Teone di Smirne
De utilitate mathematicarum ad legendum Platonis ...
Introductio
Oti mn ox on te sunenai tn mayhmatikw
legomnvn par Pltvni m ka atn skhmnon n
t yevr& tat, pw n pou mologseien: w d
od t lla nvfelw od nnhtow per tata
mpeira, di polln atw mfanzein oike. t mn
on sumpshw gevmetraw ka sumpshw mousikw ka
stronomaw mpeiron genmenon tow Pltvnow sug-
grmmasin ntugxnein makaristn mn e t gnoito,
o mn eporon od =dion ll pnu pollo to
k padvn pnou demenon. ste d tow dihmarthk-
taw to n tow maymasin skhynai, regomnouw d
tw gnsevw tn suggrammtvn ato m pantpasin
n poyosi diamarten, kefalaidh ka sntomon
poihsmeya tn nagkavn ka n de mlista tow
nteujomnoiw Pltvni mayhmatikn yevrhmtvn par-
dosin, riymhtikn te ka mousikn ka gevmetrikn
tn te kat stereometran ka stronoman, n xvrw
ox on te ena fhsi tuxen to rstou bou, di pol-
ln pnu dhlsaw w o xr tn mayhmtvn melen.
Eratosynhw mn gr n t pigrafomn Pla-
tvnik fhsin ti, Dhloiw to yeo xrsantow p
pallag loimo bvmn to ntow diplasona kata-
skeusai, polln rxitktosin mpesen poran zh-
tosin pvw xr steren stereo gensyai diplsion,
fiksyai te peusomnouw per totou Pltvnow. tn
d fnai atow, w ra o diplasou bvmo yew
demenow toto Dhloiw mantesato, profrvn d ka
neidzvn tow Ellhsin melosi mayhmtvn ka gev-
metraw ligvrhksin.
koloyvw d t to Puyou parainsei poll ka
atw dijeisin pr to n tow maymasi xrhsmou. n
te gr t Epinomdi protrpvn p t maymat fhsin:
o gr neu totvn pot tiw n plei edaimnvn
gensetai fsiw, ll' otow trpow, ath trof,
tata t maymata, ete xalep ete =dia, di ta-
thw ton: melsai d o yemitn sti yen. ka n
tow fejw tn toiotn fhsin k polln na gegonta
edamon te sesyai ka softaton ma ka makrion.
n d t Polite& fhsn: k tn ke' tn o pro-
kriyntew timw te tn llvn mezouw osontai, t te
xdhn maymata psin n t paide& genmena totoiw
sunakton ew snocin okeithtw te lllvn tn
mayhmtvn ka tw to ntow fsevw. paraine te
prton mn mpeiron gensyai riymhtikw, peita gev-
metrikw, trton d stereometraw, ttarton strono-
maw, n fhsin enai yevran feromnou stereo, pm-
pton d mousikw. t te xrsimon paradeiknw tn
mayhmtvn fhsn: dw e, ti oikaw dedinai, m
xrhsta t maymata prostttoimi. t d' stin o
pnu faloiw, ll psi xalepn pisteuynai, ti n
totoiw tow maymasin kstou oon rgnoiw t cu-
xw kkayaretai ka nazvpuretai mma tuflomenon
ka posbennmenon p tn llvn pithdeumtvn,
kretton n svynai murvn mmtvn: mn gr at
lyeia rtai.
