Pappus Alexandrinus Mathematical Collection, Book 8 1876 Berlin el pappu_coll8_01_el_1876.xml 088

*p*a*p*p*o*u *a*l*e*c*a*n*d*r*e*w*s *s*u*n*a*g*w*g*h*s *h. *perie/xei de\ mhxanika\ problh/mata su/mmikta a)nqhra/.

*(h mhxanikh\ qewri/a, te/knon *(ermo/dwre, pro\s polla\ kai\ mega/la tw= e)n tw=| bi/w| xrh/simos u(pa/rxousa plei/sths ei)ko/tws a)podoxh=s h)ci/wtai pro\s tw=n filoso/fwn kai\ pa=si toi=s a)po\ tw=n maqhma/twn perispou/dasto/s e)stin, e)peidh\ sxedo\n prw/th th=s peri\ th\n u(/lhn tw=n e)n tw=| ko/smw| stoi­xei/wn fusiologi/as a(/ptetai. sta/sews ga\r kai\ fora=s swma/­twn kai\ th=s kata\ to/pon kinh/sews e)n toi=s o(/lois qewrhma­tikh\ tugxa/nousa ta\ me\n kinou/mena kata\ fu/sin ai)tiologei=, ta\ d' a)nagka/zousa para\ fu/sin e)/cw tw=n oi)kei/wn to/pwn ei)s e)nanti/as kinh/seis meqi/sthsin e)pimhxanwme/nh dia\ tw=n e)c au)th=s th=s u(/lhs u(popipto/ntwn au)th=| qewrhma/twn. th=s de\ mhxanikh=s to\ me\n ei)=nai logiko\n to\ de\ xeirourgiko\n oi( peri\ to\n *(/hrwna mhxanikoi\ le/gousin: kai\ to\ me\n logiko\n sun­esta/nai me/ros e)/k te gewmetri/as kai\ a)riqmhtikh=s kai\ a)stro­nomi/as kai\ tw=n fusikw=n lo/gwn, to\ de\ xeirourgiko\n e)/k te xalkeutikh=s kai\ oi)kodomikh=s kai\ tektonikh=s kai\ zwgrafikh=s kai\ th=s e)n tou/tois kata\ xei=ra a)skh/sews: to\n me\n ou)=n e)n tai=s proeirhme/nais e)pisth/mais e)k paido\s geno/menon ka)n tai=s proeirhme/nais te/xnais e(/cin ei)lhfo/ta pro\s de\ tou/­tois fu/sin eu)ki/nhton e)/xonta, kra/tiston e)/sesqai mhxani­kw=n e)/rgwn eu(reth\n kai\ a)rxite/ktona/ fasin. mh\ dunatou= d' o)/ntos to\n au)to\n maqhma/twn te tosou/twn perigene/­sqai kai\ maqei=n a(/ma ta\s proeirhme/nas te/xnas paragge/l­lousi tw=| ta\ mhxanika\ e)/rga metaxeiri/zesqai boulome/nw| xrh=sqai tai=s oi)kei/ais te/xnais u(poxeiri/ois e)n tai=s par' e(/kasta xrei/ais.

*ma/lista de\ pa/ntwn a)nagkaio/tatai te/xnai tugxa/nousin pro\s th\n tou= bi/ou xrei/an [mhxanikh\ prohgoume/nh th=s a)rxitek­tonikh=s] h(/ te tw=n magganari/wn, mhxanikw=n kai\ au)tw=n kata\ tou\s a)rxai/ous legome/nwn (mega/la ga\r ou(=toi ba/rh dia\ mhxa­nw=n para\ fu/sin ei)s u(/yos a)na/gousin e)la/ttoni duna/mei ki­nou=ntes), kai\ h( tw=n o)rganopoiw=n tw=n pro\s to\n po/lemon a)nagkai/wn, kaloume/nwn de\ kai\ au)tw=n mhxanikw=n (be/lh ga\r kai\ li/qina kai\ sidhra= kai\ ta\ paraplh/sia tou/tois e)capo­ste/lletai ei)s makro\n o(dou= mh=kos toi=s u(p' au)tw=n ginome/nois o)rga/nois katapaltikoi=s), pro\s de\ tau/tais h( tw=n i)di/ws pa/lin kaloume/nwn mhxanopoiw=n (e)k ba/qous ga\r pollou= u(/dwr eu)kolw/teron a)na/getai dia\ tw=n a)ntlhmatikw=n o)rga/­nwn w(=n au)toi\ kataskeua/zousin). kalou=si de\ mhxanikou\s oi( palaioi\ kai\ tou\s qaumasiourgou\s, w(=n oi( me\n dia\ pneu­ma/twn filotexnou=sin, w(s *(/hrwn pneumatikoi=s, oi( de\ dia\ neu­ri/wn kai\ spa/rtwn e)myu/xwn kinh/seis dokou=si mimei=sqai, w(s *(/hrwn au)toma/tois kai\ zugi/ois, a)/lloi de\ dia\ tw=n e)f' u(/datos o)xoume/nwn, w(s *)arximh/dhs o)xoume/nois, h)\ tw=n di' u(/datos w(ro- logi/wn, w(s *(/hrwn u(drei/ois, a(\ dh\ kai\ th=| gnwmonikh=| qewri/a| koi­nwnou=nta fai/netai. mhxanikou\s de\ kalou=sin kai\ tou\s ta\s sfairopoii/+as [poiei=n] e)pistame/nous, u(f' w(=n ei)kw\n tou= ou)ra­nou= kataskeua/zetai di' o(malh=s kai\ e)gkukli/ou kinh/sews u(/datos.

*pa/ntwn de\ tou/twn th\n ai)ti/an kai\ to\n lo/gon e)pegnw­ke/nai fasi/n tines to\n *surako/sion *)arximh/dh: mo/nos ga\r ou(=tos e)n tw=| kaq' h(ma=s bi/w| poiki/lh| pro\s pa/nta ke/xrhtai th=| fu/sei kai\ th=| e)pinoi/a|, kaqw\s kai\ *gemi=nos o( maqhma­tiko\s e)n tw=| peri\ th=s tw=n maqhma/twn ta/cew/s fhsin. *ka/r­pos de\ pou/ fhsin o( *)antioxeu\s *)arximh/dh to\n *surako/sion e(\n mo/non bibli/on suntetaxe/nai mhxaniko\n to\ kata\ th\n sfai­ropoii/+an, tw=n de\ a)/llwn ou)de\n h)ciwke/nai sunta/cai. kai/toi para\ toi=s polloi=s e)pi\ mhxanikh=| docasqei\s kai\ megalo­fuh/s tis geno/menos o( qaumasto\s e)kei=nos, w(/ste diamei=nai para\ pa=sin a)nqrw/pois u(perballo/ntws u(mnou/menos, tw=n te prohgoume/nwn gewmetrikh=s kai\ a)riqmhtikh=s e)xome/nwn qewri/as [kai\] ta\ braxu/tata dokou=nta ei)=nai spoudai/ws sune/grafen: o(\s fai/netai ta\s ei)rhme/nas e)pisth/mas ou(/tws a)gaph/sas w(s mhde\n e)/cwqen u(pome/nein au)tai=s e)peisa/gein. au)to\s de\ *ka/rpos kai\ a)/lloi tine\s sunexrh/santo gewmetri/a| kai\ ei)s te/xnas tina\s eu)lo/gws: gewmetri/a ga\r ou)de\n bla/pte­tai, swmatopoiei=n pefukui=a polla\s te/xnas, dia\ tou= sun­ei=nai au)tai=s [mh/thr ou)=n w(/sper ou)=sa texnw=n ou) bla/ptetai dia\ tou= fronti/zein o)rganikh=s kai\ a)rxitektonikh=s: ou)de\ ga\r dia\ to\ sunei=nai gewmori/a| kai\ gnwmonikh=| kai\ mhxanikh=| kai\ skhnografi/a| bla/ptetai/ ti], tou)nanti/on de\ proa/gousa me\n tau/tas fai/netai, timwme/nh de\ kai\ kosmoume/nh deo/ntws u(p' au)tw=n.

*toiau/ths de\ th=s mhxanikh=s e)pisth/mhs o(mou= kai\ te/xnhs u(parxou/shs kai\ ei)s tosau=ta me/rh dih|rhme/nhs kalw=s e)/xein e)no/misa ta/ te lo/gw| gewmetrikw=| qewrou/mena [kai\ a)nagkaio/­tata peri\ th\n tw=n barw=n ki/nhsin kei/mena de\] para\ toi=s palaioi=s kai\ ta\ u(f' h(mw=n eu)xrh/stws a)neurhme/na qewrh/­mata suntomw/teron kai\ safe/steron a)nagra/yai belti/oni/ te lo/gw| tou= para\ toi=s pro/teron a)nagegramme/nou sunta/cai, oi(=on ba/rous doqe/ntos u(po\ doqei/shs [u(podoxh=s] a)gome/nou duna/mews e)n tw=| para\ to\n o(ri/zonta e)pipe/dw|, kai\ e(te/rou e)pipe/dou keklime/nou pro\s to\ u(pokei/menon doqei=san gwni/an u(potiqe/ntos, eu(rei=n th\n du/namin u(f' o(/shs a)xqh/setai to\ ba/ros e)n tw=| keklime/nw| e)pipe/dw| (tou=to de\ xrh/simon toi=s mhxanikoi=s magganari/ois: prosqe/ntes ga\r th=| eu(reqei/sh| duna/mei e(te/ran tina\ du/namin a)ndrw=n qarsou=ntes a)na/gousin to\ ba/ros), kai\ du/o doqeisw=n eu)qeiw=n a)ni/swn du/o me/sas a)na/logon eu(rei=n e)n sunexei= a)nalogi/a| (dia\ ga\r tou= qewrh/­matos tou/tou pa=n to\ doqe\n stereo\n sxh=ma kata\ to\n do­qe/nta lo/gon au)/cetai/ te kai\ meiou=tai), kai\ pw=s dunato/n e)sti tumpa/nou doqe/ntos kai\ tou= plh/qous tw=n skutalw=n au)tou= [doqe/ntwn h)\ o)do/ntwn] paraqei=nai au)tw=| tu/mpanon doqe\n e)/xon to\ plh=qos tw=n o)do/ntwn kai\ eu(rei=n th\n dia/me­tron tou= paratiqeme/nou tumpa/nou (tou=to ga\r xrh/simon ei)s polla\ kai\ th=| tw=n mhxanopoiw=n te/xnh| dia\ th\n para/qesin tw=n skutalwtw=n tumpa/nwn). e(/kaston de\ tou/twn e)n tw=| oi)­kei/w| to/pw| genh/setai fanero\n meta\ kai\ a)/llwn xrhsi/mwn a)rxite/ktoni kai\ mhxanikw=|, e)a\n pro/teron ta\ sune/xonta th\n kentrobarikh\n pragmatei/an ei)/pwmen e(ch=s.

*ti/ me\n ou)=n e)stin to\ baru\ kai\ to\ kou=fon, kai\ ti/s ai)­ti/a th=s a)/nw kai\ ka/tw toi=s sw/masi fora=s, kai\ au)to/ ge to\ a)/nw kai\ ka/tw ti/nos e)nnoi/as e)/xetai kai\ ti/sin a)fw/ristai pe/rasin, ou)de\n dei= le/gesqai par' h(mw=n to\ nu=n, e)peidh\ peri\ tou/twn e)n toi=s maqhmatikoi=s u(po\ tou= *ptolemai/ou dedh/lwtai, to\ de\ ke/ntron tou= ba/rous e)ka/stou sw/matos, o(\ th=s kentrobarikh=s pragmatei/as a)rxh\ kai\ stoixei=o/n e)stin, e)c h(=s kai\ ta\ loipa\ me/rh th=s mhxanikh=s a)nh/rthtai, ti/ pot' e)sti\n kai\ ti/ bou/letai lekte/on: e)k tou/tou ga/r, oi)=mai, kai\ ta\ loipa\ tw=n e)n th=| pragmatei/a| qewroume/nwn e)/stai safh=. le/gomen de\ ke/ntron ba/rous e(ka/stou sw/matos ei)=nai shmei=o/n ti kei/menon e)nto/s, a)f' ou(= kat' e)pi/noian a)rthqe\n to\ ba/ros h)remei= fero/menon kai\ fula/ssei th\n e)c a)rxh=s qe/sin [ou) mh\ peritrepo/menon e)n th=| fora=|]. tou=to de\ to\ shmei=on ou) mo/non e)n toi=s tetagme/nois a)lla\ ka)n toi=s a)ta/ktws e)sxh­matisme/nois eu(ri/sketai sw/masin u(pa/rxon, e)fo/dw| tini\ qew­rou/menon toiau/th|.

