Procli
In primum Euclidis Elementorum Liber
Prologi pars prior.
Tn mayhmatikn osan ote tn prvtstvn
n tow osi genn ote tn sxtvn enai ka par
tn pln dirhmnvn, nagkaon, ll tn mshn
xran peilhfnai tn te merstvn ka pln ka
sunytvn ka diairtvn postsevn ka tn meri-
stn ka n sunysesin pantoaiw ka poiklaiw diai-
rsesin fvrismnvn. t mn gr e kat tat
xon ka mnimon ka nlegkton tn per atn l-
gvn peranxousan atn pofanei tn n l fe-
romnvn edn, t d diejodikn tn piboln ka t
taw diastsesi tn pokeimnvn prosxrmenon ka
t p' llvn rxn lla prokataskeuzon katade-
estran at ddvsi tjin tw merstou ka n at
televw drumnhw fsevw. diper omai ka Pl-
tvn tw gnseiw direi tn ntvn taw te prtaiw
ka msaiw ka teleutaaiw postsesi ka tow mn
merstoiw tn nohtn peddou tn yrvw ka met
plthtow diairosan t noht ka t te #l& ka
t kayarthti ka t noeide prosbol ka paf
tn ntvn tw llaw perarousan gnseiw, tow d
meristow ka fsin sxthn laxosi ka asyhtow
pasi tn djan mudrw lhyeaw ntilambanom-
nhn, tow d msoiw, oa d sti t tw mayhmatikw
edh, ka tw te merstou fsevw poleipomnoiw ka
tw meristw peridrumnoiw tn dinoian. ka gr
ath no mn sti deutra ka tw krotthw pist-
mhw, djhw d teleiotra ka kribestra ka kaya-
rvtra. diejodeei mn gr ka naplo to no
tn metran ka nelssei t sunespeiramnon tw
noerw pibolw, sungei d a plin t dirhmna
ka nafrei prw tn non. sper on a gnseiw
p' lllvn diestkasin, otv d ka t gnvst
diakkritai fsei, ka t mn noht pntvn per-
plvtai taw noeidsin prjesi, t d asyht tow
psin polepetai tn prtvn osin. t d mayh-
matik ka lvw t dianoht mshn keklrvtai tjin,
tn mn t diairsei pleonzonta, tn d t #l&
proxonta, ka tn mn t plthti leipmena, tn
d t kribe& prouprxonta ka tranestraw mn
mfseiw xonta tn asyhtn tw nohtw osaw,
eknew d mvw nta ka meristw mn t mrista,
polueidw d monoeid paradegmata tn ntvn po-
mimomena ka w sullbdhn epen n proyroiw mn
tetagmna tn prtvn edn ka kfanonta tn
nvmnhn ka mer ka gnimon kenvn parjin,
opv d ra to merismo ka tw sunysevw tn
lgvn perxonta ka tw proshkoshw taw eksin
postsevw, od perdrmonta tw poiklaw ka
diejodikw tw cuxw noseiw ka ataw sunarmo-
synta taw plaw ka lhw pshw kayareuosaiw
gnsesin.
H mn d mesthw tn mayhmatikn genn te
ka edn toiath noesyv, prw ge t parn t me-
taj sumplhrosa tn te pantelw meristn o-
sin ka tn per tn lhn meristn ginomnvn: tw
d rxw tw mayhmatikw lhw osaw piskopontew
p' atw nimen tw di pntvn tn ntvn dihko-
saw rxw ka pnta f' autn pogennsaw, lgv
d t praw ka t peiron. k gr totvn tn do
prtvn met tn to nw perighton ka tow pa-
sin lhpton atan psth t te lla pnta ka
tn mayhmtvn fsiw, kenvn mn yrvw pnta
paragousn ka jrhmnvw, tn d prontvn n
mtroiw tow proskousi ka tjei t prepos tn
prodon katadexomnvn, ka tn mn prtvn, tn
d msvn, tn d teleutavn fistamnvn. t mn
gr noht gnh kat tn autn plthta prtvw
metxei to pratow ka to perou di mn tn
nvsin ka tn tautthta ka tn mnimon parjin
ka stayern to pratow poplhromena, di d
tn ew plyow diaresin ka tn gennhtikn peri-
ousan ka tn yean terthta ka prodon tw
peiraw polaonta. t d mayhmatik pratow mn
stin kgona ka peiraw, ll' o tn prvtstvn
mnvn od tn nohtn ka krufvn rxn, ll
ka totvn, a prolyon mn p' kenvn ew deut-
ran tjin, pogennn d met' lllvn jarkosi
tow msouw diaksmouw tn ntvn ka tn n atow
poikilan, yen d ka n totoiw prorxontai mn
ew peiron o lgoi, kratontai d p tw pratow
ataw. te gr riymw p mondow rjmenow
pauston xei tn ajhsin, e d lhfyew pep-
rastai, ka tn megeyn diaresiw p' peiron xv-
re, t d diairomena pnta ristai, ka kat' nr-
geian peprastai t mria to lou. ka tw mn
peiraw ok oshw t te megyh pnta smmetra n
n ka odn rrhton od logon, ow d doke dia-
frein t n gevmetr& tn n riymhtik, ka o
riymo tn gnimon tw mondow dnamin ok n
dnanto deiknnai od n pntaw exon tow l-
gouw n autow tn ntvn, oon tow pollaplasouw
tow pimorouw. pw gr riymw jallttei tn
lgon prw tn monda ka tn pr ato genmenon
jetazmenow. to d pratow naireyntow summetra
te ka koinvna lgvn ka tautthw edn ka sthw
ka sa tw menonw sti sustoixaw ok n pote
n tow maymasin faneto, od' n pistmai tn
toiotvn san od katalceiw mnimoi ka kribew.
de tonun mfotrvn tn rxn sper tow lloiw
gnesi tn ntvn otv d ka tow mayhmatikow. t
d sxata ka n l fermena ka p tw fsevw
diaplattmena pntvw atyen mfon metxonta ka-
tafanetai, to mn perou kat tn pokeimnhn
atow dran tn edn, to d pratow kat tow
lgouw ka t sxmata ka tw morfw.
All' ti mn rxa ka tn mayhmtvn atai
proestkasin, a ka tn ntvn pntvn, fanern.
sper d tw koinw rxw atn teyevrkamen ka
di pntvn dihkosaw tn mayhmatikn genn, otv
d ka t koin atn yevrmata ka pl ka tw
miw pistmhw ggona tw psaw mo tw mayhmati-
kw gnseiw n n pexoshw nalogismeya, ka
pvw farmttei psaiw ka dnatai ka n riymow
ka n megyesi ka n kinsesi yevresyai skop-
svmen. toiata d sti t te tn nalogin ka t
tn sunysevn ka diairsevn ka tn nastrofn
ka nallagn, ti d t tn lgvn pntvn oon
pollaplasvn ka pimorvn [ka] pimern ka tn
totoiw ntikeimnvn ka plw t per t son ka
nison kaylou yevromena ka koinw, o kayson
stn n sxmasin riymow kinsesin, ll'
at kay' at totvn kteron fsin tin xon
koinn ka gnsin auto parexmenon ploust-
ran. ka mn ka t kllow ka tjiw koin pn-
tvn st tn mayhmtvn ka p tn gnvrimv-
trvn dw p t zhtomena ka k totvn p'
kena metbasiw, w d kalosin nalseiw ka sun-
yseiw. te moithw ka nomoithw tn lgvn
od' tion tn mayhmatikn genn polepousin.
ka gr sxmata t mn moia t d nmoia lgo-
men ka riymow satvw tow mn moouw tow d
nomoouw. ka sa kat tw dunmeiw nafanetai
psin movw proskei tow maymasi, tn mn duna-
mnvn tn d dunasteuomnvn. d ka n poli-
te& Svkrthw taw mosaiw chlologoumnaiw n-
yhken, t koin pntvn tn mayhmatikn lgvn n
prasin rismnoiw perilabn ka prosthsmenow n
tow erhmnoiw riymow, f' n d ka t mtra tw
te egonaw ka tw nantaw prw tathn gonaw
katafanetai.
