Euclidis Opera selecta
Elementa
OROI. TERMINI
Shmen stin, o mrow oyn.
Punto è ciò che non ha parti
Gramm d mkow platw.
Linea è lunghezza senza larghezza
Grammw d prata shmea.
Estremi di una linea sono punti
Eyea gramm stin, tiw j sou tow f' autw
shmeoiw ketai.
Linea retta è quella che giace ugualmente rispetto ai punti su di essa
Epifneia d stin, mkow ka pltow mnon xei.
Superficie è ciò che ha soltanto lunghezza e larghezza
Epifaneaw d prata gramma.
Estremi di una superficie sono linee
Eppedow pifnei stin, tiw j sou taw f'
autw eyeaiw ketai.
Superficie piana è quella che giace ugualmente rispetto alle rette (che sono) su di essa.
Eppedow d gvna stn n pipd do grammn
ptomnvn lllvn ka m p' eyeaw keimnvn prw
lllaw tn grammn klsiw.
Angolo piano è l'inclinazione delle linee l'una rispetto all'altra quella che c'è nel piano delle due rette, che si incontrano fra loro, e che non giaciono in linea retta.
Otan d a perixousai tn gvnan gramma eyeai
sin, eygrammow kaletai gvna.
Quando le linee che comprendono l'angolo sono rette, l'angolo si chiama rettilineo.
Otan d eyea p' eyean stayesa tw fejw
gvnaw saw lllaiw poi, ry katra tn svn gv-
nin sti, ka festhkua eyea kyetow kaletai,
f' n fsthken.
Quando una retta innalzata su di un'altra retta forma gli angoli adiacenti uguali fra loro, ciascuno dei due angoli è retto, e la retta innalzata si chiama perpendicolare a quella su cui è innalzata.
Amblea gvna stn mezvn ryw.
L'angolo ottuso è quello maggiore del retto.
Ojea d lssvn ryw.
L'angolo acuto invece è quello minore del retto.
Orow stn, tinw sti praw.
Termine è ciò che è estremo,limite di qualche cosa.
Sxm sti t p tinow tinvn rvn periexmenon.
Kklow st sxma ppedon p miw grammw
periexmenon [ kaletai perifreia], prw n f' nw
shmeou tn ntw to sxmatow keimnvn psai a
prospptousai eyeai [prw tn to kklou peri-
freian] sai lllaiw esn.
Kntron d to kklou t shmeon kaletai.
Dimetrow d to kklou stn eye tiw di to
kntrou gmnh ka peratoumnh f' ktera t mrh
p tw to kklou perifereaw, tiw ka dxa tmnei tn
kklon.
Hmikklion d sti t periexmenon sxma p te
tw diamtrou ka tw polambanomnhw p' atw peri-
fereaw. kntron d to mikuklou t at, ka to
kklou stn.
Sxmata eygramm sti t p eyein periex-
mena, trpleura mn t p trin, tetrpleura d t
p tessrvn, polpleura d t p pleinvn tess-
rvn eyein periexmena.
Tn d triplervn sxhmtvn spleuron mn tr-
gvnn sti t tw trew saw xon pleurw, soskelw d
t tw do mnaw saw xon pleurw, skalhnn d t tw
trew nsouw xon pleurw.
Eti d tn triplervn sxhmtvn ryognion mn
trgvnn sti t xon ryn gvnan, mblugnion d t
xon mblean gvnan, jugnion d t tw trew jeaw
xon gvnaw.
Tn d tetraplervn sxhmtvn tetrgvnon mn
stin, spleurn t sti ka ryognion, termhkew
d, ryognion mn, ok spleuron d, =mbow d,
spleuron mn, ok ryognion d, =omboeidw d t tw
penanton pleurw te ka gvnaw saw lllaiw xon,
ote spleurn stin ote ryognion: t d par
tata tetrpleura trapzia kalesyv.
Parllhlo esin eyeai, atinew n t at pi-
pd osai ka kballmenai ew peiron f' ktera t
mrh p mhdtera sumpptousin lllaiw.
Hitsyv p pantw shmeou p pn shmeon
eyean grammn gagen.
Ka peperasmnhn eyean kat t sunexw p' e-
yeaw kbalen.
Ka pant kntr ka diastmati kklon gr-
fesyai.
Ka psaw tw ryw gvnaw saw lllaiw enai.
Ka n ew do eyeaw eyea mpptousa tw
ntw ka p t at mrh gvnaw do ryn lssonaw
poi, kballomnaw tw do eyeaw p' peiron sump-
ptein, f' mrh esn a tn do ryn lssonew.
Nozioni Comuni
T t at sa ka llloiw stn sa.
Cose uguali ad una stessa sono uguali fra loro
Ka n soiw sa prostey, t la stn sa.
Ka n p svn sa fairey, t kataleipmen
stin sa.
[Ka n nsoiw sa prostey, t la stn nisa.
Ka t to ato diplsia sa llloiw stn.
Ka t to ato msh sa llloiw stn.]
Ka t farmzonta p' llhla sa llloiw stn.
Ka t lon to mrouw mezon [stin].
Ka do eyeai xvron o perixousin.
Ep tw doyeshw eyeaw peperasmnhw trgvnon s-
pleuron sustsasyai.
PROPOSIZIONI
Proposizione 1
Estv doyesa eyea peperasmnh AB.
De d p tw AB eyeaw trgvnon spleuron sust-
sasyai.
Kntr mn t A diastmati d t AB kklow gegr-
fyv BGD, ka plin kntr mn t B diastmati d
t BA kklow gegrfyv AGE, ka p to G sh-
meou, kay' tmnousin ll-
louw o kkloi, p t A, B
shmea pezexyvsan eyeai
a GA, GB.
Ka pe t A shmeon kn-
tron st to GDB kklou,
sh stn AG t AB: plin,
pe t B shmeon kntron
st to GAE kklou, sh stn BG t BA. dexyh d
ka GA t AB sh: katra ra tn GA, GB t AB
stn sh. t d t at sa ka llloiw stn sa: ka
GA ra t GB stn sh: a trew ra a GA, AB, BG
sai lllaiw esn.
Ispleuron ra st t ABG trgvnon, ka sunstatai
p tw doyeshw eyeaw peperasmnhw tw AB.
[Ep tw doyeshw ra eyeaw peperasmnhw trgvnon
spleuron sunstatai]: per dei poisai.
Prw t doynti shme t doyes eye& shn eyean
ysyai.
Estv t mn doyn shmeon t A, d doyesa eyea
BG: de d prw t A shme t doyes eye& t BG
shn eyean ysyai.
