Procli
In primum Euclidis Elementorum Liber
Prologi pars posterior
T mn d koin ka p psan diatenonta tn
mayhmatikn pistmhn n tow proeirhmnoiw lgoiw
teyemeya t te Pltvni sumporeumenoi ka par
tn llvn nalegmenoi t prw tn parosan
pragmatean mn suntenonta nomata. totoiw d
petai per te atw tw gevmetraw epen ka tw
prokeimnhw stoixeisevw, w neka tn smpanta l-
gon nesthsmeya.
Oti mn on gevmetra tw lhw st mayhma-
tikw mrow ka ti deutran xei tjin met tn
riymhtikn w k tathw teleioumnh ka forizo-
mnh^pn gr, son st =htn n at ka gnv-
stn, k tn riymhtikn forzetai lgvn^erh-
tai tow palaiow ka o pollo detai lgou prw
t parn. gnoito d' n mn per atw fghsiw
kat non, e tn pokeimnhn at piskopsaimen
lhn, tna tjin laxen n tow osi, ka tn osan.
p gr tathw kalw yevrhyeshw ka tw gignv-
skoshw atn pistmhw katafansetai dnamiw ka
fleia p' atw ka t gayn, per ew tow
manynontaw paragnetai. ka gr d ka porseien
n tiw, n tni gnei tn ntvn tn gevmetrikn lhn
tiymenow ok n mrtoi tw per atn lhyeaw.
ete gr n tow asyhtow sti t sxmata, per n
gevmtrhw dialgetai, ka xrista tw lhw, pw
ti tn gevmetran tn asyhtn mw polein f-
somen ka perigein ew tn smaton pstasin ka
suneyismn enai prw tn yan tn nohtn ka pro-
eutrepzein ew tn kat non nrgeian; po d ka
teyemeya n tow asyhtow t merw shmeon tn
plat grammn tn bay pifneian tn s-
thta tn k to kntrou grammn lvw t pol-
gvna ka poledra sxmata pnta, per n gev-
metra didskei; pw d ka o lgoi tw pistmhw
tathw nlegktoi mnousi tn asyhtn sxhmtvn
ka edn t mllon ka tton pidexomnvn ka ki-
noumnvn pnth ka metaballomnvn pshw te ori-
staw likw napeplhsmnvn ka tw mn sthtow
met tw nantaw nisthtow festhkuaw, tn d
merstvn kat merismn ka distasin proelhluy-
tvn; ete jv tw lhw st t pokemena t gev-
metr& ka lgoi kayaro ka xvristo tn asyhtn,
mristoi pntew sontai ka smatoi ka megyeiw.
ktasiw gr ka gkow ka lvw distasiw tow lgoiw
di tn likn podoxn paragnetai, t mn m-
rista meristw, t d distata diastatw, t d
knhta kinoumnvw dexomnhn. pw on ti tn
eyean tmnomen ka t trgvnon ka tn kklon;
pw d gvnin diaforw lgomen ka ajseiw atn
ka meiseiw sxhmtvn, oon trigvnikn tetragv-
nikn; pw d tw fw tn kklvn tn eyein;
pnta gr tata meristn enai tn gevmetrikn
lhn pideiknousi ka ok n mersi lgoiw fe-
sthkuan.
T mn on pora toiata prw t ka tn Pl-
tvna dianoht mn prosagoreein t tw gevmetraw
edh, xvrzein d mw p tn asyhtn t toiata
ka ew non gerein p asysevw sugxvren, ka-
toi ge, per fhn, tn n diano& lgvn mern n-
tvn ka diasttvn kat tn dithta tn cuxikn
festhktvn. e d de ka tow prgmasin atow
ka t to Pltvnos fhgsei sumfnouw podi-
dnai lgouw, otvs dielntew epvmen, pn t ka-
ylou ka t n t tn polln perilhptikn n
tow kay' kasta fantzesyai (pfuken fanetai?)
ka tn parjin n totoiw xein xriston p' atn
prxon ka katatetagmnon n atow ka met to-
tvn sugkinomenon monmvw stw ka kintvw,
pr tn polln festnai ka gennhtikn enai
to plyouw mfseiw f' auto tow pollow par-
xon ka merstvw mn at protetagmnon tn met-
exntvn, poiklaw d meyjeiw ew t detera xorhgon,
kat' pnoian p tn polln morfosyai ka tn
parjin pigenhmatikn xein ka sterogenw pi-
sunstasyai tow pollow. kat gr tataw omai tw
triplw postseiw ersomen t mn pr tn pol-
ln, t d n tow pollow, t d kat tn prw at
sxsin ka kathgoran fistmena. trittn d n-
tvn w sunelnti fnai tn kayolikn edn to
metexomnou ka n tow pollow ntow ka t merik
sumplhrontow nosvmen diaforw kat tn pokei-
mnhn lhn. ka t metxonta at ditt ymenoi,
t mn asyht t d n fantas& tn pstasin
xonta^ka gr lh ditt, ka mn tn asy-
sei suzugontvn d tn fantastn, w pou ka
Aristotlhs fhsi dittn enai t kaylou^t
katatetagmnon sugxvrsomen, t mn asyhtn w
metexmenon p tn asyhtn, t d fantastn w
n tow tw fantasaw plyesin festhkw. ka gr
fantasa di te tn morfvtikn knhsin ka t
met smatow ka n smati tn pstasin xein me-
ristn e ka dirhmnvn stn ka sxhmatismnvn
tpvn ostik, ka pn gignskei toiathn laxen
parjin. yen d ka non payhtikn tiw atn
proseipen ok knhsen. katoi ge e now, pw ok
payw ka #low; e d met pyouw nerge, pw
ti now n klhyeh dikavw; pyeia mn gr t n
proskei ka t noer fsei, t d payhtikn prrv
tw osaw kenhw. ll' omai t mson atw mf-
nai boulmenow tn te prvtstvn gnsevn ka tn
sxtvn ma ka non atn prosepen w oikuan
taw prvtstaiw ka payhtikn kat tn prw t
sxata suggneian. a mn gr sxhmtistoi ka
mrfvtoi gnseiw esin n autaw xousai t noht
ka per autw nergosai ka sunhnvmnai tow
gnvstow, pantw tpou ka pyouw llaxyen f-
kontow kayareousai. a d sxatai di tn rg-
nvn nergosi ka paymata mlln esin, jvyen
esdexmenai tw gnseiw ka sugkinomenai tow po-
keimnoiw. toiata gr a asyseiw, k biavn payh-