De d ra t koin tata mte n tow pollow
ka dirhmnoiw edesi prtvw festnai nomzein,
mte strvw ka k tn polln xonta tn gnesin,
ll' w pr atn stta ka plthti ka kribe&
diafronta tyesyai. di gr toto ka gnsiw
atn prohgetai tn polln gnsevn ka ddvsi
tw rxw kenaiw, ka a polla per atn fest-
kasi ka p' atn nafrontai. legtv gr gev-
mtrhw, ti tettrvn ntvn megeyn nlogon stai
ka t nallj, ka deikntv kat tw okeaw rxw,
aw riymhtikw ok n pote xrsaito, ka a
riymhtikw, ti tettrvn ntvn riymn nlogon
stai ka t nallj, ka toto p tn tw okeaw
pistmhw rxn. tw on kay' autn gnvrzvn
t nallj ete n megyesin ete n riymow ka tn
diaresin tn sugkeimnvn megeyn riymn ka
tn snyesin satvw tn dirhmnvn; o gr d
pou tn mn meristn esn pistmai ka gnseiw,
tn d @lvn ka tw noerw yevraw ggutrv te-
tagmnvn od man xomen pistmhn. ll poll
prteron kenvn gnsiw stn pistmh ka p'
kenhw a polla tow koinow podxontai lgouw
ka mxri tosotou gnsevn nodow p tn meri-
kvtrvn p tw likvtraw, vw n p' atn na-
drmvmen tn to ntow, n stin, pistmhn. ath
gr o t kay' at tow riymow prxonta sko-
pen jio, od t koin psi tow posow, ll tn
ntvn pntvn tn man ka mnhn osan ka par-
jin yevre, ka di toto pasn st tn pisthmn
perilhptikvtth ka psai par' kenhw lambnousi
tw rxw. e gr a nvtrv taw p' atw par-
xousi tw prtaw tn podejevn poyseiw, d
teleiotth tn pisthmn psaiw f' autw nd-
dvsi taw mn likvtraw taw d merikvtraw rxw:
di' ka n Yeaitt Svkrthw paidin spoud ke-
rannw peristeraw mn peikzei tw n mn pist-
maw. ptesyai d atw fhsn tw mn kat' glaw
tw d ka xvrw p tn llvn. a mn gr koin-
terai ka likterai pollw n autaw perixousi
merikvtraw, a d tn dirhmnvn kat' edh gnv-
stn faptmenai diestkasin lllvn ka snaptoi
prw lllaw esn p diaferousn rmhmnai tn
prtvn rxn. ma tonun pistmh protetxyv
tn polln pisthmn ka mayhmtvn t koin
ka di pntvn dikonta tn genn gnvrzousa ka
psaiw taw mayhmatikaw pistmaiw xorhgosa tw
rxw.
Ka mxri tode n mn per atw fvrsyv
didaskala. met d toto t pot' n eh t krit-
rion tn mayhmtvn yevrsvmen ka prosthsmeya
ka tw totou paradsevw gemna tn Pltvna
diairomenon n polite& xvrw mn t gnvst xvrw
d tw gnseiw ka suzgvw ponmonta tow gnvstow
tw gnseiw. tn gr ntvn t mn noht ymenow
t d asyht, tn d' a nohtn t mn noht plin
t d dianoht, ka tn asyhtn t mn asyht t
d ekast, tow mn nohtow, d tn tettrvn st
genn prtista, gnsin fsthsi tn nhsin, tow
d dianohtow dinoian, tow d asyhtow pstin ka
tow ekastow ekasan. ka toton xousan tn l-
gon pofanei tn ekasan prw tn asyhsin, n
dinoia prw tn nhsin: te gr ekasa t edvla
gignskei tn asyhtn n te dasi fantazmena
ka tow lloiw katptroiw sxthn pvw n edesin
xonta tjin ka edlvn ntvw edvla gegonta, ka
dinoia tw tn nohtn eknaw yevre tw p
tn prtvn ka pln ka meristn edn ew pl-
yow ka diaresin pobsaw, di' ka tathw mn
gnsiw p' llvn poysevn rthtai presbutrvn,
d nhsiw p' atn neisi tn nupyeton rxn.
e tonun t mayhmatik mte tn mriston laxen
pstasin ka xvristn p pshw diairsevw ka
poikilaw mte tn asysei gnvrizomnhn ka pol
metbolon ka pnth meristn, pant pou katafanw,
ti dianoht mn sti kat tn osan, dinoia d
atn prosthken w kritrion, sper tn asyh-
tn asyhsiw ka ekasa tn ekastn. yen d ka
Svkrthw mudrotran mn tn totvn gnsin
enai diorzetai tw prvtsthw pistmhw, tranest-
ran d tw dojastikw pibolw. t mn gr nei-
ligmnon ka diejodikn tw yevraw pleonzon xousi
tw nosevw, t d mnimon tn lgvn ka nleg-
kton perxon tw djhw. ka t mn j poysevw
rmhmnon kat tn fesin laxon tw prthw pi-
stmhw, t d n @loiw edesin festw kat tn
teleiotran tn asyhtn edhsin.
T mn on kritrion tn mayhmtvn pntvn
toinde kat tn non forizmeya to Pltvnos,
tn dinoian djhw mn peridrsasan autn, tw
d nosevw poleipomnhn. petai d pou katiden
mw, tna tn osan proskei lgein tn mayhma-
tikn edn te ka genn, ka pteron p tn
asyhtn atn fistnein sugxvrhton ete kat
faresin, w pou lgein eyasin, ete kat yroi-
sin tn merikn ew na tn koinn lgon, ka pr
totvn at tn pstasin doton, w te Pl-
tvn jio ka tn lvn pideiknei prodow. pr-
ton mn on e p tn asyhtn t mayhmatik
edh lgomen postnai, tw cuxw p tn n l
trignvn kklvn t edow t kuklikn t tri-
gvnikn sterogenw n aut morfoshw, pyen
krbeia ka t nlegkton prxei tow lgoiw;
ngkh gr, p tn asyhtn p cuxw. ll
mn p ge tn asyhtn dnaton, poll gr n
mllon kribeaw totoiw metn: p tw cuxw
ra, tow mn telsi t tleion tow d m kribsi
t kribw prostiyeshw. po gr n tow asyhtow
t merw t platw t bayw, po d tn
k to kntrou grammn sthw, po d o sttew
e lgoi tn pleurn, po d a tn gvnin ry-
thtew; ox rmen, w n llloiw pnta t asyht
summmiktai ka w odn n totoiw elikrinw od
to nantou kayareon, ll merist pnta ka dia-
stat ka kinomena; pw on tow kintoiw lgoiw
k tn kinoumnvn ka llote llvw xntvn atn
tn mnimon osan dsomen; pn gr t p kinou-
mnvn osin fistmenon ka parjin metablhtn
xein mologetai par' atn. pw d tow kribsi
ka nelgktoiw edesin p tn m kribn tn
krbeian prosysomen; pn gr t tw kintou
gnsevw ation meiznvw stn at toioton. cuxn
ra tn gennhtikn poyeton tn mayhmatikn
edn te ka lgvn. ll' e mn xousa t para-
degmata kat' osan fsthsin at, ka esn a
gennseiw probola tn n at prouparxntvn
edn, t te Pltvni sunesmeya tata lgontew
ka tn lhy osan tn mayhmtvn erhktew n
ehmen. e d m xousa mhd proeilhfua tow l-
gouw tosoton fanei dikosmon #lon ka tosathn
pogenn yevran, pw t gennhynta dnatai dia-
krnein, ete gnima tugxnei nta ete nemiaa ka
edvla nt' lhyn, pooiw d kansi xrvmnh tn
n totoiw lyeian parametre; pw d ka m xousa
tn osan atn pogenn tosathn poikilan l-
gvn; htomatismnhn gr otvw tn pstasin a-
tn poisomen ka prw odna ron naferomnhn.
e ra cuxw stin kgona t mayhmatik edh ka
ok p tn asyhtn xei tow lgouw, n fsth-
sin cux, ka p' kenvn tata problletai ka
a dnew atw ka o tkoi menntvn es ka dvn
kfanew edn.