Epezexyv gr p to A shmeou p t B shmeon
eyea AB, ka sunesttv p' atw trgvnon spleu-
ron t DAB, ka kbeblsyvsan p' eyeaw taw DA,
DB eyeai a AE, BZ, ka kntr mn t B diastmati
d t BG kklow gegrfyv GHY, ka plin kntr
t D ka diastmati t DH kklow gegrfyv HKL.
Epe on t B shmeon kn-
tron st to GHY kklou, sh
stn BG t BH. plin, pe
t D shmeon kntron st
to KLH kklou, sh stn
DL t DH, n DA t
DB sh stn. loip ra
AL loip t BH stn sh.
dexyh d ka BG t BH
sh: katra ra tn AL,
BG t BH stn sh. t d t at sa ka llloiw
stn sa: ka AL ra t BG stn sh.
Prw ra t doynti shme t A t doyes eye&
t BG sh eyea ketai AL: per dei poisai.
Do doyeisn eyein nsvn p tw mezonow t
lssoni shn eyean felen.
Estvsan a doyesai do eyeai
nisoi a AB, G, n mezvn stv
AB: de d p tw mezonow tw AB
t lssoni t G shn eyean felen.
Kesyv prw t A shme t
G eye& sh AD: ka kntr
mn t A diastmati d t AD kklow gegrfyv
DEZ.
Ka pe t A shmeon kntron st to DEZ kklou,
sh stn AE t AD: ll ka G t AD stin sh.
katra ra tn AE, G t AD stin sh: ste ka
AE t G stin sh.
Do ra doyeisn eyein nsvn tn AB, G p tw
mezonow tw AB t lssoni t G sh frhtai AE:
per dei poisai.
En do trgvna tw do pleurw [tas] dus pleuraw
saw x katran katr& ka tn gvnan t gvn& shn
x tn p tn svn eyein periexomnhn, ka tn bsin
t bsei shn jei, ka t trgvnon t trign son
stai, ka a loipa gvnai taw loipaw gvnaiw sai son-
tai katra katr&, f' "w a sai pleura potenousin.
Estv do trgvna t ABG, DEZ
tw do pleurw tw AB, AG taw dus
pleuraw taw DE, DZ saw xonta
katran katr& tn mn AB t DE
tn d AG t DZ ka gvnan tn p
BAG gvn& t p EDZ shn. lgv,
ti ka bsiw BG bsei t EZ sh
stn, ka t ABG trgvnon t DEZ
trign son stai, ka a loipa
gvnai taw loipaw gvnaiw sai son-
tai katra katr&, f' "w a sai pleura potenou-
sin, mn p ABG t p DEZ, d p AGB t
p DZE.
Efarmozomnou gr to ABG trignou p t DEZ
trgvnon ka tiyemnou to mn A shmeou p t D sh-
meon tw d AB eyeaw p tn DE, farmsei ka t
B shmeon p t E di t shn enai tn AB t DE: far-
mosshw d tw AB p tn DE farmsei ka AG
eyea p tn DZ di t shn enai tn p BAG gv-
nan t p EDZ: ste ka t G shmeon p t Z sh-
meon farmsei di t shn plin enai tn AG t DZ.
ll mn ka t B p t E fhrmkei: ste bsiw BG
p bsin tn EZ farmsei. e gr to mn B p t E
farmsantow to d G p t Z BG bsiw p tn EZ
ok farmsei, do eyeai xvron perijousin: per
stn dnaton. farmsei ra BG bsiw p tn EZ
ka sh at stai: ste ka lon t ABG trgvnon p
lon t DEZ trgvnon farmsei ka son at stai,
ka a loipa gvnai p tw loipw gvnaw farmsousi
ka sai ataw sontai, mn p ABG t p DEZ
d p AGB t p DZE.
En ra do trgvna tw do pleurw [tas] do pleu-
raw saw x katran katr& ka tn gvnan t gvn&
shn x tn p tn svn eyein periexomnhn, ka tn
bsin t bsei shn jei, ka t trgvnon t trign
son stai, ka a loipa gvnai taw loipaw gvnaiw sai
sontai katra katr&, f' "w a sai pleura pote-
nousin: per dei dejai.
Tn soskeln trignvn a prw t bsei gvnai sai
lllaiw esn, ka prosekblhyeisn tn svn eyein
a p tn bsin gvnai sai lllaiw sontai.
Estv trgvnon soskelw t ABG shn xon tn AB
pleurn t AG pleur, ka prosekbeblsyvsan p'
eyeaw taw AB, AG eyeai a BD,
GE: lgv, ti mn p ABG gvna
t p AGB sh stn, d p GBD
t p BGE.
Elfyv gr p tw BD tuxn sh-
meon t Z, ka frsyv p tw me-
zonow tw AE t lssoni t AZ sh
AH, ka pezexyvsan a ZG, HB
eyeai.
Epe on sh stn mn AZ t AH d AB t AG,
do d a ZA, AG dus taw HA, AB sai esn katra
katr&: ka gvnan koinn perixousi tn p ZAH:
bsiw ra ZG bsei t HB sh stn, ka t AZG
trgvnon t AHB trign son stai, ka a loipa gv-
nai taw loipaw gvnaiw sai sontai katra katr&,
f' "w a sai pleura potenousin, mn p AGZ
t p ABH, d p AZG t p AHB. ka pe
lh AZ l t AH stin sh, n AB t AG stin
sh, loip ra BZ loip t GH stin sh. dexyh
d ka ZG t HB sh: do d a BZ, ZG dus taw
GH, HB sai esn katra katr&: ka gvna p
BZG gvn& t p GHB sh, ka bsiw atn koin
BG: ka t BZG ra trgvnon t GHB trign son
stai, ka a loipa gvnai taw loipaw gvnaiw sai son-
tai katra katr&, f' "w a sai pleura potenousin:
sh ra stn mn p ZBG t p HGB d p
BGZ t p GBH. pe on lh p ABH gvna
l t p AGZ gvn& dexyh sh, n p GBH t
p BGZ sh, loip ra p ABG loip t p AGB
stin sh: ka esi prw t bsei to ABG trignou.
dexyh d ka p ZBG t p HGB sh: ka esin
p tn bsin.
Tn ra soskeln trignvn a prw t bsei gvnai
sai lllaiw esn, ka prosekblhyeisn tn svn
eyein a p tn bsin gvnai sai lllaiw sontai:
per dei dejai.
En trignou a do gvnai sai lllaiw sin, ka a
p tw saw gvnaiw potenousai pleura sai lllaiw
sontai.
Estv trgvnon t ABG shn xon tn p ABG gv-
nan t p AGB gvn&: lgv, ti ka pleur AB
pleur t AG stin sh.