mtvn ginmenai, fhsn Pltvn. d' a fan-
tasa t mson kntron katxousa tn gnsevn
negeretai mn f' autw ka probllei t gnvstn,
te d ok jv smatow osa k to merow tw
zvw ew merismn ka distasin ka sxma progei
t gnvst atw, ka di toto pn, per n no,
tpow st ka morf nomatow, ka tn te kklon
diastatw noe tw mn ktw lhw kayareonta
nohtn d lhn xonta tn n at, ka di toto
ox ew n at kklow, sper od n tow asyh-
tow. ma gr distasiw nafanetai ka t mezon
ka t lasson ka t plyow tn te kklvn ka tn
trignvn. e on n tow asyhtow kkloiw st t
kaylou katatetagmnon, ka kaston atn kklon
petlesen ka pntaw moouw llloiw kay' na
lgon postntaw, diafrontaw d megyesin tow
pokeimnoiw, kn tow fantastow kkloiw st ti
koinn ka metexmenon ka kat toto pntew tn
atn xousi morfn. d diafor atow kay' n
ntaya mnon t n t fantas& mgeyow. tan gr
pollow mokntrouw fantasyw, n n mn pntew
pokeimn ka @l ka n zv tn parjin xousin
xvrst smatow plo ka t diastmati pleon-
santow tw merow osaw, diafrousi d t te me-
gyei ka t mikrthti ka t perixesyai ka peri-
xein. dittn on soi noesyv t kaylou t n tow
pollow, t mn n tow asyhtow, t d n tow fan-
tastow. ka kuklikw lgow dittw ka trigvni-
kw ka atw to sxmatow, mn p tw nohtw
lhw, d p tw asyhtw. pr d totvn n te
n diano& lgow ka n t fsei, mn tn fan-
tastn kklvn postthw ka to n atow nw
edouw, d tn asyhtn. stvsan gr o n
oran kkloi ka lvw o tw fsevw kgonoi. ka
sper merw n diano& lgow, otvw ka fusi-
kw. sti gr ka t diastat diasttvw ka t
merist merstvw ka t megyh megyvw n taw
svmtoiw ataiw, sper a npalin t mrista
meristw ka t megyh megeyusmnvw n taw sv-
matikaw. ka di toto mn n diano& kklow ew
ka plow sti ka distatow ka at t mgeyow
mgeyew ke^lgoi gr neu lhw t toiata
ka t sxma sxhmtiston^ d' n fantas& me-
ristw sxhmatismnow distatow, ox ew mnon, ll'
ew ka polw, ka ok edow mnon, ll katatetagm-
non edow, d' n tow asyhtow ka tw kribeaw
fesin xvn ka nplevw tw eyeaw ka tw kaya-
rthtow tn @lvn poleipmenow.
Tn tonun gevmetran tan per kklou ti lg
ka diamtrou ka tn per tn kklon payhmtvn,
oon fn diairsevn tn toiotvn, mte per tn
asyhtn lgvmen nadidskein^xvrzein gr p
totvn pixeire^mte to n diano& edouw. ew
gr kklow, d per polln poietai tow lgouw
kay' na kaston probllousa ka per pntvn t
at yevrosa. ka diaretow mn kenow, diaire-
tw d n gevmetr& kklow. ll t kaylou mn
at sugxvrmen piskopen, toto d' ra keno
t katatetagmnon n tow fantastow kkloiw, ka
llon mn rn ka kat' llon yevren tn n dia-
no& kklon, per llon d poiesyai tw podejeiw.
xousa gr dinoia tow lgouw, syenosa d
suneptugmnvw den naplo te atow ka pekt-
yetai ka ew tn fantasan n proyroiw keimnhn
progei ka n ken ka met' kenhw nelttei tn
gnsin atn, gapsasa mn tn p tn asyh-
tn xvrismn, tn d fantastn lhn etrep prw
podoxn erosa tn autw edn. yen ka
nhsiw atw met fantasaw a te sunyseiw tn
sxhmtvn ka a diairseiw fantasta ka gnsiw
dw mn ew tn dianohtikn stin osan, opv d
ew kenhn nadedrmhke, tw dianoaw ew t jv
bleposhw ka tata kat t ndoyen yevroshw ka
probolaw mn xrvmnhw lgvn ll' f' autw ew
t jv kinoumnhw. e d pote sumptjasa tw dia-
stseiw ka tow tpouw ka t plyow tuptvw ka
noeidw yeasamnh prw autn pistrcai dunh-
yeh, tt' n diaferntvw tow lgouw tow gevme-
trikow doi tow merstouw, tow diasttouw, tow
osideiw, n sti plrvma. ka nrgeia atw
ath tlow n eh t riston tw per gevmetran
spoudw ka ntvw tw Ermakw dsevw rgon, p
tinow Kalucow nagoshw atn ew teleiotran ka
noervtran gnsin ka poluoshw tn n fantas&
morfvtikn piboln. ka tathn de tn melthn
meletn tn w lhyw gevmetrikn, ka prw tn
gersin ka tn p tw fantasaw metstasin ew
mnhn tn dinoian atn kay' atn poiesyai t-
low, rpzonta autn p tn diastsevn ka to
payhtiko no prw tn dianohtikn nrgeian, kay'
n pnta diasttvw cetai ka n mere tn kklon,
tn dimetron, t n t kkl polgvna, ka pnta
n psin ka kaston xvrw. di gr toto ka n
fantas& deknumen n te tow polugnoiw tow k-
klouw ggrafomnouw ka n tow kkloiw t pol-
gvna, mimomenoi tn tn mern lgvn di' lllvn
dejin. di tata gr ra ka sustseiw sxhmtvn
ka genseiw ka diairseiw nagrfomen ka yseiw
ka parabolw. diti t fantas& prosxrmeya ka
taw k tathw diastsesin, pe t ge edow at
knhtn sti ka gnhton ka diareton ka pan-
tw pokeimnou kayareon. ll ka sa krufvw
stn n ken, diastatw ka meristw ew fanta-
san progetai ka t mn probllon dinoia, t
d f' o problletai t dianohtn edow, t d
n t proballmenon payhtikw otow kalomenow
now, jelttvn autn per tn mreian to lh-
yow no ka distw auto t distaton tw
kraifnow nosevw ka morfn autn kat pnta
t mrfvta edh ka pnta gignmenow, stin
dinoia ka merw n mn lgow.