Deteron tonun e ktvyen ka p tn asyh-
tn yrozomen tow tn mayhmtvn lgouw, pw
ok ngkh tw podejeiw menouw lgein, sai p
tn asyhtn sunstantai, ka o tw p tn ka-
yolikvtrvn e ka ploustrvn edn; t gr atia
pantaxo taw podejesin okea prw tn to zh-
toumnou yran enai famn. e on t merik tn
kaylou ka t asyht tn dianohtn atia, tw mh-
xan tn ron tw podejevw p t kaylou ml-
lon nafrein nt tn meristn ka tn dianohtn
tn osan pr tn asyhtn taw podejesin sug-
genestran pofanein; od gr e tiw fasn po-
dejeien, ti t soskelw duen ryaw saw xei tw
gvnaw, ka ti t spleuron d ka t skalhnn
pstatai kat trpon, ll' pn trgvnon ka
plw podejaw xei tn pistmhn kay' at. ka
plin ti t kaylou bltion to kat mrow prw
pdeijin, ka jw ti a podejeiw k tn kaylou
mllon, j n d a podejeiw, tata prtera ka t
fsei prohgomena tn kay' kasta ka atia tn
deiknumnvn. pollo ra dousin a podeiktika
tn pisthmn per t sterogen ka t mudrtera
tn asyhtn yren, ll' o t diano& lhpt ka
teleitera tn asysei ka dj gnvrmvn yevren.
Eti d t trton lgomen, ti ka tn cuxn
timotran poiosi tw lhw o tata lgontew. e
gr mn lh t osidh ka mllon nta ka tra-
nstera par tw fsevw dxetai, d cux detera
p' kenvn ka edvla ka eknaw sterogenew n
aut diaplttei prw osan timotraw fairosa
tw lhw t kat fsin atw xrista, pw ox
tn cuxn dranestran tw lhw ka katadeestran
pofanousi; tpow mn gr ka lh tn nlvn
lgvn ka cux tn edn. ll' mn tn pr-
tvn d tn deutrvn, ka mn tn prohgoum-
nvw ntvn d tn keyen fistamnvn, ka mn
tn kat' osan, d tn kat' pnoian genomnvn.
pw on no ka tw noerw osaw prtvw met-
xousa ka plhroumnh tw gnsevw keyen ka tw
lhw zvw mudrotrvn edn stin podox tw
sxthw n tow osin draw ka prw t enai pn-
tvn telestraw; ll prw mn tathn pantn tn
djan pollow pollkiw eynaw dedvkuan perergon.
e d m sti kat faresin tn nlvn t mayh-
matik edh mte kat sunayroismn tn n tow kay'
kasta koinn, mhy' lvw sterogen ka p tn
asyhtn, ngkh dpou tn cuxn par' atw
par no lambnein at ka par' atw ka par'
kenou. ll' e mn par' atw mnon, pw eknew
tata tn noern edn; pw d metaj tw mer-
stou fsevw ka tw meristw mhdeman p tn
prtvn ew t enai sumplrvsin laxnta; pw d
prvtourg paradegmata tn lvn t n n pro-
sthken; e d par' kenou mnon, pw t atenr-
ghton tw cuxw ka atoknhton dnatai mnein,
eper o n at lgoi kat tn tn terokintvn
pstasin llaxyen ew atn rreusan; ka t dio-
sei tw lhw tw dunmei mnon oshw pnta, genn-
shw d odn tn nlvn edn; lepetai d on ka
par' atw ka par no tata pargein ka enai
plrvma tn edn atn, p mn tn noern pa-
radeigmtvn fistamnvn, atognvw d tn ew t
enai prodon lagxanntvn. ka ok ra n cux
grammateon ka tn lgvn kenn, ll gegrammnon
e ka grfon aut ka p no grafmenon. now
gr sti ka cux kat tn pr ato non ne-
lttvn autn ka ekn kenou ka tpow jv gen-
menow. e on kenow pnta noerw, ka cux t
pnta cuxikw, ka e paradeigmatikw kenow, ka
cux ekonikw, ka e sunrhmnvw, cux dirh-
mnvw. d ka Pltvn edw k pntvn f-
sthsi tn mayhmatikn edn tn cuxn ka kat'
riymow atn diaire ka sunde taw nalogaiw
ka tow rmonikow lgoiw, ka tw prvtourgow r-
xw tn sxhmtvn n at katablletai, t te ey
ka t periferw, ka kine tow n at kklouw noe-
rw. pnta ra t mayhmatik prtn stin n t
cux ka pr tn riymn o atoknhtoi ka pr
tn fainomnvn sxhmtvn t zvdiak sxmata ka
pr tn rmosmnvn o rmoniko lgoi ka pr tn
kkl kinoumnvn svmtvn o fanew kkloi dedh-
miorghntai ka plrvma tn pntvn cux: ka
dikosmow otow llow atw autn pargvn ka
p tw okeaw rxw paragmenow zvw te plhrn
autn ka p to dhmiourgo plhromenow svm-
tvw ka diasttvw, ka t' n probll tow ato
lgouw, tte ka pistmaw profanei psaw ka
retw. osvtai on n totoiw cux tow edesi
ka ote tn riymn p' atw mondvn plyow
polhpton ote tn tn diastatn dan svmati-
kw kouston, ll pnta zvtikw ka noerw t
paradegmata tn fainomnvn riymn ka sxhm-
tvn ka lgvn ka kinsevn poyeton pomnoiw t
Tima psan atw tn gnesin ka tn dhmiour-
gan p tn mayhmatikn edn sumplhrsanti ka
pntvn n at tw ataw drsanti. tn mn gr
riymn pntvn o pt roi tw rxw perieilfasi
tn grammikn ka tn pipdvn ka tn steren,
tn d lgvn pntvn o pt lgoi kat' atan n
at pro#festkasi, tn d' a sxhmtvn a rxa
dhmiourgikw drsyhsan n at, tn d kinsevn
prvtsth ka tw llaw psaw perixousa ka
kinosa sunufsthken at. pntvn gr tn kinou-
mnvn kklow rx ka kkl knhsiw. osideiw
ra ka atoknhtoi tn mayhmtvn esn o lgoi
sumplhrontew tw cuxw, ow d ka probllousa
dinoia ka jelttousa psan tn poikilan f-
sthsi tn mayhmatikn pisthmn, ka o m pote
pashtai, gennsa mn e ka neurskousa lla
p' lloiw, tow d merew atw lgouw japlosa.
pnta gr proelhfen rxoeidw ka kat tn pei-
ron autw dnamin k tn proeilhmmnvn rxn
pantodapn yevrhmtvn poietai probolw.
All d met tn osan tn mayhmatikn
edn p tn man atn pistmhn nadrmvmen,
n pr tn polln deknumen osan, ka yevrsv-
men, t t rgon atw ka tnew a dunmeiw, ka p
pson diatenousai taw nergeaiw.
T mn d rgon tw lhw mayhmatikw dianohti-
kn, sper t prteron epomen, yeton ka ote toi-
oton, poon t noern n aut monmvw drumnon
ka tleion ka atarkew f' auto ka prw aut
suneon, ote oon t tw djhw ka tw asysevw.
atai gr a gnseiw prw t ktw peredontai ka
per kena nergosi ka tw ataw ok xousi tn
gignvskomnvn. d' a mayhmatik tw mn na-
mnsevw jvyen rxetai, teleut d ew tow ndon
lgouw, ka negeretai mn p tn strvn, katant
d ew tn prohgoumnhn osan tn edn, ka k-
nhtow mn atw ok stin nrgeia kayper
noer, di d kinsevw o topikw od lloivti-
kw sper a asyseiw, ll zvtikw nelssetai ka
dijeisi tn smaton tn lgvn dikosmon, tte
mn p tn rxn p t potelsmata xvrosa,
tte d npalin deousa, ka tte mn p tn
progignvskomnvn p t zhtomena, tte d p tn
zhtoumnvn p t prohgomena kat tn gnsin.
ti tonun ote w plrhw autw pshw zhtsevw
perdrutai kayper now, ote f' trvn telei-
otai w asyhsiw, ll di zhtsevw ew tn ere-
sin preisin ka p to telow ew teleithta
pneisi.