a#. *(upokei/sqw ga\r e)pi/pedon o)rqo\n to\ *a*b*g*d neu=on ei)s to\ tou= panto\s ke/ntron, e)f' o(\ kai\ ta\ ba/ros e)/xonta pa/nta th\n r(oph\n e)/xein dokei=, kai\ e)/stw h( *a*b eu)qei=a para/llhlos tw=| e)f' ou(= bebh/kamen e)pipe/dw|. e)a\n dh/ ti tw=n ba/ros e)xo/n­twn swma/twn tiqh=tai kata\ th=s *a*b eu)qei/as ou(/tws, w(/ste tetmh=sqai pa/ntws u(po\ tou= e)pipe/dou e)kballome/nou, e(/cei pote\ qe/sin toiau/thn, w(/ste me/nein a)peri/trepton kai\ mh\ a)popi/ptein. genome/nou de\ tou/tou e)a\n nohqh=| to\ *a*b*g*d e)pi/­pedon e)kballo/menon, temei= to\ e)pikei/menon sw=ma ei)s i)so/r­ropa du/o me/rh, oi(=on peri\ a)/rthma to\ e)pi/pedon i)sorro­pou=nta. pa/lin dh\ to\ ba/ros metateqe/n, w(/ste kaq' e(/teron me/ros yau/ein th=s *a*b eu)qei/as, e(/cei pote\ qe/sin peritrepo/­menon w(/ste me/nein a)feqe\n kai\ mh\ a)popi/ptein. e)a\n ou)=n pa/lin nohqh=| to\ *a*b*g*d e)pi/pedon e)kbeblhme/non, ei)s i)sorro- pou=nta me/rh temei= to\ ba/ros kai\ sumpesei=tai tw=| pro/teron ei)s i)so/rropa te/mnonti to\ au)to\ ba/ros e)pipe/dw|: ei) ga\r mh\ temei=, ta\ au)ta\ me/rh kai\ i)so/rropa kai\ a)niso/rropa genh/­setai a)llh/lois, o(/per a)/topon.$b#. *tou/twn dh\ proeirhme/nwn noei/sqw pa/lin eu)qei=a h( *a*b o)rqh\ pro\s to\ e)f' ou(= bebh/kamen e)pi/pedon, ei)s to\ tou= panto\s ke/ntron dhlono/ti neu/ousa, kai\ to\ ba/ros o(moi/ws e)pi\ tou= *a shmei/ou tiqe/sqw, oi(=on u(poqe/mati th=| *a*b eu)qei/a| xrw/menon [sth/setai dh/pote kata\ tou= *a shmei/ou w(/ste me/­nein, ei)/ ge dh\ kai\ e)pi\ tou= di' au)th=s e)pipe/dou to\ ba/ros h)remei=n e)du/nato]. e)a\n dh\ me/nontos au)tou= e)kblhqh=| h( *a*b eu)qei=a, e)napolhfqh/setai/ ti me/ros au)th=s. e)n tw=| u(pokei­me/nw| sxh/mati. noei/sqw dh\ tou=to me/non, kai\ pa/lin kaq' e(/teron me/ros e)pikei/sqw th=| eu)qei/a| to\ ba/ros w(/ste h)remei=n: le/gw dh\ o(/ti e)kblhqei=sa h( *a*b eu)qei=a sumpesei=tai th=| pro/­teron e)napeilhmme/nh|. ei) ga\r mh\ sumpesei=tai, dunh/setai/ tina di' a)mfote/rwn au)tw=n e)kblhqe/nta e)pi/peda mh\ sum­pesei=n a)llh/lois e)nto\s tou= sxh/matos, kai\ e(ka/teron au)tw=n [e)farmozo/menon tw=| dia\ th=s *a*b e)pipe/dw|] dielei=n to\ ba/ros ei)s i)so/rropa kai\ a)niso/rropa ta\ au)ta\ me/rh, o(/per a)/topon: sumpesou=ntai a)/ra ai( ei)rhme/nai eu)qei=ai e)nto\s tou= sxh/ma­tos. o(moi/ws de\ ka)\n kat' a)/llas qe/seis tiqh=tai to\ ba/ros e)pi\ tou= *a shmei/ou w(/ste me/nein, e)kblhqei=sa h( *a*b sumpe­sei=tai tai=s pro/teron e)napeilhmme/nais [o(moi/ws] eu)qei/ais. e)c ou(= fanero\n w(s kaq' e(\n shmei=on a)llh/las temou=sin ai( to\n ei)rhme/non tro/pon e)pinoou/menai eu)qei=ai: to\ de\ shmei=on tou=to ke/ntron tou= ba/rous kalei=tai. kai\ fanero\n o(/ti e)k tou= ke/ntrou kat' e)pi/noian to\ ba/ros a)rtw/menon ou) peri­traph/setai, menei= de\ th\n e)c a)rxh=s fula/sson h(ntinou=n qe/­sin e)n th=| fora=|: pa/nta ga\r di' au)tou= e)kblhqe/nta e)pi/peda ei)s i)so/rropa me/rh diairei= to\ ba/ros, w(/ste mhdemi/an ai)ti/an e)pide/xesqai peritroph=s [i)sorro/pwn au)tou= kata\ pa=san qe/­sin tw=n e)f' e(ka/tera tou= shmei/ou ginome/nwn merw=n]. *to\ me\n ou)=n ma/lista sune/xon th\n kentrobarikh\n prag­matei/an tou=t' a)\n ei)/h, ma/qois d' a)\n ta\ me\n stoixeiw/dh o)/nta dia\ tau/ths deiknu/mena toi=s *)arximh/dous peri\ i)sor­ropiw=n e)ntuxw\n kai\ toi=s *(/hrwnos mhxanikoi=s, o(/sa de\ mh\ gnw/rima toi=s polloi=s gra/yomen e)fech=s, oi(=on ta\ toi­au=ta. g#. *)/estw tri/gwnon to\ *a*b*g, kai\ ai( pleurai\ au)tou= ei)s to\n au)to\n lo/gon temne/sqwsan toi=s *h *q *k shmei/ois, w(/ste ei)=nai w(s th\n *a*h pro\s *h*b, th\n *b*q pro\s *q*g kai\ th\n *g*k pro\s *k*a, kai\ e)pezeu/xqwsan ai( *h*q *q*k *k*h: o(/ti tou= *a*b*g trigw/nou kai\ tou= *h*q*k to\ au)to\ ke/ntron tou= ba/rous e)sti/n. *tetmh/sqwsan ga\r ai( *b*g *g*a di/xa toi=s *d *e, kai\ e)pezeu/xqwsan ai( *a*d *b*e: to\ *z a)/ra ke/ntron ba/rous e)sti\n tou= *a*b*g trigw/nou. e)a\n ga\r to\ tri/gwnon e)pi/ tinos o)rqou= e)pipe/dou e)pistaqh=| kata\ th\n *a*d eu)qei=an, e)p' ou)de/teron me/ros r(e/yei to\ tri/gwnon dia\ to\ i)/son ei)=nai to\ *a*b*d tri/­gwnon tw=| *a*g*d trigw/nw|. e)pistaqe\n de\ o(moi/ws to\ *a*b*g tri/gwnon kata\ th\n *b*e e)pi\ tou= o)rqou= e)pipe/dou e)p' ou)­de/teron me/ros r(e/yei dia\ to\ i)/sa ei)=nai ta\ *a*b*e *b*g*e tri/­gwna. ei) de\ e)f' e(kate/ras tw=n *a*d *b*e i)sorropei= to\ tri/gwnon, to\ a)/ra koino\n au)tw=n shmei=on to\ *z ke/ntron e)/stai tou= ba/rous. [noei=n de\ dei= to\ *z, w(s proei/rhtai, kei/menon e)n me/sw| tou= *a*b*g trigw/nou i)sopaxou=s te kai\ i)sobarou=s dh­lono/ti u(pokeime/nou.] kai\ fanero\n o(/ti diplasi/a e)sti\n h( me\n *a*z th=s *z*d, h( de\ *b*z th=s *z*e, kai\ o(/ti w(s h( *g*a pro\s *a*e, ou(/tws h( *a*b pro\s *d*e kai\ h( *b*z pro\s *z*e kai\ h( *a*z pro\s *z*d dia\ to\ i)sogw/nia ei)=nai kai\ ta\ *d*z*e *a*b*z tri/gwna kai\ ta\ *g*d*e *a*b*g. e)pizeuxqei=sa ou)=n h( *d*e temne/tw th\n *q*k kata\ to\ *l. e)pei\ ou)=n o( th=s *b*q pro\s *q*g lo/gos sunh=ptai e)/k te tou= th=s *q*b pro\s *d*q kai\ tou= th=s *d*q pro\s *q*g, kai\ e)/stin sunqe/nti w(s h( *b*g pro\s *g*q, h( *g*a pro\s *a*k, kai\ tw=n h(goume/nwn ta\ h(mi/sh w(s *g*d pro\s *g*q, h( *e*a pro\s *a*k, kai\ a)nastre/yanti w(s h( *g*d pro\s *d*q, h( *a*e pro\s *e*k, i)/sh de\ h( me\n *g*d th=| *b*d, h( de\ *a*e th=| *g*e, kai\ w(s a)/ra h( *b*d pro\s *d*q, h( *g*e pro\s *e*k: sunqe/nti a)/ra w(s h( *b*q pro\s *q*d, h( *g*k pro\s *k*e: su/gkeitai a)/ra kai\ o( th=s *a*h pro\s *h*b lo/gos e)/k te tou= th=s *g*k pro\s *k*e kai\ tou= th=s *d*q @3pro\s *q*g. su/gkeitai d' e)k tw=n au)tw=n kai\ o( th=s *d*l pro\s *l*e [kai\ i)/sh e)sti\n h( *q*l th=| *l*k], w(s deixqh/­setai: e)/stin a)/ra kai\ w(s h( *a*h pro\s *h*b, h( *d*l pro\s *l*e. kai\ ei)si\n par­a/llhloi ai( *a*b *d*e, kai\ e)pezeugme/nai ai( *a*d *b*e te/mnousin a)llh/las kata\ to\ *z: eu)qei=a a)/ra e)sti\n h( dia\ tw=n *h *z *l: kai\ tou=to ga\r e(ch=s [ei) mikro/n e)stin]. kai\ e)pei/ e)stin w(s h( *b*z pro\s *z*e, ou(/tws h( *h*z pro\s *z*l, diplh= de\ h( *b*z th=s *z*e, diplh= a)/ra kai\ h( *h*z th=s *z*l. trigw/nou dh\ tou= *h*q*k dixotomi/a h( *h*l, kai\ diplh= h( *h*z th=s *z*l: to\ *z a)/ra ke/n­tron ba/rous e)sti\n tou= *h*q*k trigw/nou. h)=n de\ kai\ tou= *a*b*g. d#. *to\ de\ u(perteqe\n nu=n deixqh/setai. e)/stw ga\r w(s h( *g*d pro\s *d*q, h( *g*e pro\s *e*k, kai\ e)pezeu/xqwsan ai( *d*e *q*k te/mnousai a)llh/las kata\ to\ *l: o(/ti i)/sh me/n e)stin h( *q*l th=| *k*l, o( de\ th=s *d*l pro\s *l*e lo/gos su/gkeitai e)/k te tou= th=s *d*q pro\s *q*g kai\ tou= th=s *g*k pro\s *k*e. *)/hxqw dia\ tou= *g th=| *q*k para/llhlos h( *g*z kai\ sum­pipte/tw th=| *d*e e)kblhqei/sh| kata\ to\ *z. e)pei\ ou)=n du/o eu)­qei=ai/ ei)sin ai( *d*l *l*e, kai\ e)/cwqen h( *z*l, o( a)/ra th=s *d*l pro\s *l*e lo/gos su/gkeitai e)/k te tou= th=s *d*l pro\s *l*z kai\ tou= th=s *l*z pro\s *e*l. a)lla\ tw=| me\n th=s *d*l pro\s *l*z lo/gw| o( au)to/s e)stin o( th=s *d*q pro\s *q*g dia\ to\ para/llh­lon ei)=nai th\n *g*z th=| *k*q, tw=| de\ th=s *z*l pro\s *l*e lo/gw| o( au)to/s e)stin o( th=s *g*k pro\s *k*e dia\ to\ i)sogw/nia ei)=nai ta\ *g*e*z *e*k*l tri/gwna: kai\ o( th=s *d*l a)/ra pro\s th\n *l*e lo/gos su/gkeitai e)/k te tou= th=s *d*q pro\s *q*g kai\ e)k tou= th=s *g*k pro\s *k*e. kata\ tau)ta\ dh\ deixqh/setai o(/ti kai\ o( th=s *k*l pro\s *l*q lo/gos sunh=ptai e)/k te tou= th=s *k*e pro\s *e*g kai\ tou= th=s *g*d pro\s *d*q, parallh/lou a)xqei/shs th=| *e*d dia\ tou= *g th=s *g*m kai\ sumpiptou/shs th=| *k*q e)kblh­qei/sh| kata\ to\ *m. e)pei\ ga\r pa/lin du/o eu)qei=ai/ ei)sin ai( *k*l *l*q e)/cwqen th=s *l*m lambanome/nhs, o( a)/ra th=s *k*l pro\s *l*q lo/gos su/gkeitai e)/k te tou= th=s *k*l pro\s *l*m kai\ tou= th=s *l*m pro\s *l*q. a)ll' o( me\n th=s *k*l pro\s *l*m lo/gos o( au)to/s e)stin tw=| th=s *k*e pro\s *e*g dia\ to\ para/llhlon ei)=nai pa/lin th\n *e*d th=| *g*m, o( de\ th=s *l*m pro\s *l*q lo/­gos o( au)to/s e)stin tw=| th=s *g*d pro\s *d*q dia\ to\ i)sogw/nia ei)=nai ta\ *d*q*l *g*q*m tri/gwna: o( a)/ra th=s *k*l pro\s *l*q lo/gos o( au)to/s e)stin tw=| sugkeime/nw| e)/k te tou= th=s *k*e pro\s *e*g, toute/stin tou= th=s *d*q pro\s *d*g, kai\ tou= th=s *g*d pro\s th\n *d*q lo/gou, o(\s to\n th=s i)so/thtos lo/gon poiei=: kai\ o( th=s *k*l a)/ra pro\s th\n *l*q lo/gos th=s i)so/thto/s e)stin: i)/sh a)/ra h( *k*l th=| *l*q. e#. *to\ loipo\n tw=n u(perteqe/ntwn. e)/stw para/llhlos h( *a*b th=| *g*d, kai\ w(s h( *a*z pro\s *z*b, h( *g*q pro\s *q*d, kai\ e)pezeu/xqwsan ai( *a*g *b*d te/mnousai a)llh/las kata\ to\ *e shmei=on: o(/ti h( dia\ tw=n *z *e *q eu)qei=a/ e)stin. *ei) ga\r mh/, e)/stw h( dia\ tw=n *z *e *h. e)pei\ ou)=n e)stin @3w(s h( *a*z pro\s *g*h, ou(/tws h( *z*e pro\s *e*h, w(s de\ h( *z*e pro\s *e*h, ou(/tws h( *z*b pro\s *h*d, w(s a)/ra h( *a*z pro\s *g*h, ou(/tws h( *z*b pro\s *h*d, kai\ e)nalla\c w(s h( *a*z pro\s *z*b, toute/stin w(s h( *g*q pro\s *q*d, ou(/tws h( *g*h pro\s *h*d, o(/per a)du/naton: h( a)/ra dia\ tw=n *z *e *q shmei/wn eu)qei=a/ e)stin. #2#. *parallhlogra/mmou doqe/ntos o)rqogwni/ou tou= *a*g, diagagei=n th\n *g*d w(/ste tou= *a*b*g*d trapezi/ou a)rthqe/ntos a)po\ tou= *d ta\s *a*d *b*g parallh/lous ei)=nai tw=| o(ri/zonti. *gegone/tw: h( a)/ra dia\ tou= *d kai\ tou= ke/ntrou tou= ba/­rous tou= trapezi/ou a)gome/nh eu)qei=a ka/qetos e)/stai e)pi\ to\n o(ri/zonta kai\ e)pi\ th\n *b*g. e)/stw h( *d*l, kai\ te­tmh/sqw di/xa h( *d*l kata\ to\ *e, kai\ h( *a*b kata\ to\ *z, e)pezeu/xqw de\ h( *g*e*z, kai\ tetmh/sqw h( *g*e kata\ to\ *q w(/ste diplh=n ei)=nai th\n *g*q th=s *q*e, kai\ h( *e*z di/xa tetmh/sqw kata\ to\ *h, kai\ e)pezeu/xqw h( *h*q te/mnousa th\n *d*l kata\ to\ *k: to\ me\n a)/ra *h ke/ntron ba/rous e)sti\n tou= *b*d parallhlo­gra/mmou, to\ de\ *q ke/ntron ba/rous tou= *g*d*l trigw/nou: tou= a)/ra o(/lou trapezi/ou to\ ke/ntron tou= ba/rous e)pi\ th=s *h*q e)sti/n. a)lla\ kai\ e)pi\ th=s *d*l: to\ *k a)/ra ke/ntron ba/­rous e)sti\n tou= *a*b*g*d trapezi/ou. a)lla\ kai\ tou= me\n *b*d parallhlogra/mmou to\ *h, tou= de\ *d*l*g trigw/nou to\ *q: e)/stin a)/ra w(s to\ *b*d parallhlo/grammon pro\s to\ *d*g*l tri/gwnon, ou(/tws h( *q*k pro\s th\n *k*h. e)a\n ga\r a)na\ pei=ran e)pinoh/swmen tou= me\n *b*d parallhlogra/mmou [ou(/tws e)/xon] to\ ba/ros e)n e(autw=| pa=n sunh=xqai pro\s tw=| *h, tou\ de\ *g*d*l trigw/nou pa=n to\ ba/ros e)n tw=| *q sunh=xqai, gi/netai w(/sper zugo\s h( *h*q, e)k de\ tw=n a)/krwn ta\ ei)rhme/na ba/rh. kai\ e)a\n tmhqh=|. h( *h*q kata\ to\ *k, w(/ste ei)=nai w(s to\ pro\s tw=| *h ba/ros pro\s to\ pro\s tw=| *q, toute/stin to\ *b*d parallhlo/­grammon pro\s to\ *g*d*l tri/gwnon, ou(/tws th\n *q*k eu)qei=an pro\s th\n *k*h kata\ to\n a)ntipeponqo/ta tw=n barw=n e)n toi=s zugoi=s lo/gon, e)/stai to\ *k shmei=on e)c ou(= ta\ ba/rh i)sorro­ph/sei [w(/ste kai\ to\ *a*b*g*d e)k tou= *k i)sorroph/sei]. h)/xqw­san dh\ ka/qetoi a)po\ tw=n *h *q e)pi\ th\n *b*g ai( *h*m *q*n. e)pei\ ou)=n e)stin w(s to\ *b*d parallhlo/grammon pro\s to\ *g*d*l tri/gwnon, ou(/tws h( *q*k pro\s th\n *k*h, a)ll' w(s to\ par­allhlo/grammon pro\s to\ tri/gwnon, ou(/tws h( *b*l pro\s th\n h(mi/seian th=s *l*g, w(s de\ h( *k*q pro\s th\n *k*h, ou(/tws h( *n*l pro\s th\n *l*m dia\ to\ ei)s parallh/lous ta\s *h*m *e*l *q*n dih=xqai ta\s *h*k*q *m*l*n, kai\ w(s a)/ra h( *b*l pro\s th\n h(mi/­seian th=s *l*g, ou(/tws h( *n*l pro\s th\n *l*m h(mi/seian ou)=san th=s *b*l: kai\ w(s a)/ra h( *b*l pro\s th\n diplasi/an, toute/stin pro\s th\n *l*g, ou(/tws h( *l*n pro\s th\n diplasi/an th=s *m*l, toute/stin th\n *b*l: to\ a)/ra a)po\ th=s *b*l i)/son e)sti\n tw=| u(po\ *g*l*n. [e)/stin a)/ra w(s me\n h( *g*l pro\s *l*b, h( *b*l pro\s *l*n.] w(s de\ h( *g*l pro\s *l*n, ou(/tws to\ a)po\ th=s *g*l te­tra/gwnon pro\s to\ a)po\ th\s *b*l tetra/gwnon. kai\ triplh= e)stin h( *g*l th=s *l*n (e)pei\ kai\ h( *g*e triplh= e)stin th=s *e*q: diplh= ga\r h( *g*q th=s *e*q): tripla/sion a)/ra to\ a)po\ *g*l tou= a)po\ *l*b. kai\ doqe/nta ta\ *b *g: doqe\n a)/ra to\ *l, w(/ste kai\ to\ *d. dio\ dh\ th\n *b*g temo/ntes kata\ to\ *l, w(/ste to\ a)po\ *g*l tou= a)po\ *l*b ei)=nai tripla/sion, e(/comen to\ *d th=s a)rth/sews shmei=on. te/mnetai de\ h( *b*g ou(/tws. z#. *eu)qei=an temei=n w(/ste th\n mei/zona th=s e)la/ttonos ei)=nai duna/mei triplasi/an. *)/estw eu)qei=a h( *a*d kai\ tetmh/sqw tw=| *g, w(/ste th\n *a*g th=s *g*d ei)=nai triplh=n, kai\ e)pi\ th=s *a*d gegra/fqw h(miku/­klion to\ *a*b*d, kai\ pro\s o)rqa\s th=| *a*d a)po\ tou= *g h( *g*b, kai\ pepoih/sqw w(s h( *a*g pro\s *g*b, ou(/tws h( *a*e pro\s *d*e: o(/ti h( *a*e th=s *d*e duna/mei triplasi/a e)sti/n. *)epei\ ga\r h( *b*g tw=n *a*g *g*d me/sh a)na/logo/n e)stin, w(s a)/ra h( *a*g pro\s th\n *g*d, ou(/tws to\ a)po\ *a*g pro\s to\ a)po\ *b*g, toute/stin to\ a)po\ *a*e pro\s to\ a)po\ *d*e: triplasi/a a)/ra h( *a*e th=s *d*e duna/mei. *(omoi/ws kai\ ei)s to\n doqe/nta lo/gon duna/mei tmhqh/setai h( *a*d eu)qei=a kai\ pa=sa h( doqei=sa eu)qei=a. h#. *qe/sei ai( *a*b *a*g, kai\ doqe\n to\ *b, kai\ dih/xqw h( *g*d a)pote/mnousa doqe/nta lo/gon to\n th=s *a*g pro\s *b*d: dei=cai o(/ti tou= *a*g*d trigw/nou to\ ke/ntron tou= ba/rous e)sti\ pro\s qe/sei. *tetmh/sqw h( *a*g di/xa tw=| *e, kai\ e)pizeuxqei=sa h( *d*e tetmh/sqw kata\ to\ *z, w(/ste th\n *e*z tri/ton me/ros ei)=nai th=s @3*e*d: to\ *z a)/ra ke/ntron ba/rous e)sti\n tou= *a*g*d trigw/nou (tou=to ga\r pro­de/deiktai). h)/xqw dh\ th=| *a*e para/l­lhlos h( *z*h, kai\ th=s *a*b tri/ton me/ros e)/stw h( *a*q. e)/stin de\ kai\ h( *a*h tri/ton me/ros th=s *a*d, e)pei\ kai\ h( *e*z th=s *e*d: kai\ loipo\n ou)=n h( *q*h tri/ton me/ros e)sti\n th=s *b*d. lo/gos de\ th=s *b*d pro\s th\n *a*g do­qei/s [th=s de\ *a*g pro\s th\n *z*h: triplasi/a ga\r au)th=s e)stin, o(/ti kai\ h( me\n *d*a th=s *d*h h(mioli/a e)sti/n, toute/stin h( *a*e th=s *z*h, h( de\ *g*a th=s *a*e diplh=]: lo/gos a)/ra kai\ th=s *h*q pro\s th=n *h*z doqei/s. kai\ do­qei=sa h( pro\s tw=| *h gwni/a (kai\ ga\r h( pro\s tw=| *a): doqei=sa a)/ra kai\ h( u(po\ *h*q*z gwni/a. kai\ doqe\n to\ *q: qe/sei a)/ra h( *q*z eu)qei=a, kai\ e)/stin e)p' au)th=s to\ *z ke/ntron. *tau=ta me\n ou)=n kai\ ta\ toiau=ta qewri/an e)/xei, ta\ de\ kai\ ei)s xrei/an duna/mena pesei=n mhxanikh\n toiau=t' a)\n ei)/h. q#. *)epi/pedon e)kkli=nai, w(/ste to\ kli/ma au)tou= e)f' e(\n neu/ein shmei=on doqe/ntos a)klinou=s e)pipe/dou, toute/stin par­allh/lou tw=| o(ri/zonti, e)n parallhlogra/mmw|, to\ de\ kli/ma e)/stw e)n th=| doqei/sh| gwni/a|. *)/estw to\ doqe\n parallhlo/grammon pro/teron i)so/pleu­ron to\ *a*b*g*d, h( de\ doqei=sa gwni/a, e)n h(=| boulo/meqa e)k-@3kli=nai to\ e)pi/pedon, h( u(po\ *e*z*h, a)po\ de\ tw=n *a *b *d shmei/wn tw=| u(pokeime/nw| e)pipe/dw| pro\s o)rqa\s a)nesta/twsan ai( *a*q *b*k *d*l, to\ de\ *g shmei=on e)/stw o(/pou boulo/meqa th\n kli/sin neu/ein, kai\ th=| me\n *a*g e)pizeuxqei/sh| i)/sh kei/sqw h( *z*h, th=| de\ *z*h pro\s o)rqa\s h)/xqw h( *e*h, th=| de\ *h*e i)/sh kei/sqw h( *a*q. e)a\n dh\ noh/swmen e)pezeugme/nhn th\n *q*g, e)/stai h( u(po\ *q*g*a gwni/a th=s kli/sews tw=n e)pipe/dwn. h)/xqw dh\ kai\ a)po\ tou= *b e)pi\ th\n *a*g ka/qetos h( *b*m, kai\ th=| *g*m i)/sh kei/sqw h( *z*n, th=| de\ *z*h pro\s o)rqa\s h)/xqw h( *n*c, th=| de\ *n*c i)/sh kei/sqw e(kate/ra tw=n *b*k *d*l, kai\ e)pizeuxqei=­sai ai( *q*l *q*k e)kbeblh/sqwsan kai\ sumpipte/twsan tai=s *a*d *a*b e)kblhqei/sais kata\ ta\ *p *r shmei=a [o(/ti de\ sum­pi/ptousin dh=lon: a)p' e)latto/nwn ga/r ei)sin du/o o)rqw=n kai\ au)tai\ ka)kei=nai]: e)/stai dh\ to\ *q*k*l e)pi/pedon keklime/non pro\s to\ *a*b*g*d e)n th=| u(po\ *q*g*a, toute/stin th=| u(po\ *e*z*h. e)a\n ga\r noh/swmen th=| *a*q para/llhlon h)gme/nhn th\n *m*o, kai\ e)pezeugme/nhn th\n *o*k, e)/stai h( me\n *m*o i)/sh th=| *n*c dia\ to\ i)sogw/nion ei)=nai to\ *z*n*c tri/gwnon tw=| *m*o*g, h( de\ *k*o th=| *b*m i)/sh kai\ para/llhlos, kai\ parallhlo/grammon to\ *k*b*m*o o)rqo\n pro\s u(pokei/menon. kai\ e)pei\ ta\ *p *g *r shmei=a e)n$ dusi\n a(/ma e)pipe/dois e)sti\n tw=| te u(pokeime/nw| *a*b*g*d [e)n w(=| e)stin kai\ ta\ *p *r shmei=a, a)lla\] kai\ e)n tw=| *k*q*l*g, ta\ *p *g *r a)/ra shmei=a e)pi\ mia=s e)stin eu)qei/as th=s *p*g*r, koi­nh=s tomh=s ou)/shs tw=n ei)rhme/nwn e)pipe/dwn. dia\ tau)ta\ dh\ kai\ ta\ *k *o *l shmei=a e)pi\ th=s koinh=s e)sti tomh=s tou= *k*q*l*g e)pipe/dou kai\ tou= dia\ tw=n *k *o *l parallh/lou tw=| *a*b*g*d e)pipe/dw|, w(/ste th\n dia\ tw=n *k *o *l eu)qei=an para/llhlon ei)=nai th=| *p*r. e)pei\ ou)=n e)stin w(s me\n h( *a*p pro\s *p*d, h( *q*a pro\s *l*d, w(s de\ h( *a*r pro\s *r*b, h( *a*q pro\s *b*k, kai\ i)/sh e)sti\n h( *d*l th=| *b*k, i)/sh a)/ra kai\ h( *a*p th=| *a*r kai\ gwni/a h( u(po\ *a*p*r th=| u(po\ *a*r*p. e)/stin de\ kai\ h( u(po\ *p*a*g i)/sh th=| u(po\ *r*a*g: loiph\ a)/ra h( u(po\ *a*g*p th=| u(po\ *a*g*r: o)rqh\ a)/ra e)sti\n e(kate/ra au)tw=n, kai\ h( *p*r eu)qei=a di/xa te kai\ pro\s o)rqa\s te/mnetai u(po\ th=s *a*g. kai\ e)/stin au)th=| pro\s o)rqa\s kai\ tw=| *a*b*g*d e)pipe/dw| h( *m*o: kai\ h( *o*g a)/ra pro\s o)rqa/s e)stin th=| *r*p dia\ lh=mma sfairikw=n: o)rqh\ a)/ra e)sti\n e(kate/ra tw=n u(po\ *a*g*p *o*g*p: to\ *k*q*l*g a)/ra e)pi/pedon ke/klitai pro\s to\ [a)po\] *a*b*g*d e)n th=| do­qei/sh| gwni/a| th=| u(po\ *e*z*h.