Per mn on tw gevmetrikw lhw tosata xo-
men lgein ok gnoontew, sa ka filsofow Por-
frios n tow summktoiw ggrafen ka o plestoi
tn Platvnikn diatttontai, sumfvntera d
enai tata taw gevmetrikaw fdoiw nomzontew ka
t Pltvni dianoht kalonti t pokemena t
gevmetr&. sundei gr on tata llloiw, diti
tn gevmetrikn edn a mn atai, kay' w ka
dinoia probllei tw podejeiw, n at proufe-
stkasin, at d kasta t diairomena ka sunti-
ymena sxmata per tn fantasan probblhtai.
per d tw pistmhw atw tw totvn yevretikw
met tata lgvmen. gnvstik mn on sti megeyn
ka sxhmtvn ka tn n totoiw pertvn, ti d tn
lgvn tn n atow gevmetra ka tn payn
tn per at ka tn pantovn ysevn ka kin-
sevn, proosa mn p to merow shmeou, kata-
banousa d mxri tn steren ka tw polueidew
atn diaforthtaw neurskousa, ka a plin p
tn sunyetvtrvn p t plostera ka tw rxw
tw totvn natrxousa. ka gr sunysesin xrtai
ka nalsesin, e mn j poysevn rmsa ka
tw rxw p tw pr atw pistmhw lambnousa,
xrvmnh d taw dialektikaw psaiw meydoiw, per
mn tw rxw diairsesi tn edn p tn genn
ka tow ristikow lgoiw, per d t met tw rxw
podejesi ka nalsesin, na ka p tn plou-
strvn t poikiltera deikn pronta ka p'
at plin nastrfonta ka xvrw mn per tn
pokeimnvn at poioumnh tow lgouw, xvrw d
per tn jivmtvn, f' n rmhtai prw tw po-
dejeiw, ka tn athmtvn, xvrw d per tn kay'
at sumbebhktvn, ka deknusin prxonta tow
pokeimnoiw. ksth gr tn pisthmn llo mn
xei t gnow, per pragmateetai ka o t pyh
skopen protyetai, llaw d tw rxw, aw xrtai
prw tw podejeiw, lla d t kay' at prxonta.
ka t mn jimata koin psaiw, e ka ksth
xrtai prw tn pokeimnhn lhn okevw atow,
t d gnow ka t kay' at sumbebhkw diafron.
T mn on pokemena gevmetraw st trgvna
ka tetrgvna ka kkloi ka lvw sxmata ka me-
gyh ka t totvn prata, t d kay' at pr-
xonta totoiw a diairseiw, o lgoi, a fa, a
sthtew, a parabola, a perbola, a lleceiw,
pnta t toiata, t d atmata ka t jimata,
di' n podeknusin kasta, t p pantw shmeou
p pn shmeon eyean gein, t n p svn sa
fairey, sa enai t kataleipmena ka t totoiw
pmena. di ka ote pn prblhma ote pn r-
thma gevmetrikn stin, ll sa k tn gevmetraw
stin rxn, ka k totvn legxmenow lgxoito
n w gevmtrhw. sa d m k totvn, o gevme-
trik ll' gevmtrhta. ditt d ka tat stin:
gr pantelw j trvn stn rxn, sper t
mousikn rthm famen gevmtrhton, ti j llvn
pantelw poysevn rmhtai ka ok k tn gev-
metraw rxn, t taw gevmetrikaw rxaw xrme-
non, ll diastrfvw, oon e tiw lgoi tw parall-
louw sumpptein. ka di tata ra ka gevmetra
kritria paraddvsin mn, f' n dunhsmeya dia-
gignskein t te pmena taw rxaw atw ka sa
tn kenvn lyeian kbanei. o gr trpoi, kay'
ow t ceudria dielgxein dunatn, p dihmrthtai,
tathn xousi tn paggelan. lla gr petai taw
gevmetrikaw rxaw ka lla taw riymhtikaw. t
gr de lgein per tn ... pmpolu lepontai to-
tvn; kribestra gr stin pistmh llh llhw, w
fhsin Aristotlhs, te poikilvtraiw rxaw
xrvmnh tw j ploustrvn poysevn rmhmnhw
ka t diti lgousa tw t ti ginvskoshw ka
per nohtn pragmateuomnh tw tn asyhtn fa-
ptomnhw. ka kat tataw tw podseiw tw kri-
beaw riymhtik mn kribestra gevmetras^a
gr kenhw rxa t plthti diafrousin. mn
gr monw yetw stin, d stigm ysin xousa,
ka rxa gevmetraw mn stigm proslabosa tn
ysin, riymhtikw d mons^gevmetra d sfai-
rikw ka riymhtik mousiks^atai gr tw
ataw podidasi kaylou tn p' kenaw yevrh-
mtvn^gevmetra d mhxanikw ptikw, ti per
asyhtn atai poiontai tow lgouw. a mn on
riymhtikw rxa ka gevmetraw tn llvn dia-
frousin, a d atn totvn poyseiw diestkasi
mn p' lllvn, kay' n epomen distasin, xousin
d' a ka koinvnan prw lllaw, di ka tn yev-
rhmtvn tn deiknumnvn t mn stin ataw koin
t d dia katraw. t mn gr pnta lgon enai
=htn riymhtik proskei mn, gevmetr& d oda-
mw: es gr n at ka rrhtoi lgoi. ka t r-
syai kat t lasson tow tn tetragnvn gnmo-
naw riymhtikw dion: n gevmetr& gr t lxiston
lvw ok stin. gevmetraw d stin jareta t
per tw yseis^o gr riymo ysin ok xousin
^t per tw fs^n gr sunexsi t ptesyai
^t per tw lgous^pou gr p' peiron
diaresiw, ke ka t logon. koin d stin mfo-
trvn t kat tw tomw, oaw Ekledhs n t deu-
tr paraddvsi, pln tw tn eyean ew kron
ka mson lgon temnoshw. tn d' a koinn to-
tvn yevrhmtvn t mn p gevmetraw ew riymhti-
kn metgetai, t d mpalin p riymhtikw ew
gevmetran, t d movw mfotraiw prosken p
tw lhw mayhmatikw pistmhw ew atw kay-
konta. t mn gr nallj ka a nastrofa tn
lgvn ka a sunyseiw ka a diairseiw kat toton
tn trpon st koinn mfotraiw, t d tn sum-
mtrvn riymhtik mn yevre prtvw, gevmetra d
deutrvw kenhn mimoumnh. di ka t smmetra
tot tata forzetai, sa lgon xei prw llhla,
n riymw prw riymn w tw summetraw prohgou-
mnvw n riymow fistamnhw. pou gr riymw,
ke ka t smmetron, ka pou t smmetron, ka
riymw. t ge mn tn trignvn ka tetragnvn
gevmetra mn yevre prtvw, kat' nalogan d la-
bosa par' atw riymhtik: ka gr n tow
riymow sxmata kat' atan stn. k tn pote-
lesmtvn on rmhyntew p tw ataw atn tw
n tow riymow mtimen, ka pou mn parallktvw
t at sumptmata yevromen, sper ti pn po-
lgvnon ew trgvna dialetai, pou d t snegguw
gapmen, oon erntew n gevmetr& tetrgvnon
tetragnou diplsion, n riymow d ok xontew
nw dontw famen llon llou diplsion prxein,
sper to p tw pentdow p tw ptdow dipl-
siow nw dontow.