Dunmeiw ge mn xei dittw, tw mn ew pl-
yow proagosaw tw rxw ka pogennsaw tw polu-
eidew tw yevraw trapow, tw d sunagvgow tn
polln diejdvn ew tw okeaw poyseiw. diti
gr rxw proestsato t te n ka t plyow, t te
praw ka t peiron, t pokemena at prw tn
katlhcin mshn te klhrsato tjin tn merstvn
edn ka tn pnth meristn, ektvw d omai ka
a gnvstika dunmeiw tw lhw atn pistmhw
dipla pefnasin osai, ka a mn mn ew tn
nvsin spedousai ka sumptssousai t plyow, a
d diakritika tn pln ew t poikla ka tn
kayolikvtrvn ew t meriktera, ka tn n rx
lgvn tetagmnvn ew t detera ka t pollost
p tn rxn. nvyen gr rxomnh dikei mxri
tn asyhtn potelesmtvn ka sunptei prw tn
fsin ka sunapodeknusi poll met tw fusiolo-
gaw, sper d ktvyen paniosa suneggzei pvw t
noer gnsei ka fptetai tw tn prtvn yevraw.
di d ka n taw poperatsesin autw tn te
mhxanikn lhn probalen ka tn ptikn ka kat-
optrikn yevran ka llaw pollw sumpeplegmnaw
tow asyhtow ka di' kenvn nergosaw, ka n taw
ndoiw tn merstvn ka @lvn nosevn ntilam-
bnetai ka met' kenvn teleio tw meristw pibo-
lw ka tw n diejdoiw feromnaw gnseiw, t te
autw gnh ka edh taw osaiw kenaiw fomoio,
ka tn per yen lyeian ka tn per tn ntvn
yevran n tow okeoiw kfanei logismow. tosata
ka per totvn ersyv.
T d nteyen tw pistmhw tathw katdvmen
jafnhw p tn rxhgikvttvn gnsevn mxri tn
sxtvn diatenon. mn on Tmaiow kat padeu-
sin dn tn tn mayhmtvn gnsin pokale. diti
d toton xei tn lgon prw tn tn lvn pi-
stmhn ka tn prthn filosofan, n paidea
prw tn retn. mn gr proeutrepzei tn cuxn
yesin diastrfoiw ew tn telean zvn, d pro-
paraskeuzei tn dinoian mn ka t mma tw cu-
xw ew tn nteyen periagvgn. di ka n poli-
te& Svkrthw ryw epen: mma gr tw cuxw
p tn llvn pithdeumtvn potuflomenon ka
katoruttmenon p tn mayhmtvn mnvn nazv-
puresya te ka negeresyai plin ew tn yan
to ntow ka p tn edlvn p t lhy ka
p to skotdouw ew t noern meystasyai fw,
ka lvw p to sphlaou ka tn n tot gene-
siourgn desmn ka tn gktrvn tw lhw p
tn smaton natenasyai ka mriston osan.
t te gr kllow ka tjiw tn n mayhmatik
lgvn ka t mnimon ka stw tw yevraw atow
mw sunptei tow nohtow ka nidrei telvw, e
mn stsin, e d t ye kllei diaprpousin, e
d tn prw llhla tjin diaszousin. O d n t
Fadr Svkrthw trew mn paraddvsi tow na-
gomnouw, o ka tn prtiston ato sumplhrosi
bon: tn filsofon, tn rvtikn, tn mousikn.
ll t mn rvtik tw nagvgw rx ka dw
nteyen p to fainomnou kllouw panabasmow
xrvmn tow msoiw edesi tn kaln, t d mou-
sik trthn laxnti tjin p tn n asysesin
rmonin p tw fanew rmonaw ka tow lgouw
tow n tataiw metbasiw. ka t mn ciw, t
d ko tw namnsevw rganon. t d filosf
tn fsin pyen ra ka di tnvn naknhsiw tw
noerw gnsevw ka prw t ntvw n ka tn l-
yeian gersiw; de gr ka tot di t telw tw
okeaw rxw. gr fusik ret ka mma telw
ka yow laxen. ggertai mn on dh par' ato
ka pthtai per t n tn fsin toiotow, doton
d at t maymata, fhsn Plvtnos, prw
suneyismn tw svmtou fsevw ka totoiw sper
sxmasi xrmenon kton p tow n t dialektik
lgouw ka lvw w tn tn ntvn yevran.
All' ti mn prw filosofan mayhmatik tn
prvtsthn parxetai suntleian, k totvn dlon.
de d ka tn kay' kasta memnsyai, ka ti yeo-
log& mn proeutrepzei tw noerw pibolw. sa
gr tow telsi dusyrata ka nnth fanetai tw
per tn yen lhyeaw ew dignvsin, tata o tw
mayhmatikw lgoi pist ka katafan ka nlegkta
di tn eknvn pofanousi. tn mn gr per-
ousvn diottvn n tow riymow tw mfseiw
deiknousi, tn d noern sxhmtvn n tow dianoh-
tow tw dunmeiw kfanousin. di ka Pltvn
poll ka yaumast dgmata per yen di tn
mayhmatikn edn mw nadidskei ka tn Pu-
yagorevn filosofa parapetsmasi totoiw xrv-
mnh tn mustagvgan katakrptei tn yevn dogm-
tvn. toiotow gr ka erw smpaw lgow ka
Fillaos n taw Bkxaiw ka low trpow tw
Puyagrou per yen fhgsevw.
Prw d tn fusikn yevran t mgista sum-
blletai, tn te tn lgvn etajan nafanousa,
kay' n dedhmiorghtai t pn, ka nalogan tn
pnta t n t ksm sundsasan, w pou fhsn
Tmaiow, ka fla t maxmena ka prosgora ka
sumpay t diestta poisasan, ka t pl ka
prvtourg stoixea ka pnth t summetr& ka t
sthti sunexmena dejasa, di' n ka pw ora-
nw teleyh, sxmata t proskonta kat tw au-
to merdaw podejmenow, ti d riymow tow o-
keouw kst tn gignomnvn ka taw peridoiw
atn ka taw pokatastsesin neurosa, di' n
tw te egonaw kstvn ka tw nantaw forw sul-
logzesyai dunatn. tata gr omai ka Tmaiow
ndeiknmenow pantaxo di tn mayhmatikn no-
mtvn kfanei tn per tw fsevw tn lvn yev-
ran ka tw genseiw tn stoixevn riymow ka
sxmasi katakosme ka tw dunmeiw atn ka t
pyh ka tw poiseiw ew at nafrei, tn te gv-
nin tw jthtaw ka tw mblthtaw ka tn pleu-
rn tw leithtaw tw nantaw dunmeiw, t te
plyow ka tn ligthta tn stoixevn atimenow
tw pantoaw metabolw.
Prw ge mn tn politikn kaloumnhn filoso-
fan pw ox fsomen atn poll d ka yau-
mast suntelen, tow te kairow tn prjevn na-
metroumnhn ka tw poiklaw peridouw to pantw
ka tow proskontaw riymow taw gensesi, tow
te fomoivmatikow ka tow tw nomoithtow atouw,
tow te gonmouw ka teleouw, ka tow nantouw to-
toiw, tow te narmonou zvw xorhgow ka tow tw
narmostaw parektikow ka lvw forw ka foraw
ostikow. d ka n polite& tn mousn l-
gow kfanei tn gevmetrikn riymn smpanta k-
rion meinnvn ka xeirnvn gensevn tiymenow ka
tw te ltou tn diastrfvn yn diamonw, ka
tw metabolw tn rstvn politein ew tw l-
gouw ka mpayew. ti gr tw lhw st mayhmati-
kw tn pistmhn paradonai to legomnou totou
gevmetriko riymo ka o miw tinw oon riymh-
tikw gevmetraw pant pou dlon: di pntvn
gr tn mayhmtvn o lgoi tw te egonaw ka
tw gonaw dikousi.
Prw d' a tn yikn filosofan mw teleio,
tjin ka narmnion zvn ntiyesa tow yesin
mn ka sxmata prponta t ret ka mlh ka
kinseiw paraddvsin, f' n d ka Ayhnaow
jnow teleiosyai boletai tow tw yikw retw
k nvn metalhcomnouw, tn te retn protenei
tow lgouw, llvw mn n tow riymow, llvw d
n tow sxmasin, llvw d n tow kat mousikn
sumfnoiw ka tn kakin tw perbolw ka tw
ndeaw paradeknusi, di' n potelomeya mtrioi
t yow ka kekosmhmnoi. ka di toto Svkr-
thw n Gorg& mn tn Kallikla tw tktou ka
kolstou zvw atimenow, gevmetraw gr, fhsn,
melew ka tw kat' atn sthtow. n polite& d
tw turannikw donw tn pstasin, n laxen,
erskei prw tn to basilvw kat tn ppedon
gnesin ka tn steren.