*)alla\ dh\ e)/stw mei/zwn h( *a*b th=s *a*d, tw=n a)/llwn u(pokeime/nwn tw=n au)tw=n: le/gw o(/ti h( u(po\ *a*g*p o)cei=a/ e)stin.

*)epei\ ga/r e)stin w(s me\n h( *a*p pro\s *p*d, h( *q*a pro\s *d*l, w(s de\ h( *a*r pro\s *r*b, h( *q*a pro\s *b*k, kai\ i)/sh e)sti\n h( *d*l th=| *b*k, kai\ w(s a)/ra h( *a*p pro\s *p*d, h( *a*r pro\s *r*b: kai\ dielo/nti a)/ra e)sti\n w(s h( *a*d pro\s *d*p, ou(/tws h( *a*b pro\s *b*r, kai\ e)nalla\c w(s h( *a*d pro\s *a*b, ou(/tws h( *d*p pro\s *b*r. e)la/ttwn de\ h( *a*d th=s *a*b: e)la/t­twn a)/ra kai\ h( *d*p th=s *b*r: o(/lh a)/ra h( *a*p e)la/ttwn e)sti\n th=s *a*r, w(/ste kai\ gwni/a h( u(po\ *a*r*p e)la/sswn e)sti\n th=s u(po\ *a*p*r: mei/zwn a)/ra h( u(po\ *a*p*r th=s u(po\ *a*r*p. e)/stin de\ kai\ h( u(po\ *g*a*p th=s u(po\ *g*a*r mei/zwn: loiph\ a)/ra h( u(po\ *a*g*p tou= *a*g*p trigw/nou loiph=s th=s u(po\ *a*g*r tou= *a*g*r trigw/nou e)la/sswn e)sti/n: o)cei=a a)/ra h( u(po\ *a*g*p gwni/a: h( kli/sis a)/ra tw=n ei)rhme/nwn e)pipe/dwn pro/s ti sh­mei=on metacu\ tw=n *g *p qewrei=tai, a)po\ tou= *a shmei/ou e)pi\ th\n *g*p kaqe/tou a)gome/nhs. w(s ou)=n e)kkli=nai dunato/n e)stin e)pi/pedon e)n th=| doqei/sh| gwni/a pro\s e)pi/pedon, duna- to/n e)stin a)/ra kai\ e)kkeklime/nou th\n kli/sin ei)pei=n, toute/stin e)n poi/a| gwni/a| ke/klitai to\ e)pi/pedon pro\s to\ para/llhlon tw=| o(ri/zonti.

i#. *ba/rous doqe/ntos u(po\ doqei/shs a)gome/nou duna/mews e)n tw=| para\ to\n o(ri/zonta e)pipe/dw| kai\ e(te/rou e)pipe/dou keklime/nou pro\s to\ u(pokei/menon doqei=san gwni/an u(poti­qe/ntos, eu(rei=n th\n du/namin u(f' o(/shs a)xqh/setai to\ ba/ros e)n tw=| keklime/nw| e)pipe/dw|.

3@*)/estw to\ me\n dia\ th=s *m*n eu)­qei/as e)pi/pedon to\ u(pokei/menon, to\ de\ dia\ th=s *m*k keklime/non pro\s au)to\ gwni/an doqei=san th\n u(po\ *k*m*n u(potiqe/n, ba/ros de/ ti to\ *a kinei/sqw u(po\ duna/mews th=s *g e)pi\ tou= u(pokeime/nou e)pipe/dou, kai\ noei/sqw tw=| *a i)sobarh\s sfai=ra h( peri\ ke/ntron to\ *e, kai\ kei/sqw e)pi\ tou= dia\ tw=n *m *k e)pipe/dou yau/ousa au)tou= kata\ to\ *l sh­mei=on, w(s e)/stin sfairikw=n g# qew­rh/mati: h( a)/ra *e*l e)pizeuxqei=sa ka/qetos e)/stai e)pi\ to\ e)pi/pedon (kai\ tou=to ga\r de/deiktai qewrh/­mati d# sfairikw=n), w(/ste kai\ pro\s th\n *k*m ka/qeto/s e)stin h( *e*l. e)kbeblh/sqw to\ dia\ tw=n *k*m *e*l e)pi/pedon kai\ poiei/tw tomh\n e)n th=| sfai/ra| ku/klon to\n *l*h*c, kai\ h)/xqw dia\ tou= *e ke/ntrou th=| *m*n para/llhlos h( *e*q, kai\ ka/qetos e)p' au)th\n a)po\ tou= *l h( *l*z. e)pei\ ou)=n doqei=sa/ e)stin h( u(po\ *e*q*l gwni/a (i)/sh ga/r e)stin th=| u(po\ *k*m*n doqei/sh| [o)cei/a|] gwni/a|), doqei=sa a)/ra kai\ h( u(po\ *e*l*z i)/sh ou)=sa th=| u(po\ *e*q*l (i)sogw/nion ga/r e)stin to\ *e*q*l tw=| *e*l*z trigw/nw|): doqe\n a)/ra to\ *e*l*z tri/gwnon tw=| ei)/dei: lo/gos a)/ra th=s *e*l, toute/stin th=s *e*h, pro\s *e*z doqei/s: kai\ loiph=s a)/ra th=s *z*h pro\s *e*z lo/gos @3e)sti\n doqei/s. pepoih/sqw ou)=n w(s h( *h*z pro\s *z*e, ou(/tws to\ me\n *a ba/ros pro\s to\ *b, h( de\ *g du/namis pro\s th\n *d. kai\ e)/stin tou= *a du/­namis h( *g: kai\ tou= *b a)/ra du/na­mis e)n tw=| au)tw=| e)pipe/dw| e)/stai h( *d. kai\ e)pei/ e)stin w(s h( *h*z eu)­qei=a pro\s th\n *z*e, ou(/tws to\ *a ba/ros pro\s to\ *b, a)\n teqh=| ta\ *a *b ba/rh peri\ ke/ntra ta\ *e *h, i)sorroph/sei a)rtw/mena a)po\ tou= *z shmei/ou [h)\ e)pi\ u(poqe/matos kei/­mena tou= *l*z o)rqou= pro\s to\n o(ri/­zonta]. kei=tai de\ to\ *a ba/ros peri\ ke/ntron to\ *e (a)nt' au)tou= ga\r h( sfai=ra): teqe\n a)/ra to\ *b ba/ros peri\ ke/ntron to\ *h i)sorroph/sei th=| sfai/ra|, w(/ste mh\ kata­fe/resqai th\n sfai=ran dia\ th\n kli/sin tou= e)pipe/dou, a)ll' e)festa/nai a)rreph=, w(s ei) kai\ e)pi\ tou= u(pokeime/nou e(stw=sa e)tu/gxanen. e)kinei=to de\ e)n tw=| u(pokeime/nw| e)pipe/dw| u(po\ th=s *g duna/mews: kinhqh/setai a)/ra e)n tw=| keklime/nw| e)pipe/dw| pro\s sunamfote/rou th=s te *g duna/mews kai\ th=s tou= *b ba/rous, toute/stin th=s *d duna/mews. kai\ e)/stin doqei=sa h( *d du/namis.