E gr nisw stin AB t AG, tra
atn mezvn stn. stv mezvn AB,
ka frsyv p tw mezonow tw AB
t lttoni t AG sh DB, ka peze-
xyv DG.
Epe on sh stn DB t AG koin
d BG, do d a DB, BG do taw AG,
GB sai esn katra katr&, ka gvna p DBG
gvn& t p AGB stin sh: bsiw ra DG bsei t
AB sh stn, ka t DBG trgvnon t AGB trign
son stai, t lasson t mezoni: per topon: ok ra
nisw stin AB t AG: sh ra.
En ra trignou a do gvnai sai lllaiw sin,
ka a p tw saw gvnaw potenousai pleura sai
lllaiw sontai: per dei dejai.
Ep tw atw eyeaw do taw ataw eyeaiw llai
do eyeai sai katra katr& o sustaysontai prw
ll ka ll shme p t at mrh t at prata
xousai taw j rxw eyeaiw.
E gr dunatn, p tw atw eyeaw tw AB do
taw ataw eyeaiw taw AG, GB llai do eyeai a
AD, DB sai katra katr& sunesttvsan prw ll
ka ll shme t te G ka D p
t at mrh t at prata xou-
sai, ste shn enai tn mn GA t
DA t at praw xousan at t A,
tn d GB t DB t at praw
xousan at t B, ka pezexyv
GD.
Epe on sh stn AG t AD, sh st ka gvna
p AGD t p ADG: mezvn ra p ADG tw
p DGB: poll ra p GDB mezvn st tw p
DGB. plin pe sh stn GB t DB, sh st ka
gvna p GDB gvn& t p DGB. dexyh d a-
tw ka poll mezvn: per stn dnaton.
Ok ra p tw atw eyeaw do taw ataw e-
yeaiw llai do eyeai sai katra katr& susta-
ysontai prw ll ka ll shme p t at mrh
t at prata xousai taw j rxw eyeaiw: per
dei dejai.
En do trgvna tw do pleurw [tas] do pleuraw
saw x katran katra, x d ka tn bsin t
bsei shn, ka tn gvnan t gvn& shn jei tn p
tn svn eyein periexomnhn.
Estv do trgvna t ABG, DEZ tw do pleurw
tw AB, AG taw do pleuraw taw DE, DZ saw xon-
ta katran katr&, tn mn
AB t DE tn d AG t DZ:
xtv d ka bsin tn BG bsei
t EZ shn: lgv, ti ka gvna
p BAG gvn& t p EDZ
stin sh.
Efarmozomnou gr to ABG
trignou p t DEZ trgvnon
ka tiyemnou to mn B shmeou p t E shmeon
tw d BG eyeaw p tn EZ farmsei ka t G
shmeon p t Z di t shn enai tn BG t EZ:
farmosshw d tw BG p tn EZ farmsousi ka
a BA, GA p tw ED, DZ. e gr bsiw mn BG
p bsin tn EZ farmsei, a d BA, AG pleura
p tw ED, DZ ok farmsousin ll paralljou-
sin w a EH, HZ, sustaysontai p tw atw
eyeaw do taw ataw eyeaiw llai do eyeai
sai katra katr& prw ll ka ll shme p
t at mrh t at prata xousai. o sunstantai
d: ok ra farmozomnhw tw BG bsevw p tn EZ
bsin ok farmsousi ka a BA, AG pleura p
tw ED, DZ. farmsousin ra: ste ka gvna p
BAG p gvnan tn p EDZ farmsei ka sh at
stai.
En ra do trgvna tw do pleurw [tas] do pleu-
raw saw x katran katr& ka tn bsin t bsei
shn x, ka tn gvnan t gvn& shn jei tn p
tn svn eyein periexomnhn: per dei dejai.
Tn doyesan gvnan eygrammon dxa temen.
Estv doyesa gvna eygrammow p BAG. de
d atn dxa temen.
Elfyv p tw AB tuxn shmeon t D, ka fr-
syv p tw AG t AD sh AE, ka pezexyv
DE, ka sunesttv p tw DE trgvnon spleuron
t DEZ, ka pezexyv AZ: lgv, ti p BAG
gvna dxa ttmhtai p tw AZ eyeaw.
Epe gr sh stn AD t AE, koin d AZ,
do d a DA, AZ dus taw EA, AZ
sai esn katra katr&. ka bsiw
DZ bsei t EZ sh stn: gvna ra
p DAZ gvn& t p EAZ sh
stn.
H ra doyesa gvna eygrammow
p BAG dxa ttmhtai p tw AZ
eyeaw: per dei poisai.
Tn doyesan eyean peperasmnhn dxa temen.
Estv doyesa eyea peperasmnh AB: de d
tn AB eyean peperasmnhn dxa temen.
Sunesttv p' atw trgvnon s-
pleuron t ABG, ka tetmsyv p
AGB gvna dxa t GD eye&: lgv,
ti AB eyea dxa ttmhtai kat
t D shmeon.
Epe gr sh stn AG t GB,
koin d GD, do d a AG, GD do
taw BG, GD sai esn katra katr&: ka gvna
p AGD gvn& t p BGD sh stn: bsiw ra
AD bsei t BD sh stn.
H ra doyesa eyea peperasmnh AB dxa t-
tmhtai kat t D: per dei poisai.
T doyes eye& p to prw at doyntow sh-
meou prw ryw gvnaw eyean grammn gagen.
Estv mn doyesa eyea AB t d doyn shmeon
p' atw t G: de d p to G shmeou t AB e-
ye& prw ryw gvnaw eyean grammn gagen.
Elfyv p tw AG tuxn shmeon t D, ka kesyv
t GD sh GE, ka sunesttv p tw DE trgvnon
spleuron t ZDE, ka pezexyv ZG: lgv, ti t
doyes eye& t AB p
to prw at doyntow sh-
meou to G prw ryw gv-
naw eyea gramm ktai
ZG.
Epe gr sh stn DG
t GE, koin d GZ, do
d a DG, GZ dus taw EG, GZ sai esn katra
katr&: ka bsiw DZ bsei t ZE sh stn: gvna
ra p DGZ gvn& t p EGZ sh stn: ka
esin fejw. tan d eyea p' eyean stayesa tw
fejw gvnaw saw lllaiw poi, ry katra tn
svn gvnin stin: ry ra stn katra tn p
DGZ, ZGE.
T ra doyes eye& t AB p to prw at do-
yntow shmeou to G prw ryw gvnaw eyea gramm
ktai GZ: per dei poisai.
Ep tn doyesan eyean peiron p to doyntow
shmeou, m stin p' atw, kyeton eyean gram-
mn gagen.