All mn taw te llaiw pistmaiw te ka txnaiw
lkon t p tw mayhmatikw felow paragnetai,
myoimen n nnosantew, ti taw mn yevrhtikaw
oon =htorik ka taw toiasde psaiw, sai di l-
gvn kurontai, teleithta ka tjin prostyhsin ka
t k prtvn te ka msvn ka teleutavn sumplh-
rosyai kat tn prw atn peikasan, taw d
poihtikaw n paradegmatow tjei proufsthke tow
lgouw tn poioumnvn ka t mtra n at pro-
upostsasa, taw d praktikaw tn nrgeian ka tn
knhsin forzei di tn sttvn autw ka kin-
tvn edn. lvw gr a txnai psai, kayper d
fhsin n t Filb Svkrthw, riymhtikw don-
tai ka metrhtikw ka statikw, toi pasn tinn.
atai d psai perixontai n tow mayhmatikow l-
goiw ka kat' kenouw rzontai. ka gr a tn
riymn dianoma ka tn mtrvn poikila ka
tn barn diaforthw p tathw gnvrzontai.
T mn tonun felow tw mayhmatikw lhw pi-
stmhw prw te filosofan atn ka tw llaw pi-
stmaw ka txnaw stai di totvn gnrimon tow
koousin, dh d tinew tn ntilogikn pixei-
rosi kayairen tn jan tw pistmhw tathw, o
mn t kaln atw ka t gayn fairontew w
o per totvn poioumnhw tow lgouw, o d xrhsi-
mvtraw tw tn asyhtn mpeiraw pofanontai
tn n at kaylou yevroumnvn, oon gevdesan
gevmetraw, ka tn tn polln riymhtikn tw n
yevrmasin festshw, ka tn nautikn strolo-
gan tw kaylou deiknuoshw. ote gr ploutomen
t ginskein tn ploton, ll t xrsyai, ote
edaimonomen t tn edaimonan ginskein, ll
t zn edaimonikw, ste ka prw tn bon tn
nyrpinon ka tw prjeiw o tw gnvstikw tn
mayhmatikn, ll tw mpeirikw suntelen molo-
gsomen. o gr gnoontew mn tow lgouw, ge-
gumnasmnoi d per tn n tow kay' kasta peran
l ka pant diafrousi prw tw nyrvpikw xreaw
tn per tn yevran mnhn sxolaktvn.
Prw d tow tata lgontaw panthsmeya t
mn kllow pideknuntew tn mayhmtvn p to-
tvn, f' n ka Aristotlhs pexerhsen mw
peyein. tra gr tata diaferntvw ka n tow
smasi ka n taw cuxaw t kllow potelen, tn
tjin, tn summetran, t rismnon, pe ka t
asxow t mn svmatikn p tw likw tajaw ka
morfaw ka summetraw ka oristaw n t sun-
yt krathsshw parufstatai, t d [cuxikn] p
tw logaw plhmmelw ka tktvw kinoumnhw ka
narmstou prw tn lgon oshw ka tn keyen ron
o katadexomnhw, ste ka t kllow n tow nan-
toiw n xoi tn parjin, tjei dhlad ka summetr&
ka t rismn. tata d n t mayhmatik mlista
yevromen pistm, tn mn tjin n t tn deu-
trvn e ka poikilvtrvn p tn prtvn ka
ploustrvn kfnsei^sunrthtai gr e t p-
mena tow mprosyen, ka t mn rxw xei lgon, t
d tn pomnvn taw prtaiw poysesin^tn d
summetran n t sumfvn& tn deiknumnvn prw
llhla ka t prw tn non pntvn nafor^m-
tron gr sti koinn tw lhw pistmhw now, par'
o ka tw rxw lambnei ka prw n pistrfei tow
manynontas^t d rismnon n tow stsin e
ka kintoiw lgoiw: o gr llote llvw xei t
gnvst atw sper t dojast ka t asyht
atw, ll' e t at protenetai ka ristai tow
noerow edesin. e tonun t mn potelestik to
kllouw st tata diaferntvw, t d maymata
kat tata xarakthrzetai, prdhlon ti ka n to-
toiw st t kaln. ka pw gr o mllei, no mn
katalmpontow nvyen tn pistmhn, tathw d ew
non peigomnhw ka mw p asysevw ew kenon
metgein speudoshw;
T d' a felow atw ok ew tw nyrvpnaw
xreaw forntaw krnein jisomen od tw ngkhw
stoxazomnouw: otv gr ka tn yevrhtikn retn
atn xrhston mologsomen enai autn tn n-
yrvpnvn xvrzousan ka, o tata suntenei, md'
lvw ginskein aroumnhn. ka n Yeaitt
Svkrthw per tn korufavn xrhsmdn ntvw [?]
pshw mn atow fsthsi sxsevw tw prw tn
nyrpinon bon, pshw d ngkhw ka xreaw elu-
ton atn tn dinoian ew tn tn ntvn natenei
perivpn. ka tonun ka tn mayhmatikn pistmhn
atn di' atn aretn ka tn j atw yevran
enai yeton, ll' o di tw nyrvpnaw xreaw. e
d xr prw llo ti t j atw felow nafrein,
prw tn noern gnsin tn fleian tn p' atw
nenekton: ew gr kenhn mw podhge ka pro-
eutrepzei, t mma tw cuxw pokayarousa ka
fairosa t p tn asysevn mpdia prw tn
gnsin tn lvn. sper on tn kayartikn lhn
retn o prw tw bivtikw xreaw poblpontew
xrhsmhn xrhstn famen, ll prw tn n yevr&
bon, otvs ka tw mayhmatikw t tlow ew non
napmpein proskei ka tn smpasan sofan. di
ka per atn nrgeia kay' atn t stin ja
spoudw ka di tn noern zvn. dhlo d t di'
autn enai tow metiosin aretn, ka Aristo-
tlhs po fhsin, t mhdenw misyo prokeimnou
tow zhtosin mvw n lg xrn tosathn pdo-
sin tn tn mayhmtvn yevran laben, ti d t
pntaw n at filoxvren ka bolesyai sxolzein
tn llvn femnouw, soi ka kat mikrn fcanto
tw p' atw feleaw, ste o ge katafronhtikw
xousi tw tn mayhmtvn gnsevw, geustoi tug-
xnousin ntew tn n atow donn.
O d on di toto tn mayhmatikn tima-
ston, ti m prw tw nyrvpnaw mn xreaw sun-
tele^t gr sxata atw phxmata tw toi-
athw xreaw stoxzetai ka sa mey' lhw nerge^
ll tonanton yaumaston atw tn #lan ka
t n at mn t gayn xein. ka gr lvw pau-
smenoi tw per t nagkaa frontdow o nyrvpoi
per tn zthsin trponto tn mayhmtvn. ka
toto ektvw: prta mn gr t sntrofa ka mo-
fu t gensei perispodast sti tow nyrpoiw:
detera d t tw gensevw polonta tn cuxn ka
namimnskonta to ntow. otvw ra ka t nag-
kaa pr tn di' at timvn ka t tw asysevw
smfula pr tn kat non gignvskomnvn mtimen.
ka gr psa gnesiw ka n at strefomnh
tw cuxw zv pfuken p to telow ew t t-
leion xvren. tosata ka prw totouw ersyv tow
tn mayhmatikn timzontaw pistmhn.
Isvw d' n tinew k tw metraw staw rmme-
noi ka tn Pltvna proistmenoi mrtura tn
lgvn pixeirseian ew perocan gein tow pi-
polaiotrouw tw tn mayhmtvn krosevw. ka
gr atn dpou tn filsofon n polite& tn ma-
yhmatikn tathn gnsin to tn pisthmn pe-
lanein xoro ka tw rxw tw atw w gnoosan
dielgxein, ka t rx mn, m eden, teleut d
ka msa, j n ok oden, p totoiw prosysousi,
ka sa lla par to Svkrtouw n kenoiw nedh
kat tw yevraw tathw prriptai. prw d flouw
ndraw mew poiomenoi tow lgouw namnsomen
atow, ti ka atw Pltvn kayartikn tw
cuxw ka nagvgn tn mayhmatikn enai safw
pofanetai, tn xln fairosan to noero tw
dianoaw fvtw to krettonow svynai murvn sv-
matikn mmtvn kat tn Omhrikn Ayhnn, w n
m mnon tn Ermakn drvn, ll ka tn Ayh-
nakn metxousan: ka w pistmhn atn pokale
pantaxo ka w tw megsthw edaimonaw atan tow
metiosin.