Tata mn on p plon prohggomen tn koi-
nvnan tn kat tw rxw tn do totvn pi-
sthmn ka tn diaforn paristntew. gevmetriko
gr t sunorn t mn koin yevrmata, poaiw r-
xaw petai koinaw, t d dia poaiw, ka otv t te
gevmtrhta ka t gevmetrik diairesyai, ka t
mn ew llhn, t d ew llhn pistmhn gein.
nvyen d plin pintew katdvmen tn lhn gev-
metran, yen te rmhtai ka mxri tnow preisin.
otv gr tn n at dikosmon tn lgvn yeas-
meya. nosvmen d psi tow osin atn sumpar-
ekteinomnhn ka psin pibllousan tw autw dia-
noseiw ka pntvn n aut perixousan t edh,
kat mn t krtaton atw ka noertaton t n-
tvw nta periayrosan ka di' eknvn nadidskou-
san tw te tn yevn diaksmvn dithtaw ka tw
tn noern edn dunmeis^xei gr ka totvn
tow lgouw n tow okeoiw yemasi ka deknusi,
tna mn sti t yeow w proskonta sxmata, tna
d taw prtaiw osaiw, tna d taw tn cuxn po-
stsesi^kat d tw msaw gnseiw nelttei tow
dianohtikow lgouw ka japlo ka yevre tn n
atow poikilan ka tw prjeiw atn kfanei ka
t per atow pyh, tw te koinvnaw atn ka
tw diaforthtaw, f' n d ka tw fantastw dia-
morfseiw sxhmtvn n prasin rismnoiw peri-
lambnei ka ngei prw tn osidh tn lgvn
pstasin^kat d tw trtaw tw dianosevw diej-
douw tn fsin piskope ka t edh tn asyh-
tn stoixevn ka tn per at dunmevn, pvw
kat' atan n tow lgoiw atw proelhptai, para-
ddvsin. xei gr eknaw mn tn nohtn lvn ge-
nn, paradegmata d tn asyhtn, osvtai d
kat t edh t dianoht ka di msvn totvn
neis te ka kteisin f' la t nta ka t gin-
mena. gevmetrikw d per tn ntvn e filoso-
fosa ka prw tow lgoiw pasi tn retn tw
eknaw perixei tn te noern ka tn cuxikn ka
tn fusikn ka psaw n tjei paraddvsi tw tn
politein diakosmseiw ka n at deknusi tw poi-
klaw atn metabolw, ka tata mn @lvw ka
gnvstikw nergosa, tw d lhw faptomnh pol-
lw f' autw pistmaw kddvsin, oon tn gev-
desan, tn mhxanikn, tn ptikn, di' n ka tn
ynhtn bon eergete. ka gr polemistria rgana
ka fulaktria tn plevn di totvn kateskeu-
sato ka tw tn rn peridouw gnvrmouw poh-
sen ka tw tn tpvn yseiw, mtra te fhgsato
t mn tn kat gn dn, t d tn kat ylaw-
san, zug te ka trutnaw dhmiorghsen, f' n
tn kat' riymn sthta taw plesi dihkrbvsen,
to te pantw ksmou tn tjin di' eknvn mfan
katsthsen ka poll tow nyrpoiw p pstvn
nfhne ka pist psin deijen: oon d ka Irvn
Surakosiow epen lgetai per Arximdous, te
tn trirmenon kateskease nan, n pareskeuzeto
pmpein Ptolema t basile t Agupt. pn-
tvn gr ma Surakousvn lksai tn nan o du-
namnvn Arximdhw tn Irvna mnon atn kat-
agagen pohsen. kataplagew d kenow Ap ta-
thw, fh, tw mraw per pantw Arximdei l-
gonti pisteuton. t d at ka Glvn fasin
epen, nka to stefnou m luyntow, n kat-
eskeasen, tn lkn ksthn neren tn sugkra-
yeisn ln.
Tata mn on pollo tn presbutrvn ngra-
can, tn mayhmatikn gkvmizein proymenoi, ka
di tata lga p polln mew n totoiw pare-
ymeya tn tw gevmetraw pantelw gnsin ka f-
leian pideiknntew: tn d gnesin atw tn n t
perid tat met tata lekton. mn gr dai-
mniow Aristotlhs epn t at dojsmata
pollkiw ew nyrpouw fiknesyai kat tinaw te-
tagmnaw peridouw to pantw, ka m kay' mw
prton tow f' mn gnvsyntaw tw pistmaw
sstasin laben, ll ka n llaiw periforaw od'
epen psaiw taw te genomnaiw ka taw ayiw so-
mnaiw kfanna te ka fanisynai plin atw.
pe d xr tw rxw ka tn texnn ka tn pi-
sthmn prw tn parosan perodon skopen, lgo-
men, ti par' Aguptois mn ersyai prton
gevmetra par tn polln strhtai, k tw tn
xvrvn nametrsevw labosa tn gnesin. nag-
kaa gr n kenoiw ath di tn nodon to Ne-
lou tow proskontaw rouw kstoiw fanzon-
tow. ka yaumastn odn p tw xreaw rjasyai
tn eresin ka tathw ka tn llvn pisthmn,
peid pn t n gensei fermenon p to telow
ew t tleion preisin. p asysevw on ew lo-
gismn ka p totou p non metbasiw gnoito
n ektvw. sper on par tow Fonijin di tw
mporeaw ka t sunallgmata tn rxn laben
tn riymn kribw gnsiw, otv d ka par'
Aguptois gevmetra di tn erhmnhn atan
erhtai. Yals d prton ew Agupton lyn met-
gagen ew tn Ellda tn yevran tathn ka
poll mn atw eren, polln d tw rxw tow
met' atn fhgsato, tow mn kayolikteron pi-
bllvn, tow d asyhtikteron. met d toton
Mmerkos [?] Sthsixrou to poihto delfw,
w facmenow tw per gevmetran spoudw mnh-
moneetai, ka Ippaw Hleow strhsen w p
gevmetr& djan ato labntow. p d totoiw
Puyagras tn per atn filosofan ew sxma
paideaw leuyrou metsthsen, nvyen tw rxw
atw piskopomenow ka @lvw ka noerw t yev-
rmata diereunmenow, w d ka tn tn lgvn
pragmatean ka tn tn kosmikn sxhmtvn ssta-
sin neren. met d toton Anajagras Klazo-
mniow polln fcato tn kat gevmetran ka
Onopdhs Xow, lg neterow n Anajagrou,
n ka Pltvn n tow nterastaw mnhmneusen
w p tow maymasi djan labntvn. f' ow Ippo-
krths Xow tn to mhnskou tetragvnismn
ern, ka Yedvros Kurhnaow gnonto per
gevmetran pifanew. prtow gr Ippokrthw tn
mnhmoneuomnvn ka stoixea sungracen. Pltvn
d' p totoiw genmenow megsthn pohsen pdosin
t te lla maymata ka tn gevmetran laben di
tn per at spoudn, w pou dlw sti ka t
suggrmmata tow mayhmatikow lgoiw katapuknsaw
ka pantaxo t per at yama tn filosofaw
ntexomnvn pegervn. n d tot t xrn ka
Levdmas Ysiow n ka Arxtas Tarantnow
ka Yeathtos Ayhnaow, par' n phujyh t
yevrmata ka prolyen ew pisthmonikvtran s-
stasin. Levdmantow d neterow Neokledhs
ka totou mayhtw Lvn, o poll proseuprh-
san tow pr atn, ste tn Lonta ka t stoixea
sunyenai t te plyei ka t xre& tn deiknum-
nvn pimelsteron, ka diorismow eren, pte duna-
tn sti t zhtomenon prblhma ka pte dnaton.