Estv mn doyesa eyea peirow AB t d do-
yn shmeon, m stin p' atw, t G: de d p tn
doyesan eyean peiron tn AB p to doyntow
shmeou to G, m stin p' atw, kyeton eyean
grammn gagen.
Elfyv gr p t tera mrh tw AB eyeaw tu-
xn shmeon t D, ka kntr mn t G diastmati d
t GD kklow gegrfyv EZH, ka tetmsyv EH
eyea dxa kat t Y, ka
pezexyvsan a GH, GY,
GE eyeai: lgv, ti p
tn doyesan eyean peiron
tn AB p to doyntow
shmeou to G, m stin
p' atw, kyetow ktai
GY.
Epe gr sh stn HY t YE, koin d YG,
do d a HY, YG do taw EY, YG sai esn ka-
tra katr&: ka bsiw GH bsei t GE stin sh:
gvna ra p GYH gvn& t p EYG stin sh.
ka esin fejw. tan d eyea p' eyean stayesa
tw fejw gvnaw saw lllaiw poi, ry katra
tn svn gvnin stin, ka festhkua eyea ky-
etow kaletai f' n fsthken.
Ep tn doyesan ra eyean peiron tn AB p
to doyntow shmeou to G, m stin p' atw,
kyetow ktai GY: per dei poisai.
En eyea p' eyean stayesa gvnaw poi, toi
do ryw dusn ryaw saw poisei.
Eyea gr tiw AB p' eyean tn GD stayesa
gvnaw poietv tw p GBA, ABD: lgv, ti a p
GBA, ABD gvnai toi do rya
esin dusn ryaw sai.
E mn on sh stn p GBA
t p ABD, do rya esin. e d
o, xyv p to B shmeou t GD
[eye&] prw ryw BE: a ra p
GBE, EBD do rya esin: ka pe
p GBE dus taw p GBA,
ABE sh stn, koin proskesyv p EBD: a ra
p GBE, EBD tris taw p GBA, ABE, EBD
sai esn. plin, pe p DBA dus taw p DBE,
EBA sh stn, koin proskesyv p ABG: a ra
p DBA, ABG tris taw p DBE, EBA, ABG sai
esn. dexyhsan d ka a p GBE, EBD tris taw
ataw sai: t d t at sa ka llloiw stn sa:
ka a p GBE, EBD ra taw p DBA, ABG sai
esn: ll a p GBE, EBD do rya esin: ka a
p DBA, ABG ra dusn ryaw sai esn.
En ra eyea p' eyean stayesa gvnaw poi,
toi do ryw dusn ryaw saw poisei: per dei
dejai.
En prw tini eye& ka t prw at shme do
eyeai m p t at mrh kemenai tw fejw gvnaw
dusn ryaw saw poisin, p' eyeaw sontai lllaiw
a eyeai.
Prw gr tini eye& t AB ka t prw at sh-
me t B do eyeai a BG, BD m p t at mrh
kemenai tw fejw gvnaw tw p
ABG, ABD do ryaw saw poietv-
san: lgv, ti p' eyeaw st t
GB BD.
E gr m sti t BG p' eyeaw
BD, stv t GB p' eyeaw BE.
Epe on eyea AB p' eyean tn GBE fsth-
ken, a ra p ABG, ABE gvnai do ryaw sai
esn: es d ka a p ABG, ABD do ryaw sai:
a ra p GBA, ABE taw p GBA, ABD sai
esn. koin frsyv p GBA: loip ra p
ABE loip t p ABD stin sh, lssvn t me-
zoni: per stn dnaton. ok ra p' eyeaw stn
BE t GB. movw d dejomen, ti od llh tiw pln
tw BD: p' eyeaw ra stn GB t BD.
En ra prw tini eye& ka t prw at shme
do eyeai m p t at mrh kemenai tw fejw
gvnaw dusn ryaw saw poisin, p' eyeaw sontai
lllaiw a eyeai: per dei dejai.
En do eyeai tmnvsin lllaw, tw kat korufn
gvnaw saw lllaiw poiosin.
Do gr eyeai a AB, GD temntvsan lllaw kat
t E shmeon: lgv, ti sh stn mn p AEG gv-
na t p DEB, d p GEB t p AED.
Epe gr eyea AE p' eyean
tn GD fsthke gvnaw poiosa
tw p GEA, AED, a ra p
GEA, AED gvnai dusn ryaw
sai esn. plin, pe eyea DE
p' eyean tn AB fsthke gv-
naw poiosa tw p AED, DEB, a ra p AED,
DEB gvnai dusn ryaw sai esn. dexyhsan d ka a
p GEA, AED dusn ryaw sai: a ra p GEA,
AED taw p AED, DEB sai esn. koin frsyv
p AED: loip ra p GEA loip t p BED sh
stn: movw d deixysetai, ti ka a p GEB,
DEA sai esn.
En ra do eyeai tmnvsin lllaw, tw kat ko-
rufn gvnaw saw lllaiw poiosin: per dei dejai.
[Prisma
Ek d totou fanern ti, n do eyeai tmnvsin
lllaw, tw prw t tom gvnaw ttrasin ryaw saw
poisousin.]
Pantw trignou miw tn pleurn prosekblhyeshw
ktw gvna katraw tn ntw ka penanton gv-
nin mezvn stn.
Estv trgvnon t ABG, ka prosekbeblsyv ato
ma pleur BG p t D: lgv, ti ktw gvna
p AGD mezvn stn katraw tn
ntw ka penanton tn p GBA,
BAG gvnin.
Tetmsyv AG dxa kat t E, ka
pizeuxyesa BE kbeblsyv p' e-
yeaw p t Z, ka kesyv t BE sh
EZ, ka pezexyv ZG, ka dixyv
AG p t H.
Epe on sh stn mn AE t EG, d BE t EZ,
do d a AE, EB dus taw GE, EZ sai esn katra
katr&: ka gvna p AEB gvn& t p ZEG sh
stn: kat korufn gr: bsiw ra AB bsei t ZG
sh stn, ka t ABE trgvnon t ZEG trign
stn son, ka a loipa gvnai taw loipaw gvnaiw
sai esn katra katr&, f' "w a sai pleura po-
tenousin: sh ra stn p BAE t p EGZ. me-
zvn d stin p EGD tw p EGZ: mezvn ra
p AGD tw p BAE. movw d tw BG tetmh-
mnhw dxa deixysetai ka p BGH, toutstin
p AGD, mezvn ka tw p ABG.
Pantw ra trignou miw tn pleurn prosekblh-
yeshw ktw gvna katraw tn ntw ka penanton
gvnin mezvn stn: per dei dejai.
Pantw trignou a do gvnai do ryn lssonw
esi pnt metalambanmenai.