All t boletai di tn n polite& lgvn
fairn atw tn tw pistmhw pvnuman; g
frsv suntmvw: prw gr edtaw parn stai
moi lgow. pistmhn Pltvn pollaxo mn
prosagoreei psan w epen otv tn tn kaylou
gnsin, ntidiairomenow atn prw tn asyhsin
t kay' kasta gnvrzousan, kn texnikw kn mpei-
rikw tw toiathw gnsevw trpow. ka kat to-
ton omai tn lgon n te politik ka n sofist
fanetai xrmenow t tw pistmhw nmati, ka
atn tn gennaan tn sofistikn pistmhn tiy-
menow, n n Gorg& Svkrthw mpeiran pfhnen
osan, ka tn kolakikn ka pollw llaw, mpeiraw
osaw ll' ok pistmaw lhyew. tathn d atn
tn kaylou gnsin dielmenow ew te tn tw ataw
gnvrzousan ka tn neu ataw gnvstikn, tn mn
tran pistmhn jio kalen, tn d loipn mpei-
ran. ka otvw d taw mn txnaiw metaddvs
pou to tw pistmhw nmatow, taw d mpeiraiw
odamw. logon gr prgm fhsin n sumpos,
pw n eh pistmh. ka psa ra gnsiw lgon
xousa tn gnvstikn ka atan pistmh tw stin.
plin tonun tn pistmhn tathn p' ataw gnv-
ristikn tn pokeimnvn diaire, ka tn mn sto-
xastikn ka meristn, tn d tn kay' at ka w-
atvw xntvn e gnvstikn [diakrnei?] ka kat
tathn tn dikrisin atrikn mn ka psan tn per
t nula pragmatean xvrzei tw pistmhw, tn d'
a mayhmatikn ka lvw tn tn dvn yevrhtikn
pistmhn prosagoreei. tathn d' a tn pist-
mhn, n tn texnn forzomen, diairn tn mn
nupyeton enai boletai, tn d j poysevw r-
mhmnhn, ka tn mn nupyeton tn lvn enai
gnvstikn mxri to gayo ka tw nvttv tn
pntvn ataw nabanousan ka tw nagvgw tlow
poioumnhn t gayn, tn d rismnaw rxw pro-
sthsamnhn p totvn deiknnai t pmena ataw
ok p' rxn ll' p teleutn osan. ka otvw
d tn mayhmatikn te poysesin xrvmnhn tw
nupoytou ka teleaw pistmhw polepesya fh-
sin. ma gr ntvw pistmh, kay' n t nta
pnta ginskein pefkamen, ka f' w psai a
rxa taw mn ggutrv tetagmnaiw, taw d porrv-
trv. m d tonun lgvmen, ti tn pisthmn
Pltvn pelanei tn mayhmatikn, ll' ti miw
pistmhw atn tw krotthw deutran pofanei,
mhd' ti tw okeaw rxw gnoen atn fhsin, ll'
ti par' kenhw labosan ka napodektvw xousan
k totvn t fejw podeiknnai. ka gr tn cu-
xn tn k tn mayhmatikn lgvn postsan pot
mn kinsevw rxn enai sugxvre, pot d p tn
nohtn genn dxesyai tn knhsin. ka sundei
tata llloiw. tow mn gr p' llou kinoumnoiw
ata tw kinsew stin, pshw d ra kinsevw ok
ata. kat t at d on ka mayhmatik tw
mn prvtsthw stn pistmhw deutra ka w prw
kenhn telw, pistmh d mvw, ox w nup-
yetow, ll' w tn dvn n cux lgvn gnvristik
ka w tw ataw podidosa tn sumperasmtvn
ka lgon xousa tn pokeimnvn aut gnvstn.
tosata ka per tw Pltvnos gnmhw pr tn
mayhmtvn ersyv.
Tna d' n tiw paitseien tn mayhmatikn,
ka pw n dnait tiw atn krnein ryw met
tata lgvmen. mn gr plw pepaideumnow per
pnta kritikw, fhsn Aristotlhs, d per t
maymata pepaideumnow tn n totoiw lgvn stai
kritikw tw rythtow. de tonun rouw proseilh-
fnai tw krsevw ka gignskein prton mn f'
n de kat t koin poiesyai tw podejeiw, ka
f' n ew tw kstvn dithtaw poblpein. poll
gr prxei t at tow kat' edow diafrousin,
oon trignoiw psin a do rya. poll d tn mn
atn xei kathgoran, diafrei d kat' edow f'
kstvn t koinn, oon moithw n sxmasi ka
riymow. o de d man pdeijin p totvn pai-
ten tn mayhmatikn: o gr a ata rxa sxhm-
tvn ka riymn, ll diafrousai kat t poke-
menon gnow. e d t kay' at sumbebhkw n, ka
pdeijiw ma: t gr do ryw xein gvnaw ta-
tn n psi trignoiw, ka t sumbbhken tatn
n pasi, t trgvnon, ka trigvnikw lgow. sper
d ka t tttarsin ryaw saw xein tw ktw o
tow trignoiw mnon, ll ka psin eyugrmmoiw
prxei, ka pdeijiw p pnta farmttei, kay-
son eygramma. ka gr kastow lgow suneisfrei
tin pntvw dithta ka pyow, o metxei pnta
kat tn lgon kenon, oon tn trigvnikn tn
eyugrammikn lvw tn to sxmatow.
Deteron tonun, e kat tn pokeimnhn lhn
poietai tw podejeiw, oon e nagkaouw poddvsi
lgouw ka nelgktouw, ll m piyanow mhd to
ektow napeplhsmnouw. moion gr fhsin Ari-
stotlhs =htorikn podejeiw paiten ka mayh-
matiko piyanologontow podxesyai. de gr ka-
ston pistmona ka texnthn proskontaw podidnai
tow prgmasi, per pragmateetai, tow lgouw.
otvw ka Pltvn n Tima tn mn fusiolgon
ektaw paite lgouw w n per toiotvn pragma-
teumenon, tn d per tn nohtn nadidskonta
ka tw stshw osaw nelgktouw ka kintouw.
eyw gr t pokemena taw pistmaiw taw
txnaiw poie diaforw, oon e t mn knhta, t
d kinomena, ka t mn plostera, t d sunyet-
tera, ka t mn noht, t d asyht. ot' on
psan tn mayhmatikn tn atn krbeian pait-
somen^e gr mn asyhtn fptoit pvw, d
nohtn eh gnsiw pokeimnvn, ox movw mfte-
rai kribew, ll' tra mllon. di tn riymh-
tikn tw rmonikw mllon krib famen^oy'
lvw tn mayhmatikn ka tw llaw pistmaw taw
ataw podejesin jisomen xrsyai. t gr po-
kemena diaforn ok lghn parxetai.
T d trton lgomen, ti ka per tautthtow
ka terthtow peskfyai de tn mllonta krnein
ryw tow tw mayhmatikw lgouw ka per to
kay' at ka to kat sumbebhkw ka per tw na-
logaw ka per pntvn tn toiotvn. sxedn gr
a martai psai kat tata sumbanousi tn oo-
mnvn podeiknnai mayhmatikw, o mntoi ka
deiknntvn, tan t tatn w teron kay' kaston
edow podeiknvsin, t teron w tatn, tan
t kat sumbebhkw prxon w kay' at paralam-
bnvsin, t kay' at w kat sumbebhkw, oon
ti perifreia kallvn tw eyeaw, t spleu-
ron to soskelow. o gr mayhmatiko tata dio-
rzein.
Ttarton on, ti tw mayhmatikw mshn xo-
shw tjin tn te nohtn ka asyhtn ka pollw
mn eknaw tn yevn n aut deiknuoshw, poll
d paradegmata tn fusikn lgvn, de ka tw
podejeiw triplw p' atw yevren, tw mn noerv-
traw, tw d diejodikvtraw, tw d ka djhw fapto-
mnaw. de gr kat t problmata tw podejeiw
diaferosaw enai ka okevw tow gnesi diairesyai
tn ntvn, pe ka at psi sunufanetai atow
ka prw pntaw sunarmzei tow autw lgouw.
