo 6qtrd,rooupe 6tr 0o npdnet vo €Xet

rdnoro ld0oq. Iltitg 0o pnopotloov vo undp;1ouv r6oot tetpoytrtvt-

dpr0po( 6oor ror quorxoi 6tov on6 to onetpootivolo tov r€rpoYo-

oprOptirv tre(nouv t6oo noltro( quotro( opr0pro(; Nrti,r0oupe 6roro011-

6tr or rpuorro( opr0po( 0o npCnet vo e(vot (fiol0 fisptoo6tepot> on6

;TouS r€rpoyovrroriq opropo0q. I6prpovo 6ptrl5 F€ tov nopondvtrl opt-

opd, to 6rio o0votro €Xouv tov (6ro ntrrl0trd opt0p6 rot, enop€vog, e(vot

bo oe op€yeOog". Aur6 ro on}6 nopd6erypo tov((et 6vo o$oorlpe(trtto

;1opoftqprouK6 turv opr0grjorptrrv onetpoouvdtrtrtv. Mnopodpe vo urlo-

oqpqoupE to (6ro yro 6trouq touq 0eurodq rot opvqttro0g ordpotoug

oprOpo0q, roOtiq xot yro rous quotxorlq opt0godq, pe Bdoq rqv nopord-

to onerx6vrorl:

1 2 3 4t t t fV Y Y V

0 1 - 1 2

8 9t l

4 4

5 6t l

- 2 3

7

t-3

1 0 1 1f t

5 - 5

E6rir, trorndv, €Xoupe rdvet pto oprptpovootipowrl onetx6vtol, Iono(o neprtrotrrBdver rriOe or€poto rot rdOe quotr6 opr0p6. Iuven6q, ot

quorxo( oprOpro( e(vor t6oot 6oot rot ot ordpotot opt0poi. flpdypott,

dloutrre to nopordro topo(o 0e6pqpo.

OEQPHMA: Kri0e opr0pqotpo onetpootlvolo €yet rov 6ro nlr10tx6

opr0p6 1r€ rouq rpuorro0q opt0po0q.

Aurd to Oetirpqpo tr€er ontrdrq drt 6tov prtrdpe yro to piye0os rov

opr0prlortrurrv onttpoouv6trtrtv, 6tro to o0votro outo6 rou e(6ouq €Xouv

to 6ro p€ye0oq, 6qtro6ri 6Xouv tov (6ro nlr10rr6 opr0yd. To ev l6yo 0e-

rilpqpo ogrlver vo evvorl0e( 6tt dvo dnctpo unoodvotro ev6q opt0ptiot-

pou ouv6trou dyer to (6ro trrdytOoq pe ro opXrrd o0votro. Mto ott(o yto

rqv ono(o q €woro ouui poq Kdv€t vo vrtirOoupe nopd(evo e(vot pto

r6r6qro rurv rurr€pootrrivrrrv oprOptiv nou 6ev qv €Xouv ot dnetpot

opr0pol l-ro nopdberypo, ov npooO€ooupe ro I oro 10, noipvouye

6vov oprOgd peyol0tepo on6 ro 10.'Opurg, ov npoo06ooup€ ro 1 o'

dvo onerpoo0votro, 6tv no(pvoupe 6vo upeyotrUtspo) o0votro, no(pvou- 269

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X E O A A A I O I O

pe €va ouvolo rou (6rou peyd00u5. Enopcvulq, ov A e(vor o nlq0rrdqoprOpdq ev6q onerpoouv6}ou, tdre A + 1 = A. Aut6 rpo(vetor evrertiqlovOoop€vo yror( pe pdoq qv roO4peprvrl poq epnirpfq 6rov npo_o'droupe 6vo ov'reipevo o' ivo orivolo ov.rerpdvtrrv, ro orivoro y(ve-ror yeyohltepo. Oo np€ner dptrrq vo owrtrqg0otiye dtr e6dr €1oupe vordvoupe pe onerpoorivorq ro ono(o 6e oupnepupdpowor ndwo dn'qro fi€r€pqop€vo orjvotro. ,Evoq rp6noq yro vo oupgl]urr0oripe pe ounjvrrl ouFnrprqopd e(vor vo oregr.poore dtr €vo onerpoo0votro e(vor td_oo peydtro rou, ov npooo€ooupe oe qutd ro 1 ({ onorovo{nor€ n€rr€_poop€vo opr0trrd), o opr0p6q out6q nopoe(vor ooripovtoq yio vo oMd_{er to p€yeOoq rou qn€tpoouvd}ou. lro vo oltrri{oupe npdypotr to p€_ye'og ev6q onerpoouvdtrou, 0o np.ner vo rdvoupe rdnoro oneipou pre-yC0ouq olloyr]. Av ooq gnep6erier 11 r6io dn 4 np6o0eor; evdq nenepo_op€vou opr'pod o, dvo onerpoorivolo 6ev otrtrd(er ro pCye'dq tou, 6eve(ote or pdvor. Or poO4pouro( npoptr4potr(ovrov yr, ourd ro O€po en(6eroer(eg. 'Orov prldpe yro to yiyeOoq ev6q onerpoouvdtrou, 0upr;0e(te6tr op(ouge ro p€yeOog pe Bdoq uq onerrov(oerg. Xpqortrronouivroqqv €woro u1q onerr.vrorlg, trrnopo,pe vo .oripe d. ro onerpoo'volo6xouv npdyyotr 'rocpoperrrti peyc'r1, our6q dpurq or 'rogopiq 6ev e(vqrnenrpoog€veq olld e(vor 6roqop€q pe tinerpo p6yegoS.

Ttirpo nou €Xoupe 14 p€0o6o yro rrlv onerr6vror;, pnopori;"re vo npo_Xoprjooupe o, ivo ond to nro ernlqrurd oupnep<iogoro turv po'rlgo_ur6v. fvop(oupe 6tr r<iOe onerpoorjvolo nou pp(oreror oE ogglpovo_o{povr4 onrrrdvror; pr rouq guorrodq oprOpo0q iXer tov 6ro ntr40rrdopr0p6 pe to orivolo rurv guorKdrv opr'pdrv. n600r pqtor opr'po(undpXouv; Mnopo0pe vo qnerrov(ooupe dlouq rouq p4roUg oprgpo0qen( tov rpuorrdlv opr'ptirv; Me pro nptirq potrd, <po(veror dtr ro orivo_Io drurv tt'rv pqtriv opr'ptirv 00 npiner vo e(vor notr' peyo'itepo on6to orivoro r,v rpuorrtilv opr'ptirv. Mdvo pero{ri rov guorrtiv opr'pdlv1 ror 2 un<ipXer dnerpo nhjOoq prlrrirv oprOpdrv. Mnopodp€ vo ro y€vr_refooupe ourd ror vo noripe dtr pero{6 0norurv6{nore 6rio rpuorrtivopr0ptirv undpxer dnerpo n'j00q pqtdrv oprgytirv. Iltirq gnoporiv, ror-270 n6v, ro 60o orjvolq vo iXouv tov (6ro nlq0rrd opr0p6;

j

l

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H I I T O P I A T O Y f ,

Iregte(re u1v onerr6vroq tou fl(voxo 8, 6nou ronoOerofpe rdOe pq-id opr0pd pr6oo o' €vo tetpoytrrvrr6 n(voro. O n(voroq out6q ouveX(e-r0r €Tr'dTr€rpo npos ro 6e{rd xor ilpoq ro Kdro, Kor nepr}oppdver rd0ep4r6 opr0g6, ogoU rdOe prp6q opr0p6q dXer tq Fopqrl p/q, 6nou or pKqr q t(vor ordporor oprOpo(. Ilepvdpe on6 6lo ro rldogoro (errvdrvrogon6 rrlv ndvo oprorep{ yov(o ror op{voupe toug oprOgodq Trou €Trovo-lopBdvovtor, €tor tiote vo ndpoupe tov l(voro 9. Ie our6 rov nivoro,to pCtrr1 6efivouv n6q vo npoXrrrprioouge on6 rov 6vov oprOgd mov dtr-lo yro vo oBtjoouge roug oprOpodq nou enovoloprprivovror. H ondrpoo4yu tq 6roypog{ ev6g r}dogotoq 6ev e(vot 60orotrr1. Av ro rtrdopo eivor

{6q ovdyoyo t6te ro xpotdpe, ov 6ev e(vor ro oBr]voupe. Itov n(voxoundplouv 6tror or 0etrro( prlro( opr0po(. Av 06troupe vo ppodpe ro p/q(uno0€rovrog du our6q o ldyoq eivor ovdyr.rryo rtrdogo), rdre onl6q ro-reBo(voupe oq q oetpd p Kor npoxopdpe 6e(rd rrrq q mrjlq q. Enop€-vurq, ouyfisprtroppdvovror 6tror or Oelro( pqto( opr0go(. flo(pvovrdqrouq rotd oerpd, pnopodpe trorn6v vo O€ooupe o€ opqtpovoorjgovqonetxdvrorl outoriq rouq oprOpofq F€ rouq rpuorrorjq opr0poriq:

5I . . .

tv

1 0

E6tir 0o pnopo0oote vo ovtrtd{ere 6rt trdBope un'd{xv poq g6voroug Oettro0g pqro0q opr0godg. Aur6 e(vor otr{Oero, otrtrd prnopo0trre vopnd(oupe tov (6ro n(voro Kor p€ opvrllxd rtrdoporo. T6te ronoOero0-ge ro 0 og np6to otorXe(o xor, no(pvovroq evotrId( rtrdogoro on6 rov0eur6 ror rov opvqrrrd n(voro, iXoupe:

1 1 2 3 r r 2 3 4t Z T T 3 V i Z Ti t f f f f + + t9 * * V + 9 + + +

1 2 3 4 5 6 7 8 9

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Ttirpo 6Xoupe rqv onerr6vrorl fiou Odlouge. Kd0e pqr6g oprOpdq,0ettxdq, opvqttxdq xor ro prq6iv, ougnepilogBdveror p6vo pro rpopdrot 6tror ot opr0po( onerrov((ovror en( rov guorrdv opr0pdlv. Korol{- 271

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l

flINAKAI 8. fl(vorog trrrv pqrtirv oprOptirv

nINAKAI 9. flivoroq ttrlv pqttirv oprOprirv

l l

7 + t " '

38

58

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/ , / /+ - + + + +

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272

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H I f T o P I A T o Y f i

youpe, enop€voS, oro evtunoorord outrrnipoopo dtr o nlnorrdq opr0_pdq rtrrv rpuorrdv opr0pti,v e(vor (ooq pe rov ntr4Orrd oprOpd 6r'v ttrrvp4ttirv opr0pdrv. Tr ouppoiver, dpoye, ge ro odvolo 6trov trrrv otryrppr_rtiv opr0ptirv; 'H dtrrrrv rov uneppour6v oprOptirv; ,looq vo undpXerdvo6 p6vo nl40rrdq opr0pdq yro 6lo to onerpoorlvotrol

And t6re nou o lotrrlo(oq un€6er{e t4v onerrdvroq pero{0 rujv guot_rdv oprOptirv Kor ru,v rerpoyovrrtiv opr0prir4 or poOrltrrouro( npoplrlpo-tf(owov yro u( ouv€nereq ourod rou yeyovdroq. ,Enpene vo neprpCvou_pe nep(nou 6rordoro nev{vto Xpdvrq p€Xpr ro t6loq rou 6Cxorou Cvo_rou ouirvo, yro vo orovrqOe( nlrjpoq ro eprirtrlgo ourd.

O o(r6Aoyog fr<dopyr Kdvrop

'0Io oXe6dv doo np6ner vo pd0oupe ytq rous unrpBourorlg oprOporiqoqeitrovror o' €vov o{r6loyo dvOpono, rov lr€opyr Kdwop, o ono(ogi(r1oe pro e{orpetrrd ev6rogipouoq qtrtrd rpoyrrrl (orj (Errdvo 43). fev_v{0r1xe or4v Ayio lletpo0nolq, orq poo(o, onq 3 Mopr(ou 1845.3 [1ori_po( rou rirov o fr€opyx fou6trwepov Kdwop, ivoq Aovdq €pnopog, rorpr;tdpo tou r; Mop(o Mnetrr Kdwop. To 1856,6rov o frdopyr rirovdwero Xpovrirv, q orroydvero peroirqoe oq Opovrgorlpq rou Mdrv,,oro rpoti6ro tqq 'Eoqq, ylo vo onorp0yer o friopyr o npeopritepoqrouq 6prpe(q ptiorrouq Xerprirveq.

.Otov o lrEopyr o ve6tepoq €yrve 6e_rondwe Xpovtirv, en€6er(e peyrilo totr€wo mo po04pourd ror ypdqrn_re oro Avtirepo flolweXve(o frpovr_Ntourdl oto Nrdpporor, dnou 0oonoU6o(e pr;Xovrrdg rotd nopdrlrlorl rou notipo tou. To 1863 erorjX0rloro flovenrorrjpro tou Bepotr(vou, 6nou en6]e{e vo onouddoer po0rlpo_nxd, quorr{ ror rpilooorp(o. ,Htov Cvoq Oeooeporipevoq veop6q, Xopo-nqprotrr6 nou rlqpovdprloe on6 rouq yove(q rou.

3. Mu i r , a .217.

. - . q. Ewoe( tr1 Opovrrpodptrl qg 6uurrig fepyov(oq, nou pp(oretor orov notoprd

Mdrv, rr dXr q Opovrgoripq nou pp(oretor orov notopd ,Ovtep. (r.r.u.) 273

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EIKONA 43. fK€opyK Kdvrop, 184s-191 8.(Ytrrrd ond roug o6etrrpo0q Mnpdouv, h€ptrrvyr)

Iro flovenrorrlgto rou Bepotr(vou 66oorov 60o ond roug onou6or6te-poug po0rlgourofq rou 6irotou €votou ouirvo: o KopI Bdreporpoq (tSts-'1897) Kor o Adonotrw Kp6verep (1323-1891). Auro( or 60o 6rdoqpor dvrpeq0o 6nor(ov onorpootour6 p6tro otq (rrrq tou Kdvrop, o €voq rotr6 ror o dl-trog roxd. Evtir o Bdreporpoq 0o ytvdrov ivoq ond rouq uroorrlprrtdg rou,o Kpdverep 0o notreproOoe oe dtrr1 rou rq (rrrri trg r6€eq rou Kdwop.

'Otov nrlpe ro 6r6oruoprxd rou magna cum laude (pe inorvo) ond274 ro flovenroujpro rou Bepotr(vou ro 1867, o Kdvrop 6ev pnopodoe vo

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Fro rov€Tnorqprorf Oioq ovdtroyrl F€ rrq rrov6trltdq tou. Og erO€Xrrlre vo 6r6d{er poO4portrd o'ivo r6rouxd oXotreio 0qtr€trrv.

1869, o Krivrop pp{re pro rev4 06o4 oro prrpd flovenrouipro tou0ioq nou e(orolouOorio€ vo pQV e(vor rod(ro r4g e(o(peqq er-

tou oro llovenror{pro rou Bepotrivou.Eyroroord0qre, Iorn6v, oro Xdtre ror dyrve ovontrrlptrrrrjq roO4yrprlg1872. To (6ro €roq, yvrrrpior4xe ror €yrve qiAoq U, €vov dllo veopd

po0r1pour6. To 6vopd rou {rov prXopvt Nr€vrerrvt - o dv-rou UoS i6ooe ro o0yXpovo oprop6 rov dppqrtrrv opr0ptirv,

14 6rdor1pr1 epyoo(o rou rrl Xpovrd nou yvriproe tov Kd-Auro( or 60o po0qporrro(, nou 0o ouvrrodrpepov rdoo noMd oto

oro t6trq rou 6€rorou dvorou ouirvo, €yrvov otevo( rpitrorouppepi(ovtov ror unoor{pr(ov o ivoq uq r6€eg tou dtr}ou.

E ivor opr0pf iorpot or oAy€pprr<oi opr0pro i ;

"0r olyepprro( oprOpo( neprtrogBdvouv 6trouq rouq prlrorig opr0godq, ro-0tig rr ere(voug ioug dppqrouq opr0porjq nou onotetro0v trdoerq nolutrr-vriutrrv. E(vor oprOp{orgo ro orivo}o turv otryrBprrdrv oprOptirv; Aur6 rpo!vetor nohj 60orolo np6Blqgo, otrtrd o Kdvrop ro €luoe edrotro. Aut6nou dxove {rov vo rono0et{oer oe oXiorl 6rdro(r1q touq otryeBprrodqopt0poUq, pelertirwoq ro notru6vupo tov ono(urv onotelo0v hioerq.

fro rdOe nolutitvupo, o Kdvtop dproe €vov oriporo opr0p6, rov H,rov ono(o ov6pooe rirpoq rou noluovdpou. Oupq0e(te 6u €vo nolut0vu_po dXet q popgli ooxu + o,xu-' + orxu'* ... * ou-rX + ou = 0. Op(oupe rodQoq tou oq e{rjq: nptirrov, unotroy[(oupe ro v-1, ro ono(o e(vor ontrrirqo peyotr0repoq rr0€rqq rou x pe(ov 1. Kordnrv, npooO€touge oe ourdqv ondlurr; trgrj 6trurv rurv ouweleottirv. Aiyovroq ondtruq lp{ evvo_oUpe q Oeurq trprj rou ouvtetreonj, ove(dprqto ond ro ov ou{ 11 trpqefvor Oerrr{ { opvrlnrri oto no}urirvulio. Aurd rdver ro 0rfrog rou no_lutr tv0pou vu roodtor p€ H = (v-t) + lool + lo, l +. . .+ lo" l , dnou orypopp€q p€oo otrq ono(eq pp(orovrot or ouvteleot€q onorrofv vo Xpn_ 275

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l ( E o A A A t O 1 0

orponorodpe trq ondtrureq rrp€q rouq. fro nopd6erygo, to ontrd nolurir-vupo x + 1 = 0 6Xer 6$oq 2, yror(oo = 1, or = I ror v = 1. Aurd poq 6(verH = (1-1) + 1 + i = 2. Mnoporipe vo 6oripe op6ourq 6u, yro rdOe lprjrou H, dXoupe 6vo pdvo nenepoop€vo oprOgd nr0ovrirv notrurrrv0pr.uv. Ofl(voroq 10 nopouord(er ro ovt(ororXo nolutirvugo yto uS tperq np6teqrrp6g tou H.

lto H = 1, €Xoull€ 6vo p6vo notrurirvugo, ro x = 0. 'Otov ro H e(vot 2,dXoupe t€ooepo notrurirvullo ror 6rov e(vor 3 no(pvoupe dvtexo 6roqo-petrrd notrutirvupo. Kord ouv6n€ro, o opr0g6q tov notruov0polv nouovrrororXoriv oe rd0e rirpog ou(dver ypriyopo. Oordoo, oe xd0e ouyre-rprpdvo H ovTlororXe( 6voq g6vo nenepoop€voq opr0p6g notru0v0pov.Ttirpo Xprlorponoro0pe to 6rdor1trro Oedrpqpo tou Koptr frdouq, to Oepe-trui6eg Oedlpqpo r4g AtryeBpoq. A6er dtr rdOe notruulvuprr{ e{iotrrorl6Xer g(o routrdXrorov l0or1 ( p(o. 'Evo n6propo rou 0errrprjgor6g toue(vor 6n €vo notrudrvugo €Xer (6ro orprBtirg opr0p6 }0oeov pe ro po0pdrou nolurrrv0gou. Enop6vrrrq, rd0e notrutirvupo iXer ouyrexprpdvo oprO-p6 trUoeurv' yro nopd6erypo, ro notrutirvullo x' + 2x - 5 = 0 iXer rperq trrj-orrq, evtir to notrutirvugo x' - 7x + 1 = 0 dXer 6rio trrjorrq. fro rdOe no-Iutirvugo rou fl(voro 10, pnoporjpe vo roroypdr[oug€ uS ]0oerq oprivo-vroq rouq oprOpofq nou enovotropBdvovror. Tdre, yro rd0e o0votro no-truovfprrrv rou ovrrorotXoriv oe rdOe rrpri tou H, €Xou1le ivo oro0ep6opr0p6 govo6rr6v trdoeov, 6r1lo6q otryeBprrdrv opr0gdrv.

nINAKAI 10. flotrurrlvulurd 0gr1

I'Y{os

Io]utitvul.ro Movo6rr{ trOor1

I

)

3

x = 0

x + 1 = 0 , x - 1 = 0 , 2 x = 0 , x 2 = 0

x + 2 = 0 , x - 2 = 0 , 2 x + 1 = 0 ,

2 x - 1 = 0 , 3 X = 0 , x t + 1 = 0 ,

x' - I = 0, xt + x = 0, X2 -x =0,

2 x ' = 0 , x 3 = o

0-1, 1-2,2,-+, +,'�,[., -'�,["

276

A

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H I I T O P I A T o Y ' t

, An6 rov fl(voro t0 BtrCnoupe 6u undpXer p(o povo6rr{ hio4 yro H = 1,4 ono(o e(vor ro 0. l-ro H = 2 eXouge 6fo trrioerq, ro I Kot to _1. fro H = 3{oupe i{r povo6rr€q l0oerq. l-lpoq to nopdv, 0o oyvoriooupe ro yeyov6qdu 6ev {oupe 166o u r6oug oprOgo( e(vor ro y'J ror to _y'J lopyjapo0o pd0outrre to ndwo yr, owo0q rouq oprOpo0q). Mnoporip. .,;rpo uocnerrovfooupe nq povo6rr6g trfoerq poq en( rov rpuorrtiv oprOptirv pe tovnopordro rp6no:.

0++I

- I

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8 9

Iqv nopondvtrr onerrdvro4 neprtropBdvovtor dlor or olyepprro(opr0poi yrot( rdOe nolutirvullo €Xer pro ovt(ororX4 trg{ H. Ie ounj qvrtp{ tou H ovtrororXoriv or llovo6rroi otryeBprro( opr0poi nou 0o oner_rovrororjv en( rov guorr6v oprOgtirv. An6 ounj rrlv opglpovoorjpovrrlonerrdvro4 ovulopBovdpoore, 6noq orprprirq ror o Kdvtop, dtr or otr_yepprxo( oprOgo('np€ner vo e(vur opr0priorpor. Oo ppofpe noti dvoonerpoorivolo nou vo 1.r4v eivor opr0prjotgo;

Idoor s i ' vo t o t unsppor t r<o i opr0po i ;

O Krivrop rjrov rrilpo irorpoq vo O€oer ro nro orlpowrrd eptirtrlyo rqq(o{q rou: e(vor oprOprjorpor or uneppattro( oprOgo(; O€towoq ourO toepdlqyo ovoprrltrdrov ov or Trpoytrtouro( oprOpoi otouq ono(ouq oupn€_prloppdvovror or olyepprro( ror or unepponro( opr0poi e(vor oprOprior_1.tot. Av or uneppotrro( ror or olyepprro( opr0po( e(vor opr0pr]orpor, r6reto d0poropti roug 0o eivot xt ourd oprOpr]orpo.Itrq 29 Noeppp(ou 1823,o Kdvtop dyporpe dvo ypdgpo oto giAo rou, rov prXopvr Nt€vterrvr.

Mnoptir vo oou rdvo pro ep6qo4 nou dXer 6vo rdnoro Oeopqnrd ev_6roq6pov yro p6vo, olld otrlv ono(o 6ev gnopdr vo onovqour; ,lourgpnope(q eori vo tqv onovtrjoerq ror 0o ritov nolU euyevrr6 ond pdpouqoou ov Fou 6ypo<peq oXetrrd. ,EXer trrq e(r]q: ndpe to orivolo dtrov tu_rv ZTT

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guotrt0v oprOptirv v rot ovdproo€ to N. Enrnldov, ndpe, oq no0ye, to ori_votro dltov tov 0etrr6v npoypotrrtiv opr0ptilv x rqt ovduooli to R. Aor_n6v, 4 ep6t4o1 e(vor ontrtiq I e(r1q: pnope( ro N vo ouv6uoote( ue to Rpe t6toro tpdno, drore oe rdOe otorxe(o tori evdq ouvdtrou vo ovrrotorxe(6vo ror g6vo €vo otorXe(o rou dllou ouvdtrou; Me puo np6q proudqnovtder rove(q: <'OXr, our6 e(vor oOtlvotov, yror( to N onotele(ror on6(extrrptotd tyritrroro evdl to R E(vor dvo ouvex€q orivolon. Autrj 6ptrlg rg€votooq 6ev qnooetrvuer t(noto. Kr doo rr ov nrore0o d'to N roi R6ev enrtp€nouv 6vov tdtoto ouv6uoopr6, e{orotrou0dr vo prrlv pnopti vopprrr r4v orr(o. Kor outr] q ort(o e(vor nou ;.re onooXotrei pnope( vo e(vornotrri qntrf .5

O Kdvrop, e66, iXer oyvor]oer rouq opvrlrrKoUg npoygorrrodg optO_poUq, oltrd out6 6ev onotde( npdptrrlpo. Av pnope( vo pper Uro oilsr_r6vroq yro dlouq roug 0euroriq npoypourotiq opr0po6g en( rov rpuor-r6v opr0pdrv, rdr€, €rr€Kr€(vovtog tqv (6ro reXvrr{, pnope( vq oupfi€pr_ldper ror 6louq rouq opvquroUg npoypotrroUq oprOporlq. O Nrdvrerrvr€ypoQe orov Kdvrop dtr 6ev rj(epe vo onovrqoer oro epdrrrlgd tou. Iulouv€Xero, orrg Z Aereppp(ou 1873, orrdr prdvo rlpip€S perd ro np6toypdppo tou, o Kdvrop iypor[e ndtrr orov Nriwerrvr.

Teleuto(o efio ro Xp6vo,vo npoXrrrp{ou_r trtyo neproo6tepo rqv etroo(ofiou oou ovErpepo. orlyepo nroteUu_r 6tr tetre(trroo ;re out6 to O€po. Avoutonotc0pror, 6e 0o pptrr enrerr€otepo rcprtrj on6 eo€vo. flo(pvtrl lotndvto Odppoq vo unoBdlrrr otrlv rp(oq oou ourd nou dXrrr ypd{rer, nopd trgotdAeteg nou pnopei vo riXer to nptirto npooX€6ro.u

Ie 6rdorqpo tr(yo neproo6repo on6 pro ep6opd6o, o Kdwop o16_woge ndvtrr otrlv ovordluQq rrlq (orjg tou _ 6u or npoygolro( oprO_po( 6ev e(vor opr0prlorpor, yrotf e(vor unepBolrrd notrIo(. oo rorrd{ou-pe 66o ond trq ono6e(etq tou. H npdrq e(vor 11 ond6er{r1 Trou lropou_o(ooe orov Prxopvt Nt€vte*vt ro Aeripppro rou rgz3 ror 6qpoo(euoe

5. To on6onqopo nopot(Oetor oto: Sherman K. stein, Mathematics: The Man-Made University (N6o Ydprq, W.H. Freeman and Company, 1963), o.252.

6. 'O.r. , o. 253.278

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ro 1874.7 H 6efrepq on66er[q e(vor pro €(unvq erpopgoyrj rou 6erq6t-ro0 ouott]poroq, q ono(o undpXer oro neproodrepo eyXerp(6ro Trouovorp6povror oq 0eop(o trrlv ouv6}trrv. Kor or 6rio ono6e(erq e(vorunCpoXeq, otrtrd q nptitq efvor rdnoq nro rcopQrj.

Irqv nptirrl on66er{ri rou o Kdvtop Xp4orpono(,qoe q pri0o6o nourrporoTropouo(ooov or 'EtrIqveg, t1 reductio ad absurdum (erq dronovonoyutytj), 6qlo6q rqv €ggeorl oro6errrrr{ p60o6o, tqv ono(o dXoupetj64 ouvovr{o€r os Trporlyo0pevo rerpd}oro. Ynd0eoe ouorootrrd. 6tr or0erro( npoypouro( opr0po( 0o pnoporioov vo orr€rKovrororiv o1.r<prtrrovo-o{pouro en( rov rpuorr6v opt0ptirv rr €6er[e du ourd o64yoUoe oe owfqooq. Av ro o0volo rov 0eurtiv npoypolrdv opr0ptirv e(vot pq opr0pq-ot;.t0, t6te e(vor pq opr0pqorpo ror ro odvolo dtrcuv rov Oelrtiv Kor op-vrlttr6v npoygolrtiv opr0truitv. Aut{ 11 on66er(r1 enopre( en(o4q yro voono6e(er 6tr or uneppottro( oprOpro( e(vor pr1 opr0pr]orpor, o<poU dtro rodtrtro e64 npoyprotrrdv oprOgtirv e(vor opr0;.u]orpo. Av ro orjvotro dtrtrlvrov Trpoygorrxtirv opr0ptitv e(vot pq oprOprjotpo, out6 0o npiner vo orpe(-Ietor moug unepBouxodq optOporig.

Av uno0€ooupe 6rr or Oetrro( npoyprortro( oprOpo( e(vor opr0pqor-por, r6te 0o np€net vo undpXrr g(o routrdXrorov onerrdvtoq ovdpeoooe outo6q Kdr rous rpuotrofq oprOgotig. Yno0€ore 6rt gro tdtoro oner-rdvrorl 6(6eror ond u1v nopordtur oxotrouO(o, dnou ro rdOe o e(vqr€vog 0eur6g npoytrrotrr6g opt0p6q:

U)t, U)2, tD3, 0J4, U)s, U)6, trj7, U.ls, U)e, ...

Iqv nopondvru orotrouO(q, or 6e(rteq 6eiXvouv touq rpuorrofg oprO-podg en( tov ono(ctrv onerrov((ovror or rpoyporrro( opr0po(, 6qtro6{ roOrdrpopo o. An6 rqv und0eoq rou Kdvrop, 11 nopondvur orotrouO(o ne-priXer 6trouq touq Oetrrodq npoygourodq opr0porlg.

Ag ndpouge 6vo rp{po rqg euOe(oq rov npoypourdrv oprOpriv ror ogovopdooupe to drpo rou o Kor p. Iqv npoygorrrdrqro, 6ev €Xer oqpooio

7. joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy ofthe tnfinite(t1p(votov, Princeton University Press, 1979), o. 50. 279

Page 265: Ο Ταξιδευτής των μαθηματικών

Trorous 6rio opr0pofq 0o 6rotri{ougc r.uq drpo rou eu06ypoppou tprjpo-rog. To gdvo nou uno0€rouge e66 e(vor 6rt or o ror p e(vor 6rorpopelrofoprOpo( rot 0o noUge ouOo(peto dr o o e(vor o prrp6repoq ond rouq 60o,iror 6me o < B. Arpod 6lor or npoyponro( opr0go( nrpr€Xovtor oulv oKo-louO(o pog, {6poupe 6u ror 6}ot or npoyponrof oprOpo( ovdgeoo mougo xor B 0o npdner vo nepr€Xovror orqv nopondvo orotrou0(o. Aq npoXur-prlooupe or4v oxotrou0(o g€Xpr vo ouvovr{ooupe 6do opr0po6g, 0o touqovopdooupe oo Kor op or ono(or neprdXowor oro euOdypoppo rgr]po pog,ro (o, B). Oo noripe en(oqq 6rq oqv nep(nroorl Foq, oo ( o)u. Auto( ot611o oprOpoi nou iXouv enileye( on6 tqv orotrouOfo goq, op(ouv 6vo dtr-Io eu06ypogpo rp{po p6oo mo nptirro tprlpo poq (Err6vo 44). IuveXr(ou-pe t6po oqv orolou0(o p€Xpr vo ppo0ge 66o ordprl opr0poUq, nou ou-r{v q rpopd pp(orovror ovdgeoo orouq 0.}o Kor urp ror 0o ovopdooupaowo0q rouq v6ouq oprOgo0g uly Kor trl6. Auro( or 6rjo v€or oprOpo( op(ouv€vo rp(to rp{po p€oo mo tprjpo (ur", trlp). 'Onog pnope(re vo 6e(u, pno_podpe vo ouvrX(ooupe ouujv 14 6ro6rxoo(o en, dnerpo. Kd0e <popd op(-(oupe 6vo v€o eu06ypoppo rpr]po, Bp(orovroq 60o ox6pq oprOporig orrlvaxolou0(o poq Kor, mr1 ouvdXero, pnopoU;rc vo Xprlorporoujooupe ourdro euOriypcpgo rpripo yro vo op(oouge rivo dtrtro rprjpo. Meprrd on6 runpdrro eu0Uypoptrro tprjporo rpoivovror orqv Err6vo 44.

Trirpo 0o ouppo0v 6fo npdyporo ro0tirq ouveXi(eror ourrj q 6ro6rro-o(o. flptirov, 0o pnopo0oop€ vo unoO€ooupe 6n ourrl 4 6ro6rroo(ootopordet ror 6u notri 6e pp(oroupe dtrIouq 66o oprOpo0q nou vo ppl

@, 0, 0( o e P

EIKONA 44. H np6t4 ondOerlq rou Kdvrop yro rn Un opr0prrlorp6rrlto rov npoypo-urdrv oplOpuirv. Av 4 trl,, oz, oy... e(vor l.rro opr0priotgq orotrouO(o npoygolxtirvoptOprirv, r6re e(vor 6uvordv vo Orlgroupyrjooupe €vo onerpoo0votro rrpolopivov6toot4prdrov ({ur., oJ, {,1,, ou}, {rrl., o",}) nou vo 6Xer €vo oprord oqpeio, to ono(o6ev ovrjrer orrlv orolouO(o. Enop€vurg, ropr<i oprOgrjorprrl orotrou0io 6ev pnopr( voneprdXer 6trouq rouq npoypourofq opr0pro0g.

Opro

l ' . , . u -

28G

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H I f T O P I A T O Y ' T

orovror p€oo mo tetreuto(o euOriypoppo rgripo nou op(ooge. AMd out6

e(vor ovr(rpoor1, ytor( {€poupe 6n peto[ri onotov6rlnote 6do npoyport-

xdrv opt0ptirv undpXer 6vog dltroq opr0g6q. Iqv npoygottx6rqto, pno-

po0pe vo ndpe or6po nopaxdro: ovdpeoo ot 60o onotouo6{note

npoylrotrrofg oprOgorjq undpXer dnetpo ntrriOoq npoypotrrtiv opt0ptitv.

Enoprdvtrtg, ourq q nptirq un60eoq 6ev gnope( vo ouppe(. Ave(dpqro

ond to n6oeg <pop€q €Xoupe ppet dtro xot ptrpdrepo euO0ypoglto rptlgo-

ro, flou dlo torprd(ouv ro 6vo g€oo oto dMo, pnopotige vo ouveX(oou-

p€ oTrlv orotrouO(o rurv npoYgourdrv optOptitv Foq Kot vo ppo0ge dtr-

louq 66o optOporig nou pp(orovtot p€oo oto retreuto(o tprlpo poq.

Enop€vtrtg, 0o npCner vo roXdet 11 6efrepq npoUnd0eoq: pnopotlptg

6qtro6{, vo op(oouge 6vov dnerpo opt0p6 rprlpdrov, ro €vo p6oo mo

dtrlo. Enrnldov, outd ro tprigoto 0o 6Xouv 6vov dnerpo opr0pd drpurv,

6qlo6q opr0ptirv, nou 6trot pp(orovrot gioo oto opXrr6 rgqpo (q P).Ieour6 ro oqpeio qq on66er[r1q rou, o Kdvtop Xprlorpono(r1oe ivo Oetiprl-

po nou e(Xe ovorotrurpre( vop(tepo to 6€roto €voto qtdvo ond 60o go-

Oqporrrofq: rov To€Xo repio Mndpvopt Mnotrtodvo (1781-1848) rot €vov

on6 rouq roOrlyrlrdq tou Kdwop mo floventouipro tou Bepotr(vou, tov

Koptr Bdrepotpoq. O Bdteporpog qtov out6g nou nopouo(ooe qv ono

6er[q rou Oetrrpriporog Mnotrtodvo-Bdrepotpog to 1865, notr6 perd to

Odvoto rou Mnotrtodvo, oot6oo ovoyvop(oqKe rl ouvctoqopd rot rov

60o oe d,tt orpopd rtq notr6 orltrrovttrdg t6deq rou 0eurpriporog. To 0etiprl-

po Mnotrtodvo-Bdrepmpoq nou Xperdorqre o Kdvrop e(vot ro effg:

OEQPHMA MnoATrANo-BAiEPilPAI: Av x,, xz xr,'.. €(vot gro of-

(ouoo orotrou0(o oprOpdrv, 11 ono(o e(vor rppoyp€vq on6 rd-

norov opt0p6 B peyotrfrepo un6 rdOe x, r6te undpXet €vo

6pro orrlv orotrou0(o, ro L.

'Orqv trotn6v rotrd(oupe to onetpootivotro tgrlpdtu,v nou fXouge

op(oer on6 tqv orotrouO(o gog p€ touq Ontro0g npoypolrodg opt0-

po0q, Btrinoupe og€or..lq drt to gtrp6repo drpo rrrrv tgrlptdttrtv Foq oxrl-

pot(ouv pto v€o orotrouO(o opt0ptitv, pto dnetp4, ori[ouoo orotrou0(o

nou e(vor dvur <ppoyp€vq on6 to p. Eno;rdvurq, ovdpeoo oroug o ror p 281

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K E O A A A I O 1 O

undpXer 6vo opror6 orgpe(o oe ourr] r4v orotrouO(o, ro L. H orotrouO(oe(vor on}6g rl: o, oo, uJ,,, (r).,..., 4 ono(o onoretre( tp4po rrlg opXrrrjqorotrouO(oq o. Tt e(6ouq oprOpdg e(vor o L; Aev prnope( vo e(vor dvogon6 roug npoyporrrodg oprOpo0g rqq orolou0(oq u)r, 02, trt3, o4,... yro_r(, ov r]rov, 0o rprdvope tctrrrd oe ourdv roOdrg 0o ovo(qrorioope opr0_po6g yro vo op(ooupe enrnl6ov rprlporo, Kor ro L 0o r]rov ro drcpoev6q on6 ourd ru rp{poro. Aurrl r4v 16€o Xperd(eror vo rrl orerprodpetrryo yro vo ulv ovrrtrrlrpOotipe. Av ro L ovrjr<e or4v orotrouO(o rovnpoyporrr6v optOptirv, rdre ro L 0o onerrovr(6rov m( ev6g ouyrrrprp6-vou rpuorrof opr0po6, rou v, Kor 0o efie trrg 6e(rrq rou our6 rov oprO_p6. Av 4 ovo(rpr1o4 poq yro enrntrdov rprjporo evr6g rou opXtrod rpr]_;rot6g pog ouveXr(6rov ordrvto, r6rE oe v prjporo 0o grdvope oro \ rorro L 0o {rov rdre ivo on6 ro drpo rou rpq;.ror6q goq Kor, enop€vtrrg,6e 0o {rov ro dpr6 pog.

Iuventig, orpod ro L 6€v Unop€( vo ovrlrer orqv orotrou0(o o ror rou_roXpdvoq vo e(vor dpro, €Xoupre Bprr ivov npoyporrx6 opr0pd (o<po6 toL e(vor opr0p6g r4q euOeiog poq) nou 6ev neprtroppdveror or4v orolou_0(o o. Enop€vog, 11 orotrouO(o ur 6ev nepr6Xer 6louq roug 0ertro6g npoy_gorrrofg oprOporjq. lr.ri n ovr(rpoorl ono6e rrvrjet drt 6ev pnope( vo e(_vor ntrr1p4g ro;.rrd orotrou0(o npoyportrdrv oprOpdrv nou r(0eror oe ou_rpr;.tovoor]powr1 onerr6vtoq Fs rouq guorrodg opr0po6g, 0o tre(nouv 611_tro6rj op€rp4ror npoygolr(oi oprOpo(. Iuven6g, o ntrqOrrdg opr0g6g rovnpoypolr6v opt0;rdrv r(vor npeyotrJr€poq> on6 rov ntrr1O116 opt0pd rurvquotrtiv oprOpdrv (r1 rurv prpdrv r] ordpo ror rtrlv olyeBprr<tirv opr0prirv).

M r o d e 0 r € p n o n 6 d e r ( n

H np6rq on66er{q rou Kdvrop ro t8Z3 yro ro 6rr ot Oerrro( npoyFol_rco( opt0po( 6ev e (vor opr0prlorpor e(vor o(youpo Kotr{{, orpo6 6e orqp(-(erot oe r(noro ndpo ond rqv ivvoro rqg onrtrdvrorlg rurv opr0pdrv rorro 0e6pr1po Mnotrtodvo-Bdreporpoq. H 6erireprl on66et(q, rou ovo_nr6X0qre geprrd Xp6vro opy6repo, orrlpr(erot oro 6ero6rr6 pog o0or4_a&a

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H I I T O P I A T o Y f i

Aunj q on66er(q e(vqr ouorootrrd (6to p€ rnv nptirrq ond6er[r1Xprlorponore( qv ippeoq p60o6o, o0prptrrvo pe qv ono(o unoO€_

toupe 6l pnope( vo y(ver pro opqlpovoorjpovq onetr6vroq pero[d rurv-npoygotrxtitv Kor rov guorrtilv aprOprirv. Ie oun] trlv nep(moor1, 0o'uno0Cooupe 6tr or opr0po( poq ougpol(ovror F€ rn 6ero6rrrj touqIopgri.Oo uno0iooupe, entnl€ov, dtr €Xoupe onerxov(oer gdvo rouqnpoytrrotrrofq oprOpodq nou Bp(orovtor ovdgeoo oro O Kor to 1. AvpnopCooupe vo ono6e({oupe dtr ot npoygouroi oprOgo( ovdpeoo oro0 ror to 1 e(vor 1rrl aprOprlorgor, tdrr e(vor oryoupo 61 6tror or fipoypo_nroi opr0po( e(vor p4 opr0prjorgor.

Aq uno0Cooupe 6tr iXoupe yprirlrer oe nivoro dloug rouq Trpoypol_roUq oprOpo0q ovdpeoo oro 0 Kor ro 1, iXovroq xdver qv nopordruronerr6vroq:

'I e 0,1 792038409827...

2 e 0,3755500000000...

3 e 0,0001788345441...

4 + 0,4998783333333...

5 e 0,8455967928739...

O nfvoxoq, peBo(urq, ouveXi(eror en, dnerpo, ogod undpXouv dnrr_por guorro( opr0po( ror dnetpor npoyportro( opr0po( ovdpeoo oto 0Kor ro 1. Kdnoror on6 rouq npoygo.roriq oprOgoriq tou n(vqro 00 e(vorpqro( oprOpo( pe nenepoop6vo rj nepro6rrd 6ero6rrd {nq(q, evdr <itrtror0o e(vor dpp4ror, pq nepro6rro( oprOpo(. H un60eo4, BeBoirrrq, e(vor drr6}or or npoypotrrcoi oprOpo( ovdpeoo oro 0 ror to 1 0o neprlotrrBdvo_vror orov n(voro.

fro vq ono6ei,[oupe du undpXer ovtupoorl, ro p6vo nou Xperd(erore(vor vo O4proupyr]ooupe ivov oprOpd Trou vq pp(oreror ovdlrroo oro0 Kor ro 1, o onoioq vo Frlv undpXer orov n(voro. Aurd ro rdvoupe cuqeffq: yro t4v np6t4 6ero6rrrl 0co1 6e{ui ond qv uno6roorotrrr enrr€- zg3

Page 269: Ο Ταξιδευτής των μαθηματικών

t ( E o A A A t O I O

youpe €vo qrlrp(o 6rorpopetrrd on6 ere(vo nou undpXer orov opr0pdnou one rrovf(eror oro 1. Enil€youpre 6qtro6rj €vo {rqrp(o Ororpopetrxdond to t. Ag enrtri(oup€ rov oprOpd 3, npdypo nou oqpo(ver 6rt oopt0p6q nou 6qproupyoUge opXr(er U€ ro 0,3... fro ro Oefrrpo rfqgiopog nqyo(vouF€ oro oetirepo npoy;-rotrr6 'ptopd tou n(voro xor enili-youpe 6vo r!4rp(o 6rorpopetrx6 ond to 6edrepo rfrqg(o rou 6erirepouopr0poU. Enil€youpr, ouven6q, €vo {rqrp(o 6rorpopeux6 ond ro Z. Aqnodpe 6rr e(vor to 4. O oprep6q nou 64groupyo6ge y(vetor trirpo 0,34...IuveX[(oupe ge ourd rov rp6no, enttr€yowoq ndvro ro vroor6 rfrqrp(o6rot tiore vo e(vor 6rorpopelr6 on6 to vroor6 rlrqg(o rou vrooroUnpoytrrortxod op109o6. 'Orov 6qtruoupy4Oe( ourdq o opr0p6q, Fnopo0psvo 6odpe opdourg 61 6ev pnope( vo e(vot €voq on6 rouq oprOpo0q roun(vord pog. fror(; Ag unoO€ooupe 6u rdnotoq unooqpr(et noq o opr0_poq nou 6qproupyrjoope e(vor o vroot6g opr0p6g rou nfvoro. Mnopori_p€ vo roF€ oro vroor6 $r1rp(o ourod rou vtoorou opr0trrori rot vo 6e(-[oupe drt our6 e(vor 6ro<popnrrd ond ro 6116 pog vroor6 rf4rp(o. Og errofrou, o n(vorog ruJV npoypolrdrv opr0prilv 6ev pnope( vo e(vot ntrr{-pr1g, 6noq orprpriq 6ev rirov nlripnS 11 orolouO(o rov LU orrlv Trporl_yoUpevq on66er(4. Aqtro6rj, undpXouv ndpo notrtro( npoyporrro( optO-

. po( yro vo on€trovtorotiv en( r'v rpuorr6v opt0pdrv. 'Exoutrre ro nopo-rdro undpoXo Oedrprlgo, ro ono(o orpefioupe orov Kdvrop. fror( e(vorundpoXo; ftor( e(vor rdoo ofwogo ror, rDor6oo, rdoo toXup6.

oEopHMA: To orjvolo dlov rov npoypolrtirv opr0ptirv 6ev e(voropr0prjorpo.

O Kdvtop i6ooe ov6proro orouq ntr40rro6g oprOpodg rou ouvdtrourov rpuotr6v opr0p6v, otrtrd rqr rou ouvdlou rov fipoygortxrirv opr0_prirv. Or rpuotro( oprOpo( (rot, Kor, enirroo4, dtro ro oprOprlorpo orivo_lo) €Xouv rov nlr10rr6 opr0p6 No, o ono(oq e(vor ontrdrg ro npurroypdppo rou eppoirod okpdpr;tou, ro dlerp, ge 6e(rr4 ro p46€v. Tovnl40rr6 optOpd rov rpoyportrdrv opr0gtiv rov ovdpooe N,. Xoprl orovKdvrop, €Xou;.re ovorotr0rpet rdrpo pro peyd}4 otr4Oero yro roug opr0_

x&4 gor1g. fvurpt(oupe 6rr ro odvolo 6ltrrv t,v quorK'v oprOpuirv, drov rov

Page 270: Ο Ταξιδευτής των μαθηματικών

t l l fToPIA TOY t l

.rloopdtov rqt 6trtDv ttrrv otryeBprrtirv oprOgriv 6Xouv tov 6ro ntrq0tr6

opr0p6, rov No, uot6oo or d}tror oprOpo( qg euOe(oq tov optOptitv, ot

uneppouroi opt0po(, €Xouv peyotrfteprl nurv6tqro Kot oxrlpor((ouv €vo

ofvolo nou 6Xet N, orotXe(o.

E6tir pnopo0pe vo mopottiooupe trQo, Yto vo ovotroytorofpe rt oKpl-

ptirq uno6qtrdlverot 6tov }€pe 6tr ot unepponxo( oprOpo( e(vor pq oprO-

prjorpor. Ixegte(re 6tr Otiroope €vo ouyrerptpdvo optop6 Yto rov unsp-

porrr6 opr0g6 e. E(noge 6rt e = lim(t + 1/v)". fleptypdrpope ouven6q to

e p'dvov nenepoop6vo opr0p6 ouprpdtrrrrv. Mnopo0pe vo op(oouge 11 vo

neprypdrpoupe xoO€vov on6 touq uneppotrrorig oprOpofq g'6vo nenepo-

opdvo odvotro oupp6trrrrv; H ondwrlorl e(vor 6Xr. Av pnopotioope, r6te

0o pnoporlo-ape vo 6rotdfoup€ ro odvotro 6trtuv rov nenepoop€vov ne-

pryporprirv o' €vo oprOpr]orpo o0votro. Enop€voq, 0o pnopofooge vo

lprlorponor{ooul.lt uq nenepoopdveq nepryporp€g rrrg Cvo e(6og onetr6vt-

olq rov unepponrtiv opt0ptirv en( tutv rpuotrtiv oprOptirv, xot r6re 6e 0o

{rov pq opr0priorpot. Aur6 o6r1ye( oro nopd[evo oupndpoopo 6rt 0u

np6ner vo undpXer €voq pr1 opr0gqorgoq opr0p6g uneppotrr6v opt0ptitv,

roug ono(oug 6ev pnopo0ge not€ vo neprypd{routrre p' 6vov nenepoop€-

vo opr0p6 oupp6trurv. Oq er to0rou, undpXet 6vog prq oprOpriorpoq opt0-

p6g oprOprirv nou 6ev pnopofpe not€ vo neprypdrloupe. Iqpo(veq dpo-

yg our6 6tr undpXouv ouyrerptpdvor oprOpo( nou 6ev pnopotiv vo nept-

ypoqodv; 'OXr onopo(rr1to. Touq ouyrerptpdvouq optOpofq pnopoUpe

ndvro vo rouq petretriooups Kot vo rouq neptypdrloupr€, €^n((ouFle, p'

dvo nenepoopdvo o6votro oupp6lrrrv, Kot vo touq npooO€ooup€ orov

opr0priorgo n(voxd Uoq U€ trq nepryporp€g turv unepBotrrtirv opt0gtitv.