Estv trgvnon t ABG: lgv, ti to ABG trig-
nou a do gvnai do ryn ltto-
nw esi pnt metalambanmenai.
Ekbeblsyv gr BG p t D.
Ka pe trignou to ABG ktw
sti gvna p AGD, mezvn st
tw ntw ka penanton tw p
ABG. koin proskesyv p AGB: a ra p AGD,
AGB tn p ABG, BGA mezonw esin. ll' a p
AGD, AGB do ryaw sai esn: a ra p ABG,
BGA do ryn lssonw esin. movw d dejomen,
ti ka a p BAG, AGB do ryn lssonw esi
ka ti a p GAB, ABG.
Pantw ra trignou a do gvnai do ryn lw-
sonw esi pnt metalambanmenai: per dei dejai.
Pantw trignou mezvn pleur tn mezona gv-
nan potenei.
Estv gr trgvnon t ABG mezona xon tn AG
pleurn tw AB: lgv, ti ka gvna p ABG me-
zvn st tw p BGA.
Epe gr mezvn stn AG tw AB, kesyv t
AB sh AD, ka pezexyv BD.
Ka pe trignou to BGD ktw sti gvna p
ADB, mezvn st tw ntw ka
penanton tw p DGB: sh
d p ADB t p ABD, pe
ka pleur AB t AD stin
sh: mezvn ra ka p ABD
tw p AGB: poll ra p
ABG mezvn st tw p AGB.
Pantw ra trignou mezvn pleur tn mezona
gvnan potenei: per dei dejai.
Pantw trignou p tn mezona gvnan mezvn
pleur potenei.
Estv trgvnon t ABG mezona xon
tn p ABG gvnan tw p BGA: lgv,
ti ka pleur AG pleurw tw AB
mezvn stn.
E gr m, toi sh stn AG t AB
lssvn: sh mn on ok stin AG t
AB: sh gr 'n n ka gvna p ABG
t p AGB: ok sti d: ok ra sh
stn AG t AB. od mn lssvn stn AG
tw AB: lssvn gr 'n n ka gvna p ABG tw
p AGB: ok sti d: ok ra lssvn stn AG
tw AB. dexyh d, ti od sh stn. mezvn ra stn
AG tw AB.
Pantw ra trignou p tn mezona gvnan me-
zvn pleur potenei: per dei dejai.
Pantw trignou a do pleura tw loipw mezonw
esi pnt metalambanmenai.
Estv gr trgvnon t ABG: lgv, ti to ABG
trignou a do pleura tw loipw mezonw esi pnt
metalambanmenai, a mn BA, AG tw BG, a d AB,
BG tw AG, a d BG, GA tw AB.
Dixyv gr BA p t D shmeon,
ka kesyv t GA sh AD, ka p-
ezexyv DG.
Epe on sh stn DA t AG,
sh st ka gvna p ADG t
p AGD: mezvn ra p BGD
tw p ADG: ka pe trgvnn sti
t DGB mezona xon tn p BGD
gvnan tw p BDG, p d tn mezona gvnan me-
zvn pleur potenei, DB ra tw BG sti mezvn.
sh d DA t AG: mezonew ra a BA, AG tw BG:
movw d dejomen, ti ka a mn AB, BG tw GA
mezonw esin, a d BG, GA tw AB.
Pantw ra trignou a do pleura tw loipw me-
zonw esi pnt metalambanmenai: per dei dejai.
En trignou p miw tn pleurn p tn pertvn
do eyeai ntw sustaysin, a sustayesai tn loi-
pn to trignou do pleurn lttonew mn sontai,
mezona d gvnan perijousin.
Trignou gr to ABG p miw tn pleurn tw
BG p tn pertvn tn B, G do eyeai ntw sun-
esttvsan a BD, DG: lgv, ti a BD, DG tn loi-
pn to trignou do pleurn tn BA, AG lssonew
mn esin, mezona d gvnan perixousi tn p BDG
tw p BAG.
Dixyv gr BD p t E. ka pe pantw trignou
a do pleura tw loipw mezonw esin, to ABE ra
trignou a do pleura a AB, AE tw BE mezonw
esin: koin proskesyv EG: a ra BA, AG tn
BE, EG mezonw esin. plin, pe to GED trignou
a do pleura a GE, ED tw
GD mezonw esin, koin prow-
kesyv DB: a GE, EB ra
tn GD, DB mezonw esin.
ll tn BE, EG mezonew
dexyhsan a BA, AG: poll
ra a BA, AG tn BD, DG
mezonw esin.
Plin, pe pantw trignou ktw gvna tw ntw
ka penanton mezvn stn, to GDE ra trignou
ktw gvna p BDG mezvn st tw p GED.
di tat tonun ka to ABE trignou ktw gvna
p GEB mezvn st tw p BAG. ll tw p
GEB mezvn dexyh p BDG: poll ra p
BDG mezvn st tw p BAG.
En ra trignou p miw tn pleurn p tn pe-
rtvn do eyeai ntw sustaysin, a sustayesai
tn loipn to trignou do pleurn lttonew mn
esin, mezona d gvnan perixousin: per dei dejai.
Ek trin eyein, a esin sai tris taw doyesaiw
[eyeais], trgvnon sustsasyai: de d tw do tw
loipw mezonaw enai pnt metalambanomnaw [di t
ka pantw trignou tw do pleurw tw loipw mezo-
naw enai pnt metalambanomnas].
Estvsan a doyesai trew eyeai a A, B, G, n a
do tw loipw mezonew stvsan pnt metalamban-
menai, a mn A, B tw G, a d A, G tw B, ka ti a B,
G tw A: de d k tn svn taw A, B, G trgvnon
sustsasyai.
Ekkesyv tiw eyea DE peperasmnh mn kat t
D peirow d kat t E, ka kesyv t mn A sh
DZ, t d B sh ZH, t d G sh HY: ka kntr
mn t Z, diastmati d t ZD kklow gegrfyv
DKL: plin kntr mn t H, diastmati d t HY
kklow gegrfyv KLY, ka pezexyvsan a KZ,
KH: lgv, ti k trin eyein tn svn taw A, B, G
trgvnon sunstatai t KZH.
Epe gr t Z shmeon kntron st to DKL k-
klou, sh stn ZD t ZK: ll ZD t A stin sh.
ka KZ ra t A stin sh. plin, pe t H shmeon
kntron st to LKY kklou, sh stn HY t HK:
ll HY t G stin sh: ka KH ra t G stin
sh. st d ka ZH t B sh: a trew ra eyeai a
KZ, ZH, HK tris taw A, B, G sai esn.