n d t bdm tw Politeaw per riymhtikw
lgvn w stin nagkaiotth pasn fhsin, peita w
de psaiw mn txnaiw, psaiw d dianoaiw ka pist-
maiw ka t polemik. paggloion gon strathgn
Agammnona n taw tragdaiw Palamdhw kstote
pofanei. fhs gr riymn ern tw te tjeiw
katastsai t stratopd n Il ka jariymsai
naw te ka t lla pnta, w pr to nariymtvn
ntvn ka to Agammnonow w oiken od souw exe
pdaw edtow, ege m pstato riymen. kinduneei
on tn prw nhsin gntvn fsei enai, ka odew
at xrtai lktik nti prw osan ka nosevw
paraklhtik. sa mn gr plw kine tn asyhsin,
ok stin pegertik ka paraklhtik nosevw, oon
ti rmenow dktulw sti, ka ti paxw leptw
mgaw mikrw. sa d' nantvw kine asyhsin,
pegertik ka paraklhtik sti dianoaw, oon tan t
at fanhtai mga ka mikrn, kofon ka bar, n
ka poll. ka t n on ka riymw paraklhtik
ka pegertik sti dianoaw, pe t n pote poll
fanetai: logistik d ka riymhtik lkw ka gv-
gw prw lyeian. pton d logistikw m divtikw,
ll' w n p yan tw tn riymn fsevw fkvn-
tai t nosei, od prsevw xrin mprvn kap-
lvn meletntaw, ll' neka cuxw tw p' lyeian
ka osan do. toto gr nv gei tn cuxn ka
per atn tn riymn nagkzei dialgesyai, ok
podexmenon, n tiw at smata a t rat
xonta riymow prosfermenow dialghtai. ka plin
n t at fhsin: ti o logistiko ew panta t
maymata jew fontai, o te bradew ew t jte-
roi ato atn gensyai. ti n t at fhsi: ka
n polm d' a xrsimon prw tw stratopedeseiw
ka katalceiw xvrvn ka junagvgw ka jetseiw
stratiw. n te tow jw painn tn per t toiata
maymata spoudn, gevmetra mn, fhsn, st per
tn to pipdou yevran, stronoma d per tn to
stereo forn: ath d' nagkzei ew t nv rn ka
p tn nynde kese gei. ka mn d per mousi-
kw n t at fhsin, ti duen detai tn ntvn
yevra, stronomaw ka rmonaw: ka atai delfa
a pistmai, w o Puyagoriko. o mn on tw
kouomnaw sumfvnaw a ka fyggouw llloiw na-
metrontew nnuta ponosi. televw parabllontew
t ta, oon k geitnvn fvnn yhrmenoi, o mn
fasin koein n ms tin xon ka mikrtaton enai
disthma toto, metrhton, o d mfisbhtosin w
moion dh fyeggomnou, t ta to no prosths-
menoi. taw xordaw prgmata parxousin p tn koll-
bvn streblontew. o d gayo riymhtiko zhtosin
piskopontew, tnew smfvnoi riymo riymow ka
tnew o. ka toto xrsimon prw tn to gayo
ka kalo zthsin, llvw d xrhston. ka totvn
pntvn myodow n mn p tn lllvn fkhtai
koinvnan ka jullogisy stin llloiw okea, f-
rei atn pragmatea karpn. o d tata deino
dialektiko: o gr m dnvntai laben te ka po-
djasyai lgon. ox on te d toto m di' kenvn
lynta tn mayhmtvn: dw gr sti di' atn p
tn tn ntvn yan n t dialgesyai.
plin te n t Epinom poll mn ka lla pr
riymhtikw diejrxetai, yeo dron atn lgvn, ka
ox on te neu tathw spoudaon gensyai tin. po-
bw d ntikrw fhsin: eper gr riymn k tw n-
yrvpnhw fsevw jloimen, ok n pou ti frnimoi
genomeya, od' n ti pot totou to zou, fhsn,
cux psan retn lboi: sxedn totou lgow
eh. zon d ti m ginskoi do ka tra mhd pe-
rittn mhd rtion, gnoo d t parpan riymn, ok
n pote didnai lgon, per n asyseiw ka mnmaw
mnon eh kekthmnow: stermenow d lhyow lgou
sofw ok n pote gnoito. o mn od t tn llvn
texnn legmena, nn dilyomen, odpote totvn
odn mnei, pnta d poletai t parpan, tan
riymhtikw tiw mel. djeie d' n svw tis braxvw
riymo desyai t tn nyrpvn gnow, w ew tw
txnaw poblcasi: katoi mga mn ka toto. e d
tiw doi t yeon tw gensevw ka t ynhtn, n ka
t yeosebw gnvrisysetai ka riymw ntvw, ok
n ti pw mntiw gnoh smpanta riymn, shw mn
dunmevw atiow n eh sugginmenow, pe ka mousi-
kn psan di' riymo met kinsew te ka fyggvn
dlon ti de. ka t mgiston, gayn w pntvn a-
tion: ti d kakn odenw sti, toto gnvston. sxe-
dn d lgistow, taktow, sxmvn te ka rruymow
nrmostw te sfdra ka pny' sa kako kekoin-
nhk tinow, stiw lleiptai pantw riymo. n d tow
fejw fhsin: stin xon mhdew mw pote peiytv
tw esebeaw enai t ynht gnei. k gr totou
fesyai ka tw llaw retw t maynti kat trpon.