Yndpxer rdrr ovdptoct oro No tcol N' ;

'Onulg ouppoivet ouXvd oro poOrlpottrd, tr1 ottypri nou 0o onowrl0e(

Cvo epdrqpo eprpov(owor [ogvrrd xr dtrtro epurqpoto nou onottofv

onowrloetg. An6 q ortYpli nou o Kdvtop on€6er(e du undpXouv 6t1o

6rorpopeuxd onetpootivotro, ro No Kot N,, to endpevo eptirqpo ritov 285

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K E O A A A I O r O

ov undpXrr rdnoro onetpoorivolo rou vo Bp(oreror ovdprod rouq.YndpXer, 64Io6{, ivo onerpoo'volo nou vo e(vor unepBoltrd peydloyro vo on€rKovrore( en( rov rpuorrtirv opr0prdrv ror uneppolrrd grrp6 yrovo orr€rKovrore( en( rov rpoygorrrdrv opr0pdrv; Aur6 ro eptirrr;go iyrveyvoor6 ug q un60uq rou ouveyolg. O Kdvrop 6ev n(oreue nurq unap_Xer €vo tdroto orjvotro, otrtrd 6ev pnopo6oe vo ro ono6e([er. ftq vo ono_vrqOe( ourd ro epdr4po inpene vo neptp€voupe p€Xpr ro .1963. Erce(vr114 Xpovrd, o veop6g po0rlpour6q l1otr Koiv, nou yrvvr]0r1K€ ro 1934,oni6er(e drr ro epdrqpo rjrov avorogdooto. OFoq, u orprBdrq o4_po(ver out6; Kord rovdvo orerpt6poorr epurr4;.roro rou ovor, ror roun6Xt,, epor{poto 64lo6rj nou dlouv 6do nrOov€q onovr{oerg, e(re(vor> €(Ts n6Xtr. AIld ge ro eptr.ru]trroro ruJV po0q;.rorrrdrv, ro npdypo_ro 6ev e(vor onopo(r4ro €ror. Iroug enror{povrg rrlg loyrrrjq optoer vo64proupyodv rouq 6rdrpopouq tope(q rtrlv po04porrrtiv ond odvotrqo(ropdrurv nou ono6iXovror oq olq0rj. Aurd orprp6q €rove o Eurtrei_64q drov ov€nru{e rqv onoyoyrrrj yrurgerp(o rou. roxup(orrlre rdnoronpdyporo, ro ono(o 6ev on€6er(e otrtrd onldrq to onodiyrrlre og 6ror_oOqrrrd oI40{. It4 ouvdXero, ro XprlorporoQoc yto vo ouvoydyrr yrtlr_petprrd 0etoprjporo.

Tr oX6oq iXet ourd F€ ro ovoroqdororo; To n€pnto o{iurpo tou Eu_rtre(64 l6er drt yro rdOe euOe(o ror €vo oqpe(o ert6q euOe(og, un<ipXerp(o ror p6vo p(o euOe(o nou nepvdsr ond ro oqge(o ror e(vor nopdtrI4_trr1 pe r4v opXrrrl ruOe(o (Err6vo 45). l-ro notrIo 1p6vro, or proO4porrro(n(otruov du our6 ro o((urpo 6ev r]rov outond6errro ror drt 0o dnoenevo ouvoX0e( qnd ro dllo ofr6poto. AniruXov 6purq dtreq or npoondOer_€g roug vo ouvoydyouv to ndpnto o{(urpo. Irjpepo yvop(oupr 6n e(_vor o60vqtov vo o6qyrl0odp€ ogqrpcrrKd oe our6, npdypo nou o4po(_vrr drr yro to nipnro of(urtrro rou Eurtre(64 roXUer ro i6ro dnoq *o,'y,orrlv un60eor1 rou ouvexodq - e(vor ror to 6rio ovonorpdoroto. Av odlou_Fr Fro ycoprrp(o nou vo nepr6Xrr ro n€pnto o{(opo, 0o np€ner vo rooupnepildBouF€ oq pro ovon66errr4 np6rooq (o{(urpo).

H Oeurp(o rcuv ouvdlov, 6ror dnoq qv ovimu{e o Kdwop, ovopd(e_rat on)oi'rq 1cupfu ruv ouv6)uv ror poo((eror ottq 6roro0qtrr€q 1616_

286

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H I I T O P I A T O Y f i

euOe(o nopdtrlrltrq ou'1v opltrqeu0e(o

\

\\oqpe(o

oppr{ euoe(o

45. To n€prnto o((or;.ro tou Eurtre(6r1. Ao0e(or1q grog euOe(oq rot ev6q orlpel'

nou 6ev ovrjreroe ouui tqv euOe(o, on6 qut6 to orlpe(o nepvdet p(o rot g6vo

eu0e(o nopdtrtrqtrr1 otqv opXtrri eu0e(o.

rtrlv ouv6ltrtv. 'Opoq, q o0yXpovq Oetrrp(o ttrtv ouv6trov poo(etot

ro'dvo o0votro o(rurpdrurv to ono(o ovonr0X04rov on6 6do paOrlpott-

rofg, rov Epvot Zdppetro Kot rov Avrotrrp Op€vretr. Autd ro o(ttitgoro

ovoprd(ovtot tdrpo o(rtirgoro rov Zdpgetro-Opdvretr r1 onldrg o[ttitgoro

ZOK (Z6pgetro-Op€vretr-Kdvtop). An6 ro ZoK 6ev pnopodpe vo o6qyr;-

0ofge orgotpettrd oro ov eivot olnOriq n dXt n undOeoq rou ouveXodq.

Av O€troupe 11 un60eor1 rou ouveXodg vo onoretre( tprlpo rrlq 6trr1g poq

0eurpiog rov optOgtirv, 0o npdnet vo Bporipe €vo o((urpo nou 0o tqv

eyyudtor Kor vo ro npooOiooupe oto o{ttigoto zoK. Yndpxouv er6oXdg

qq Oeurp(oq tulv ouvdtrcuv nou e(te neprtrogBdvouv eite 6ev neprlotrrpd-

vouv illv undOeoq rou ouveyotlg. To epdrqpo rtipo e(vot: ntig }ettoup-

ye( o rdogoq poq; E(vot, dpoye, rol0tepo Yto Uqq vo Xplotporotrloou-pe trlo Ortrtp(o ouvdtrtuv dnou 11 un60eoq rou ouvelo0g e(vot olr10r1q {6nou e(vor rfreubrlq; Tr rorprd(et rotr0repo pe lg ovdyreg gog; flpog ro

nop6v 6e yvtrrp(oupe rrlv.ondvrqoq outo0 rou eporrlgoroq.

H 6rdrproq ovdgeoo oto optOgrlotgo o0votro ( No) rot to gr1 optO-

trrrlorpo orivotro ( N,) 6ev e(vot ooripovq. To opr0priotpo o0volo ttrtv

opt0ptirv, ro ono(o ouvriOtrrg rout(oupe yto eurotr(o p€ rouq rpuotroUg

opr0porlq, {tov ro nptirro onetpootivo}o nou 6ptoov rot petr€tqoov ot

dvOponor. Itrl gorpoXp6vto npoondOeto rotovdrlorlq rLuv enerrdoetlv 287

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t ( E o A A A t O I O

opr0priorpo euOrjypappo tgri;roto

pq opr0prjorpo euOriypopgo r;rripoto

EIKoNA 46. Apr0gr]orpo rot yr1 oprOp{orpo odvotro. Ie pro eu0e(o dnerpou prlrouq,pnopoUpe vo Xopiooupre €vo dnerpo oMd oprOtrrrioryo nlriOoq ure,iypo1l1rrulfnlr,i-ttr-rv. 'Oprrrq, orrlv f6ro euOe(o, 6e 0^o Xrrrporioe 6vo'p4 opr0'rio,po ntrrj0og.rOilili,

Fov rgnpdrurv, 6r1lo6rj ov ro ntrriOoq rouq {tov prl oprep4orpo, .u yip., e";lii;,ITTOV.

oro x6po Kor r,v ouv6e6rp€vtrrv ge our€q peyeotirv dxoupe ouvovtrioer€vo dltro, 6ropopetrrd onetpoorivolo, to odvolo N,, nou ;reprr€g go_piq oupBotrr(eror Kor ,q c (nou ovtrororxe( or4 tri{q continuum, outvt-Xiq).'Xpero(6poote ourd ro peyolftepo orjvolo c yro ro po04potrrd[.roq, Kor Xpero(6poote. ro po04porrrd pog yro rlq rpuorr€q enronjge5pog. Mro onlrj ond6er{4 rou 6tr ro N0 Kol c e(vor 6rogopetrrd go(verortog efiq: Mnoporipre vo tono0et{ooupe €vo opr0;rr]ortrro, dnerpo nlrjOoqeuO0ypoppurv tp4pdttrrv ndvo oulv eu0e(o rov oprOptirv, brXtoq ro €vovo oyyt(er to dllo (Err6vo ae. Av dpurq ro enrXerprlooupe ourd p, ivop4 opr0prjorpo, dnerpo ntrr]0oq tprlpdrov, rdre rdnoro qnd ourd 0onprinrr vo oupn(ntouv. flpdypotr, ov o opr0gdg e(vor p4 oprOpr]orpog,r6re ev piper oupninrouv.n

'EXoupe neprypdrlrer rdrpo ol6rlrlprl rnv euOe(o rov npoypo'xdrv

oprOptirv. fvtrrp((oupe 6tr qnoretre(tor ond guorrorig opr0goriq, rfuiogo_to, olyepprrodq oprgpo0q - 6lo opr0p{otFo _ xoOtirq ror ond rouq no_pd(evouq unepponro0q oprOporiq, or ono(or 6ev e(vor oprOprjor;ror. Trdllo 0o €npene vo yvurp(oup€ yro rouq oprOpoOg;

8. Continuum hypothesis: ynd0eo4 tou ouveXodg. (I.t.v.,19' Leo Zippin, Llses of rnfinity (oudotuy*-*, r'r,. Mathematicar Association olAmerica. 1962), o. 56.288

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rcrriYorruog ro pooilerooprOpoi

Mr pro nptirq porrd, pnope( vo rpo(veror 611 dXoupe onovrqoet oro po_oud eporqporo yro roug opt0po6q. yndpXer Fro opqlpovoo4povrrl0vlororX(o ovdpeoo oro orlpe(o r4g euOe(oq rov oprOpdrv Kclt orousnpoypolro6g opr0po6g. 'EXoupe e{orrero0e( pc ro 6rdrpopo e(6r1 rurvnpoypolrdrv oprOpdrv ror yvurp(ouge to ndvro yro ro opr0pr]orpro rorio pr1 opr0prjorpo orivotro, F€ rouq nopd{evoug unepBorrrofg oprO-po0g vo onotelodv 6vo potrtr6 nopd6erypo ev6g pr1 optOpr]orpou ouv6_trou. Tr dtrIo undpXer;

'Orov e{erdoops rlv €(oXr1 on66er(r1 rou Kdvrop oXertrd pe ro 6rror oAyeBprro( optOpo( r(vor opr0pr1or1lor, ouvovrqoopr ruyo(o rivo nq_pd(evo orippolo, ro ono(o or6nrpo oyvor]oope.'Hrqv to V-i, to ono(oeprpov(or4re rSoo orrlv opvrlrrrq 6oo rot orq Oertrrl rou proprprl. Tr ororotr6 e(vor n rfipoyovrK4 p((o rrlq opvqrrrrjq trrovd6oq; Oo np€ner voe(vor €vog opr0p6q o ono(oq, ov rov urfdrooupe oro rerpdytrrvo, poq 6(_vtr €vov opvqrtrd opr0p6. 'OFrq, on6 ng r6t6r1reg rrlq optOpqrrrrjgyvr,rp(oupre 6Tr, ov notrtrqntrqordooupe 6Uo opr0go6g nou €Xouv ro (6ronp6or1po, ro yrv6pevo e(vor ivog 0errr6g opr0pdg. Og er ro6rou, ovurfriroou;re oro terpdytr_rvo rivov opvqrrrd opr0pd, ro onordtrrogo e(vor0errr6q opt0p6g, dnurq rr 6rov urlrtirvoupre oro rerpdy,vo €vo Oetrrd 3&9

I

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opr0pd. Av urprirooupe oro tetpdyovo ro pq6€v, ro onor€treopo e(vot

ndtrt p46€v. Oo(veror, enop6vog, 6rt 6ev undpXer rovdvoq rpoyportr6g

opr0g6q otqv euOefo ttrtv opt0pdrv ilou, ov tov utptitooulte oro rerpdytrt-

vo, 0u gog 6tioet dvo opvrlrtr6 ytv6gevo.

Mnope( vo pnotlpe orov netpoog6 vo onptir(ouge rdrtl on6 ro Xotr(ro V-r Kor vcr to {eXdooupe. Aut6 €rovov notrtro( po0qportro( en( ottir-

vrq. 'Opulg, ro np6Btrr1go 6ev t{orpov(orr1re. Iuvovrrjoo;.re ruXo(o ivo

Oerrrd ror €vo opvqrrr6 {-, nou onote}ofoov trdostg tou notrutlvtipou

x' + 1 = 0. Aurd e(vor dvo oprerd ontrd notrutitvuFo, rou ono(ou xot ot

66o l0oerg e(vor out] q nopd(evq rerpoyovrr{ p((o ev6g opvrlttrof

opr0pro6, rov ono(o 0o ovopdoouyt apvrprc6 pfir6.To noludrvupo 6i

0o dXer roptd tr0or1 ov 6ev e(rlyqooupe ge rdnoto rp6no to y'-r. tt ro-

rdotoorl y(veror ox6go nto 66orotrr1, ytor( out6 to e(6oq ldorlg dXet

. notrlig popq€q.To notrutirvu;.ro x'+ 2 = 0 €Xer 6do trdoetq, rrg y'- rot-{j. np(rret, trotn6v, vo ppo0pe €vov opt0;.r6 TIou, ov tov urlrdooupe

oto rerpdytrtvo, 0o toofrot ge -2! Avriloppov6poore opdotrtg 6rt yto

rd0e Oetrxd npoygottr6 opr0p6 r 0o np€ner vo undpXet 11 rerpoyovtrrl

p((o rou Ut(ov r, tr6yur rou noluovdpou x'+ r : 0. fto vo enilfooupe

our6 ro yevrx6 nolutirvupro np€net vo perorpipougs ro r oro 6e(( ox€-

Ioq rrlg e((ouror1g, rdvovtdg to opvrptr6, rot perd vo ppodpe rrlv r€-

tpoyovrrq p(o ouro0 rou opt0pto6. Aurd uno6qtrtivet drt on6 outd to

nopd(evo opvrprrd pr(rd undpXouv t6oo 6oo rot ot Oettro( TrpoYpott-

ro( oprOpo(! lto vo y(vouv ro npdyporo or6go nto notrUntroro, dyoupe

ro notrutirvul.ro xo + 1 - 0. Enildovroq outr] rqv e[(ooor1 pp(oroupe 6u

x =4,[i, np6rerrot 6qlo6( yto 6vov opr0p6 o ono(og, drov notrtrontro-

otoore( ge rov eour6 rou tp€tq qopdq, ro onotdtreopo e(vot €vog opvq-

rrr6g opr0p6g. Og er to6tou, undpXouv rt dtrtro pt(rd n€po on6 rrlv

rerpoyovrrrl p(o, ro onoio nepilcpBdvouv ourofq roug nopd[evoug

oprOpodq. flo0 orqv euXrj pp(orovtot, orpo6 6e pp(orovtot ndvo otqv

Eu0e(o ttrtv npoygottr6v cpr0p6v; YndpXouv, dpoye, orqv npoYUorF

r6rqro;En( notrtro0q ot6veg, ot goOqprottro( onldrq oyvoodoov ro pr(rd tutv

299 opvqrrrc0v opr0puirv. Av 6vo notrutirvupto efXe urq trdoetg rou p6vo opvrllrd

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E N E K T E I N O N T A E T O B A E I A E I O

r6re ot poOrlgouro( 6rjtrurvov 6u ro notruritvupo qrov dluro. O pre-'Etrtrr1voq olyepprmrlq, o Ardrpowog, ondpptrne otrorlrlpolrd

opvqtrrd pt(rd urq pr(eq!' 'Evog on6 rouq npdrtouq nou or€<prrlrov 6rtrerpoyovrr€g pi(eq turv opvqrrdrv oprOptitv 0o pnoporioov vo e(vot 16-

notruovdgtrrv rjrov o Algn€p Ztpdp (1590-1633), roOdtg rot triyot dtr-

nou d(1oov to 66xuro €B6ogo ourrvo.' Xptrdm4re vo o}trd(er ndpo16oo 11 dtryeppo 6oo ror q yeoperp(,q nporof our€g ot nopd{eveg

pnop€oouv vo Bpouv pro oo<potrrj po0qgortrq Bdoq. Aur6 ou-l6yur rurv npoonoOerdrv 60o owptitv toug ono(ouq eXoupe {6q ovo-rou Pavd Nrerdpt (Kopr€orou) Kor rou Koptr l-rdoug. O Ntexdpt

ro onopo(rr1ro epyole(o, evdr o frdoug 66ooe qv retrrr{ tr0oq.

y6veon rnq ovoAurKnq y€t,r . tp€rpioq

rr1 6rdprero rnq (onq rou, o Pev€ Nrexdpt (Err6vo 47) 6e ouver-nori 6tr to dpyo rou 0o €Oere ro 0ep€trro proq evtetrtitg vi-

rtrdor1g oprOptirv. Iqv npoytrronr6rqro, onfpprnre rrq rerpoytovrr6grov opvquxdrv oprOydtv og ntOovdq trUoetg nolutrtvfpov rt €rpro-

pdtrroro, oro oqpe(o vo enlvofoet to odyXpovo 6po cpowaouxo(yr' our€q uq p((eq.

O Nrerdpr yew{Oqre mrq 31 Moprfou 1596, orq Ao Ar, ou1 fotrtr(o, oeoprmorporrrq yotrlrrq orroydvero.' Aprerd erinopoq 6ort vo Urlv ovrl-

yro 14 6oulerd, elehX0rlre o'€vov euyevrl rou 6€rotou dp6ogou or-

dvo nou rou dpeoov ro ruXepd notXv(6to rot ot yuvo(req, o ono(og 6ptrtg'6d0ere e[o(pet4 rloorxrl not6e(o. ho vrdto tou 6toor€6o(e unqpertivrog

mo mpor6 xor notreptivroq mo ntreupd btdrpoptrtv Eupono(utv npryr(nov.

Ilopdtrtrr1lo, r]rov 6vog elofperoq po04potrrdg rot rprtr6oo<pog, o ono(og

'1. Kl ine, o. 143.2 . 'O. r r . , o .253.3. Eric Temple Bell ,

1965), o. 3s.Men of Mathematics (N€o Y6prr1, Simon and Schuster,

Zg7

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K E O A A A I O 1 I

E IKoNA 47. Pw€ Nr€Kdpr , 1s96-1650.(Ytrrx6 on6 touq o6ekporig Mnpdouv, hiptrtvyx)

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E N E K T E I N O N T A E T O B A E I A E I O

en(or1q Xp6vo yto vo ouppdtret oug <puotr€q entonipeg. O r6opoq

yvop[�(er neptoodrepo on6 ro nep(rpqpo enrXe(pqpd rou Yto qv Unop-

nkdrpropot, dpo undpXur". H nto orlpowrrrl ouvetorpopd rou €(vot ro

pe to ono(o ooXolo6pome e6dr: q ovotrunrri yeognp(o.

Ynor0etor 6tr otqv qtrtr(o rov e(root tpttitv ettiv, rt evti unqperof-

mo pouoprrd orpotd, o Nterdpr e(6e rp(o rpoullottrd 6vetpo, to

epprlveuoe og npoet6ono(qor1 yro vo eyxorotre(q,st rrl Fn Tropo-

(orj tou Kor vo ouvttorp€pet oto poOqgortrd, orq rpilooorp(o rot

quorrdq enrorrlgeg. Arrlyorivrov 6rr €trope 6vo goyrr6 rtretO( oto

Tou, ro ono(o 0o tou dvotye ro 6p6go yto ndpo notrI€q otr{-' flot6 6ev onordtrurpe rt rltov out6 to proytx6 rlet6L atrtrd ot Bto-

tou und0erov 6tt qnotelodoe u1 pdorl qq yeopetp(oq rou.

Kotd touq Klootrodg Xpdvouq orrlv opXo(o Eltrd6o, 11 ovordtru{r1

oo0gpetpou tqq 6royrrrv(ou (6r1tro6ri, 6tr ro y'7 e(vot dppqroq oprO-

xot 6ev ptnope( vo ovoilclpooro0e( oq rldogo) irove touq rope(q

d\yeppoq Kot rrlq yetlpetp(og vo orotrouOrioouv Dtogopertrotig

. H dtryeppo ooXotrri0qre p€ rous opr0godq, ev6 q yeopetp(o

K€ Fr uS entrrdoetq oro Xrirpo. O Nrerdpt pn6peoe vo rp€pet

rovtd outorig touq 66o €ntorrlgovtKo0g rld6oug. Aur6 to rororpe-

pe pro ontrri rorooreurl. fl{pe nprirro rqv euOe(o tulv npoyportrfv

(nop' 6Io nou ere(vo ro Xp6vro 6ev pn6peoe vo errtpqoet

rov ntrofro ourdg tqg eu0e(og) Kot rrlv rono0€rqoe opr(6vtto.

nrlpe pro 6e0tepq eu0E(o ttrtv opr0gtirv ror q oXe6(oo€ Kd0sro,

6ote vq ripver rqv nprilrq eu0e(o und opO{ yurvio (Err6vo 48). Kd'

eu0eio ovopd(tror d{ovag xot ro o4ge(o 6nou ot euOe(eg ouvovrdt-

t(vot q appy tuv a{6vuv, otqv ono(o erXope(ror 11 ttp{ gr16€v. H

rd0e orlpe(ou tou entn66ou gnope(, enop€vog, vo roOoptore( pe

Bdorl 6do opr0po0q: tqv on6otoot] tou rotd pqrog rou opt(6vrtou

rd[ovo on6 qv opXrl Kot ulv on6otooq tou rqrd griroq tou rd0etou

d[ovo on6 rqv opXq. fto ouvtop(o, o opt(6vttoq dfovoq ovogd(erot

d{ovaE ruv x, evdr o rd0etog d{ovog ovogd(etot d{ovaE tuv y.

4. Holl ingdale, o. 126. 293

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K E O A A A I O I I

/eu0e(q

rov opr0truirv

(:1,5, -2)

eu0e(qtt lv opt0ptirv

EIKONA 48. Kopr€orovd oriorqpo ouwetoypdvov. Kd0e oqpe(o tou enrnC6ou op((e-ror ;rdvo on6 6Uo npoygotrro0g opr0po0q (x,y), 6nou to x e(vor 11 opr(6vtro on6-moorl Kor to y r(vor 4 rdOeq ondoroorl ond rqv opXrj tu_rv o{6vurv.

fto nopd6erypo, orqv Errdvo 48 undpXer dvo oqge(o ndvo ond rovoprOpd 2 rou d(ovo rov x Kql oro 6e(r<i rou opr0go0 3 rou d{ovo rovy. M6vo our6 ro orlge(o npooOrop(eror ond rouq 6rio oprOpofq (2,3).Mr our6 tov rp6no, rd0e oqpe(o npoo6rop(erqr ond €vo orjvotro 66oopr0ptirv. To 6e0repo oqpe(o orqv Etx6vo 48 pp(oK€rol oro (-1,5, -2).Aur6 o4po(vet 6tr yro vo evton(oouge to oqpe(o ond rrlv opXrj rrrrvo{6vrlrv, npoXrrrpdge 1,5 povdOeq Trpo( ro oprotepd ror, nd}r ond rqvoppj rov o{6vrrrv, 2 govd6eq npos ro xdto. Or 6fo optOpo( rou ouv-6€ovtor pe rdOe oqge(o tou mrn€6ou ovogd(owor ouvttroylt€veE rouorlpe(ou. l-lpoq upqv rou Nrsrdpt, ovogd(ouye our6 to odor4po rap=tnnv6 otiot4 pa ouvteruyp€vuv.

O Nterdpr npdogrpe notr0 ncproodrepo ond qv onhj erpedpeorl294 outorj rou e(o(perou ouor{gorog yro rov npoo6roprop6 rov o4pe(r.uv

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enrrd6ou trtg ouv6trutv opr0ptirv. Iuvet6rltono(r1oe, el(oqg, rrl Xprlot-ouro0 rou ouot{poroq Yto ulv nepryporp{ rov yetlpetptr6v

trrq olyepprrdlv oX6oeov. Aut6 e(vot to rtret6( rrlg ovotruttrqg

H otryeBprrri e((otrrorl ev6g rdrtrou, tou ono(ou ro x6vrpo

otqv opyrl rov o(6vtrtv, e(vot 11 x' * y' = r', 6rtou ro x Kot y

uq ouvrstoypdveq rrrrv oqge(urv rqg neprrp€petoq tou rU-

Kor ro r e(vot n ort(vo rou rfrtrou. flop'6tro nou o r0rtrog eiXe ge-

lnq0e( rotrd mqv opXor6trlto, q ovolutrrri yetopetp(o gdg nrlye notr0

nopondpo. Mog €6uroe rr1 6uvor6trlro vo ooyotrqOodpe ge rtg xopnU-

leq, trq onoieg Xerpr(6tov 66orotro q eurlef6eto yeurperp(o. H e{(otloq

yro qv €trtrtupr1, tov entprirq rdrtro nou neptypdq€t rtq tpoXt6g ttrtv

nlov4r6v, e(vor 11 x'lo2 +y'lp'= 1,6nou ro o Kot p e(vor o ptydtroq rot

o pup6g d(ovog. H nopoBotrr] eivot dvo odvotro oqge(rrrv nou pp(oxo-

wor oe oroOeprl on6orooq on6 6vo oqpe(o Kot l"tto euOe(o. H e((otlorl

rrlq e(vot q y2 = 4mx, dnou m efvot 4 ondoroo4 tou oto0epol oqge[ou

ond rqv opXri tutv o(6vtrtv.

Or ro;.rn6treq, 6ntrrg 11 nopopotrrl rot q €trtrerrlrq, poqOodv oulv r€pt-

ypoqrj notrtrdrv rpuotrtirv rpotvog€vtrtv fiou noporlpodge y6ptrt poq' To

drt pnopofpe vo ovonoptotoripe our€q ttg oriv0ereg yeurpetptr€g ro-

pnrltreg urq olyeBptxdq oX€oetq rotdorqoe 6uvorq rq ofyXpovq yeo-

perp(o. 'Onurq 0o 6o0pe, ro v€o o0orqpo tou Nterdpr Yto ulv ovoro-

pdoroorl rov orlpe(ov rot€orqoe en(oqg 6uvorrl qv rotov6qorl rulv

opvqrtrdrv pr(rtirv. krlv enoXq tou 6ptlg, o Nrerdpt 6ev rltov €rotpoq

vq rdver our6 ro Frlpo. H Xp{oq tou ouorrlgotog ouvteroypfvtrtv tou

Ncrdpr yto rrlv rorov6rloq tulv opvqrtrdv pr(rdrv €npene vo neptg€-

vet to dpyo tou Koptr l-rdouq oXe66v 6tor6oto Xp6vto.O Ntexdpt pprlre tuyo(o ro rlet6( rou Yto tqv ovotrurtrrl yerrrpetp(o

ro 1619, oor6oo 6e 6qpoo(euoe to onotetr€opoto g€Xpt ro'1637 - pto

xo0uotdprlorl 6erooxrti ettitv. Ie our6 to Xpovtr6 6tdorrlpo, nolipqoe

oe Ordrpopeg erorpote(eg xor notrtrdg rpop6g 6tonv60veuoe ooBopd.'Onrrrq dXet entoqpdvet o Mnetr oro prptr(o rou Ot dv\ptE tuv pa9qpau-

ruiv, yw o6ionorq o<go(po on6 pouor€ro 0o pnopoUoe vo 6Xet r€pgq-

r(oer rr1 oto6to6pop(o tou Nte rdpr notr6 nptv on6 ro 1637, Kt o r6opog 295

Page 281: Ο Ταξιδευτής των μαθηματικών

0o dnpene vo neprpCver rr dtrtro yro ulv notrd onou6o(o ovordtrug{rou.'To teleuto(o Xp6vro rrlq (or]q rou Nrerdpr r]tov orptepopivo mr1rpilooog(o, oro trroOqpourd rot orrq rpuorriq enrorrlpeq. To 1650 opptit.orrlo€ Kor n€Oove oe qtrrr(o nev{vto tpuiv ertiv.

O l - rdoug npoo6ropi ' (€ lrouq prYcr6rKoog opropooq

O Koptr Op(vrprX frdouq yevv{0qre otrq 30 Anprtr(ou 1777 un6 ronet-vodg oyp6req yove(q.'Hrov €vo nor6[-0ot1po xor q pqrdpo rou, q Nro-po06o Mnev( frdouq, nop6rpuve ro yro rrls vo onou6doet yto vo ofit-treu0epr.u0e( on6 rqv oyponr( rotoyoyri rou. Aurd ro €rove ge q po{-

0ero tou 6ofro rou MnpdouvopoiX, o ono(og Xop{yr1oe orov frdoug

€vo en(6opro Trou rou €6rrroe rr1 6uvor6rqro vo eyyporpe( oto KoldytoKopotr(ve oe qtrrr(o 6erondvre ertiv ror oro flovenroulltto tou fx€ttv-yKrv oro 6eroorrti. flqpe ro 6r6ortoptrd tou on6 to floventoujpto rouXitrporevr ro '1799,6vroq p6trrg e(roor 6rio ertirv. Ena6ri 6ev pnopo0oe

vo e{oorpotr(oer pro O€oq poOqpouxof ilou vo rou op€oet, dytve 6teu0u-vr{q tou oorcpooKon€(ou rou flovenrotqgfou tou lrfttvyrev rot, opy6-repo, xo0rlyqujq oorpovop(og. To po0qgortxd 6pog nopigervov rl po-

vo6rrrj €gtrrovq oydnq tou. E{ort(og tov po0qportKdv qvoKotrd{edv

rou, 11 rp{pq rou oq €vog ond rouq peyotr0repouq po0rlpottroUq tourdopou e{ontrdrOqre yprlyopo orqv Eupdrnrl. H npooonrrrj tou (urri

trrot6oo 6roronr6rov ouXvd on6 rpoyo6(eq, dnoq o Odvorog - tr6ytrtooOiveroq - 60o ou(tlytrrv (r1 np6rq op€ooq g6trtq yivvrloe) rot ev6qnot6ro0.u Aur6 €rove tov fxdouq vo rpdper ro Odvoro ror, oro t€treu-ro(o Xp6vro rnq (urtiq rou, vo pqv epnroteUetor ro0dtrou roug yrorpo0g.

O kdouq 64goo(euoe oprerd oqpovrrrd €pyo, ouvetorpdpowoq 6totoroug rotrre(g rqg 0eop(og rrrrv oprOptirv, rrlq pryo6rrrlq ovdtruoqq, qq

5. Eric Temple Bell, Men of Mathematics, a. 43.6 . Mu i r . o . 172.296

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E N E K T E I N O N T A E T O B A E I A E I O

,optrqg yeur;rerpiog, rrlg tonotroy(oq, qq €qopgoYrlq rov ;ro0rlgort-

o.r1u'oo.pouo;r(o, roOtirg Kot ortq enror{geg rou UoYvrlrtolli:jll

xpuototrtroyporp(og, rqg omrxrlg Kclt rou qtrerrprogo0' To npooontr6

orlperurpordpr6 rou' ro ono(o 6e 6rlpoote0qre rrio* Ut:" -"::l,1tJ:

ro0dvot6rou(n€Oovero1855),onordtrur|re6tre(xeen(oqgnpoptr€$etq gq eurtre(6eto yeopetp(o. 0o16oo, epdg e6tir pog ev6torpdpouv ot ou-

vdneteg rqg 6r6onoptrrlq 6torptpqg rou"Onog ovorp6pope npoqyougd-

vuJqr orrl 6totptprl rou ro 1799 on€6et(e yro nptirrq rpopd 6tr rdOe no-

lutirvupo 6Xet touloXtorov p(o tr0or1''Optlg' Yto vo ro ono6e(et our6' o

Ixdoug fnpene vo orergte( pro yeprj 0egetr(ooq rou npoptrqgoroq lov

pt(uirv rurv opvqrtr6v optgpdrv' Arpof ro notrudrvupo x' + 1 = 0 €Xet og

iOor,q [,Ouo pr(rd opvrprrdrv optOprirv' ro otpstrrtir6eq Oe6pqpo rqq

ffVrppoS 0o rpov trovOoop€vo ov 6e pp(oroge rdnotov tp6no vo ro

xuptoTouu€.O frdoug Xpqorpono(qoe ro o6orrlgo ge ttg 60o tepv6peveq opr0-

pqrrrdg euOe(eg rou Nterdpt' Oq opr(6vrro d[ovo tov x 6ptoe rrlv EU-

Oe(o rov npoyporrrdv opr0ptitv, U€ rrlv ono(o e(poore e(otretop€vot'

0g rdOeto d(ovo Xpqotpono(r1oe pto dtrtrr1 oprOpqrtrri euOe(o' rrlv €u-

0e(o ttlv rpovroortrtlv oprOgtitv, rl ono(o e tXe oq pdoq to {-t ' f to eu-

rotr(o, ouXyd ovrtKoOtoro0ge ro y'-t pe to ypdgpo i' Kd0e orlptfo rou

d[ovo rov x e(vot 6vog npoyportr6q opr0p6g' Kd0e oqpe(o rou d(ovo

tov y e(vot 6vog npoyportx6g opt0p6q nou €Xet notrtrontrootoore( pe ro

iri* {-.t An6 out6 ptrdnouge 6tt ro {J pp(oreror gto povd6o ndvt''t

on6 rqv opXq rou d(ovo rurv Y (Etr6vo 49)' evtir rc ztfi e(vot to oq-

pe(o nou pp(orerot 60o povd6eg ndvtrl on6 qv oppl rou dfovo rov y'

*o, ro -{a pp(orerot ontrtirg gro povdbo rdro on6 rrlv opXq rou d[o-

vo rtov y. Tr e(vot 6tro ro orlpe(o tou entn€6ou nou 6e Bp(orovror ndvtrl

mov d(ovo turv x 11 ndu,,r otov d(ovo rov Y; E(vot ot 1'ttyo6rro( optOpof'

€vog ouv6uoop6g npoyportxdv rot qowoortKt0v oprOptirv' Kd0t oqpe(o

ouppotr(etor pe 6rio oprOpo0g, rouq o rot p' rt ouro( ot 60o optO;roi

opiqouu ro pryo6rr6 opr0g6 o + 9i, 6nou ro o e(vot ro npoypotrr6 96-

poq Kot to p e(vot ro qovroortK6 gepog'

Me our6 tov rp6no o l-rdouq 6ptos pro 6teupup€vq rldoq oprogdrv, 297

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t ( E o A A A t O I I

EIKONA 49. To en(ne6o rou frdouq. Or trg6g orov opr(6vtro d[ovo oupBo]((ouvnpoyuotrrodg oproprorlq, evtir or opr0go( otov xdoeto <i{ovo ouppotr(ouv noMontrd-oro rou y'-1. Kd0e orlpe(o oro en(ne6o rou frdoug ouppotri(eror ge 60o npoypol_rodgrpr0poUq oe onoro6{norc on6 ng nopordrur 66o gopg€q: (o,F) ri o + pi, dnoui = V -1. Iuxvd ovopd(eror pryo6rr6 en(ne6o.

roug ptyo6rro0q opr0gofg, p€ rous npoy;.rourodq opr0po0q vo otror€_lorjv €vo unoo0votro rov gryo6trtirv opr0ptirv. Or npoypouro( oprOpo(efvor ere(vor or pryo6rro( oprOpo( rou ro govrooux6 prdpog rouq roo0_ror pe p46iv. Avrrorpdrpurg, o d(ovoq rurv y ou;.rpotr(er exe(vouq rougro0opd rpovroourodg opr0po6g, ot ono(or e(vot pryo6rro( oprOpo( nouro rpoyl.louK6 rouq gdpog roo0ror pe pq6€v. O notrtrontrooroopdg tounpoyl.rottrod opr0pori 1 lrs ro y'J neprorpdrper our6 rov opt0p6 opr_orep6orpogo, 6r1tro6r1 on6 rov d{ovo rtov x npos to ndvo otrlv ovr(_orotX4 06oq y'l rou d(ovo rurv y. Av notrtrontroordoouge ndlt pe toy'-t, o opt0p6q neptorp€<petot (ovd optotep6orpo<po npog ro rdrur,oro ovr(ororXo opv4rrr6 rprlgo rou d(ovo tov x (Err6vo 50). Aurri r16e0repr1 nrprorporprl pdq 6(ver ,n.rt-t = -1. Oq er ro6tou, o noltrontro_298

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T E I N O N T A I T O B A E I A E I O

oroop6g rou 1 p€ ro y'J 6do rpopiq 6Xer urg onotdleopo ro -1 nou rlrd-

poupe. Ev ouvropio, 6Xoupe: t / - t l . f / - t l = i ' i = i '= - t .

E(vor onopoirrlto, r6po, vo 6tootpol(ooupe 6rt or ouvriOetq npd[etg

rqg opr0pquxrlg €Xouv v6r1go pe rouq gryo6tro0g optOpo0q. fto vo

npooO€oou pe 6 6o gryo6rro0 g opr0pto0q, €Xou trts:

( o + B i ) + ( y + 6 i ) = o + 9 i * Y * 6 i = ( o * Y ) * ( P * O ) i

Enop€voq, yto vo npoo06ooupe 6rio pryo6rxodg oprOgodg, npoo06-

roupe ontrdrg ro 6fo npoypottrd p€pr1 (to o Kot ro y) xot ro 6do rpo-

woorrrd p€pq (ro p rot to 6). H orpo(peor1 e(vot e(ioou erfuotrr1:

( o + B i ) - ( y + 6 i ) = o + P i - y - 6 i = ( o - V ) + ( B - 6 ) i

EIKONA 50. To en(nE6o tou frdouq. I1otrtrontrootd(ovtoE ro 1 pe to { -1 €[vot oov

vo neprotpdrpoutrre to 1 optorep6otporpo xqtd 90' otov xd0ero d(ovo yto vo nd-

pougs ro {1. o notrtron}ooroopdg rou {l pre to {T neptotp€rper to yrvdptevo n<i-

h oprotepdotpogo rord 90o, otov opt(6vtro d[ovo, 6nou no(pver qv ttpti -1. Og

er rodtou, ^ti i: : -1. 299

fJ=i=(o,r)

Page 285: Ο Ταξιδευτής των μαθηματικών

E6ti, ndtrr, ontrrirq ogorpofpe ro ovr(ororXo npoygonrd pdpq ror,or4 ouv6Xero, ro ovr(ororXo govroorrrd p€pq. O notrtrontrooroopdq 6Xerev6rorp€pov, yror( no(pvoupe:

(o + pi).(y + 6i) = oy + o6i + pyi +p6i'�

'Opoq, i' = -1, rr €tor gnoporip€ vo ovuKorclotrlooupe ro -'l orov r€-IEuro(o 6po ror vo ndpoupe:

(o + Bi) .(y+ 6i) = oy + o6i + Fyi - 96 = (oy -96) * (o6 + py) i

H 6ro(peoq 6do pryo6rxdrv opr0pdrv e(vor oprerd nro 60orotrr1:

(o + pi)/(y + 6i) = [(oy + p6) + (py - o6)i]/(y'+ 6'�;

0o16oo, drov erretrdooup€ rrq t€ooeprg npd(erq rqg opr0pqrrrrjg,oro r6troq no(pvoupe ndvrq dtrloug puyo6rrofg opr0po0q. Enop€vurg,or pryo6rrof oprOpo( e(vot r]eroro( ulq npoq rrq tiooeprg npd[erg rqqopr0pqlx{q, en6q o16 qv er6rrrj nep(nrooq qq 6to(peoqq g€ ro pl-6dv. Enrnldov, or prryo6tro( oprOgo( pnoporiv vo utptrr0o0v oe 6uvd-

F€rS Kor vo unotroyrorofv ro pr(rd rouq, ror or oprOpo( nou npox0-ntouv e{oro}ou0odv vo e(vor pryo6tro( opr0po(. Oq er rodrou, or Fr-yo6tro( oprOpo( e(vor rtreroto( ror oq npoq our€g rtg 6rjo enrnl6ovnpd{etg.

Enopdvulq, q eniAuorl tov nolurrrvtjpurv 6Xer Betrlur0e( notrrj. Av ore-rprodpe to nolurivullo x' - 6x + 11 = 0, €Xoupe ug 66o trfoerq: 3 + {1iKo( 3 - lDi. fro vo npoo6rop(ooupe to o4ge(o ourtiv rov 6rio trfoeurvoro en(ne6o rulv gryo6rr6v opr0p6v (to en(ne6o rou frdoug), F€roKr-vofpoore 6e[rd on6 rqv opX( rou d(ovo rurv x rord rpetq povd6eg (Er-r6vo 51). Merd, on6 qv opXrj turv o(6vr.rrv peroxrvofgoors ilpoq rondvto on6orooq (oq pe y'7 (nep(nou 1,414 povd6eq) ror npoo6rop((ou-pre ro rptirto poq orlpe(o, ro 3 + tf1i. fro ro 6edrepo oqge(o ;lerorrvori-goore ndtrr rord priroq rou d{ovo rov x rp€rq povd6eq, petd perorr-voUpoore on6otooq (o4 pe y'7 npos ro rdro yro vo npoo6rop(oouyero r - iDi. Iqv npoypour6rqro, pp(oxoupe 6tr or pryo6rro( oprOpo(nou €Xouv rpovrootrxd p6pr1 onovrdlvtor rord (eriyq, ro ono(o iXouv trl300

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E N E K T E I N O N T A I T O B A f , I A E I O

(3, 1e)ri/ 3+ t [ i i

\ (:, -^tilnt -,[Ti

E|KONA51. I lpoo6top((ovtog trg 66o trUoerg tou notruovrigou x'- 6x + 1l = 0 oro

en(ne6o tou frdoug.

Uopq{ o + pi rot o - Bi. Aurod rou e(6oug ro (efyq ovopd(ovtot ou(u-

ydq pryo\xo( aPfiPo(.

o Koptr kdoug enrxe(pr1oe oto r€trog out6 nou ot dllot trroorlportro(

efiov onorpdyer. loXup(orrlre 6rt ot <povroortro( opt0go( (oqv npoy;to-

lr6tr1ro, dtrot ot gryo6txo( opt0poQ 4rov rdoo unoprrof ovrtretgevtrd

6oo rot ot rpoYportKo( optOpo(. Av unripXov ot opvrptro( xot ot dpprl-

ror optOpo(, t6re unrlpXov xot ot gtyobtro( opt0po(' flpoxotre( irntrrllq

ro 6n, evtir o frdouq 6pr(e toug pryo6tro0g oprOtrrorig urg orlpe(o ev6g

enrn€6ou, 60o dtrtror goo4gotrxo( npoonooo0oov vo rdvouv to (6to

npdypo. O 6vog qtov o ouro6(6oxtog Noppqy6q po0qgolr6g Kdonop

Bdoetr (t 74s-1s22) xot o 6efrepog rlrov o Etrpet6g troytouiq Zov-Popndp

Apyrdv (1765-1822). Kot ot 66o npdretvov 6vo yturyporptrd rp6no ovo-

nopdotooqg rov pryo6rrttv opr0ptirv, oltrd ro 6pyo roug oe yevtreg

ypoppdq oyvorloqre. 301

.a

Page 287: Ο Ταξιδευτής των μαθηματικών

f l6oor undpxouv;

flporpov6q iXouge xdver ivo repdorto B{po npoq rrlv roredOuvorl qqen€rtoorlq qq dvvorog turv opr0p6v. Kortd(ovtog rqv eu0e(o rov npoy_porrr6v opr0prirv rou enrn€6ou tou frdouq, pnope( vo dXoupe ro €vro_vo 6roro0qur6 ouvo(o0r1po dtr undpXouv notrtj neptoodtepo orlpe(ooro en(ne6o on'doo undpxouv oe g(o pdvo eu0e(o. To (6ro 6roro0r1trr6ouvo(oOqpo erXov ror or neproodrepor pro0qpolro(. flptiypotr, ond rd_0e oq;.re(o rou d{ovo rov y pfiopoUpe vo rpopr]{oupe gro euOe(o no-pdll4}q npoq rov d{ovo rov x. Aunj 4 euOe(o nepr6Xer 16oo orlpe(o6oo ror q euOe(o rov npoypourdrv oprOpdrv. Arpo6 undpXer yq opr0prj_orpo nl{0og oqpe(ov orov dlovo rov y, propoOpe vo oXe6rdooupr pr1opr0gr]orgo ntr{Ooq euOeu0v, roOep(o on6 ng ono(eq 0o nepriXer pulopr0pqorpo ntr{0oq orlge(trrv. Ao<potrdg, ourd ro gq oprOgrjorgo orjvotropq oprOg{otputv oqpe(ov e(vor, ond p6vo tou, no}6 peyotrrltepo ondro orivolo ttrrv o4pe(ov proq pdvo euOe(oq.

fro vo onovr,iooupe oe outd ro rptirtqpo, np€ner yro dllq pro gopdvo otpogofp€ orov fx€opyr Kdvrop. Kor, yro dMq trlo rpopd, Bp(orou_pe 6n to eptitqpo iXer teOe( o'€vo ypdppo rou Kdvrop npog ro rpiiorol,, rov PrXopvt NrCvterrvr, ro ono(o ypdrpqre orrq 5 lovouop1r..:lu ltZ+.

E(vot, dpoyg 6uvot6v vo qnetrov(ooupe ;r, 6vov xor govo6rr6 tpdno proenupdvero (og nodpe 6vo tetpdycrrvo, ougneprlogBovoy€vqq qq neprg6_rpou tou) en( puog euOe(og (oq noUpe ;uo eugeio ypoppr4, ougneprtropBo_voptdvtrlv trov drpulv trlq), dtor tirmE oe rdOe oqpe(o qq entgdveroq voovtrotorXe( dvo oqpe(o tqq euOe(oq ror, owrotp6rprrlq, oe rdOe orlpre(o qgeu0e(oq vo owrotorXe( €vo oqye(o r_qq enrcpdvetoq;7

Aur{v rq rpopd o Kdvrop Xperdoqre neproodtepeq on6 orrdr 4gd_psE yro vo pper rq tr0oq. 'Hrov rp(o Xpdvro opy6tepo, to t8ZZ, 6tov6oreile otov Nrivrexrvr ro npoox€Oro prog trfoqg. Aut6 nou 6rove oKdvtop rjtov vo neprypdQer ivov rpdno oprprtrrovoorlpovr4q onerr6vr_

302 7. Dauben, o .54 .

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T E I N O N T A I T O B A E I A E I O

oqg ovdpeoo oro or1;rt(o tv6g mtn€6ou Kot rcl oqpe(o prog g6vo euOe(-

og. Kd0e orlge(o rou entn66ou ovonop(ororot on6 66o opt0po0q (o,F)

nou oupBol(ouv ro npoYllottr6 rot ro (povroortK6 p6pog ev6g gryo6t-

ro0 oprOpo0, rou o + pi. Kor ot 60o ouro( opr0po( €Xouv 6vo pdvo 6e-

xo6rr6 ovdnruypo. fto euxotr(o, 0o uno0€ooupe 6tt t6oo o o 6oo rot o

p pp(orovror peto{6 0 Kot 1.

o = 0r0ro203o4.. . Kot

I = o,F'9,F,90"'

Mnopo0pe vo Xprlotgollotqooupe ourd ro 66o 6erq6trd ovomdy-

UoTo yto vo 6qproupyqooupe 6vov rot pt6vo €vov npoyportr6 opt0p6

rrlg euOt(og rov TlpoYgortrdrv optOpdrv. Enttr€yovrog evoltrd( ro {rq<pio

rou on6 rov o Kor rov B, rorooreud(oupe our6 tov opr0p6, tov p'

r - o,o, p,orpro,Froopo...

O opt0pr6q rou KorooK€udooge, o p, e(vot 6vog npoyportr6g opt0p6g,

o ono(og onetrov((erot ;t' 6vo g6vo tp6no en( rou opt0po6 q + pi' Aurof

rou e(6oug 11 rorooreu{ o6r1ye( ot Uto opq(Uovoo4povrq onetrovtorl

dtrov rov ptyo6trdrv optOptirv rou pryobtrof entn€6ou en( qg eu0e(og ttrtv

npoypotrrtirv opr0ptirv. Aur6 ono6strv0Et 6rt o ntr40tr6g opt0p6g rrlq tu-

0r(og rutv npoygottrtiv opt0;rtirv, N', e(vot rot o ntrr10tr6g opt0g6g 6ltlv

rov puyo6tr6v opt0gdrv tou pryo6tro0 entn€6ou. An6 our6 ovutroppov6-

poote n6oo ntroUoto rot nurvq oe optOpodg t(vor 11 euOe(o rov npoYport-

rtlv opr0gtitv, orpoU o oopt0pr6q, rulv orlpe(trtv u1g tuOt(og e(vor (6tog pe

rov oopt0trt6o rcuv orlpe(tr-rv rou entn€6ou. Aur6 o(youpo ovttrpdoret ge rq

6to(o0qor1 poq, Kot ro (6to ouvdBq otov Kdvrop' 'EXovrog oro X€pto rqv

on66et(ri rou, or6po rt out6g 6uoroleu6tov vo rrlv ono6eXre(''

YndpXouv,tlor6oo,orlpovrtr€g6to<popdgovdpteoootol-ttyo6tr6enfte-6o rot orrlv Euoe(o ru-rv npoygortrtiv oprogtirv. 'Otrot ot opt0po( qg euot(-

og ttrrv opropdrv e(vot 6toreroypdvot on6 rr1 oxdorl (PsYo^Urepog on6"'

Ar1tro6q, ctv ot o rot p e(vot opr0go(, Kclt o o 6ev e(vot foog pe tov B, r6te

8. O.n., o. 55. 3*3

Page 289: Ο Ταξιδευτής των μαθηματικών

E[rE o > p e(re B > o. Mr d]lo ]dyro, or 6do opr0po( efvqt 6roteroypdvoroUprpovo pe ro pdye06g rouq. Aur6 6ev toXrier oto pryo6trd en(ne6o. Aevpnoporipe vo nofpe pe rdnoro oro0ep6rqro 6rt, ov €Xoupe 6rjo pryo6r-roUg optOpotjq, rouq o + pi Kor y + 6i, o €voq e(vot peyotrfrepoq on6 rovdtrlo. Enopdvurg, evtir or pryo6rro( opr0trrof oXrlpor((ouv dvo oprOprllrd nr-6b, our6 ro ne6(o 6ev e(vor btoreroypr€vo. Ile6(o oprOpdrv e(vor onoro6q-nore ofvotro npoypourc0v q pryo6rxtirv opr0p6v rdroro tirore ro dOpor-opo, 11 6rorpopd, ro yrv6pevo Kor ro nql(ro (pe e[o(proq rq 6to(peor1 ;rero pr166v) e(vor rdnorog dtrlog opr0p6g rou ouvdlou. Enop€vurg, 6vo ne6(oopt0ptirv e(vot rtretor6 oq npoq ug rEooeptg npd(erg qg oprOpqur4g. Ornpoyporrro( opt0go( oyqpor(ouv (vo Snraoypivo n$b ap0ptlv.Onog ouppofver ror Fr rouq npoypouro0g oprOpodq, or pryobrro( opr0-po( eivor e(re pqto( oprOpo( (6rov rdoo o o doo rot o B orov o + Bi e(vorprlroi opr0;roQ e(re dpprpor opr0poi ror e(vqt e(re otryepprro( e(re unep-polro( opt0;ro( (6rov e(re o o efrr o B e(vor unepBolro( opr0poQ. Av rpo-vrooro6pe ro prryo6tr6 en(ne6o ror ouvet64tonorrlooupe 6u o opeyotrtire-poq 6yrog" rtr;v optO;uirv 16oo orov d(ovo rov x doo ror orov d{ovo rovy t(vor uneppouro( oprOgoi our6 poq rdvet vo ovltrrlrpOo0pe n6oo nurvdroroveprl;.rdvor oro en(ne6o e(vur or pryo6rro( unrpBourof oprO;ro(.