Edojos d Kndiow, Lontow mn lg neterow,
tarow d tn per Pltvna genmenow, prtow
tn kaylou kaloumnvn yevrhmtvn t plyow
hjhsen ka taw trisn nalogaiw llaw trew prow-
yhken ka t per tn tomn rxn labnta par
Pltvnow ew plyow progagen ka taw nalsesin
p' atn xrhsmenow. Amklas d Hraklethw,
ew tn Pltvnow tarvn ka Mnaixmos kroa-
tw n Edjou ka Pltvni d suggegonw ka
delfw ato Deinstratos ti televtran poh-
san tn lhn gevmetran. Yedios d Mgnhw n
te tow maymasin dojen enai diafrvn ka kat
tn llhn filosofan: ka gr t stoixea kalw sun-
tajen ka poll tn rikn [?] kayoliktera poh-
sen. ka mntoi ka Kuzikhnw Aynaios kat
tow atow gegonw xrnouw ka n tow lloiw mn
maymasi, mlista d kat gevmetran pifanw g-
neto. digon on otoi met' lllvn n Akadhm&
koinw poiomenoi tw zhtseiw. Ermtimos d
Kolofniow t p' Edjou prohuporhmna ka
Yeaittou progagen p plon ka tn stoixevn
poll nere ka tn tpvn tin sungracen. F-
lippos d Mendaow, Pltvnow n mayhtw ka
p' kenou protrapew ew t maymata, ka tw zh-
tseiw poieto kat tw Pltvnow fhgseiw ka
tata proballen aut, sa eto t Pltvnow fi-
losof& suntelen. o mn on tw storaw nagr-
cantew mxri totou progousi tn tw pistmhw
tathw televsin. o plu d totvn neterw stin
Ekledhs t stoixea sunagagn ka poll
mn tn Edjou suntjaw, poll d tn Yeaittou
televsmenow, ti d t malakteron deiknmena tow
mprosyen ew nelgktouw podejeiw nagagn. g-
gone d otow nr p to prtou Ptolemaou:
ka gr Arximdhs pibaln ka t prt mnh-
moneei to Ekledou, ka mntoi ka fasin ti
Ptolemaow ret pote atn, e tw stin per gev-
metran dw suntomvtra tw stoixeisevw: d
pekrnato, m enai basilikn trapn p gevme-
tran. neterow mn on sti tn per Pltvna,
presbterow d Eratosynouw ka Arximdouw. otoi
gr sgxronoi llloiw, w po fhsin Eratosy-
nhs. ka t proairsei d Platvnikw sti ka t
filosof& tat okeow, yen d ka tw sumpshw
stoixeisevw tlow proestsato tn tn kaloumnvn
Platvnikn sxhmtvn sstasin. poll mn on ka
lla to ndrw totou mayhmatik suggrmmata
yaumastw kribeaw ka pisthmonikw yevraw mest. toiata gr ka t ptik ka t katoptrik, toiatai d ka a kat mousikn stoixeiseiw, ti d
t per diairsevn biblon. diaferntvw d' n tiw
atn gasyeh kat tn gevmetrikn stoixevsin
tw tjevw neka ka tw klogw tn prw t stoi-
xea pepoihmnvn yevrhmtvn te ka problhmtvn.
ka gr ox sa nexrei lgein ll' sa stoixeion
dnato parelhfen, ti d tow tn sullogismn
pantoouw trpouw, tow mn p tn atvn lamb-
nontaw tn pstin, tow d p tekmhrvn rmhm-
nouw, pntaw d nelgktouw ka kribew ka prw
pistmhn okeouw, prw d totoiw tw meydouw
psaw tw dialektikw, tn mn diairetikn n taw
ersesi tn edn, tn d ristikn n tow osi-
desi lgoiw, tn d podeiktikn n tow p tn
rxn ew t zhtomena metabsesi, tn d naluti-
kn n taw p tn zhtoumnvn p tw rxw na-
strofaw. ka mn ka t poikla tn ntistrofn
edh tn te ploustrvn ka tn sunyetvtrvn ka-
nw stin n t pragmate& tat dihkribvmna yev-
ren, ka tna mn la loiw ntistrfein dnatai,
tna d la mresi ka npalin, tna d w mrh
mresin. ti d lgomen tn sunxeian tn ersevn,
tn okonoman ka tn tjin tn te prohgoumnvn
ka tn pomnvn, tn dnamin, mey' w kasta para-
ddvsin. ka t tuxn prosyew feln ok pi-
stmhw lanyneiw popesn ka ew t nanton cedow
ka tn gnoian penexyew; peid d poll fan-
tzetai mn w tw lhyeaw ntexmena ka taw pi-
sthmonikaw rxaw kolouyonta, fretai d ew tn
p tn rxn plnhn ka tow pipolaiotrouw
japat, meydouw paraddvken ka tw totvn di-
oratikw fronsevw, w xontew gumnzein mn dunh-
smeya tow rxomnouw tw yevraw tathw prw
tn eresin tn paralogismn, nejapthtoi d dia-
mnein. ka toto d t sggramma, di' o tn para-
skeun mn tathn ntyhsi, Ceudarvn pgracen,
trpouw te atn poiklouw n tjei diariymhsmenow
ka kay' kaston gumnsaw mn tn dinoian pan-
tooiw yevrmasi ka t cedei t lhyw parayew
ka t per& tn legxon tw pthw sunarmsaw.
toto mn on t biblon kayartikn sti ka gumna-
stikn, d stoixevsiw atw tw pisthmonikw
yevraw tn n gevmetr& pragmtvn nlegkton
xei ka telean fghsin.