peita paradeknusi yeosbeian t trp tiw mayse-
tai. lgei d den mayen prton stronoman. e gr
t katacedesyai ka nyrpvn deinn, pol deinte-
ron yen: katacedoito d' n ceudew xvn djaw
per yen: ceudew d' n djaw xoi per yen mhd
tn tn asyhtn yen fsin peskemmnow, toutstin
stronoman. gnoesyai d fhsi tow pollow, ti so-
ftaton ngkh tn lhyw stronmon enai, m tn
kay' Hsodon stronomonta, oon dusmw te ka na-
tolw peskemmnon, ll tw peridouw tn pt,
m =&dvw pot psa fsiw kan gnoito yevrsai.
tn d' p tata paraskeuzonta fseiw oaw dunatn
pollw prodidskein xrea stn yzonta pada nta
ka neanskon di mayhmtvn: n t mgiston enai
riymn pistmona atn, ll' o smata xntvn,
ka atw tw to peritto te ka rtou gensew te
ka dunmevw, son parxetai prw tn tn ntvn
fsin. totoiw d fejw maymata mn kalosi, fhs,
sfdra geloon noma gevmetran: sti d tn ok
ntvn movn llloiw fsei riymn movsiw prw
tn tn pipdvn moran. lgei d tina ka tran
mpeiran ka txnhn, n d stereometran kale, e tiw,
fhs, tow trew riymow j n t ppeda enai ajh-
yntaw moouw ka nomoouw ntaw, w proepon, ste-
re poie smata: toto d yen te ka yaumastn sti.
ka n Polite& d per sumfvnaw tw kat mou-
sikn fhsi: kallsth ka megsth tn per plevn
sumfvnin stin sofa, w mn kat lgon zn
mtoxow, d poleipmenow okofyrow ka per plin
odam svtriow, te t mgista mayanvn.
ka n t trt d tw Politeaw, didskvn ti
mnow mousikw filsofow, fhsn: r' on prw yen
otvw od mousiko prteron smeya, ote ato ote
ow famen mew paideuton enai tow flakaw, prn
n panta t tw svfrosnhw edh ka ndreaw ka
megaleithtow ka megaloprepeaw ka sa totvn
delf ka t totvn penanta pantax perifer-
mena xvrzvmen ka nnta n ow stin asyanmeya
ka at ka eknaw atn ka mte n mikrow mte
n megloiw timzvmen, ll tw atw omeya txnhw
enai ka melthw; di gr totvn ka tn pr atn
t te felow k mousikw dhlo, ka ti mnow ntvw
mousikw filsofow, mousow d kakw. t mn gr
ehye& ntvw, tiw stn ret t e t yh kate-
skeuasmna xein, pesya fhsin elogan, toutsti t
e lg xrsyai, t d elog& tn esxhmosnhn ka
eruyman ka earmostan: esxhmosnhn gr per
mlow, earmostan d per rmonan, eruyman d
per =uymn: t d kakohye&, toutsti t kak yei,
fhsn pesyai kakologan, toutsti kako lgou xr-
sin, t d kakolog& sxhmosnhn ka rruyman ka
narmostan per pnta t genmena ka mimomena:
ste mnow n eh mousikw kurvw eyhw, stiw eh
n filsofow. dhlo d ka t erhmna. pe gr
mousik t eruymon ka ermoston ka esxhmon
mpoie t cux k nou esduomnh di t t fele&
memigmnhn xein blab donn, dnatn fhsi tleon
mousikn gensyai m edta t n pant esxhmon ka
t tw esxhmosnhw ka leuyerithtow ka svfros-
nhw edh m gnvrzonta, toutsti tw daw. mlei pi-
frei: n pant perifermena^toutsti t edh^ka
m timzvn at mt' n smikrow mt' n megloiw.