O oprop6g ru.rv ;rtyo6rrrirv opt0ptirv np6orpepe notr6 neptoo6tepoon6 rrlv onlrl rorrono(qoq ev6g prnetrd qg dtryeppoq. 'Hrov 11 oq€rrl-p(o ev6g evrel6g v€ou rtrd6ou rulv po0qpronr6v, nou e(vor yvoor6g urgptyo6tr4 ovdtruo4. Aur6 endrpe{r€ orouq poOrlporrrodg vo 6rero6Uoouvnto onoretreoportrd oro guorrrd rov Oeoprlrrrtirv poOrlporrr6v ror voop(oouv pro peyolfreprl yrdpo provr€trurv oro €qoppoop€vo poOqpol-rd. Ot ptyo6rro( optOpo( rdvouv notr6 nrproo6repo on6 ro vo orore-Iofv tr0oerg notrurrrvtil,urrv, ytor( e(rlyoriv rdnoreg dtrIeg nopoordoergnou 6ev 6Xouv v6r1go, clv Trsproprotofpe pdvo orouq npoyprorrro0goprOpodq. ho nopd6erypo, q e[(or.rtorl e' = -t 6ev dXer ]6o11 nou vo e(-vor npoyporrr6g opr0p6g. 'OUoS, ov trdpoupe un' 6r1tv poq rouq Fryo-Orrodg opr0po0g, 1116o11 e(vor 11 ri rl n/-t. Aur6 ro pryo6tr6 opr0p4l-16 oqpe(o pp(oreror orov d(ovo rov y Kor oe on6orooq n ndvo on6rrlv opX4 rr.rrv o{6vujv. hrlv npoypour6r4ro, q rour6rrlro e^' + 1 = 0304

Page 290: Ο Ταξιδευτής των μαθηματικών

E N E K T E I N O N T A I T O B A I I A E I O

u1t0 ourr] 0eope(tor ond rouq po04potrro0g dtrou rou rdopou urg pioond trq nro Bo0uotdXooreq ror unripoXeq oXioerq dlov rov po0rlpou_rdv. IupneprlopBdver o€ Fro e{(oooq qv npri{q tqq npdoOeo4q, u1q€oq qg ro6rqroq, rouq opproriq opr0po6q I Kol 0, rouq 66o unepBo_rroUq oprggoriq e Kot ?tr, Kor ro pryo6rr6 opr0pd i,i /_.1..Otro ourd 1ro_

0v0yvopiorrlK€ np6rr1 gopd ond rov A€ovopvt ,orlep ro 1/48, rolf

nprv o lrdouq Oepeirdroer oroOep<i toug pryo6rro0q opr0poUg. H rourd_

(( oe pro tdoo onlrj nopdoroo4 proq r<ivouv vo ooor((oupe .

Or re rpodr r<o i oprOpo i rou XdprArov

Agorl o fxdouq ene(€rerve rouq opr0poriq ond qv euOe(o rov npoypo_uxtiv oprOptilv oro gryo6rrd en(ne6o, {tov pdvo 06po Xp6vou ro voe(ttdorr xdnoroq ov ourrj 4 6ro6rrooio gnoporioe vo ouvcXtore( orovrpto6rdototo lipo. Aqlo6rl, undpXer dpoye 6vo odvotro opt0pdrv nouvo onoretre(tor ond opr0p4rtriq rptd6eq (o, g, y), or ono(eg vo ovrr_ororXoriv oe or;pe(o ev6q rpro6rdororou Xripou Kol vo trertoupyorjvu(oqq ulg oprOpoi Aurdq o rdnoroq r]rov o foufirop p6ouov Xdprltov,o geyoldrepoq (ooq po0qpour6q r4g lplov6(oq. fwv{04re oro Aou_BXvo ro 1805 Kot ono6e(yr4re 6n r]tov dvo npdtrrpo oventuytrrivo nor_6(, nou €poOe noMiq {6veq ytrtirooeq nprv ond rrlv erp4pe(o rou on6€vov notrrjytrrrrooo Oe(o. Apydrepo toXup(otrlre 6rr oro 6erotp(o tou{{epe 6erorpeiq ytrdrooeq. EuruXtirg, o veop6q lptrov66q trovono(4oe t4yluloooloyrrrj nepripyerd Tou Kor €orperfe rlv Trpoooxrl rou oro poOq_

Ie 4trrx(o 6eroentd erriv, o Xdprkov pn{re oro Ko}€yro Tp(vrn rouAouptr(vou. I1€vre Xp6vro opydtepo, {tov oxdpo gorqrrlq 6tov 6ropGoqre Booilrr6g oorpovdpos rnq lptrov6(oq ror ro04yqrrlq oorpovo_pr(og oro Tp(vru.'

To 't 828, o Xdptlrov dpXroe vo or€cpretor t4v en€rrooq qg dvvotog

potrrd.

, . i

r it lr i

305 rii l] lilil

iil,|tr

9. Holl ingdale, o. 337.

Page 291: Ο Ταξιδευτής των μαθηματικών

rov pryo6rrrirv oprOptirv orov rprobrdororo Xtirpo. Xpcro(drov €vov rp6no yro vo op(oer trg npd{erg rqq npdoOeorls Kor rou notrtrontrooroopo0tov rptd6ov rou, 6ror tiore vo pnv Koroilorodvror ot poorriq r6t6r4regrqg dlyeBpog. Aur6q or r6r6rrpeg nepilogBdvouv rouq nopoKdro rovdvegyro onorouobrlnore rperq opr0po6g o, B ror y.

1. o + p = P * o (ovuptroOelrcr] r6r6rr1to rrlg np6o0eorlg)2. o.P - p.o (ovupno0etrrrj r6r6rqto rou notrtrontrooroopof)3 . (o+ 0) *V=o+ (F+y) (npoor ro tp ror r r r j t6 r6 r r l ro rqqnp6o0e-

onq)+. (o.F).V = o.(F.y) (npooerorpronrq r6rdrrlto rou nollontrooro-

opori)5. o'(B + Y) : o'B + o'Y (ilrp€ptortrrl r6tdrrlro)

O Xdpilrov, 6pulq, 6ev ;rnopo0oe vo pper rouq rordtrIqtroug opr-o;.rodq nou 0o 6rorrlpo6oov rrq nopondvur ndvre r6r6rqreg. O notrtro-ntrooroopr6g rv6g pryo6tro6 oprOpof rou entn€6ou op(oqre oq 6vo e(-6oq neprorporprlg. Oo16oo, drov o Xdpiltov npoond0qoe vo op(oet tovnoltrontrooroopd orrq rperq 6roordo€rq oq r€prorporp4, on6ruXe. 'Onurg

ono6etrv0eror, 6ev e(vot 6uvorq n Xprion rprd6urv yro vo oprore( proneprotpogrl ortq rp€rq 6rootdosrg. Tr oupBo(ver ge rrq r€oorprq 6roord-oerg; Mnoporiv vo oprorofv oprO;ro( otrg r€ooeprg 6rootdoetq rou vo6rorqpo0v rtg t6r6rqrtg rqq dtryeppoq; H dpeuvo tou Xdputrtov l.r€rcrro-n(orrlre tiorr vo pstrerrjoet 6vo e(6og ouneppryobtroU" opr0trroU, nou€Xer rq poprprj q + pi + yi + 6k.

O Xdpilrov ndleue pe our6 ro np6p}4po en( 6eron€vre Xp6vro. Tetrr-rd, q 166o yro 14 tr6or1 rod r1trOe [o<pvrrd roOdrg nepnoroUoe po(( pe rrlyuvo(ro rou. H rlpepogrlv(o rlrov 16 OrtoBp(ou'l 843 Kor, yro pro or6porpopd, yvrrrp((ouF€ Trlv orprprl qpepoprlv(o puog onou6o(og po0qpotrrrjgovordlugrlg. O Xdpilrov ouver6rpono(r1oe 611 dnpsne vo eyrorole({.rerp(o on6 rrg r6r6rrpeg rrlg dtryeBpoq, rqv ovrrp€ro0eurr] t6r6rqro tounotrtrontrooroo1loU. Av 6ev en€peve 6rr o.B = p.o, ;rnopofoe vo op(oet ugrfooeptg npd(rtg rrlg opt0prlrrrrlg yto roug vdoug oprOporig rou, rouq

306 ono(oug ov6pooe rcrpa\rcoAE apilpoiq. Or optOtrro( ouro( op(orqrov trrg

Page 292: Ο Ταξιδευτής των μαθηματικών

E N E K T E I N O N T A E T O B A f I A E I O

npoypotrrtirv oprOptitv rrlq Uopqrlqo + p i + y j + 6 k , y r q 1 q

g roXriouv or nopordtur rour6rqrtg:

i ' = i ' = k ' = - 1 r o t i i k = - 1 .

0 Xdpilrov, trorndv, rordrpepe vo op(ott trq npd[ctg 6rot tiore vo

6treg or r6t6rqreg, err6g on6 rqv ovrtpsro0ettrrl t6t6rr1To rou

po0. Iriprpo yvurp((ouge 6rt p6vo ot dtryeBpeq nou opG

pdXpr ror oe 60o 6tootdoetg pnopotiv vo rpototiv rnv ovrtl.l€ro-

t6r6r4ro rou notrtrontrootoopo0. Ooeg dtryeBpeg op((ovror yto

Xdrpouq, np€net vo tyrototre(nouv ouul rrlv t6r6qto.

0 Xdpilrov 0eopo6oe 6tt outo( ot tetpo6tro( opr0yoi oloretroUoov

onou6ordreprl ouvetorpopd rou Kclt ndpooe ro peyolUtepo p6poq qq

rou epgqve0ovrdg touq. flop'6tro nou tct o0yXpovo po0rlpottrd

Xprlorpronorofv ourorjq roOourodg touq terpo6tro0g opt0pofq, rl tU-

roug o6rlyqoe npdypotr oe or1;rovttrri np6o6o. 'Evog rerpo6t-

oprOpdg dXet 6fo tptt];roro, to roOopd npoyprottr6 rprlpo, ro o, ro

ovoprd(eror Bo01tut6, Kctt ro (qovroortr6> rprlpo, ro pi + Yi + 6k,

ono(o ovopd(erot 6nvuopaur6. To 6tovuollottr6 rp{go e(vot 6vo

roreu0uvr{proq ypoppnq o€ rpttq 6toordoetg. Xprlotponoteftot oe

tope(g rqg entoulgqq orrg pdpeg Uoq Kot 11 getrdq tou e(vot

oro po0qprorrrd oq 6tovuoprotrrrj ovdtruor1.

Or rerpo6rro( optOpo( tou Xdprltov €rovov touq otryepptor€q vo ou-

dtt trrnoporiv vo ovonruX0o0v v€eg dtryeBpeq, ov oro-

6eXro6pe otrtroy€g olg pootr€g t6t6tqreq rqg dtryeppog. Aur6 tfvot ovd-

loyo pe rqv ovdnru(q rnq Fn eurtre(6erog yeulperp(oq, Xpqorponotrir-vrog evotrtromtx€q tr0oetq yto ro ndpnro o((tlgo rou Eurtre(6r1. Og er

rofrou, q Oeop(o rov rerpo6trrirv oprOptirv rpov 11 opxn rrlq odyXpovqg

orp4 pr1trr€vr1q dtryeppoq.llprv orprloou;.re roug rerpo6tro0g optOpofg rou Xdptilrov, npfnet vo

Odooupe dvo on6 to oyon4ptdvo pog eporr]potq: n6oot orptptig rerpo-

6uo( oprOpo( undpXouv; E6tit, yto dtrtrr1 pto rpopd, o Kdvtop ltog 66tloe

qv ondvrqorl. Oo enovotrdpoupe rrlv (6to 6to6troo(o oretr6vto4q rov

rerpo6tr6v oprOpdrv rou unepXtitpou orrlv Eu0e(o rov npoyporrr6v 3&?

Page 293: Ο Ταξιδευτής των μαθηματικών

oprOpdrv. lpdrpoupt rouq r€ooeprg optOpofq nou onotetro6v €vo ouyre-rprpdvo rerpo6rr6 opr0p6 ge ro 6ero6rrd ovonrdypord rouq. fto euro-tr(o, unoOdroupe ndh 6tr or opr0go( pp(orovror ovdgeoo oro 0 Kor ro1. 'EXouge, Iotndv:

0 = 0,oro2o3o4.. .

F = o,F'F'F,9,'..Y = 0,Y,YtYrYo...6 = 0,6,616160...

An6 ourodg toug r6ootptg opr0po0g, nou op((ouv tov ouyrerprp€vorerpo6rr6 opr0p6 pog, 6rlproupyoO;re tov ovt(ororXo npoyporrr6 opt0-p6, rov p.

p - 0,o,B,y,6,orBryr6ro,Bryr6rooBoyo6o...

Iuvendrg, ond 6trouq toug ovop(Ogqrouq rerpo6rrodq oprOporig oro

Xdrpo rurv reoodpov 6roordoeov, no(pvoupe pro ogqrpovoorlpovrrlunerr6vrorl rn( rcuv npoyprorrrdrv oprOgtirv. O ntrr10rrc6q opr0p6g rov rr-tpo6rrtirv opr0puiv, 6noq xor rov pryo6trtirv oprOpuirv, e(vor (6ro9 pe tovntrr10rr6 opr0pd rov npoyporrrtiv oprOptirv. Aur6 oqpo(vrr 6tr oe grop6vo euOe(o undpXouv rdoo orlpeio 6oo rqr oro en(ne6o, r1 oto Xtirporov rptdrv Oroordoeov, { ordpo Kor oro Xtipo rov reoodpov 6roord-oetrrv. Irrlv npoygortr6qro, o "opr0p6q" ttrrv oqpe(ov 6ev e(vor 6vognpoo6roprour6q nopdyovroq rrl( 6roorortrdrrlrog. O Kdvrop on66et(e6rr yro opr0prlorpo ntr4Oog 6roordoeov, o ntrq0rr6q oprOpdg rov orlpe(-tr-rv roofrqr p€ rov ntrq0rr6 opt0p6 rov o4pe(ov prroq euOs(og, 6qlo6qrov N,. Aur6 e(vor 66orotro vo ro ono6eyrofpe 6toto0qurd ror no}trofpo0q;rortro( endpevov 6rr o k6opyr Kdvrop 0o npfner vo 6rove trd0oq.

'EXoupe e(erdoer, trorn6v, ere(voug roug optOpo0q nou Xprlorponoro0gemrg ro0rlpeprv€q Opomqpr6qrdq pog ror, enrntrdov, dXouge e{erdoet rouqpryo6rro0g opt0profq, opr0pofg nou e(vor onopo(rrpor yro ulv roOrlpepr-vq nponrrq rov enroulptirv. Irq ouv€Xero 0o e{erdooupe nopd[evoug rore(ururo0g oprOpo0g, nou ro evev{wo ewdo rorg eror6 rou yevrro0 ntrr1Ou-opo6 6ev rouq €Xer orodorr nor6. flpoerorpome(re yro neprn€rerol308

Page 294: Ο Ταξιδευτής των μαθηματικών

rous uTr€pBouKo0q opt0pofq, e{erdoopc n6g o Kdvtop

611o dnerpouq ntrq0tro0g optOpo0g, rouq No Kot N,, yto 66o ei6q

, rov ouvdtrtrtv nou e(vot opr0trrrjorpo Kot rov ouvdtrtrtv

e(vor pq oprOgriotpo. Av N e(vot to onetpootivotro 6ltrtv rov guot-

opr0ptirv, t6re op(ouge ro N oq tov ntrr10tr6 opt0p6 rou N. Me ou-

1d rov rpdno onorperiyouge vo ouyXrioul.te ivo o0votro F€ rov ntrq0trd

iou opr0g6. Me nop6poro tp6no, doto drt ro R ouppotri(et to oOvotro

dtrov rov npoyporrrdrv oprOtrrdrv. Tdte ro R e(vot o nlq0tr6q tou optO-

y6q, o ono(oq ouXvd ouprpol(etor ontrti,rq oq c. idpoupe 6rr N = No. O

Kdvrop n(oreue 6tt c = N, otrtrd 6ev pnopo0o€ vo ro ono6e(et. :€pou-

ge en(or;q 6tr €Xer teOe( ro eptirq;ro ov undpXouv rdnotot dnetpot nlq-

0rro( opr0po( ovdpeoo oto No Kot rov c, otrtrd 11 ondvrqoq e(vot ovono-

rprioroul.'Otrulq, o Kdvrop npoXtilprloe nol6 nto n€po on6 ouro0q touq 6Uo

nptirroug dnerpoug ntr4Orro0q opt0po6q, ytot( oulv npoyltottr6t4ro

6proe pro dnerpq orotrou0(o tdtottrtv opt0pdrv, dnou ot No rot N, t]tov

or 6tio npdrror opt0pro(. Auto( ot ntrqOrro( oprOgo( ovotrrdoqrov unep-

nenepaopivot ap0po( - opt0po(, 6r1tro6r1, nou {tov n6po ti 4tov peyo-

trdrepor on6 roug nenepoop6vouq opr0gorlq. H g€0o6oq rotooreu{q 309

Page 295: Ο Ταξιδευτής των μαθηματικών

t ( E o A A A t O 1 2

toU rdOe 6to6oXrro6 un€pnflrspooprdvou oprOgod e(vor ontro0otorq. Av€Xoupe 6va nenepoopivo orivotro pe v orotXe(o, r6re 6repeuvofpe pendooug Ororpopetrxo0g rp6nouq ;.rnopo0ge vo opoOonoujooupr ourriro v ororye(o. 'Otreg our€q or nrOov€q opo6onor{oerq y(vovror ororXe(oev6g v€ou ouvdtrou, nou o nl40rrdq rou opr0pr6g e(vor peyotr0repoqon6 rov ntrq0tr6 opr0p6 rou opXrro0 ouv6trou. Oot6oo, 6rov orerpr6-poore 6tro ro e(64 rurv nrOovtirv ouvdtrtov, rot t6ro(repo ro onetpoorivo-tro, ntig (€poupe du pnopo0ge vo oXqpor(ooupre 6trouq roug ntOovo0gouv8uooprodq otorXe(trrv; [r' our6v ro tr6yo dXoupe dvo o{(trrpo ond u1Oeop(o rov ouv6lov:

A:IOMA TON IAHOIKON APIOMQN: frq rd0e odvotro undpXer prooutrtroyri ouvdtrurv, 11 ono(o nepr6Xer ovdpeoo oro ororXe(oqg 6Io ro unoofvotro rou 6o0dvtog ouv6trou.

Aur6 ro o((opo eyyudror 6rr pnopo0pe ndvro vo oXqpor(ooupe roo0votro p€ rous ouv6uoogodq rov otorXe(ov. To o{(rrrgo ourd e(vornpo<pov€q 6rov np6rerrol yro n€n€poop€vo orivotro, otrtrd 6Xr ror 16oonpo<pov6q 6rov np6rerror yro on€rpooUvotro.

iro nopd6erypo, orerpre(re ro odvolo nou dXrr rp(o orolXe(o, to 1,2,3. Aur6 to otivolo ro ou;rpotr(oup€ oq {1 ,2,3}. Me ndooug 6rorpo-pelxo0g tp6nouq pnopofye vo ouppotr(ooupe ourd ro rp(o ororXe(o; Hondvrrloq ono6errvderot drr e(vor orrti, ror ro orrd 6rorpopeurd o0vo-tro e(vot m: {A\, {11, l2}, {3}, {1, 21, 11,3\, {2,3}, {1,2, 3}. To npt irroo6volo, w {A\, ovopd(etot rev6 odvotro ror ouXvd ypdrperor ontrdrq rrrgO, ywptq ro dyrrotpo. To rev6 o6votro e (vot dvoq on6 rouq nrOovofgrp6nouq yro vo ouppol(ooupe ro rp(o orotXe(o, 6qlo64 ro vo Url oup-po}(ooupr rov6vo on6 ourd. To o0votro rrirpo nou nepr€Xet ro ortdl o0-votrd pog e(vor ro lA, f i \ , l2\ , {3l1, {1,2\, 11,3L 12,3}, { i ,2,3l1\ ror 6Xernlq0rr6 opr0pr6 8. O rfnog nou Uoq trder 6tr dvo o0votro ge rp(o orot-

Xe(o gnope( vo oXqprot(oer o116 6ro<poperrrd oUvotro e (vor o: 2r = 8. Au-16 pnopori;le vo to yevrredooug€: €vo oUvolo pe v orotXe(o pnope( vooXqpor(oer 2" 6rorpopeurd odvotru. O opr0p6g rcuv ouv6trov ilou pTro-

310 potipe vo oXqpor(ooupe on6 ro v e(vor ndwo geyotrdrepoq on6 ro v.

Page 296: Ο Ταξιδευτής των μαθηματικών

Enopr6vtrtg, o nenepoopr€vog ntrr10rr6g opr0pt6g rou 2" e(vot geyotr0tepog

on6 ro v. O Kdvtop, oro nootyvtrtoro ypdgpo rou orov Nr€vrertvr ro

AerdpBpro rou 1873, on€6er(e ourri orptpdrg rq oXdorl yto to onetpoo0-

votro' ond6er[e 6llo6n 6tt, ov v E(vot o ntrq0tr6q opr0;r6g ev6g onetpo-

ouv6trou, r6te v < 2". Aur6 ovopd(erot 0e6pqpo rou Kdvrop.

OEOPHMA TOY KANTOP: f to xd0t ntrq0tr6 opr0pr6 v, v < 2".

Av ferrvr]ooupe ond rov No, pnopo0pe npdypott vo 6qgtoupyrloouge

rnv oxotrou0(o rov dnetpcttv ntrq0rrdrv opr0p.rrv no(pvovroq 6to6oXtr6g

6uvdpcrg rou 2 Us rov nopordrtl rp6no: Nr = 2no. Me nopdpoto tp6no,

N, = 2Nr rot N, = 2n'�. Kde€ rpopd, ur{rtirvouge oirtrtitg ro 2 orov rp6Xovro

uTrspTrtnspoog€vo opt0p6 yro vo 6rlgtoupyrlooupe tov en6pevo UTI€pTI€-

nepoog€vo opr0pr6. Mt our6 tov tp6no 64gtoupyofpe €vo onetpoofvo-

tro unepnenepoogfvtrlv oprOpdrv: No, N', Nr, N,, N., Nr, Nu, Nr,"'

Xprlorponotdrvroq ro 0edrprlpo tou Kdvrop, pnopoUpe vo rpovrootorj-

ge ndrg ro No 6rlgroupye( rov c. Oo ono6e(oupe 6rt 2no = 10N0. O N0 rt0-

po a(vor ontrtirg to oprOprlorpo o0votro {1 ,2,3,...\, rt €tot ro 10n0 ouppo-

trGt 6louq roug nt0ovotig rp6noug ps rouq ono(oug ro xdOe {rqrp(o rdOe

ouv6trou pnope( vo opo6onotq0e( oe ogd6eg rcov 66ro rlrqrp(ov.' O rdOe

ouvbuoopr6g 6rlgtoupye( 6vo ptovo6rr6 odvotro {o, F, Y,...}, ro onofo pno-

pe( vo ougBotri(et ro 6ero6tr6 ovdnruypo ev6g optO;rof ovdpeoo oto 0

ror ro 1. Enop€vog, dtrot ot ouv6uool.to( TIou ovctiloptordlvrot pe ro 10n0

0o npinet vo nepdoppdvouv rdOa 6ero6rr6 ovdmuypo ovdpeoo oro 0

Kor ro 1, rot o ntrq0tr6g opt0;r6g ourof rou ouv6trou e(vot o c. O c 6trttrtg

e(vot en(or1g o ntrq0tr6g opr0g6q o}6rtrqprlg rqg euOe(og rov TlpoYport-

rdrv opt0pdrv, n€po ond ro drt E(vot o ntrr10tr6g opr0p6g rov orlpe(trtv

ovdpeoo oro 0 rot ro'1.'EXoupe, ouvendtg,2xo = lgxo = c.'Evo dtrtro no-

pd(evo onordtreopo e(vot 6rt: No! = |'2'3'4... = c.

Og opr0po(, ol uilspfisiltpoog€vot opr0po( np6net vq unoroUouv o'

6vo odvotro t6torirulv, 6nog rot ot r€r€poogdvot opt0go(. lro nopd-

6rrypo, yto rouq un€prl€Tlspoopdvoug opr0go0g tiXouv optore( q np6-

t. E. Kamke, fheory of Sets (N€o Y6prr1, Dover Publications, 1950), o' 47 3? r

Page 297: Ο Ταξιδευτής των μαθηματικών

oOeoq ror o notrtrunlooroopdg, o}trd e(vot pdtrIov poperd. r-ropo6e(y-porog Xdptv, ov u < w r6re N" a N* = N* ror Nu.N* = N*. Enop€vrrrg, or}40rr6q opr0p6q rou o0po(ogoroq Kor rou yrvop€vou e(vor ontrdrg opeyotrfrepog ntrr10rr6g opr0p6q rov 66o nopoy6vrov. Autd oqpo(ver6il N,2 = N, Kor, ornv npoyprorrr6rqro, N," = N, yto rd0e nenepoopr€vov. Iqpo(ver m(oqg 6u N, + N, = 2 N, = v.N, = N, yro rdOe nenepoog€_vo v. Enop€vog, drov or ur€pn€rspoogdvot opr0pof npoor(Oevror tjnoltrqntroord(ovror, 6ev olld(ouv noltrd npdyporo. .Orov 6poq Xp4or_ponorouvror og er0€req no(pvoupre evbrorpripovro onoretr€opoto, 6nogrr1 6q ptou py(o v€urv unepnenepooprEvrrrv opt0pdrv.

Err6g on6 trrro Oetrrp(o yro rouq urspnEr€poopdvoug nl40rro6q opr0_pouq, o Kdvtop ov€ntu{e pro nopdtrtrqtrq Oetrrp(o ylo rous ouvreroy;.ui_voug opr0trro6g. And roug opro;-ro0q nou dxoupe 6tiroer yro rous ouvr€-roygdvoug Kor rous n}40rxo6q opr0poUq, 0o 0up6poore du o ntr4Orrdgopr0p6g oxer((eror pe to ntrr]Ooq ororxe(,v ev6g ouv6trou, evdr o ouvre-royp6voq oxer((eror pr pro orotrouO(o opt0ptirv nou e{etr(ooeror, €vo pr]-po rd0e rpopd, ;r€xpr tov retreuro(o oprOpd. H orotrou0(o rurv <puorr6vopr0pdrv 11 ,2,3, 4,5,. . .1 efvot gro rdroro orotrou0(o. To dvo 6pro nouno(pvoupe npooOirowo( ro 1 o€ rdOe ouvreroypivo opr0p6 yro vo 611_proupyrlooupe rov en6pevo ovopd(eror ur, rr dXoupe drr o = No. Ar1tro64,o nptirrog u[spTr€n€poop6vog ouvretoypdvoq opt0g6q, o ur, roo6ror perov nptiro unEprsr€poop€vo nlq0rr6 opr0p6. Kordnrv, o Kdvrop 6proepeyotrdtepoug ouvreroyp6voug oprOporig, nou ovnorotxoUoov otouqnlqOrr<oUq rou opr0goriq.

E(vor 6dorolo vo ovrrtr4<p0oripe n6or1 o6(qo4 €Xoupe 6rov nqyoGvoupe on6 rov ivov unrpr€rspoopdvo opr0p6 orov endpevo. Otovnqyo(voupe ond tov No orov N,, n4yo(voupe on6 ro opr0prlotpo orj_votro rov rpuorrdrv oprOptirv oro urr oprOprlorpo orivotro 6rov rovnpoyportrdrv opt0pdrv, Kor or npoyForrro( oprOgo( e(vor enoprdrg nu-rvo( { noludpr0por yro vo yep(oouv ro Xdrpo o}6rtrqpou rou orjpno_vrog. A<porj o N, e(vor p4 oprOpqorpog, r6re dtrot or peyoldrepor unep_nenepoopr€vor oprogo( e(vor en(oqg pq opr0prjorpor, rdrr nou rdver

3€e rov N, rov nptiro pr4 oprOgrjotpo nlqOtrd opr0p6. ftdoo peyolUrep4

Page 298: Ο Ταξιδευτής των μαθηματικών

]m*mf

id 6e pog 6(ver vo roroldpoups rov tepdotro ntroUto tou Nr. Kor rr

! oupBo(ver p€ Tov N, ror rov N,o, rj or6po Kclr rov N,oooooo; Auro(

I oorpotrtiq e(vor oprOpro( 16oo peydtror, tirorc 6ev dXoupe rovdvov rp6no-I y,o vo ovrilqg0o6pe rqv notrlontr6t4td touq. Oot6oo, e(goote p6hg

I 0Tnv opxn!

! l " lpoxrrrpovroq npoq KdnorouqI npoy t - ,o rKd p€ydAouq oprepo0qI

I Mnoporlpe vo ouveX(ooupe vo op(oupe v€oug, geyotr0repoug un€pTrr-

! nepoopdvoug optOgorig. Mnopo0pe vo rono0et{ooupe 6troug rouq quor-

I xorjg opr0porig tog 6e(rreg rou N, retrerdvovroq Fs ro uJ tilote vo nd-

f pouur rov N,, o ono(og oqv npoyportrdrnro e(vot (6tog pe rov Nno.

a MnopoUpe vo ouveX(ooup€ [-l€ rov (6ro rp6no ronoOercitvtog dtro ror

I nro peydtroug 6e(rreg, ror 6e(rreq 6errrtiv, qltrd dtro ourd 0o e(vor pdtr-

a }ov poperd. Iro retrog 0o €Xoupe ontrdrg pro opd6o oupB6trov, ro 6vo

I 6(nlo oro dtrtro, Xopig vo ovrttroppov6poote neproo6repo ro 6reupuprd-

I vo ntrr10og orotXe(cov. fto vo ouppe( our6, 0o np€ntr vo e(erdooupe

I np6ro ro ano)uto anapo.

I lprv on6 rov Kdvrop, or rpil6oorpor ror or poOqgolro( dpr(ov ou-

I vrjOog 6rio g6vo rorrlyop(eg: rqv nenepoop€vq Kor rrlv dnerprl. 011611-

! nore ovqre orrlv nenepoop€vr1 rorrlyop(o pnopoUoe vo y(ver ovrilqnr6

I on6 ro vou. An6 r4v dtrtrq, q dnerprl rorrlyop(o 6ev rpov rorovoqrrl,

I ti.ou n€po ond rr1 troyrrrl ror ouXvd rourr(drov ge rov Oe6. Aur{ r1 ro-

J qyop(o onordtreoe ro on6truro dnerpo. o Pofvu Pdrep, oro 6pyo rou

I To dnapo Kat o vougt entoqpo(ve t:

r 3 r3Ii

Page 299: Ο Ταξιδευτής των μαθηματικών

Me pdorl rr1 troyrrr], to dnerpo e(vot obtov6rlto. Aev undpXer rdnotog pr1

rurtrrrdg rp6nog yto vo ro rprdoouge on6 rdtul. Onoto6rlnore npoYpo-

ttrr] yvtilorl rou one(pou npdnet vo e(vot guortrtottr{, ov pepo(tlg e(vot

6uvot6 rdrr 6nurg 11 puortrtottr{ yvtloq... Ardpo rt ov 11 ntr{pr1g yvtiorl

rou onr(pou e(vor 6uvotrl gdvo ;r6our rou FuortKtopo0, eforotrouOe( vo

e(vor 6uvor6 ror vo o{((er rov r6no vo pttrdpe yn pepwfi yvr.0or1 tou

oneipou gt Boorl rq troytr<r].)

O Kdvrop n(orEue oro on6}uro dnerpo rot ro rorirt(e pe tov Oe6.' H

np6roorl rou rpov 6rr unrjpXov rdto on6 qur6 ro ondtruro dnetpo loyt-

rd dnerpo, yro ro ono(o prnopo0oopre vo ptlrlooupre. Mdltoro, n(oreue

6rt o Oe6g rof €orertre rr1 Oeop(o rou yto rouq ur€pr€nepoo;tfvoug

opt0potig. To on6}uto dnetpo €Xtt m(o4g tourtore( ;re 14 oulloyrl 6trtlv

rov one(purv. Aev pnopo6pe vo ovogdooup€ Uto rriroto oultroyrl ofvo-

Io, yror( ro odvotro 6Xet €vo ouyrerptp€vo rot, enopfvtrlg, 6vov rr€ptopt-

orrr6 opropd. To orivotro E(vot 6vo nlrl0og orotXe(trrv nou pnope( vo yi-

ver ovrrtrqnr6 oq pro ev6qro. To dnetpo 6ev pnope( vo y(ver ovtilqnr6

og evdrqro. Iuppotr((oupt ro dnerpo l.l€ ro Q. Mnopei ro o vo E(vot

€vog ouvreroypdvog opr0p6g; 'OXr, yror( ov r1rov, r6re 0o pnopotioope

vo npooO€oougs ro 1 yro vo 6rlprroupy4oou;re 6vo peyotr0repo ouvre-

roypr€vo opr0p6, ror 6ev undpXer r(noro ;reyolfrepo o.n6 ro Q. Opo(utg,

pnopo0pe vo no6pe 6n to o 6ev e(vqt onldrg dvog n}q0tr6g opt0g6g

yrorr, ov rpov, rdre 0o ;rnoporioopt vo oxrlpor(ooupre rov en6pevo ntrq-

0116 opr0p6 tug 2n, Kr our6 0o o64yo0oe en(or1g oe ovt(rpoor1, o<po0 r6re

0o unrlpXe rdrr preyoldrepo on6 ro o. Enoprfvoq, ro o 6Ev e(vot oUrE

ouvreroypr€vog o0re ntrr10tr6g opr0p6g, ytor( e(vot ndpo on6 6trouq roug

ouweroyp€vouq Kor rouq ntrrlOtrofg opt0go6q rot e(vot, og ev6rrpo,

dyvtrroro. Aurd 6pulg 6Ev onortre(et rqv nt0ov6rrpo ro Q vo dXer 6vrr.rrg

rdnoreg t6r6qreg rorv€g pre rouq urspr€repoopr€voug opt0gofg.

T6po nou fXouge op(oet ro o (rdrr rou, orrlv npoygorrr6rrlto, 6tv

pnopofpe vo ro rdvouprs ornv npdEn) og our6 nou t(vot nfpo on6

2. Rucker, oo.48-503. Dauben, o . 232.3€€

Page 300: Ο Ταξιδευτής των μαθηματικών

6loug rouq un€pnsrspoopdvoug oprOpofg, pnoporipe vo npoXoprloou-

Fr Kor vo opfooupe rdnoroug gdtrtrov nro ev6to<pdpovr€S ur€pn€Tl€pcr-

op€voug opr0proOg. A0o r6r6rqreq rou o 0o poq Boq0rloouv. flptirov,yro rdOe r6r6tqto nou €Xer ro o, r6re ourq qv t6t6tqto 0o np€net voqv 6Xer ror rdnotoq un€pn€nspoopdvog opr0p6q. ftor(; Av ro ft dXerpuo r6r6tr1ro, rrlv p, rrlv ono(o 6ev dXet rov€vo dtrtro dnerpo, r6rE 0ognopodoope vo neprypdrpoupe pr6vo ro o oq ere(vo ro dnerpo nou 6Xetqv r6r6rrpo p. T6re 6poq ro a 6e 0o rpov on6luro rot 0o pnopo0oe

vo oprote(. Enop€vtrrg, yro rdOe r6rdrqto nou 6Xet ro O, rrlv (6to t6t6rr1-

ro €Xer rr 6vog toutrdXrorov uTr€pr€nepoop€voq opt0p6g. lloporrlp{ore

dtr on6 ro (6ro out6 enrXe(pqgo pnoporige vo ndpe oe gto 6r1truto4 pe

notr0 preyotr0repq Bopfrqto, ytor( ov ro Q rlrov ro €vo p6vo on6 to

600 dnerpo nou efXov rrlv t6r6tr1ro p, tdre 0o e{orotrou0o0oope vopnoporipe vo op(ooupe ge govo6rrd rp6no ro Q, trdyovrog 6tt to Q eGvor to peyotrfrepo dnerpo nou 6Xer u1v r6r6qro p. Arpo6 ofte rov out6

6ev pnopodge vo rdvoul.re, pnoporige vo nodpe 6rt undpXet dnetpontrq0og ufi€prrsnspoo;rivurv oprOptirv nou €Xouv qv t6tdrqro p. Aurq q

r6r6qro rou o, 6u 6qtro6ti 6treq ot r6tdrqtdq rou 0o np€net vo e(vqt

rorvdg gr dtrtroug ur€pn€n€poogfvouq optOgodg, ovopd(erot apm r1e

avdr)aor1g.o A11lo611, ro o ovortrd rtg t6r6qr€g rou npoq to rdttt,

orouq un€pnrnepooprdvoug opr0po0g.H 6e0repr1 ev6ro<p€pouoo r6t6rqro tou Q e(vot drt 6ev pnopotipe

not€ vo rprdooupe oro Q 6qptoupydrvroq 6Io rot geyotrdtepouq unep-nenepoop€voug opr0pofq. fto vo e(goore nto orptpe(q, tr€ge 6rt 6evgnopodpe vo <prdooupe oro o ovofirtiooovrog opt0potlg gtrp6repoug

on6 ro Ct. Av our6 6ev (oyue, 0o pnopo0oope vo op(ooup€ ro o oq ou-td nou npordnret ond rr1 6to6oXrrq rorooxeuq opto;t€vttv uTIspTIsn€-poopr€vov oprOptirv. Enop€vutg, tr€ge 6rt 6ev gnopo6pe vo rprdooupe

oro O on6 rdto, r1 drt ro o {vot anp6otro. Ttirpo, trdyo qq opxtjq qq

ovdrtroorlg, yvtrrp((oupe dtr 6vo dretpo ntrqOog unepnenepoog€vtrtvopr0;lirv 0o np6ner en(or1g vo €Xet rqv t6tdrrlto rrlg onpootr6r4rog. Au-

4. Rucket, a.276. 315

Page 301: Ο Ταξιδευτής των μαθηματικών

to( onotelorjv rq vdo pog rldoq yryovtto(trtv opt0;rrilv, roug anp6ot-

rou1 urtpnil€paoptvouq ap0poAq.'Ooo rt ov npoonoOrioouge, 6ev

pnopo6pt nor€ vo rqrooreudooupe ourodg roug optOpotjg ovonrfooo-

vroq 6tro ror peyotr0tepoug ouv6uoogo6g rcov N' E(vor ontrdrq uneppo-

trrrd peydlor - e(vot onPootrot.

flporrtrd, o No e(vot o npdrrog onpdottog ur€pn€r€poop€vog optO-

pr6g, ytor( 6ev pnopofp€ vo rov rprdooupe npoo0€tovrog 6tro rot ne-

ptoodrepoug on6 toug nenepoog€voug opt0po0g nou pp(orovrot rdrtrt

on6out6v.ovopd(erorpr l6tvx6Eanp6ot toEopt0 l l6E,ytot (ouvt joutg

opX(oupe vo gerpdge on6 ro 0 ou1 Oeop(o rurv ouv6trtrrv (el ou rot to

0 orov No). Atrtrd ot 6to6oXtro( N yto roug ono(oug €Xouge prilqoet ovo-

nrfooovrot [errvdrvrog on6 rdtut, on6 rov No, rot enop€vog 6tv efvot

onpdotrot oprOpo(. O npdrrog onp6otrog ufi€pr€n€poog€vog opt0g6g

petd rov No ovopd(etot 0. Aur6g o 0 e(vot o np6rog geydtrog ntrr10rr6g

op t0 p6q gero(0 rtuv untpTltTlepoopr€vov optO gtilv'

l16oot onp6orrot optOgo( undpXouv getd rov 0; Mnopofge vo €TIt-

Xetpqporotroyrlooupe 611 0o np€net vo undpXouv o onp6otrot uTIspTIt-

nepoopdvot optOpo(. Oo pnopotioog€ vo (ertvriooupe ge rov 0' Kot U€-

r d v o o p ( o o u p r e r o v 6 e 0 r e p o , T o V 0 , , r o t g e r d r o v e n 6 g e v d l o u | K c l l u €

our6 rov rp6no vo ovom0(ouge ;rto otr6rtrr1pr1 orotrou0(o onp6otttrtv

ur€pr€nepoog€vov optOpdrv. AtrId pnopodpe vo rdvou;re rdrt notrU

rotrUrepo.Ooopfoougeontr t i lg t ' lgonp6otrodvovonp6otrouTl tp i lsr€-

poo;rdvo opr0p6, o po0p6g rrlg onpoorrdqroq rou onofou 6e 0o ntrr1-

otd(eror on6 rdrtl, (ertvdrvrog 611lo611 on6 tov 0' E(vot 16oo onpoot-

rog, t0ore 11 06oq rou ovdtrreoo otoug onp6otroug opr0pro6g e(vor onp6-

otrrl. Ovopd(erct urcpanp6orcog opt01t6E' Tdrpo oroprpoltitvoupe

npdyport otq orpot6oqotpo rov peydtrurv oprOpdrv' Bepoitrlg, propo0-

pe vo ouveX(ooup€ Xprlotponotdlvrog rrg (6req opX€g ror vo op(oou1'tt

rouq (ur€p-uneponpdotrouq" optogorig, rot petd roug oe(otpertro-

un€p-uTl€ponp6otrougo opr0po0g, rot ofttrt ro0efrig, otrtrd r6re y(verot

Boper6.l-lop'6tro outd, ot un€pnsnspoop€vot optOgo( undpXouv e66 ror t6-

3T6 oo rorp6, tiore ot ptoOqpolro( op(ouv pe (r1lo 6tro rot peyotrUrepoug

Page 302: Ο Ταξιδευτής των μαθηματικών

N P A I I T I A T I K A M E T A A O I

uTrrprrrr€pool"rivouq nlqOrro0q opr0gorjq. Il€po ond touq uneponp6or_rouq opt0podq, undpXouv or nl4Orro( oprOpo( M<ilo, or onep(ypontornlq'rroi opr'poi or ov€rgpooror ntrq'rrof opr'po( or ntr4'rro( op,elror6ropCpro4q, or nl4'rroi opr'yo( pdpor, or perprlorpor nl''rroi opr'poi,ot roXupd oupnoye(q n}40rro( oprOproi ot unspoupnoye(g nlq0txo(opr0trro( ror or enerrdorpor nlq0rro( opt0po(. Or enertdorpor nlrlOrro(oprOpoi e(vor or peyo.trrirepor. rdnoror poO4potrroi unoorr;p((ouv 6tr6ev undpxouv rov, ev6 dlror tr6ve dtr e(vqr or peyo^irepor opr'prof nou00 Bporipe nor€'' To orlyovtrrd e6ri e(vor dtr rd'e v60g ror peyol,te-pog nlq0tr6q oprOpdg 6ev orolou0e( onfuirq tov npor;yo0pevo nlqOrrdoprOpd, o}trd onordei pro v6o ror urfrlldreprl nlorg6ptrro rpos roondluto dnerpo. ,lotrrq ourd nou €Xoupe e6rir vo eivor ovdloyo p€ rrlvrotdotoorl nou undpXer otrl guorrri: 11 6rdonoorl tou ot6trrou, ror prerdTov ulooroprrdrv ourpo'6rrrlv, ror ndvro 0o pp(orougc 0.. ,napX., 1r.oor6po uno6roipeo4 ror yrdUo nro ororXeuir6rl ourllot(6to. ,looq

0o ou_veX(ooupe vo op(oupe 6lo ror peyoldtepouq ntrrlOrroriq op,epo,iq, .gJ_oov e{oro}ou0oripe vo enrOupo0pe rriu 16roro.Mnoporlpe vo ouvorf(ooup€ rous un€pil€n€poop€vouq opr0porigotov noporottrr fl(vqro 11, ioto rr ov e(vor qprtelrjq. ynoo€toupe drre(vor qprtelriq yrori ov 6rlfuirooupe 6tr e(vor otrortr4purp€voq, rdnorocpo04ponx6q 0o proq ono6e(er 6rr rdvoge ldOoq.

" r'-' -)'

.H _npoocr lntKr i npoondOero rou Kdvropy to rouq un€pn€n€poop€vouq op topor jq

H ond6er(q tou Kdwop yro qv unepoprOpqorp6trlto tov npoypour6vopr'prirv .rlpoorcrir're ro 1874. Ootdoo, or r6€eq ror or 'eop(eq rou 6ev

:::i":::l::iT"u:*,q. rlpriypou, ro €pyo tou Kdvrop rivorfe sur_tt6q 6rev6{eulv oXeurd F€ rnv dvvoto tou one(pou. Onurq cvogipopevtrtpfrepo, o peyd]oq owoy'vror{q tov r6eriv'.", K;;;;;.J:;;

5. 'O.Tr.,

oo. 281-285317

Page 303: Ο Ταξιδευτής των μαθηματικών

nlNAKAI 11. Meprr6g n(vorog rov urr€pn€Tlepoopr€vulv opt0ptitv

'Ovo;ro Ligpolo(o) IX6trto

Atrerp-p116dv

Atreg-€vo

Atrerp

Anpdorrot opt0po(

Yneponp6ortot opt0po(

Meydtror nlrl0tro( optO;to(:

Mdtro, onepryponrot,ov€r<ppooror, 6toptdptorlq,Pdpror, perpqotpot, toxupq

ouprnoye[q, unEpouUnoYE((,erertdor;..tot.An6truro dnttpo

- u-r, o opt0g6g ttrtv guotrdlv

oprOprdrv, rov rtrooptdtov rot

ru-rv otryeBptrdlv opt0pttitv.= (;) c, o opt0pt6g ttlv oqpe(trtv

otqv eu0r(o rov npoygortrtlv

opt0puirv.

H orotrou0(o ttrtv dtrerp nou

ovqntriooetor (ertvdrvtog on6

Tov No.

Apt0po( nou e(vot unepPotrrrd

peydtror yro vo touq qtdooupe

on6 r<dro rot oqp((ovtot otqv

opx{ qq ovdrtroorlg tou o.

Anp6ottot optOpto( nou e(vot

uneppotrrrd geydlot yto vo touq

rprdooupe on6 tov 0'.

Ilp6o0eto e(6q

ufi €prr€fi €pooU€vrrlv opt0ptitv

nou 6ev pnopotipe vo rouq

rptdooupe qn6 rdto.

fl6po ond 6trouq toug opr0ptodq,

1-uo outrtroyr1 6trtlv ttrtv ouvdtrtrtv,

yro rdnoroug o Oe6g.

No

N 1

N6, N1, N2, Nr

e

on6 roug ro0qyrpdg rou oro lloventorrlpro rou Bepotr(vou, o A€onotrvr

Kp6verep. 'Hrov our6g nou entv6rloe rr1 (oupepq rppdoq: <O Ot6g

€qtro(e roug or€potoug optOpo0g' ro un6trotno t(vor €pyo rou ov0ptit-

nouo. O Kpdverep ovqre orq o0yXpovq oXotrr] tou 6toto0qttol-tof, 11

3t* ono(o ondpprrrr€ ro dnerpo 'g rdtr otrorlrlpog€vo Kor ro geoporjot

Page 304: Ο Ταξιδευτής των μαθηματικών

onltiq oq rdu 6uvqnr6. Aurrl orprBtig rjrov 4 O€oq nou ov€nru(e oAprmor€trr1q, nprv on6 nep(nou e(roor 60o ouiveq. Me our4 rqv €woro,o Kpdverep {tov €voq o0yXpovoq nou ov{re oro orpor6ne6o tou flu0o-y6po, rou l-ltrdrrrrvo Kor rou Aproror€trq. Au{ q ovr(oroorl ovorpoptrdpe to dnerpo enextd0rlre, n€po on6 rous urrEpnsnepoop€voug opr0-po0g, orouq dppqroug Kor rouq unrppouro0q opr0pofg. 'Erntrr1(4 npo-rotrE( to 6rr, 16oo o Kpdverep doo rqt o Kdvrop {rov Opr]oror, otrtrd rorntrortrrvrordg. O Kp6vrrep 6poq n(oreue drr ot unepnenepoop€vor oprO-po( otporporioov rdtr on6 ro dnerpo peyotre(o rou Oeo6, orpoO p6vov oOe6q pnoporioe vo €Xer tqv r6r6rqro vo e(vor dnerpog. Av orr6(nore dI-tro efie ouul rqv r6t6r4ro, out6 0o ge(ove ro peyotre(o rou OeoU. O Kd-vtop 6prrlg n(oteue du or unepnelepoop€vor optOpo( enod(ovov ro pe-yole(o rou Oeof. Or un€pn€r€poop€vor oprOtrro( rpov orpdvrooro peyd-tror ror q dnerprl <p0or1 tou Oeod rpov or6go nro n€po rr on6 ourofg.

O Kpdverep 6ev {rov o g6vog trou €vovrrov6rov oroug un€pn€rr€-poopfvoug oprOporig, yror( rr dtrIor po0qpolrKo( nfor€uov 6tr to voenupdrfoupe oro dnerpo vo eroXurpr]oer oro poOrlporrrd 0o nporotro0-oe peydlq (qprd orrlv oorpdtreto nou nopefie q or€peo, Ioyrrrl 0egeXo-o{ touq.

Or 6rorpov(eg rou Kdvtop pre rov Kp6vrrep e(Xov roroorporprrdg ou-v6nereg yro rov Kdvrop. O Kpdverep 6Xt p6vo npoondOqoe vo ro0uore-p{oer { vo rpno6(oer rrl 6r1poo(euoq rdnorurv on6 ug npdrreq €nrorrl-govrr€g ovororvtioerq rou Kdvrop, otrtrd Xprlorgono(r1oe rr1 Oioq ror roxripog rou oro ilovenrorr]pro rou Bepotr(vou yro vo enrre0e( o0evopdouq Oerrrp(eg rou Kdvrop. 'Hrov prto dvrorl pdXq. O Kp6verep qrov 6voqrj6q ovoyvurpropdvoq ror ro0re popdvog ;roOqgour6q. OeurpoUvtovivog on6 roug rotr0repoug fepgovorlg po0qponro0q. O Kdvrop, on6rqv dltrq, {tov ivog veop6g, Url ovoyvopropdvog go0rlgortr6q ev6g pr-rpof novrnrorrlp(ou. Or enr06oetg rou Kpdverep €yrvov rdoo 6pr;.re(eg,tborE rdnotor dltror po0qportro( ro np6oe{ov rr dpXroov vo errppd(ouvrr1 oupndOerd rouq Trpoq rov Kdvrop.u 0o16oo, o Kpdverep 6ev (rXdrpr-

6. Muir, a. 237 3 tg

Page 305: Ο Ταξιδευτής των μαθηματικών

oe p6vo rov Kdvrop, o}trd qrov np60upoq vo rorobtrdoet dvOeppo

onorov6{nore unoor4pr(e rrlv dnor[r1 nep( po0rlpottr6v rou one(pou.