*(h me\n ou)=n gewmetrikh\ tou= problh/matos a)na/lusis u(po­de/deiktai, i(/na de\ kai\ e)pi\ paradei/gmatos poihsw/meqa th/n te kataskeuh\n kai\ th\n a)po/deicin, e)/stw to\ me\n *a ba/ros tala/ntwn, ei) tu/xoi, s# a)go/menon e)n tw=| parallh/lw| o(ri/zonti e)pipe/dw| u(po\ th=s *g kinou/shs duna/mews, toute/stin oi( ki­nou=ntes e)/stwsan a)/nqrwpoi m#, h( de\ u(po\ *k*m*n gwni/a, tout­e/stin h( u(po\ *e*q*l, dimoi/rou o)rqh=s: kai\ loiph\ a)/ra h( u(po\ *z*l*q tri/tou o)rqh=s. kai\ e)/stin o)rqh\ h( u(po\ *e*l*q: dimoi/rou a)/ra kai\ h( u(po\ *e*l*z: oi(/wn a)/ra ai( d# o)rqai\ tc# toiou/twn c# h( u(po\ *e*l*z, kai\ tou= perigrafome/nou a)/ra peri\ to\ *e*z*l tri/gwnon o)rqogw/nion ku/klou h( me\n e)pi\ th=s *e*z perife/reia$toiou/twn e)/stai rk# oi(/wn o( ku/klos tc#, au)th\ de\ h( *e*z toi­ou/twn rd# e)/ggista oi(/wn h( *e*l tou= ku/klou dia/metros rk#: tau=ta ga\r dh=la e)k tou= kano/nos tw=n e)gkukli/wn eu)qeiw=n tou= kata\ *ptolemai=on [o)/ntos] keime/nou e)n tw=| a# tw=n maqhma­tikw=n. lo/gos a)/ra th=s *e*l, toute/stin th=s *e*h, pro\s *e*z, o(\n rk# pro\s rd#: kai\ loiph=s a)/ra th=s *h*z pro\s *z*e lo/gos o(\n i#2# pro\s rd#. tou/tw| de\ o( au)to/s e)stin o( tou= *a ba/rous pro\s to\ *b, kai\ th=s *g duna/mews pro\s th\n *d, kai\ e)/stin to\ me\n *a ba/ros tala/ntwn s#, h( de\ kinou=sa du/namis a)ndrw=n m#: e)/stai a)/ra kai\ to\ me\n *b ba/ros tala/ntwn #22at#, h( de\ *d du/­namis a)nqrw/pwn sc# (w(s ga\r i#2# pro\s rd#, ou(/tws s# pro\s #22at# kai\ m# pro\s sc#): tou= a)/ra *a ba/rous tala/ntwn s# ki­noume/nou e)n parallh/lw| tw=| o(ri/zonti e)pipe/dw| u(po\ tw=n m a)ndrw=n, to\ au)to\ ba/ros kinhqh/setai u(po\ sunamfote/rwn tw=n proeirhme/nwn a)nqrw/pwn, toute/stin u(po\ t# o(/lwn, e)n e)pi­pe/dw| keklime/nw| pro\s to\n o(ri/zonta, th=s u(po\ *k*m*n gwni/as dimoi/rou o)rqh=s u(pokeime/nhs. ia#. *th=s au)th=s de/ e)stin qewri/as to\ doqe\n ba/ros th=| doqei/sh| duna/mei kinh=sai: tou=to ga\r *)arximh/dous me\n eu(/rhma [le/getai] mhxaniko/n, e)f' w(=| le/getai ei)rhke/nai: do/s moi/ (fhsi) pou= stw= kai\ kinw= th\n gh=n. *(/hrwn de\ o( *)alecandreu\s pa/nu safw=s au)tou= th\n kataskeuh\n e)ce/qeto e)n tw=| kaloume/nw| baroulkw=|, lh=mma labw\n o(/per e)n toi=s mhxanikoi=s a)pe/dei­cen, e)/nqa kai\ peri\ tw=n e# duna/mewn dialamba/nei, toute/stin tou= te sfhno\s kai\ moxlou= kai\ koxli/ou kai\ poluspa/stou kai\ a)/conos e)n tw=| peritroxi/w|, di' w(=n to\ doqe\n ba/ros th=| doqei/sh| duna/mei kinei=tai [kaq' e(ka/sthn du/namin]. e)n de\ tw=| baroulkw=| dia\ tumpa/nwn o)dontwtw=n paraqe/sews e)ki/nei to\ doqe\n ba/ros th=| doqei/sh| duna/mei, th=s diame/trou tou= tum­pa/nou pro\s th\n dia/metron tou= a)/conos lo/gon e)xou/shs o(\n e#@3 pro\s a#, tou= kinoume/nou ba/rous u(pokeime/nou tala/ntwn xi­li/wn, th=s de\ kinou/shs duna/mews u(pokeime/nhs tala/ntwn e#. *)/estw dh\ h(ma=s e)pi\ diplasi/ou lo/gou to\ au)to\ deiknu/nai, kai\ tala/ntwn rc# o)/ntos tou= kinoume/nou ba/rous a)nti\ xili/wn, kai\ th=s kinou/shs au)to\ duna/mews u(pokeime/nhs tala/ntwn d# a)nti\ e#, toute/stin o( kinw=n a)/nqrwpos duna/sqw kaq' au(to\n a)/neu mhxanh=s e(/lkein ta/lanta d#, kai\ e)/stw to\ ei)rhme/non u(p' au)tou= glwsso/komon to\ *a*b*g*d, kai\ e)n au)tw=| ei)s tou\s makrou\s kai\ parallh/lous toi/xous e)/stw a)/cwn diakei/menos eu)lu/tws strefo/menos o( *e*z, tou/tw| de\ sumfue\s e)/stw tu/m­panon w)dontwme/non [a)kti=sin o)dontwtoi=s] to\ *h*q, e)/xon th\n dia/metron diplasi/an th=s diame/trou [th=s *e*z diagwni/ou] tou= a)/conos th=s kata\ ko/trafon [gi/netai ga\r tetra/gwnos me\n peri\ me/son e)pi\ tosou=ton mh=kos, o(/son e)sti\n to\ pa/xos tou= tumpa/nou ei)s o(\ e)narmo/zetai a)sfalw=s, stroggu/los de/ pws h)\ leloifwme/nos e)k tw=n e)f' e(ka/tera tou= tumpa/nou merw=n]. e)a\n a)/ra ta\ e)k tou= ba/rous tou= e(lkome/nou dedeme/na sxoini/a [kalou/mena de\ o(/pla] dia/ tinos o)ph=s [ma=llon de\ a)natomh=s@3 platei/as] ou)/shs e)n tw=| *a*b toi/xw| e)peilhqh=| peri\ to\n *e*z a)/cona [e)f' e(ka/tera tou= *h*q tumpa/nou] kai\ strafh=| to\ *h*q tu/mpanon, tou=to e)pistre/yei kai\ to\n sumfuh= a)/cona kinou/- menon peri\ ta\ a)/kra e)n daktu/lois xalkoi=s kai\ puci/sin o(moi/ws xalkai=s [kinoume/nais], keime/nais d' e)n toi=s ei)rhme/­nois *a*b *g*d toi/xois. e)peilou/mena de\ ta\ e)k tou= ba/rous [o(\ kalei=tai forti/on] o(/pla kinh/sei to\ ba/ros. i(/na de\ kinhqh=| to\ *h*q tu/mpanon, deh/sei du/namin parasxei=n tala/ntwn plei=on p# dia\ to\ th\n dia/metron tou= tumpa/nou th=s dia­me/trou tou= a)/conos ei)=nai diplasi/an: tou=to ga\r pro/blhma/ e)stin u(po\ *(/hrwnos deiknu/menon e)n toi=s mhxanikoi=s. [kai\ a)/lla plei=sta problh/mata tw=n xrhsimwta/twn kai\ biwfe­lw=n ge/graptai]. *)epei\ ou)=n ou)k e)/xomen th\n doqei=san du/namin tala/ntwn p#, a)lla\ tala/ntwn d#, gegone/tw e(/teros a)/cwn parakei/menos para/llhlos tw=| *e*z o( *k*l, e)/xwn sumfue\s tu/mpanon w)don­twme/non to\ *m*n, w(/ste tou\s o)do/ntas au)tou= e)narmo/zein toi=s o)dou=si tou= *h*q tumpa/nou: tou=to de\ gi/netai, e)a\n h)=| w(s h( dia/metros tou= *h*q tumpa/nou pro\s th\n dia/metron tou= *m*n, ou(/tws to\ plh=qos tw=n o)do/ntwn tou= *h*q pro\s to\ plh=qos tw=n o)do/ntwn tou= *m*n (pw=s de\ tou=to gi/netai dia\ tw=n e(ch=s dh=­lon e)/stai): doqe\n me\n a)/ra e)sti\n kai\ to\ *m*n tu/mpanon. tw=| d' au)tw=| a)/coni tw=| *k*l sumfue\s e)/stw tu/mpanon to\ *c*o, e)/xon th\n dia/metron diplasi/an th=s tou= *m*n tumpa/nou dia­me/trou. dia\ dh\ tou=to deh/sei to\n boulo/menon kinei=n dia\ tou= *c*o tumpa/nou to\ ba/ros e)/xein du/namin tala/ntwn m#, e)peidh/per ta\ p# ta/lanta dipla/sia/ e)stin tw=n m# tala/n­twn. *pa/lin de\ parakei/sqw tw=| *c*o tumpa/nw| [o)dontwqe/nti] e(/teron tu/mpanon w)dontwme/non to\ *p*r sumfue\s e(te/rw| a)/coni, tw=| d' au)tw=| a)/coni e(/teron sumfue\s tu/mpanon to\ *s*t, e)/xon me\n o(moi/ws diplasi/an th\n dia/metron th=s tou= *p*r tumpa/nou diame/trou, tou\s de\ o)do/ntas mh\ sumplekome/nous toi=s o)dou=si tou= *m*n tumpa/nou: h( a)/ra dia\ tou= *s*t tumpa/nou kinou=sa to\ ba/ros du/namis e)/stai tala/ntwn k#. h)=n de\ h( doqei=sa du/namis tala/ntwn d#: deh/sei ou)=n pa/lin e(/teron me\n tu/m­panon w)dontwme/non to\ *u*f parakei=sqai tw=| *s*t [o)dontw­qe/nti], tw=| de\ a)/coni tou= *u*f tumpa/nou sumfue\s gene/sqai to\ *x*y w)dontwme/non, ou(= h( dia/metros pro\s th\n tou= *u*f tumpa/nou dia/metron lo/gon e)xe/tw o(\n ta\ b# pro\s a#: h( a)/ra kinou=sa to\ ba/ros du/namis dia\ tou= *x*y tumpa/nou e)/stai tala/ntwn i#. pa/lin dh\ parakei/sqw me\n tw=| *x*y tumpa/nw| e(/teron tu/mpanon w)dontwme/non to\ #3#5, tw=| de\ a)/coni au)tou= tu/mpanon e)/stw sumfue\s *m$14a$*m$14b$w)dontwme/non o)dou=sin lo­coi=s, ou(= h( dia/metros pro\s th\n tou= #3#5 dia/metron lo/gon e)xe/tw o(\n e)/xei ta\ i# ta/lanta pro\s ta\ th=s doqei/shs duna/­mews ta/lanta d#. *kai\ tou/twn kataskeuasqe/ntwn e)a\n e)pinoh/swmen to\ *a*b*g*d glwsso/komon mete/wron kei/menon a)metasta/tws, kai\ e)k me\n tou= *e*z a)/conos ba/ros e)ca/ywmen, e)k de\ tou= *m$14a$*m$14b$tumpa/nou th\n e(/lkousan du/namin ta\ d# ta/lanta, ou)dopo/te­ron au)tw=n katenexqh/setai, eu)lu/tws strefome/nwn tw=n a)co/­nwn kai\ th=s tw=n tumpa/nwn paraqe/sews a)kribw=s a(rmozou/­shs, a)ll' w(/sper e)pi\ zugou= tinos i)sorroph/sei h( du/namis tw=n d# tala/ntwn tw=| ba/rei tw=n rc# tala/ntwn: e)a\n a)/ra e(ni\ au)tw=n prosqw=men o)li/gon ti ba/ros, katarre/yei kai\ e)nex­qh/setai e)f' o(po/teron me/ros h( pro/sqesis gege/nhtai: ei) ga\r lo/gou xa/rin th=| tw=n d# tala/ntwn duna/mei mnaiai=on pros­teqh=| ba/ros, katakrath=san e)pispa/setai to\ ba/ros tw=n rc# tala/ntwn. a)nti\ de\ th=s prosqe/sews parakei/sqw koxli/as tw=| *m$14a$*m$14b$tumpa/nw| o( *w*a#22 e)/xwn th\n e(/lika a(rmo/zousan toi=s locoi=s o)dou=si tou= tumpa/nou tou= *m$14a$*m$14b$. tou=to de\ w(s dei= poiei=n, e)n toi=s au)toi=s mhxanikoi=s *(/hrwnos ge/graptai, kai\ h(mei=s de\ tou=to safe/steron e(ch=s gra/yomen. strefe/sqw de\ o( koxli/as eu)lu/tws peri\ to/rmous e)no/ntas e)n trh/masi strog­gu/lois, w(=n o( e(/teros u(perexe/tw ei)s to\ e)kto\s me/ros tou= glwssoko/mou kata\ to\n *g*d toi=xon, kai\ h( u(peroxh\ tetra­gwnisqei=sa labe/tw xeirola/bhn th\n #2#22*b, di' h(=s e)pilabo/menoi kai\ e)pistre/fontes to\n koxli/an e)pistre/yomen kai\ to\ *m$14a$*m$14b$tu/mpanon, w(/ste kai\ to\ #3#5 sumfue\s au)tw=|. dia\ de\ tou=to kai\ to\ parakei/menon au)tw=| to\ *x*y strafh/setai, kai\ to\ sumfue\s au)tw=| to\ *u*f, kai\ to\ parakei/menon au)tw=| to\ *s*t, kai\ to\ tou/tw| sumfue\s to\ *p*r, kai\ to\ tou/tw| parakei/menon to\ *c*o, kai\ to\ tou/tw| sumfue\s to\ *m*n, kai\ to\ tou/tw| pa­rakei/menon to\ *h*q, w(/ste kai\ o( tou/tw| sumfuh\s a)/cwn o( *e*z, peri\ o(\n e)peilou=ntes ta\ e)k tou= forti/ou o(/pla kinh/somen to\ ba/ros. o(/ti ga\r kinh/setai dh=lon e)k tou= prosteqei=sqai e(te/ran du/namin th\n th=s xeirola/bhs, h(/tis perigra/fei ku/klon th=s tou= koxli/ou perime/trou mei/zona: a)pedei/xqh ga\r e)n tw=| peri\ zugw=n *)arximh/dous kai\ toi=s *fi/lwnos kai\ *(/hrwnos mhxanikoi=s, o(/ti oi( mei/zones ku/kloi katakratou=sin tw=n e)lasso/nwn ku/klwn, o(/tan peri\ to\ au)to\ ke/ntron h( ku/lisis au)tw=n gi/nhtai. ib#. *ta\ me\n ou)=n ma/lista sune/xonta th\n mhxanikh\n qewri/an tau=t' a)\n ei)/h. th=s de\ o)rganikh=s polla\ me\n ei)/dh kai\ me/rh: ta\ me\n ga\r u(po\ th=s mhxanikh=s kai\ gnwmonikh=s kai\ th=s peri\ u(drei/wn pragmatei/as lo/gw| qewrou/mena di' au)tw=n tw=n o)rga/nwn u(po\ tau/ths kataskeuazo/mena dei/knutai, polla\ de\ kai\ xwri\s tw=n mhxanikw=n e)/cwqen u(p' au)th=s e)pi­telei=tai, kai/ tina tai=s gewmetrikai=s e)fo/dois dusxei/rista metalabou=sa toi=s o)rga/nois ei)s r(a|dieste/ran h)/gage kata­skeuh/n. au)ti/ka gou=n to\ kalou/menon *dhliako\n pro/blhma th=| fu/sei stereo\n u(pa/rxon ou)x oi(=o/n t' h)=n kataskeua/sai tw=| gewmetrikw=| lo/gw| katakolouqou=ntas, e)pei\ mhde\ ta\s tou= kw/nou toma\s r(a/|dion e)n e)pipe/dw| gra/fein h)=n, toi=s d' o)r­ga/nois metalhfqe\n ei)s xeirourgi/an kai\ kataskeuh\n e)pith/­deion [ma=llon th=s u(po\ tw=n a)/llwn e)kteqeime/nhs ou(/tws] a)\n$ a)naxqei/h [to\ prokei/menon], le/gw de\ to\ ku/bon ku/bou dipla/­sion eu(rei=n. ou) mo/non de\ dipla/sios eu(ri/sketai dia\ tou= u(pokeime/nou o)rga/nou, a)lla\ kai\ kaqo/lou lo/gon e)/xwn to\n e)pitaxqe/nta.

*kateskeua/sqw ga\r h(miku/klion to\ *a*b*g, kai\ a)po\ tou= *d ke/ntrou pro\s o)rqa\s a)nh/xqw h( *d*b, kai\ kinei/sqw kano/­nio/n ti peri\ to\ *a shmei=on ou(/tws w(/ste to\ me\n e(\n pe/ras au)tou= perikei=sqai tuli/w| tini\ kata\ to\ *a shmei=on e(stw=ti, to\ de\ loipo\n me/ros w(s peri\ ke/ntron to\ tula/rion kinei=sqai metacu\ tw=n *b *g. tou/twn de\ kateskeuasme/nwn e)piteta/xqw du/o ku/bous eu(rei=n lo/gon e)/xontas pro\s a)llh/lous doqe/nta, kai\ tw=| lo/gw| o( au)to\s pepoih/sqw o( th=s *b*d pro\s *d*e, kai\ e)pizeuxqei=sa h( *g*e e)kbeblh/sqw e)pi\ to\ *z. parage/sqw dh\ to\ kano/nion metacu\ tw=n *b *g, e(/ws ou(= to\ a)polambano/menon au)tou= me/ros metacu\ tw=n *z*e *e*b eu)qeiw=n i)/son ge/nhtai tw=| metacu\ th=s *b*e eu)qei/as kai\ th=s *b*k*g periferei/as: tou=to ga\r peira/zontes ai)ei\ kai\ meta/gontes to\ kano/nion r(a|di/ws poih/somen. gegone/tw dh/, kai\ e)xe/tw qe/sin th\n *a*h*q*k, w(/ste i)/sas ei)=nai ta\s *h*q *q*k: le/gw o(/ti o( a)po\ th=s *b*d ku/bos pro\s to\n a)po\ th=s *d*q ku/bon lo/gon e)/xei to\n e)pitax­qe/nta, toute/stin to\n th=s *b*d pro\s *d*e.

*noei/sqw ga\r o( ku/klos prosanapeplhrwme/nos, kai\ e)pizeuxqei=sa h( *k*d e)kbeblh/sqw e)pi\ to\ *l, kai\ e)pezeu/xqw h( *l*h: para/llhlos a)/ra e)sti\n th=| *b*d dia\ to\ i)/shn ei)=nai th\n me\n *k*q th=| *q*h, th\n de\ *k*d th=| *d*l. e)pezeu/xqw dh\ kai\ h(/ te *a*l kai\ h( *l*g. e)pei\ ou)=n o)rqh/ e)stin h( u(po\ *h*a*l e)n h(mikukli/w| kai\ ka/qetos h( *a*m, e)/stin a)/ra w(s to\ a)po\ *l*m pro\s to\ a)po\ *m*a, toute/stin w(s h( *g*m pro\s *m*a, ou(/tws to\ a)po\ *a*m pro\s to\ a)po\ *m*h. koino\s proskei/sqw lo/gos o( th=s *a*m pro\s *m*h: o( a)/ra sugkei/menos e)/k te tou= th=s *g*m pro\s *m*a kai\ tou= th=s *a*m pro\s *m*h, toute/stin o( th=s *g*m pro\s *m*h, lo/gos o( au)to/s e)stin tw=| sugkeime/nw| e)/k te tou= tou= a)po\ th=s *a*m pro\s to\ a)po\ th=s *m*h kai\ e)k tou= th=s *a*m pro\s *m*h. o( de\ sugkei/menos e)/k te tou= tou= a)po\ th=s *a*m pro\s to\ a)po\ th=s *m*h kai\ tou= th=s *a*m pro\s *m*h o( au)to/s e)stin tw=| lo/gw| o(\n e)/xei o( a)po\ th=s *a*m ku/bos pro\s to\n a)po\ th=s *m*h ku/bon: kai\ o( th=s *g*m a)/ra pro\s th\n *m*h lo/gos o( au)to/s e)stin tw=| lo/gw| tou= a)po\ th=s *a*m ku/bou pro\s to\n a)po\ th=s *m*h ku/bon. a)ll' w(s me\n h( *g*m pro\s *m*h, ou(/tws h( *g*d pro\s *d*e, toute/stin h( *b*d pro\s *d*e, w(s de\ h( *a*m pro\s *m*h, ou(/tws h( *a*d pro\s *d*q, toute/stin h( *d*b pro\s *d*q: kai\ w(s a)/ra h( *b*d pro\s *d*e, toute/stin w(s o( doqei\s lo/gos, ou(/tws o( a)po\ th=s *b*d ku/bos pro\s to\n a)po\ th=s *d*q ku/bon. *pro/blhma o)rganiko\n e)pi\ kuli/ndrou.

ig#. *ta\ d' o)rganika\ e)n toi=s mhxanikoi=s lego/mena pro- blh/mata/ [e)stin o(/ti] gi/netai th=s gewmetrikh=s e)cousi/as a)fai­rou/mena, oi(=a/ e)stin kai\ ta\ e(ni\ diasth/mati grafo/mena kai\ to\ e)pi\ tou= ta\s ba/seis a)mfote/ras lelwbhme/nou kuli/ndrou proteino/menon u(po\ tw=n a)rxitekto/nwn. a)ciou=si ga\r me/rous e)pifanei/as o)rqou= kuli/ndrou doqe/ntos, ou(= mhde\n me/ros u(gie\s fula/ssetai tw=n e)n tai=s ba/sesi perifereiw=n, eu(rei=n to\ pa/xos tou= kuli/ndrou, toute/stin tou= ku/klou th\n dia/me­tron a)f' ou(= th\n ge/nesin e)/sxen o( ku/lindros. eu(ri/sketai de\ meqodeuqe\n ou(/tws.