Ek trin ra eyein tn KZ, ZH, HK, a esin
sai tris taw doyesaiw eyeaiw taw A, B, G, trgv-
non sunstatai t KZH: per dei poisai.
Prw t doyes eye& ka t prw at shme t
doyes gvn& eyugrmm shn gvnan eygrammon
sustsasyai.
Estv mn doyesa eyea AB, t d prw at
shmeon t A, d doyesa gvna eygrammow p
DGE: de d prw t doyes eye&
t AB ka t prw at shme t
A t doyes gvn& eyugrmm t
p DGE shn gvnan eygrammon
sustsasyai.
Elfyv f' katraw tn GD, GE
tuxnta shmea t D, E, ka peze-
xyv DE: ka k trin eyein,
a esin sai tris taw GD, DE, GE, trgvnon sun-
esttv t AZH, ste shn enai tn mn GD t AZ,
tn d GE t AH, ka ti tn DE t ZH.
Epe on do a DG, GE do taw ZA, AH sai
esn katra katr&, ka bsiw DE bsei t ZH sh,
gvna ra p DGE gvn& t p ZAH stin sh.
Prw ra t doyes eye& t AB ka t prw at
shme t A t doyes gvn& eyugrmm t p
DGE sh gvna eygrammow sunstatai p ZAH:
per dei poisai.
En do trgvna tw do pleurw [tas] do pleuraw
saw x katran katr&, tn d gvnan tw gvnaw
mezona x tn p tn svn eyein periexomnhn,
ka tn bsin tw bsevw mezona jei.
Estv do trgvna t ABG, DEZ tw do pleurw tw
AB, AG taw do pleuraw taw DE, DZ saw xonta
katran katr&, tn mn AB t DE tn d AG t
DZ, d prw t A gvna tw prw t D gvnaw me-
zvn stv: lgv, ti ka bsiw BG bsevw tw EZ
mezvn stn.
Epe gr mezvn p BAG gvna tw p EDZ
gvnaw, sunesttv prw t DE eye& ka t prw at
shme t D t p BAG gvn& sh p EDH, ka
kesyv potr& tn AG, DZ sh DH, ka peze-
xyvsan a EH, ZH.
Epe on sh stn mn AB t DE, d AG t DH,
do d a BA, AG dus taw ED, DH sai esn ka-
tra katr&: ka gvna p
BAG gvn& t p EDH sh:
bsiw ra BG bsei t EH
stin sh. plin, pe sh stn
DZ t DH, sh st ka
p DHZ gvna t p DZH:
mezvn ra p DZH tw
p EHZ: poll ra mezvn stn p EZH tw
p EHZ. ka pe trgvnn sti t EZH mezona
xon tn p EZH gvnan tw p EHZ, p d tn
mezona gvnan mezvn pleur potenei, mezvn ra
ka pleur EH tw EZ. sh d EH t BG: mezvn
ra ka BG tw EZ.
En ra do trgvna tw do pleurw dus pleuraw
saw x katran katr&, tn d gvnan tw gvnaw
mezona x tn p tn svn eyein periexomnhn,
ka tn bsin tw bsevw mezona jei: per dei dejai.
En do trgvna tw do pleurw dus pleuraw saw
x katran katr&, tn d bsin tw bsevw mezona
x, ka tn gvnan tw gvnaw mezona jei tn p
tn svn eyein periexomnhn.
Estv do trgvna t ABG, DEZ tw do pleurw
tw AB, AG taw do pleuraw taw DE, DZ saw xonta
katran katr&, tn mn AB t
DE, tn d AG t DZ: bsiw d
BG bsevw tw EZ mezvn stv:
lgv, ti ka gvna p BAG
gvnaw tw p EDZ mezvn stn:
E gr m, toi sh stn at
lssvn: sh mn on ok stin
p BAG t p EDZ: sh gr 'n n ka bsiw
BG bsei t EZ: ok sti d. ok ra sh st gvna
p BAG t p EDZ: od mn lssvn stn p
BAG tw p EDZ: lssvn gr 'n n ka bsiw BG
bsevw tw EZ: ok sti d: ok ra lssvn stn
p BAG gvna tw p EDZ. dexyh d ti od
sh: mezvn ra stn p BAG tw p EDZ.
En ra do trgvna tw do pleurw dus pleuraw
saw x katran kter&, tn d bsin tw bsevw
mezona x, ka tn gvnan tw gvnaw mezona jei tn
p tn svn eyein periexomnhn: per dei dejai.
En do trgvna tw do gvnaw dus gvnaiw saw x
katran katr& ka man pleurn mi pleur shn toi
tn prw taw saiw gvnaiw tn potenousan p man
tn svn gvnin, ka tw loipw pleurw taw loipaw
pleuraw saw jei [katran katr&] ka tn loipn
gvnan t loip gvn&.
Estv do trgvna t ABG, DEZ tw do gvnaw tw
p ABG, BGA dus taw p DEZ, EZD saw xonta
katran katr&, tn mn p
ABG t p DEZ, tn d p
BGA t p EZD: xtv d ka
man pleurn mi pleur shn,
prteron tn prw taw saiw
gvnaiw tn BG t EZ: lgv,
ti ka tw loipw pleurw taw
loipaw pleuraw saw jei katran katr&, tn mn AB
t DE tn d AG t DZ, ka tn loipn gvnan t loip
gvn&, tn p BAG t p EDZ.
E gr nisw stin AB t DE, ma atn mezvn
stn. stv mezvn AB, ka kesyv t DE sh BH,
ka pezexyv HG.
Epe on sh stn mn BH t DE, d BG t EZ,
do d a BH, BG dus taw DE, EZ sai esn katra
katr&: ka gvna p HBG gvn& t p DEZ sh
stn: bsiw ra HG bsei t DZ sh stn, ka t
HBG trgvnon t DEZ trign son stn, ka a loi-
pa gvnai taw loipaw gvnaiw sai sontai, f' "w a
sai pleura potenousin: sh ra p HGB gvna t
p DZE. ll p DZE t p BGA pkeitai sh:
ka p BGH ra t p BGA sh stn, lssvn
t mezoni: per dnaton. ok ra nisw stin AB t
DE. sh ra. sti d ka BG t EZ sh: do d a AB,
BG dus taw DE, EZ sai esn katra katr&: ka
gvna p ABG gvn& t p DEZ stin sh: bsiw
ra AG bsei t DZ sh stn, ka loip gvna p
BAG t loip gvn& t p EDZ sh stn.
All d plin stvsan a p tw saw gvnaw pleu-
ra potenousai sai, w AB t DE: lgv plin, ti
ka a loipa pleura taw loipaw pleuraw sai sontai,
mn AG t DZ, d BG t EZ ka ti loip gvna
p BAG t loip gvn& t p EDZ sh stn.