To Mdto rou 1884, o Kdvrop €noOe 6vo ooBop6toro vtuptrd rlovt-

op6 nou rpdrrloe nep(nou 6vo pr1vo. An€6utoE ourrl rqv ooO€veto oe

unepr6noorl ond rqv evooX6lrlori rou l.ls rrlv un60eoq rou ouveXofg

(e(vot o N' = ci) Kot orrlv euotoO4o(o rou e(otrfog ttlv entO€oeov rou

Kp6verep. flop' 6tro nou rpotv6rov 6u e(Xe ovopptioet ntr{ptlg 6rov

en€orper{re orrl 6outretd, ro endgevo Xp6vro or rp(oerg rfuXtrrlg oord0et-

oq enovotrotrtpdvovrov pe ou(ov6gevr1 6wooq. O Kdvrop rpotv6rov nog

€nooXe ond rdnoto e(6oq poOrdg rordOtrtrfqg. To 1899 n€Oove o ptrp6-

rspoq ytoq tou. Aut6, oe ouv6uoop6 pe dtrtro yeyovdto rlou tou nporo-

Ieoov €woor1 rot dyXog, rov €rovov vo Ulrct orrlv tUuXtorprrrl Ktrtvtrq

tou Xdtre. H rotd0trtrfr] rou €ytve nto €vrovq Kot ro X€tg€ptv6 tp(pr1vo

rou 1902-1901 onotrIdXtrlre on6 ro 6t6orrtrd rou roO{rovro rot eto{-

X0r1 oe rtrtvrrq.An6 rdtt og ro tdtrog rnq (orlq rou, ro 19,|8, o Kdvrop ;'rnotvdpyorve

oe rtrtvrr€g. Ot Broypdrpot rou ovo<p€pouv 6tt ot ent06oetg tou Kp6verep

rov o6rlyrloov orrlv rlruXrxrl ootd0eto.' Aurq q un60eor1 e(vot pepo(t-rg

ou(r1tr1otgr1, ytor( nt0ov6v vo pr1 pd0oupt nor€ notot nopdyovteg on6

ntreupdg rpuotoloyfog, ov unrlpXov Kdnotot, ouv€potrov orqv ooO€vetd

rou. Me pdoq ro rotvdrg ono6errd, o Kdvrop t(Xe dvov neruXqp6vo €y-

yogo p(o pe q yuvo(ro rou, rrl Bdtrt, rot rltov nor€pog 6(t not6ttitv.

Enop€vrltg, o' dvov ropdo rqg (tlr1g rou pndpeoe vo trerroupytio€t €Iltru-

Xtig. Evtir ot po0rlptottrig ovtilqryerg rou 6€Xovrov ouveXe(g mt06oetgqn6 rov Kpdvtrep, dtrtrot eupdurg yvtrloto( po0qporrrof rov unoorrlpt-

(ov, evOopp0vovrdg rov oro €pyo rou. O Kp6verep n€Oove ro 1891,

oorooo q oo06veto rou Kdvrop 6ev nopouo(ooe 6<peoq riorepo ono

our6. Apy6repo, o Kdvrop r€p6roe ouotoortr{ ovoyvtitptorl yto ro 6pyo

rou, €yrve en(rtpto p€trog trlg Mo0rlgourrlg Erotpe(og rou Aov6(vou, pt€-

trog r4q Erotpe(og Entorqptitv rou lrdttvyrtv (€vo ond ro onou6ot6repo

rdvrpo poOrlportrdrv rou 6€rorou Evorou ottitvo), rot 6trope 6vo perdtr-

}?G 7. Dauben, o .285.

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N P A I M A T I | ( A M E I A A O I

fuo ond rr; Boorlrrrl Erorpefo tou Aov6(vou. AuoruXdrq, dtreq our€g or6rorp(oerg 6ev on6rperlrov to 0dvord rou or rfruyrorprrrj rtrrvrr{, oe r1}r_r(o eB6ogrjvto 66o erdrv.

Evd ro peyoldtepo rp{go tou ipyou rou Kdvrop €yrvr ono6errdon6 rouq poOqpolrofq rou erroorod ouirvo, rdnoro npoptrrjpoto e{o-rotrouOo0v vo undpXouv. f1ptirov, undpXouv orcdpo rdnoror unoorlpr_nfq rou 6roro0quopo6, or ono(or onoppfnrouv r4v 166o ev6q n]{pougone(pou. 'Onoro po0qporrrd onorto0v vo teppotr(eror rdnog pro dnrr_pr1 6ro6rroo(o, dnoq xdvouv ot <puorro( opr0po( drov rprdvouv oro rD,Oeoporivror 6nonro. Enrnl6ov, 6Xer ono6erXre( du ivo rprlpo rou ipyoutou Kdvtop poor((etot ond troyrrdq ovurpdorrq. lvurp(oupe dl yroonorouo6{notc 6do nenepool,rivoug oprOporiq, o ror p, np€net vo roXri-er pio on6 uq nopordro oXioerq: o > p i 0 > o ri o = B. loXrier ndvroout6 yto rous unrpn€nepoopdvouq opr0podq; O Kdvtop on€6er(e 6tr,o9o6 ro v < 2", r6re yro ro dlerp nou 6qpro0pyrloe (6r1tro6{ rouq No,N,,...) roXue p(o on6 ug nopondvo oXdoerq. yndpXouv 6poq dtrIotun€prsn€pooplEvor oprOpo( nou 6ev s(vor rdnoro ond to dtre<p nou 6pr_oe o Kdvrop; E(vor npdy;rotr ro c dvo on6 ro dtrerp rou Kdvrop;

To np6Btr4;ro ov yro onorouo6r]nore 6fo unepnenepoopivouq ntr4_Orxodq opr0po0q roXrja p(o ond nS Tp€rq oXioetq (peyotrtirepo, grrp6te_po rj (oo) e{oprdror on6 r4v undOeo4 r4q rotr4q 6rdro(4g.

OPTIMOI: 'Evo ofvotro e(vor rotrd 6roretoyp€vo ov yro rd0r uno_o6volo, outrrnepdopBovoprdvou tou (6rou tou ouv6trou, undp_Xer €vo npc0ro ororXe(o. To rev6 odvotro, ro A,}twptfror ro_ld 6roreroypdvo.

Av ro ofvotro dltov rov nlrl0rrtirv oprOprirv e(vot rcotrd 6rot..oy;.r€-vo, r6te e(vot 6uvor6v vo ono6e(oupe dl outo( e(vor ouyrp(orpol ps_ro(U rouq. O Kdwop n(oreue 6rr or unepnenepooprivor nlqOrro( opr0-trro( tou oX4pdtr(ov dvo rotrd 6roreroyp6vo o0volo, oltrd 6ev gnoporl_or vo ro ono6e(er. To Ordrprl;ro rqq rolr]q 6rdtofqq, dl 64]o6rj rdOeorivolo pnope( vo e(vor rold 6roreroyg€vo, ono6erprlre reltrd on6rov Epvor Z€pgelo (1s71-1956) ro 1904, ror rdte pn6peoe vo ono6er- 3r1

Page 307: Ο Ταξιδευτής των μαθηματικών

K E O A A A I O I 2

1re( 6rr, yro onotouo6r]nore 6fo ntrrl0txofq optOgo0q, roXrier Pr(o ond

rrq rp€tg oX€oetq.To Oedrprlpo rqq rotrrig 6tdto(qq 6ev 66tve ondvtrloq oe 6Io ro

npoBtrrlgoro nou unrlpXov orq Oeurp(o rov uil€pn€nepoopdvtrtv opt0-

1-uirv rou Kdvrop. Ot rpuotro( opr0po( rorolr]youv orov np6ro uil€pns-

nepool.tivo opr0p6, rov o. Ot unepnenepoo;tfvor opr0trro( rorotr4youv

dpoye oro o; To vo uno06ooupe rdtt r€roto o6r1ye( oe ovr(rpoor1, rot o

Kdvtop nopo6dyr4re 6tt ot unepnenepoo;r€vot oprOpo( 6e rprdvouv no-

t6 oe rdnoto 6pto, 6nurg ot <puotro( opr0go(' Aurd ro 6totdntloe oro

nopoxdttrt 0etirprl po:'

oEoPHMA: To o0oqtrro Q 6ltlv ttlv opr0pdrv e(vot pto onotrritutg

dnetprl, ovordtrou0q outrtroYq.

Oo(vetot e(otpeurd nopd[evo vo grtrdpe Yto ro o oov vo e(vot 1-ttoev6qro 6tov, otrlv npoypoux6rrlto, on6 tov (6to rov oprop6 rou, ou-

t6 e(vor ovor6trou0o. Auui 11 16€o pooovt(er notrtrofg entorqpoveg, ytot(

rr1 Oeurporlv pootrd rleyd6r rqg 0eulp(og turv ouv6trtrtv.

llop6l e(orotrouOotiv vo undpXouv npoBtrriporo pE rrl Oecup(o rcuv

ouv6trtrtv Kot rouq dnetpoug opr0porig tou Kdvrop, 11 Oeulp(o rov un€p-

nenepoogdvtrtv oprOptirv np6o0eoe 6vo ouvopnoortr6 rot orlpovttr6

topdo oro o0yXpovo goOqpottrd. 'Orov O€troupts vo oK€qro6pe rdrt

Trou vo (enepvder rord nol0 tpdg roug (6toug, pnoporjpe ttitpo vo pe-

treulooupe outo0g touq otrqOtvd tepdottoug opt0trtodg.

322 8.'O.n., o. 243.

Page 308: Ο Ταξιδευτής των μαθηματικών

I[ - , r ^ ^ A r o

1 3

[0, .o,ogus(g oproFoFvnFovts

ttEF

'EXoupe e(ndoet ndrg ot dvOponot oytuv(ovrov vo ovorotrdr'pouv 6tro

ro e(6r1 opr0priv nou tt0po rorotroppdvouv rov proOrlportr6 pog r6opo'

Ilop' 6tro nou peydlo rprl;ro ourdrv rov ovorotrd$eov orpefierol orrl

pe0o6rrrj, ronroorrr4 boutrerd rorvr.rrvuitv nou onoretro0vrov on6 ouvrl-

0roydvoug dvrpeg rot yuvo(reg, orlptovrrrrl np6o6og oqefierot ortq npo-

ond0eteg geprovuttrtEvtrtv ot6trttrtv. tldrg 11 ovrfiqr|ln UoS yto roug opr0-

go0g oXer(erot U€ rlv ov0pt0ntvq tu<puto pog, t6to(tepo pe rrlv eurputo

opropdvtrtv ovOptirnov (aXrrrprortirv; Av 6Ev unqpXov rdnotot e[otpertrd

rolovrodXot dvOptunor, oE rt en(nE6o 0o e(Xov rprdoer to ;ro0rlgortrd

pog; E(vot 6vrtlg tbto(tepo €Trovoorortrr] q ouveto<popd rov e(otpertro

rurputitv or6purv; Mnopotiv, dpoye, ouro( ot (eXurproro( dv0ponot vo

Xerprotofv touq optOpofg ge tbto(repoug rp6noug, rotovotdvtoq rouq

or Bd0og pe rp6no nou epe(g ot un6trotnot 6ev pnopodpe; Oo npoono-

0rlooupe opXrxd vo e(erdooupe rdnoteg neptntc0oetg ordgutv nou qoi-

v€rot vo 6Xouv pto trov6rrpo yto unotroytopo6g notr6 geyotrfrepq on6

ro p6oo dvOptlno.'HOq on6 rtg opX€g rou 6€rorou 6y6ilou ottitvo, outrIfYovrctv orot-

Xt(o yto dropo nou pnopodoov vo rdvouv 66orotroug rct yprlyopouq

voepoUg unotroytopotig, noltrd on6 ro ono(o qrov no16rd-0ofporo' Kd- 3EE

Page 309: Ο Ταξιδευτής των μαθηματικών

notor ond ouro6g roug ovOptirnoug ;.reyoldrvowoq €yrvov tropnpo( snt-or4poveg rot proOq;rortro(, evcO dllor rlrov 6rovorlrrrd ro0uorepqg6vor.Aur6 ro eupd rpdopo 6rovorlrrrcilv rrovor{tcov uno6rltrdrver 6rr 11 rrov6-rrlro yro voepoUq unotroyropoOq 6ev onotetre( XopoKrrlprorrr6 prroq yevr-r4g eurputog, otrtrd r(vor 6vo e(er6rreuprdvo rotr€vro.

Irtq p€peq pog, ondvro vuirOouprr 6€oq on6 rov yprjyopo voepd Xer-prop6 peydtrulv opr0pdrv. Aurd e(vot nr0qv6v vo 6e(Xver €vov r6opo nronoldntroro ror Bopreorrlpivo, r] prnopr( vo orpeileror orrlv roXdroulovdmu{q rov unotroyrotirv, Tr.,Jv pqXovtirv 6nlo6n nou rdvouv rougunoloyropodq rorvorol(o. [ror( vo e{oorodpoore noltron}ootd(ovroge[orf{rproug opr0goUq pe ro puotr6 proq, drov or uno}oyror€g nou €Xou-pe oro on(rr pog gnoporiv edrotro vo notrtrontrootd(ouv opr0trro6g pe 6e-ron6vtE { e(root CInq(o; O€}ur vo nrorEful 6rr dXoupe onoporpuv0e(on6 rqv ntrlrrrrrj rupovvfo our6v rov Xpovop6purv unotroytogrirv, yrovo orpo<pofprE or rro orprlprlp6vo ror peord vorlporog epurrr]poro. Ko-rd rr1 6rdprero rLrJV Trporlyodpevtrrv ouirvrrrv, or voe pof unotroyropo( rlrovrdl 6to6e6opr€vo orrl 6r6oor6pevq 0tr4 rurv oXole(ov. fro vo e(ooroUvTrl FVnUn rouq, or dv0purnor ouXvd o<pt6purvov Xp6vo or4v onoprvrlprd-v€uorl For(pooretr6v norqpdrurv ror rpoyou6rtirv. 'lourg q Orovorptrrlenr6e{r6rr1ro vo e(Xe peyotr0reprl o((o ere(vo ro }ty6repo nep(ntroro

Xp6vro.llop' 6Io rou orrq pdpeq prog 6e 6(voupe €prpooq oroug voepoUg

unotroyto;rodg, prnopr( vo O€troupe vo e(erdooupre neprnrdoerg e(orpert-rtiv vorprtiv trovorrpov, (oog onldrg ror p6vo yro vo rorovor]oouperotrUrepo n6g epre(q, c,rg dvOponor, npooO€roupe trg 6rr€9 Foq npoKo-rolr]{etg orrlv €vvolo rcuv opr0pt0v. fro rqv €peuvd pog undpXouv ru-ruX6q 60o e(orpenrfg n4ydg. fro roug r{����€orpeurorig opt0popv4povegovdpeoo oro ouvr10r1 ntrr10uog6 dXoupe ro €pyo rou lrfBev Mn. Ipr0,roOqyr1r4 QuXoloy(og oro flovenrorrlpto Mnpdouv, Ot onouSabt vot-po( ap0poltvriltovsg.' Mro dpeuvo yro rouq onou6o(oug oprOpopvrjpo-

1.Steven B.Smith, The 6reat Mental Calculators (N6o y6prr1, Columbia Univer_344 sity press, 1e83).

Page 310: Ο Ταξιδευτής των μαθηματικών

' Y E I E A P I O M O M N H M O N E E

veg prero(0 rov rpuXtrd oo0evtirv €Xoupe oro BtBtr(o tou Ntdpotrvr A'

Tpdrpepr, torpoU, 6teu0uvt] ev6g K€vrpou tUuXrrqg Yye(og rou Oov vrt

Aor, oro Outordvotv, E{atptttxo( dv1punot.'

Apr0propvdpoveg psroq0 r t r lv ouvn0rop€vt-rvov0pr,0nov

Or unotroyropo( ge peydtroug optOpo0g on6 ouvqOtop€voug 11 npotrt-

op€voug ovoptirnoug xaporrrlpi(ovror ouvr]or..-rg ond 6vo oovotro t6ro(-

tepov 6e(torqturv rot ouv0qr6v' Kor'opXdg, ot neptoo6repot opt0po-

pvripoveg {rov not6rd-0oUpoto rot €Xouv, 11 etXov, e(otpertrd ov€nruY-

pfvq pvr1pr1. Ie notrIdg neptnrdoetg, ot dv0ponor ouro( preydtrrrroov

onogovopdvot rot, pt€oo orq povo(td toug, €go0ov vo oyono0v roug

opr0po0g. IuXvd, rpo(vovrot ovoYKootrt€vot vo perpo0v rut vo rdvouv

unotroyropo0g. floltr€g qopiq, 6tov ovorotroTrr€Iot Io toro(repo ro}€vro

ttrrv t6totputitv opr0gogvrlp6vtrtv, ot (6tot 11 dtrlor npoonoOotiv vo no-

oouordoouv ro rutr€vro louq oro rorv6. Enopdvtog, notrtro( opt0gogvrl-

poveg €6tvov rot 6(vouv nopoordottg 6toore6d(ovroq orpooqpto

otqv Eupdrnq Kot rrlv AgePtrrl.

To e(6q rov unotroytoptirv nou rdvouv epn(nrouv o' €vo neptoptoptdvo

orivotro roqyoptdrv. flptOrov, o notrtrontrootoopdq e(vot dvo rup(opXo 0d-

Fo rov nopoordoetiv rouq. Mnopodv rot notrtrontroord(ouv pero(6 roug

66o geydtroug opr0pto6g 11 urlrdrvouv €vov opr0p6 oe plo 60vopq' Aurdg ot

enr6e(erg orpopodv ouvqOurg t(re oprOgodg ge tp(o 11 rEooepo Unq(o,

nou utlttitvovrot oro rerpdytlvo 11 mov rfBo, e(re Uovotlliqtouq 11 6tQqrpt-

ouq oprOgofg nou utfttitvovror oe geyotrrltepeq 6uvdperg. Entntr€ov, 6qgo-

rprtrrig e(vor ror 4 e(oyrrryri rurv pt(tirv geydtrrrrv opt0ptitv, nop' 6tro TIou ro

nep(epyo e(vot 6u q t(oyuryri pr(tirv e(vot pro 6to6troo(o trty6repo notrU-

ntrorq on, 6oo ouver6qtonore( ouvr]otrrg ro rorv6 flou rouq noporotrouoef.

2. Darold A. Treffert,

shers ,1989) .

Extraordinary People (Nio Y6prr1, Harper & Row Publi-325

Page 311: Ο Ταξιδευτής των μαθηματικών

Or opr0gopvrjFovsq nou 6(vouv nopootdoerg gnope( en(or1q vcr €Tn-6errv6ouv tqv rrovdtqtd toug vo pp(orouv peydlouq npdtoug oprO-pofq n vq qnlonotodv peydtroug o6v0erouq aprOprofg, nop6l 11 rrovd-rrlro ourrl 6ev enrdetrvtieror rdoo ouXvd 6oo o notrtrontrooroopdg ror 11e[oyury( pr(tirv. Kdnoror opr0popv(goveg enr6errv0ouv sn(or1g rrlv rKo-v6rrlrd rous vo onopvrlpoveJouv peydloug oprOporig 4 geydtreg outrtro-y€q optOptirv, 11 vo pp(orouv U€ Uro porrd rov ntrr10rr6 oprOgd ev6q ou-vdtrou ovrrrer;ldvov.

Or optOpopv{poveq ondvro nopouotd(ouv nopobstyporo np6o0e-oqg 11 orpofpeons. H 6ro(peor1 Xprlorgonore(ror yro rnv ontrono(qoq, oI-ld ourrl ro0ourrl 6ev nopouord(eror. Or noltrontrooroour€q oXdoerg pe-ro{U tov oprOptirv rpo(vetor ntoq yorlteriouv rouq oprOgopvripov€q Kor,enog€voq, e(vor nro e6rolo vo rrq Ouprofvror.

lldoo onoubo(eq e(vor our€g ot rrov6r4reg; Av rrg 6oripe €flrqovsro-rd, rdvouv ro puotrd FoS vo ooor(er. lro nopd6erypo, o T(evrevrdtoMnd(rov, nou yevvrl0qK€ ro 1 702 aro 'Elprov, orqv Ayy}(o, ovorpdpe-rot 6rt rpov retre(rr-rg ovotrrpdpqrog rqr nr0ov6v btovorlrtrd roOuorep4-p€vog. Oordoo, U€ To 1.ruotrd rou, tirfrooe oro rerpdyrrrvo dvov opr0p6pe rprdvro evv€o $r1<p(o. Xperdorrlrr 6u6pror pqveg yro vo ro rdv€t, otr-Id ov ouver6rlronou]oou1.re 611 6ev pnopo0oe vo 6toBdoer 11 vo ypd{eropr0porig, npdrerror yro €vo ernlrlnrr6 rot6p0ogo. O Mnd(rov 66rvenopoordoerg orouq yt(rovdg rou, rdvovroq 66orotroug voepodg unotro-yroporig yro vo rov repvodv pn(ptg. MnopoUoe vo 0ugqOe( n6oeg pn(-peg rlnte pre our6 rov rp6no oe rdOe noprn on6 rqv qtrrr(o tov 6tir6eroertiv ror perd.

O fr6Xov Mdpnv ZoXop(og Nrd(e yevv{OrlK€ oro Appofpyo, orrlfeppov(o, to 1824. Kord rq 6rdpreto proq e(€roorlq rov rrovor{rovrou, notrtrontroo(ooe 6tio orroQ4rproug oprOpo0q no(pvovtog og onot6-treopo rivo ooord 6eroe(o{t{rpto opr0pd oe nrvrlvro rdooepo 6eurep6-trenro.

flop' 6lo rou or voepo( unotroytogo( e(vor ondvror gero(ti TuJV yu-vorrdrv, 6Xouv undp(er geprrdg onou6o(eg yuvo(req opr0gopvrlpoveg. To

326 1980, rl lorouvrdtro Nr€pr on6 r4v lv6(o en66er{e ug rrov6qrdg rrlg

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notrtrontroord(ovroq ooord 60o opr0pofq ge 6erorp(o t|4rp(o o ro06voq

ror no(pvovrcg og onordtreopo €vo bcroe{orfqrpto opr0g6 gdvo oe e(-

xoot ort6 6eurep6trento. Ot 60o optOpo( enel6yrloov tuXo(o on6 uno-

troyrorr1. O lt(pev I;.10 noporqpe(: <Aur6q o Xp6vog e(vot 16oo notrf

rotrtirepog on6 ott6qnote €Xet ovo<pepOe( oto nopetr06v, tiote g6vov

og on(oteuro pnopoti;re vcl rov neptypotltougeo.r

Mto Oqporprtr{g nopdoroorl rurv opt0gogv4p6vtrlv e(vot q e[uyt'tyti

pr(tirv on6 peydtroug optOgodg. YndpXouv 60o tr6yor yt' ourd: npdrov,

e(vor rdn nou ouppdtrtr€t os Uto rotrq en(6et{r1 rot, 6e0repov, e(vot trty6-

repo 66orotro on' 6oo rpovto(dpoore. fro nopd6€tYpo, q ruBtr{ p(o

evdg orrorfrrjrptou optOpod 0o e(vot onldrg dvoq 6tQryptoq opt0pt6g. Eno-

p6vog, q e(oyrrryrl pr(tirv nepilopBdvtr rov npoo6topto;rd tou retreuroG

ou Kor rou opXrro0 {rryp(ou qq p((og. Ko0drq ou(dvet o opt0g6g $r1-rp(rrrv rrlg 60vopr1g, 16oo nto 6rjoxotro e(vot vu unotroyiooupe ro Qrlrpioqg p((og. Ot optOpo( Trou XprlotgoTrotodvrot Ylo rqv e(oytrtyti pt(tiv e(-

vot ndvro r€trereg buvdpetg, 6rot 6ore or p((eq vo E(vot ndwo or€potot

opr0go(. fro ttg reptrtdg 6uvdpetg, 10 r€trsuro(o (6e(0 rlnq(o qg p(og

pnope( vo npoo6toptote( onortretortxd ond ro tEtreuro(o qrlrp(o r4g 66-

voprlg. 'Evo rotvdtono nopd6etypo e66 e(vor ro yeyov6g 6tt 6trot or

opr0po( nou trrlyouv oe 5 €Xouv urg 6torp€tr1 ro 5. Enop€vtrtg, 1-tto 66vo-

pn Trou trriyer oe 5 np€net vo €Xet yro p((o Trou vo trr1yet oe 5.

O Brp Kldtv yevvriOrlre ro 1912 oro Aportpvropr. Ie otr6rtrr1p4 rqv

enoyyel;.rortrq tou (rrrri €6tvc nopootdoetg novroU orrlv Euptlnq rot te-

trrrd n4ye vo epyoore( oro CERN (Euptrrnoir6 K€vtpo [luprlvtrdv Epeu-

vtiv). To 197a e(riyoye rqv etrootq rp(tr1 p(o ev6g optOpo0 pe 6tor6-

oro q,rlq(o oe 6eroorrtil trenrd rot entd 6eurep6trenro' Aur6 dgurg 6ev

rpov ro nro svrulloloror6 rotdpOtrtpd rou ytor(, opydrepo, e(rlyoye trlv

epOog4roorrj tp(rr1 pr(o tv6g opr0;ro6 pe nevror6oto {rryp(o oe 6tjo tre-

nro rot evv€o 6eursp6trenro.'Or opr0po;.rv{goveq 6to0€rouv, en(oqq, e(orpeurri pv{Un. O Xovq

3. S teven B.smi th , o .97 .4. OJr., a.289. 32V

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K E O A A A I O T 3

'Epneporopr, nou yevvrl0rlK€ orrl Btdvvq ro 1929 o}trd peydltloe or4

Iovyrdrl, 6Xer onopvq;rovefoet to np6ro €vrero Xilrd6eq $qrp(o rou 6e-

ro6rrof ovorufyporoq rou fi. To onorel€opclro IOV tpuXotroytrtilv reor'

6e(Xvouv 6rt ot neptoo6repot dvOprrrnot €youv pto gdtrIov neptoptog€vr1

pvrlprl oe 6,rt orpopd roug oprOpo0q. O pr€oog opt0pr6g Qqrp(tlv nou

prnopotipe vo ovoroldoouft€ ov to orofooul-te p(o p6vo <popd e(vot

enrd Qrlrp(o, ov ot optOpto( tr6yovrot pov6rovo, rot og evvdo {qrp(o ov

tr€yovrot puOptrd. Mnope( vo rprdooupte oro 6dr6ero rfrqrp(o, ov ot

opr0po( rou r€or tr€yovrot pu0ptrd otrtrd rot oe (e6yr1. O lrotr6g opt0-

popvrlgovog Zor lvdouvtt (1s67-1950) unoptrriOrlrt ot r€or on6 to 6td-

orlpo llruXo}6yo rou 6€rorou 6votou ortirvo Atrrppdvr Mntv€, o ono(og

onorpdvOrlre drt o lvdouvrr gnopoUoe vo Ougdror oopdvto 66o {tr1q(o

ov ro drouye gio p6vo <popd. Aur6g o opt0g6g tpryp(tlv e(vot rpttg pe

6(r rpop€g peyotr0repog or' 6,rt roro<p€pvouv or ouv4Otopivot dvOptrt-

rror - €vrurI0.)otord, o}trd 6Xt on(oreuro. O Idtro O(vretroroiv, TIou Ysv-

vrjOrlre ro 1896 | 1897 oro Aour( orrlv llotrov(o, pnopouoe vo trctvo-

tropBdver optOpodq pe e(root 6tug e(root orrdr {rr1rp(o ov rouq dp}ene

pr6vo yro €vo 6eutep6trenro. Anopv4p6veuoe dvov opr0g6 pt rptdvro

evvdo {rr1rp(o 0otepo on6 p6vo r€ooepo 6eurep6trenro.

H pvnUn no((er nporpovdrg oqpovrtr6 p6}o otoug unotroyropotig pe

peydtroug optOporjq. O Btg Ktrorv, orodyovrdg rr1 ouv€Xeto, onogvrlg6-

vruot rnv npono(beto p€Xpr ro erot6 en( erord (ovr( yto ro 6ti6ero en(

6tir6ero rov poo4rtiv oro oXotre(o), ro terpdyovo 6trr.uv rov oxepo(lrtv

p€Xpt ro Xfito ror 6troug roug 1.229 nptitroug optOpodg nou undpXouv

p€Xpt to 6€ro Xrltd6eq. 'Evo on6 to rol€vrq rdnottrtv optOpopvqp6vtrtv

e(vot rot q trov6qro vo ;rerpofv YpnYopo. O fr6Xov Ntd(e pnopofoe

vo prerprlott pe pto p6vo portd rov opr0g6 rurv gnt(etrttitv nou Tt€ro0ot

rdnotog ndvut o' €vo rpon6(r. Meptr€g qopdq, Xprlot;ronoto0oe ro rq}6-

vro rou yto vo unotroy(oet rov opt0p6 turv ptptr(trtv nou unrlpXov ou1

BrBtrto0(rr1 rdnotou onrttoti, 6nou 6ev e(Xe [ovondet.

3eE 5. O.n., o. 245.

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;

) Y E I f A P I O M O M N H M O N E f

l - lpt i l ror Kol o0ve€rot opr0Poi

To releuro(o rotr€vto nou 0o petreqooupe e(vot q ontronoQo4 nrrv ofvOerov

oprOptirv rot o npoo6toptogdq rov npdtttrtv opt0ptirv. A0vowog optog€vo

npoBtrriporq rdnotot e(otperrrd ypqyopor oprOpopvqpoveq no(pvouv €vo

peyotro opr0p6 Kot rov ovotr0ouv moug npdnouq nopdyow€q rou. fto nopd-

6E,yFq o opr0trr6q 30 gnope( vq "ovo^ue€i, ri vo nopoyovronorl0e( mouq

oprOgo0g 2'3'5. ftor( or opr0go( 2, 3 Kot 5 ovopd(owot nptirot optOpoi

oPlIMoL flptitrog ap0p1E e(vor rivog guorr6q opt0p6q nou props(

vo 6totpe0e( r6trsro p6vo on6 rov eourd rou Kot rov opt0g6 1'

Atrtror opt0pto(, nou 6ev e(vot nptittot, pnopodv vo 6torpe0o0v ritreto

on6 neptoo6rspouq opr0go0g on6 ro 1 Kot rov eour6 roug. l-ro nopd-

6ery;ro, ro 4 pnope(vo 6totpe0e(tdtrero on6 ro l, ro 4 Kot ro 2. O optO-

969 6 gnope( vo 6totpe0e( on6 roug 1,6,2 Kat 3. Auro( or optOpo( ovo-

pd(ovror o0v0erot oprOpof.

oPlIMoI: IAv1nog apfip6q e(vot €vog rpuorrdg opt0p6g nou

pnope( vo 6torpe0t( tdtreto rot pt d}trouq optOgotlg n6po on6

To 1 Kot rov eour6 rou.

An6 rouq nopondvtl opropo0q E(vot erirotro va 6or1;re 6rt 6tror or rpu-

orxo( oprOgo( nou E(vot ptyoldrepot on6 ro 1 XuJp(ovrot oe 6do rtrdoetg:

toug npdrroug opt0gotig Kot rous o0vOeroug opt0po0g. Aev gnopodpe vo

ovotr0ooupe €vov npdro opt0pr6 oe ptrpdrepoug opt0pofg, otrtrd dtrot ot

o0v0eror optOpo( gnoporiv vo ovonclpooro0o0v oq pro outrtroyr1 np6rov

oprOgdrv nollonlootoopivurv peto{d rouq. Autd o64yef o' 6vov on6 rouq

0ep€trroug tr(Ooug turv poOrlporrrdrv, o ono(og trdet drr undpXtt dvog pdvo

rp6nog pe rov ono(o ovotrUetot €vog oprOgdq'6r1lo6q, rdOe opr0p6g €Xet

€vo xor pdvo €vo oOvotro np6ttlv nopoydvrov.

OEMEAIOAEI OEOPHMA THI APIOMHTIKHI: Kd0e rpuorr6g opt0p6g

trreyotrUtepoq on6 to 1 gnope( vo €Kqpoors( tog ytv6pevo

np6rov opr0gdrv g'€vov ror povo6trd rp6no. 329

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Kdnoror rbtoque(g oprOpopvrlpovrs, yrcl vo notrtrqntroordoouv 66opeydtroug optOprodg, ovolfouv npdrro touq oprOpotiq ror perd opX((ouvvo rdvouv npd{erg [-l€ rous npdrrouq oprOpoOq, 6orE vo o6r1yr10o6vorrlv ondvrqorl roug. Oordoo, or po04prouro( ovoyvop((ouv rr €vo Bo-O0repo np6ptrqpo oE 6,1 orpopd roug nptirroug oprOpodq, n€po on6rrlv ontrr1 Xprjorl rouq oroug unotroyroporig. YndpXouv dnerpor rpuotxo(oprOgo( nou e(vor npdrror, otrtrd or npd.rror opr0po( oportirvouv ro0tirg orrpuorro( opr0po( y(vovror 6Io rot peyoltirepot. To npdptrrlpo rou vcr

Bp(orouv ypnyopo noror geydlor opt0po( i(vor npdlror Kor noror o6v0e-rot rtvE( ro ev6rorp€pov rurv po0rlporrrdrv e6tir ror ot6veg. Tr ouprpo(verpr rrlv rrovdrnro rou pr€oou ovOptirrou vo ovoyvtrrp((er roug npdrrougopr0trrofg; O opydprog, o<po0 pdOrr npdrro rr e(vor o nptirrog opr0p6g,6e 6uorole6eror vcl rr€r 6rr ro Z e(vot nptirrog opr0g6q Kor ro 9 ofv0e-roq opr0p6g. Onoroo6rlnore €Xer e(oorrlOe( pe roug opr0po6q pnope(vo ovoyvtrrp(oet opr€ooq 6u ro 51 e(vor o6v0erog opr0p6g (3.12) evdr ro53 e(vor np6rog opr0p6g. Ot enoyyetrpor(eq poOqpolro( pnoporiv ou-

Xvd vo rouv ov e(vor nptirror 11 o0vOeror rpr{rrlqror 11 ror rerporprlrproropr0po(.

H ontrono(4oq peydtrov oprOprtirv Kor ro vo ovoyvtrrpr(er rovr(g ovrdnotor peydtror upt0go( e(vor np6rot iXer ev6rorpdpov, yror( 6ev undp-

Xer rov€vog yvoor6g oly6ptOpoq { rdnoro orprBrlg orotrou0(o npd(eovyro vct pnopo6pe vo 6rorp(voupe yp{yopo 6vo peydtro odvOero ono6vo peydlo nptirro opr0pr6. 0o16oo, rdnotot oprOpopvr]poveq 6e(Xvouvvo 6Xouv rolfvto oE our6. O Ztpo K6trpnoupv (i804-i840) rlrov €vogApeprrov6g r6rorpuqg opr0popvrlpovoq, rou yevv{0r1re oro Bepp6vr.MnopoUoe vo ndper onorovOrlnore e(orfrlrpro r] enrog4rpro opr0pd rorvo n€r ov 4rov oriv0etoq r1 npdrrog. Av rltov ofvOerog, pnopodoe vorrr rouq nopdyovr€g rou pr€oo oe tr(yo Deurep6trenro. O otrydpt0p6grou 4rov nporpov6q unoouve(6r1roq' trror6oo, o<pof ro ordrprrlre notr0npooerrtrd, ror€Ir1(e oro oupn€poopo 6rr r(Xe onopv4povedoer dvoerreropdvo otjvotro rorolrl{eov oprOptirv, tirore vo ;rnope( vo pp(oretroug nopdyovreg. fro nopd6etypro, pdvo opropivo rfqrp(o orrq 66o 6e-

33* (r6reprg 0€oerg rrrrv nopoy6vrtov 0o 6Xouv og onorftreopo rqv ovr(-

l

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ororXq rordlq[r1 oro o0v0ero oprOpd. Aut{ q teXvrrrl e(vor nop6goro 1.r'ere(vq nou Xpqorponore(ror yro rrlv e{oyoyq pr(tirv.

fro nopd6erypo, o xotdtroyos rou K6trpnoupv pe to nrOovd rorolq-rlxd tpqq(o rov nopoy6vrov rou orjv0erou oprOpo0 2.983 onoretre(torond e(roor (e0yq oprOprirv. Aqlo6rl, ro 83 gnope( vo oxll.rouorei p6vo

ond e(roor 6rorpoperrrd 6up4rpro (efyq. E(rtd(ovroq ourd to (efyq, e(-vor Ouvotd vo ovorotr0{roup€ rous nopdyoweg rou 2.983. To np6ro(eUyoq rorotrqrcrrtiv Qqq(ov e(vor ro 01 Kor 83. Arpori ro 1 6ev e(vornapdyovroq (tr6yo rou 6rt e(vor nopdyovroq rdOe opr0pori), nrOovo(nopdyovreg prnopo0v vo e(vor ro 101 Kol ro 83. To yrv6yev6 rouq 6poqe(vot €vog opr0p6q U€yoldr€poq on6 ro 8.000. Av enrouvdrlroullr ofioro-6r]nore dtrlo {qrp(o oro 01 Kor 83, ro yrv6pevo e(vor or6po peyotr0re-por opr0go(. Enog€vog, pnoporlpe vo 6roypdrlouge our6 ro nptiro (eri-yog. To 6e0tepo (efyoq e(vor ro 03 Kor 61. Mnopodpe vo 6o0pe op6-ooq 6rr to 3 6ev e(vot nopdyovroq rou 2.983 (ov npoo0iooupe ro Vrl-rpio 0o ovorotr0rloupe 611 6e 6rorpodvror ge ro 3), rr dpo q endpevqnrOovdrqro e(vor ro l03 x 61. Aroypdrpoupe 6poq rr our6v ro ouv6uo-op6, yro rouq (6rouq ldyouq lre rov np6to ouv6uoop6. Telrrd, drovrptdvoupe oro 6y6oo (edyoq, ro 19 Kor 57, pp(oroupe roug nopdyovrdqpoq.O opr0p6g 19 6ev gnope(vo y(ver 119, yror(r6te to yrv6pevo nou0o npor0Qer 0o e(vor uneppohrd peydlo. Iuvet6qronotdtvtoq dputg 6tt20 x 150 tooUror p€ 3.000, rorotroBo(voupe opdooq 6rr ro 19 x 157rooriror ge ro 6tr6 FoS 2.983. fro pog, prpo(og, n Xpt]on evdq r€roroun(voxo ontronoiqoqq e(vor opyrl ror dXoprl, otrtrd oto X€pro evdq 16ro-quo0q oprOgopvrlgovo, r6ro(tepo ev6g or6;rou nou €Xet eooreprredoetr4v 614 6ro6troo(o, o unoloyro;.r69 e(vor eUrolog ror rpo(vetor oye66vourdporoq.

O 6uf{rprog n(voroq nou orrogvnp6vEuoe o Kdtrgnoupv ror dtrtror6ev enopre( ror Xperd(eror enrnldov po{0ero yro rlv ontronoiqoq. OT(opr( fldprep Mn(vtep (1806-1878) rjrov 6voq rOrorpu{g oprOgopvrlpo-voq on6 ro M6pergv rqg Ayytr(oq. Bprlre tuXo(o dvov oly6pt0po yro uSontronor{oetq, nou {rov q6r1 yvoor6g orouq poOrlpottrofq. M6vo otneprrro( oprOgo( iXouv ev6rorp€pov oto npoptrrjtrroro onlono(qoqq, 331

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orpou onoroo6rlnore dprroq opr0pdq brorpe(ror U€ ro 2 rot pnope( voonlonorrlOe( rot vo ovoX0e( o, €vo prrp6repo o0v0ero opt0p6. ,Evoq

neprrr6g orivOeroq opr0pr6g pnope( vo yporpe( u;g yrv6prevo rov ropo_roro: (o n F)(o - F) ro ono(o, drov notrtrqntrooroote(, pog 6(ver: q, _B'. Enop€voq, rd0r neprrrdg oUvOerog opr0p6q pnope( vo yporpe( og6torpopd 6rjo oprOpdrv uqrop6vov oro rerpdytrrvo, rov o ror 0. O Mn(_vrsp Xprlorpono(r1oe ourd ro yeyov6g yrct vcl pp(orer roug nopdyovreg.Av o x e(vor o oprOpdq nou qrdxvougr, r6re x - o, _ B, r] o, _ x = p..Enopdvrrrg, ovo(nrdge onltirg €vov opropd (<o), rou ono(ou ro rerpdytrr_vo 6torp€per qn6 rov x rord ivo r€lero rerpdyovo.

Ag erpopp6ooupe rq ;160o6o ourrl orov opr0p6 5.25.t, or 6fo nopd_yovr€q rou onoiou e(vot ot oprOpo( 59 Kor g9. -errvdpe on6 ro prrp6re_po rftrero rerpdyovo nou r(vor geyotr0tepo on6 ro 5.251, rou e(vor roretpdyu-rvo rou 23. Oo poq0oUoe, Bepo(trrg, ov efiope onopvqpovrUoerro rerpdycuvo 6trtuv rcov oprOpdrv on6 ro 1 €og ro 100! yrftirvoupe tror-n6v oro rerpdytrrvo ro /3 Kor no(pvoupe 5.329, an6 ro ono(o orpotpori_pe ro 5.25 1. To onor€treogo roofror pe zg - nou 6ev efvot r€lero rerpd-yrrrvo. Au{dvoupe rtipo ro Z3 rord p(o povd6o ror no(pvou1.rt t4, roono(o u{drvoup€ oro rerpdyurvo ror no(pvoupE 5.416. Arporptirwoq ro5.25.1 ro(pvou1t 225, nou e(vor ro rerpdytrrvo rou 15. Enop€vtrrg, pno-po0pe vo ypdgoupe 6u 5.251 - (t+ + 15).(t4 _ t5) = 89.59, ror 16o6 or66o orprpdrg nopdyovreg nou q,dXVoFs.

Iqv npd(r1, 6e Xperd(etot vo ouveX(ooupe vo npoo06roup€ ro I Korvo u{rdrvoupre oro rerpdytrrvo yro vo 6ortpdooupe rov opt0p6 <o>. flo_poqprlore 6rr (o + t)'= o' + 2a+ 1. Enop€voq, yro vo ou(r]oougs Ko_rd .l ro rerpdyovo rou cl, ro pr6vo nou Xperd(eror vo rdvouue e(vor vonpoo06ooupr oro 0'To 20 + l.

Xprlorponordrvros rrq 66o reXvrr€q rurv rorotrr;(eov rov oprOptirv rorrov brorpoprirv rov r€rpoytirvov, or opr0popvqpoveq pnopotiv vo onlo_nororjv ypfiyopo opt0;rodg nou rprdvouv €urg ro dvo erorogprtlpro Kor,peprr€g qop€q, 6o9 ro 6€ro eroroppripro.

333

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O O Y E I E A P I O M O M N H M O N E f

H nepinroon rou T(ov rcor rou Mdrr<A

Mro er6trrl rorqyop(o oprOpopvqg6vov, nou ovoprd(owot tSrcrput( 1trrpdvo4,6Xouv mr6e(et tqv rrov6rnro vo ovoyvurp((ouv ov unepBotrr_rd peydlor oprOpo( e(vot npdror r] o0vOeror. Or r6ro<pue(g prrp6voeq e(_vol dropo nou 6Xouv orlgovlrd XoUlIn yevrrrl euguio, dnurg gerpdrorpe ro ouvq0rop6vo teor voqpooriv4g, oordoo 6ro06rouv 6vo ofroon_ge(oro rotr€vto e(te ot4 pouorr{, e(re oqv rdXvq, e(te orouq unotroyt_opofq. Or r6rogue(g grrpdvoeg e(vor ouXvd dropo ourrorrrd 11 dv0pot_ror rou dXouv unoore( rorurorrq oro oprorepd rp{Fo rou eyrerpd}ouTouS oulv nor6trr] rouq r1trrr(o. H e(otpettrrj rrovdqrd rouq orrl Fouor_r{ { qv ovoyvtirproq oprOpdrv ouv6ud(eror ouvrlOog ;_re ;.rro notrU roXu_pn FVnFn. Kdnotor r6rorpue(g ptrp6voeq trrnoporiv vo Ou;.rorjwor rr rorp6efXe rd0e p€po r4g (orjq roug. To rotrdvto oroug unotroytopo0g er6rlltir_veror ouvrlOurq orouq rlpepotroyrorofg unotroyroprofq, 6nou o r6rorpu4qprrp6vouq prnopr( vo r€r rq ooorrj qg€po rqq ep6opd6og ov tou 6r.0_ooup€ rct un6}orno ototXe(o rdnoroq rlpepoprlv(og tou nopel06vtog rjrou ;rillovroq.

'Evo dtrtro on(oteuro rot ondvro rotrdvro rdnorov on6outorig r(vor 4 trov6r4rd rouq vo Bp(orouv roug nptirrouq opr0po6g.Aurrl r(vot q nep(nrtooq rrr_rv T(ov xor Mdrrtr.

O 66rrtrrp OtrrBep Io[ e(vor ro0qyrlr4q rtrrvrrrjg veupo]oy(og oto lo_rprr6 Kotr€yto Alpnepr Arvoroiv, or4 N6o y6pr4. Iuvdwrloe yro npdrrrlrpopd rouq 6(6ugouq r6ro<pue(g prrpdvoeg T(ov ror Mdrrtr ro 1966 o, tvarporr16 voooxope(o. Ere(vr1 rqv enoXrl, or 6(6u;ror {rov e(roor €{r ertirvror e(xov rlerore( oe doutro on6 r4v qtrrr(o rcuv enrd et6v. 'Hrov rot oL6do qurtotrrof rfuXururo( ror 6tovoqrrrd rco0uorepr1trrdvor. Oordoo, ro_069 noporolou0o6oe rrg roOqpeptvdg 6poorr1pr6r4r€q roug, o 6drropIo{ ovordtrurfre drr or 6(6upor erXov €vo nopd{evo tol€vro.

Mro pipo, ro0tirq o 6drrrrlp Io( rouq noporolou0o0oe, noporr]pq_oe dtr rltov onopporp4;rdvor os Uro nopd{evq ou(riqoq, dnou o roOi_voq pdo0oe 6ro6oXrrd ror orotrou0orjor €vo 6tdotrlpo ouln{g. Kot or66o Xopoyelo0oov, oov vo onotrdpBovov rdnoro 6trd roug oore(o.fllqo(ooe orcunqtrd ror rd0roe rdro vo ororioEr. frpdrro o T(ov dleye 33i

' , .,i*+,

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€vov e(otfrlgto opt0p6. o Mdrrl drouye rov opr0pd ror ouyrevrptr.rv6-rov. Mto ouyp4 opy6repo, onovrofo€ p€ rov 6116 rou e(orprlgro oprO_p6. Merd 4rov 4 oerpd tou T(ov vo oreqre( nprv onovrrloer. And ron6oo Xopo0pevo rpo(vovrov ro np6oond rouq, o 66rrurp Io{ r1(epr drronoldpBovov ro prrp6 roug norXv(6r.

Tr rlrov ouro( or oprOpo(; O 66rrtrrp Io{ rordyporfre peprrorig rorroug n{pe on(l rou yro vcl rouq pretrerqoer. IupBoutreu6gevog 6rdrpopoprBl(o po0rlporrKrirv, ovordtrupe 6rr rlrov 6lor npc0ror opt0poil ,Evoq

e{oq{rproq opr0p6q pp(orerot ovdpeoo oro erord Xttrrd6sg ror ro 6voeroropprripro. lldrq gnoporioov ouro( ot 6tio veopo(, rou Kor ot 6dorlrov 6tovorlrtrd ro0uorep4pivot, vo (6pouv, 11 vo ovoyvtrtpf(ouv, rdoopeydloug np6roug oprO;rotiq; H rorop(o 6purg 6ev retrerdver e6tir, yror(o 66rr'p ro{ en€orperp€ oro voooKope(o qv endpevq gdpo roupotrtir-vrog po(( rou ro 6rrd rou Brptr(o pe npdrroug oprOgorig.

Aur4v rr1 rpopd rd0roe rdrur po(( p€ rous 6i6upoug, tiore vo gnope(vo oul]trrerooxrr oro notxv(6r rouq. rrrlv opxr] ere(vor 6(oro(ov vo ror-{ouv pnpoord oro yrorp6, otrtrd odvropo {ovdpXtoov. Arpo6 rqpoKo_Ao60r1oe orurnntrd yro l(yo, o 6drrurp ro( npdrerve rov 6116 rou oKTo-qrnqlo npdro opr0p6, rov ono(o 6tdBooe ond ro prptr(o rou p€ rouqnpdrroug opr0poUq. Kor or 66o r<dp<pu_roov ro pdrro rouq F€ €rntrr1(r1orov vsoq€pp€vo. 'Yorepo on6 prro no0oq nou rpduloe pro6 trent6 rjhyo neproo6r€po, Kor or 6Uo v6or Xogoy€trooov (orpvtrd ror dyvegovrororpourd ge ro re<pdlto rouq. o r(ov ouyrevrptirorlre yro nollrj 6poror reltrd e(ne 6vov opr0p6 pe ew6o Snq(o.

,Enerro on6 pra no6oq yrovo orerpte(, o Mdrrtr e(ne €vo 6e6repo opt0pd pre evv€o q4rp(o. O 6d-rrop ro{ ro(ro(e oro Brptr(o rou p€ roug nptirrouq optOporig ror r(ne€vov np6ro oprOpd pe 6€ro Sqq(o.

llp6ror oprOgo( pe evv6o ror pe 66ro rlryp(ol .Evog opr0p6q p€ €v_v€o $q<p(o pp(oreror ovdpeoo oro erot6 eroropgripto ror to dvo 6roe-roropp0pro. 'Evog opr0g6g ge 6€ro rlrqrp(o pp(oreror ovdpeoo oro 6vo6rosroroppr0pro Kol ro 6dro 6roerotopprjpro. To norxv(6r ouvex(orr1re,oltrd ofvropo or vdor {en€pooov rqv rrovdrqto rou yrorpof vo roug

334 noporolou0rloe r.