d tn den gnsiw per tn filsofon: od gr
edeh tiw n t ksmion ka sfron ka esxhmon
atw n sxmvn ka klastow: t d' n b esxh-
mon ka eruymon ka ermoston eknew tw ntvw
esxhmosnhw ka earmostaw ka eruymaw, toutsti
tn nohtn ka den eknew t asyht.
ka o Puyagoriko d, ow pollax petai Pltvn,
tn mousikn fasin nantvn sunarmogn ka tn
polln nvsin ka tn dxa fronontvn sumfrnhsin:
o gr =uymn mnon ka mlouw suntaktikn, ll'
plw pantw sustmatow: tlow gr atw t non
te ka sunarmzein. ka gr yew sunarmostw tn
diafvnontvn, ka toto mgiston rgon yeo kat
mousikn te ka kat atrikn t xyr fla poien.
n mousik, fasn, mnoia tn pragmtvn, ti ka
ristokrata to pantw: ka gr ath n ksm mn
rmona, n plei d' enoma, n okoiw d svfrosnh
gnesyai pfuke: sustatik gr sti ka nvtik tn
polln: d nrgeia ka xrsiw, fhs, tw pist-
mhw tathw p tessrvn gnetai tn nyrvpnvn,
cuxw, smatow, okou, plevw: prosdetai gr tata
t tssara sunarmogw ka suntjevw.
n d t Polite& Pltvn pr tn mayhmtvn
ka tde fh: gayw d nr stiw diaszei tn ryn
djan tn k paideaw at ggenomnvn n te lpaiw
ka donaw ka piyumaiw ka fboiw ka m kbllei.
d moi doke moion enai, ylv peiksai. o nn
bafew, peidn boulhysi bcai ria st' enai
lourg, prton mn klgontai k tosotvn xrvm-
tvn man fsin tn tn leukn, peita prokata-
skeuzousin ok lg paraskeu yerapesantew, pvw
djhtai ti mlista t nyow, ka otvw bptousi: ka
mn n tot t trp baf, mo ti t bafn ka
fsiw, ka ote neu =ummtvn ote met =ummtvn
dnatai atn t nyow fairesyai: d' n m,
osya oa d gnetai, n m proyerapesaw bpt,
kpluta ka jthla ka o deusopoi. toioto d kat
dnamin rgzesyai gesyai xr ka mw: paideomen
gr tow padaw n mousik te ka gumnastik ka
grmmasi ka gevmetr& ka n riymhtik, odn llo
mhxanmenoi, pvw mew proekkayrantew ka pro-
yerapesantew sper tis stuptikow tow maymasi
totoiw, tow per pshw retw n n kmanynvsin
steron lgouw ndejointo sper bafn, na deuso-
poiw atn dja gnoito, di t tn fsin ka
trofn pithdean sxhknai, ka m kpln atn
tn bafn t =mmata tata, dein nta kklzein,
te don, pantw streblo deinotra osa ka koinv-
naw, lph te ka fbow ka piyuma, pantw llou
=mmatow.