All totvn mn dhn, per d tn edn tw
mayhmatikw met tata dioriston, tna te ka psa
tn riymn stin. met gr t lon ka pantelw
atw gnow de d pou ka tw tn merikvtrvn pi-
sthmn kat' edh diaforw nalogsasyai. tow mn
on Puyagoreois dkei tetrax diairen tn
lhn mayhmatikn pistmhn, t mn atw per t
posn, t d per t phlkon forzousi ka totvn
kteron dittn tiyemnoiw: t te gr posn kay'
at tn pstasin xein, prw llo yevresyai
kat sxsin, ka t phlkon stw kinomenon
enai: ka tn mn riymhtikn t kay' at t po-
sn yevren, tn d mousikn t prw llo, gevme-
tran d t phlkon knhton prxon ka tn sfai-
rikn t kay' at kinomenon: piskopen d' a t
phlkon ka posn ote mgeyow plw ote plyow
ll t kay' kteron rismnon: toto gr fe-
losaw tn pervn tw pistmaw katanoen, w ok
nn tn kay' kteron peiran gnsei perilaben.
tan d tata lgvsin ndrew ew pan sofaw lh-
laktew, ote t posn t n tow asyhtow koein
mew jisomen ote t phlkon t per t smata
fantazmenon. tata gr omai yevren tw fusio-
logaw stn, ll' o tw mayhmatikw atw. ll'
pe tn nvsin ka tn dikrisin tn lvn ka tn
tautthta met tw terthtow ew tn tw cuxw
sumplrvsin dhmiourgw parelhfen ka prw ta-
taiw stsin ka knhsin ka k totvn atn tn ge-
nn psthsen, w Tmaiow mw neddajen, lekton,
ti kat mn tn terthta tn atw ka tn diare-
sin tn lgvn ka t plyow dinoia stsa ka
nosasa autn n ka poll osan tow te riymow
probllei ka tn totvn gnsin, tn riymhtikn,
kat d tn nvsin to plyouw ka tn prw aut
koinvnan ka tn sndesmon tn mousikn. di'
ka riymhtik presbutra tw mousikw, pe ka
cux diairetai prton dhmiourgikw, ey' otvw
sunddetai tow lgoiw, w Pltvn fhgetai. ka
a plin kat mn tn stsin tn n at tn nr-
geian drsasa gevmetran f' autw jfhnen, ka
t n sxma t osidew ka tw dhmiourgikw rxw
tn sxhmtvn pntvn, kat d tn knhsin tn sfai-
rikn. kinetai gr ka at kat tow kklouw,
sthken d e satvw kat tw ataw tn kklvn,
t ey ka periferw. ka di toto kntaya pro-
#fsthken gevmetra tw sfairikw sper stsiw
tw kinsevw.
Epe d' ok ew tn peirodnamon autw fo-
rsa tn edn nlijin tw pistmaw gnnhsen
tataw, ll' ew tn kat gnh to pratow perioxn,
di d tot fasin atw p te to plyouw ka
megyouw felosaw t peiron per t peperasmnon
dh tn pragmatean xein. rxw gr n at pn-
tvn drusen now ka to plyouw ka to meg-
youw. pe ka moiomerw stin lh prw autn
ka ma ka diaretow ka a plin dirhmnh ka
tn tn edn kfanousa ksmon, peratw te metxei
ka peiraw osidouw p tn nohtn. ll noe
mn atn kat t praw, genn d zvw ka lgouw
pantoouw kat tn peiran. a tonun noseiw
atw tw pistmaw tataw psthsan kat t praw
t n ataw, ll' o kat tn peiran tw zvw.
no gr ekna frousin, ll' o zvw.
Tn mn tonun Puyagorevn lgow otow
ka tn tettrvn pisthmn diaresiw, kat' llon
d' a trpon tn mayhmatikn tmnein tinw jiosin,
sper ka Gemnos, ka poiosi tn mn per t
noht mnon, tn d per t asyht [nergosan?]
ka totvn faptomnhn, noht dpou kalontew,
sa kay' autn cux yemata nakine, xvrzousa
tn nlvn autn edn. ka tw mn per t noht
pragmateuomnhw do t prtista ka kuritata mrh
tyentai riymhtikn ka gevmetran, tw d per t
asyht tn nrgeian xoshw j, mhxanikn, stro-
logan, ptikn, gevdesan, kanonikn, logistikn.
t d' a taktikn ok jiosin n ti tn mern tw
mayhmatikw lgein, sper teroi, ll prosxrsyai
tte mn logistik, kayper n taw jariymsesi
tn lgvn, tte d gevdes&, kayper n taw diai-
rsesi tn xvrvn ka taw nametrsesin, sper d
poll plon ote t storikn ote t atrikn mrow
enai mayhmatikw, e ka prosxrntai pollkiw o te
tw storaw grfontew tow mayhmatikow yevrma-
sin, yseiw klimtvn frzontew megyh plevn
ka diamtrouw periblouw ka diamtrouw peri-
mtrouw sullogizmenoi, ka o atro poll tn o-
kevn di tn toiotvn fdvn safhnzontew. t
gr p tw strologaw felow ew atrikn ka
Ippokrths dlon poie ka pntew soi ti per
rn ka tpvn erkasi. kat t at d on ka
taktikw xrsetai mn tow yevrmasi tn mayh-
matikn, o mntoi mayhmatikw stin, e ka pot
mn lxiston dejai t plyow boulmenow ew kklon
sxhmatzoi t stratpedon, pot d pleston ew te-
trgvnon pentgvnon llo ti polgvnon.
Totvn d tn edn ntvn tw lhw mayhmati-
kw mn gevmetra diairetai plin ew te tn p-
pedon yevran ka tn stereometran. per gr sh-
mea ka grammw dizous tiw ok sti pragmatea,
kayson od sxma gnoito n n totoiw neu pi-
pdvn steren. pantaxo ge mn rgon tw gev-
metraw n te pipdoiw ka stereow sunistnein
sugkrnein diairen t sunestta. tw d riymh-
tikw satvw diaresiw ew te tn tn grammikn
riymn yevran ka tn tn pipdvn ka tn tn
steren. ka gr t edh to riymo kay' at
skope pronta p mondow, ka tw genseiw tn
pipdvn tn te movn ka tn nomovn, ka tw
ew trthn ajhn prodouw. gevdesa d ka logistik
tataiw nlogon, o per nohtn riymn sxhm-
tvn, ll per asyhtn poiomenai tow lgouw.
o gr klindron knon rgon tw gevdesaw me-
tren, ll svrow w knouw ka frata w kuln-
drouw, od di' eyein nohtn, ll di' asyhtn,
tte mn kribestrvn, w di tn ktnvn tn lia-
kn, tte d paxutrvn, oon di sprtvn ka sty-
mhw. od' a logistikw at kay' aut yevre
t pyh tn riymn, ll' p tn asyhtn, yen
ka tn pvnuman atow p tn metroumnvn t-
yetai, mhltaw kaln tinaw ka fialtaw. ka lxi-
ston mn odn enai sugxvre kayper riymhti-
kw, w mntoi prw ti gnow lambnei t lxiston.
gr ew nyrvpow mtron at gnetai to plyouw
w monw. plin ptik ka kanonik gevmetraw
es ka riymhtikw kgonoi, mn taw cesi gram-
maw xrvmnh ka taw k totvn sunistamnaiw gv-
naiw, diairoumnh d ew te tn dvw kaloumnhn
ptikn, tiw tn ceudw fainomnvn par tw po-
stseiw tn ratn tn atan poddvsin, oon tw
tn paralllvn sumptsevw tw tn tetragnvn
w kklvn yevraw, ka ew tn katoptrikn smpa-
san tn per tw naklseiw tw pantoaw pragma-
teuomnhn ka t ekastik gnsei sumplekomnhn,
ka tn legomnhn skhnografikn deiknsan, pw n
t fainmena m ruyma morfa fantzoito n
taw eksi par tw postseiw ka t ch tn
gegrammnvn. d' a kanonik tow fainomnouw
lgouw tn rmonin skopetai, tw tn kannvn
katatomw neurskousa ka t asysei pantaxo
prosxrvmnh ka w fhsn Pltvn ta to no
prosthsamnh.