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...petd o T(ov, €netto on6 6vo tepdotto 6tdotrlpo ouroouYK€vrpoonq'

ovoro(vtrloe 6vo 6ul6ero{rrl<pto opt0g6' Aev eiXo tp6no vo etr6y{o ov

rltov q 6ev rltov ourot6g xot Oev pnoporioo vo onovtrlotrl' ytot( to 6tr6

you ptptr(o 6ev npoXtlpo0oa n€po on6 toug 6erorlrlqroug nptirroug

opt0potig. O Mdtrtr 6;ro9 rdtt ordpove": rot pto titpo opy6tepo' ot 6f

Oupor ovtdtrtroooov etrooorlrrl<ptoug npdrtoug opt0po0g' q €tot un€Oeto

toutrdXtotov, yrot( 6ev e(Xo rov€vov tp6no vo to etrdy(tr-t'"

Erroou{rrlrproug npdrroug optOporig; Mnopotioov npdypotr o T(ov rot

o Mdtrtr vo npoo6top(ouv t6oo repdottoug npdrroug opr0po6g; Iltitg

pnope( rdnotog, n6oo gdtrtrov rdnotog pe Xopqtr6 6e(rrr1 voqgoodvqg'

vo npoo6top[(et €vov srroooqrrlqto np6ro opr0p6; 'EXtrt €vov unotroyrorrl

386-25 MHz. Ar6po rot pe npd(erg orp(peroq Drntrof t|lryp(ou orq ytrdto-

oo BASlc, 6ev pnoptir pe rov unotroytorri gou vo etrdy(ur opt0po6g n€po

ond rqv neptoX{ ttlv 66ro rptotKoroppup(trrv (6erorp(o Qq<g(o) yto to

ov €(vot ri 6ev eivot nptirrot opr0po( (err6g rt ov e(got npd0ugog vo

ovo(o rov unotroytorq pou Kot vo rov oqrlocrr ovotXr6 yto pdpeg rdoe

rpopd). t-ltirg quto( ot 60o r6rorpue(g prrp6voeg pnopotioov or6po rot vo

0uuofvror €vov etroootprirpro oprOpd, n6oo pdtrIov vo rov etr6yXouv ov

e(vor 11 6ev e(vot npdttog; Av or ovorpop€g e(vot otrqOtv€g, rdre ourd ro

dropo xordrpepov rdtt npoygorrrd o[roorlpe(rrrro - o(toorlpe(rrrro dXt

ontrdrg enet6q e(vot 6tovoqrtrd roOuoteprlg€vot, otrtrd o(tooqps(oro ov

ouyrpr0e( ge ro rotrdvro tou onotou6tinore ovOptirou'

Au0evr(eg nou €Xouv petrerrioet to toroptr6 rurv T(ov ror Mdtrtr oug-

roovo0v 6rt or 66o veopo( 6ev pnopoUoov rot 6Ev dpptorov roug nput-

roug optOtrrorig ttrEyXovrog toug peydtrouq optO;ro0q pe rov nopo6ooto-

16 rp6no. Aqtro6r1, 6ev €pptorov roug nptiroug oprOtrro0g 6rotptivrog

voepd gaydloug opr0gorig pe ptrp6repouq nptirtoug optOgotig' Atv

pnopotioov vo exteldoouv r6rotoug voepotlg unotroytopo0g' ytor( 6ev

;rnopotloov o0r€ Kov vo npooO€oouv otlord prrpofg opr0po0g' Enrntr6-

ov, o opt0p6g ttrtv voepdv unotroytoptitv nou e(vot onopo(rqrot Yto rov

6. Oliver Sacks, Ihe Man Who Mistook His Wife for a Hat (N€o Y6prq' Harper

perennial, 1985), o. 203.

I vvIIU 'Y"r 'vv^ 335

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6leyXo or6po rr ev6g e{or]rrigrou npdrrou opt0porl 0o r]rov n€po ondtrq 6uvordr4rdq roug. E(vor, en(or1g, on(0ovo vo Xprlorponotorjoov ou_ver6qtd rdnoto ;.r€0o6o oov rr our{ nou egdpgo(e o T(opt( Mn(vrep, oono(og nopouo(o(e neprrrorig orivOeroug opr0po69 urq 6roqop€q rerpo_ytirvtr-rv optOptirv, o<pod ourrl q pd0o6oq Xpqorponote( errev6g rrlv np6-oOeor1, r4v ogo(peo4 Kor rov notrIontroorooud.

fl6q trorn6v to rordrpepvov o T(ov rot o Mdrrtr; Antrc0g 6e yvtop(oupe.Mro evOrogdpouoo 166o e(vor q e(rjq: or npc0ror opr0po( €Xouv pro r6r6u1_ro nou ge rdnorov rpdno pnope( vo onarovrore( voepd, npoo6rop(ovrdgroug €rol oq npdrrouq oprOpo0g. 'lotrrg or 6(6ugor pnoporioov vo oTurKo_v(oouv voepd rouq optOpodq Kor vo 6ouv our4 rrlv r6rdrqro. Epe(q or ou-vqOtopdvot dvOponor pnopoUpe, ntOovtirq, vo pdOou;re vo optr€noupe,ourrl rqv r6tdr4ro. Ag eln(oou;.re 6tr rdnoro p€po 0o rotoldpoupr nrirg oT(ov ror o Mdrrtr €rovov ourd to erntrqnrrd rorop0drpotd roug. Auoru_Xtirg, 0o np€nrt vo or4prXro6pe oe dtrlouq rOrorpue(g prrrpdvoeg r] oe dtrtregrotvor6peq dpeuveg yto vo trdooupe ro o(vrypo. To 1977, or rrl6eg6vegroug Xdrproov ru 66o ob€trrpto, yto vo evOopptjvouv rnv KorvovrKono(qo4roug. flop' 6lo nou ror or 66o rrirpo enr6etrv0ouv 1.rro petrtropdv4 rrov6_rrlro vcr trerroupyoriv un6 enonre(o oe neprpdMov r6pugdturv enov€vro-{qg ooOevtirv, rov€voq on6 rouq 60o 6rv rdver ntr€ov oqpovtrro6g npspo_Ioyroro6g unoloyroporjg ri 6ev npoo6ropi(rr nptirrouq oprOpo0q.,

T r onpo i ' vouv 6Ao ourd ;

Kdnotor 0o unoorqp(ouv 611 6ev undpXer 6rorpopd ovdpeoo oe €vqvr6rorpuq opr0popvrlpovo rt dvo ouvrlOropdvo dvOpono n6po on6 rrqtipeg e[dorrlonq, rrlv oydn4 yto toug oprOporjq Kor rnv rotrr1 pvr1pr1. Ol.l. Pdoug Mnol (1850-1925), rotoprr6q, l€rropog proOr1;.rorrrtiv rotnprirqv en(roupoq ro0rly4rqq oro Kotr€yro Tpfvtrr, oro K6pnprr( uno-orrlpr(et outrl rrlv dnorf4:

336 /. Treffert, o. 41.

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Or enr6doerq 4tov t6oo o(roorlpe(oteq, tiore rdnorot nopoulpqt6q Oeur_porioov 6u ro dropo ourd erXov 6rogopeuro0 e(6oug rrov6trlteq onderefueq rov ovOprilnulv qq enoXrjg touq. Autr] q dnot[q 6e oqp((etornouOevd. Onoroo6{note dvOptonoq pte e{orpeurr] prvripq ror p€ Uro gu_orr{ rXorl yro rlv opr0prlttrr], ov ouveXdrq otpdger 614 tqv npoooXr]Tou orrl gel€rr1 trrrv optOgtitv ror npoorpCper otov Eour6 rou oto0eprje(dorr1or1, onortd trreydtrr1 rrovdtnro orouq voeporig unoloytoporiq ror,peBo(urg, or enr6doerq outriv nou e(vor r6ro(tepo nporrroprdvor nporo-troUv peydlrl irnlq(1.8

O lt(Bev lprO nrorrder nep(nou ro (6ro. unooqpr(er dtr or vorprrdqrov6trlrrq nou Xpnorponoro0vror e(vot (6req yr ere(veg fiou Xprlortlo-rororjvror oulv nep(ntoon rrls ytrtirooog. Enop€vog, onoro6{nore dro_po Trou ro enr0upe( oprerd, pnope( vo pdOer nrirq vo y(ver oprOpopvrj_Fovoq.

Av 6;ulg eiXote opretd r(vr1tpo... rr ov yvcup(ote rrlv npono(6ero, 0opnoporioo vo oclq 6r6ti(ur ntirq vo e{dyere ruprrdq p((eq oe trrydrepoon6 ;uo ep6oyd6o ror 0o Xpero(dpouv €vo_6uo Unv€S yro vo ooq €K_nqr6eUoo ottq pootrdg opX6g yro vo y(vete €voq s{orpetrrdq opr0po_gvrjprulv.'

'EXouv 6(rro, dpoye, ot MnoI ror lgr0 6rov unoor4pi(ouv 6u 6)ot or,opr0popvrjgoveq r(vor ouorootrrd (6ror pe toug ouvqOropr€vouq ovOptir_noug; 'H g{nurq undplouv rdnoror opr0popvrjyoveq, 6noq o T(ov ror oMdrrl, nou 6ev undyovtor oto o0vr10eq cpdopo ttrrv ovOptinrvov rKovo_t{ttrrv; 'Ooov orpopd touq p€ooug rj 6{unvouq oprOgopvrj;roveq, o lpr06Xer pdtrIov 6(xro orrlv rono06tqori rou. Av 6o0o0v ro rotd}trqtro rivq_Tpo Kor undpXouv or rordllrlleq ouv0{req, or neproodrepor dvOponorpnoporjv vo y(vouv oprOpopvrjpoveq. Or 6e{rdrrlreE nou enr6errvdouve(vor ouvopnoottr€q, otrtrti rpo(vetor 6u ov{rouv o, €vo orjvo}o 6e(ron]_

8' w.w' Rouse Ball, "Calcurating prodigiesu, oto The wortd of Mathematics,t6v.I , o . 4 6 7 .

9. Steven B. Smith, o. XV. 337

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| ( E O A A A I O I 3

rulv ro ono(o nepiloppdver rqv e{otpetrrri pvripn (q ono(o gnopei voonorr;Oe( pe erno(6euon), rrlv oydnrl yto rouq opr0poUg ror nold ypd_vo yro vo no((rr rove(g pe rrq oX€oerq rcrrv oprOpdrv. IXdoe rg rou ovoKo_trfntovrqr oe veop{ rlltr(o evot'prortirvovror o, ,rroioy,o.rro,q otryo_p(0poug, ot ono(or opydrepo y(vovror unoouve(6r;ror ror trertoupyotivoXe66v oropto(o.

Orov 6purq np6rertor yro rdnoto outtortr6 rj 6rovo4trrd rqOuore_p4pivo dropro, 11 un60eoq rou rpro 6ev enoprr(. o odrcrurp NrdpolvrTpdrpepr unooqp((er du €vog ouv6uoop6g nopoy6wtrrv 6rlproupye(fvo pouorrd r] opr0;ropv4povo rOrogurl prrpdvou _ ro oUv6pogo rour6torpu{ prrrpdvou. Av orerptoti;.re drr or r.rorpur(g prrpdvoeq ouXvd 6evpnopo6v vo notrIontroordoouv 660 povo{rlrprouq opr'po6q, o,te ror vonpoo'ioouv r] vo ogorpdoouv, rorotroBo(voupe op€ourq d'to norxvf.rur rous opr'poriq ror prerd rl €o,rspiK€uorr rou notxvr6rod e(vor proe64ynon ovenopr{q. Or opr0popvrjgovrq t6torpur(q prrp6vorq €Xouv ou_v{Ourg rp(o oprOp4trrd ro}€vro: (j) qv onopvqpdveuo4 oprOptirv, r1ono(o oupneprtroppdver rrlv onopvrlF6veuo4 qpepdrv ror 4prepopr4vuirv,dnu;g oqv nep[nturoq rulv 4pepotroyrortirv unotroyro1.u.0v, (2) qv rrov6_rrlrcr vo ovoyvu.rp((ouv rouq npdrrouq opt0porlg Kor rous nopdyovreg,(3) rqv rrov6rqto vo perpoUv ypriyopo rl or6po ror oropro(o. Aw6qnr0ov6v vo e(vot unoouve(64reg rrovdrrlreq, yrot( or t6ro<pue(q FrKp6vo_eq (€pouv nofuj rold rt rdvouv, otrId 6rv prnopofv vo neprypd{rouvndrg ro rdvouv.

Ie notrtrorjg opt0popvqpoveq r6rogue(q prrp6voeg, to oprorep6 4prr_o<po(pro tou eyregdlou rous txer petop€veg rrov6rr'eq. Aur6 pnopr(vo npo{l0e ond eyregotrrr4 xdrooq }6yu.r rdnotou rporiporoq rord 146tdpreto rqq eyrupooriv4q r] otqv npcOr4 nor6rrrl r1trrrro. Iro oy6pto,pnopei en(o4q vo e(vot onorritreopo ovrooppon(og rqq teorootep6vr;q,rdrr nou enrBpo6rivrr r4v ovdntu{r1 r'v veuprrtOv ouv6ioeov oro oor_orepo 4prorpo(pro rou eyrerpdlou roug. O Tpdqepr unooqp((er 6u ou_t4 4 npdrtpq BtrdBq oro oprorep6 4progo(pro rou eyregdlou o6qye( orpro peror6nto4 rurv trerroupyuirv oro 6e{f rlprorpo(pro. orov nir;ord(rr

338 o xpdvoq t4g ydvv4o4q, 6roerotoprpfpro veuprrd rrjnopo rou eyrerpri-

E*:

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A P I E M O M N H M O N E f

onotuyXdvowos vo ouv6e0orjv pe dtrtro rUttopo, rotoorpigovrotonopporpdvror ndh ond rov eyr€rpolo. H e{€itt(4 go(veror vo €Xer

oorlYrlofl oe nopoytrryr] unepBoltrd notrtrdrv veuptrriv rurdpurv, yro voDroogol(oer dtr ndvro 0o undpXouv oprerd yro vo y(vetor o onopoitrl_roq oprOpdq ouv66oeov. ro npdoOero, oorivOeto rrjrropo orp{vovrorvq KoroorpqqoUv. 'looq, drov or trertoupy(eq rou oplor€pori tprlpotogrou eyreqdtrou peroron((ovror oro 6e(( ryrjpo, rdnoro ond outd toenrnl6ov rdttopo 6vtrrrq oXqpot(ouv ouvbio€rq, F€ onordtreopo vod;1oupe dropo ge nro roXupd ro 6e(i qprorpo(pro.

To rup(opXo oprotepd 4prorpo(pro orouq ouvqOrop€voug ov0ptirnougt(vor une0Ouvo yro nq ypopprriq 6ro6rroo(eq nou epnepr€Xovror oroldyq oto poOqpotrrd, otouq oultroyropodq Kor oro oXe6roogd. To 6e[(rlprogo(pro rupropXr( onq oroproieq 6pooqpr6trlteq, dnoq e(vor r1dpeoq onurrj ovtiiqrfq. Aur6g o rotopepropdq rov trerroupyuirv otooprorepd ror to 6e{( qgrogo(pro po40d vo eppqvedooupe uq 6rorpop6qovdpeoo oro guoroloyrxd dtopo Kot orouq oprOpogvqpoveq r6rorpue(qgrrpdvoeq. Or r6rorpue(q prrpdvoeq 6uorotredovror ps uq ypopprrEq 6ro_6rroo(eg (trq 6ro6rroo(eq, 64lo6rj, nou onorelo0wor on6 otd6ro, ro

i olrofo ouppo(vouv 6vo rdOe rpopd), pe e{oipeoq rq pouorr{, oqv'ono(o, nope;lnrnt6vroq, t(uptopxe( ro 6e{( rprjgo rou eyregd}ou. otrOtorpueiq prrp6vosE e(vor 6uvoro( oe npoBlr]poro ilou onorroUv dpeo4ovrii4rf4, dnurq q oropro(o op(0trrqoq. Enopivoq, oqv nep(ntoorl rurvovOpdrnurv our6v dXoupe rdtt neptoodrepo ond rrlv onog6voon, To €v_6rogipov yro rous oprOpodq Kot rnv e[dorqor1. Or oprOpopvr];roveqr6rogue(q prrp6vorq 6ro0€rouv dvov eyrirpolo nou lerroupye( 6rogope_urd, 6vov eyrirpolo otov ono(o ruptopXe( ro 6e{( rr dy ro oprorep6qgrorpo(pro. O Tp€rpepr enro4po(ver, eniorlq, 6tr €Xouv 6rogopmrori e(_6ouq ovogvqorrq, trdyo Oropprj[errrv oe neproX€q rou unorptrororj roueyrerpdlou. Ot ovopv{oerq rouq e(vor ouv{Otrlq notrrj €vtoveg ol}d ne_propropdveq. Aur6 e{r1ye( w p6per yror( 16oo notrtro( rOtorpue(q prrpdvo_eq 6Xouv eppoviq orov rpdno or6rlqq rouq. Oot6oo, ordpo rr qur6 6eve(vor oprerd, yror( o TpC<pepr l€et:

339

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K E O A A A I O T 3

O totrovtoriXog rOrorpurlg trlrpdvoug, pre ouvel{ enovdtrq$q rot e(dorr1-

or1, ;rnopt( vo 6qpuoupy{oet enoprq ro6tronoqorl €tot tilote vo ovo-

nruX0o6v rdnotot oouve(6r1tot otry6prOptot (nop' 6tro nou 6e 0o touq

ovtttroppdvetot). 'Opog, otrlv nep(nrulorl tou ;rtrp6vou nou 6to0dtet

erntrrlrttrdg trov6trlteg, 11 np6oBoorl otoug rovovtg qg pouotrr]g q

tov ;ro0q;rortrdrv, yto nopd6erypo, e(vot t6oo ertetog6vrl nou 0o np6-

ner vo undpXouv rdnoteg npoyovrrdg (rtrqpovopqp€veq) ovopvqoetg

nou vo e(r1yo0v outrl trlv np6oBoorl. Iro ev tr6yo dtopro, outoti tou et-

boug ot clvopvrlottq rtrqpovopo0vtor ove(dptrlto ond rr1 yevrrr]

eu<ouio.to

O Tp€rpepr eroriyoye €vo Entntrdov orotXe(o: rqv rtrrlpovopnpdvn

ovdpvrloq pouotrtiv ri po0qportrtirv rqv6vuJv' 0o16oo, tr€yovrog ovd-

Fvlol, o Tp6qepr 6ev evvoE( rqv ovdrtrrloq ouyrtrptgfvov Y€YovO-

rov rou nopetrO6vrog. Xpqorgonore( rr1 trd(q ourri yto vo ovorpep0ef

oe rctrr1povo;rr1p€veg veupotroyrx€g 6e(t6rr1reg Yro rrlv err€treoq po0r1-

porrr6v rot pouotrtiv npd[eurv. Aurdg or t6to(reptg 6e(t6rqrtg, uno-

orqp((er, e(vot 6rorpopertr€g ond q yevtrd euquto Uqq Kot' enog€vtrtg'

6rov q yevtr( eurpuio rou t6torpuorig ptrpdvou €Xet unoore( Fldpn,

our6g ot e{tt6rxeugdvtq 6e(t6rqrtg evtoXtiovrot' Me pdorl rrlv €peuvd

rou, o Tp€<pept unoorqp((et 6rt o oprOgopvrlpttrrv t6ro<pur1g ;rrrp6voug

e(vot oqpovrtrd 6to<popertr6g cn6 dvo rpuototroyrr6 dropo' Me ru-

p(opXo ro 6e(( rlprorpo(pto rou tyrarpdtrou, 6Xovrog nt0ov6v 6vo nto

notrdntroro 6e(( rlptorpo(pto e(otr(oq rt 'rv entnl€ov veuptr6v ouvb€oE-

ov, U€ np6oBooq ortg <puototroytrd rtrqpovoprlpdveg 6e(r6rrpeg nou

oXer((ovror ]r€ rous rov6veg rr.rrv po0rlpottKtitv, ot ouv6uoop6 U€ rouq

neprBotrtrovttro0g nopdyovr€S rrlq rlruXotroytrrig onopr6vooqq ro0tig rt

€vo ev6torp€pov Yto rouq optOpodg, o optOpopvrlpurv t6torpur]q gtrp6-

voug rprdvet oe v€o Bd0q 6oov o<popd orqv dpeoq ovrfir1r|lq ttrtv

opt0gtitv."

10. Treffert, o. 220.

11. 'O.f i . , a. 222.34S

Page 326: Ο Ταξιδευτής των μαθηματικών

To pfAAov rov unoAoytopL(-tv

E6troyo, gnope( rdnotog vo porrloet: ytor( 6Iq ourrl 11 rpooop(o yto

roug unotroyropo0g; Mnopo0pe ontrtirg vo ovo(ouge tov unotroytorq

Uoq Kor lnop!- 6Xou;.rt rqv ondvqoq nou O€Ioupe. AItrd ro vo Uno-pofpe vo unoloy(oupe (otoq e(vot rlto orlpclvrtK6 on6 rq gqXovtrrl no-

poyoyn prog dpeorlg ondwrlorlq oE rdnoto onou5o(o opt0prlrtr6 np6-

ptr1po. Mnope( vq onoretre( rq pdoq gtog en(rrqrqq goOrlpotrrrlg o{u-

66prerog Kor Ftoq Bo0frepqq ovtfir1{rqg rov t'ptOptritv.

I oroprrd, op retof 8tonpene(g entorrl ptoveg unrl plov e(otpertro( optO-

po;rv4poveg, 6ncog ot olou6o(ot goOqtrrottro( Adovopvr 'Ortrep, Koptr

kdouq ror lprvtBdoo Popdvour(ov. O T(ov rpov N6tpov, o not€pog rov

o0yXpovurv unotroytor6v Kot rnq Oetlp(oq rov notyv(tr-rv (tqg po0qptott-

rqg 0eulp(og rov rotrfrepcrtv enrtroydrv), {rov rr our6g yvtuor6q Yto ulv

rrov6rqrd rou orouq unotroyropodq.

Or rotrovroriXot optOpopvrqgovtg dXouv dtrot dvo rotv6 Xoporrrlptort-16: trorpefouv vo ord<provror oprOpo0q Kot vo no((ouv pe optOptofg.

Aurrl 11 enr0up(o vo no((et rove(g ge oprOpo0g €Xet undp(et €vo onorpo-

ororrrrlg oqpoo(og orotXt(o oe no}I€g go0qportr€q ovorcotrriQerg. Iroug

npdrtouq l-luOoyopeioug, e(6oge ndg ro notXv(6to rouq U€ outrtroy€g Xo-trrrr6v rofg un66et[ov oprop€veg opt0prltrrfg oX6oetg. To norXv(6t pe

oprOpofg ro0q o6rlyqoe oe rdrotq qnd ro npdrro poOrlportrd Oeopq-

poto. [1ro np6orpoto, dvog on6 Touq ilto onouOo(oug poOrlprotrrodg, o

Koptr frdouq, drove pto e(otpelrd oqpovrtrri ovordtru{rr1 orrl Oetlp(o

rov oprOgtirv ro 1792, 6rov rltov p6vo Oerondvrt etrrtv." Ynoorrlpt(e

6tr ro ntrr]Oog rov nptirov optOptirv nou undpXouv gttofd Tov quot-

rtiv opt0ptirv ou(dvet ge Bdoq rr1 ouvdpqorl v/(log v). Aur4 11 ouvdp-

rr1or1 €yrve opy6repo to 1ttip4pa ruv npdtuv ap0ptitv. l-ltig or€rpulre

o lrdouq ourqv 14 ouvdprrlo4; M€tp4oe n6oot nptiror cptOpo( undp-

Xouv o€ 6to6oXrr€g opd6eg rurv Xrtr(utv rpuotrtiv opt0ptilv. Auq 11 roro-

12. A.E. lngham, The Distribution af Prime Numbers (Kdprnptr( Cambridge Uni-

versity Press, 1990), o. 3. 34t

Page 327: Ο Ταξιδευτής των μαθηματικών

voprl rod un€6er[r ro nopondvtrt 0e6pr1po. Enop€vog, o lrdouq, 6voon6 ro nro tropnpd pruold rou r6o;rou, €rove rqv ovordtruQrl rounpoo6 ro p ((ovrog enr pretrdrg ro r unotroy((ovrog nptirrouq optO pro 6 g, rdvo-vrog 6qtro6r1 rerptppdvouq unotroyroporig pr pdorl ro rorvd pfrpo rototo0ud.

Ae 0o pdOoupe norri n6oeg on6 ug geydtreg ovorohirferg rov po0r1-porrrtiv €ytvov on6 rdnorov nou {tov np60upog vo rdtoet rct vq rd-vsr unopov€lrd ro0rlpreprvodg unotroyropodg, otrtrd o opt0p6g roug 0onp€ner vo e(vor oqprovrtr6g. H (6ro unopovq ror enr;rdtreto evundpXerKor orrs rpuotrdg entor{geg, 6nou or epeuvrp€g orpt€ptrtoov Xp6vto outr-trEyovrog ror ovoltjovroq 6e6op€vo nprv y(ver rdnoro orlprovrrrrl ovo-rdtrurf4. Enopdvr.r-rg, or ontro( unoloyro;ro( e(vor ouoroorrro( yro ro po-

0qpotrxd. Kdnoror ouor4po( rprr€g rou ttroororj ouitvo pnope( vo I6vEpre on6yvurorl 6rr or urotroyrordg €rtre$ov rrlv entOup(o FoS vo rdvoupeunotroytopotig ror, rord ouvfntro, o6e0oupe npog €vo v€o peooitrtvopo0rlporrrrlg r0rpkroqq, orpo6 ot pellovrrrdg ycvr€g 0o Xdoouv yprlyo-po rnv rrov6r4rd rouq vo rdvouv unotroytoprodg. Ynooulp((o 611 0oouppe( ro ovr(Oero. Or unoloyrordg 6e pog orepodv rrg 6t(r6rqreg noue(vor onopo(rrlr€q yro rrq opt0prptr€q npriferq. Avrt0drog, onoretrodv6vo oro6, yror( Ierroupyo0v og pro en6rroorl rou vou prog 6oov orpopdoroug unotroytopodg. Otrot rripo pnopo6pe vo y(vou;re ou0evt(eg orougunotroyropo6g, p€ Touq unotroyrordg nou €Xouge oro on(rr pog. O notr-Iontrootoopr6g q 11 ortrono(qor1 6ev e(vor on6 p6vo roug 16oo oqgovrr-rcd, doo ro 6u poq o6qyo6v o€ Uro vdo Bo0rd yvtiroq rov oprOtruirv.Epor4poro nou oyet((owcrr F€ rouq opt0;rorig, ro ono(o o p6oog dv-0ptrrnog prnopef vo prq orerpr6rov nord nptv, ytor( ouvendyovrov no]U-ntroroug unoloytopro6g, pnopo0v rtirpo vo onovrrl0oriv pt l(yo trenrd q

tr(yeq 6peg Trpoypopporropori. To p6vo nou onorte(ror e(vot ro evbto-rp€pov yto roug oprOpo0g.

348

Page 328: Ο Ταξιδευτής των μαθηματικών

K E O A n A l o " l 4

Tr o4pa(vouv rilrr ourti;

t-lpoodropi(ovrog ro e PdlrnPo

'EXoupe 6tov6oet notr6 6p6po orqv €peuvd prog ooov orpopo oroug

opr0po6q. :ertv6vrog on6 roug rpuotrodg oprOptotig, e(erdoope rrlv

npodtreuor] rouq Kot Oteup6vopre rtg t66eg Uoq Yto vo oupntpttrdBouge

ru rtrdoptoro, roug dpprltouq, rouq pryo6tro0q Kot rouq urtprtr€po-

opr6voug oprOpo6g. Kdvoge en(oqg unoO€oetg yto ro ntig ptrpouv ro

dtrtro 6vro. Tr.0po e(vot drpo vo ovopurrrl0o6pe: rt e(vot ot opt0po( pe

pro yevtrdre prl fvvoto; hqv npoyportr6rqro, ourd 6ev e(vor 6vo ;ro0q-

portr6 otrtrd €vo qttrooo<prr6 eptirrqpo rot yt' ouro 0o nptnet vo e(po-

or€ noootKrtKo(, riore vo pq poutrtd(oupe o' dvo rdtrpo ordtretorov rpt-

trooorp t r6v ou(r1tr1 octrtv.

Ot neproo6repeg dvvoreg rurv po0rlportr6v e(vot eUrotrq Korovorlr€q'

yror( pooi(ovtot oE ontrofg optoporig Kot ouYKsKptptveg oXdoetg rou U€

eurol(o pnopofpe vo ovrttrrlrpOofpe' H <pttrooo<p(o rov ptoOrlpottr6v e(-

vor notr6 nto 66orotrq, ytor( pog ovoyrd(tt vo oropoKpuv0ofge on6 rtg

go0rlportrdg onoSe((etg Kclt rouq po0qgortro0g unotroytopotjg Kot vo

e(erdooupe rtg 6toto0rptr6g t66eg poq 06tovrog Bo06repo epurrrlporo'

To npulropXtr6 evbto<pdpov prog 6oov orpopd oroug optOptodg e(vot to

e(r1g: pe notov rp6no undpXouv or optOpo( rqt rt ouvenqyerot ouro yto

ro of pnov ;t€oo oro ono(o (odpe; Av t1 ovtoAoyio E(vot q patr€tq ro0 e(- €€E

Page 329: Ο Ταξιδευτής των μαθηματικών

vot 4 r4g Unop(qq, r6re 0€troupe vo roroldpoup€ rrlv ovrotroy(o turvoprO;uirv. E6dr rprdvoup€ orov orpoy'vro(o tr(Oo rou 6trou 0€gorog: ovor opt0po( undplouv, ntirg undpXouv, ror undpXouv dpoyr ove{dprrlroon6 roug ovOpdrnoug; Bepo(oq, 6ev evvoodpe du or optOpo( undpXouvpe r4v nu0oy6pero €vvoro, 611}o6{ u-rg rpuorrd ovrrre(prvo nou enrntrE_ouv yrlpo Foq oro XurpdXpovo. Ae 0o ppoUpe nord rov oprOpd 3 vo (r_roupd(eror rdro ond dvo BpdXo 11 vo rpriBtror n(oo on6 6vo 6€vrpo.Aur6 nou evvoodpe e(vot ro r(qg: undpXouv dpoye ro poOqgourd ovtr_re(prvo (6,rt rr ov e(vor orrlv npoyportrdrrlro ourd ro ovrrre(pevo) oeoXdoerg nou e(vot ove{dprqreq ond rq o16g11 pog yr, ourd to ovure(pe_vcl Kclr rrq oX6oerq; ME dtrtrq tr6yro, undpXouv or otrr]Oe reg rov ;ro0qporr_r6v oq otrr]Oereq yro ro o0prnov p€oo oro ono(o (odpe, ove{dpqroond rr1 6rrrl pog dnop(r1;

Mnope(re vo rcorotrdBtre op€oog drt dXoutrre ovo(er r4v n6pro or46dorotrq dvvoro r4g 6nop{r1g. Ardrpopor rprtr6oorpor goxOotlv e6tir ror e[-roor nevre or6veg vo rorovo4oouv rdnog rrlv €vvoro rqq 6nop[r1g, roroldrlqpo grlooorptrd rrvrlporo Orpetrtdr0rlrov ndvo oe ouyrerptpr6vegepp4ve(eq u1g ovroloy(oq. Ae 0o npiner vo pnorjpe orov nerpootrr6 vooneUooupe vo 6c0ooupe pro ol4Oorpovr] ondvr4oq oe 6tro ro epurr{po_To rrou ouvorlovror p€ rnv ovrotroy(o, yror( ov rdvouge rd. rdroro 0oe(vol oov vo pno(voupe o'6vo vopron€6ro roUrprurv unoOioeov. re ou-r4v rr; o0vropq petrdr4, 6ev pnopo0ge vo eln((oupre 611 0q onovrfoou-pe oe 6Io ro rpttrooogrxd eprrrujporo nou orpopotiv otqv 6nop(4 rurvopr0p6v. To rotrdrepo, (otuq, nou pnopo0pe vo rdvouge efvor vo orro_yporp{oouge ro eporqporo rou oqoporjv oroug opt0porig pe r€rotorp6no, tiore vo (eroOop(oouv ot or€perg poq. Me ouu] r4v €vvoro, 0oouFqLov{oouge 611 6ev npdrerror vo rorotr4(ougs o€ rdnoro retrrrriondvrrlo4 6oov ogopd orouq opr0poUg. oordoo, 61 0o e(uoore 16oodrotrpor drorg vo p4v rororpdpoup€ vcl ore<prodpe ele60epo oXerrrd pe14v ovroloytrrj rouq un6oroorl.

Og un6Bo0po rov unoO€oedrv poq 0o op(ooupe rrlv eldXtorq rot t4pdyror4 ovroloyrrr] 0€or1 yro roug opt0po6g, etrn((owog drr q retrrrr]ondvrqorl 0o noyr6eure( ovdpreoo oro 6do ourd drpo. flpdrrov, q etrd-

l

344

Page 330: Ο Ταξιδευτής των μαθηματικών

pmq 0€oq: or optOgo( undpXouv pt6vo oro vou rov ov0p6nov. Av 6ev

unrjpXov or dvOponor, rdre 6e 0o unrjpXov rot ot opt0pto(.'H, ytc vo e(-

Foor€ nro orptpe(q, ov 6ev unrlpXe rdnoto outrtroytortr{ ovr6rqro, r6reor opr0poi 6e 0o unrjpXov. Or oprOgo(, pc(( pe 6treq rtq dtrtreg po0rlpott-

xdg olriOereq, e(vor ontr6q or€Qetq nou rdvoupe' 11 0nop(r1 rouq oq or€-tperg, trorn6v, 6e ouv6detor 1r€ rrlv 6nop{q rou orr6qnote e(vut ove{dp-

tqro ond ro vou gog. To rpuotr6 ofgnov e(vot oetred0epo ond opt0-po0g, rot or dvOponor enrpdtrtrouv touq opt0potig ndvut oe out6 to,

Xurp(g opr0gofg, odpnov. Aurrl 4 etrdXtorrl 0ioq ovopd(erot oro po0r1-

1tor6 rovotpampop6g Kcu orrl rpilooo<p(o peokolt6g (npoypororpo-

r(o). Tovr(er 611 Orlgroupyori;re rouq oprO;rodg voqrtrd - 6ev onoretrodv

rp{po tou pq ovOptinrvou odpnovtog.Kor rdrpo q piyrotrl Odoq: ot opt0po( e(vot ovrtre(p€vo rqq or6{qq.

Aev e(vor or or€Qerg our€q roOourdg, otrId €(vot ro ovttrefpevo ro

ono(o e{erd(ouv or or6{rerg. 'O}o to ovttxe(pevo rrlg or6{qq (ougnept-

tropBovopdvurv rov oprOgrirv) undpXouv ove{dprrlto on6 rtg outrtroyr-otrrdg evdpyersq rov or6gurv. Aurd ro ovrrrt(psvo tqq or€tpqg pp(oro-

vror €{ur ond ro vou rt dXouv pro ondlurrl dnop{q ov6tepq ond to ge-

ropotrtrdtrrevo rpuorr6 odpnov. Enog€voq, ot opt0po( rot ro dtrtro ovrt-re(pevo tqq or6rfqq oXrlptor((ouv ivo t6ovtr6 o0gnov otrr10et6v nou

ovortrdror oto ror6repo, ouveXtiq geropotrtrdgevo, utrtrd o0pnov. hqvnpoyporrr6qro, ro rpuorr6 oUpnov 6ev e(vot nopd pro {reu6o(o0r1o4 rt

our6 nou undpXer npoygotrrd e(vot o rdogog rov t6etiv. Aur{ 11 p6yt-

orrl 06oq e(vor o x)ootx6E 6edop6q f ilaruwop6E.Op(oope ontrd.rq ev ouvropr(o rov r6eotrropd rot to peotrtop6 ror 6ev

rdvope xogrd npoond0ero vo orrrotroyqooupe e(re rov Usv e(re rov 6e.

llprv ouveX(ooupe, e(vot Xp(otpo vo etooydyoupe xdnotoug entntrdov

6pouq yro vo onorpdyoup€ rov errputrtog6 r4g ou(tjrrlorlg pog, rot 0o

Xpr;orponorrlooug€ rrlv opotroy(o nou Xprlotponoiqoe o onou6o(og rpt-

I6oorpog ror goOrlporrr6q rou errooto0 ouitvo, Mn€prpovr Pdoetr.'

1. Bertrand Russell, Ihe Problems of Philosophy (Aov6(vo, Oxford University

Press, 1959), o. 12. 345

Page 331: Ο Ταξιδευτής των μαθηματικών

K E O A A A I O I 4

. Aolqqpnrd 5t6o1tiva: 11 oro0rFrlprorq ttooytrtyrl ntrqporpopttitv

on6 rov x6opo ytiptrl prog, 6r1tro6r1 ot qXot, ro Xpdtporo, ot oogeg,

q urp4 rot odrur roOe(qg.

o Ewanuon: 11 ouver6rlrono(r1or1 rov oroorFrlprorutv 6e6opt€vtttv'

611tro6r1, 11 dpeorl ovtitrrlrfrq tov epnetpttilv rou K6KKtvou, rou

ortrqpo0, rou ytruro0 rot o0ro roOeffg.

c ouotxd avttxtfittvo: ourd e(vor ro uno0errrd ovrtre(pevo rlou TIt-

orsdouptt 6rr undpXouv oro tpuotr6 r6opo rot unoO€roupe 6rt

qnorEtrofv rrlv otr(o rov oto0rlqptortiv 1-tog 6e6op€vov' To gr1tro

e(vor €vo rpuotrd ovtrre(gevo, nou oupr€po(voupte 6rt pp(orerot

pnpoord prog, enttbq gog 6rlproupye(rot pto evrdnoorl on6 ro

16ro(repo otoOrlqptord 6e6op€vo rou orpoyyulo6, rou rdrrtvou

Kcrr rou ortrrlpoti.. l6ta: n t6€o e(vqt €vo ovrtre (pevo rnq or6rfrqg. Ot r6deg 6ev npo-

fpXovtot on6 ro oto0rlrrlprord 6e6o;t€vo, o6re 6Xou;re evrun6-

oerg ord ourdg. Ttg ort<propoore eooreptro.

c fri4q: np6retrot Yto rrl ouYK€Kprgdvrl en(yvooq ptog t6€og'

t Ka1o)txi iwon: ovrtKt(Utvo nou undpXouv olov r6ogo rov ro-

Ootrrrtirv €vvordtv, rc ono(o unoOdtoupe 6rt nporotrofv rtq ouYKs-

rprp€veg r6€eg rov or€r.lredrv pog. Aur6q ot roOotrrrdg €vvoteg nept-

tropBdvouv roug opr0pofg, rtg proOrlgorrrEg oXeottg, dnog ro ru,

ro06q rot, ntOovdv, dtrtreg yevtr€g t6t6tqteg, 6nog 11 opsrrl, rl

etreu0ep(o, q on6Xpuroq rou npdotvou, q €vtoorl rou r]Xou rot ou-

ro roOe[dq.

t l opouo td (ouge rd rpo rougop togo6qo r r l vE t rdvo52 ,qono (o6e ( -

xvrr ul ouve(6r1or1 pog p6oo o' €vo geydtro rour(. M6oo orrl ouve(6r1oq

pog, pttirvou;re dpteoo rtg evruntioetg on6 ro otoOrprlptord 6e6opdvo

ror perd uno0€toupe 6rt ourd ro oroorFrlptord 6e6ogdvo uno6r1trdt-

VOUVTnVfnop ( r l r puo t r t i vov t t r e tpdvu rv6 (u ron6 ro rou r ( .Ko r ,ov t ( .

orotXo rp6no, €Xoupe or6$etg nou orpoporiv t6deg' YndpXouv, opoye,

rooo t r t r dg€vvo teg6 ( t , r on6 ro rou r ( r4gouve (6 r l o r l q ,nouVc lovT toTo t -

346 Xo6v o.g eooreptr€g gog 16619; Bepo(urg, t(poore e((oou 6rrototroyr1p6-

Page 332: Ο Ταξιδευτής των μαθηματικών

T I f H M A I N O Y N O A A A Y T A ;

vor vo ovopulq0oUpe: undpXouv rpuorrd ovtrre(gevo Trou vo ovuoror_Xorlv oto eooreprrd oroOqt4prord poq 6e6og€vo; Or rpil6ooqor uro_orqp((ouv 61 6ev pnopo0trre nor6 vo e(lroore npoygourd B€Botor yroro tr undpXrr 6[ur on6 to xour( r4q ouvei6qoqq, yrot( otr6rjnore pui_voull€, ro otoOqqprord 6e6op6vo Kor uS or€rfrerq, ro prdrvou;re p6oooro rour(. [r' ourdv ro tr6yo €Xoupe Xpqorponoujoer pro 6wovq, evro(oypolltri yro vo 6roXurp(ooupe tdoo tov r6ogo rov ro0otrrr6v rOedrv6oo ror rov utrrr6 r6opo on6 rov 6116 pog ouver6qtd rdopo. O 6roXur-progdq rov oroOqrqprortiv 6e6opdvov on6 rrq or€rlrerq yivetor Fs Fro6rorerop;.rivr1 ypotrgri, yro vo uno6qtrtirort rqv oBeBordrqrd poq oXel-rd pe r4v orprB4 6rdrproq ovdgeoo oro oro0qrrlprord 6r6op€vo rororrg or€{erq. To (6ro pnopo0pe vo no6pe yro rrl 6rdrptoq ovdpeoooroug 66o e[rrrreprrodg rdopouq. E(vot npoypolrd 6roXrlrprog€vor, xorov vor, ntiq; Bepo(rrlg, q Err6vo 52 e(vor ivo onlororqp€vo 6rdypopporot orpqvet ovondvtqro notrtrd eporrjgoro. fro nopd6erygo, e(vor orovopvrloerg or6Qerq q e(vor eooreprrd oro0qrqproxd 6e6og€vo; Tr e(vqrro ouvoroOqporo;

Ot neptoodrepor on6 epdq €Xoupe pro roopoOeop(o nou ovogd(eror

YAIKOT KOIMOT KOIMOT TON KAOOAIKON ENNOION

6€wpo

oor€pro

pprilor

dMor dv0ponot

ouroK(vqro

crogrxri otopot(6ro

opr0po(

yeoperprrri opiporo

o1€oerq

loyrrrj ouyyivero

III

EIKONA 52. Iugpotrrr{ ovonopdorooq qq ouve(6qoqq.

ENTYnOIETE ! IKrvrrr

KOKKlVO

orpoyyu16

oxl4p6

olgup6

rIpdolvo

rpro

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r0rloq

347

Page 333: Ο Ταξιδευτής των μαθηματικών

on)oi'r6q pedop6g. Aur{ 11 roopo0eop(o tr€er 6rr ro utrrrd ovrrre(pevooro ofpnov undpXouv orprpdrq dncuq ro ovrtlopBovdpoore. To 66vrponou ptrdnuL on6 ro nfou-r nopd0upo rou onrrro0 pou e(vor npoyportrd

Q4trd rot rutrrv6prr6, pe roorovdXpopo rptror6 ror npdorvo rpUMo.Optrrg, ov rorrd(oupre nro rpoo€Krrrd rov ontroir6 peotrrop6, ppror6po-ore op€otuq ovrrp6ronor ge npopl{poro ror o6rlyo0poore oro ougnd-poopo 6rr ro ultrd ovrrre(prvo 6Ev efvor orprprirg 6nog ro owtloppo-v6poore. Ie our6 ro enrXe(pr1po 660qre prro rtroorrrj eppqve(o on6 roven(orono Mndprtrr (1685-i253), rpildoorpo rou 6frorou 6y6oou ouitvo,o ono(og oro epyo rov Tpaq \d)oyot ptra{i'Y)a rat Od1vou ova-nruooer rqv dno$r1 6rr ere(veg or t6r6rrpeq nou ono6(6oupe ouv4Ourgoro utrrrd ovrrre(pevo r(vor, orrlv npoyporrr6rqro, rbt6tqreg ruJV or-oOqtqprortirv opydvurv rou ov0p6nou. An6 our6, o Mndprh ou;rn6po-ve 6rr 11 rpdorl rou r6opou 6rv e(vor uhrr1, o}trd nvruporrr{.'

O feppov6g r6eotrror4g lgdvouetr Kovr prog np6orpepe rqv opotroy(onou 0o pog Boq0{ort vo neprypd{roupe ourrl qv ooupqov(o ovdpeoooro oro0rlrrlprord 6e 6o;rdvo ror ro uno6rltrorjpevo ovrrre (F€vo, roono(o unor(0€rclr roq nporolo0v ourd ro oroOrlrrlprord 6r6opivo.Ov6prooe ro otoOqrrlprord 6e6op€vo qatv6lttva ror ro ovrrxe(U€vo Trouro 6qptoupyoiv voo1trttvo. Ynooqpr{e 6rt, ogod npoyllortrcd prnopoti-pe vo yvop((oupe p6vo ro rporv6pevo, ro vooripevo 0o nopoprfvouvndvro 6vo puoulpro yro Foq.

Me r4v nro ouorrlprl rpprrlve(o qq Kovorpomrprorrrcr]q Ordrp4o4g,0o l6yope 6rt 6Iq doo pp(orowor €(ur on6 ro rour( oqv Err<6vo 52 e(-vor rpuotrcd ovrrre(prevo. Orov ro dropo neOofvel, ro rour( roroppdrt rrorprlver 6vov rdopo ooteprdrv, ntrovqrtirv ror BpdXov. Aur6q e(vor ro0o-p6g uhop6g. Me rov rtrootr<6 rbeoltopr6 rj ntrorovropr6, trdpe 6rr dtro6oo pp(orovror €(ur on6 ro rour( rrlg ouve(6r1or1g e(vor ro0otrrr€q 6v-vorrq, Kctr ourdg or ro0otrtr€g dvvoreg 6rlproupyoriv rdoo tq oto0rlrrlpro-rd 6e6op6vo rttrv rvrunc0oetirv pog, 6oo ror rrg r6deg rov or6Qer.0v pog.

z. Ceorge Berkeley, nThree Dialogues Between Hylas and phi lonouso, oto phi loS€S sophers Speak for Themselves: Berkeley, Hume and Kant, oa. 1-9s.

Page 334: Ο Ταξιδευτής των μαθηματικών

II

, I E H M A I N O Y N O A A A Y T A ;

To npdptrrl;ro Us ro vo onorpoo(oou1't€ floto Koogo0eurp(o ono6eX6-

poore e(vot dtt, e(re orq p(o ntotdtfoupte e(re orqv dtrtrq, out6 €yet og

onot€Ieopc rprtrooorprrdg Ouorotrieg. Mnopo0pre vo enrtr€(oupe to p€oo

6p6po xor vo 6epo0pe dtt undpXouv 16oo to rpuotrd ovrtre(gevo doo

ror or roOotrtr€g €vvoteg - gro dnor{rr1 nou €(vot yvuro{ wg 5ui'o1t69''Opog, €vog t€rotog etrryp6q 6ev onogorpdvet rtg buoxotr(eg' Oordoo' ot

neproo6repot dvOponot 0o €vtrD0qv nto dvero pe our6v to g€oo 6p6-

go. Or neptoodrepot on6 eydg onopp(ntoutrre tlg ovdqrrl tqv nopor{-

prloq 6tr ro rpuotr6 o6gnov 6ev undpXer' An6 rqv dtrtrq, vrtirOoupe dpo-

tro pr6vo p€ Uto utrrottxq dnorlq Ylo rrl q0or1' Enop€vutg' tpo(vetot

oorgotrdg rot potrrrd Yto Uqq vo ono6eyto6pe ro rpuotr6 rdopo ev6 nt-

orefouge rcor oe €vov dtrIo r6otrto, rov r6opo rov roOotrtrtiv evvottitv'

IuXvd, orerpr6trroore out6 rov r6ogo rov roOotrrrtirv Evvottilv tlg €vo

nveupotrr6 Boofiero nou 6ev nepr€Xet p6vo rtg ro0otrrrdg dvvoteg' oltrd

Kor rtq nveupottr€g ovr6rqteg, nopo6e(yporog Xdprv rtg {ruX6q 6otlv

6Xouv ne0dver, or6Po rot tov Ot6.

0o16oo, 11 rotdotoorl pnopt( vo e(vot ox6po nto notrfntrorr1' Kottd[-

re rrlv Etr6vo 53. E66 dXoupe tonoOerrlott 6vo e(6oq rpitrrpou ovdpeoo

oto e(orsptx6 ror to eooreptr6 rou rourto0' Aurd onetxov((er rrlv t6€o

6rt ro ntiq ovrilopBov6poor€ rov K6oUo, t6oo tov r6opo rov roOotrt-

rtiv evVouitv 6oo rot tov r6opo rov rpuotrtiv ovttretpivt'tv, e(oprdtot

on6 to ndrq e(poote rprroygdvot trtg ov0pdrntvo 6vto' Aqtro6r1' Btr€noupe

npdyporo rot orerpr6poor€ Yto npdygoro pe rov rp6no TIou ro rdvoupe

tr6yto tou ntirg e(goote ouyrporrlgivot t'rtg 6vro (e(te trrg rpuorrd e(te oq

r6eolrortrd 6vro). Aut6 e(vot yvoor6 aS EYuKtvrpw1 SAqppa' Mt l1

protroyrxq €woto, rorovoofge ro €YoK€vrptr6 6fAqppo ouvet6qronottil-

vroq 6tt 11 (6ro q 6o911 tou eyreqdtrou Foq, ps rouq 6to€Koroppfpro otr-

trqloouvbe6pevoug veupdtveg, por1Od otov ro0optog6 to6 ntig ovrrtropt-

Bov6poore rrlv TlpoYportr6trpo. [16oo, trotn6v, to ntig Btrdnouge ro

npdypoto ro0op(eror on6 ro ntirg e(vot grroyg€vq 11 ouve(6r1o{ pog;

flrotefoupe dpoye 6tt ro rpuotrd ovttre(pevo undpXouv ge ouyrerptp€-

voug rpdnoug, rup(trtg ener6ri e(poore 6rot gttoypdvor; E(vot ot optOgo(

ovtrxe(pevo qq oK€qinq pog, orptptirg ener6r] e(poore dtot rpttoypdvot; 349

Page 335: Ο Ταξιδευτής των μαθηματικών

K E O A A A I O T 4

YAIKOI KOIMOT

6ivtpo

oor€pro

pprilor

KOIMOI TON MOOAIKON ENNOION

opr0po(

yeogerprrci oXriporo

ox€o€rq

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loyrr{ ouyydvero

oToFtKo

EIKONA 53. Orkpoptopivq ouppotrrrr] ovonopdotoorl rrlq ouvr(6qor1g.