E gr nisw stin BG t EZ, ma atn mezvn
stn. stv mezvn, e dunatn, BG, ka kesyv t EZ
sh BY, ka pezexyv AY. ka pe sh stn mn
BY t EZ d AB t DE, do d a AB, BY dus taw
DE, EZ sai esn katra katr&: ka gvnaw saw
perixousin: bsiw ra AY bsei t DZ sh stn, ka
t ABY trgvnon t DEZ trign son stn, ka a
loipa gvnai taw loipaw gvnaiw sai sontai, f' "w a
sai pleura potenousin: sh ra stn p BYA
gvna t p EZD. ll p EZD t p BGA stin
sh: trignou d to AYG ktw gvna p BYA
sh st t ntw ka penanton t p BGA: per
dnaton. ok ra nisw stin BG t EZ: sh ra.
st d ka AB t DE sh. do d a AB, BG do taw
DE, EZ sai esn katra katr&: ka gvnaw saw
perixousi: bsiw ra AG bsei t DZ sh stn, ka
t ABG trgvnon t DEZ trign son ka loip
gvna p BAG t loip gvn& t p EDZ sh.
En ra do trgvna tw do gvnaw dus gvnaiw saw
x katran katr& ka man pleurn mi pleur shn
toi tn prw taw saiw gvnaiw, tn potenousan
p man tn svn gvnin, ka tw loipw pleurw taw
loipaw pleuraw saw jei ka tn loipn gvnan t
loip gvn&: per dei dejai.
En ew do eyeaw eyea mpptousa tw nallj
gvnaw saw lllaiw poi, parllhloi sontai lllaiw
a eyeai.
Ew gr do eyeaw tw AB, GD eyea mpptousa
EZ tw nallj gvnaw tw p AEZ, EZD saw ll-
laiw poietv: lgv, ti parllhlw stin AB t GD.
E gr m, kballmenai a AB, GD sumpesontai
toi p t B, D mrh p t A, G. kbeblsyvsan ka
sumpipttvsan p t B, D mrh kat t H. trignou
d to HEZ ktw gvna p AEZ sh st t ntw
ka penanton t p EZH: per stn dnaton: ok
ra a AB, GD kballmenai sumpesontai p t B,
D mrh. movw d deixysetai, ti od p t A, G: a
d p mhdtera t mrh sumpptousai parllhlo esin:
parllhlow ra stn AB t GD.
En ra ew do eyeaw eyea mpptousa tw nallj
gvnaw saw lllaiw poi, parllhloi sontai a
eyeai: per dei dejai.
En ew do eyeaw eyea mpptousa tn ktw gv-
nan t ntw ka penanton ka p t at mrh shn
poi tw ntw ka p t at mrh dusn ryaw saw,
parllhloi sontai lllaiw a eyeai.
Ew gr do eyeaw tw AB, GD eyea mpptousa
EZ tn ktw gvnan tn p EHB t ntw ka penan-
ton gvn& t p HYD shn poietv tw ntw ka p
t at mrh tw p BHY, HYD dusn ryaw saw:
lgv, ti parllhlw stin AB t GD.
Epe gr sh stn p EHB t p HYD, ll
p EHB t p AHY stin sh, ka p AHY ra
t p HYD stin sh: ka esin nallj: parllhlow
ra stn AB t GD.
Plin, pe a p BHY, HYD do ryaw sai esn,
es d ka a p AHY, BHY du-
sn ryaw sai, a ra p AHY,
BHY taw p BHY, HYD sai
esn: koin frsyv p BHY:
loip ra p AHY loip t p
HYD stin sh: ka esin nallj:
parllhlow ra stn AB t GD.
En ra ew do eyeaw eyea mpptousa tn ktw
gvnan t ntw ka penanton ka p t at mrh shn
poi tw ntw ka p t at mrh dusn ryaw saw,
parllhloi sontai a eyeai: per dei dejai.
H ew tw paralllouw eyeaw eyea mpptousa
tw te nallj gvnaw saw lllaiw poie ka tn ktw
t ntw ka penanton shn ka tw ntw ka p t
at mrh dusn ryaw saw.
Ew gr paralllouw eyeaw tw AB, GD eyea
mpipttv EZ: lgv, ti tw nallj gvnaw tw
p AHY, HYD saw poie ka tn ktw gvnan tn p
EHB t ntw ka penanton t p HYD shn ka tw
ntw ka p t at mrh tw p
BHY, HYD dusn ryaw saw.
E gr nisw stin p AHY
t p HYD, ma atn mezvn
stn. stv mezvn p AHY:
koin proskesyv p BHY: a
ra p AHY, BHY tn p
BHY, HYD mezonw esin. ll a p AHY, BHY
dusn ryaw sai esn. [ka] a ra p BHY,
HYD do ryn lssonw esin. a d p' lassnvn
do ryn kballmenai ew peiron sumpptousin: a
ra AB, GD kballmenai ew peiron sumpesontai:
o sumpptousi d di t paralllouw atw pokesyai:
ok ra nisw stin p AHY t p HYD: sh ra.
ll p AHY t p EHB stin sh: ka p
EHB ra t p HYD stin sh. koin proskesyv
p BHY: a ra p EHB, BHY taw p BHY,
HYD sai esn. ll a p EHB, BHY do ryaw
sai esn: ka a p BHY, HYD ra do ryaw sai
esn.
H ra ew tw paralllouw eyeaw eyea mpptousa
tw te nallj gvnaw saw lllaiw poie ka tn ktw
t ntw ka penanton shn ka tw ntw ka p t
at mrh dusn ryaw saw: per dei dejai.
A t at eye& parllhloi ka lllaiw es par-
llhloi.
Estv katra tn AB, GD t EZ parllhlow: lgv,
ti ka AB t GD sti parllhlow.
Empipttv gr ew atw eyea HK.
Ka pe ew paralllouw eyeaw tw AB, EZ eyea
mpptvken HK, sh ra p AHK t p HYZ.
plin, pe ew paralllouw eyeaw tw EZ, GD eyea
mpptvken HK, sh stn p HYZ t p HKD.
dexyh d ka p AHK t p HYZ sh. ka p
AHK ra t p HKD stin sh:
ka esin nallj. parllhlow ra
stn AB t GD.
[A ra t at eye& parllhloi
ka lllaiw es parllhloi:] per
dei dejai.
Di to doyntow shmeou t doyes eye& parllh-
lon eyean grammn gagen.
Estv t mn doyn shmeon t A, d doyesa eyea
BG: de d di to A shmeou t
BG eye& parllhlon eyean gram-
mn gagen.