*ei)lh/fqw e)pi\ th=s doqei/shs e)pifanei/as du/o shmei=a ta\ *a *b, kai\ ke/ntrois au)toi=s e(ni\ diasth/mati seshmeiw/sqw e)pi\@3 th=s e)pifanei/as prw=ton to\ *g, kai\ pa/lin ke/ntrois au)toi=s toi=s *a *b diasth/mati tou= prote/rou mei/zoni seshmeiw/sqw to\ *d, kai\ a)/llw| diasth/mati to\ *e, kai\ a)/llw| to\ *z, kai\ a)/llw| to\ *h. e)/stai dh\ ta\ e# shmei=a ta\ *g *d *e *z *h e)n e(ni\ e)pi­pe/dw| dia\ to\ kai\ th\n e)pizeugnu/ousan e(/kaston au)tw=n w(s ko­rufh\n i)soskelou=s trigw/nou th=| dixotomi/a| th=s e)pizeugnou/shs eu)qei/as ta\ *a *b w(s ba/sews koinh=s tw=n trigw/nwn o)rqh\n ei)=nai pro\s th\n *a*b [kai\ e)n e(ni\ gi/nesqai e)pipe/dw| ta\s e# eu)qei/as, kai\ dh=lon o(/ti ta\ *g *d *e *z *h shmei=a]. tau=ta de\ ei)s e)pi/pedon e)kqhso/meqa ou(/tws: e)k triw=n me\n eu)qeiw=n tw=n e)pizeugnuousw=n ta\ *g *d *e tri/gwnon e)n tw=| e)pipe/dw| sunesta/tw to\ *q*k*l, e)k triw=n de\ tw=n e)pizeugnuousw=n ta\ *d *e *z to\ *k*l*m, e)k triw=n de\ tw=n e)pizeugnuousw=n ta\ *e *z *h shmei=a tri/gwnon sunesta/tw to\ *l*m*n: e)/stai a)/ra e)kkei/mena ta\ *q*k*l *k*l*m *l*m*n tri/gwna a)nti\ tw=n *g*d*e *d*e*z *e*z*h trigw/nwn. a)\n dh\ peri\ ta\ *q *k *l *m *n sh­mei=a gra/ywmen e)/lleiyin, o( e)la/sswn au)th=s a)/cwn dia/metros e)/stai tou= ku/klou tou= to\n ku/lindron a)pergasame/nou.

id#. *zhtoume/nou dh\ peri\ pe/nte ta\ doqe/nta shmei=a e)n e(ni\ e)pipe/dw| kei/mena ta\ *q *k *l *m *n e)/lleiyin gra/yai, perigegra/fqw, kai\ e)pizeuxqei=sai ai( *q*n *m*k pro/teron e)/stwsan para/llhloi, kai\ di/xa tetmh/sqw e(kate/ra au)tw=n toi=s *a *b, kai\ e)pizeuxqei=sa h( *a*b e)kbeblh/sqw e)pi\ ta\ *e *z th=s e)llei/yews shmei=a: h( *e*z a)/ra dia/metro/s e)stin th=s e)llei/yews dia\ to\n i# o(/ron tw=n kwnikw=n, qe/sei dedome/nh: doqe\n ga\r kai\ e(ka/teron tw=n *a *b shmei/wn th=| qe/sei. h)/xqw dh\ dia\ tou= *l th=| *e*z para/llhlos h( *l*c, kai\ e)pizeuxqei=sai ai( *c*k *l*m sumpipte/twsan th=| *q*n e)kblhqei/sh| kata\ ta\ *p *h: doqe/nta a)/ra ta\ *g *h (doqe\n ga\r e(/kaston tw=n *l *m *q *n). kai\ e)pei\ w(s to\ u(po\ *c*d*l pro\s to\ u(po\ tw=n *m*d*k, ou(/tws to\ u(po\ *c*g*l pro\s e(ka/teron tw=n u(po\ *h*g*p *n*g*q, e)/stai a)/ra i)/son to\ u(po\ *h*g*p tw=| u(po\ *n*g*q. kai\ e)/stin doqe\n to\ u(po\ *n*g*q (doqei=sa ga\r e(kate/ra): doqe\n a)/ra to\ *p. a)lla\ kai\ to\ *k: qe/sei a)/ra h( *k*p*c. a)lla\ kai\ h( *l*g*c: doqe\n a)/ra to\ *c. kai\ e)/stin e)pi\ th=s e)llei/yews. e)pizeux­qei=sai dh\ ai( *n*c *l*q sumpipte/twsan th=| *e*z diame/trw| e)kblhqei/sh| kata\ ta\ *r *s: e)/stai dh\ pa/lin w(s to\ u(po\ *n*g*q pro\s to\ u(po\ *c*g*l, ou(/tws to\ u(po\ *n*a*q pro\s e(ka/teron tw=n u(po\ *r*a*s *e*a*z, kai\ dia\ tou=to i)/son to\ u(po\ *r*a*s tw=|@3 u(po\ *e*a*z. kai\ e)/stin doqe\n to\ u(po\ *r*a*s (doqei=sai ga/r ei)sin ai( *r*a *a*s): doqe\n a)/ra kai\ to\ u(po\ *e*a *a*z. tw=| d' o(moi/w| tro/pw| deixqh/setai kai\ to\ u(po\ *e*b*z doqe/n. kai\ doqe/nta ta\ *a *b: doqe/nta a)/ra kai\ ta\ *e *z, w(s e(ch=s deix­qh/setai: w(/ste h( *e*z dia/metros de/dotai tw=| mege/qei. dh=lon d' o(/ti kai\ h( suzugh\s au)th=|: de/dotai ga\r o( th=s *e*z plagi/as pro\s th\n o)rqi/an au)th=s lo/gos o( au)to\s w)\n tw=| tou= u(po\ *e*a*z pro\s to\ a)po\ *a*n.

ie#. *to\ u(perteqe/n. e)/stw doqe\n e(ka/teron tw=n u(po\ *a*g*b *a*d*b, kai\ doqe/nta ta\ *g *d: o(/ti ta\ *a *b doqe/nta e)sti/n.

*)/estw ga\r tw=| me\n u(po\ *a*g*b i)/son to\ u(po\ *d*g*e, tw=| de\ u(po\ *a*d*b i)/son to\ u(po\ *g*d*z: e)/stai a)/ra w(s h( *g*e pro\s th\n *e*a, ou(/tws h( *a*z pro\s *z*d (dia\ ga\r th\n kataskeuh\n e(ka/teros lo/gos o( au)to/s e)stin tw=| th=s *g*b pro\s *b*d): i)/son a)/ra to\ u(po\ *e*g *z*d tw=| u(po\ *e*a*z, w(/ste kai\ to\ *a sh­mei=on doqe/n. o(moi/ws kai\ to\ *b.