Av 06troupe vo e(poore ouvr4prlrrro( oXnrrd U€ rnv ovrotroyfo rovopt0pr6v, 0o prnopo6oope vo uroOer4ooupe rrlv dnorpq 6rt pnopodpevo efpoore oryoupor gdvo yro ro oro0qrrlprord 6e6ogdvo Kor rrq Eoor€_prr6g r6€eg rurv or6$edv pog, 6r1tro6q p6vo yro ro npdyproro ro ono(oovrttroppov6poore dpreoo. Kovdvog 6ev oprproprpe( ro 6rr Btrdnur npd-orvo Xptirl.ro, 6rov dvrtrrg €Xo rrlv epnrrp(o rou npdorvou Xpdrgorog. TootoOrprlptor6 6e6opdvo rou npdorvou Xptirporog e(vqr rdrt ouyrerptprd-vo. Orr6rlnore 6{ur on6 14 ouve(6qor1 (ov 6vrurg €Xer v6qpo vo grtrdpeyro ro €ordrsprxd ror ro e(oteptr6 qg ouve(6r1or1g) dXer pr6vo 6vo 6eu_repedov e(6oq 6nop{qq' pro 0nop(r1 nou e(oprdrot on6 rrlv dpeorl en(-yvoorl pog. Ae l6pe Oqlobrl 6rr 11 dpeorl en(voorl Foq, ro oro0rlrrlpro-rd 6e6op6vo ror or orc€rferg, rdvouv outd rq e(oreprrd ovrtre(pevo voundpXouv, olld 6rr onorro0pe yv6o4 rurv e{oreprrtiv ovrrrerpdvov pepdorl ro oto0rlrrlptord 6e6o;rdvo ror rtg or€Qe rq.

ftor( rdrE 0o np€ne r vo nrorefoupre E(re oro rpuotrd ovrrre(pevo r(reoroug opr0pro0g (roug opr0po69 oq nporopxrrdg ovonopoordoetq rulvrq0ohr6v evvordrv); O tr6yog yro rov ono(o 0o npdner vo nrorrrjoupe

35* oro <puotrd ovrrrr(pevo efvor nporpovrlg: ov nrorerjoupe, r6re no(pvou-

ENTYnorErr ! xr,r,rtrOqqprord 6e6.1 (16€eq)- D l <

16rrrvo

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I E T I M A I N O Y N O A A A Y T A ;

pe onorpdoerg fiou goq enrrp€nouv vo tretroupyofge pe entruX(o. Av

npoono0{ooupe vo oyvorjooups ro rpuotrd ovttre(pevo, r6te petdvetot

q rrovdr4ro enrB(utotig goq. To vo uno06roupe 6tt undpX€t ovrtKttl-t€vt-

rd ro rpuorxd o6pnov rpo(vetot vo trettoupye( rotrd p€oo o€ Fto Koogo-

0eop(o nou oufdver qv trov6rqro enrB(utotig 1tog.Tr ouppo(ver p€ rouq opr0po0g; YndpXet rdnotoq trdyoq vo ntote6-

oupe 6rr or oprOpo( undpXouv i(trr ond qv oro;rtx{ poq ouve(6r1oq;

E6dr enrorp6rpougt yto dtrtrr1 gro rpopd orov orpoyutvto(o tr(Oo rrlg

dpeuvdq gog.

loroprrr i e(€Ar(n

E(vor pogovrrr6 vo uno0€roupe drt ot dv0ponot ordrptovrot qurd ro

epotrlporo on6 t6te nou on6rrqoov oprerd toXupotig eyrerpdloug

dlote vo prnoporiv vo rdvouv rdtt tdroto. Mnopotipe vo <povrooto6pe

ov0pdrnouq vo rdOovrur rdro on6 6vo ortep6 6dvtpo, 6erd6eg Xrtrtd6eg

Xpdvro Trprv, Kqr vo ou(rltotiv ov to 6dvtpo unrlpXe r1 6ev un{pXe oqv

npoyporrr6qto exe(.'Ofroq, 4 nptirrrl poprup(o dtt y(vovrov dvttrtg t€-

roreq rpilooorprr6g ou(r1tr]oerg npodpXetot on6 roug opXo(oug 'E}lqveg,

or oilo(or e(Xov tqv npovoqttr6trlro vo KoroYpdrpouv rrg ordrletq toug,

dror tiore vo pnopo0pe rdnoto pdpo vo trg 6ropdooupe.'EXoupe {6116er 6u or flu0oy6peror n(oteuov noq to rpuotrd ovrtre(pevo rotooreud-(ovtov oreuOe(oq ond roug opt0;to6g. Enop€vtlg, to rpuotrd ovrtre(pe-

vc €{ur on6 q 6rrq rouq (rot rr1 6trq poq) ouvt(6r1oq e(Xov trtto orogt-

nj 6ogrl nou poor(drov otouq rpuotroriq optOgodq. Aut6 orlgotve 6tt ot

opr0po( rorotrdgpovov 6Xt pdvo tov r6opo ttlv ro0otrtrtiv evvottitv, otr-

trd rqr to rpuotxd rdogo. O lltrdrov n(oteue o' €vov r6opo t6e6v, o

ono(og neprefie ondlureq, ouivteq, ro0otrtrig t6€eq nou pe rdnotov rp6-

no ovogeryv0ovrqv Ut to Xtirpo Kot l.loq 66tvov ro rpuotr6 r6opo.

An6 qv enoXrl tou fltrdrovo, or rprtr6oorpor npooXopo0oov oe 6to-

rpopetrxd otpot6ne6o, dtrIot unooulp((ovtoq to peotrtop6 rot dtrIot rov

r6eotrrop6. koug notrrttopofq 16oo qq A0or1g doo rot rqg Avototrrig, q 351

Page 337: Ο Ταξιδευτής των μαθηματικών

Opqore(o unoorrlpt(e ouvq0urg ondrferg undp rou t6eoltopo6. ruv6ud-(ovrog on6rferq rou ltrdrovq Kor rou Aprororitrq, or xpro'ovo( 0eotr6-yor rou Meoo(trrvo unoorr]pr(ov 6xr p6vov 6rr o oe6q 6qprorlpyrloe rorpuorro o6pnov, otrId rr 6rr q dnetprl eurpuio Tou ouvrrlpe( u1v 0nop{r1dtrov rov ro0olrrurv svvouirv, mer6r1 rrg or€rpreror 6roprdrg. Enopdvoq,4 6vvoro rou rdrlou 6e 0o oropornoer vo undpxer ov ne0dvouv dtror ordvOpconot, yror( 11 roOotrrr{ 6vvoro qq rurtrrrdrqrog 0o nopope(ver orovou rou Orori.

Ar6po ror orov errootd ouivo, Btrdnouprr our6 ro orevd ouoxe'opdovdpeoo otqv t6€o evdg novro66vo;rou oeo6 Kcrr ors roOotrrr€g dvvoreg.o Kdvrop on66cr{e 6rt 6ev trrnope( vo undpxer rdnotog ur€pnrrspo-op6vog opr0p6g nou vo eivot pryotrrirepoq ond 6trouq roug dtrtroug.olor or urspr€rr€poopdvor oprOpo( eivor grrp6repor on6 rrlv dnerpqKor prl n€prypdrftp4 rpdo4 rou oeo6. o oedg e(vor ro Andluro Anrrpo,ror ro prrp6repo orerpo rov un€pn€ilrpoopfvrrrv opr0pdrv pp(orovrorpero(6 Oeod ror nenepoopdvou rdogou. Or neproo6repor 1.ro04trrortro(rou b€rorou dvorou Kclr rov opxtiv rou e rrooro0 orrirvo {rov r6eotrror6gror n(oreuov orrlv ovrrKsrprvrrrl g0o4 rurv proo4porrrcilv ovrrrerp€vr.rrv,oupnepttropBovopr€vrrrv rov opropdrv. Tr ouppo(ver rdrpo; Idrg B]€nouvot po0rlprorrro( rodq opt0po6q, nou qnoretrodv rov nupqvo turv poOq_porrrtiv;

Av oroporodoops ro p€oo dvrpo 11 rq p6or1 yuvo(ro oro 6p6po rorroug prlro6oope: "yndpXouv or opr0po( ove(dprrpo on6 rouq ov0pdr_rrous;>/ ro nr0ov6tepo e(vor vo no(pvope opvrpr4 ondvrqorl. Av 6ro_rundrvope er vdou ro epdrr4po und 14 Fopqrl: uyndpXer o trdyoq r4g6rop€rpou rou rfrtrou Trpoq qv nepupdprrd rou ovr(dpr4ro on6 rougovOpr.i-rnoug;o, ro nt0ov6repo eivot vo noipvope pro ovdperrrq ofiovrn_or1. Av rdvop€ Kor rrq 60o e purrrloerq orouq Oeurprprrodg po0qporrro6g,or neproo6rrpot 0o onowodoov: oNot, undpXouv oq olr]Oereq nprvon6, 11 ove[dpqro ond, roug ovOptinougn. Enopdvoq, wd o p€oogdvrpog n n p6ol yuvo(ro oro 6p6;ro rctr(vouv neproodrepo npos rovuhop6, or po0rlporrro(, ror r6ro(repo or Oeop4uro( goO4porrror, rll

359 vouv npoq rov r6eohopd. fror(;

Page 338: Ο Ταξιδευτής των μαθηματικών

T l I H M A I N O Y N O A A A Y T A T

Tr evvoo0pe A€yovrog r jnop(n;

Av Odtroupre vo roroldpoup€ ov undpXouv or opr0po(, rdre npdner voe{etdooupe tr e(vor ouuj r1 r6rdtqto qq Unop(1g. 'Orov }€pe du rdrrurdpXer, u orprprirg evvooripe; Ag orerproripe to rpuorrd ovtrxe(pevo.frot( nrote0oupe 6tr ro ultrd ovure(pevo undpXouv n€po on6 trl ouve[-0qo{ paq; flp6rov. undpXer q oto0ep6qto rov 6rr6v gog oto04r4pro-rdv 6r6og€vov. Kd0e rpopd nou ;lno(vo oto ootrdvr pou undpXer o ro-voniq. Av orpirfrur rrlv nldtq pou npos tov rovon€ ror gerd (ovoyup(-0u, vo rov rortd(trr, ndtrr sre( e(vor. H egnerp(o e(vor oroOeprj. Av o ro-von€g pprox6tov ere( q pro onypri otrki, o(cpv4g, qv endpevq otryprle(oqovr(6rov, ev6 eytir e(orotrouOorioo vo rorrd(ur Trpoq ro (6ro g€poq,0o dpXr(o vo 6reportirpor ov o rovoniq undpXer 4 6ev undpX€r orrlvnpoypotrr6tqro. To (6ro ouppo(v€r Kol l-r€ ro dtrlo ulrrd ovrrre(pevomov r6opo prog. YndpXouv ere( yrc poq. Av e{orpovrotodv, neprpdvou-pe vo undpXer rdnoru e{iyqon nou vo iXer v6qgo. Aev eprpov((ovrorofte e(orpov(ovror oilpoypoppdrroru ond tov 16o;ro poq. Enop€voq,ro uhxd ovtrre(pevo eprpov(ovtor ge otoOepd oroOqrqprord 6e6ogivo.

'EXouv to poOqgourd ovure(pevo our{ u1v rbrdrqro; To po0qpourdovtrrefgevo 6ev rorotropBdvouv rdnoro ouyrerprg€vo tdno ror Xpdvo,rrdtor 6ev neprpivouge vo to Bpodge gero(ri rtrlv oroOqrqprordv 6e6o-

U6vov Foq (nop'dtro ilou ouyK€Kprgdvo oro0rprlprord 6e6op€vo pno-po6v vo uno6qtrtirvouv roOolrr€q €vvoreq). Oot6oo, ro poOqpolrd ovu-re(1revo e(vor oro0epd. H qrotrouO(o rov rpuorrtfv opr0p6v nopog€veryv{oro 6roreroypivq. Or norriAeq oXioerq tulv rpuorrtiv opr0p6v nopo-pdvouv otoOep€q. O trdyoq rqg 6rop€rpou ev6q r0rtrou ilpo( rrlv nepup6-perd rou e(vor 6voq opr0p6q o onofoq 6ev otrtrd(er. (Aut6 npoiinoOitet6tr 6ev undpXer ivoq ror6g 6tdBoloq nou otrId(er ouveXtirq qv ugr] rou4 ofre Kor ro noto 0updpor eydr 6n e(vor outrl q ugrl.) Enop€voq, orotr{0ereg turv poOqgotrrdlv nopouord(ouv dvttrrq oro0ep6rqro, dnoqorprp6q ror ro oroOrlrqprord 6e6op€vo tov uhr6v ovlrery€vtrrv.

'Evo 6edtepo XopoKrrlprour6 trlq 6nop{qq e(vor 4 ove{cpqo(o. Toovure(pevo rou rpuorro0 rdo;rou 6e rpo(veror vo poo(ovror ol4 or€r{rq 353

Page 339: Ο Ταξιδευτής των μαθηματικών

Foq yro re(vo. Aur6 <po(vetot €vtovo 6tov onorotrpdpe vo erodtr0oupe

oe dyvoorq neptoyrl. llpoXtupdpe o'€vo povondrt oro 6doog, yepdrot

6{or.f4, ro0tirg e}n(oupe vo ern}oyotige on6 our6 nou undpXet perd

r4v endpevq orporprl. Ae yvurpi(oupe {64 noro ovrtre(pevo gog nepttrtd-

vouv, yror( ourd ro ovrtre(gevo 6ev orpefiouv rrlv fnop{r1 roug o'epdg.

Eno;rdvog, our6 gog rdvet vo ntore0oupe drr ov e;re(g 6e otp(tloupe

orrlv en6prtvrl orporprl, outd ro o06oro ovrtre(pevo 0o t(orotrou0otiv

vo undpXouv.l-lporotre( €rntrr1(q ro yeyov6g 6rt ro poOqportrd epnept€Xouv rrlv

(6ro o(o04or1 ovordtru{rqg'11 epnerp(o ourrig u1g ouveXofq ovoxd}uQr1g

rdver notrtrorjg poOrlprortrofg vo ntoreriouv 6rt ot go0qportr€q otr{0eteg

nou ovo(rpodv undpXouv nptv ond rrlv ovordtrurlrq roug rot 0o ouveX(-

oouv vcl undpXouv rot gerd. fto nopd6etypct, ov ro n e(vot pto entv6rl-

on rnq 6trr1g pou or€gr1q, t6re ytot( 6e yvtrtpi(rrr ro eroroppuptoor6

Qrlrp(o tou 6ero6rroU ovonr0ypordg rou; f to vo gd0ur noto e(vot ro rfrl-

rp(o out6, 0o np6net vo rdvo pro oetpd on6 po0r1;rorrr6g npd(etg rot

vo ro <ovoKcltr0qoo. Bepo(trrg, €voq rovorporutptorqg 0o 6leye 6rt ro

sKoroppuproor6 rf4rp(o rou ovonr0yForoq rou rc 6ev undpXet, tionou

vo ro unoloy(oet rdnotog. M€Xpt r6re, to {rr1rp(o our6 e(vot pr6vo €vo

6uvql16 Snq(o.'EXovroq qv (6ro 6rd0eor1 yro ovordlurlrq, rotp6 nptv, rdnotog flu-

0oy6perog ovordtruQe 6rt q 6roy6vtog ev6g terpoytitvou e(vot oo0prpte-

rprl rpoq rrlv nleupd rou. Aev rrlv dptoe 6rot - rrlv ovordtrurlre €rot.

flop' 6tro rou ot po0r1;rortro( npoonoOotiv vo ovrtl-l€rtrlTl(ouv rptrtrd

roug unop[tororjg roXupto;rorig oro go0rlgortrd, ouvovrodv 6uorotr(eg

tr6yur ourqg rrlg rfuXoloyrrrlg o(o0r1or1g 6rt ovorotrfnrouv Kot 6ev erpeu-

p(orouv. Av rot e(vot olrl0eto drt ot 0ttoprlrtro( po0qportro( eqeup(-

orouv vdo poOrlpourd ouorrlpoto, out6 ro rdvouv srpeuplorovrog

o(rdrgoto Kot, orn ouv€Xeto, ovorotr0nrouv ro 0eurprlporo flou ouvend-

yovror outd ro o(ttiporo.YndpXer, dpoye, rdnoto nopd6etypo roOop{q erpefpeoqg, ilou vo

pnopodpe vo ovrtnopopdloupt p€ rov rdopo ttlv po0qgorrr6v nou

dXer novorotru<p0e(',; kerpre(re rrlv oyytrtrq ytrcOooo, pto roOopd ov0ptit-354

Page 340: Ο Ταξιδευτής των μαθηματικών

I f H M A I N O Y N O A A A Y T A T

nrvrleqe0peoq. Oeopo6ge ro tre(rrd og oultroydg oproptirv rr 6Xr urq 16-

FouS Ll€ olqOereg. Av [orpvrrd ouvovro0oopre 6vo ov pe ovtirrepq eu-quio nou 0o pog enrorenrdrov ond ro rdvrpo rou l-otro[io poq, 0o pogrordnl4ooe ov prtro0oe oyytrrxd. Autd orpefietor oro 6rr yvtrrp(oupenog enrvo{oope ro oyytrud xor 0o rjrov onpdopevo vo nroredouge dtrror xdnoroq dtrtrog 0o pnopo0oe vo enrvo{oer ove{dpt4to qv (6ro

ytrtirooo.Oot6oo, 4 rordotoorl 6ev e(vor t6oo (erdOopq drov np6rerror yro

ro po0r1trronrd. Oo poq e(dntrque, dpoye, ov out6 to dtrtro troyrr6 ov6ev ovtrtroprpovdtov qv €vvoro tou opr0gori; flrorefoupe 6l or oprO-po( e(vor t6oo roOolrriq €vvoreg, 6ore 6tro ro 6vro ge ovdrreprl eurpufo0o €npene vo rouq yvcrrp(ouv; ho pu0roropriporo r] orrq rorv(eg enr-oqpovrrriq rpovtoo(og, dXouge rptdoer oto o4ge(o vo unoO€toupe 6u'r1titrero gi0o6oq yro vo opX(oouge vo €nKorvuJvo0pe pe e{crryr]rvo 6vtoe(vor g€otrt rrlq (noyK6ogroq> ytrtitoooq rov po0r1porrr6v. Av 6poqroXder ourd, t6te or otr{0ereq tov po0qpottrdv 6e orrlp((ovror e{ otro-r}{pou ot4v ovOptinrvq enrvorltrrdtqto.

Aur6 6prrrg 6e ouvr'1yope( onotr0rog undp rou r6eotrropo0. H npoy-gotrxdtqto e(vor dtr 6ev 6Xouge ouvovr{oer 6vto ond dtrlouq r6opoug.Av eiXope ouvovrrloer, gnope( vo {o<pvro(6pooc ov 6ev e(Xov rdnoro6e(r6qto ouyrpiorpq ge ro go0qgorrrd. 'lorrrg, dtrulq, pdvo ere(vq rudwo nou egeup(oxouv poOqpotrxd nop6poro pe ro 6rrd Foq vo Uno-po0v vo ovom0{ouv pro enror{pr1 rou vo enoprei ylo vo to{r66Qouvoro Xdrpo Kor vo poq enrorerpro0v! 'EXoupe pil{oer yro rouq geydlougeyre<pdtroug trrrv <potrorvtirv Kor rov 6etrrprvuirv. Mnope( q pntre rpdlotvo,pe rov ryrdrpold qq nou (uyr(rr entd rild, vo petprloer ro rpdpro nouTrlv Trpoon€pvoUv rotrugn6vtoq; 1ooq ourd ro oK€dvro 6vro vo €Xouv6vov no}dntroro ror eurpu{ notrrtrogd, Xop(q 6poq vo ovrilopBdvovrorro poOrlpotxd ovtrre(pevo. Mnope( vo oxeqtdpoore 6rt or poOrlpol-rCq otr{Oeteg e(vor ro0o}rr6q ror, enopivoq, pnoporiv vo ovorotrurpOodvon' 6tro ro enopxtiq eugur] dvro, evdr otrlv npoypotrr6trlto pnopefonltirg vo onerrov((ouv tov tp6no pe rov ono(o trerroupye( o ov0pdnr-voq vous. 355

Page 341: Ο Ταξιδευτής των μαθηματικών

H rp(rr1 tbt6rrlto qg 0nop(r1g nou np€net vo trdpoupe un' 6{;tv pog

e(vot q not6rqro rou n€pt€Xog€vou ttrtv oroOqrrlptoxtilv 6e6opr6vtrtv'

Avoyvop((oupe 6rr ro orooqrrlprord 6e6op€vo 6ev e(vqt ord{retg. 'otov

p^€no npdorvo xptirpro, our6 fxer 6vov notortrd ntrofto nou 6ev rov

6xouvotor€{ re tgpoudoovorpopdoronpdotvoXp6u0.Tooroor l t r lp ro -rd pog 6e6op€vo e(vor €vtovo xot onotetreo;rortxd pe rpdno nou 6ev

e(vot ot or€rfetg pog. And qur6 rot p6vo ro yeyov6g, 0o ;rnopodoope

vo oupnepdvoupe 6rt ro ovrtre(gevo nou pp(orovrot n(out on6 ro ot-

oOrprlptord 6e6og€vo undpXouv npoyportrd, evti ro ovrtre(gsvo nou

pp(orovrot n(oo on6 trg or6{retg pog 6ev undpXouv' Oo16oo, out6 6ev

E(vot €vo o6rog<proprprFo ororxe(o €vdvTro orov t6eotrtog6. Ot poOqpo-

rtr€g r6€eg pog rpnepr€xouv prct ooqr]vero nou tre(net on6 ro otoorFrl-

prord 6e6ogdvo. H ovrfi4{rr1 gou yro rov rdrtro (u-rq ro yeogetptr6 16-

no rr0v orlpe(u.rv nou 6tro roon€xouv on6 €vo p6vo oqge(o ev6g turtre(-

6etou entn€6ou) E(vot r6trEro rot orptBrlg, 1-t' €vov rp6no nou 6ev ;lno-pe( vo e(vot rovfvo otoorFqptor6 6e6og€vo. 'Orov ordrptopot rov 16-

rtro, pnope( eooreptrd vo rpovrd(opot €vo otpoyyutrd o0votro rourrG

60v, otrtrd or rourr(6eg ontrdrg pe por10o0v. Aev onetxov((ouv u1v ovr[-

lqr.[rl pou yrcl rov r0rtro, yror( o r0rtrog prou e(vot prto rooo]trq €vvoto

nou 6e ouv6€etot pe rov rdno q ro yp6vo.

Mto 6toore6oourrl eptirrrloq e(vot no6 ndve ot ordr'lretg Poq Yto rov

r0rtro 6rov 6ev tov orerpr6poort. 'Otov orp6rpoupe rrlv nldrrl gog

orov rovon€, oropordet ro oto0rlqpror6 6e6opr€vo rou rovon€, oltrd

nrorefoupl€ 6rr o utrrr6g rovon€g og utrrrd odrgo e(orotrouoe( vo unop-

Xet, dtotpog vo Uoq nopdXet rt dtrtro oroOrlqptord 6e6o;r€vo p6trtg (o-

voyup(oou;re npoq q trreptd rou. I1o0 nrlyo(vet 11 dvvoto rou rfrtrou

6rov 6ev rov orerpr6poore; Tov e<peup(oroupe { rov ovorotrdmouge

Eovd rd0e rpopd nou rov rp6pvoupe ou1 ouve(6lo{ Uoq; Yno0froupe

6rt n rivvoto rou rfrtrou ono0qreferor orq pvrlgrl poq, Yto vo rqv ovo-

rotr€oou;re 6rov rq xpetootodpe. Av 11 fvvoto tou rdrtrou onoorlreferot

orrl pvqprl, r6re 0o np€nst vo undpXet pro optoptdvq oultroyri veupdr-

vov oTov eyrd<potrd poq, or ono(or e(vor €rot 6topoprpopdvot tiore vo

3SS 6torqpo6v ourq rrlv 6vvoro. 'Orov ot vtuptirveg nupo6oto6vrot, 11 ev-

Page 342: Ο Ταξιδευτής των μαθηματικών

T l f , H M A I N O Y N O A A A Y T A T

voro rou rdrtrou dpXeror otq ouve(6qoti lroS. Enop€voq, q €vvoro rourfrlou iXer 6vo Broloyrrd ovdtroyo otov utrrr6 rdogo. Aurd to ovdtroyonpCner vo e(vor orprBiq, yrotf 6rov orerpt6poor€ rov *r6rtro, 6e <po-wo(6poote €vo orpoyyutr6 npriypo nou pord(er rord npoo€yyron p€rtirlo ror, otq ouvdXero, ge pdoq outr]v rq pv4povrrrj 6ro6rroo(o, ore-pr6poore qv orprprj dvvoro rqq rurtrrrdrqroq. 'EXoupe 6vroq qv ovd_Uvqon rnq arprBoUq ivvoroq rj rou oxprpo0q opropo0 rqq xurtrrrdrrltog.Av 11 €vvoto nou iXoupe yro rov r0rtro 6Xer €vo orprpiq protroyrr6 ovd_troyo oro rpuorx6 rdopo, rdte dtreg or ivvoreq 6Xouv €vu orprBig ovdlo-yo oro quorK6 r6opo.

'Evo otiyXpovo nopd6erygo 6uiopo0 nopouord(er o Mn€prpovr pd_oetr oto ofvrogo BrBtr(o rou Ta npoB)riparo rnE gt)ooogfuq. fn vaoqp(er q Oerrrp(o tou, 6e(Xver nptirto du or roOolrrig ohjOereq undp_Xouv Kor pilder yr' ourdg oov vq e(vor oXioerq: oXdoerq e(re pero[6 rpu_orr6v ovtrretp€vtrrv, efte peto[0 d]trov rqOotrrrdrv evvou6v. Kor6nrv,unooqp(er du ro endpevo npdptr4go e(vqr vu onorpoo(ooupe ov our6qor oX€oerg undpXouv ove{dpr4to ond ro vou ruJv ov0p6nov - orprBtirqro 6ilqppd Foq. O Pdoe} onorpdv0qre 6rr, npdypotr, out6 roX6et.

Mnopofpe, enogdvtrlq, vo uno0€ooupe ttirpo ntrlg e(vor otrrj0ero drt t(no_ro voquxd 6ev undpXer rrrq npo0ndOeoq oro yeyov6q dtt ro E6tgporipyoe(vor Bdpero rou Aov6(vou. Atrtrd out6 to yeyov6q egnepr6Xer tq oXdorg"pdpero rou>, Trou e(vor pro ro0otrrrrl 6vvotq. rqt 0o 4tov o6rivoto ytoro 6tro yeyovds vo Un ouvendyetor rdtr voqtrr6 ov q oX€oq <B6perorou), nou onotele( ouorottr6 ototXe(o tou yeyov6rog, neprdXer 6vturqrdl vorprr6... Enopivoq, 11 oX6or1 "p6pero tou, efvqr pr(rd 6rorpopetr-x{ on6 outo0 rou e(6ouq to npdyproto [to quorrd ovure(yevo]. Aev efvqr odre oto Xtitpo oUte oto Xpdvo, 6ev e(vor o0te utrrrr] orlte vor1nrr1.trrotdoo, efvqr xdtr.r

O Pdoetr rorol{yer oro ougndpoopo du, orpo0 ro oUgnov e(vor ye_pdro npdyporo, ourd to np<iygoro oXeri(ovror pero{ri rouq. Or oX€-oerg out€g 6ev e(vor rpuorrd ovure(psvo, o6re vo4trriq oordQerg>, ror

3 . Russe l l , o .98 357

Page 343: Ο Ταξιδευτής των μαθηματικών

enop€voq 6Xouv pto 0nop(r1 rdnog ove(dprrlq Kor on6 ro 6rio. Yndp-

Xouv orov r6opo rov ro0otrrr6v evvouirv. AtrId o Pdoetr rorotrqyeruneppotrrrd ypnyopo oro oupnipoopd rou ror ond ro enrXs(pr1pd routre(ner 11 orp(pero nou pnope( vo poq ne(oEt. 'Evo np6ptrr1go pr ro no-pd6erypo Aov6(vo-E6rppodpyo, nou ovorp€prt, r(vor nog feXvder 6l oeourq qv nep(nroorl 11 voqrtr{ 6poorrlpr6rqro npoqye(ror rqq oXdorlgop6pero rou>. Auul q 6poorrlpr6rrpo e(vol o oprop6g rou Aov6(vou rorrou E6rpBoripyou Loq 6rorpopeurdrv on6 dnor.pq rpuorrr]. Arpo0 ro op(.-ooups oq rdroro, pnopoOpe gerd vo egopp6ooupe rr1 oX€or1 uBdperorou). To np6p}4po e6dr e(vor vo nror6rpoulle 6rl ro odpnov e(vor yepd-ro on6 6rorpoperrrd rot [eXulprord npdygoro. To ofgnov gnope( npdy-

frorr vo e(vor lrro ouveXqg norrrtropoprp(o, q ono(o 6roXurp((etot oe 6ro-rpopenrd ovrrrefpevo p6vov 6tov rdnoro eurpuig ov 6rorpoponou]oerourrl qv norrilopoprp(o. Enopdvr,ug, or oX€oerg e{oprtirvror ond ro ntigy(verot q 6rorpopono(r1oq rot, rord ouvdnero, undpXouv pr6vo Uorepoon6 prro rdroro 6poorqpr6rqro rr 6Xr nptv.

O T(dpr Krvyr rpo(veror vo E(vor 6vog o0yXpovoq KovorpoKrtBrotrlg.Iro €{oXo BrBtr(o tou, H riyvry ruv pa1rlpartrtiv, \ttt:

Or go0rlponro( rdvouv po0qpotrrd. Kr dtov rdvouv po0qprotrrd, ooXo-Aotivtor pre to ovuxr(gevo nou €Xouv 6qproupyr]oer. Autd ro ovure(pevoe(vor ocpqprlp€veq dvvoreg xor 6ev undpXouv 6{ur on6 rr1 govtoo(o touproOrlprorrrod. 'EXouv nporrtote( on6 ro 6qproupyd rouq, rov go0rlprorr-r<6, pe opro;r6veg r6r6rqreg. An6 ourdg ug erXcrrpqpr6veq r6r6tr1teg, Xprlot-ponouirvtog roug v6poug qg troyrrr]g Kqr rouq rov6veq turv proOrlpott-rtrrv, o prooqprotrrog ouvdyer citrtreg r6r6rqrrg.o

O Krvyr, oo1600, 6e Odtrer vo eyrorotre($er rele(ulg rrlv r6eotrrourr]06oq, ytor( orprlvet ovotrrq pro oveno(o0qrr1 Xopopd6o orov r6eotrrop6.Milder yro ro nopd[evo yeyovdg 6rr ro poOrlgortrd, 6nulq ro erpe0pe ovouq rov po0r1porrr6v, provrelonoro0v rdoo rotrd rov utrrr6 r6opo.

358 4. Jerry P. King, The Art of Mathemaflcs (N€o Y6prq, Plenum Press, i992), a.29.

Page 344: Ο Ταξιδευτής των μαθηματικών

llop'6tro outd, 11 dnoQrl 6orrrv unootrlp((ouv fflv €K rou gr16ev6g 6r1pt-

oupy(o rou rdogou on6 tov Oe6 onotuyXdv€t vo tpprlv€uo€t rrlv tTtovo-

IogBov6prevrl egoppootp6qto rdnotutv ond ttq nto Oetoprlttrdq Kot qqrl-

pqpdveq 6oprdq ttrrv poOqpotrrtirv. H yeopetp(o tou Piprov, yro nopd6sry-

po, orotrouOe( pe po0qtrrottrrj pepor6trlto 6vq o0votro o[ttrlprdtutv, orpt-

prirg 6nog rot 11 otre(o yeulpretp(o tou Eurtre(6r1.s

Ie dtrtro oqtrre(o, o Ktvyr etodyet qv r6do 6tt, orqv npoyportr6tqto,

undpXer rdnoro e(6og puorrrtottrr]q opXriq nou letroupye( npog 6rpetroq

Tov nro ertremuop6vtrtv poOqgortrtirv rpopptilv, Yto vo rtq ouv66oet pe

rq 6rrrl roug xoOotrtx6rqro.

Enrntrdov, rpo(vetot vo lettoupye( prto ovtiteprl Kqt lluottKtortKri oto0qtt-

Kn opxi, 11 ono(o nopdyer pro Oettrrl ouoXdtrorl ovdgeoo oqv ropt|t6q-

to trlq pro0rlyourrjg 6wotoq Kot rrlv op06qto rot onou6qt6tqtd qg -

rj, 6nurq to 06tet o Xdpvtt, gro Oettrri ouoXfttorl ovdpreoo otqv opop<ptd

tqg ivvoroq rot.trl oopopdr4rd tr1q...o

To ndrg orprBdrg 0o roroldpoupe our{v rrl FuortKlortrrl oppi 6ev

ovorpdperot, ytor( ourd ro0out6 ro v6qpo rrlq trd[rlq "puortrtortx{"

uno6qtrtirver pto otrr]0ero nou e(vot yvuroul pe dgeoq en(yvtlorl rt 6Xt pe

ovdtruor1. Av ourrj rl UuortKtortKr], oroOqrrrri opXri nopdyet pto ouoXdtt-

or1 ovdpeoo oe xop{r€g poOrlportr€q dwoteq rot r4 goOqpottxt] op06q-

to rot onou6otdrrpo, erpeUpov dpoye ot poO4gurrro( ou{ r4v opm, rl

pqnurq rpov rdrt nou npotinrlpXe rqg erpefpeorlq; Ebtit rpo(verot vo et-

odyoupre pto oX€oq ovdpeoo oe ovrtre(trrevo Kot, rot6ntv, vo gtldpe

oov vo unrlpXe ouui 11 oX€oq ove(dptqro on6 roug ov0ptinoug.

Oo enrXerpriooup€ vo 6totuntiooul.ts rnv Kovotporrtprorrrq dno{q

ror etrn((ouge vo qq ono6tiroouge rqv ovoyvrirptorl nou tqq o[r(et. To

o0pnov e(vor pro ouveXrlg notrrlopop<p(o, Ttou 6ev 6Xer npoxo0optop6-

veg uno6rorpioetq. Yn6 p(o €vvoto, 6ev €1et poq6q. 'Orov epe(q, og eu-

quq 6vto, Buirvoupe ro o0pnov, to 6torpoponoto0pre oe [eXoptord

5. 'o.Tr., o. 43.6. O.fi., a. 139. 359

Page 345: Ο Ταξιδευτής των μαθηματικών

onpdyporo> pe Bdoq u1 Brotroytrrl rorooreurl pog. Arpori ro rdvoupeour6, op((oupe oy€oerg ovdgeoo oro Ororpoponotrlp6vo ovlrs(pevo.Kord ouv€neto, or oXioetg nou dXoupe op(oer (or ro0otrtr6g €vvoreg) e[-vor rrbrrd yro epdg. 'Evo dllo eurpu6q ov pnope( vo 6rorpoponoujoer ol-Itdg ro o6pnov ror, enogdvctrg, vo 6rlgroupy(orr Oro<poperrr€g oXdoergror, dpo, Oro<poperrr€g ro0otrrr€g €vvoreq. Enop€vtrrg, or ro0olrrriq €v-voreg 6ev undpXouv nopd p6vo oro vou rov ouyK€Kpr;r€vcuv eurputirv6vrov nou rrg op((ouv.

To nopondvur rpo(veror og rotrr1 ovdnru{q rrlq KovorpoKrrprolrr]q06o49. tror( 6pog 6ev €Xo nerorr( ondluro; Eoefg pnope( vo €Xere net-orr(, ey6 6pog ntrdur or6po oe pro Odloooo opepot6rqrog. Av uno0€-ooupe 6tr rdnoro eu<puig ov np€ner npdrro vo brorpoponotrloer r4v not-rtlopoprp(o rou o0pnowoq, yrcl vo pnopfoer rdnog vo ro yvurp(oet(ovr( vo ovrilrlrpOe( opdourg ro odgnov cug €vo opyovrrd 6tro - ;rto no-tr0 urpo(o err6vo), r6re unoorqp(oupe rqv t66o 6rr 11 6rogopono(r1or1epnepr€Xer rrlv ovoyvtirproq {eXurptortiv owtrerpdvcrrv. Ae6opdvou 6rrqurd ro ouyrerprpdvo ovrrre(pevo undpXouv trrdvo yro poq, rouq ov-Opr.0noug, pnopo6pr enrntr€ov vo ro opobonorrlooupe oe outrloydg rjofvotro. Arpod oXqpor(ooupe odvotro, trrnopo0pe vo ovoprrrr40o0;.re ytoro ntrr1Oog rov ororXe(ov rov ouv6lov - 6r1trobr1 roug oprOpo0g! Aev6Xer orlgoo(o yto rr e(6oug ovtrre(pevo prildye, ro pr6vo nou €Xer orlpo-o(o E(vor drr €Xoupe 6toXop(oet ro o0pnov oe ourd ro owrre(pevo. Oger rourou, €vo eu<pu€g ov on6 rdnorov dtrtro yotro{(o pnope( vo 6toXtrr-p(oer rov r6opo rou pe evretr6g 6rorpopeurd rpdno. rr ov rdver rd-norou e(6oug 6rorpopono(qoq, tdre 6rlproupye( odvotrq ror optOpodg,ove(dprrlro on6 ro ov ro ctvctyvop((er { 6ev ro ovoyvrrrp((er r.ug 16roro.Enopdvrrrg, rpo(veror drr 6ev pnopo0ge vo {erpUyoupe ond rqv rporop-

Xrr4 166o rov ouv6trrrrv Kor ruJV oprOpdrv. Oo(veror vo undpXouv oq Fropoorrcrl opXrl rou o0pnovrog, yror( oe rdOe oufirqorl yro ro o0prnovovorpep6poore oE our6 oq rdrr nou nepriXet ovrrre(gevo. fr'ourdv toA6yo 6ev 6Xo nerore( on6truro on6 rqv €nrx€tpnporotroy(o rov rov-orporrtprotdrv. Oo16oo, ror ro ovr(0ero drpo rou rtrqotrod t6eohopro6pte rdver vo vuirOr.rr e{(oou dBotro. 1ourg o oorpo}r1g 6p6pog vo E(vor o36S

Page 346: Ο Ταξιδευτής των μαθηματικών

I I H M A I N O Y N O A A A V T A T

6uiopdg - q ono6oX{ 1600 rou rpuotroU o0pnovroq, 6oo rot rov Ko0o-

trrrdv ewouitv.O B.X. Beprpdrorep, oto dpyo rov Mn rptAoooqfu ruv €motnpdv,

unootrlp(er rt outdg rqv dno$r1 6tt or oprO;ro( e(vot poorro( yto q 6to-

qoponoi4oq rou o0trrnowoq.

Kd0e <popd nou ovofyoupe to ;rdtro FclS, ro ontrr6 poq ne6[o pdq ono-

rotrrjnter (eXoptotdq Uopq€q ror oXqpoto, ro06q ror ;rto notrrtr(o Xptt-prdtov, rporoorrdoeov rot onoXptioeu-rv' ror Dtdrpopeq t66eg ouv06ovtot

pre outd nou ptr6noupe. ME dtrtro tr6yto, €Xoupe enQvooq tov {eXtrtpt-

otdrv ovtrrerpr6vrrrv. Aut6 to yeyov6q uno6rltrtivet 6tt pnoporige vo oTto-

provtirooupe rdnoro ouyrerptp€vo ovtrre(1rtvo oTI'dtrq tq dtrtro ovttre(;le-

vo. To onopovur;rfvo ovtrre(ptevo e(vqt 6vq outo-toutt(6gevo <Kdrt> Kot,

ge ou{ qv 6vvoto, e(vot pto u1"tovd6oo.t

H e(eArKrKri dnoqUn

Aev dXoupe onovrrloer oto opXrr6 poq epdrqpo: undpXouv ot optOtrto(

ove(dpqro ond toug ov0ptirnouE; Oot6oo, orrlv npoYJlortr6rqro, 6ev

nrore0oge 6tr 0o onovtqoouge. To rotrdrepo nou pnopofpe vo no0pe

e(vqr 6tt npoonoO{oope vo oproOeu]ooutrre ro eptitrlpo pe €vov rp6no

Trou vo dXer v6qpo. Meprr6q qop€S, e(vor Xprlotpo vu e(erd(ouge 6vo

eptirqpo gdoo o' dvo e(etrtntr6 ntro(oro. lltirq €Xouv e{etrtX0e( to poOq-

golxd oq rg4go rou notrtttogoti poq, Kot tt ouvendyerot out6 Ylo rrlv

ovtotroy(o rov optOptitv;To poOqponrd (tr(v4oov drqv ot dvOponot dpXtoov vo perortvotjv

e6dr rr ere( ro ovrrK€(Utvo Kot vo nopoqpo0v 6rr unrlpXe pto t6t6rqro

rorv{ oe dtreg ug ou)\troy6q ovrtretgdvov, q ono(o rltov ove[dprrpq ond

ro (6ro ro ovurs(ptvo - q r6r6rqro ouul r]rov ro ntr{Ooq orotXe(tlv rtlv

oultroydrv. Enopdvoq, or opt0po( yewr]Or1rov (tj ovorotr0rprrlrov). ApXt-

7. W.H. Werkmeister. ,4 Philosophy of Science (A(vrotrv, University of Nebraska

Press, 1940), o. 141. 361

Page 347: Ο Ταξιδευτής των μαθηματικών

rd, or oprO;.ro( {rov onltirg nepryporpdg nenepoop€vtrrv outrtroydrv ovrt-rerprivr-r-rv 6ncug, yro nopd6e rypo, ev6g o(euyoprorin yovruirv q ev6q n(eri-yougo ov0ptirnov. H ovdnru(r1 rov po0qporrrtiv Boor(6rov orrlv oqor-perrrrl 6to6rroo[o. Ko0tirg or oprOpo( dyrvov nto orp4prlg€vor, on€rrr1-oov yevrr6repq roXU - pnoporioov, 6r1tro6r1, vo €qoppoorodv oE 6troror ne proo6rrprs Kor 6rorpopert169 oultroy€g ovrrre rpdvtrlv. Aurrl r1orpotpeurrj 6ro6rroo(o ouve$(eror oe 6trr1 rq 6rdprero rrlg e(elrrrrrrlqpog rorop(og. llfpo on6 ulv oqotp€urr1 6ro6troo(o, or oprOpo( Enerrd-0qrov. Iulv opXrl unrlpXov pr6vo nenepoopr€veg outrtroydg ovrrrergdvovKor lrro nenepoo;.r€vq orotrou0(o opr0ptirv.'Enrrro owt]q<p0(ropr 6rt orguorro( opr0po( e(vor dnerpor - 1.rto €vvoro nou 6Ev uno6rllov6rov on6ro oro0rlr4ptord 6e6op€vo (rot pto t6€o q ono(o ouvovro0oe orpo6pfgovn6pdoerg en( Xilrer(eg).

AIId q en€rrooq ouveX(or1re. Or oprOpo( enerrd0rlrov, riore vo ou-pnepildBouv ro rtrdoporo ror perd rouq opvrpKo0g opr0godg. Ilpoor€-Oqrov or dpprlror ror ot uneppouro( oprOpo(. hr1 ouviXero orerprr]roperoug ptyo6rro6g, roug uneppryoOrro0g rot rouq urspTr€nepoop€vougopr0porig. Aurq 11 611111 Orobrroo(o qq orpo(peoqg Kor rrlq endrroorlg

Xoporrqp((er rrlv e[€trr{q rov poOqpolrcOv pog evvou.0v. MnopoUpedpoye vo uno06ooupe nog 6Xou;re rptdoer oro r€trog; BeBo(oq, oe rd0eyevrd rdnoror dv0ponor entg6vouv 6u q e{dtrrfq 6Xer otror]r1pur0e(. tldrgpnope( rdnorog vo qovrclor€( vo nder nopon€po; 'Otrulq, ov Bpror6po-ore pdvo oro np6ro ord6ro qq Trv€uForrK{q ror no}rrtoprrrlg pog e{€trt-(r19, r6re pnopo6pe vo uno0€ooupe 6n ror 11 ovriArlgri poq yro rouqopr0;rorig ror ro poOqportrd poq e(vor oretrr1. Ot dworeg nou orpopo6vrouq optOgorlg 0o npoXrrrprjoouv Kr dtrIo oe dyvooreg neptoX6g.

Aurrl 4 eupfreprl errdvo rqg ovOptinrvrlg <p6or1g uno6qtrtiver pro6rorpopenrrl €rpoorl rou npoptrqgorropo0 poq yro rrlv ovrotroyrr{ ro-rdoroorl rov oprOpdrv. Ag uno0€ooupe 6rr, ev pdpet, q ovrfir1$r1 pogyro rous oprOpo0g ovrovqrtrd ro roro( efpoore og dv0ponor. E(oprd-ror, 6qtro6r], on6 14 ouyrerprtrrdv4 Brotroyrrr] rorooreurl gog. To eptirrrl-po lotn6v r(0erot og e[49: n6oo c(vor roOopd ovOpdrntvrl 11 ovrfir1{rr1

Uoq Kor n6oo e(vor orroupevrr{; Mnopo0;re vo 6o0pe prro ovd}oyr1 ro-362

Page 348: Ο Ταξιδευτής των μαθηματικών

rdoroorl orrl yeurperp(o. hqv opXt], unoO€oopte 6rt rl qfon rou otipno-

vrog npdner vo e(vot eurtre(beto. Kotdntv ovorotrdtfoge pq euxtre(6eteq

yeuperpfeg. Atdnerot, ]otn6v, ro ofgnov on6 rrlv eurtre(6eto yetlperp(o

rj on6 gro on6 ttg pr1 eurle(6ereg yeurperp(eg; O Atvoroiv un€Oeoe pto

gr1 eurtrs(6ero yeoperp(o yro 14 ytvrrri 0etrrp(o rrlg oXertr6rrlrog. 'Optrtg,

11 yeurperp(o tou <puotro6 o0pnovrog gnope( vo Frlv e(vot pt(o on6 rtg

ouyrerprp€veg yrur;rerp(eg nou dXoutrre ovonr0(et. O Xtitpog pnope( vo

dXet pro nro yevrreup€vr1 notriloptoprp(o on' 6,rt onotoOqnort ouYKsKpt-

pdvq yeurperp(o, 6rot 6ore vo pnopodpe 6wulg vo enil6(oupe 6noro

yeoperp(o mrOupodpe.Opo(trrg, ot otrr1Oereg rou odpnovrog gnope( vo e(vqt nto yevtrdq on6

rouq oprOpofq Kot rouq rdrtrouq rov po0qgorrrdtv pog. Av ouvovroU-

oop€ Uro dllrl opd6o e[orpertrd eurpudtv 6vrtltv, gnopt( vo pntrsr6po-

or€ Kor vcl clvoororov6goort qn6 ro 6trd toug, ouyrerptpivo go0rlpo-

trrd. fl(otrt on6 ro 6rrd roug rot ro 6trd gog poOrlportrd pnopei vopp(orerot pro po0Uteprl otrr1Oeto, €vo e(6oq petopo0rlportrt0v nou yevt-

rrdovror o'6vo ov6repo en(le6o. E6dr 6E Xpqotponortir 14 I€[4 opero-

poOrlprortrd, ovorpepdprevoq orrlv ovotrurrrq petr€rr1 rov rov6vtlv rot

rov opXtirv nou 6tdnouv rn Xprion rov po0qporrrdrv ouppdtrurv, otrtrd

npore(vur dvo Bo06repo Kot nto yevtreup€vo v6r1po. T6oo ro 6trd pog

go0qpolxd, 6oo rot ro po0qportxd rulv dtrtrurv dvrov 0o onoretrofv et-

6rr€g neprnrrioerg our6v ttuv petopto0rlpottr6v.

hqv npoond0etd gog vo onorol6Qoup€ rrlv undoroorl rov optO-

pdrv, orqprpqroge ortg troytrdq 6to6troo(eg rqg ovdtruorlg. Aur6 e(vot

orlo16, orpoU or oprOgo( e(vot ovttre(p€vo rrlq or€$r1q. OUurS, ov np6-

Krrror vo pdOoupre rdrt on6 rr1 porpoXpdvto lorop(o FoS, oC pd0ouge

drr e(vot ouXvd entr(v6uvo vo unoO€roupts nog o utrtrdg r6opog tret-

roupye( ndvro oriprpovo F€ rouq rov6veg qq e{oyoydg oupnepoopd-

Turv l-rs pdorl rq troyrrq. Ot t66eg ouv66ovrot pero[6 rouq Ut ]oytr6 rp6-

no ror ofvotro t6etirv prnopo0v e(te vo ou;lrpovodv pero[0 roug e(re vo

e(vor ovrupottrd. Orov npoono0odps vo erpopp6ooup€ rl troytrq oe

rd0e ntreupd rou rpuotrod odgnovroq ge oro0ep6 tp6no, nopaoup6po-

ore ouXvd oro povondrt rou orpdlporog. lto vo rorotrdBoupre ro ytor(, 3S3

Page 349: Ο Ταξιδευτής των μαθηματικών

0o npiner vo Oupq0o0trre 6tr or r6€eg pog yro ro <puorrd r6opo oy4pror(-(ouv €vo povrdtro ror, pr6oo oe our6 ro povritro, e(dyoupe troyrrd ou-pnepdoporo. 'En€rro, neptprdvoupre vo oup<pr.ove( o rpuorr6q r6opog pero oupnepooporo ourd. 'Orov ro oro0rlqprord goq 6e6op6vo 6r ou1.r_rpovorlv pe ro troyrrd oupnepdoporo, pno(vou1.rr ouyvd orov nrrpoop6vo {erpoprtrrOo0pr ro oroOqrrlprord 6e6op€vo, ovr( vo npoono0{ooupevo peltu.0ooupe ro povrdtro poq.

O tlopgsv(6qq Kor o Zrjvov n(orsuov, orqpr(6grvor oro loyrrd ent_Xctp{pord roug, 6l o r6oproq rjrov operdBtrlroq Kor drr 11 r(vr1oq {tovonl6g pro orp0otrpondrrl. To otoOqrqprord 6e6og6vo 6trr.rrv rov rrvo0_p€vov guorKtirv ovrtrerp€vrrrv oyvorl0qrov. Kdnoto enoXrl rlrov troyrrdvo orerpr6poote drt o rdopoq rjrov enine6og yror( ro povr€lo nou sLXoF€ yro qv npoyporrr6r4ro pog enfrpene vo ouvdyoul_re loyrrd rcdrrr€roro. Av 11 fq {rov orpoyyu}q, ov {rov oov Fro pndtro, 0o €nerpre p€_oo o' rivo peydtro revd. 'Hrov otroyrr6" vo nroredoupre 6u unr]pXov trr6_vo 6dro oupdvto otirgoro, orpod ro 6€ro r]rov rep6q opr0p6q, Kor orovopr0pd outd nepilol.rBdvovrov 6tro ro oupdvro otirporo. Enop€vtrrg, 6r_rorotroyrlp6vo <putrdrroov rov lotrilofo, o<pori toXup(orrlre 6u e(6e rpey_ydpro y0pur ond rov A(o. To oro0qrrlprord 6e6op€vo rou fotrrtro(ouenpene vo 4rov eorpolp€vo, orpof 11 rlnop{q neproodrepov on6 6€rooupdvrcuv orrlpdrtrtv 0o onoretrofoe ovr(rpoo4.