Elfyv p tw BG tuxn shmeon
t D, ka pezexyv AD: ka
sunesttv prw t DA eye& ka t prw at sh-
me t A t p ADG gvn& sh p DAE: ka
kbeblsyv p' eyeaw t EA eyea AZ.
Ka pe ew do eyeaw tw BG, EZ eyea mpptousa
AD tw nallj gvnaw tw p EAD, ADG saw ll-
laiw pepohken, parllhlow ra stn EAZ t BG.
Di to doyntow ra shmeou to A t doyes eye&
t BG parllhlow eyea gramm ktai EAZ: per
dei poisai.
Pantw trignou miw tn pleurn prosekblhyeshw
ktw gvna dus taw ntw ka penanton sh stn,
ka a ntw to trignou trew gvnai dusn ryaw sai
esn.
Estv trgvnon t ABG, ka prosekbeblsyv ato
ma pleur BG p t D: lgv, ti ktw gvna
p AGD sh st dus taw ntw ka p-
enanton taw p GAB, ABG, ka a n-
tw to trignou trew gvnai a p ABG,
BGA, GAB dusn ryaw sai esn.
Hxyv gr di to G shmeou t AB
eye& parllhlow GE.
Ka pe parllhlw stin AB t GE, ka ew atw
mpptvken AG, a nallj gvnai a p BAG,
AGE sai lllaiw esn. plin, pe parllhlw
stin AB t GE, ka ew atw mpptvken eyea
BD, ktw gvna p EGD sh st t ntw ka
penanton t p ABG. dexyh d ka p AGE t
p BAG sh: lh ra p AGD gvna sh st dus
taw ntw ka penanton taw p BAG, ABG.
Koin proskesyv p AGB: a ra p AGD, AGB
tris taw p ABG, BGA, GAB sai esn. ll' a p
AGD, AGB dusn ryaw sai esn: ka a p AGB,
GBA, GAB ra dusn ryaw sai esn.
Pantw ra trignou miw tn pleurn prosekblhye-
shw ktw gvna dus taw ntw ka penanton sh
stn, ka a ntw to trignou trew gvnai dusn ryaw
sai esn: per dei dejai.
A tw saw te ka paralllouw p t at mrh pi-
zeugnousai eyeai ka ata sai te ka parllhlo
esin.
Estvsan sai te ka parllhloi
a AB, GD, ka pizeugntvsan
atw p t at mrh eyeai
a AG, BD: lgv, ti ka a AG, BD sai te ka
parllhlo esin.
Epezexyv BG. ka pe parllhlw stin AB
t GD, ka ew atw mpptvken BG, a nallj
gvnai a p ABG, BGD sai lllaiw esn. ka pe
sh stn AB t GD koin d BG, do d a AB, BG
do taw BG, GD sai esn: ka gvna p ABG
gvn& t p BGD sh: bsiw ra AG bsei t BD
stin sh, ka t ABG trgvnon t BGD trign son
stn, ka a loipa gvnai taw loipaw gvnaiw sai
sontai katra katr&, f' "w a sai pleura pote-
nousin: sh ra p AGB gvna t p GBD. ka
pe ew do eyeaw tw AG, BD eyea mpptousa
BG tw nallj gvnaw saw lllaiw pepohken, parl-
lhlow ra stn AG t BD. dexyh d at ka sh.
A ra tw saw te ka paralllouw p t at mrh
pizeugnousai eyeai ka ata sai te ka parllhlo
esin: per dei dejai.
Tn parallhlogrmmvn xvrvn a penanton pleu-
ra te ka gvnai sai lllaiw esn, ka dimetrow
at dxa tmnei.
Estv parallhlgrammon xvron t AGDB, di-
metrow d ato BG: lgv, ti to AGDB parallh-
logrmmou a penanton pleura te ka gvnai sai
lllaiw esn, ka BG dimetros
at dxa tmnei.
Epe gr parllhlw stin AB
t GD, ka ew atw mpptvken e-
yea BG, a nallj gvnai a p
ABG, BGD sai lllaiw esn. plin, pe parllhlw
stin AG t BD, ka ew atw mpptvken BG, a
nallj gvnai a p AGB, GBD sai lllaiw esn.
do d trgvn sti t ABG, BGD tw do gvnaw tw
p ABG, BGA dus taw p BGD, GBD saw xonta
katran katr& ka man pleurn mi pleur shn tn
prw taw saiw gvnaiw koinn atn tn BG: ka tw
loipw ra pleurw taw loipaw saw jei katran ka-
tr& ka tn loipn gvnan t loip gvn&: sh ra
mn AB pleur t GD, d AG t BD, ka ti sh
stn p BAG gvna t p GDB. ka pe sh stn
mn p ABG gvna t p BGD, d p GBD t
p AGB, lh ra p ABD l t p AGD stin
sh. dexyh d ka p BAG t p GDB sh.
Tn ra parallhlogrmmvn xvrvn a penanton
pleura te ka gvnai sai lllaiw esn.
Lgv d, ti ka dimetrow at dxa tmnei. pe
gr sh stn AB t GD, koin d BG, do d a AB,
BG dus taw GD, BG sai esn katra katr&: ka
gvna p ABG gvn& t p BGD sh. ka bsiw ra
AG t DB sh. ka t ABG [ra] trgvnon t BGD
trign son stn.
H ra BG dimetrow dxa tmnei t ABGD parallh-
lgrammon: per dei dejai.
T parallhlgramma t p tw atw bsevw nta
ka n taw ataw parallloiw sa llloiw stn.
Estv parallhlgramma t ABGD,
EBGZ p tw atw bsevw tw BG
ka n taw ataw parallloiw taw
AZ, BG: lgv, ti son st t ABGD
t EBGZ parallhlogrmm.
Epe gr parallhlgrammn sti t ABGD, sh stn
AD t BG. di t at d ka EZ t BG stin
sh: ste ka AD t EZ stin sh: ka koin DE:
lh ra AE l t DZ stin sh. sti d ka AB
t DG sh: do d a EA, AB do taw ZD, DG sai
esn katra katr&: ka gvna p ZDG gvn& t
p EAB stin sh ktw t ntw: bsiw ra EB
bsei t ZG sh stn, ka t EAB trgvnon t DZG
trign son stai: koinn frsyv t DHE: loipn
ra t ABHD trapzion loip t EHGZ trapez
stn son: koinn proskesyv t HBG trgvnon: lon
ra t ABGD parallhlgrammon l t EBGZ paral-
lhlogrmm son stn.
T ra parallhlgramma t p tw atw bsevw
nta ka n taw ataw parallloiw sa llloiw stn:
per dei dejai.
Copyright 2000 Giovanni Ferrero