i#2#. *mh\ e)/stwsan dh\ ai( ta\ *n *q *m *k dedome/na e)pi\ th=s e)llei/yews shmei=a e)pizeugnu/ousai para/llhloi, kai\ e)pi­zeuxqei=sai ai( *n*k *m*q temne/twsan a)llh/las kata\ to\ *t, kai\ dia\ tou= *l para/llhlos h)/xqw th=| *m*q h( *l*u*f: e)/stai @3dh\ lo/gos tou= u(po\ *n*u*k pro\s to\ u(po\ *l*u*f doqei\s$(o( au)to\s ga\r tw=| tou= u(po\ *n*t*k pro\s to\ u(po\ *m*t*q). kai\ doqe\n to\ u(po\ *n*u*k: doqe\n a)/ra kai\ to\ u(po\ *l*u*f: kai\ doqe/nta ta\ *l*u: doqe\n a)/ra to\ *f: a)ph=k­tai ou)=n ei)s to\ progegram­me/non, peri\ pe/nte shmei=a ta\ *n *m *l *f *q gra/yai e)/lleiyin th\n *n*m*l*f*q parallh/­lwn u(pokeime/nwn tw=n *m*q *f*l. iz#. *(ra/|dion de\ suzugw=n diame/trwn e)llei/yews porisqei­sw=n w(ntinwnou=n tou\s a)/conas au)th=s o)rganikw=s eu(rei=n. me­qodeu/etai de\ to\n tro/pon tou=ton. *)ekkei/sqwsan ai( proeureqei=sai th=s e)llei/yews dia/me­troi suzugei=s ai( *a*b *g*d di/xa te/mnousai a)llh/las kata\ to\ *e, kai\ dia\ me\n tou= *a th=| *g*d para/llhlos h)/xqw h( *z*h, tw=| de\ a)po\ *d*e i)/son kei/sqw to\ u(po\ *e*a*q, kai\ h( *e*q di/xa@3 tetmh/sqw kata\ to\ *k: e)/stai dh\ to\ *k metacu\ tw=n *a *q (mei/­zwn ga/r e)stin h( *d*e th=s *e*a), kai\ th=| *e*q pro\s o)rqa\s a)po\ tou= *k h)/xqw h( *k*l te/mnousa th\n *z*h kata\ to\ *l, kai\ peri\ ke/ntron to\ *l dia\ tou= *e grafome/nh ku/klou perife/reia tem­ne/tw th\n *h*z kata\ ta\ *z *h, kai\ e)pezeu/xqwsan ai( *e*h *e*z, kai\ ka/qetoi h)/xqwsan e)p' au)ta\s ai( *a*m *a*n, kai\ tw=| me\n u(po\ *h*e*m i)/son kei/sqw e(ka/teron tw=n a)po\ *e*o *e*p, tw=| de\ u(po\ *z*e*n e(ka/teron tw=n a)po\ *e*r *e*s: e)/sontai ou)=n eu(rhme/noi th=s e)llei/yews a)/cones oi( *o*p *r*s, w(=n o( e)la/xi- stos i)/sos e)/stai tw=| tou= kuli/ndrou pa/xei, kaqw\s e)n a)rxh=| proei/rhtai. ih#. *sfai/ras metew/rou doqei=san qe/sin e)xou/shs pro\s to\ u(pokei/menon, eu(rei=n to/ te shmei=on e)f' o(\ pi/ptei kaqeti­kw=s e)nexqei=sa [kai\ kaq' o(\ pi/ptei shmei=on] kai\ th\n e)la­xi/sthn a)potemnome/nhn a)po\ th=s kaqe/tou metacu\ tw=n du/o shmei/wn tou= te kata\ th\n e)pifa/neian th=s sfai/ras kai\ tou= kata\ to\ e)pi/pedon. progra/fetai de\ to\ ku/klou doqe/ntos me­tew/rou mh\ e)n o)rqw=| e)pipe/dw| pro\s to\ u(pokei/menon eu(rei=n th/n te koinh\n tomh\n tw=n e)pipe/dwn a)mfote/rwn kai\ th\n kli/sin. *)/estw mete/wros ku/klos, kai\ ei)lh/fqw e)p' au)tou= tri/a shmei=a ta\ *a *b *g, kai\ h)/xqwsan a)p' au)tw=n e)pi\ to\ u(po-@3kei/menon e)pi/pedon ka/qetoi. a)xqh/sontai de\ ou(/tws: a)po\ tou= *g prospesou=sa eu)qei=a pro\s to\ u(pokei/menon e)pi/pedon w(s h( *g*d perienhne/xqw kai\ yaue/tw tou= e)pipe/dou kaq' e(/tera du/o shmei=a ta\ *e *z, kai\ ei)lh/fqw tou= peri\ ta\ *d *e *z ku/klou ke/ntron to\ *k: h( ou)=n a)po\ tou= *g ka/qetos e)pi\ to\ *k shmei=on pesei=tai, kai\ doqe\n e)/stai to\ *k. h)/xqwsan kai\ a)po\ tw=n *a *b ka/qetoi o(moi/ws ai( *b*q *a*l: e)pizeuxqei=sai dh\ ai( *k*l *q*l e)kbeblh/sqwsan, kai\ pepoih/sqw w(s me\n h( *g*k pro\s *a*l, ou(/tws h( *k*m pro\s *m*l, w(s de\ h( *b*q pro\s *a*l, ou(/tws h( *q*o pro\s *o*l [doqe/nta a)/ra ta\ *m *o e)f' h(mi=n ga/r e)sti toiau/tas kaqe/tous labei=n w(/ste e)laxi/sthn e)n au)tai=s ei)=nai mi/an, w(s th\n *a*l]: eu)qei=ai a)/ra ai( *m*a*g *b*a*o. kai\ e)/sontai e)n tw=| e)pipe/dw| tou= *a*b*g ku/­klou: h( a)/ra koinh\ tomh\ au)tou= kai\ tou= u(pokeime/nou e)pi­pe/dou e)sti\n h( *m*o. h)/xqw a)po\ tou= *l e)pi\ th\n *m*o ka/qetos h( *l*n, kai\ e)pezeu/xqw h( *a*n: kai\ h( *a*n a)/ra ka/qetos e)/stai e)pi\ th\n *m*o: pepo/ristai a)/ra kai\ h( u(po\ *a*n*l gwni/a, tw=n e)pipe/dwn h( kli/sis. iq#. *tou/tou prodeixqe/ntos e)/stw sfai=ra mete/wros, kai\ prokei/sqw to/ te shmei=on eu(rei=n, e)f' o(\ pesei=tai kaqeti­kw=s e)pi\ to\ u(pokei/menon e)pi/pedon e)nexqei=sa, kai\ th\n e)laxi/­sthn a)potemnome/nhn a)po\ th=s kaqe/tou metacu\ th=s e)pifa­nei/as kai\ tou= e)pipe/dou. *)/estw h( sfai=ra mete/wros keime/nh peri\ ke/ntron to\ *e, kai\ e)n au)th=| me/gisto/s tis e)ggegra/fqw ku/klos o( *a*b*g: h)/toi dh\ e)n o)rqw=| e)/stai e)pipe/dw| pro\s to\ u(pokei/menon h)\ ou)/, @3gnwso/meqa de\ ou(/tws: labo/ntes e)pi\ th=s periferei/as au)tou= tri/a tuxo/nta shmei=a kaqe/tous a)/comen e)pi\ to\ u(pokei/menon e)pi/pedon, w(s memaqh/kamen, ka)\n me\n ta\ shmei=a e)f' a(\ pi/ptousin ai( ka/qetoi e)p' eu)qei/as a)llh/lois w)=sin, o)rqa\ pro\s a)/l­lhla e)/stai ta\ e)pi/peda, e)a\n de\ mh/, keklime/na. *)/estw dh\ pro/teron o)rqa/, kai\ h)/xqw­san a)po\ tw=n *a *g shmei/wn ka/qetoi ai( *a*d *g*h: h)/toi dh\ i)/sai, e)/sontai h)\ ou)/. *)/estwsan i)/sai, kai\ tetmh/sqw h( *d*h e)pizeuxqei=sa di/xa tw=| *z: e)/stai dh\ to\ *z to\ zhtou/menon shmei=on e)n tw=| e)pipe/dw|, h( de\ dixotomi/a th=s *a*b*g peri­ferei/as to\ *b e)pi\ th=s e)pifanei/as e)farmo/zon tw=| *z, kai\ h( *b*z e)laxi/sth ka/qetos, w(s proei/rhtai. k#. *mh\ e)/stwsan de\ i)/sai ai( ka/qetoi, a)lla\ e)laxi/sth h( *a*d, kai\ pepoih/sqw w(s h( *g*h pro\s *a*d, ou(/tws h( *h*q pro\s *q*d, e)kblhqei/shs th=s *h*d: e)/stai dh\ to\ *q, kaq' o(\ h( a)po\ tou= *g e)pi\ to\ *a sumpi/ptei tw=| u(pokeime/nw| e)pipe/dw|, kai\ doqei=sa e)/stai h(/ te *a*q eu)qei=a kai\ h( u(po\ *a*q*d gw­ni/a. tou/twn genome/nwn e)kkei/sqw ku/klos i)/sos tw=| megi/stw| peri\ dia/metron th\n *k*l, kai\ proskei/sqw h( *l*m i)/sh th=| *a*q, kai\ th=| u(po\ *a*q*d gwni/a| i)/sh sunesta/tw h( u(po\ *k*m*n, @3kai\ a)po\ tw=n *k *l ka/­qetoi ai( *l*o *k*n, kai\ a)po\ tou= ke/ntrou h( *s*p, kai\ th=| me\n *l*r peri­ferei/a| i)/sh a)peilh/fqw h( *a*b, th=| de\ *o*p eu)­qei/a| i)/sh h( *d*z [to\ de\ au)to\ h)=n le/gein di/xa h( *d*h tw=| *z]. e)/stai ou)=n to\ me\n *z shmei=on, e)f' o(\ h( sfai=ra katafero­me/nh pesei=tai, to\ de\ *b to\ e)pi\ th=s e)pifanei/as, h( de\ e)la­xi/sth ka/qetos h( *b*z. ka#. *mh\ e)/stw de\ o( *a*b*g ku/klos e)n [e(ni\] e)pipe/dw| o)rqw=| pro\s to\ u(pokei/menon, kai\ ei)lh/fqw h( koinh\ tw=n e)pipe/dwn tomh\ h( *d*q, kai\ ei)lh/fqw e)pi\ tou= *a*b*g ku/klou shmei=a ta\ *a *g kata\ dia/metron a)llh/lois kei/mena ou(/tws w(/ste th\n e)p' au)ta\ e)pizeugnume/nhn th\n *g*a sumpi/ptein th=| koinh=| tomh=| th=| *d*q [e)/stin ga\r e)p' e)moi\ dia\ to\ th\n *d*q e)n tw=| tou= *a*b*g ku/klou e)pipe/dw| ei)=nai]. sumpipte/tw kata\ to\ *q: doqei=sa a)/ra h( *a*q kai\ h( *q gwni/a. h)/xqw a)po\ tou= *e ke/ntrou ka/­qetos e)pi\ th\n *d*q h( *e*b*d. a)xqh/setai ou(/tws: e)kkei/sqw ku/klos o( *h*z*l i)/sos tw=| megi/stw| tw=| *a*b*g peri\ dia/metron th\n *z*h, kai\ proskei/sqw h( *h*k i)/sh th=| *g*q, kai\ th=| u(po\@3 *a*q*d gwni/a| i)/sh sunesta/tw h( u(po\ *z*k*m, kai\ a)po\ tou= *o ke/ntrou ka/qetos h( *o*l*m, kai\ th=| me\n *h*l periferei/a| i)/sh a)peilh/fqw h( *g*b, th=| de\ *k*m eu)qei/a| h( *q*d: h( *d*b a)/ra i)/sh e)sti\n th=| *m*l kai\ ka/qeto/s e)stin e)pi\ th\n *d*q kai\ e)k­ballome/nh e)pi\ to\ *e ke/ntron pi/ptei: tau=ta ga\r dh=la e)k th=s o(moio/thtos. h)/xqw dh\ th=| *d*q pro\s o)rqa\s e)n tw=| u(po­keime/nw| e)pipe/dw| h( *d*i: h( *d*q a)/ra o)rqh\ pro\s to\ dia\ tw=n *e *d *i e)pi/pedon, w(/ste kai\ o( *a*b*g ku/klos o)rqo\s pro\s to\ dia\ tw=n *e *d *i e)pi/pedon: e)kblhqe\n a)/ra to\ dia\ tw=n *e *d *i e)pi/pedon ku/klon poih/sei e)n th=| sfai/ra| me/giston o)rqo\n pro\s to\n *a*b*g dia\ tw=n po/lwn au)tou= pi/ptonta kai\ dia\ tw=n *b *o shmei/wn, w(/ste, e)a\n tou= *a*b*g to\n po/lon labo/ntes to\n *p dia\ tou= *p kai\ e(kate/rou tw=n *b *o gra/ywmen ku/- klon, ou(=tos e)/stai o( gino/menos me/gistos e)n th=| sfai/ra| [u(po\ tou= dia\ tw=n *o *d *i e)pipe/dou]. gegra/fqw o( *b*p*o, kai\ @3e)kkei/sqw pa/lin ku/klos o( *r*n*t peri\ dia/metron th\n *r*t, kai\ pros­kei/sqw h( *r*f i)/sh th=| *b*d, kai\ th=| u(po\ *b*d*i gw­ni/a| i)/sh h( u(po\ *r*f*c, kai\ a)po\ tou= *l ke/ntrou ka/qetos h( *l*n*c, kai\ th=| me\n *r*n periferei/a| i)/sh a)peilh/fqw e)pi\ tou= *p*b*o ku/­klou h( *b*u, th=| de\ *f*c i)/sh h( *d*i, kai\ e)pezeu/xqw h( *i*u: h( *i*u a)/ra i)/sh e)/stai th=| *c*n kai\ e)kballome/nh e)pi\ to\ *e ke/n­tron pesei=tai kai\ e)/stai ka/qetos e)pi\ to\ u(pokei/menon e)pi/­pedon, e)pei\ kai\ e)pi\ th\n *i*d: to\ me\n a)/ra *i shmei=on e)/stai e)f' o(\ pi/ptei h( sfai=ra, to\ de\ *u kaq' o(\ pi/ptei, h( de\ e)laxi/sth ka/qetos h( *i*u. kb#. *sfai/ras u(pokeime/nhs kai\ shmei/ou doqe/ntos e)kto\s au)th=s, eu(rei=n to\ shmei=on kaq' o(\ h( a)po\ tou= doqe/ntos e)pi\ to\ ke/ntron e)pizeugnume/nh te/mnei th\n e)pifa/neian. *)/estin de\ fanero/n: a)\n ga\r h(tisou=n a)po\ tou= doqe/ntos eu)qei=a prospesou=sa pro\s th\n e)pifa/neian perienexqh=|, kai\ au(/th gra/yei ku/klon kai\ po/los au)tou= to\ zhtou/menon e)/stai shmei=on. *(upokei/sqw pa/lin h( sfai=ra, kai\ du/o shmei=a dedo/sqw th=s e)pifanei/as e)kto\s a)mfo/tera, kai\ prokei/sqw ta\ sh­mei=a labei=n kaq' a(\ h( e)pi\ ta\ doqe/nta e)pizeugnume/nh te/mnei th\n e)pifa/neian. *kei/sqw ga\r h( sfai=ra peri\ ke/ntron to\ *b, kai\ ta\ do­qe/nta shmei=a e)kto\s e)/stw ta\ *a *g, kai\ kaq' a(\ sumba/llousin th=| e)pifanei/a| ai( a)po\ tw=n *a *g e)pi\ to\ *b e)pizeugnu/menai ei)lh/fqw shmei=a ta\ *d *e, di' w(=n gegra/fqw me/gistos ku/klos o( *d*e*z*h: doqei=sai a)/ra ai( *a*d *g*e (lh=mma ga/r): kai\ dia\ to\ dedo/sqai th\n e)k tou= ke/ntrou th=s sfai/ras kai\ o(/lai do­qh/sontai ai( *a*b *g*b. e)/stin de\ kai\ h( ta\ doqe/nta e)pizeu­gnu/ousa h( *a*g doqei=sa. e)k triw=n ou)=n tw=n *a*b *a*g *g*b tri/gwnon sunesta/tw to\ *q*k*l, kai\ peri\ ke/ntron to\ *q ge­gra/fqw ku/klos i)/sos tw=| *e*d*z*h o( *s*m*n*o. e)a\n me\n ou(=tos te/mnh| th\n *k*l, dh=lon o(/ti kai\ h( e)pi\ ta\ *a *g e)pizeugnume/nh te/mnei th\n sfai=ran, ei) de\ mh/, ou) te/mnei. temne/tw ou)=n o( ku/klos th\n *k*l kata\ ta\ *m *n, kai\ th=| me\n *s*m periferei/a| i)/sh a)peilh/fqw h( *d*h, th=| de\ *o*n h( *e*z. fanero\n dh\ o(/ti ta\ *h *z shmei=a e)/stai kaq' a(\ te/mnei h( e)pizeugnu/ousa ta\ *a *g shmei=a th\n th=s sfai/ras e)pifa/neian. kg#. *xrh/sima kai\ ta\ e)n toi=s i)di/ws legome/nois o)rgani­koi=s kai\ ma/lisq' o(/tan e)pi\ to\ eu)/kolon u(po\ th=s a)nalu/sews xeiragwgou/mena th\n a)na/logon pei=ran diafeu/gein du/nhtai, oi(=on ei)s to\n doqe/nta ku/klon e(pta\ e(ca/gwna e)ggra/yai, to\ me\n peri\ to\ au)to\ ke/ntron tw=| ku/klw|, ta\ de\ loipa\ e(\c a)po\ me\n tw=n tou= me/sou pleurw=n a)nagegramme/na, ta\s de\ a)nti­keime/nas pleura\s e)/xonta e)nhrmosme/nas e(ka/sthn ei)s th\n tou= ku/klou perife/reian. *)/estw o( doqei\s ku/klos peri\ ke/ntron to\ *h, kai\ kei/sqw peri\ to\ au)to\ ke/ntron e(cagw/nou pleura\ h( *q*k, w(/ste e)/stai to\ a)po\ th=s *q*k a)nagrafe\n e(ca/gwnon th\n *m*n pleura\n e)/xon e)nhrmosme/nhn th=| tou= ku/klou periferei/a|, kai\ e)pezeu/xqw h( *h*k: e)p' eu)qei/as a)/ra e)sti\n th=| *k*l pleura=| tou= e(cagw/nou, dia\ to\ dimoi/rou me\n ei)=nai th\n u(po\ *h*k*q, o)rqh=s de\ kai\ tri/tou th\n u(po\ *q*k*l. e)pezeu/xqw h( *h*n. e)pei\ i)/sai ai( *h*k *k*l, diplh= e)sti\n h( *h*l th=s *l*n. kai\ doqei=sa h( *l gwni/a (o)rqh=s ga\r kai\ tri/tou): doqe\n a)/ra to\ *n*l*h tri/gwnon tw=| ei)/dei: lo/gos a)/ra th=s *h*n pro\s *n*l doqei/s. kai\ doqei=sa h( *h*n: doqei=sa a)/ra kai\ h( *n*l pleura\ tou= e(cagw/nou. *to\ de\ o)rganiko\n ou(/tws: e)kkei/sqw th=s e)k tou= ke/ntrou tou= ku/klou tri/ton me/ros h( *a*g, kai\ e)p' au)th=s tmh=ma ku/­klou to\ *a*b*g gwni/an dexo/menon dimoi/rou o)rqh=s, kai\ oi(/wn e)sti\n h( *a*g e#, toiou/twn d# a)peilh/fqw h( *g*e, kai\ h)/xqw e)faptome/nh h( *b*e: le/gw o(/ti h( *a*b e)pizeuxqei=sa i)/sh e)sti\n th=| *q*k tou= e(cagw/nou pleura=|. *)ekbeblh/sqw h( *b*g, kai\ th=| *a*b i)/sh a)fh|rh/sqw h( *b*d: i)so/pleuron a)/ra to\ *a*b*d. kai\ th=| e)k tou= ke/ntrou tou= ku/­klou i)/sh h( *a*z. e)pei\ h( *a*e pro\s *e*g lo/gon e)/xei o(\n ta\ q# pro\s d#, e(/cei kai\ to\ a)po\ *a*b pro\s to\ a)po\ *b*g to\n au)to\n lo/gon: h(mioli/a a)/ra h( *a*b, toute/stin h( *b*d, th=s *b*g: diplh= a)/ra h( *b*g th=s *g*d. a)lla\ kai\ h( *z*g th=s *g*a: kai\ h( *b*z a)/ra e)pizeuxqei=sa th=s *a*d, toute/stin th=s *a*b, e)sti\n diplh=. h)=n de\ kai\ h( *h*l th=s *l*n diplh=, kai\ i)/sas perie/xousin gw­ni/as: o(/moion a)/ra to\ *a*b*z tri/gwnon tw=| *n*l*h trigw/nw|. kai\ e)/stin i)/sh h( *a*z th=| *n*h: i)/sh a)/ra kai\ h( *a*b th=| *l*n h)\ th=| *q*k. *to\ au)to\ a)/llws safe/steron. kd#. *)/estw th=| e)k tou= ke/ntrou tou= doqe/ntos ku/klou i)/sh h( *a*z, kai\ a)peilh/fqw au)th=s to\ g# me/ros, kai\ e)/stw h( *a*g, e)f' h(=s tmh=ma ku/klou gegra/fqw to\ *a*b*g dexo/menon gwni/an dimoi/rou o)rqh=s, kai\ oi(/wn e)sti\n h( *a*g e#, toiou/twn d# a)pei­lh/fqw h( *g*e, kai\ h)/xqw e)faptome/nh tou= tmh/matos h( *e*b, kai\ e)pezeu/xqw h(/ te *a*b kai\ h( *z*b, kai\ e)/ti e)pizeuxqei=sa h( *b*g e)kbeblh/sqw e)pi\ to\ *d, kai\ kei/sqw th=| *a*b i)/sh h( *b*d, kai\ e)pezeu/xqw h( *a*d. e)pei\ ou)=n ei)s ku/klon dih/xqhsan h(/ te *e*g*a kai\ h( *e*b, kai\ h( me\n te/mnei to\n ku/klon h( de\ e)fa/ptetai, to\ a)/ra u(po\ *a*e*g i)/son e)sti\n tw=| a)po\ th=s *e*b: e)/stin a)/ra w(s h( *a*e pro\s *e*b, ou(/tws h( *b*e pro\s *g*e: i)sogw/nion a)/ra to\ *g*b*e tri/gwnon tw=| *a*b*e trigw/nw|. e)/stin a)/ra w(s h( *e*a pro\s *a*b, h( *e*b pro\s *b*g: kai\ w(s a)/ra to\ a)po\ th=s *a*e pro\s to\ a)po\ th=s *e*b, to\ a)po\ th=s *a*b pro\s to\ a)po\ th=s *b*g. a)ll' w(s to\ a)po\ th=s *a*e pro\s to\ a)po\ th=s *e*b, ou(/tws e)sti\n h( *a*e pro\s *e*g dia\ k# tou= #2#. kai\ w(s a)/ra h( *a*e pro\s *e*g, ou(/tws to\ a)po\ th=s *a*b, toute/stin to\ a)po\ th=s *b*d, pro\s to\ a)po\ th=s *b*g: to\ a)/ra a)po\ th=s *b*d pro\s to\ a)po\ th=s *b*g lo/gon e)/xei o(\n ta\ q# pro\s d#: h(mioli/a a)/ra h( *b*d th=s *b*g: diplasi/a a)/ra h( *b*g th=s *g*d. e)/stin de\ kai\ h( *z*g th=s *g*a diplasi/a: w(s a)/ra h( *z*g pro\s *g*a, h( *b*g pro\s *g*d. kai\ i)/sai ei)si\n ai( pro\s tw=| *g gw­ni/ai: i)/sh a)/ra kai\ h( me\n *d gwni/a th=| u(po\ *z*b*g, h( de\ *z th=| u(po\ *g*a*d: e)/stin a)/ra w(s h( *z*b pro\s *b*g, ou(/tws h( *a*d pro\s *d*g. e)nalla\c w(s h( *z*b pro\s *a*d, ou(/tws h( *b*g pro\s *g*d. diplasi/a de\ h( *b*g th=s *g*d: diplasi/a a)/ra kai\ h( *z*b th=s *a*d, toute/stin th=s *a*b. kai\ e)/stin dimoi/rou h( *d: dimoi/rou a)/ra o)rqh=s kai\ h( u(po\ *z*b*g. o(/lh de\ h( u(po\ *a*b*z mia=s o)rqh=s kai\ g#. e)a\n ou)=n e)/xwmen ku/klon, ou(= ke/ntron to\ *h, i)/shn e)/xonta th\n e)k tou= ke/ntrou th=| *a*z eu)qei/a|, kai\ diaga/­gwmen a)po\ tou= ke/ntrou au)tou= th\n *h*c eu)qei=an, kai\ i)/shn qw=men th=| *z*b th\n *h*l eu)qei=an, kai\ pro\s th=| *h*l eu)qei/a| kai\ tw=| *l shmei/w| i)/shn gwni/an susthsw/meqa th\n u(po\ *h*l*n th=| u(po\ *z*b*a, kai\ e)pizeu/cwmen th\n *h*n, i)sogw/nion gi/netai@3 to\ *h*l*n tri/gwnon tw=| *a*z*b trigw/nw|. kai\ e)/stin h( *a*z i)/sh th=| *h*n: i)/sh a)/ra kai\ h( *n*l th=| *a*b. kai\ fanero\n o(/ti a)po\ th=s i)/shs th=| *a*b eu)qei/as gi/netai h( tw=n z# ei)s to\n ku/klon e(cagw/nwn e)ggrafh/. ke#. *pw=s de\ kai\ h( tw=n proeirhme/nwn tumpa/nwn gi/netai para/qesis, nu=n e)rou=men. *)/estw ga\r du/o tu/mpana e)/ntorna kai\ parakei/mena a)l­lh/lois ta\ *a *b, kai\ e)/stw w(s h( dia/metros tou= *a pro\s th\n dia/metron tou= *b, ou(/tws to\ plh=qos tw=n o)do/ntwn tou= *a pro\s to\ plh=qos tw=n o)do/ntwn tou= *b: ou(/tws ga\r h( para/­qesis tw=n tumpa/nwn sw/|zetai dia\ to\ ei)=nai w(s th\n peri/­metron tou= ku/klou pro\s th\n peri/metron, ou(/tws th\n dia/­metron pro\s th\n dia/metron (tou=to ga\r e(ch=s). u(pokei/sqw dh\ to\ me\n *a o)do/ntwn c#, to\ de\ *b o)do/ntwn m#: le/gw o(/ti e)sti\n w(s to\ ta/xos tou= *a pro\s to\ ta/xos tou= *b, ou(/tws to\ plh=qos tw=n o)do/ntwn tou= *b pro\s to\ plh=qos tw=n o)do/ntwn tou= *a. *)epei\ ga\r para/keitai a)llh/lois ta\ *a *b, o(/sous a)\n o)do/ntas kinhqh=| to\ *b, tosou/tous o)do/ntas kinhqh/setai kai\ to\ *a: o(/tan a)/ra to\ *b strefo/menon mi/an a)pokata/stasin poih/shtai, to/te to\ *a m# o)do/ntas kinhqh/setai, w(/ste kai/, o(/tan to\ *b c# a)pokatasta/seis poih/shtai, o(/son e)sti\n to\ plh=qos tw=n o)do/ntwn tou= *a, to/te to\ *a o)do/ntas kinhqh/­setai #22bu#, o(/son e)sti\n to\ plh=qos tw=n o)do/ntwn tou= *a e)pi\ to\ plh=qos tw=n o)do/ntwn tou= *b. o(moi/ws de\ deixqh/setai kai/, o(/tan to\ *a m# a)pokatasta/seis poih/shtai, o(/son e)sti\n to\ plh=qos tw=n o)do/ntwn tou= *b, to/te to\ *b o)do/ntas kekinh­me/non #22bu#, o(/son e)sti\n to\ plh=qos tw=n o)do/ntwn tou= *b e)pi\ to\ plh=qos tw=n o)do/ntwn tou= *a: o(/tan a)/ra to\ *a a)pokata­sta/seis poih/shtai m#, o(/son e)sti\n to\ plh=qos tw=n o)do/ntwn tou= *b, to/te kai\ to\ *b a)pokatasta/seis poiei=tai c#, o(/son e)sti\n to\ plh=qos tw=n o)do/ntwn tou= *a: e)/stin a)/ra w(s to\ ta/xos tou= *a pro\s to\ ta/xos tou= *b, ou(/tws to\ plh=qos tw=n o)do/ntwn tou= *b pro\s to\ plh=qos tw=n o)do/ntwn tou= *a. k#2#. *(/oti de\ ai( tw=n ku/klwn perife/reiai pro\s a)llh/las ei)si\n w(s ai( dia/metroi, nu=n dei/comen. *)/estwsan ga\r du/o ku/kloi oi( *a*b *g*d, kai\ dia/metroi au)tw=n ai( *a*b *g*d: le/gw o(/ti e)sti\n w(s h( tou= *a*b ku/klou perife/reia pro\s th\n tou= *g*d ku/klou perife/reian, ou(/tws h( *a*b dia/metros pro\s th\n *g*d. *)epei\ ga/r e)stin w(s o( *a*b ku/klos pro\s to\n *g*d ku/klon, ou(/tws to\ a)po\ th=s *a*b tetra/gwnon pro\s to\ a)po\ th=s *g*d tetra/gwnon, a)lla\ tou= me\n *a*b ku/klou tetrapla/sio/n e)stin to\ periexo/menon o)rqogw/nion u(po/ te th=s *a*b diame/trou kai\ th=s tou= *a*b periferei/as, tou= de\ *g*d ku/klou tetrapla/sio/n e)stin to\ u(po\ th=s *g*d kai\ th=s tou= *g*d periferei/as (to\ ga\r u(po\ th=s e)k tou= ke/ntrou tou= ku/klou kai\ th=s perime/trou tou= ku/klou periexo/menon o)rqogw/nion dipla/sio/n e)stin tou= e)m­badou= tou= ku/klou, w(s *)arximh/dhs, kai\ w(s e)n tw=| ei)s to\ prw=ton tw=n maqhmatikw=n sxoli/w| de/deiktai kai\ u(f' h(mw=n di' e(no\s qewrh/matos), kai\ w(s a)/ra to\ u(po\ th=s *a*b kai\ th=s periferei/as tou= *a*b pro\s to\ u(po\ th=s *g*d kai\ th=s tou= *g*d ku/klou periferei/as, ou(/tws to\ a)po\ th=s *a*b tetra/gwnon pro\s to\ a)po\ th=s *g*d. kai\ e)nalla\c w(s to\ u(po\ th=s tou= *a*b ku/klou periferei/as kai\ th=s *a*b pro\s to\ a)po\ th=s *a*b, ou(/tws to\ u(po\ th=s tou= *g*d ku/klou periferei/as kai\ th=s *g*d pro\s to\ a)po\ th=s *g*d: kai\ w(s a)/ra h( tou= *a*b ku/klou pe­rife/reia pro\s th\n *a*b, ou(/tws h( tou= *g*d perife/reia pro\s th\n$ *g*d (tou=to ga\r prw=to/n e)stin e)n tw=| #2# lambano/menon), kai\ e)nalla\c w(s h( tou= *a*b perife/reia pro\s th\n tou= *g*d peri­fe/reian, ou(/tws h( *a*b pro\s th\n *g*d.

kz#. *tumpa/nou doqe/ntos kai\ tou= plh/qous tw=n o)do/ntwn au)tou=, e)piteta/xqw paraqei=nai au)tw=| tu/mpanon doqe\n e)/xon to\ plh=qos tw=n o)do/ntwn kai\ eu(rei=n th\n dia/metron tou= para­tiqeme/nou tumpa/nou.