Prw d tataiw mhxanik kaloumnh tw per
t asyht ka t nula pragmateaw mrow pr-
xousa, p d tathn te rganopoik tn kat
plemon pithdevn rgnvn, oa d ka Arxim-
dhs lgetai kataskeusai tn polemontvn tn Su-
rkousan muntik rgana, ka yaumatopoik t
mn di pnn filotexnosa, sper ka Kthsbios
ka Hrvn pragmateontai, t d di =opn, n tw
mn kinsevw tn nisorropan atiaton, tw d st-
sevw tn sorropan, sper ka Tmaiow dirisen,
t d di nervn ka sprtvn mcxouw lkw ka
kinseiw pomimoumnvn. p d tn mhxanikn stin
ka tn sorrpvn lvw ka tn legomnvn ken-
trobarikn dignvsiw, ka sfairopoia kat mmh-
sin tn oranvn periforn, oan ka Arximdhs
pragmatesato, ka lvw psa tw lhw kinhtik.
loip d strologa per tn kosmikn kinsevn
dialambnousa ka per megeyn ka sxhmtvn tn
oranvn svmtvn ka fvtismn ka postsevn
tn p gw ka tn toiotvn pntvn, pol mn
polaousa tw asysevw, pol d ka prw tn
fusikn pikoinvnosa yevran. tathw d ra m-
row st ka gnvmonik per tn rn katamtrh-
sin sxoloumnh di tw tn gnvmnvn ysevw, ka
metevroskopik tn te jarmtvn tw diaforw
ka tn strvn tw postseiw neurskousa ka
poll lla ka poikla tn kat' strologan yevrh-
mtvn nadidskousa, ka dioptrik tw e poxw
lou ka selnhw ka tn llvn strvn kataman-
ynousa di tn toiotvn rgnvn. toiata ka
per tn tw mayhmatikw mern p tn palain
nagegrammna pareilfamen.
Een d on. plin kena yevrsvmen, pvw
Pltvn yrigxn tn mayhmtvn n polite& tn
dialektikn proserhken, ka stiw sndesmow atn,
n (?) tn pinomda sunyew paraddvsi. ka lgo-
men, ti kayper now perdrutai tw dianoaw
ka xorhge tw rxw nvyen at ka teleio tn
dinoian f' auto, kat t at d ka dia-
lektik, filosofaw osa t kayartaton mrow,
prosexw perplvtai tn mayhmtvn ka perixei
tn lhn atn nlijin, ka ddvsi dunmeiw f'
autw taw pistmaiw atn pantoaw, telesiourgow
ka kritikw ka noerw, tn nalutikn lgv ka
tn diairetikn ka tn ristikn ka tn podeikti-
kn. f' n d xorhgoumnh ka teleioumnh ma-
yhmatik t mn di' nalsevw erskei, t d di
sunysevw, ka t mn diairetikw fhgetai, t d
ristikw, t d di' podejevw katadetai tn zh-
toumnvn, sunarmzousa mn tow pokeimnoiw aut
tw meydouw tataw, xrvmnh d kst prw tn
yevran tn msvn lgvn, yen d ka a nalseiw
p' atw ka o rismo ka a diairseiw ka a po-
dejeiw okea t esi ka kat tn trpon tw mayh-
matikw gnsevw nelssontai. yrigxw on ektvw
stn dialektik tn mayhmtvn, pn t noern
atn teleiosa ka t kribw nelegktteron per-
gazomnh ka t knhton mnimon satvw diafu-
lttousa ka t #lon ka kayarn ew tn plthta
tn to no ka tn #lan nafrousa ka tw te
rxw atn tw prtaw forzousa di tn lgvn
ka tw tn genn ka edn tn p' at diakr-
seiw kfanousa, tw te sunyseiw tw k tn rxn
t met tw rxw proagosaw ka tw nalseiw tw
t' p t prta ka tw rxw paniosaw nadi-
dskousa.
Ka mn ka tn sndesmon tn mayhmtvn o
tn nalogan, sper Eratosynhs oetai, yeton.
gr naloga tn koinn tow maymasin n ti ka
lgetai enai ka stin. poll d' a ka lla dikei
di pntvn w epen t kay' at prxonta t
koin fsei tn mayhmtvn. ll' w mew n fa-
men, prosexw mn stin atn sndesmow ma ka
lh mayhmatik tw pasn tn kay' kasta pisth-
mn rxw plosteron n aut perixousa ka tn
te koinvnan atn ka tn diaforn peskemmnh
ka sa t at n psaiw nadidskousa, ka sa
pleosin prxei ka sa lttosin. ka p tathn
p tn polln nodow tow kat trpon many-
nousin. nvtrv d' ti tathw dialektik tn ma-
yhmtvn n eh sndesmow, ka yrigxn atn, w
fhn, n polite& proserhken. ath gr ka tn
lhn mayhmatikn teleio ka ew non napmpei taw
autw dunmesi, ka pistmhn ntvw pofanei
ka mnimon ka nlegkton pergzetai. trthn d'
n xoi tjin n tow sundsmoiw now atw p-
saw tw dialektikw dunmeiw n aut monoeidw
perixvn ka tn poikilan atn di tw plthtow
ka tn merismn di tw merow gnsevw ka t
plyow di tw nsevw sungvn. atw d on
sumptssei mn tw neljeiw tn dialektikn mey-
dvn, sunde d nvyen psan tn dijodon tn ma-
yhmatikn lgvn, tlow d' st t riston tw na-
ggou poreaw ka tw gnvstikw nergeaw. tat
moi ka per totvn divrsyv.
T d' a noma at toto t tw mayhmatikw
ka t tn mayhmtvn pyen n famen p tn
palain taw pistmaiw tataiw podedsyai ka
tna n xoi proskonta lgon; doke d moi m tn
pituxntvn enai, kayper d t poll tn nom-
tvn, toiath tw pistmhw tn dianohtikn lgvn
proshgora, ll' sper d ka lgetai tn Puya-
gorevn kateidtvn mn ti psa kaloumnh m-
yhsiw nmnhsw stin ok jvyen ntiyemnh taw
cuxaw, sper t p tn asyhtn fantsmata
tupotai n t fantas&, od' peisodidhw osa,
kayper dojastik gnsiw, ll' negeiromnh mn
p tn fainomnvn, proballomnh d ndoyen p'
atw tw dianoaw ew autn pistrefomnhw, katei-
dtvn d' a ka ti tw namnseiw e ka pollax-
yen deiknnai dunatu, ll diaferntvw, ti ka
Pltvn fhsn, k tn mayhmtvn. n gr tiw p
t diagrmmata g, fhsn kenow, ntaya ra
safstata kathgore, w stin myhsiw nmnhsiw.
yen d ka n t Mnvni Svkrthw k tode to
trpou tw pixeirsevw pdeijen, ti t manynein
ok llo t stin namimnskesyai tn cuxn tn
autw lgvn. ation d ti t mn namimnhskme-
nn sti t dianohtikn tw cuxw. toto d n tow
lgoiw osvtai tn mayhmtvn ka tw pistmaw
atn n aut proelhfen, kn m nerg kat'
atw. xei d' on psaw osivdw ka krufvw, pro-
fanei d ksthn, tan fairey tn mpodvn tn
k tw asysevw. a mn gr asyseiw sunptousin
atn tow meristow, a d fantasai morfvtikn ki-
nsevn napimplsin, a d rjeiw perispsin ew
tn mpay bon. pn d t meristn mpdin sti
tw ew autow mn pistrofw, ka pn t morfv-
tikn piyolo tn mrfvton gnsin, ka pn t
mpayw kluma tw payow nergeaw stn. tan
on tata tw dianoaw flvmen, tte kat' atn
gignskein tow n at dunmeya lgouw, ka pi-
stmonew enai kat' nrgeian ka gnsin tn osidh
probllein. desmtai d ntew ka t mma tw cu-
xw montew o m pote tn proskousan mn telei-
thta sxohmen. ath tonun stn myhsiw tn
dvn n cux lgvn nmnhsiw, ka mayhmatik
di tathn prw tw namnseiw mn tw kenvn
suntelosa gnsiw diaferntvw ponomzetai. ka
t rgon ra tw pistmhw tathw poon d t stin
k to nmatow dhlotai kinhtikn tw mftou gn-
sevw ka gertikn tw nosevw ka kayartikn tw
dianoaw ka kfantorikn tn kat' osan mn
parxntvn edn, lyhw te ka gnoaw faireti-
kn, n p tw gensevw sxomen, ka polutikn
tn k tw logaw desmn, kat tn yen ntvw
tn tw pistmhw tathw foron, w progei mn ew
t mfanw t noer dra, plhro d pnta tn
yevn lgvn, kine d tw cuxw p non, ka sper
k krou bayow negerei, di zhtsevw d pistr-
fei prw autw, ka di maieaw teleio, ka di' er-
sevw to kayaro no perigei prw tn makaran
zvn. d ka mew nayntew tout t sggramma
tn per tw mayhmatikw pistmhw yevran peri-
grcomen.
Prologi pars posterior
.
Copyright 1999 Giovanni Ferrero