Tw on skopw tw pragmateaw tathw svw
rseta tiw, g d ka prw toton epoimi n, ti
dioriston stn tn pryesin kat te t prgmata,
per n a zhtseiw, ka kat tn manynonta. ka
prw mn at t pokemena blpontew lgomen, w
ra per tn kosmikn sxhmtvn stn smpaw t
gevmtr lgow, rxmenow mn p tn pln, te-
leutn d ew tn poikilan tw totvn sustsevw,
ka xvrw mn kasta fistw, mo d tw ew tn
sfaran atn ggrafw ka tow lgouw ow xei
prw llhla paradidow. di ka tn kay' kasta
biblvn tow skopow tinew p tn ksmon naf-
rein jvsan ka tn xrean atn, n parxetai
prw tn to pantw yevran ngracan. prw d
tn manynonta diorizmenoi tn skopn at toto,
lgetai, stoixevsin at prokesyai fsomen ka
televsin tw tn manyanntvn dianoaw prw tn
smpasan gevmetran. p gr totvn rmmenoi
ka t lla gnnai dunhsmeya tw pistmhw ta-
thw mrh, ka tn poikilan tn n at perilaben
neu totvn dnaton mn stin ka lhptow tn
llvn myhsiw. t gr rxoeidstata ka plostata
yevrmata ka suggenstata taw prtaiw poyse-
sin ntaya sunyroistai tjin labnta tn prpou-
san ka a tn llvn podejeiw totoiw w gnvri-
mvttaiw xrntai ka p totvn rmhntai. kayper
d ka Arximdhs n tow per sfaraw ka ku-
lndrou ka Apollnios ka o lloi pntew fa-
nontai tow n at t pragmate& dedeigmnoiw [s]
rxaw mologoumnaiw xrmenoi.
Skopw mn on otow, stoixeisa te prw tn
lhn pistmhn tow manynontaw ka tn kosmikn
sxhmtvn divrismnaw paradonai sustseiw. at
d toto t tw stoixeisevw noma ka t to stoi-
xeou, par' ka stoixevsiw, tna n xoi lgon;
na d ka per tw pigrafw ti zhtsvmen. tn
tonun yevrhmtvn t mn eyasi stoixea kalen,
t d stoixeidh, t d jv tw totvn forzetai
dunmevw. stoixea mn on ponomzontai, n
yevra diiknetai prw tn tn llvn pistmhn,
ka f' n paragnetai mn tn n atow prvn
dilusiw. w gr tw ggrammtou fvnw esin
rxa prtai ka plostatai ka diaretoi, aw t
noma tn stoixevn pifhmzomen, ka psa ljiw
k totvn fsthken ka pw lgow, otv d ka tw
lhw gevmetraw st tina yevrmata prohgomena
ka rxw lgon xonta prw t fejw ka dikonta
di pntvn ka parexmena polln podejeiw sum-
ptvmtvn, d stoixea prosagoreousi. stoi-
xeidh d' stn sa diatenei mn p plev ka t
plon xei ka t xaren, okti mn ka tn tn
stoixevn [jan] t m prw psan atn tn pi-
stmhn koinn enai tn yevran, oon tow trignoiw
tw p tn gvnin kaytouw p tw plagaw kay'
n shmeon sumpptein. sa te mte ew plyow xei
dikousan tn gnsin mte a glafurn ti profa-
nei ka xaren, tata ka tw tn stoixeivdn jv
pptei dunmevw.
Plin d t stoixeon lgetai dixw, w fhsn
Mnaixmos. ka gr t kataskeuzon st to kata-
skeuazomnou stoixeon, w t prton par' Ekled
to deutrou, ka to pmptou t ttarton. otv
d ka lllvn enai poll stoixea =hysetai: kata-
skeuzetai gr j lllvn. deknutai gr ka k
to ttrasin ryaw enai saw tw jv tn eyu-
grmmvn gvnaw t plyow tn ntw ryaw svn
ka npalin k totou keno. ka oiken lmmati
t toioto stoxeon. llvw d lgetai stoixeon, ew
plosteron prxon diairetai t snyeton: o-
tvw d o pn ti =hysetai pantw stoixeon, ll
t rxoeidstera tn n potelsmatow lg tetag-
mnvn, sper t atmata stoixea tn yevrhmtvn.
kat d toto to stoixeou t shmainmenon ka t
par' Ekled stoixea sunetxyh, t mn tw per
t ppeda gevmetraw, t d tw stereometraw. otv
d ka n tow riymhtikow ka n tow stronomikow
stoixeiseiw pollo sungracan.
Esti d toto xalepn ka t kljasyai ka
tjai kat trpon t stoixea kay' ksthn pist-
mhn, f' n t lla progetai pnta ka ew t
lla naletai. ka tn pixeirhsntvn o mn plev,
o d lttv sunagagen dunyhsan, ka o mn bra-
xutraiw podejesin xrsanto, o d ew mkow p-
ranton jteinan tn yevran, ka o mn tn di'
duntou trpon jklinan, o d tn nalogan, o
d prokataskeuw mhxansanto prw tow nairon-
taw tw rxw, ka lvw pollo tinew erhntai tr-
poi tw stoixeisevw kstoiw. de d tn toiathn
pragmatean pn mn peskeusyai t perittn^
mpdion gr toto prw tn myhsin^klgein
d t sunxonta pnta ka sungonta t prokemenon
^nusimtaton gr toto prw tn pistmhn^
safhneaw d' ma ka suntomaw polln pepoisyai
prnoian^t gr nanta totvn piyolo tn
dinoian mn^tw te tn yevrhmtvn n prasi
kayolikow perilcevw nteilfyai^t gr ew
t meriktera temaxzonta tn didaskalan dusper-
lhpton pergzetai tn gnsin. kat pntaw d to-
touw tow trpouw eroi tiw n tn Ekledou
stoixevsin tn llvn diafrousan: t mn gr
xrsimon atw ew tn per tn rxikn sxhmtvn
suntele yevran, t d safw ka dihryrvmnon
p tn ploustrvn p t poikiltera metbasiw
pergzetai ka p tn koinn nnoin katabol
tw yevraw, t d kayolikn tw podejevw di
tn prtvn yevrhmtvn ka rxoeidn p t zh-
tomena metbasiw. ka gr sa paralimpnein doke,
taw ataw fdoiw ggnetai gnrima [tow erhm-
nois?], sper sstasiw to skalhno ka soske-
low, w mxanon esgonta ka pranton poiki-
lan lltria tw tn stoixevn stn klogw, sper
t per tn tktvn lgvn, Apollnios p
plon jeirgsato, w atvn tn paradedomnvn
xei tn sstasin, sper t edh tn gvnin t
poll ka tn grammn. tata gr paralleiptai
mn ka par' lloiw tuxe lgou pleonow, xei d tn
gnsin p tn pln. tosata per tw lhw stoi-
xeisevw exomen nagrfein.