Av pnopodpe vo e(poore o(youpot yro rdtr, out6 e(vor ro e{r1g: roofprnov 6ev e(vor ovoyroop€vo vo orotrou0e( ro oupnep<iopctrct rulvtroyrr6v onoytrryrirv poq. 'Orov ptdrvoupe oro0rpqprord 6e6oprdvo noudpyovror oe ovr(<pooq pe ro povr€tro FoS yro qv npoyporrrdrqro, 0onpdner vo (ovoorerprotipre ro govr€tro trroq. Av ro ofpnov rjrov onotrf_rrrrg op0otroyrr6, r6re o l6yog rqg 6topdrpou rou xdrlou npos qv n€pr_rpdperd rou nr0ovdv vo rlrov 3 - €vo {erd0opo, orptpdq ror rorlrd 3.Av eytir rouldXrotov erXo erpe0per ro o0pnov, e(vor nr0ov6v vo etXo ent_tr€(er ro n vo rooUror ge 3. Xo(popor 6pog nou 6ev erpedpo ro o0pnov.fro pdvo, 11 ovordtrurlrrl trlg otr40tvrjq rtpqg rou n e(vor notrd nto ouvop_noorrrc{ ror evbrorp€pouoo on6 onorov6rlnore oprOgd 0o gnoporioo voe(Xo oroprprore(.364

Page 350: Ο Ταξιδευτής των μαθηματικών

Otroxtrrlprirootrre qv runtrrl ovdmu[r1 turv opr0gdrv pe ro ipyo rou fr6-

opyr Kdvtop yto rous ull€pn€n€poopr€voug oprOpo0q oro t€trq rou 66-

xorou ivorou ottitvo. Aurd onoretre( u1v tetreuro(o en€rrooq rqg dvvotog

trrrv oprOptirv, oltrd 6Xr ro t€trog rrlg torop(oq rouq. fl€po on6 tov opt-

opd ttrrv dppqturv oprOpdrv on6 tov Ntfvrertvt Kot rov ufltpn€n€po-

op€vov oprOtrrdrv ord tov Kdvtop, orlpertirOrlrov n d}treg onou6o(eq go-

0qpotrr€q e[el[[etq, rotd q 6rdpreto tou 6edtepou 1-ttooti tou etroorof

ouivq. H p(o rjrov q ovordtrur|4 ruJV prl eurleOetov yetlperpttiv rot r1

6e0tepq, ouuj nou pog ev6rorp6pet o' ttotjto to ptptr(o, {rov 11 npo-

ondOero Oepetr(uroqq 6trov rov po04potrrdrv oe o{rtirgoro.

To trrowdtro yr' outrl qv npoond0eto rlrov ro €pyo rou Eurtrebq, o

ono(og Xpqotgono(,r1oe dvo o0volo olrurpdturv rot ofrogortr6v npotdoe-

ov yro vo ouvoydyer to 0etrrpr]poto qg rtroorrr]q yeurpetp(oq. Or poOq-

potrro( r1trnr(ov 6tr 0o ppeOei ivo ofvotro o[rurpdturv on6 to ono(o 0o

pnopo0oov vo ouvoX0o0v troyrrd 6treq or otrq0eteg rqq optOprlttr{q. Av

out6 rirov 6uvot6, r6te 6trot or (eXurproro( toge(g ttttv poOqportrtiv 0o

gnoporioov vo ouvoX0o0v ond qv opt0pqlrrl, ror otr6rtrr1po to otro66-

Urlpo rov poOqgotrrdrv 0o pnopoOoe vo mqprpei pto Yto ndwo, oe ye-

pd 0ep€ho. To orjvotro tutv o[topdrov 0o np€ner vo dXet 6Uo r6t6qreg: 365

Page 351: Ο Ταξιδευτής των μαθηματικών

ro o(uirporo 0o np€ner vo e(vot ntr(pn, enoprrl 6qlo6rj yro vo nop€Xouv6treq rrg oXerrr€g rbr6r4req rrlq optOpqrtrrlq rot, en(oqg, 0o np€ner vo oup-rpovodv pero(0 roug. lro vo oup<prrrvotiv pero{d rouq, ro 0eoprlporo nouono6errvUovror pe pdorl ro o(uiporo 6e 0o np€net nor6 vo ovlrpodouvro €vo ro dtrtro. Av rlrov 6uvor6v vo ouvdyoul.re ovnrgotrrd Oeop(poro,ndrg 0o pnopo0oe rove(g vo €Xer epnroroodvrl oro poO4polrd;

To 1889, o lro}69 poOrlportr6g T(ou(€ne fledvo brordnooe €vq odvotrondvre o(ulpdTUJV nou 0o pnopo0oov vo Xprloigonorq0o0v yro vo ouvoyd-youps ro Oeopqporo rrlq oprOtrrqlrr1g. Aurd to ontrd o{r6proro rltov:

1. To 1 e(vor opr0p6q.2. Av ro x e(vor opr0p6g, o en6pevoq rou x efvor en(or1g oprOpdg.3. To 1 6ev e(vot o erdpevog rdnorou opt0poU.4. Av 6do opr0po( €Xouv (ooug en6pevoug, 16rr e(vor (oor.

5. Av dvo o0votro oprOpdrv nept€Xer rov opr0p6 'l ror neprdXe r 6trougroug mdpevouq rov ororye(ov rou, r6re ro orjvotro nept€Xetdtroug toug opr0podg.'

Ie ourd to o(r6proro, o fkdvo 6ev dptoe rqv €vvoro rou opr0;-rotloUre rqv dvvoro rou er6pevou. Oot6oo, e(vor rpovep6 drr ot oprOpo(6noq rciug Xpqotpronote( o flsdvo oro o(uirpord rou, e(vor or <puorro(oprOpo(. Me pdorl ourd ro ontrd o{uir;rorcr, Kclr {errvtirvroq on6 rovnp6ro rpuorrd opr0p6, ro 1, rordrpepe vo op(oet dtroug rouq rpuorrotigopt0porig. An6 q orryprl nou EXoutrre rouq rpuorroUg oprOpotlg, pno-podpre vo ouvoydyoupe roug undlotnoug opr0podg roOdlg ror rrg oX€-orrq rouq. E(vor erntrrlrrrr6 ro drr rdoo ontrd o{r6poro e(vor 16oo toXu-pd. Aev e(vot pnep6eprdvo o0te ooorprl, otrtrd nrptrtrr(ouv our6g orpr-p6g rrg dvvoreg nou e(vot poorr€g yro rouq rpuorro0q oprOgodq: ro 6rr(ertvdpt on6 ro 1 Kor, orn ouv€Xero, npoXrrrpdpe 6ro6oXtrd os rd0Een6pevo opr0g6. To n€pnro o((opo t(vot ro o(ftrrgo rqg enoyuryrjg. Iq-go(ver onltirq 61, ov pnopo6pe vo ono6e(oupe pro r6r6qro yro ro 1Kor yro 6loug roug endgevoug opr0po0g, r6re ono6errv6eror yro dloug

366 l. Boye r , o . 645 .

Page 352: Ο Ταξιδευτής των μαθηματικών

roug <puotxorig opr0po0q. Aurd to o{uir1loto, 6nurg Xpqorponor{04rovon6 tov fledvo ror dllouq poOrlporrrodq, rjrov ypogp6vo ge oupBotrrrdrp6no rot ouvo6edovrov on6 ouotqpoUq rovdvrq yro t4 loyrrr] e{oyur_y4 oupnepoopdrov. E6dr 6rorundroops ro poorxd o{rdr;.roro F€ Tnvontronorlp€vr1 goprp{ roug.

H 6ro6rroo(o o(urrpourr]g Oegetr(oor;q rtrrv poOqpourdv ouveX(oq_re on6 rouq Mn€prpovr Pdoetr ror Atrgpevr Nop0 OudrrXevt (1861_1947) pe rrlv oud rorvo0 npoond0erd rouq vo ouvOdoouv 6Io to o(urt_potrrd poO4pottrd pe qv runtr{ troyrrrj oro principia Mathematica. Avour6 prnopoUor vo €rnrEux0e( tdre 6tro ro poOqgorrrd 0o pnopodoovvo ovoX0ofv oe Boorrd o{uirtrroto qq }oyrrrjg. Tt 0o pnopotjoe voepno6(oer r6po ro go0qpourd;

To nopdSo{o

AuoruXdrg, ro np6ypoptrro qq ovoyoyriq 6lurv rov go0qprolrrilv oeo{uiporo 6ev npoXriprlo€ Jr€ rov rpdno nou ro eiXov oXe6rdoer or [led_vo, Pdoel, OudrrXevr ror d}tror. Eprpov(or4rov pruyp€S oro onou6o(oorxo66pqpo nou 6pr(ov. To nptirro npdptrrlpo rjrov 4 ovordlurfr4 rtrrvnopo66{ov nou npoipXovrov on6 ug 6rotoOqrrris Foq ovtr}r]{rerg ytoro oUvotro - q dvvoro rov ouvdlrrrv Bprordrov oro sn(rcevrpo rov v6ovo(u.oprorrr6v goOrlpotrrtirv, ytor( ro odvotro Xpqorponototivrov yro voop(oouv rqv dvvoro rulv oprOprirv.

'EXouge {6q ovogiper ro np6ro nopd6ofo: e(vor ro nopd6ofo rou

Kdvrop, 6tr 6r1tro6{ 6ev pnope( vo undp{er €vog gdyrotoq uTr€pr€rspo-opfvoq optOpdq. Ag unoOdooupe drt ro O e(vor o nlq0tr6q opt0p6qev6q ouv6}ou, ororXe(o tou ono(ou e(vor 6lo ro dtrtro odvotro. Autd royryovrro(o o0votro 0o np€net vo nepr6Xer dXr pdvo rov eour6 rou, oltrdror rdOe nr0ov6 unoo0votro tou eourod rou. Enogdvoq, o ntrr10rr6qopr0p6q rou ouvdtrou npriner vq e(vqr geyotrdrepog ond ro O. Itqvrpoyl-rorrr6qro, 0o npEner vo e(vot routrdXrorov 20, rdu nou ovurpd-oK€r 1r€ rov roxuproFd drr o ntrr10rr6q rou opr0p6q e(vor to o. B€po(oq, 367

Page 353: Ο Ταξιδευτής των μαθηματικών

qur6 ro ovrtperon(oope opvoripevor ontrdrg 6rr undpyer dvo 16oo ;reyd-tro o0votro, 11 €vog 16oo peydtrog ntrr10rr6g opr0p6g.

O Pdoel ovordtru{re €vo nopd6o{o nou rpov rpupp€vo oqv on}oi'-rrl 0eop(o rtov ouv6}ulv, ro ono(o rlrov nro bdorolo vo npoono0{oou-p€ vcl ro orrrotroyrlooupe. Mnopo0gr vo 6rlproupyrlooupe o6volo, nouro orotXe(o roug e(vor rr ourd odvolo. Av unoOdooupe 6rr ro A e(vot roorjvotro dlov rtr-rv oprorep6Xetpcuv norXrdrv rou gn€r(pnotr ror 6rr ro Be(vqr ro ofvolo 6lov rov opoevrrtiv nonrtirv, r6re ro {R,a} e(vor dvoorjvolo nou onoretre(ror un6 60o ofvotro, 6qlo64 ro odvotrq A ror B. OPdoel oXrlpduoe 6vo o0volo, ro I, rou ono(ou ro orotXe(o {rov ror ro(6to orjvotro. E(nr 6rr dvo orivotro 0o e(vor ororXe(o rou I ov 6ev neprd-

X€r ulq ororxsio rou tov equr6 rou. Av I = {o'', az, o2,...1, r6re rdOe o, eLvot dvo otjvolo nou 6ev nepriXer u.rg ororXr(o rou rov eour6 rou. Oonp6net trorn6v, ovopurrrl0rlre o Pdoetr, vo oupnepildpoups ro I urgororXe(o rou I; Av to I 6ev neprdXrr tov eour6 rou, 1611 0o npdnet voefvor ororXe(o rou I. Atrtrd, ov e(vor ororXe(o rou I, r6re neprdXet rovEour6 rou rot 6,: 0o npinet vo e(vor ororXefo rou I. Atrtrd, ov 6ev e(vororotXe(o, r6re 0o npfner vo e(vor! E{ ou ror ro nopd6o{o. An}drq 6eve(vor 6uvor6v vo dXoupe €vo odvolo oov ro I.

To nopd6o(o rou Pdoetr on66er(e 6rr or Ororo0rlrrrdg ovrrtr4rf€r( Foqyro ro orivotro o6rlyo6oov oe ovrupdoerq ror 6rr or poOrlpotrrof 0o€npene vo orerprofv vo npooeyy(oouv rov optop6 rr.rrv opr0gtirv ge dtr-loug rp6noug, n6po on6 ouroriq rqg ontroirqg Orurp(og rov ouvdtrtrrv.Aur6 o6{yr1oe, rord rr1 6rdpreto rou etroorod ou.0vo, orqv ovdruu(4rqg runrrrlq o(ulporrrrlg 0eurp(oq rtrrv ouv6trrrrv.

To nopd6o(o rou Pdoetr ev0dppuve rr dltroug vo rpd{ouv yro nopd-6o(o orrg 6roro04rrrdq ovrrtrrlrfrrg ;.rog. Kdnoro on6 ourd ro nopd6o(oerntr{ooouv ror e(vor btoore6ootrrd. kerpre(rr ro nopd6o(o rqg trd-

UroS rou T6;roov, nou ovorotrJrprrlrr on6 ro Bpetov6 rprtrdoorpo T(€rpgT6proov ro 1970.'� Oetrrpo6pe 6l €youge €vo rd}eto plXdvrlpo yto vo

i?n 2. E.l. Borowski Kor J.M. Eorwein, The HarperCollins Dictionary of Mathematics$e66 (Neo Yopxrl, HarperColl ins Publishers, ig91), a. 5Bg.

Page 354: Ο Ταξιδευτής των μαθηματικών

A P I O M O I

ovdpoupe ror vo oprlvoug€ glo ldpno. Iqv opXq ro qurq ovdBet ytoivo lenr6 ror perd oprlver yro trrrod trenr6. .Entrro,

ovdper ndh yro dvoritopro rou trenrod ror opqver yto 6vo dy6oo rou trenrod. Aurd ouvcX(-(etor pe ro gog vo ovdper Kor vo oBrlvet fotepo on6 to pro6 rrlqnporlyorjpevrlg Xpovrrrjq nepr66ou. .yorepo

on6 60o ol6rlqpo Imrd,0o €Xer ougBe( pro dnetprl oxotrou0(o ovoppdrov ror oBqorpdrov.'Yorepo

on6 60o treurd, ro rprrlq 0o e(vot ovoppdvo { oB4op6vo;'Evo dllo urpo(o nopd6o{o Boor(etor oto opXtrd nopd6o{o rou pd_oetr. kegre(re pro n6trr1 nou €Xer dvo g6vo roup€o. Aur6q o roup6oq(upr(rr 6notov 6ev {up(,(eror p6voq tou. floroq {upftr rov roup€o; Av oroupfog 6ev {upr(erot trrdvoq tou, rdre {upr(er rov eourd rou. Av {upr(e-ror trr6voq tou, rdrr 6e 0o np€ner vo {upr(er rov eout6 rou.'Evo dltro 6{unvo nopd6o{o ovorolUrprqre qnd ro Bperovd prB}ro04_rdpro T(. T(. Mn6pr. Ag unoO€ooupre 6n ro[tvopo6;re dtrouq roug Oeurotiqoriporoug opr0godE pe pdo4 ro prrpdrepo opr0p6 oulloptirv nou e(votonopo(tr;teg, oro oyytrrrd, yro vo rous neprypdrfoupr. Enog€vurq, o opt&p6q oep-ev-nv' onorre( rperq outrtrop€q, evdr o opr0p6g vdrv_l_oeB_evordooeptq. Ag oregrodpe rtirpo ro o0voro dl,v rov or.pourrv oprOptirvnou onolrodv routrdXrorov 6sroEwdo outrtropdg yro vo fi€prypo<potiv. Ieour6 to odvolo neprdXeror o prrcpdrepog or€poroq opr0p6q nou qnorts(6rrorvv€o oullopig. 'Opurq, pdhg neprypdrfoge out6 rov ordporo oprO_p6 utg to "prrp6rrpo ordporo opt0p6 o olo(oq 6w neprypdrpror pte trty6_tepeq on6 6eroevv€o oulloBiq,, nou on6 gdvo rou onoretre( neptypogrjrov 6eroorrri outrlopdrv! Enop€voq, 6ev undpXet rdnoroq prrpdtepoqor€porog rou vo onorre( yro ulv neptyporpr] tou 6rroevv€o outrlopdq.;

llotrtrd on6 to nopd6o{o y(vovror nopd6o{o drov gro nepryporpr]or4pt(eror orrlv ouroovorpopd. 'EXoupe,

enopdvoq, ro nopd6o(o rovouvdlov nou nepr€Xouv rov Equr6 roug, rov neprypogtilv nou neprypd_qouv rov rour6 rouq Kor rov roup€rrlv nou (up(ovror trrdvot roug. Aurd

3. Seventeen (sev'en,teen): 6eroenrd. (I.r.M.)4. Ninety-seven (nine,ty-sev,en): evev{vro enr<i. (I.t.M.)5. 'O.n., o. 49. 3Sg

Page 355: Ο Ταξιδευτής των μαθηματικών

ro nopd6o(o qno6etrv0ouv 6tt, oe ewototroytr6 en(ne6o, e{orotrou0o6-

pe vo nolefoups F€ ng poorr6g ovtrtrrirlrerq nou Xplotgonotodge yto vo

6toruntivoupe rtq €vvoteg Yto rous oprOpo0g ror ro poOrlgortrd.

Mro 6e0rtpq 6uoxotr(o ot 6,tt orpopd r4v npoond0eto vo Bpo0peo[uirptoro yro 6tro ro ptoOrlprotrxd ouv6€erot ps rlv 6vvoto u1g ntrr1p6-

qrog. 'EXet unoreOe( 6rr dvo ntrrjpeq odoqpo o{ttlgdttrtv, 6noq out6

rou fledvo, 0o np€net vo e(vot oprerd toXupd, tiorE vo onoDerrvfet 11

vo 6rorfrtf6et rdOe toXuprop6 nou 6totuntir0rlre evt6q rou ououlporog.

Mnope( vo Xpetoorofv Xp6vro 6outrerdg yto vo pnop€ooutrt€ v'onoqov-

OoUpe yro onotov6rlnore ouyrerprp€vo roXuprog6 otrtrd, Oeopqrtrd rou-

IdXtorov, 0o np€net vo gnopo0pe vo ono6ei{oupe ov e(vot otrqOe(q {rlreu6e(g 6tror or roXupropo(. Aurq q dno{q dtrtro{e on6 ro 6pyo rou

Koupr friwel (1906-1978), o ono(og to 1930 onC6erle dtt, trt€oo oe

onoro6{nore o6orrlgo o{topdrov, nou t(vot opretd toXup6 t0ore vo

epprlve0er rqv oprOpqrtr{, e(vot buvordv vo 6toruntitooug€ toxupt-

opo0g nou 6ev pnopofv vo ono6etyto0v. Aur6 e(vot ro looiyvtooro 0r-

d,pqpa u1q oti)tng tou fr€vretr. Aut6 orprpdrq to Oetirpqpo ror66et(e

611 6ev e(vot 6uvot6v vo onoOe(oupe ov undpXet r] 6ev undpXtt €vog

ntrr10rr6g opt0p6g ovdgsoo otov No Kot rov N'

H ovordtru{rq rou Oeurprlporoq qq or€trtrog rord<ptpe €vo pfnqpo

oro odyXpovo po04porrxd, oov ro Xrdnqpo nou rordrpepe orouq [1u0o-

ydperoug 11 ovordtrurlrrl rou oodp;rerpou qq 6toytlv(ou. An6 rrlv onrrrrl

yov(o rrlg rpilooorp(og, trroq tr€et 6tt 11 troyrr4 rot proOqgortrri ovdtruorl

6ev enopro0v yto vo onoKotrUrlrouv dtreq rtg otrr1Oereg Yto rous optOpo0g.

Enop€vtrtg, 0o e{orotrou0ofv ndvro vo undpyouv otrr]0eteq nou 4 ovord-

Iu{rq roug nopop€vet d(rrr on6 ourd roOoutd to poOqportrd.

Merd Tov aKoor6 CI lovo

Meydtro tprlpo outtirv nou dXou;re ratrUrpet neptypdger ro 6pyo 0eop4-

rrrdrv poOr1;rortrtitv, 6noq our6 ouvO€erot Jl€ rq 6trq pog dvvoto yto

37O roug oprOpo0q.'Opoq, 11 e(dtrt{r1 rurv optOgtirv e(vot nto notrdn}orr1 on6

Page 356: Ο Ταξιδευτής των μαθηματικών

A P I O M O I

ng Oeurp(eg nou ov€mu(ov outo( or enoyyetrpor(eg go0rlpottxo(. OroprOtrro( onordo6v rprlpo tqg (6roq rrlg 6o;1119 rou notrtrtogoU pog.

Enop€vog, evtir 11 Oeurp(o rrrtv opr0gdtv e(etrtoo6rov oro vou trrrv po0q-porrxtirv, o notrrrrop6g rrrrv opr0p6v ovonruoodrov novtotj orrlv Kotvo-v(o gog. Av ro poOqporrrd efiov ev6torpdpov p6vo yro roug enoyyelpro-r(eg po04potrro6g xor ro oyvootioe q undlotnrl rotvutv(o, rdre o 16-

opog 0o {rov 6vrog 6vog (o<pep6q r6nog. Or opt0po( xot ro poOqport-

rd onoretrodv r4v rrvqrrjpro 60vo;rq qg reXvotroy(oq gog ror ouv6io-vror Kor ro 60o orevd pre rqv ro04geprvri Uoq 0nop{q.

'Evo p6rpo yro to n6oo notrdnlorq fXouv y(ver to poOrlpottrd pog

e(vor 11 Xp4o4 rtrrv poOrlporrrtiv ond ro gdoo notr(rq. Ifgrpovo Us rovroOqyqrrl poOrlpourtiv T(dpt Ktvyr, undpXouv nep(nou nev{vro yiltd-

6rq enoyyelpor(eg po0rlporrro( olq HnA, ov perp{ooupe ro p6}q rovrpuirv e0vrr6v po0qprourdrv opyovtitoEurv. Auro( ot nevqwo Xttrtd6eg;ro0qpotrro( 6rlpoore6ouv rd0e Xpdvo, oe Xfiro newqr6otq 6rogoperr-rd nepro6rrd, nep(nou efroor n€vre Xilrd6eg po0qportrig ovorotvtioetgnou orrlp((ovror oe ipeuvo.u h4 ouvdXtro, o Krvyr entoqpo(vet 6rt oXe-

6dv rov€voq 6e 6roBd(er ourrig rtg e(root n€vre Xrtrrd6tg entorqpovtrdgovororvtioe lq. Og en( ro nle(orov, 6ropd(ovrot on6 rrlv rprlrrl €ntrpo-nrj rov nepto6rrtirv Kcrr rouq (6roug touq ouyypoqe(q. Enop€vog, nop'6tro nou dXouge ndpo nollodg enoyyetrpor(eq poOqportro0g, rt ouro( otpo0rlgorrro( rd0e Xp6vo nopdyouv €vo 6pyo peydtro oe 6yro, out6 ro

6pyo 6rv €Xer dproq en(6pooq o'epdg rouq un6}ornoug.Aur6 6purq 6e orlpo(vet 6rr ot vdeg ovorolf{rerg rov po0qpottr6v 6e

rprdvouv oro eupf rotvd yror(, puor4pul66q, rptdvouv. Aur6 nou <prd-ver og epdg e(vor ro rotrdrepo roppdrr rov poOqprortrdtv, our6 bqlobrlnou e(vqr oqpovrrr6 yro Uoq orn Fn poOqgortrrl (ttti ltoS.'EXouv undp-(et oprer€g rpdotrg-rtrtr6rd oqv ov0p6ntvrl torop(o, 6rov eiXope pto

€rpn[n Xpqorprono(r1onq rov goOrlpourtiv on6 ro p€oo dv0pono. Aur6

6e orlpro(ver 6tr 6tror orrlv rorvov(o eno6(r1oov ug yvtioetg roug 6oovo<popd orouq opr0go0g, otrtrd drt ro drovov peydlo rpqporo tou ntrr1-

6 . K i n g , o . 6 377

Page 357: Ο Ταξιδευτής των μαθηματικών

0uopo6. M(o ond rrg npurreq qop€g rlrov 6rov ovonrdX0rlrov or oypo-rotrhdpyertg orrl 6urrrrl Ao(o, y0ptrr oro 8500 n.X., 6rov ouv40rop6vordvrpeg ror yuvo(reg dnpene vo prdOouv vo prerpofv ror vo rdvouv troyo-ptoolrotig.

tlpotirOqoq rurv proOrlgolrtiv efiope rr 6rov e(etrtX0qrov or ndtrerg,yfpur oro 3500 n.X., ror or rupepvrlreg Xperd(ovrov ypoge(q ror repe[gyro vo rdvouv o0v0eroug unotroyropotig yro ro ndvro, on6 rqv nopo-rotro604oq turv enoXc0v pdXpr ro po(poopo qq ylq Kor rov Korop€pr-opd rrlg epyoo(og. Auq q e(dlt(r1 enrroXtivOrlre ond rqv erpe0peoq r4gypoqqq orrl loupcp(o, y6pur oro 3100 n.X..

flop'6tro nou 4 EtrIdbo €(4oe pro on(ouurq dv0rloq oro poOqgol-rd ror orrl rpilooo<p(o ovdpeoo oro 500 n.X. rqr ro .l00 p.X., yro rougouvrl0rogdvoug notr(teg 6w dltro(ov noltrd. O dporog ror ro unotroyt-ortrd ntvdrro dyrvov 6r1porpili oq Bo40nporo yro roug unotroyropo0g,utrId o p€oog dvOptrrnog 6e ouppererXe orov ntrofto rov ;roOqporrrdrvrou ovorotrfrrrovrov.

Aurd dllo(e ro 1455, 6rov o loXdveq frofrevprnepyr (lourepBdp-yroq) nopouo(ooe Xurd perotrtrtrd rrvrlrd runoyporprrd orotXe(o yro rnverrUnoorl prptr(ov. An6 ouul rrlv enoXrl rot perd, gnopo0oov vo cKTU-ntivovror peydtror oprOgo( oXerrrd rp0r1v6v BrBtr(trrv yro po(r<4 rqrqvd-trooq. Aur6 porlOr1oe orqv npo60qorl rurv tbetirv rqg Avoy€vvqor1g,orrq ono(eg oupnepilogpdvovrov or €Trrorlllovrrfq ovorotr0qr€rq, Korrcuv rv6oopoBrr6v opr0prlrrrdrv {rr1rp(ov, ro ono(o etor]X0qoov ond rrlvEyyfq Avorotrr1. Aur6 eyrotv(ooe en(o4q rqv reXvoloyrrrl enoX( nouoxotroppdvoupe oqpepo. Or entorr]prov€q Kor or reXvrro( gnoporlv vonpooeyy((ouv rrq r6deg rov ;loOqprotrrtiv ror vo erpoppd(ouv rrq ovo-rolf gerq rouq orcr ro0rlpeprvd npoptrrlporo. Auto( ot er6tro(, U€ rn osr-pd roug, dpXroov vo errrporjv ro poOrlporrrd yro rrlv oro0rprrq rougo[(o.

O etroordg ouirvog pdg npdo<pepe 66o peydleq e(etr(erg oro ;ro0r1-portrd, rou F€ rr1 oerpd roug 0o ouppdtrouv orrl 6rd6oor1 rurv ;ro0r1;ro-rrr6v orov eupti ntrr1Ouopd. H nptiul, rpuorrd, e(vor q erpedprorl rcrrv

372 unolovtor6v.

Page 358: Ο Ταξιδευτής των μαθηματικών

MoOnpor r r<d on6 Pnxov fq

Or neptoo6repot enoyyetrpot(eg poOqgolro( dpyrloov vo onobeXrofv

roug unoloytor€g. Evdr ot iloverltorqptoro( p4Xovtro( €nor(ov n€ptxo-

pe(q ge trg v6eg ouoreu€q oro t6lr1 rrlg 6eroer(og rou nevrlvro Kot ortq

opXdg qg 6eroer(og rou e(tivto, ot poOqportro( nouvo0oov to rerpdtrt

rouq Kt €treyov drt or prlXovdg 6ev pnopodoov vq rdvouv ono6e([etq,

rot or ono6e((etg rirov ourd pc ro orlo(o ooXolo0vtov ot po0rlportxo('

M6vo ot enroujpoveg nou ooXotro0vrot P€ ltr sqopFooUdvo goOqport-

rd ovrrtr{rp0qKov rrl peydtrrl Xprlorg6rrpo ttlv unotroytor6v rot o top€-

og qg oprOprlrtxrig ovdtruoqq €rove 6vo ;rrydtro dtrgo npog to egnpdg'

Kotd q 6tdpreto ttlv retreuro(tlv rptdvro €rtiv, ot neptoo6repor go0r1-

pottxo( npooXtirprloov retrtrd oq Xprloq rov unotroytottitv, ox6po rt ov

notrlo( on6 ouroUg eforotrou0odv vo vtt(lOouv 6tr ro npoypotrrd po04-

pottrd y(vovror pe gol0pr rot Xopr(. 'EXer oprptopqrrlOt( or6po rot 4

ovtitrrlrlrq 6tt ot unotroyror€g 6ev ;rnopo0v vo rdvouv ono6e(etq' To no-

o(yvooro Ottirpqpo tov teoodptlv Xprrrpdrov ovorp€per 611 6e Xperd(o-

vtor neptoo6tepo on6 r€ooepo Xptirporo Ylo vo Xpopor(ooupe oflotov-

6rjnore Xdpq, 6rot 6ote vo Urlv undpXouv nouOevd 60o yetrovtrdg Xtl-

ptq U€ ro (6ro Xpdrgo. To 0edrpqpo npord0qre ro 66roro dvoto ottitvo,

otrId 6ev ono6eftrqre p€Xpr ro 1976, Ve rn Xpnoq rofupdrv unotroyt-

ottirv. H on66er[q rou Oeurpriporoq pdvo ge gotr0pr ror Xopt( 6ev e(vot

Ouvoul en( rou noPdvrog.

0o16oo, undpXouv rt dtrtreq 66o ntreup6g ttlv unoloytott0v nou e(vot

oXerrr6g pe rr1 ou(qrrlori Uoq. flptirov, ot unotroytot€q, orrlv ouo(o, e(-

vor unotroytoux€q prlXovdq 6uo6txtilv oprOgtirv' To ndvro oroug unolo-

yrordg (ertvo0v q ovdyovrot orrlv onoO{reuor1, trlv np6o0eon n rrlv

orpo(peoq 6uo6trtiv oprOptirv. Enopr€vutg, orov nuprlvo orptptitg tt'lv

unotroytordv PP(orovtur oPt0Po(.'OUoq, ro nto orlpovrtrd XopoKrlpiortK6 rov unotroytottitv Yto ro

p€oo dvOprrrno e(vot 6rt o unotroyrorrlg gdg entrp€ntt vo no(ouge pe

opr0po0q. 'EXoupe q6q ovc<p€pet 16oo orlpovrrxd e(vot to notXv(6t pe

opr0poriq, xdrt nou ouXvd nopoptrdnetot ortg ou(rpqostq Yto ro rt t(vot 373

Page 359: Ο Ταξιδευτής των μαθηματικών

ro po0r1trrortrd. Oo16oo, q 6poorqprdrrlro nou npo4ye(rot rqq oopo-p4g evooXdtrqolq rou Oeurpquxori poOqpolrori 1.1€ rrlv on66et(q 0eo-prlpdrurv e(vor ro ouvoproolK6 norXv(6t pe rtq 6vvoteg rov optOptitv.llotrtrdg poOrlpotrr€g ovorotrrirperg €yrvov enet6rl rdnotog poOrlportrdg

6nor(e ;.re oprOpo0g rot forpvtrd nopor4prloe rdrt 6rorpopettr6. O p6-

ooq dvOpurnog iXer rtirpo orq 6td0eor1 rou Ftct onou6o(o prlXovq yto vo

no(er pe roug oprOpo0q. Aev e(vot ontrdrq 6rr €Xoupe €vo rolfrepo pto-

tr0Br ror Xopr(, dXoupe pro prlyovrj nou poq entrp€net vo no(oupre pte

roug oprOpodg, pe rpdnoug nou rlrov o6rivotot Xutp(g ourr1. Mnopofpe

ttirpo vo 0€roupe oroug eouro0q gog epurrrlporo nou orpopotiv opt0-pofg ror perd vo XpqorgonoroUpte rr1 ptqXovrj yto vcl no(oupe, roOti.rgovo(rltodpe rrg onovr4oerg. 'lotrtg ot onovujoetq vo ourorporoOo0v.Aur6 e(vqr dvo ntreov€rtnl.ro Trou 6ev etXe roprd yevtd nprv on6 epdg.M6vo r6po, pe q peydtrq 6td6ooq ttr.tv otrtortiv unoloytottitv, pnope(

o p€oog dvOponog vo epeuvrloer ror vo pretreuloet pe euXdptoro rp6no

rr1 Bootrrl <p0oq rov oprO;rdrv. fro vo rdver ourod rou e(6oug rtg 6peu-veg p' 6vov unotroyror{, o Xprlorqg np€net vc yvopr(et pto ytrtitooo

npoypolrForropo0. Enop€vr,-rg, or vdot dvOpurnot np€ner vo ev0oppdvo-vror vo pro0o(vouv npoypopgorrollo, yro vo pnopo0v vo Xpnotponot-ofv ntrripog out€q rtg roronlqrrtr6g p4Xov6g.

Oo obqyrloer, dpoye, oE rdrr rotvoripyro ourq q eurotp(o, rdoo rov

;roOqporrrc0v, 6oo rol rov prl proOqpottrtirv, vo oololofvrcrt F€ rdrt 16-roto; 'EXouge 116l (noer rqv np6rq peydtrrl ovordtrugrl nou Boo(oqreoro norXv(6r F€ rous unoloytordq - rr1 0eop(o rov rppdrrol.

Korr t r rvrog rov 0€6 KordUoro

Or nsproo6repot dv0ponor dXouv ouvovr4oet to rppdrrol. E(vot ere(vegor novfproprp€q Xpol.lorror6g err6vtg nou (6poupe 6rt ouv6dovrot, pe

rdnoro poyrr6 rp6no, 16oo pe ro po0r1;rorrrd 6oo rot pe rov rotp6.Ito ouyrerprprdvo, ro rppdrrotr e(vor otivOereg yeurprerprr€g ropn6]eg,

3F€ nou 6e perooXrlpori(ovror o€ onld oXrlpotcl clv rtq peye0rjvoupre rot 11

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III

:

lA P I E M O I

6rdoroorl roug Bp(oreror ovdpeoo oe ordporouq oprOgofg. yndpXouv

6t1o poprp€q outrlg r4q ovonruoodpevrlg neptoXr]q: q Oeurp(o rou Xdougoug rpuotr€g entor(peg ror 11 0eurp(o rurv rppdrrotr oro poOrlportrd. Orouv€neteg, 16oo yro rrg rpuorrdg entou]peq, doo xor yro ro go0rlpourd,e(vor repdorreq ror 0o otrId(ouv ro np6oono rov rpuotr6v entorrl;ltirvKor rov po0qporrrtirv poq orouq en6;levoug ouiveg.

Oo e{erdooupe nptirro rrg rpuorr€g entorr]peg. Mo04portrd povr6tro

Xp4otyonotorlvrov e66 ror oui_rveq yro vo neprypd{rouv ror vo npoB}d-tfouv <puorrd rporv6pevo. Or ntrovrlreq rrvodvror oe Ellerntrr6g rpoXr6gytipo on6 tov'Htrro ror or pndtreq rov rovovuirv orolou0o6v nopcBotrr-r6q rpoXr€g otov odpo. H entorr]pr1 npoptrdner pr prydtr4 enrruX(o ourdrq ontrd rporv6pevo. 'Opoq, 6rov ro <porv6pevo y(vovror etrdXroto nronotr0ntroro, ro poOrlporrrd pog povrilo rpo(veror vo onoruyXdvouv.Itor( dpoye, nopd rouq pnecupotroyrrofg 6opurpoprrodq oro0po6q rorrouq un€punotroyror€q, 6ev gnopo0v or ;rerer.opotrdyor vo npoBtr€{rouvov odpro 0o Bp€(er 11 ov 0o dXoupr trrord6o; Ae (qrdpe vo ;rdOoupre rrrorp6 0o rdver €vo Urlvo n ordpo rot pto eB6ogd6o opy6repo. O€lou-pe onl6g vo yvurp((oupe pe rdnoro oryouptd rov rotp6 yro lq en6pe-veg 6uo-rperg nFdp€q. 0o16oo, r; rrov6rrpo ruJV F€uopotr6yov vo npo-ptrdnouv rrg ronrrig rorprriq ouv0rjrrq yro uq en6peveg pro ri 6uo 4p6-peq e(vot, 6nog (6poupe 6tror, ovenoprrlq.

frqr( 6ev ;rnopodpe vo Xprlorponotrlooupe roug unotroyrordg trrogyro vcl npoptr6rfoupe ro Xprlporrorr]pro ror vo y(voupre nlo0oror; Tr y(-v€ror F€ ro qntr6 €pyo r4q neptyporprlq rtrrv 6tvtirv rou vepo6 yfptrr on6roug BpdXoug o'€vo yeipoppo; H olrlOero e(vor 6tr noltrd ouorrlporor4g <pUorlg 6ev pnoporiv vo Trsprypoqotiv pe ro rloorrd po0qporrrd.Kdnore, n(oreuov drt ro pdvo nou dXoupe vo rcdvou1lr e(vot vo enil6-(oupe ro ooor6 ofvolo e(rorioeov ror vo prprloougs rrq rordtrtr4legopXrrdg ouvO{reg ror, re}trd, 0o enrtr0ooupe 6tro ro npoBtrrlporo ou-roU rou e(6ouq. Oprop6vo ond ro ouorr]poro rqq <priorlg nou pelerdperpo(vovror pdtrIov oro0epd ror 11 nope(o rulv yeyov6ttrrv 6e 6toropdo-oErol ono prrp6q ovopotr(eg rord rr1 6tdprero rrlq e{€}r{nq roug. Agndpoupre, yro nopd6rrypo, ulv rpoXrd Tnq fnq. Aev neprp6voupe drt n €FS

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| ( E O A A A I O I 5

ir1 0o {erpriyet on6 r4v rpoXrd rrlq Kor 0o eyroro}e(tper ro rltrtord ori_orqpo rord 14 6tdprero ev6g oerogoU rj rnq €rprl(rlg ev6g 4rpororerou.En[or1g, or (u,vrovo( opyovropo( <po(vovror on6truro ovOerrrro( orrg otr-Ioy€q. Mtrpd (tiro rou (6rou e(6oug, nou yevvrofvror oe 6rorpoperrr6neprBdtrtrov, peyotrrrrvovroq e{oro}ou006v vo 6xouv u1v (6ro yevrrripopqrl. flpdy;rorr, our6q e(vot o rp6nog rqrd rov oro(o trerroupye(ol6rtrr1p4 n qUon.

fro vo ovrrperon(ooupe 6rjorotro ouorrlporo, dnog o rorp6g, ovo_nr6(oge o0volq e(ro6oetr.rv 16oo nohjnloro, riore trrro dpeoq miluor16ev 4rov nporlrd 6uvor{, rr 6ror Xploryonou]0qrov yryovrro(or uno_troyrordg yro vo prnop€ooupre vq ro rrpoo€yy(ooupe. Evrofrorg, noildouor4poro rqg rpdorlg ovrrordrovror enrpdvog ouq ;rryotrdt€p€S npo_ondOerig Foq vo rdvoupe npoptr€{erg. Ot entorqpoveg, }otndv, dpXr_oov vq ovopurrroUvror pr]nurg 6ev unripXouv ouor{poro orrl rp6or1 nouvo e(vor ryyevdlq ooro0{, Trou vct otrtrd(ouv 6qlo6rj nope(o FE rnv ro_poprrp{ oltroyrj ouv04rdrv. Aur6 nou ovordlurfov qrov notrudpr0poouorqporo e{orpeurd euo(o0r1ro otrg opXrr€q ouv0rlreg. Ilpdyporr, er_vol 16oo euo(oOr1to, tiore 6ev e(vqt 6uvor6v vo perprlooups uS opxr_rdg ouvO4req gr enoprr] orp(Beto, nporerp€vou vo rdvouge npoptrd_Qerg. Eroxtrrp{or€ oro xdog. AvrrtropBovdpoore rdrpo 6rr rdnotrg n}eu-p€g rou ofprnovrog e(vor rdoo unepruo(o0r1uq, rou rpo(vetor vo ou_gnepup6povror g'€vov evretrtirg xoortr6 rp6no. Aev €xer oqproo(o n600pehr6vovror ro po04porrrd pog, ro ouorr]poro ourd 0o rororp€pvouvndvro vo otrld(ouv nope(o npoq roreuOdvoerg nou 6ev pnoporjpe vonpopl6rf oupe .

Aev e(vor dprrrq ro ndvro onoyoqreurrrd. AvorolUnrovroq ro Xdog,ovoroldrpope en(or1q 6tr dvo odoqpo nou e(vor euo(o0qro ror letroup_ye( odprptrrvo pe tr(youq p6vo ontroUq vdpoug gnope( vo nopdyer e[o_rppevrrd no}dntroro onoretrdoporo. Enopdvog, o f6rog o nloUrog rrlgg0o4g e(vor onordtreopo qq €goppoyrlg ontrrirv rov6vurv nou erpopp6_(ovror {ovd rot (ovd o'6vo euq(o04ro o6or4pro. To tetrtxd onordtreopro6ev pnope( vo npoBtrerp0e( on6 r4v ontr6r4ro rov rov6vtrrv Kcrr €pn€_

EF* pr6xer pto on(orrurq nolunlor6r4ror rrou 6ev r4 ptr€noupe otoug ev 16-

Page 362: Ο Ταξιδευτής των μαθηματικών

yo rov6veq. Aur6 €Xtr ouv€nereq yto ro ndvro, on6 ro ntig peyotrtirvouvor opyovropo( ge Bdorl ontro0g yevertrodg r66rreg og ro nrirg trerroup-ye( q e0vrrq orrovop(o. H ovordlu$r1 rou Xdoug Kor rov euo(oOrpovouorqpdrov qrov ro dpeoo onor€treopo rrlg npoondOerog povretrono(-qoqg rdrorov ouorrlpdrov pe unotroyrordq.

'Otro 6oo dXoupe ner g€Xpt rdrpo qo(veror vo ouv6dovrclr [r€ roerpoptrroopdvo poOqporrrd. tkirg pnoporiv or unotroyror€q Kor ro ouorrl-gorq rrlq <p6or1g vo enrlped(ouv tov ro0op6, notrugo0rl r6ogo rov 0e-oprlrrrc0v poOrlporrr6v; Ilporole( €rntr4(r1 ro 6rr o rdopog Foq ruJV Ko-0otrrrdv €vvouitv, 6nog errppd(eror fr€ To o0or4po opr0prirv, neprfXer ro6tr6 rou ydog. Orov oXe6rd(oupr pro errdvo outo0 rou Xdouq orovuno)\oytorq, 6rlproupyorjpe €vo rppdrrotr. il69 gnope( vo ouprBofver ou-16; tl6q pnope( q orptpqq poOrlporrrrj opXrj rurv opr0;rdrv vo neprdXetetx6veg Xdoug; To vo Buiroouge dpeoo ro Xdog nou wundpXer orougopr0po0g e(vor oov vo rorrd(ou;re rov Oe6 rordporo.

E(vor 16oo on}6 vo 6qproupyrlooup€ orov unotroyrorrl gro ovonopd-orooq ev6g po04gortrod rppdxrotr xor vo 6o6pe our6 ro Xdog, t6ore pno-pe( vo ro rdver o xoOdvog nou xorovoe( oe pdOog u1v dlyeBpo n 6Xetoro on(rr tou €vov unotroytorri.'Ag ndpouge, yro nopd6erypo, ro oltr6notrutirvul,to x' + 8 = 0. H e{(ouroq ourrj nepr€Xet dnrtpo ntrr10oq notrdntro-rov tremogepeuiv, rrou 6w e(Xope nor€ rpovroore( nprv ond ro retreuro(oe(roor Xp6vro. Arpo6 to nolutirvutrro e(vor rp(rou po0po0, €Xer rpetg 16-o€rq, lq xl, x2 Kcn x,. M(o on6 lg tr0oerg e(vor o prp6g oprOpdq -2. Avovurotoor{oouFs ro xr F€ ro -2, iXoupe: (-z)' * a = -8 + 8 = 0. Or dtrtreg66o l6orrg e(vor ot pryobrro( opr0go( x2 - 1 + y'ji ror Xi = 1 - i/f i. nurdqnq rp€rq trfoerg pnopodpe vo nS ovorcrpoorqooupe yporptrd oro en(ne-6o rov ptyo6rrtiv opr0puirv, dntr-rg qo(verot orqv Err6vo 54.

Mto 6r1porpilrlg pd0o6og enfiuorlq nolurrrv0purrv e(vor 11 enovolqnrr-rrl npoo€yyror1. fto vo Xprlorponor{ooupe ourrjv rr1 pd0o6o opi(ouge6vov unotroytorrr6 oly6pr0pro, €ror drore drov errd(ouge rr1 tr0or1 ror

7. Roger T. Stevens, Fractal: Programming in Turbo Pascal (P6vryouvt I(1, M&T E€€Pub l ish ing , te90) . Sgas

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t ( E o A A A t O l 5

ElKONA54. Or rperg trrioerg rou x'+ 8 = 0 €Xouv ovonopooro0e(yporprrd oro en(ne-6o rou frdouq (( en(ne6o trlv yryo6rrtirv opr0pdrv) oq xl, x2 ror xr. En(oqq, iXouvovonopooto0e( ypogrrd or npdrteg r€ooeprg ugdq (r,, E2, E3 Ko( Er) nou no(pvoupe6rov Xprlorpronorodpre rr1 pr€0o6o rou Ne0tovq yro vo pporipre ro xr.

ronoOerotipe our{ r4v erroo(o otov oly6pr0po, ro onori}eopo vo e(vorpro rol0rep4 Erroo(o. 'Enerro ypqorponoto0;re ourrj rrlv rolUrepq erro-ofo otov otry6pr0go, yro vo ndpoupe prro 6rUrepr1, orprBdorepq errco-o(o. IuveX(ouU€ pr our6 tov rpdno, 6onou vo nerorofpe 6rr 11 ond-vrqo4 poq pp(oretor rnoprtirq rovrd orrlv npoyportr4 tr0or1. E(vor ev-6rorp6pov vo enroqprdvoupe drr dnoroq otry6pr0poq Xpqorgonore(rorotrlv enovdtr4{q 6Xer dvo e(6og evoopotop€vnq opxiq ouroovorpopdq,rrlv oro(o ouvovr4oope drov ou(qrodoop€ yro ro nopd6o{o. H tr0oqrou otrydpr0pou e(vut pro dtrtrq trUor1 rou otry6pr0pou ror ofro ro0e{r19.Enop€vtug, 11 enovdtrq{4 ovortrdrot nd}r orov eour6 rr1g.

378 'Evo e(6og enovotrrlnlrr]g pe066ou ovom6X0rlre ond tov loodx

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I

ii?

A P I O M O I

Nedrovo (1642-1t2t), nfiov6v ro onou6or6T€po rTuor4povo o}trd rorpo0qpottr6 nou €(qoe not€. Ie ourrjv t4 p60o6o iXoupr tov nopord_rul olydpr0go yro vo enovotroppdvoupe ouveXtirq ug }doerq Foq yro rox' + 8 = 0, dnou q rd0e etroo(o oupBoh(eror urg E.

r, = r, -Ei' J-3 E, '

Aq (ertvr]ooup€ F€ ulv npdrq ert(pqo4 Uog nou tooUror p€ ro pryo_6116 opr0p6 4 + 4i. Aurd to E, €Xer ovonopooroOe( ypogrrd orqv Err6vo54. 'Orov ovonoprotorige ypogtrd roug pryo6rro0q opr0porjq Foq rousouppotri(oupe oq 6toreroyg€vo (edyq, 64lo6rj wg (+,+) yro rov 4 + 4i.Ttitpo, ogoti ro E, Bp(orceror nro rovrd or4 tr6oq x, pnoporipe vo uno06_ooutrre 6n, (envtirwoq ond ourq rrlv err(pqoq, 0o rorolr]{oup€ vo lto-pdyoupe 6ro6oXrriq erroo(eq nou ntrrlord(ouv dlo ror neproodr€po oulxr. Kor, npdyporr, our6 orprpdrq ougBo(ver. Avrtro0rordvros ro 4 + 4iorov nopondvtrr oly6pr0po, no(pvoupe 6rr E, = 2,67 + 2,75i, ro ono(o6Xet eniorlg ovonopooroOe( yporprrd orqv Err6vo 54. Awrro0rordwoqlorndv ro E2 orov oly6pt0po no(pvoupe ro E3, To ono(o pe rq oerpd rounopdyer ro Er. Ttirpo pptor6;.roote notr6 rovrd or4 ooorrj lfoq t + y'J;,q ono(o roodror nep(nou p€.1 + 1,73i. Mtypt rtipo, dtro trerroupyoriv6ntr;g nepr;rdvope. Xprlorponotdrvroq q p60o6o tou Nrrjrovo, (ertvdpepe pro erroo(o nou pp(oreror rowd orn x2 Kor rorotr4youpe o€ Fro oKo_louO(o ertrprloecuv nou ouyrcI(ver orn xr.

Oo prnopodoops vo uno06ooupe 6rr rd0e rpopd nou I npdrr4 ercr(_Fnori Foq, 4 E,, pp(orerot nto rovrd os Frcl ouyrerprpdv4 tr6o4, rr1 x,r6re 11 pet€nerro orotrouO(o errrpu]oeov 0o ouyrl(ver oe ourqv rr1 x,. HErrdvo 55 Oeryver ro en(ne6o rov ptyo6trdrv opt0ptirv, ro ono(o €Xet 6r_orpeOe( pe dwoveg ypopp€S o€ rpsrq neproXdq yUpur on6 lg tpsrq tr6_oerg. Oo uno0€ooupe 6rt 6treg or nprireq eruprloerg y0pur ond rr1 x, 0oouyrtr(vouv orn x2, ror 6rr ro (6ro 0o oupBe( pe rrq dtrtreq 6fo neproXig _6u or np6req errrpqorrg 0o ouyrtr(vouv ndvro orrlv rovrrvdreprl trUo4.Oo ;.rnoporioe vo undp{rr €vo p6vo epdrr4po doov orpopo oro oqpe(onou Bp(orowor oro ev6rdpeoo orprprirg rcr_rv 6rio ltiorurv. Aur6 ro eo6_ EF*

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K E O A A A I O I 5

oqpe(o nou ouyrl(vouv

. o-n *'

orlpe(o nou ouyrl(vouvoTl x]

onui,o nou ouyxl(vouvorl x3

EIKONA 55. To pryo6rrd en(ne6o dXrr 6rorpe0r( os rpsrs (oeg neproX6g. 'Orov Xpqor-

gonoro0ge tr1 p60o6o enfiuorlc rou Ne6rulvo, q troyrrrj un60eoq e(vor 6tr exe(vo rooqpreio nou pp(orovtor nro rovtd oro orlpe(o rd0e tr0or1q ouyxtr(vouv oe our6 ro o4-geio. Iuyrtr(vouv 6gurg;

rrlpo €0eoov otov T(ov Xrorignopvr ro 1976 or nporoers(q rporqr€goto pdO4go rou on€rpoortro0 }oyropo0.u fltrqord(ouv ndvru or opXrriqerroo(eg npoq qv rovlv6reprl tr0or1; Aev rjrov o(youpoq yro rqv ond-wqon Kor toug e(ne 611 0o touq onowo0oe rrlv endpev4 ep6opd6o.