*)/estw tu/mpanon to\ *a, ou(= to\ plh=qos tw=n o)do/ntwn e)/stw o( *b a)riqmo\s [mona/dwn c#], kai\ parakei/sqw tw=| *a to\ *g tu/mpanon, ou(= to\ plh=qos tw=n o)do/ntwn e)/stw o( *d a)riqmo\s [mona/dwn m#]: dei= dh\ tou= *g th\n dia/metron eu(rei=n.

*)epei\ ou)=n o( *b a)riqmo\s plh=qo/s e)stin o)do/ntwn tou= *a, o( de\ *d plh=qo/s e)stin o)do/ntwn tou= *g [kai\ e)/stin to\ me\n plh=qos tw=n o)do/ntwn tou= *a h( peri/metros au)tou=, to\ de\ plh=­qos tw=n o)do/ntwn tou= *g h( peri/metros au)tou=], e)/stin a)/ra w(s o( *b a)riqmo\s pro\s to\n *d, ou(/tws h( peri/metros tou= *a pro\s th\n peri/metron tou= *g. w(s de\ h( peri/metros pro\s th\n pe­ri/metron, ou(/tws h( dia/metros pro\s th\n dia/metron. lo/gos de\ tou= *b a)riqmou= pro\s to\n *d a)riqmo\n doqei/s [e)/stin ga\r o( tw=n c# pro\s ta\ m#]: lo/gos a)/ra kai\ th=s diame/trou tou= *a pro\s th\n dia/metron tou= *g doqei/s [o( tw=n c# pro\s ta\ m#]. kai\ e)/stin doqei=sa h( dia/metros tou= *a: doqei=sa a)/ra kai\ h( dia/metros tou= *g [dei= ga\r poiei=n w(s to\n c# a)riqmo\n pro\s to\n m#, ou(/tws th\n dia/metron tou= *a pro\s a)/llhn tina/, kai\ o( peri\ dia/metron e)kei/nhn grafo/menos ku/klos i)/sos e)/stai tw=| zhtoume/nw| tumpa/nw|].

*)organikw=s de\ ou(/tws: e)kkei/sqw tis eu)qei=a h( *e*z te­tmhme/nh ei)s i)/sa, i)/sa to\ plh=qos toi=s o)dou=si tou= *a tum­pa/nou [toute/stin c#], kai\ pro\s o)rqa\s au)th=| a)xqei=sa kei/sqw diame/trw| tou= *a tumpa/nou i)/sh h( *z*h, kai\ e)pezeu/xqw h( *e*h, kai\ [oi(/wn h( *e*z c#, toiou/twn m#] a)peilh/fqw h( *e*q tou= plh/qous tw=n o)do/ntwn tou= *g ginome/nh, kai\ dia\ tou= *q para/llhlos th=| *z*h h)/xqw h( *q*k: kai\ e)/stai a)/ra h( *q*k i)/sh th=| diame/trw| tou= *g tumpa/nou (fanera\ ga\r h( a)po/deicis).

kh#. *pw=s de\ kataskeua/zetai koxli/as th\n e(/lika a(rmo­sth\n e)/xwn toi=s locoi=s o)dou=si tou= doqe/ntos tumpa/nou, fa­nero\n ou(/tws e)/stai.

*noei/sqw ku/lindros i)sopaxw=s tetorneume/nos o( *a*d*e*z, pleura\ d' au)tou= h( *a*e, kai\ ei)lh/fqw monostro/fou e(/likos e)p' au)th=s dia/sthma to\ *a*b, kai\ lepi/dion xalkou=n gegenh/­sqw, ou(= to\ me\n *h*q*k me/ros tri/gwnon o)rqogw/nion e)/stw o)rqh\n e)/xon th\n *q gwni/an, to\ de\ loipo\n parallhlo/grammon o)rqogw/nion to\ *q*k*l, i)/sh de\ kei/sqw h( *q*h th=| *a*b, h( de\ *q*k th=| perime/trw| tou= *a*d*e*z kuli/ndrou, kai\ perikampte/­sqw to\ lepi/dion peri\ to\n ku/lindron, i(/na kai\ to\ *q*k*l parallhlo/grammon ku/lindros ge/nhtai a(pto/menos tou= *d*e, o(/tan ei)saxqh=|, kai\ kei/sqw to\ me\n *q e)pi\ to\ *a, to\ de\ *h e)pi\ to\ *b, kai\ ou(/tws gra/yomen dia\ th=s *h*k u(poteinou/shs kamfqei/shs [de\] th\n kaloume/nhn mono/strofon e(/lika w(s th\n *b*a. kai\ pa/lin metaqe/ntes to\ lepi/dion, w(/ste to\ me\n *q kata\ to\ *b ei)=nai to\ de\ *h kata\ to\ *g, gra/yomen dia\ th=s *h*k e(te/ran e(/lika mono/strofon, w(/ste th\n o(/lhn ei)=nai di/­strofon. e)n w(=| ga\r xro/nw| to\ *a e)pi\ to\ *b paragi/netai o(malw=s kinou/menon, e)n tou/tw| kai\ h( *a*b kata\ th=s e)pifa­nei/as tou= kuli/ndrou kinhqei=sa ei)s to\ au)to\ a)pokaqi/statai kai\ to\ ei)rhme/non fe/resqai shmei=on kata\ th=s *a*b eu)qei/as gra/yei th\n mono/strofon e(/lika: tou=to ga\r *)apollw/nios o( *pergeu\s a)pe/deicen. [e)a\n ou)=n kai\ e(kate/ran tw=n *a*b *b*g kai\ ta\s e(ch=s a)/xri tou= *e di/xa te/mnwmen kai\ dia\ tw=n sh­mei/wn tw=| lepidi/w| gra/ywmen monostro/fous e(/likas a)p' au)tw=n kata\ to\ ba/qos th=s e(/likos o(\ boulo/meqa la/bwmen kai\ a)po\ tou= ba/qous loipo\n kai\ th=s grafei/shs e(/likos, r(a|di/ws th\n e(/lika fakoeidh= r(inh/santes e(/comen a)phrtisme/nhn.]

kq#. *pa/lin noei/sqw e)n th=| e(te/ra| e)pifanei/a| tou= doqe/n- tos tumpa/nou peri\ to\n ko/trafon ku/klos, ou(= perife/reia h( *r*u*t ke/ntron de\ to\ *c, kai\ ta\ *r *u *t i)/son a)p' a)llh/lwn a)pe/xonta, lo/gou xa/rin tou= pano\s ku/klou ei)s ei)/kosi te/s­sara dih|rhme/nou, kai\ a)po\ tw=n *r *u *t e)pi\ to\ *c ke/ntron neu/ousai dih/xqwsan a)/xri tou= peri\ to\ *c ke/ntron gegramme/­nou ku/klou tou= *m*n*p*f ai( *r*o *u*o *t*o, kai\ a)po\ tw=n di­xotomou/ntwn ta\s *o*o periferei/as shmei/wn dih/xqwsan e)pi\ ta\ *r *u *t shmei=a ai( *m*r *n*r *n*u *p*u *p*t *t*f, kai\ a)po\\$ th=s *o*r eu)qei/as proh/xqw e)n th=| kurth=| tou= tumpa/nou e)pi­fanei/a| h( *r*s me/xri th=s periferei/as ou)=sa tou= e)n th=| e(te/ra| e)pifanei/a| tou= tumpa/nou peri\ to\n ko/trafon o(moi/ws grafo­me/nou tou= *x*w ku/klou, kai\ a)po\ tou= *s th=| me\n h(misei/a| th=s *r*u periferei/as [w(s locw/sews] i)/sh kei/sqw h( *s*x, th=| de\ *r*u h( *x*w, kai\ ou(/tws e(ch=s i)/shn qe/ntes th=| *u*t th\n *w*y kai\ ta\s loipa/s, kai\ e)pizeu/cantes ta\s *r*x *u*w *t*y e(/comen ta\s tw=n o)do/ntwn locw/seis. kai\ e)pei\ i)/sos e)sti\n o( *r*u ku/klos tw=| *x*w ku/klw|, gra/yomen ka)n th=| e(te/ra| e)pifanei/a| tou= tum­pa/nou peri\ ke/ntron to\ a)ntikei/menon tw=| *c shmei/w| ku/klon i)/son tw=| *m*n, kai\ a)po\ tw=n *x *w a)gago/ntes e)p' au)to\n eu)qei/as neuou/sas e)pi\ to\ ke/ntron au)tou=, kai\ ta\ au)ta\ poi­h/santes toi=s e)pi\ th=s *r*u*t periferei/as [tou= ku/klou] e(/comen kai\ th\n a)/llhn pleura\n tou= tumpa/nou katagegramme/nhn. kai\ loipo\n e)kko/yantes ta\ metacu\ tw=n grammw=n sxh/mata w(s ta\ *n*r*u *u*p*t kai\ ta\ a)ntikei/mena e(/comen to\ tu/mpanon w)dontwme/non o)dou=sin locoi=s. e)mbai/nei de\ e(/kastos ei)s th\n tou= koxli/ou e(/lika, e)pei\ kai\ to\ metacu\ dia/sthma to\ *r*u i)/son e)sti\n tw=| *a*b diasth/mati th=s tou= koxli/ou e(/likos. kai\ dh=lon w(s kaq' e(ka/sthn strofh\n tou= koxli/ou ei(=s o)dou\s parenexqh/setai: tou=to ga\r *(/hrwn a)pe/deicen e)n toi=s mhxa­nikoi=s, grafh/setai de\ kai\ u(f' h(mw=n, i(/na mhde\n e)/cwqen e)pizhtw=men.

l#. *noei/sqw ga\r koxli/as o( *a*b, h( de\ e)n au)tw=| e(/lic h( *a*g*d*e*z*b [noei/sqwsan de\ mono/strofoi ai( ei)rhme/nai e(/likes], tu/mpanon de\ e)/stw [to\] parakei/menon kai\ w)dontwme/non to\ *h*g*e*q o)do/ntas e)/xon tou\s *h*g *g*e *e*q a(rmo/zontas th=| e(/liki [oi( a)/ra loipoi\ ou)k e)narmo/sousin ei)s ta\s loipa\s e(/likas]. e)a\n ou)=n e)pistre/fwmen to\n koxli/an, w(/ste to\ *e shmei=on parwqei=sqai e)pi\ ta\ *g me/rh, pare/stai to\ *e e)pi\ to\ *g, o(/tan o( koxli/as a)pokata/stasin mi/an poih/shtai, kai\ e(/cei o( me\n *g*e o)dou\s th\n tou= *g*h qe/sin, o( de\ *e*q th\n tou= *g*e, kai\ pa/lin o( *e*q qe/sin e)sxhkw\s th\n *g*e e)n mia=| tou= koxli/ou peristrofh=| o(/los paraxqh/setai. kai\ e)pi\ tw=n e(ch=s o)do/n­twn ta\ au)ta\ e)pinoei=n xrh/, w(/ste, o(/sous a)\n o)do/ntas e)/xh| to\ tu/mpanon, tosauta/kis o( koxli/as kinhqei\s mi/an a)pokata/­stasin tou= tumpa/nou poih/setai.

la#. *tosau=ta me\n ou)=n peri\ tou= baroulkou=, tw=n de\ proeirhme/nwn e# duna/mewn e)k tw=n *(/hrwnos th\n e)/kqesin e)pitomw/teron poihso/meqa pro\s u(po/mnhsin tw=n filoma­qou/ntwn, prosqe/ntes e)/ti kai\ ta\ peri\ th=s monokw/lou kai\ dikw/lou kai\ trikw/lou kai\ tetrakw/lou mhxanh=s a)nagkai/ws lego/mena, mh/ pote kai\ tw=n bibli/wn e)n oi(=s tau=ta ge/graptai a)pori/a ge/nhtai tw=| zhtou=nti: kai\ ga\r h(mei=s kata\ polla\ me/rh diefqarme/nois e)netu/xomen a)na/rxois te kai\ a)tele/si bibli/ois. pe/nte toi/nun ou)sw=n duna/mewn di' w(=n to\ doqe\n ba/ros th=| doqei/sh| bi/a| kinei=tai, a)nagkai=o/n e)stin ta/ te sxh/mata au)tw=n kai\ ta\s xrei/as e)/ti de\ kai\ ta\ o)no/mata e)kqe/sqai. a)pode/dotai de\ u(po\ tou= *(/hrwnos kai\ *fi/lwnos kai\ dio/ti ai( proeirhme/nai duna/meis ei)s mi/an a)/gontai fu/sin, kai/toi para\ polu\ dialla/ssousai toi=s sxh/masin. o)no/mata me\n ou)=n e)stin ta/de: a)/cwn e)n peritroxi/w|, moxlo/s, polu/­spaston, sfh/n, kai\ pro\s tou/tois o( kalou/menos a)/peiros koxli/as.

*(o me\n ou)=n a)/cwn o( e)n tw=| peritroxi/w| kataskeua/zetai ou(/tws: cu/lon dei= labei=n eu)/tonon tetra/gwnon (kaqa/per do­@3ki/da) kai\ tou/tou ta\ a)/kra si­mw/santa stroggu/la poih=sai kai\ xoiniki/das periqei=nai xalka=s sunararui/as tw=| a)/coni, w(/ste e)mblhqei/sas au)ta\s ei)s trh/mata stroggu/la e)n a)kinh/­tw| tini\ ph/gmati eu)lu/tws stre/­fesqai tw=n trhma/twn tribei=s xalkou=s e)xo/ntwn u(pokeime/nous tai=s xoiniki/si: kalei=tai de\ to\ ei)rhme/non cu/lon a)/cwn. peri\ de\ me/son to\n a)/cona periti/­qetai tu/mpanon e)/xon trh=ma tetra/gwnon a(rmosto\n tw=| a)/coni, w(/ste a(/ma stre/fesqai to/n te a)/cona kai\ to\ peritro/xion.

*(h me\n ou)=n kataskeuh\ dedh/lwtai, xrei/a d' e)sti\n h( me/llousa le/gesqai. o(/tan ga\r boulw/meqa mega/la ba/rh kinei=n e)la/ssoni bi/a|, ta\ e)kdedeme/na e)k tou= ba/rous o(/pla periqe/ntes peri\ ta\ sesimwme/na tou= a)/conos, kai\ e)mbalo/n­tes skuta/las ei)s ta\ e)n tw=| peritroxi/w| trh/mata, e)pistre/­fomen to\ peritro/xion kata/gontes ta\s skuta/las, kai\ ou(/tws eu)ko/pws kinhqh/setai to\ ba/ros u(po\ e)la/ssonos duna/mews tw=n o(/plwn peri\ to\n a)/cona e)peiloume/nwn [h)\ kai\ diamhruo­me/nwn u(po/ tinos pro\s to\ mh\ a(/pan to\ o(/plon perikei=sqai tw=| a)/coni]. tou= de\ ei)rhme/nou o)rga/nou to\ me\n me/geqos a(r­mo/zesqai dei= pro\s ta\ me/llonta kinei=sqai ba/rh, th\n de\ summetri/an pro\s to\n lo/gon o(\n e)/xei to\ kinou/menon ba/ros pro\s th\n kinou=san du/namin, w(s e(ch=s deixqh/setai.