Tn d smpasan okonoman tn n at lgvn
de pvw nadidjomen. peid tn pistmhn ta-
thn tn gevmetran j poysevw ena famen ka
p rxn rismnvn t fejw podeiknnai^
ma gr nupyetow, a d llai par' kenhw po-
dxontai tw rxs^ngkh d pou tn tn n
gevmetr& stoixevsin suntttonta xvrw mn para-
donai tw rxw tw pistmhw, xvrw d t p tn
rxn sumpersmata, ka tn mn rxn m did-
nai lgon, tn d pomnvn taw rxaw. odema
gr pistmh tw autw rxw podeknusin, od
poietai lgon per atn, ll' atopstvw xei per
atw, ka mlln esin at katafanew tn fejw.
ka tw mn oden di' atw, t d met tata di'
kenaw. otv gr ka fusiolgow p' rxw riw-
mnhw progei tow lgouw poymenow enai knhsin,
ka atrw ka tn llvn pisthmn [?] ka texni-
tn kastow. e d tiw ew tatn sumfrei tw te
rxw ka t p tn rxn, otow pitarttei tn
smpasan gnsin ka sugkuk t mhdn proskonta
llloiw. rx gr ka t p' atw fsei diristai
lllvn.
Prton mn on, per fhn, dei diastelasyai
tw te rxw ka t pmena taw rxaw, d ka
poie Ekledhs kay' kaston w epen biblon
ka pr pshw tw pragmateaw tw koinw tw pi-
stmhw tathw rxw ktiymenow. peita ka atw
diaire tw koinw rxw ew te tw poyseiw ka t
atmata ka t jimata. diafrei gr tata pnta
lllvn ka ok stin tat jvma ka athma ka
pyesiw, w po fhsin daimniow Aristotlhs,
ll' tan mn ka t manynonti gnrimon ka
kay' at pistn t paralambanmenon ew rxw
tjin, jvma t toiotn stin, oon t t t at
sa ka llloiw sa enai. tan d m x mn n-
noian kovn to legomnou tn atpiston, tye-
tai d mvw ka sugxvre t lambnonti, t toioton
pyesw sti. t gr enai tn kklon sxma toon
kat koinn mn nnoian o proeilfamen didktvw,
kosantew d sugxvromen podejevw xvrw. tan
d a ka gnvston t legmenon ka m sugxv-
rontow to manynontow mvw lambnhtai, thni-
kata, fhsn, athma toto kalomen, oon t psaw
tw ryw gvnaw saw enai. dhlosi d o per tinow
tn athmtvn katapragmatesasyai spoudsantew,
w p mhdenw atyen sugxvresyai dunamnou.
ka kat mn tn Aristotlous fghsin toton
diristai tn trpon jvma ka athma ka pye-
siw. pollkiw d ka pnta tata kalosin poy-
seiw, sper o p tw Stos jvma psan p-
fansin pln, ste kat mn totouw ka a poy-
seiw jimata, kat d tow trouw ka t jimata
poyseiw.
Plin d' a t p tn rxn ew problmata
diairetai ka yevrmata, t mn tw genseiw peri-
xonta tn sxhmtvn ka tw tomw ka tw fair-
seiw prosyseiw ka lvw t paymata t gignmena
per at, t d kay' at sumbebhkta kstoiw
deiknonta. kayper gr a poihtika tn pisth-
mn yevraw metxousin, kat t at d ka a
yevrhtika t problmata taw poisesin nlogon
proseilfasin. dh d tn palain o mn pnta
yevrmata kalen jvsan, w o per Spesippon
ka Amfnomon, gomenoi taw yevrhtikaw pi-
stmaiw okeiotran enai tn tn yevrhmtvn prow-
hgoran tn tn problhmtvn, llvw te ka per
dvn poioumnaiw tow lgouw. o gr sti gnesiw
n tow doiw, ste od t prblhma xran p
totvn n xoi, gnesin paggellmenon ka pohsin
to mpv prteron ntow, oon soplerou trignou
sstasin, tetragnou doyeshw eyeaw nagrafn,
ysin eyeaw prw t doynti shme. meinon
on fasi lgein, ti pnta tat sti, tw d gen-
seiw atn o poihtikw ll gnvstikw rmen w-
ane gignmena lambnontew t e nta, ste ka
pnta yevrhmatikw romen ll' o problhmatikw
lambnesyai. o d npalin pnta problmata l-
gein dikaoun w o per Mnaixmon mayhmatiko,
tn d proboln enai dittn: te mn porsasyai
t zhtomenon, te d perivrismnon labntaw den
tw stn, pon ti, t pponyen, tnaw xei
prw llo sxseiw. ka lgousi mn ryw mfte-
roi: ka gr o per Spesippon kals^o gr
toiat sti t problmata gevmetraw, oa t mh-
xanikw: asyht gr tata ka gnesin xonta ka
pantoan metaboln^ka o per tn Mnaixmon
^o gr neu tw ew lhn prodou ka a tn
yevrhmtvn esn erseiw. lgv d lhn tn noh-
tn. ew kenhn on o lgoi prontew ka morfon-
tew atn ektvw dpou taw gensesin oiknai l-
gontai. tn gr tw dianoaw mn knhsin ka tn
proboln tn n at lgvn gnesin tn n fan-
tas& sxhmtvn ena famen ka tn per at
payhmtvn. ke gr a sustseiw ka a toma ka
a yseiw ka a parabola ka a prosyseiw ka a
fairseiw, t d n t diano& pnta sthken neu
gensevw ka pshw metabolw.
Esti mn on ka problmata gevmetrik ka
yevrmata, diti d yevra t pleonzon stn n
at, sper p mhxanikw poiseiw, ka t probl-
mata pnta metxei yevraw, o mn npalin: lvw
gr a podejeiw yevraw esn rgon. pnta d t
n gevmetr& t met tw rxw di' podejevw lam-
bnetai, ste kointeron t yerhma. o pnta d
t yevrmata detai tn problhmtvn, ll' stn
ka atyen xei tn pdeijin to zhtoumnou. o
d diorzontew t yerhma to problmatw fasi
pn mn prblhma pidxesyai tn kathgoroumnvn
tw n at lhw, at te kaston ka t ntikeme-
non, pn d yerhma at mn pidxesyai t kat-
hgoromenon, o mntoi ka t ntikemenon. lgv
d lhn mn atn t gnow, per o zthsiw, oon
trgvnon tetrgvnon kklon, smptvma d
kathgoromenon t kay' at sumbebhkw, oon son
tomn ysin llo ti toioton. tan on pro-
ten tiw otvw, ew kklon ntenai trgvnon s-
pleuron, prblhma lgei. dunatn gr ew atn
ntenai ka m spleuron: ka plin, p tw do-
yeshw eyeaw peperasmnhw sustsasyai trgvnon
spleuron, prblhma t toinde: dunatn gr sust-
sasyai ka m spleuron. tan d tn soskeln
saw enai tw prw t bsei proten tiw, yerhma
faton atn protenein. o gr dunatn ka m
saw enai tw prw t bsei tn soskeln: ste e
tiw problhmatikw sxhmatsaw epoi, ew mikklion
ryn ntenai gvnan, gevmetrtou djan n lboi.