ka gr a tn filosofan mhsin fah tiw n
lhyow teletw ka tn ntvn w lhyw musthrvn
pardosin. musevw d mrh pnte. t mn prohgo-
menon kayarmw: ote gr pasi tow boulomnoiw met-
ousa musthrvn stn, ll' esn ow atn erge-
syai proagoreetai, oon tow xeraw m kayarw ka
fvnn jneton xontaw, ka atow d tow m ergo-
mnouw ngkh kayarmo tinow prteron tuxen. met
d tn kyarsin deutra stn tw teletw pardosiw:
trth d <> ponomazomnh poptea: tetrth d,
d ka tlow tw popteaw, ndesiw ka stemmtvn
pyesiw, ste ka troiw, w tiw parlabe teletw,
paradonai dnasyai, d&douxaw tuxnta erofantaw
tinow llhw ervsnhw: pmpth d j atn peri-
genomnh kat t yeofilw ka yeow sundaiton edai-
mona. kat tat d ka tn Platvnikn lgvn par-
dosiw t mn prton xei kayarmn tina, oon tn n tow
proskousi maymasin k padvn suggumnasan. mn
gr Empedoklw krhnvn p pnt' nimnt fhsin tei-
ri xalk den porrptesyai: d Pltvn p pnte
mayhmtvn den fhsi poiesyai tn kyarsin: tata
d' stn riymhtik, gevmetra, stereometra, mousik,
stronoma. t d telet oiken tn kat filosofan
yevrhmtvn pardosiw, tn te logikn ka politikn
ka fusikn. poptean d nomzei tn per t
noht ka t ntvw nta ka t tn den prag-
matean. ndesin d ka katstecin ghton t j
n atw tiw katmayen on te gensyai ka trouw
ew tn atn yevran katastsai. pmpton d' n eh
ka teletaton k totvn perigenomnh edaimona
ka kat' atn tn Pltvna movsiw ye kat t
dunatn.
poll mn on ka lla xoi tiw n lgein para-
deiknw t tn mayhmtvn xrsimon ka nagkaon.
to d m doken peiroklvw diatrbein t tn
mayhmtvn pan trepton dh prw tn pardosin
tn nagkavn kat t maymata yevrhmtvn, ox
sa dnaito n tn ntugxnonta riymhtikn telvw
gevmtrhn mousikn stronmon pofnai: od
gr sti toto prohgomenon prokemenon pasi
tow Pltvni ntugxnousi: mna d tata paradso-
men, sa jarke prw t dunhynai sunenai tn
suggrammtvn ato. od gr atw jio ew sxaton
graw fiksyai diagrmmata grfonta ka meldan,
ll paidik oetai tata t maymata, propara-
skeuastik ka kayartik nta cuxw ew t pitdeion
atn prw filosofan gensyai. mlista mn on
xr tn mllonta ow te mew paradsomen ow te
Pltvn sungracen ntejesyai di gon tw prthw
grammikw stoixeisevw kexvrhknai: =on gr n
junpoito ow paradsomen. stai d' mvw toiata
ka t par' mn, w ka t pantpasin mut tn
mayhmtvn gnrima gensyai.
prton d mnhmonesomen tn riymhtikn yevrh-
mtvn, ow sunzeuktai ka t tw n riymow mousikw:
tw mn gr n rgnoiw o pantpasi prosdemeya, kay
ka atw Pltvn fhgetai lgvn w o xr sper
k geitnvn fvnn yhreuomnouw prgmata parxein
taw xordaw: regmeya d tn n ksm rmonan ka
tn n tot mousikn katanosai: tathn d ox
on te katiden m tw n riymow prteron yevrhti-
kow genomnouw. di ka pmpthn Pltvn fhsn
enai tn mousikn, tn n ksm lgvn, tiw stn n
t kinsei ka tjei ka sumfvn& tn n at kinou-
mnvn strvn. mn d' nagkaon deutran atn
tttein met riymhtikn ka kat' atn tn Pltvna,
peid od' n ksm mousik lhpt neu tw jariy-
moumnhw ka nooumnhw mousikw. ste e mn sun-
zeuktai t per cilow riymow yevr& n riymow
mousik, deutra n taxyeh prw tn tw metraw
yevraw emreian. prw d tn fusikn tjin prth
mn n eh per riymow yevra, kaloumnh riy-
mhtik: deutra d per t ppeda, kaloumnh gev-
metra: trth d per t stere, tiw st stereome-
tra: tetrth per t kinomena stere, tiw
stn stronoma. d tw tn kinsevn ka diasth-
mtvn poi sxsiw st mousik, tiw ox oa t sti
lhfynai m prteron mn atn n riymow kata-
nohsntvn: di prw tn metran yevran met' riy-
mhtikn tetxyv n riymow mousik, w d prw tn
fsin pmpth <> tw to ksmou rmonaw yevrh-
tik mousik. kat d tow Puyagorikow presbeuta
t tn riymn w rx ka phg ka =za tn
pntvn.
Copyright 1999 Giovanni Ferrero