O Xto6pnopvr 6rv pn6peoe vo Bper pro rord}trqtrrl ondvrrlorl oe proep6opd6o. 'Enpene vo nder otov unotroyrorrl tou ror vo opX(oet vo no(-(er. To norXv(6r rou rov o6rlyqoe orqv ovoxdtrurp( rou drr rdrr notrd no-pd(evo ouvdporve oro ;rryo6rr6 En(nr6o. H opXrrq un60eor1 yro rrq €r-

8. lames Gleick, Chaos: Making a New Science (N€o ydprrl, Viking penguin,38G $s7\ . 'o .217.

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A P I O M O I

Koorrq rou ouyrl(vouv ot4v rovtrv6rep4 trrio4 iByo(e vdrltrro ror r]rovloytrrl' otrid 'rov en(orlg evter6q trov'oop€v4. aefre tr ouppo(ver orrlvErr6vo 56, 6rov nopouord(oupr rqv nqy{ 6}trrv trrrv nputrov eruprloe_urv nou o64yotiv ot4 x, l Aev no(pvoupe €vov (erdOopo 6roXurprop6rtrrv 6rogoperrr6v trUoeov, no(pvoupe pro nofuj nep(nlorc4 rrr6vo' _no(pvoupe dvo rppdrrol. Tr ouppo(vet ov ndpoupe trrro prrp{ neproX4nolfntrorov lenropepertirv rcor 14 pryeOdvoupe; H lenropdpero 6ev (e_

EIKONA 56' H ef€roor1 rov oqpre(trrv oro pryo6rxd en(ne60 pe 14 p€0060 rou Ne6rul-vo nopdyer pro rrrdvo rppdrrol' ot poripeg neprox€g rrrq €rKovoq eivor exeiveq ornptireg exlpfo€rq U€ rrl p60o6o tou Nefrovo, o, or,olq o64yoriv or4 x, 1Oe ,pofve_tor). Or <ionpeq neproX€g r(vor.or extr;ujoerq nou o6qyotiv orq x2 rj orq x,. Melerdr_vros .q po'peq neproxiq 6e(r<i on6 rov rd'ero ri{ovo, pnopef vo ovoprrrrrl'0ripe:ytoti or np6req er<rtpr]oerg, nou e(vor notr6 nro rovrd e(re orq x, ritr;_n _;;n_Youv oIn xr; "! !r / !r / vv'r-

B€E

Page 367: Ο Ταξιδευτής των μαθηματικών

roOop((er, otrtrd nopopdv€r oq e(Xe (Err6vo 57). Aqtro6r1, ove(dprqro

on6 ro n6oo 0o peyeO0voupe rrlv etx6vo ovdgeoo otrg 6tdrpopeq tr6-o€rq Trou rpo(vovror pe trsnrop6peteg, 6e 0o 6Xouge nor€ 6vov (erd0opo

ror ontrd 6roXoprop6. Or ntro0oreg lenrop6peteq rou rppdxtotr ouveXi(ouv vo undpyouv, yto ndvro!

Err6g on6 rov Xro6pnopvr, rr dtrtrot dXouv epeuvqoet trg nopd{eveg

err6vrg rou 1.rryo6rro0 entn€6ou. Or nptiror rlou ro 6ruvov rpov 60ofd}tror go0rlgorrro(, or froor6v Zttrtd ror fltcp Ooto6, ot ono(ot Enpenevo rdvouv roug unotroytopoUq toug Xurp(g rq Borj0eto unotroytorrl, rotd14 6rdprero rou A' floyr6oprrou flotr€pou. A]]d ro €pyo toug, rot ot

enor6}ou0eg novdgop<peq err6vtq nou ovopd(ovtot otjvotrq rou Ziltd,

oe peydtro po0p6 oyvor10r1rov. Irrl 6eroet(o rou 19l0, o Mnevoud Md-

wetrprnpot dpXtoe vo prelerd ro o6votro tou Zilrd Kot, ro 1979, ovar6-tru{re rov nonno0 rov <ppdrrotr rou ptyo6rroti enrn66ou, to o0votro Md-vrelpnpor.'

To o[rooqpre(rlro oXertrd pe ug errdveq <ppdrtotr, 6nurq ourrl orqvErr6vo 56, e(vor ro 611 6t orqp((ovror Fs rov€vov rp6no endvcrl oro Xd-oq rou ouordgorog rou npoyportrod r6ogou. Ot p6veg fvvoteq nou

Xprlorponoro0vrot dpXovror oneuOe(og on6 ro Oeop4rtrd po0rlpottrd.

Iqv Etr6vo 56 Btrdnoupre €vo 0epetrrdr6€q XqpoKrlprorrrd tou ouorrlpo-roq rov opt0pdrv poq. l-ror( undpXer 16o11 lenropr€pero; Kd0e notrutitvu-po pe Bo0p6 66o r1 geyolUrepo 0o nopdyer pto etr6vo rppdrrotr Xprlor-ponouivrog rrlv (6ro yevtr{ reXvtrrj. Xp4otponouioope ro x' + 8 : 0 p6-

vo ener6rj e(vor 16oo onl6. Av 6youge erpedper 6leq rtq t6ieg nou oXsr(-(ovror pe ro po0rlprorrrd, pe notov tp6no eqedpop€ rrlv ovonopdoto-orl rnq Err6vog 56;

Mro eprnerp(o, nou 6e 6torp€per on6 ro vo oxotig 1-tto onou6o(o oult-rptuv(o, e(vor ro vo rd0eoor pnpoord orqv o06vr1 rou unotroyror{ rot vo

Btrfnerg vo (r.uvrove6er Uro Xpoportoul ovonopdoroo4 ev6g noluovul.tt-ro6 rppdrrotr. Evd errul(ooovror or trenropdpeteg, ivo qtrexrptrd <popr(o6ronepvd ro otirl.ro pog roOtirg ouvsr6rlronorofpe 6rt rotrd(oupre, p€oo

38? 9.'O.f i . , o. 222

Page 368: Ο Ταξιδευτής των μαθηματικών

EIKONA 52. H preyi0uvo4 tc

;frJ:.I?:i*#**:#'[-i1]J'.xJffi[::':"u '�o x'+ B = 0 pos 6(ve

6o d pr t'v ovonopoyuryrr raklffffi ;; ;:ffi #,yff

-J*ff;f i;'l;

It

" , 1

ss3

Page 369: Ο Ταξιδευτής των μαθηματικών

on6 xdroro peyd}4 oxtopli, to utrrrd qS (CIrlq yro vo otev(ooupe rrlvouo(o trlq dnop{r1E. To rppdrrotr e(vor dvo ox6po nopd6€rypo 1uoq ovo-rdtru$qq nou rdver rouq po0qpotrro6q vo nrote0ouv or4v on6trurqdnop[q rurv po0rlpour6v cvurerpivrrrv. Ito e{orperrr6 prptr(o rou Xriog;

Ar1lttoupytitvraq pn v€a cmotr1pr1, o T(irpq f rtrirr onorotrpd vo ne r:

To o6votro Mdvtetrpnpot [...] un<ip1er. YnrjpXe nporoO [...] o Xtodpnopvrror o Dauady rototrdBouv q poOqpottrr] ouo(o rou, nprv ordpo roovorolUQer o Mdvtetrpnpot. YnripXe on6 q otrypri nou 11 enro{pq 64-ptotipyqoe dvq nlo(oto - pto 6opuj ;uyo6trtiv opr0ptirv ror pro ovtilqrlrrlyro rtq enovoloypovdpeveg trerroupy(eq. Metd nep(geve vo onoroluqtei.'lor.uq dpoq ror vo unrlpXe or6go vurp(repo, ond tq otry;rr] nou n qdorldpXroe vo outoopyovtirveror ll€ rrl por]0ero onl6v rpuor(rirv vdpru-rv, orono(or enovoloprpdvovrov ge ot€treurrtr1 unoprovrl Kor F€ rov (6ro tp6nonqvto0.to

To reAeurofo Oec0pnpo rou Oeppd

flptv retreuirooups p€ trq enrruX(eg rou aroorod ouirvo, 6ev pnopo0pevo pqv ovorpepoupe 6vo np6ogoto oupBdv, to ono(o pnope( vo Oeo-pq0e( trlq €vo ond ro oqpovrrrdrepo enrredyporo rov goOqporrrdrv.Oupdore 6n or nu0oy6perot optOpo( e(vor rprd6eq or€pourrv oprOgtirvrrou rKovororo0v tqv e{(ouroq: o'* F' = y'. =ipoupe drt undpXar dnerponlrl0oq rdrorurv rprd6cov ror dl r]tov yvurordg otoug opXo(ouq Boputrdr-vtoug. Mro guorr{ en€rroorl tov nuOoy6peurrv opr0pdrv e(vor vq ovq-poqOo0pe ov undpXouv onororO{note rp€rq guorKo( opr0po( Trou orr0por toug vo ouv66ovror pe rq oX€or1: o'+ F'= y'.

O flrep we Oeppd yewrj0qre ro 1601 oro Mnog6v vre Aoptiv, or4fotrtr(o. Kdnoror 0etrrporiv 6tr {rov o ro}dtspoq poOrlporrr6q rou 6iro-tou €p6opou ouivo ror ofuoupo rlrov €voq o16 rouq onou6ordrepougtpoorrdXveg po0rlporrro0g rou r6otrrou, npoorp€povrog o4 govttrd 6pyo

384 10.'O.n., o. 239.

Page 370: Ο Ταξιδευτής των μαθηματικών

II5

Iili

I:I

j

oe 6,rr orpopd q Oepetr(oo4 rou orr€rpoorrrod troytopof, erpsup(oro_vroq rrlv ovotrurrr4 yeoperp(o ovefdprqto ond rov Nrerdpr rqr 0€ro_vros rcl 0e pdtrro rrlg odyXpovrlq Oerr;p(og rt.rv opr0pc0v.,,

O Oeppd, ro i637, svdr petrnorioe 6vo ovr(ypoqo rou 6e6repou pr_ptr(ou r,v Apr9prlurdtv rou At6<povrou, orlpr(,oe oro nepr0rirpro roo16lou0o re(pevo:

Avn06rurq, eivor o6dvorov vo Xurp(ooupe 6vov rdpo oe 6rio rdpoug, pror€roprq 6dvogrl oe 6do r€topreg buvdperg r1, yevrrd, onoro6lnou 6U_voprl novo ono ro 2 oe 6Uo 6uvdgerg rou (6rou poOpo0: €yur ovorotrri_{er pro oArl0rvd un€po;1 ond6er(r1, otrtrd ourd to nepr0tirpro e(vor unep_Botrrrci otvo yro vo rl Xoprocr.' '

E6rir o Oeppd ondvr4oe oe no}6 neproo6repo on6 ro epdrr4gd pog.OXr p6vo s(vor o6rivorov vo ppod;r€ rp€(q rpuorroUg opr0po6g rou votrqvororo0v rr1 oX€o4 o' * g' = y', otrId e(vor o60vorov vo BpoUperperg opr0podq nou vo rKovororodv q oX6o4 o" + 0" - y" 6rov ro v E(_vor peyolfrepo on6 ro 2. Aur6g o on}6g roXuptoprdg e(vor yvoor6q ctrgnro retrguro(o 0e6pr1po rou Oeppdo. O Oeppd, 6prog, 6ev ror€ypo{renor€ rrlv on66e r(4 rou Kcrr nollo( 0eu-rpo0v 6rr eorpotrp€vo n(oreue nogrrlv erXe Bpe r, yror( ro epyole(o nou 4rov onopo(rqro yro ulv ond6er{4rou 0roprlpor6g rou 6ev rlrov 6ro0dotgo ro 16j1 . Oot6oo, noroq Uno_pef vo e(vor oryoupog; Aurd nou f6poupre E(vor drr on6 ro 163/ o( ro'l 993, pro nep(o6o 356 ertirv, rovdvoq 6ev prn6peoe vo ono6e({et ro re_treuro(o Oetirpqpo rou Oepprd - ror npoond0rloov o;rdrprpor enoyyel_Forisq Kor epoorr€Xvrg poOqporrro(. Mnop6 vo Oupr10rir 6rr, orq rporrq_rrlg oro lloventoull.rro rrlg ftodro, or roOrlyrlriq frou rov proOrlporrrdrvgdg npoer6ono(r1oov vo pq Xdvoupe rov rorp6 pog npoono0dvrog voono6e({oupre ro retreurq(o Oerirprlpo rou Oeppd, yror( oe ndpo notr}odgrporrrlr6g erXe y(vet dppovrl 16€o, 6uorotredovrdg roug dror orrg onou66groug. Aev oro6oope 6tror ourrl rrlv npoer6ono(r1or1.

11. Bell, Men of Mathematics, o. 5t.12 . 'Or t . , o . 71 . ES5

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Kord tq 6rdprero rpr6v qpeptirv, on6 rrq 21 cuq rrg 23 louv(ou 1993,

6voq vrponotr6g Ayytrog po0rlpolr6g ond ro flovenrorrlpro rou llp(v-orov, o Avrptou loudrlg, nopouo(oor tpetg opro(eg 6rol€(erg ge 0€po:olXrjporo p€rpou, etrtretnrrrdg ropndtreg Kor ovonopoordoerg rou fro-troudo." An6 qv npdrq p€po, ro oxpoor4pro gdvrerpt npoq ro nod66eue o foudrtrg. ho r€Ioq rrlg rp(rrlq 6rdtre(qq, o ioudilq dypoge orovn(voro ro releuro(o Oedrprlpro rou Oeppd Kor ro oKpoorrlpro (6onooe

oe 0uetrtrti6r1 Xerporpor4poro. Tetrrrd, dorepo on6 356 Xp6vro, rdnoroge(Xe nopouordoer pro ond6er(q. Oo ovrd(er 6poq n on66er{r] Tou orouorrlp4 e(droo4; Oo np€ner vo nepr;r€voup€ yro vo to 6tonlor6oou-pe, yror( o foudrlg 6ev €Xet or6po 64poorefoet rrlv on66er{r1 rou o€rdnoto enrotrlpovrr6 nepro6rr6, 6ore vo pnopoUv rr dtrIor poOqpouro(vo e(erdoouv rr1 6outrerd rou. Ogtrrg, ere(vor nou noporolouOo0v rrg6rotr€(erg rou e(vor nenerog€vor drr ro rordrpept. Av s(vot 6ror ro npdy-pcrrcl, o Avtprou loudttrg €Xet e(oorpol(oer 14 0€or1 rou oro roroprrd pr-

Btr(o, ener6rl €truor €vo on6 ro 6uorol6repo npoBtr{poro rov po0qpo-rtrtiv, evtir 11 on66er(r1 rou 0o 0eopr10e( 6vo on6 ro e(orperrrd po0rlpo-

rrrd enrreriyporo rou etroorof ouirvo.

f l6oo pnopoUp€ vo npoxurpr iooup€;

T6po nou €Xoupe roug 6rro09 pog unotroytor€g rot 6rrd pog BrBtr(o po-0rlpolrtiv, n6oo prnopofpre vo npoXtrtptlooupe ovorolUnrovtog v6opoOqprorrrd; Av o eyr€rpo}6q Uoq e(vot rdroroq nou 11 p€or1 eurputo e(vornep(nou eror6 orqv rtr(poro rou 6e(rrrl voqgoodvrlq, ndoo porprd prno-pe( vo gog o6rlyrloer ourrl q dp<purr1 eurpuio prog, ro0drq ovo(4rofpe6tro ror nro nep(reXvo poOrlporrrd; BeBo(og, ov poq enrordnrovroverr(vor or e{oyrlrvor on6 ro rdvrpo rou lotrol(o rqr ono6errvu6rov 6rre(Xov 6e(rreg vo4poofv4g y6prrr oro 66o Xrtrtdbeg, pnope( vo vuiOope

13. Michael D. Lemonick, ,<Fini to Fermat's Last Theoremo, I ime (5 loutr(ou3 8 6 B % \ , a . 4 7 .

Page 372: Ο Ταξιδευτής των μαθηματικών

noq oneilo0poore rdnoq on6 qv e{orpeurrl outr}oyrorrrrj rrovdrrlrdrouq. Mnopofpe, dpoye, vo totrpriooupe vo nrotirpoupe dtr 0o e(Xopeqv rrov6qro vo Korovorjooupe ro go04potrrd rouq; Mrjnurq 0o vrdl-Ooge 6nrrlq vrdrOouv ro ror6noulo q ro orutrrd orodyovroq Qtotr€(erqyto q 0eop(o rou orr€rpoouroti troyropori, ener6r] 6e 0o rorotropo(voper(noro n€po qn6 to 6rr rdnoroq trroq pilder; YndpXer rdnoro 6pro 6oovorpopd oro no0 pnoporipe vo rprdooupe, onl6q ror p6vo }dytrr qq ne-proprour{q 6op{g rou eyrerpdtrou pog;

Ag unoO€ooupe 6u rdnolo pdpo (erler6tirvoups dtro to rtrer6rd touyevetrroU poq 166rro. Ag uno06ooupe, enrntriov, 6rt pnopofpe vo X€rpr-otodpc outd tov rdr6rro €rot 6ote vo 6tirooupe oro nor6rd Uoq ro €Tu-0upqrd, npoypol.lponop€vo Xoporcrrlprourd. Oo pnopofoope, 16r€, vonpoypoppot(ooupe epnpo0drog ro nor6rd poq c0ore vo dXouv 6e(rrrlvorlpoodvqq Orordoro, r] tnpordoro, { or6go ror 6rjo Xilrd6eq; Tr e(-6oug goOqpotrrd 0q ovoxdtrunrov;'O,rt rr ov {rov, 6oor ond epdq ovrj-Koup€ ornv nolord yrvrd e(vor nr0ov6v vo Frlv to roroloBo(vope not€.Oo [6pope, 6;.u.,r9, 6u oorpoldrg 0o {rov evrunoorord ror nep(reXvo.

387

Page 373: Ο Ταξιδευτής των μαθηματικών

dparag. opXo(o unotroyrortr{ ouoreur] nou onoretre(rot on6 otporp(6ro nepoop6vooe pdp6oug, ot ono(eq pp(orovror rono0er4p6veg p€oo o,ivo {dtrrvo ntro(oro. Orunotroyropo( y(vovrot gerorrvcirvrog ro orporp(6ro rord puirog rov pdp6trrv.

a)yeppu6g ap0p69. opr0p6q nou e(vor trrjor1 evdg notruovripou, 6tror or ouwrtreordqtou ono(ou e(vor pqro( oprOpro(.

ak{ov6ptv6 oiorqpa op01ttiu. ro etrlrlvrrd o0otrlpo opiOptrlorlq nou Boo((eror oee(roor enrd 6ro<popetrrd opr0prltrrd q,qq(o Kor Xpqorponou]0qre rup(og prtri ro100 n.X. Ovopd(etor enioqq u,-rvrr6 o0orr1po.

arEporoq apfip6g. onoroo6{nore opt0p6q rou ouvdtrou rurv oprOpdrv nou onoretreGrot on6 toug 0etrroriq Kot rouq opvrltrro0q rpuotrodq opr0go0g ouv ro pr16iv.

oro)ou9ia @qnovdrot: oxotrou0(o optOgdrv nou opX((et pe ro 1, 6nou rdOe 6ro6o1_K6q 6poq eivor to dOporopo tu.rv 66o nporlyofpevuiv 6prrrv. Or npdrtor enrd 6porr4g oroAouo(oq e(vor or: i , 1, 2, 3, 5, 8, i 3,. . .

dpeo4 errQ4o4: q dpeorl en(yvtrlorl. rou ntrrjOoug tov ororye(rrlv cv6g ouv6trou ovl-rrrp6vov.

ova)unrri yeupetpfu: q yrurpetp(o orrlv ono(o or 06oerg onerrov((ovror pe opr0prltt-r€g ouvtetoypdveq ror e<poppd(ovror or otryepprr€g pd0o6or.

avanogdootry: prdoo oe 6vo tunrr6 ;"ro0qpolr6 r] loyrr6 orjorqgo, gro npdtoorlnou 6ev pnope( vo ono6erXre( r] vo grlv ono6eryre( pe pdo4 ro o(rc0goto rououorr]protog.

a6poteg e{o6oeq. e[rodroerg prr dnerpo oprOpd tr0oeurv, dnoq, yro nopd6erygo, r1e[(ooo4 x + f - 7. Autdq or e{rotiroerg e(vor Xp4orpreg yto rq petr€rq noltrtiv rpuor-r6v ouor4pdrcov.

anttr6vro4 Evo npoq iva (aprptltovoo4povt4 antwdvto4): ercXdrpqo4 ev6g ror p6vo 3SS

Page 374: Ο Ταξιδευτής των μαθηματικών

r A o f f A P t o

ev6q orotle(ou ev6q ouv6lou (n.X. turv oprOprrltrrdrv tri{eulv) oe rd0e orotXe(oevdq 6edrepou ouvdtrou (n.X. rtrrv 6oXrrltrov).

dnetpo: yup( qpdypo, pq rppoypr€vo, gr1 nenepoo;"r€vo. Irqv nep(nrooq ev6q ouvd-trou, ro oOvotro pnope( vo onerrovrore( en( ev6g yvrlo(ou unoouvdtrou tou, 6nog,yro nopdOetyyo, or guorro( oprOpo(, nou onerrov((owor eni rov tetpoyrirvrrrvrov quorr6v opt0gtirv.

arAd 6nrerayytva. tvo o0volo oprOprirv e(vor qntrd 6toteroyp€vo 6rov yto onorouo-6r]nore rperq oprOgorig x, y, z rou ouvdtrou roXtiouv or nopor<ito 60o ouvOrjreg:(1) x =y r i x <y { y < x Kor (z) ov x <y Ko( y < z r6te x < z. Or npoypotrro(opr0po( e(vor onld 6roretoypCvor, evtir or pryo6rxo( oprOpo( 6ev e(vor.

on66e(r:l: gro oxotrou0(o loyrrtirv prlpr<irov nou ono6errv0et qv otrr]0ero ev6q ou-pnepdoyotoq pe pdo4 €vo orivolo o(rurprdtu_rv. Ot npdrreq ntrrjprrrq rotovoqr€gono6e({erq ovomdX0qrov on6 rouq'Etrtrr1veq, yro vo ono6e(ouv xdnoreq yeope-rprr€q oX6oerg.

an6r)o4: q rordorooq gtog dnerprlq orotrou0(og opr0ptirv ri opr0pqlxrlg oerpdq 11ono(o 6ev 6Xer rpp<lypo rj 6pro.

an6Aut4 ng{: 4 Oenrq ugri evdg oprOprori ove{dptrlto on6 to opXrxd npdoqpro touopr0pro6.

an6)uro dnetpo: ovt6vlto q ono(o rout((eror pe ro dOporopo dtrov rrrrv one(purv,peprr€q rpop€q ouoXer((eror pe tov Oed ror ougpotr(erol U€ ro o. Mro €vvoronou 6ev pnope( vo y(ver rotovorltrj loyrxd. olld pdvo puonrrotrrd.

aftp1orroe unepnnepooptvog apfip6q. unepnenepoopdvoq opr0pr6q nou 6ev pno-pe( vo oprore( pe pdorl purpdrepouq unepnenepoop6vouq oprOpodq.

apfipr1o4: erlpeorl rou ntrrlOrrori opr0prori evdq ouv6trou.ap(0yr1o4 pe rAa6d: Evo e(6og op(0prr1o4q nou 6ev onorre( rr1 Xpr]o4 ytrtiooog. To

ovlre(pevo pe to ono(o perpdpe, rlo6rd r] Xol(rro yro nopd6erypo, onerrov((o-vror opqrpovooripovro eni rou ouv6trou tov ovurerp€vov nou perptirvror.

ap(0p4o4 pe ta 6dXtuAo: 11 lp4orgono(qoq rulv 6oXr0l0iv yro rrlv onerr6vro4 en(xdnorou ouv6lou ovrtret;"rdvtr_rv nou np6rertor vo pretpq0e(.

ap(01t4o4 pt ro otitlto: pro en€rroorl qg opfOprloqg pre to 6d;rutro, 6nou 6to<pope-trrd pdprl rou odrprotoq ovonoprororiv rpuotrodq oprOpro0q.

ap6p4otpo oivo)o: ivo onerpoo0votro nou prnope( vo te0ei oe opuprprovoor]povqonerx6vroq F€ rouq quorKorig oprOpo0q.

ap0pqurd (t49fu kopndp: to oprOprrltrrd rlrrlrp(o nou Xpqorponorodvrov on6 touqApopeg tov 90 ordrvo g.X. ror onorelo0oov geroyporp€q ru_rv opXor6tepov rv6o-ppoXpovrrdrv opr0prpxrirv rf qq(urv.

ap6p4trx( aro)ou1fu: €vo odvotro oprOpdrv: A1, A2, A3,... A",... oXrlprou(er Uro or(o-louO(o oprOptirv ov or oprOpro( e(vor rotrd 6roreroyp€vor, 6qtro6rj ov or unore(gr-vor 6e(rteg orotrou0o0v tr1 oerpd turv guorrtirv oprOprdrv.

ap0p4nrfi avd)uoq: I Xp{on unotroyrottirv ror unotroyrotrrdrv otryop(0ptrtv yto r4vnpoo6yyroq qq l0oqq odv0ertrrv npoplrlgdrov.

ap0p4nni oetpd: to outrtroyrr6 d0poropo ev6q ouv6trou oprOpdrv.

1j

390

Page 375: Ο Ταξιδευτής των μαθηματικών

Ii.

r A o f f , A P l o

ap&pr1tu| r!4qb:to ypont6 odppotro TIou ovofloptotd €vov opt0p6 11 ovonoptorq

ro rlrryp(o ev6g oPt0Poti.

aptllto( Atouph: uneppottro( opt0po( qg gopqrlg:

q,/10' ' + orl lo' t +o,/10'r + . . '

dnou ro o e(vot oxCpotot opr0po( pe ttp€q 0T16 0 €oq 9'

apt0y6E-etrxtm: opt0pr6g nou Xprlorponoteitot oq 0ior1 ev6g ov6ptotoq'

ip,eyAS tuv apillttitv (number-number): o 6ero6tr6q op0pdq nou oXqpot(etot ypdgo-

wog 6to6oXrrd 6loug roug rpuorroriq opt0po09 yro nopd6erygo 0,12345678910111213"'

dpp4tog apr0trt6E opt0g6g qg eu0e(og ruJV TlpoYpotlKdrv opt0pdrv nou 6ev pnopa(

vo errppoore( rrrg tr6yog 6tio ordporov opt0gtitv'

apy4: to oqpe(o trlg euOe(oq tov opr0ptirv nou ouvO€etot pe tov opt0p6 0 q pe to

oqge(o tou lttyo6tro0 entn66ou dnou tdpvovtot ot 60o d{oveq'

aoitrpttpa,, dtov 6rlo yey€0q 6ev pnopo0v vo errppootofv r'rg tr6yog 6rjo or6pottrlv

opt0ptitv.attrx| otlotqpa apfi1ttiu. €vo opXoio, yponr6, etrtrr1vrr6 oUorrlpo oprOgdrv nou Bo-

o((etot oe i(r opX€tuno oprOpqttrd rlrryp(o xor lprlorponotqOrlre nep(nou pfXp'

To 100 n.X.ppayyovrcd optllt4ttrd tp4Efu: wloutottrd optOpqttrd {rypio TIou qvttorotxouv'

*orq optOprotig qn6 to 1 d{xq ro 9, to ono(q Xprlotponoto6vtov ri6q on6 tov 3o

ouirvo n.X. To o0pBotro outd o6riyrloov tetrtrd oto ofyXpovo rv6oopoprrd optO-

pqurd rlrryp(o fiou Xprlotponoro0vtot or1pepo'

ypo1ttrrrc4 t{buoq: effooorl 6nou 6trot or dyvt'totot €Xouv er06tq q povd6o''6,oio0r1uopaq.

q n(otq 6tr to po6r1fottrd 6ev np€net vo ooXotrotivtot pre onetpoo6-

votro ror dtr np€ner vo y(vovrot ono6ertCq p6vo ere(veq or ono6e(etg rlou II€pl-

troppdvouv nenepoop€vo ptipoto ti 6og6q'

6ua6rxrl ap@1tr1or1: gto opXo(o poprprl op(Opr1onq flou Xpqotponore( opt0prlttrdq tr6-

[etg too6tlvopeg pe to.6vo' xot to '60o', xo0drg ror npoo0aur€g ou(e0(etq ou-

t6v rulv 60o ouPtP6trtrtv.

ewooa6w6 oiot4pa apfiptitu €vo o0orrlpo opr0gtirv pe pdoq ro 20, ot ovt(0eor1 ge

to rv6oopoptr6 orlotqpo nou €Xet og Bdorl to to'

exltr4E opt0p6q nou ypd<perot og unepre(gevog 6e(xtr1g o' 6vo 6e0tepo opt0p6

(opr0p6 pdoeurq) xor 6efiver tq 60voprq oulv ono(o npdner vo urpurOe( out6g o

6e0tePog oPt0P6g'

tyl lolano6elxtwript0o6ogono6etrurrr]p€0o6oqnouovoxotrr i<ptr lreondnotrtr€gopXo(egro tv t l v (eg ,6nourdnotoguno06ter toov t (0e toon6our6nounpoKt t ro tvo ono6erXte( rot petd 6etXvet 6tt undpXer rdnoto ovt(rpooq' Ovopd(etot en(or1q

reductio ad absurdum (etg dtonov onoyutyti)'

e{4rovta6tr6 otot4pa apfiptitu o0otqpo oprOprdrv pe pdoq ro 60' os ovt(0eoq pe

ro 6116 poq tv6oopoptr6 oriorrlpo nou €Xet og pdoq to 10'

en*tdotltog nA40w6q ap0y6E. o pretlotr0tepog ntrr1Otrdq opr0p6g nou yvurpi(ouge

npog to nop6v '

t v *w\ ' v FsYu"vrLvv) " " ' r " " ' - ) - r ' - r - ) 391

Page 376: Ο Ταξιδευτής των μαθηματικών

r ^ n f f A P l o

u\efu tuv ap&ytiu pto ypoppri one(pou prlrouq, 6nou rdOe orlpa(o ouoXet((etotpe dvov xor pr6vo 6vov npoyportr6 opt0g6.

euild\en ynpttpfu: 11 yeu-rptetp(o nou ovonr0X04rE on6 tov Eurle(6r1 Kot tKovo-

nore( to o((topo t1g nopotrtrqtrou, to ono(o tr€Et: ov 6o0e( pto euOe(o xot 6vo oq-pr(o rrrdg eu0e(oq, on6 ourd ro orlpe(o ptnopei vo nepdoet g(o p6vo dtrtrr1 tu-

Oe(o nopdtrtrqlq npoq rrlv opXtxrl euOe(o.1tti.tp41to: puo np6roorl rj Cvog r0noq nou ouvdyetot ond €vo ofvotro o{rurpdturv

ror/rj titrtrov 0eurp4pdrov.*rl:rt:, apfiptiu 11 pe0o6rrri getr€q turv rpuotxtitv optOgtitv Kot rov oX€oetiv

cpauni ypagfl: opyoh otyumroxr] yporp{, nou Xprlotponototivtov on6 toug ypo-

rpe(q yro qv roOrlpeprv4 rr1pr1o4 opXeiov.ttpoy)urpwd: opXo(o oryumtoxti yporp(, nou Xpqorponotodvrov ortq en(orlgeg nept-

ordotrg ror oto pvrlpeio.u6oapapw6 otjorqpa apfiptittt to rup(opXo o0orrlpo opt0ptitv nou Xp4ottrronote(ror

otrg pr€peg poq, ro onoio poor(eror oro 6dro ptovoOtrd oUppotro 0,1,2,3,4,5,

6, 7, 8 Kor 9, Kot xprloruoTrorr( ivo oriotrlpo 0€oqq-npr1q.xa0opt(6ytvo an6 t4 0to4 oiot4po apfiptitu odotrlpo oprOprtiv nou ouv6ud(et qv

trgrj evdq oprOprltrrod rlrqq(ou pe rr1 Odoq tou evr6g trlg nopdotoorlq tou oprO-prori, yro vo 6tiroer qv tetrrrrj lgrj tou. To otiyXpovo rv6oopoprr6 odoqpo opt0-gdrv e(vor €vo odotrlpo roOopr(6trrevo ond tr1 06oq.

xaAd \tdtetaypivo: (vo otjvotro e(vot rotrd 6totetoyp€vo ov yto r<iOe unooUvotro,oupneprtroppovogdvou rou (6rou tou ouv6trou, undpXer €vo npdto ototXe(o. To

rev6 orjvotro, ro O, 0eope(tor rotrd 6toreroyp€vo.ravowr6g ap0trt6E opr0pdg nou to 6ero6tr6 ovdnruygd tou nept6yet oe (or1 ovotro-

y(o ror ro 6€ro rpqrp(o on6 to 0 6oq to 9. Anotrdturg rovovtrdg opr0g6g e(vot o

opr0prdq nou to 6ero6rr6 ovdmuyprd tou rdtco ond onoro6rinore pdoq nept€Xetoe (oq ovotroy(o rd0e Qryp(o.

ropteonvd oiotqpa ouvtetoylttvuu n Xpion 6ilo rd0etov optOptrlttrdrv eu0etdvyto rov npoo6roptopd r<iOe oqpe(ou ev6q enrnd6ou pre 6fo npoygottxodg opt0-

uouq.rinou: 6rdrpopo rop66vro pe r6pnouq, to ono(o Xprlorptonototivtov on6 touq 'lvro

oto N€o K6opo yro t4v roroypoqri tov unopX6vttrtv Kot rov ouvotrtroytitv touq.

rAaotd oivo)o: 6vo odvotro opr0ptitv e(vot rtretord yto pto npd[q, ov rd0e €qoppo-yr] outrjg qg np<i[qq oe opt0pro6g nou ovtixouv oro o0votro €yet og onot€Aeopo€vov dtrtro opr0pd rou ouvdtrou.

xovotparnpop<iq 11 n(otr1 6tr to proOq;,rottrd ovttre(gevo 6ev undplouv ove(dprr1-

to ond rov ovOodlntvo vou.roun6vt (r1 ltdpxa): ot4v op(0pqor1, ivo gtxp6 n{trtvo teXvodpYnpo, Trou Xpnotpo-

noro0vrov orq Dutrrrj Ao(o peto(0 8500 Kqt 3500 n.X., yto rqv KoroYporp{ tou

ntrr10rrorl opr0pori 6rd<poptrtv ouvdtrrrrv ovttxetg€vtrtv.Aa1ep€v4 napa6opi: p60o6oq eniiuorlg e(todroeurv Xpqotponotdtvtog gto etroo(o r1392

Page 377: Ο Ταξιδευτής των μαθηματικών

ono(o ovlroO(ororor orrlv e((oororl yro vo nopdyer pto ovtiorotXrl npoooppoy4nou o6r1ye( oe gro petrloprivq e rroo(o.

Aoydpr1poq. o ex0€rqg orov ono(o np€net vo urlro0e( ivoq opt0pt6g nou ovopd(etorpdoq, yro vo €youge oq onot6treo;ro €vo ouyrerprg€vo opt0p6. O loydpt0pogrou v p€ pdorl ro o ypd<petor log"v. tto nopd6etypo, log,o100 = 2, { 10'� = 100.

)oytortxri: q opXoio etrtrqvrxri unotroyrotrr{ tdXvq, 11 ono(o Oeurpo0wov epnoptrrl ou-votrtroy( rr 6Xr eyreqotrtr{ ovo(tr1oq.

1-tt0o6og qq e{dvrA4oqq. trXvrrrl nou ovont0X0qxe ond toug 'Etrtrr1veq yto vo ppl

oKouv ro rprpo66v roprn0trov yetrrgerprrdrv oXrlprdrurv. Me rqv telvtr{ ouul ppi-

oKovtor 6ro6oyrrd olo€vo ror xotr6ttpeg npooeyy(oerg, unotroy((ovrog to tppo-6dv opOoyov(ov ror rpryrirvrrrv evr6g tqg ropn0lrlg.

1tt0o6oE tuv Ptitouv yuprraiu gd0o6og nou opXtxd Xpqotponot{0qre on6 toug op-

Xo(oug Ary0ntrouq yro rov notrtronloorooprd opr0pritv, 6rntrootd(ovrog 6to6o1trd€vov opr0p6 xot npooO€rovtoq orrl ouv6Xero to rotdtrlrltro notrtrontrdoro. Yro0e-

fi0r1re opy6trpo ond roug Kevrpoeuponofoug.

1t46iu o ntrq0rr6g opr0g6g tou revof ouvdtrou' ro o0ppotro nou 6ev dXet rtg{, otrtrdprnope( vo Xpqotptonorrl0e( urq opr0p6g 06o119 o' dvo xo0opt(dprevo on6 tq O€oqoriorrlpo opr0ptirv' to opr0prlttr6 q,rlgio 0.

pq ur)e6en ytuperpb'. xd0e yeoprerp(o nou poo((etor otrlv ovttrordoroorl roueurtre(Oerou o[tdrporog qq nopo]tr(]ou g' 6vo btorpopettx6 o{itrrpo. Btr. eurtre(-6eto yttrtperp(o.

pr1 ra0opt(61ttvo an6 tq 0ior1 otjot4pa ap0ptitu. onoto6{note o6orrlgo opt0puitv,6nou q 06o11 ev6g ouyrerprp€vou opr0prrltrroU Qrlq(ou 6t Boq0d orov npoo6to-ptopr6 trlg rtprrlg rou {rqrp(ou. Enop€vtrrg, to opt0prlttxd Qqrg(o gnopo0v vo ypd-qovro( pe onoro6qnote oerpd, Xcop(g vo otrtrd(et q rtp{ tov optOgtitv.

lttyo\rxdE ap0p6E opr0p6q qS Uopqris (o,P) ri o + pi, 6nou to o Kot p e(vot npoy-prolro( optOgo( rot to i e(vor ro fi. Or npoyirotrro( Kot ot qovroortro( opr0-go( e(vor unoo0volo urrv gryo6txtirv opt0gtitv.

pova6nb x)dopa: xtrdopo rrlg Fopqrlq 1/v, xtr<io1io nou o optOgrlt{q tou t(vot to 1.veo6uo6rxrl ap(0pr7o4: opXo(o goprp{ opiOpqoqg, nou XprlotUorlote( optOprlttr6g tr6[etg

roo6rivogeg pe ro <rivo> xor ro.60on. ro0drg rot ou(eUfeqg outtirv tutv 6tio ougB6-lulv, ror nrprtroppdver qv np6o0eoq, rqv orpoipeoq ror rov notrtrontrootoop6.

ovto)oyh: q pretr€tq tou vo(triorog rrlg dnop{r1q.dpro: q orotrouO(o rtrlv 6prrrv Ar, A2, Ar,... A",... €X€l €vo dpro I ov yto rd0e 0ettxq tt-

pr1 e undpXer ivoq opr0p6q N r€rorog tirote, yto rd0e v > N, 11 on6truq ugrj rouL - A " < € .

napayovur6: ro yrvdpevo 6trttv turv <puotr6v optOgtirv nou e(vqt ptrp6tepot 11(oor pr '€vo ouyxerprp€vo rpuotx6 opr0p6 ror 6qtrdrveror ge to o0ppotro o!,, toono(o rono0ere(ror petd tov ouyrerprg€vo rpuotr6 opt0g6. fto nopd6etypo,5 ! = 1 ' 2 . 3 . 4 . 5 = 1 2 0 .

napd6o{o: €vo rnrXe(prlpro on6 to ono(o npor6mouv ovrtqottrd oupnepdoptoto,p€ot,r onoyurydv nou roXrlouv pe pdoq broroOqrtxd ono6errorig oulAoyropoUq. 393

Page 378: Ο Ταξιδευτής των μαθηματικών

ne6b ap0ptiu xdOe odvolo npoypotrrt0v { ptyo6rrtiv opt0ptirv r6toro tiote tod0poropo, q 6ro<popd, to yrvdpevo ror ro nr1tr(ro rouq (pe e[o(peo4 to gq6€v)e(vot rivog dtrtroq oprOpdg rou ouvdtrou. Enogdvtrrg, ro ne6(o opr0truirv e(vor rtrer-o16 yro trq r€ooeprq np<i(erq qq optOptqtrrqq.

nevtaiul ap(9pqo4: €vo opXo(o o0oqpo opr0gtirv pe pdorl ro 5.il40tx6g ap6p6g opt0y6g o ono(oq ro0op((er n6oo ororXe(o undpXouv o'dvq oUvotro.noAuuvupwf e{buo1: e{(oooq p' dvov r] neproo6tepoug oyvtirotoug, nou urftirvo-

wor oe 6uvdperq ror notrtrontroord(ovror pe opr0podq, or ono(or ovopd(ovtorouvteleot€q.'Evo notrurirvupo p'6vov dyvtrroto. rov x, 6Xer q yevtr{ pop<pri:oox" + o,xu-t * ... * ou_rX + o" = 0.

npoypaux6g ap0p6q. or opr0po( nou ouoXer((owor pe 6tro ro oqpe(o u1q eu0e(oqtrrrv optopr0v. Anotetrodv €vooq tov npoypotrr6v xor tov unepBottrtiv oprogtirv.

npopoAwl yeuyerpfu: rtrdOoq rulv po0rlpouxdrv nou ooXotre(tor pe ere(veg trq t6rd-rnr€S Tov yeoperprr6v oX4pdtov nou 6e peroBdtrtrowqr drov to oXripoto npo-pdllovrot o' dvo 6rorpopetrrd Xtirpo.

npdtog apt01t6g guorrdq opr0p6q nou prnope( vo 6rorpe0e( r€trero gdvo ond toveour6 rou ror tov opr0p6 1.

nu0ay6peo 0edp4pa: ro terpdyovo trlg unote(vouooq ev6q opOoyov(ou rprytirvouroor3tor pe ro dOporogo rov rerpoydrvov trrrv 66o npoore(pevov ntreupdrv rj, otr-truirg, o' * P' = y', 6nou y e(vot to prjroq trlq unote(vouooq Kor ro o ror p e(vorro p{xr1 rov 66o npoore(pevov ntreuptirv.

nu0ay6petot opr9pot onoto64note odvotro tprtirv rpuotrtiv opr0ptirv nou rrovonore(ro nu0oy6peto 0e6pqpro. fto nopd6etypo, 4 tprd6o 3, 4 Kor 5 e(vot nu0oydperoroptOpo(, orporj 3' + 4' = 5'.

nurv6E q r6rdqro 6rt ov<ipeoo oe onorouo6{note 60o optOpo0q undpXer 6voq d}-loq opt0g6q. Or pqto( or dppqror, or npoygouro( rot or truyo6rro( oprOpo( e(vot6lor nurvo(.

pdp6o9 pe eyrontg. pdp6oq ouvrjOrrrq Xoptopdvq rord g{roq oro 6rio, 11 ono(o iXer

Xopoyp€veq eyronig yro rrlv xoroyporp{ rrrrv orrovo1uxtirv ouvotrtroydrv. 'Otov

evtitvovtor ro 66o tpriproto, or eyrondg dpXovtor 11 p(o on€vovtr oqv dltrr1.pqtopwrl d)yeppa: pro opXo(o popcpri dlyeppoq, 6nou to npoplrlptoto 6rotuntivo-

vrov offl ouv{Or1 yltirooo Xtop(q q Xp{or1 orprpo0q ouppotrropoti.p4t69 ap6p6q. rdOe opr0p6q nou pnope( vo er<ppoote( oq trdyoq 6Uo pr1 p46evrr6v

orCpourlv opr0prirv.p((a: tvog opr0p6q nou rrqvonore( pro e{(otooq' drov, 6qtro6rj, ovttrotootqoer tov

dyvtrroto orqv e((orrlorl ror to 60o or6trq rrlq e((otr-ror1q 6e(r<i ror oprotepd ondto (oov CXouv qv (6ro ttprj.

o4yeb ouootip€uon;. Vto ritrtrq ovotrroo(o yro to 6pro, er6rr<i 6tov undpXouv nepro-o6tepo on6 6vo 6pto.

ototytb: Cvo ouyxexprpCvo ovure(pevo 4 Filoq ev6q ouv6trou.ouyrerprytv4 apfip4o4:6rov to ovtrre(pevo nou petpdrwor pp(oxovtot o€ opqtgo-

voo{gowr1 onerr6vtoq pe roundvto (p<ipreq) { o0ppotro op(0pr1or1q.394

Page 379: Ο Ταξιδευτής των μαθηματικών

orjyx)to7: q rotdotooq puog dnetpqg orotrouO(oq oprOptitv t] optOgqttr{g oetpdg

nou npooeyy((er ivo 6pto.ouyront6ptvq dyeppa: dtryeppo nou pp(orerot ovdpreoo otq pqtoptr{ Kot rn oug-

Botrtrrj dtryeppo'Xprlorponote( €vo ouv6uoop6 oupp6trtrrv xor ouvrlOtog€vulv tr6-

(eurv orrg otryepprrig npordoetg rqg.

ouypokni dAytppo: 11 rllytppo nou Xp1otgonore( rotrd npoo6roptog€vo otippotro

ror 6Xr t4 ouv{Or1 ytrtitooo.

ouwyiq rkioya: opr0pt6g nou qnoretre(tot ond 6vov ordpoto rot dvo rtrdopo r€toto,

tiore o nopovogoorr]q rou rtrdo;.rotog vo eivot en(or1g €vog orEpotog rot 6vo

xtrdopo tou [6rou r6nou. Kd0e dpprltoq opt0pt6q prnope( vo ovonopooto0e( pe

rrl Uopq{ ev6q dnetpou ouveXodg rtrdoptotog.

oAvlerog opt01t6g. rpuorr6q opr0g6g nou pnope( vo 6rotpe0e( rileto xqt on6 dtrloug

oprOpo0q, n€po ond q povd6o rot tov eout6 tou.

oAvo)o: Suo outrtroyri ovttretpdvtov ri otorXe(ov.ouvt*aypiveE or 60o npoygortro( optOpro( nou ouv66ovrot ge xd0e orlpe(o rou ne-

6(ou rqg ovotrulr{g yeulpttrptoq.

ouwetayytvog apfip6q. opr0p6q nou npoo6top((er tq oXerrrri 0€or1 ev6g orotXe(ou

oe €vo orivotro.olotqpa ypapptxtitv t{rot6oeuu.6rio r] ne ptoo6repeg e[totiroe rg, 6nou 11 rd0e e((our-

o4 nepttrotrrBdver €vov q neptoo6repoug oyvrioroug rot o xd0E dyvoorog €Xet tr-tg

er0€tn ro 1.tfAtroE apfi1tdg. <puotr6g opt0g6g nou e(vot to dOpotopo ttrtv 6totpertiv rou. To 6

e(vor o npdrroq r€trerog opr0pr6q, orpo0 e = 1 + 2 + 3, Kot ro 28 e(vor o 6e0tepog,

o 0 ( 0 o 0 2 8 : 1 + 2 + 4 + 7 + 1 4 .tetpayuvtx6E ap0p69. dvog rpuorrdg opt0p6q, o ono(oq eivot to tetpdyurvo ev6g dtr-

lou rpuorroU optOpo0. fro nopdbttypro, to 4 Kot to 9 e(vot tetpoyu-rvtro( opt0po(,

ener6rj e(vot to tetpdytrtvo rou 2 Ko( Tou 3.

tttpo\uo( ap0porl ;rtyo6rxo( oprOpo( ge q popqrl o + pi + Yj + 6k, dnou to o, p, y

ror 6 e(vor npoypottro( opt0po( Kot ro i, i rot k e(vot uneppryo6tro( optOpto( 16-

roror drotr i ' = l ' = k'= i ik = -1.

tptyuvoyttpfu: 11 pd€rq tulv oXdoeov peto(0 ru-rv prrlrc0v ttrlv ntreupdtv ev6g rptytit-

vou Kot rov pr6rpov rov totlttptxtr.tv ytrtvttitv tou.

unepanp\orcoq unepnmepaoytvoE ap6y6E unepnenepool.tivog opt0y6g nou eivot

16oo onp6ottog, riore 11 onpootrdtltd tou 6ev pnope( vo optore( {errvdrwoqon6 pnrp6repouq ufi€prt€TT€pooprdvoug oprOpodq. Btr. onp6ottoq ufi€pr€n€po-

op€vog opt0p6g.unepap0p4orl.to: ivo onetpoo6votro, tou ono(ou to ototXefo 6ev gnopotiv vo te-

0odv oe oprprpovoorigovtq onetr6vtorl F€ touq quo(Ko0q optOgodg.

uneppanr6g ap0p6q. npoyprottr6g opt0g6q, nou 6ev onoretre( pt(o ti ldoq ev6g no-

truurv0gou pe pqtorig ouvtetreot6g.'Evog npoygortr6g opt0pdg nou 6ev e(vqt otr-

yeppr16g.unepyrya6x69 aptlltd1. opt0g6g nou nopdyerot on6 rrlv en€rtoorl rrlq €vvorog tou 395

Page 380: Ο Ταξιδευτής των μαθηματικών

opr0poti oe broordoerq n6po on6 toug 6to6tdototouq gryo6rro0g optOpo0g. Btr.

terpo6rro( opr0po(.unepnmepoopfuoE opt0trt6q. o nlq0rr6q { tornrdg opr0p6g ev6g onetpoouv6trou.

un60eo4 rcu ouvtyoiq. q un60eo4 6tt undpXet €voq unepnenepootrt6vog opt0p6g

ovdpeoo orov No xot tov N,. H un60eor1 ouu] 6ev prnope( vo ono6erXte(.

uno)oytour1 nudrrc:. opXo(o ouoreur] op(0pqoqg, nou onotetre(tot on6 pto ntvox(-

6o pe orlgd6ro rqt xtro6td ri 1otr(rto yto tr1v op(Opr1or1.

unotefuouoa: o' ivo rpryovo, 11 ntreupd on€vovtt on6 trlv op0ti ytlv(o.

gawaour6g ap6p6g, opr0pr6q orov rd0eto d[ovo tou ne6(ou tttv gtyo6trdrv opt0-

pdrv' opr0pr6g nou €Xer rn Uopqri oi, 6nou to o e(vot npoyprourdg opt0p6q rot

ro i e(vor to /J.qpdrtaA: gro odvOetrl yeurgetptxrj ropnritrrl, nou 6e ptetooX4pot((etot oe onl€q pop-

qCq pe pey60uvon Kor fiou q 6rdotoor] trlg pp(orerot ouvrlOutg ovdpeoo oe or€-porouq opr0go0g.

ypuo6q ptooE 11 ypuo6E )dyoq o tr6yog tou (1 + ft npoq ro 2 rl to rtrdopo ( + lfs)12.

Avorotrfrptqre ond rouq'Ellrlveg rot undpXet oe noMdq po0qgottr€g oX€oetq.

rp4qrb: onoro6rjnote on6 to 6dro optoprltrrd t l ,nq(o 0, 1, 2,3, 4, 5,6,7,8 Kot 9 tou

rv6oopoprrori ouotrjpotoq op(0gqor1g.

396

Page 381: Ο Ταξιδευτής των μαθηματικών

Ittrfii: :r i

' :i .

i , :

1 .

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Bell, Eric Temple, Mathematics: Queen and Servant of Science. N€o Y6prr1, Mccraw-

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Borowski, E.J., Kor Borwein, 1.M., The HarperCollins Dictionary of Mathematics. Nia

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