psa gr n mikukl ry stin. f' n tonun
t smptvma kayolikn sti ka ps t l par-
omarton, tata yevrmata lekton, f' n d m
kaylou mhd t pokeimn pntvw pmenon, pr-
blhma t toioton yeton. tn doyesan eyean pe-
perasmnhn dxa temen: ka gr ew nisa dunatn
^psan gvnan eygrammon dxa temen: sti gr
ka ew nisa diaresis^p tw doyeshw eyeaw
nagrcai tetrgvnon: dunatn gr ka m tetr-
gvnon^ka pnta, sa toiata, tw tn problh-
mtvn stn tjevw. o d per Zhndoton tn
proskonta mn t Onopdou diadox, tn mayh-
tn d Andrvnos, dirizon t yerhma to pro-
blmatow, t mn yerhma zhte, t sti t sm-
ptvma t kathgoromenon tw n at lhw, t d
prblhma, tnow ntow t stin. yen ka o per
tn Poseidnion t mn fvrzonto prtasin, kay'
n zhtetai t e stin m, t d prblhma prta-
sin, n zhtetai t stin pon ti, ka tn mn
yevrhtikn prtasin legon den pofantikw sxh-
matzein, oon pn trgvnon mezouw xei tw do tw
loipw, ka pantw soskelow a prw t bsei sai,
tn d problhmatikn, sper zhtontaw, e stin p
tsde tw eyeaw sustsasyai trgvnon. diafrein
gr, plw te ka orstvw zhten, e sti prw
ryw p tode to shmeou tde [t] eye&, tw
stin prw ryw yevren.
All' ti mn sti tiw diafor to te problma-
tow ka to yevrmatow, dlon k totvn, ti d ka
Ekledou stoixevsiw xei t mn problmata t
d yevrmata, fanern stai toto di tn kay'
kaston ka ato prostiyntow p tlei tn deiknu-
mnvn pou mn t "per dei poisai" pou d t
"per dei dejai", w tn yevrhmtvn xarakthri-
stikn, katoi, kayper epomen, oshw ka n tow
problmasin podejevw, ll' mvw, pou mn ka
pdeijiw tw gensevw xrin^na gr dejvmen,
ti pepohtai t protaxyn, tn pdeijin paralam-
bnomen^pou d at di' autn stin spoudw
ja tn fsin to zhtoumnou paristnein duna-
mnh. eroiw d' n tn Ekledhn tot mn sum-
plkonta t yevrmata tow problmasi ka par
mrow atow xrmenon, w n t prt bibl, tot
d pleonzonta kat t tera. t mn gr ttarton
lon problhmtvn sti, t d pmpton yevrhmtvn.
Tosata ka per totvn mn ersyv: met
d tata tn skopn forismenoi to prtou
biblou ka tn diaresin parastsantew rjmeya
tw per tow rouw pragmateaw. mn on prye-
sw stin n tot t bibl tw rxw paradonai
tw tn eyugrmmvn yevraw. e gr ka fsei
krettvn kklow ka per atn pragmatea tw
tn eyugrmmvn osaw te ka gnsevw, ll' mn
proskousa mllon per totvn didaskala tow
telestroiw ka p tn asyhtn p t noht
metgein tn dinoian spedousin. ka gr tow mn
asyhtow okea t eygramma sxmata, tow d
nohtow kklow, diti d t mn plon ka mono-
eidw ka rismnon proskei t fsei tn ntvn,
t d poiklon ka orstvw ajanmenon t plyei
tn periexousn pleurn diafrei tow asyhtow.
n tot d on t bibl t prtista ka rxo-
eidstata sxmata tn eyugrmmvn paraddotai,
t te trgvnon lgv ka t parallhlgrammon: n
gr totoiw w n gnei perixetai ka t atia tn
stoixevn, t te soskelw ka t skalhnn ka t
sunistmena k totvn, t te spleuron trgvnon
ka tetrgvnon, f' n t sxmata tn tettrvn
stoixevn sxen tn sstasin. ersomen on ka
soplerou trignou ka tetragnou gnesin, to mn
p tw doyeshw eyeaw, to d p tw doyeshw.
t on spleuron trgvnon prosexw atin sti
tn trin stoixevn purw row datow, t d te-
trgvnon tw gw. sunrthtai d on skopw to
prtou biblou ps t pragmate& ka suntele
prw lhn tn tn kosmikn stoixevn yevran. ti
d ka stoixeio tow manynontaw ew tn per tn
eyugrmmvn sxhmtvn pistmhn tw prtaw atn
rxw kalw neurn ka di' kribeaw katadsa-
menow.
Dirhtai d t biblon ew tra mgista tm-
mata, ka t mn prton tn trignvn tw genseiw
ka tw dithtaw mfanzei kat te gvnaw ka pleu-
rw ka poietai sugkrseiw atn prw llhla ka
kasta f' auto yevre. ka gr n trgvnon la-
bn pot mn p tn pleurn piskope tw gvnaw,
pot d p tn gvnin tw pleurw, sthtw te
pri ka nisthtow, ka do poymenow t at
plin di poiklvn erskei. t d deteron tn
per tn parallhlogrmmvn jufanei yevran, tw
te dithtaw tn paralllvn ka tw genseiw tn
parallhlogrmmvn nagrfon ka ti t sumpt-
mata t n atow podeiknw. t d trton tn
koinvnan tn te trignvn ka tn parallhlo-
grmmvn nafanei, n te tow sumptmasi ka taw
prw llhla sugkrsesi. ka gr t p tn atn
bsevn ka svn trgvna parallhlgramma deknu-
tai t at peponyta, ka met sumplokw mfot-
rvn p miw ntvn bsevw, ka pw n gnoito son
trign parallhlgrammon, ka tlow per tn na-
grafomnvn n tow ryogvniaoiw trignoiw tetra-
gnvn p tn pleurn tna xei lgon t p tw
poteinoshw tn ryn prw t p tn periexou-
sn atn. toa d tiw stv ka pryesiw to pr-
tou biblou tw stoixeisevw ka diaresiw.
Arxn d poiomenoi tw tn kay' kasta zht-
sevw proagoreomen tow nteujomnoiw, m tata
par' mn paiten sa diateyrlhtai tow pr mn
lhmmtia ka ptseiw ka e ti toioto. totvn mn
gr diakorew smn ka spanvw atn facmeya.
sa d pragmateivdestran xei yevran ka suntele
prw tn lhn filosofan, totvn prohgoumnhn
poihsmeya tn pmnhsin, zhlontew tow Puyago-
reous, ow prxeiron n ka toto smbolon "sxma
ka bma, ll' o sxma ka tribolon" ndeiknum-
nvn, w ra de tn gevmetran kenhn metadikein,
kay' kaston yerhma bma tyhsin ew nodon
ka parei tn cuxn ew cow, ll' ok n tow a-
syhtow katabanein fhsin ka tn snoikon tow
ynhtow xrean poplhron ka tathw stoxazomnhn
tw nteyen periagvgw katamelen.
Copyright 1999 Giovanni Ferrero