*)=hn de\ deute/ra du/namis h( dia\ tou= moxlou= [kai\ ta/xa h( proepi/noia th=s peri\ ta\ u(pera/gan ba/rh kinh/sews]: proelo/­menoi ga/r tines mega/la ba/rh kinei=n, e)peidh\ a)po\ th=s gh=s e)/dei prw=ton metewri/sai, laba\s de\ ou)k ei)=xon dia\ to\ pa/nta ta\ me/rh th=s e(/dras tou= forti/ou e)pikei=sqai tw=| e)da/fei, u(po­ru/cantes braxu\ kai\ cu/lou makrou= to\ a)/kron u(pobalo/ntes u(po\ to\ forti/on kath=gon e)k tou= e(te/rou a)/krou, u(poqe/ntes tw=| cu/lw| par' au)to\ to\ forti/on li/qon, o(\ dh\ kalei=tai u(po­mo/xlion. fanei/shs d' au)toi=s th=s kinh/sews pa/nu eu)ko/pou e)no/hsan o(/ti dunato\n kinei=sqai mega/la ba/rh dia\ tou= tro/pou tou/tou. kalei=tai de\ to\ cu/lon moxlo/s, ei)/te tetra/gwnon ei)/h ei)/te stroggu/lon. o(/sw| d' a)\n e)ggute/rw tiqh=tai tou= forti/ou to\ u(pomo/xlion, tosou/tw| eu)xere/steron kinei=tai to\ ba/ros, w(s e(ch=s deixqh/setai.

*)/estin de\ h( tri/th du/namis h( kata\ to\ polu/spaston. o(/tan ga\r boulw/meqa/ ti ba/ros e(/lkein, e)ca/yantes o(/plon e)c au)tou= e)pispw/meqa tosau/th| bi/a|, o(/sh tw=| forti/w| i)so/r­ropo/s e)stin. e)a\n de\ e(lku/santes e)k tou= forti/ou to\ o(/plon th\n me\n mi/an au)tou= a)rxh\n e)kdh/swmen e)/k tinos me/nontos xwri/ou, th\n de\ e(te/ran ba/lwmen dia\ troxi/lou e)kdedeme/nou e)k tou= forti/ou kai\ tau/thn e)pispw/meqa, eu)xere/steron ki­nh/somen to\ ba/ros. pa/lin de\ e)a\n e)k tou= me/nontos xwri/ou e)ca/ywmen e(/teron troxi/lon kai\ th\n a)gome/nhn a)rxh\n diaba­lo/ntes dia\ tou/tou e)pispw/meqa, e)/ti ma=llon eu)xere/steron kinh/somen to\ ba/ros. kai\ pa/lin e)a\n e)k tou= forti/ou troxi/­lon e(/teron e)kdh/swmen kai\ th\n a)gome/nhn a)rxh\n dia\ tou/tou diabalo/ntes e)pispw/meqa, pollw=| ma=llon eu)xere/steron ki­nh/somen to\ ba/ros %2%2 a)ei\ troxi/lous e)/k te tou= me/nontos xw­ri/ou e)ca/ptontes kai\ e)k tou= forti/ou kai\ diaba/llontes e)nalla\c th\n a)gome/nhn a)rxh\n ei)s tou\s troxi/lous eu)xere/ste­ron kinh/somen to\ ba/ros. [o(/sw| d' a)\n ei)s plei/ona kw=la to\ o(/plon ka/mpthtai, to\ ba/ros eu)kopw/teron kinhqh/setai: dei= de\ th\n e)kdennume/nhn a)rxh\n e)k tou= me/nontos xwri/ou e)ca/pte­sqai.] i(/na ou)=n mh\ kaq' e(/na tou\s troxi/lous e)/k te tou= me/­nontos xwri/ou kai\ e)k tou= forti/ou e)ca/ptwmen, oi( me\n ei)rh­me/noi ei)s to\ me/non ei)=nai xwri/on ei)s e(\n cu/lon e)nti/qentai peri\ a)/conas kinou/menoi, o(\ kalei=tai ma/gganon, tou=to de\ e)ca/ptetai e)k tou= me/nontos xwri/ou dia/ tinos e(te/rou o(/plou, oi( de\ pro\s tw=| forti/w| ei)s e(/teron ma/gganon tou/tw| i)/son, o(\ dh\ pa/lin e)ca/ptetai e)k tou= forti/ou mo/non. ou(/tws de\ dei= kata­teta/xqai e)n toi=s magga/nois tou\s troxi/lous, w(/ste ta\ kw=la mh\ e)mpleko/mena pro\s a)/llhla duspeiqh= gi/nesqai. di' h(\n d' ai)ti/an pleio/nwn tw=n kw/lwn ginome/nwn eu)kopi/a para­@3kolouqei=, dei/comen, kai\ di' h(\n ai)ti/an h( e(te/ra a)rxh\ e)k tou= me/nontos e)ca/ptetai xwri/ou.

*(h de\ e(ch=s du/namis h( dia\ tou= sfh­no\s kai\ au)th\ mega/las xrei/as parexo­me/nh pro/s te ta\s mureyika\s pie/seis kai\ ta\s dia\ th=s tektonikh=s u(peragou/sas kol­lh/seis, to\ de\ pa/ntwn me/giston, o(/tan tou\s e)k tw=n latomiw=n li/qous a)pospa=n de/h| th=s kata\ to\ ka/tw me/ros sunexei/as, ou)demi/a tw=n a)/llwn duna/mewn e)nergei=n du/natai, ou)d' a)\n a(/ma pa=sai suzeu­xqw=sin, mo/nos de\ o( sfh\n e)nergei= dia\ th=s tuxou/shs, kai\ a)/nesis me\n ou)d' h(tisou=n gi/netai kata\ ta\ dialh/mmata tw=n e)rga­zome/nwn, kartera\ de\ h( e)pi/tasis. tou=to de\ fanero\n e)k tou= kai\ mh\ plhssome/nou tou= sfhno\s e)ni/ote yo/fous kai\ r(h/gmata gi/nesqai dia\ th=s tou= sfhno\s e)nergei/as. o(/sw| d' a)\n h( tou= sfhno\s gwni/a e)la/sswn gi/nhtai, tosou/tw| eu)xere/steron e)nergei=, toute/stin di' e)la/ssonos plhgh=s, w(s dei/­comen.

*ta\ me\n ou)=n proeirhme/na o)/rgana fanera\s kai\ au)totelei=s e)/xei ta\s kata­skeua\s pollaxou= e)n tai=s xrei/ais faino­me/nas, o( de\ koxli/as e)/xei ti peri/ergon peri/ te th\n kataskeuh\n kai\ th\n xrh=sin. o(te\ me\n [ou)=n] ga\r au)­to\s kaq' au(to\n mo/nos e)nergei=, o(te\ de\ kai\ proslamba/nwn e)/ti du/namin, plh\n o(/ti ou)de\n e(/tero/n e)stin h)\ sfh\n ei)lhme/nos, a)po­leipo/menos th=s plhgh=s, dia\ moxlou= de\ kai\ strofh=s th\n ki/nh-@3sin poiou/menos. tou=to d' e)/stai dh=lon e)k tw=n mello/ntwn le/­gesqai. fu/sis me\n ou)=n u(pa/rxei th=s peri\ au)to\n pragma­tei/as toiau/th: e)a\n kuli/ndrou pleura\ fe/rhtai kata\ th=s tou= kuli/ndrou e)pifanei/as, pro\s de\ tw=| pe/rati tau/ths shmei=o/n ti a(/ma kata\ au)th=s th=s pleura=s fe/rhtai, kai\ e)n tw=| au)tw=| xro/nw| h(/ te pleura\ mi/an a)pokata/stasin poih/shtai kai\ to\ shmei=on to\ pa=n th=s pleura=s diece/lqh|, h( genome/nh u(po\ tou= shmei/ou e)n th=| kulindrikh=| e)pifanei/a| grammh\ e(/lic e)sti/n, h(\n dh\ koxli/an kalou=sin. katagra/fetai de\ e)n tw=| kuli/ndrw| ou(/tws: e)a\n e)n e)pipe/dw| du/o eu)qei/as e)kqw/meqa o)rqa\s a)llh/­lais, w(=n h( me\n mi/a i)/sh e)sti\n th=| tou= ei)rhme/nou kuli/ndrou pleura=|, h( de\ e(te/ra th=| tou= ku/klou periferei/a|, o(/s e)stin ba/sis tou= kuli/ndrou, kai\ e)pi\ ta\ pe/rata tw=n ei)rhme/nwn eu)qeiw=n e)pizeu/cwmen eu)qei=an u(potei/nousan th\n o)rqh\n gw­ni/an, teqh=| de\ h( i)/sh th=| tou= kuli/ndrou pleura=| e)pi\ th\n tou= kuli/ndrou pleura/n, h( de\ e(te/ra tw=n peri\ th\n o)rqh\n e)pei­lhqh=| kata\ th=s tou= ku/klou periferei/as, ei)lhqh/setai kai\ h( u(potei/nousa th\n o)rqh\n kata\ th=s kulindrikh=s e)pifanei/as, kaq' h(=s e)/stai h( ei)rhme/nh e(/lic. e)/cestin de\ dielo/nta th\n tou= kuli/ndrou pleura\n ei)s i)/sa, o(po/s' a)/n tis proairh=tai, kaq' e(/kaston au)th=s me/ros perigra/fein e(/lika, w(s proei/rh­tai [w(/ste e)n tw=| kuli/ndrw| plei/onas e(/likas gra/fesqai, ka­lei/sqw de\ h( a(/pac ei)lhqei=sa e(/lic mono/strofos, toute/stin h( peri\ ta\ para\ e(ka/stou me/rous ginome/nh grammh/]. kata\ au)th=s ou)=n th=s grammh=s swlh=na e)ntemo/ntes ei)s to\ ba/qos tou= kuli/ndrou kai\ e)kko/yantes, w(/ste e)n tw=| swlh=ni tu/lon@3 e)narmo/sai stereo/n, xrw=ntai tw=| koxli/a| ou(/tws: ta\ a)/kra au)tou= stroggu/la poih/santes e)narmo/zousin ei)/s tina dia­ph/gmata e)n stroggu/lois trh/masin, w(/ste eu)ko/pws au)to\n stre/fesqai, u(pe\r de\ to\n koxli/an kano/na diatiqe/ntes par­a/llhlon au)tw=| swlh=na e)/xonta me/son e)n th=| a)/nw e)pifanei/a| e)narmo/zousin ei)s tou=ton to\n swlh=na to\n ei)rhme/non tu/lon, w(/ste to\ me\n e(/teron a)/kron tou= tu/lou me/nein e)n tw=| tou= ko­xli/ou swlh=ni, to\ de\ e(/teron e)n tw=| ei)rhme/nw| e)te/rw| swlh=ni tw=| e)n tw=| kano/ni. o(/tan ou)=n bou/lwntai forti/on kinei=n dia\ tou/tou tou= o)rga/nou, o(/plon labo/ntes tou/tou th\n me\n mi/an a)rxh\n e)ca/ptousin e)k tou= forti/ou, th\n de\ e(te/ran e)k tou= pro­eirhme/nou tu/lou, kai\ trhma/twn o)/ntwn th=| kefalh=| tou= ko­xli/ou skuta/las e)mbalo/ntes kata/gousin, kai\ ou(/tws u(po\ th=s e(/likos o( tu/los parago/menos e)n tw=| swlh=ni e)pispa=tai to\ o(/plon di' ou(= kai\ to\ forti/on. e)/cestin de\ a)nti\ tw=n sku­talw=n xeirola/bhn tina\ periqei=nai tw=| a)/krw| tou= koxli/ou u(pere/xonti ei)s to\ e)kto\s tou= diaph/gmatos kai\ ou(/tws stre/­fonta to\n koxli/an e)pispa=sqai to\ forti/on. h( d' e)n tw=| ko­xli/a| e(/lic o(te\ me\n tetra/gwnos gi/netai o(te\ de\ fakoeidh/s, tetra/gwnos me/n, o(/tan o( e)n au)tw=| swlh\n o)rqa\s e)/xh| ta\s e)ntoma/s, fakoeidh\s de/, o(/tan loca\s kai\ ei)s mi/an sunago- me/nas grammh/n. kalei=tai de\ o( me\n tetra/gwnos, o( de\ fa­kwto/s.

*(/otan me\n ou)=n au)to\s kaq' au)to\n o( koxli/as e)nergh=|, tau/thn lamba/nei th\n kataskeuh/n, gi/netai de\ kai\ e(te/rws: @3proslabo/ntes ga/r tina e(te/ran du/namin th\n dia\ tou= a)/conos tou= e)n tw=| peritroxi/w| kaloume/nou [kataskeuh\n] noh/somen to\ peri\ to\n a)/cona tu/m­panon w)dontwme/non ei)=­nai, koxli/an de/ tina parakei=sqai tw=| tum­pa/nw| h)/toi o)rqo\n kei/­menon pro\s to\ e)/dafos h)\ para/llhlon tw=| e)da/­fei, e)/xonta th\n me\n e(/lika e)mpeplegme/nhn toi=s o)dou=si tou= tumpa/­nou ta\ de\ a)/kra e)n strog­gu/lois trh/masin poleuo/­mena e)/n tisin diaph/gma­sin, kaqa/per kai\ proei/­rhtai, kai\ u(peroxh=s ou)/shs tou= a)/krou tou= koxli/ou ei)s to\ e)kto\s tou= diaph/gmatos me/ros, h)/toi xeirola/bhn tina\ perikei=sqai, di' h(=s e)pistrafh/setai o( koxli/as, h)\ trh/mata, w(/ste skutalw=n e)mblhqeisw=n o(moi/ws e)pistre/fesqai: au)to/n. pa/lin ou)=n ta\ e)k tou= forti/ou o(/pla peribalo/ntes peri\ to\n a)/cona e)f' e(ka/tera tou= tumpa/nou kai\ e)pistre/fontes to\n koxli/an, di' ou(= kai\ to\ w)dontwme/non tu/mpanon, e)pispaso/meqa to\ ba/ros.

*ai( me\n ou)=n kataskeuai\ kai\ ai( xrh/seis tw=n proeirh­me/nwn pe/nte duna/mewn dedh/lwntai, ti/s de/ e)stin h( ai)ti/a, di' h(\n di' e(ka/sths au)tw=n mega/la ba/rh kinei=tai mikra=| panta/pasi duna/mei, *(/hrwn a)pe/deicen e)n toi=s mhxanikoi=s. e)n de\ toi=s e(ch=s e)k tou= g# tw=n *(/hrwnos mhxana\s gra/yomen pro\s eu)kopi/an kai\ lusite/leian a(rmozou/sas, di' w(=n pa/lin mega/la ba/rh kinhqh/setai.

*ta\ me\n ou)=n a)go/mena e)pi\ tou= e)da/fous, fhsi/n, e)pi\ xelw/nas a)/getai. h( de\ xelw/nh ph=gma/ e)stin e)k tetragw/nwn@3 cu/lwn sumpephgo/s, w(=n ta\ a)/kra a)nasesi/mwtai. tou/tois ou)=n e)piti/qetai ta\ ba/rh, kai\ e)k tw=n a)/krwn au)tw=n h)/toi polu/spasta e)kde/nnutai h)\ o(/plwn a)rxai/. tau=ta de\ h)/toi a)po\ xeiro\s e(/lketai h)\ ei)s e)rga/tas a)podi/dotai, w(=n peria­gome/nwn h( xelw/nh e)pi\ tou= e)da/fous su/retai u(poballome/nwn skutali/wn h)\ sani/dwn. e)a\n me\n ga\r mikro\n h)=| to\ forti/on, skuta/lais xrh=sqai dei=, e)a\n de\ mei=zon, tai=s sani/sin dia\ to\ tau/tas mh\ eu)ko/lws su/resqai: ai( ga\r skuta/lai kulio/­menai ki/ndunon e)/xousin tou= forti/ou o(rmh\n labo/ntos. e)/nioi de\ ou)/te skuta/lais ou)/te sani/si xrw=ntai, a)lla\ troxou\s na­stou\s prosqe/ntes tai=s xelw/nais a)/gousin.

lb#. *)epi\ de\ tw=n ei)s u(/yos bastazome/nwn forti/wn, fhsi/n, mhxanai\ gi/nontai ai( me\n mono/kwloi, ai( de\ di/kwloi, ai( de\ tri/kwloi, ai( de\ tetra/kwloi. ai( me\n ou)=n mono/kwloi ou(/tws: cu/lon eu)/tonon lamba/netai u(/yos e)/xon mei=zon h)\ ou(= boulo/­meqa to\ forti/on metewri/sai, ka)\n me\n au)to\ kaq' au(to\ i)sxuro\n h)=|, o(/plon ba/llontes peri\ au)to\ [kai\ sfi/ggontes] kai\ diamhruo/menoi kata\ e)pei/lhsin a)posfi/ggousin. tw=n de\ e)peilh/sewn to\ metacu\ dia/sthma ou) plei=on gi/netai palai­stw=n d#, kai\ ou(/tws eu)tonw/tero/n te gi/netai to\ cu/lon kai\ ai( tou= o(/plou e)peilh/seis w(/sper baqmoi\ toi=s e)rgazome/nois kai\ boulome/nois ei)s to\ a)/nw metewri/zesqai eu)/xrhstoi gi/non­tai. e)a\n de\ mh\ h)=| eu)/tonon to\ cu/lon, e)k pleio/nwn sumblh­to\n gi/netai. [stoxa/zesqai dei= tw=n mello/ntwn basta/zesqai forti/wn, o(/pws mh\ a)sqene/steron to\ kw=lon u(pa/rxh|.] i(/sta­tai ou)=n to\ kw=lon o)rqo\n e)pi/ tinos cu/lou kai\ e)k tou= a)/krou au)tou= o(/pla e)kde/nnutai tri/a pou h)\ te/ssara kai\ a)pote­qe/nta a)podi/dotai pro/s tina me/nonta xwri/a, o(/pws to\ cu/­lon, o(/pou a)/n tis bia/zhtai, mh\ paraxwrh=| katexo/menon u(po\ tw=n a)potetame/nwn o(/plwn. e)k de\ tou= a)/nw me/rous au)tou= polu/spasta e)ca/yantes kai\ a)podido/ntes ei)s to\ forti/on e)pispw=ntai h)/toi a)po\ xeiro\s h)\ ei)s e)rga/tas a)podo/ntes, ei)s o(/tan metewrisqh=| to\ forti/on. ka)\n de/h| to\n li/qon e)kteqh=­nai e)pi\ tei=xos h)\ o(/pou bou/letai/ tis, e)klu/santes e(\n tw=n e)kdennume/nwn e)k tou= a)/krou o(/plwn to\ e)pi\ ta\ e(/tera me/rh tou= forti/ou kei/menon e)gkli/nousin to\ kw=lon, h)\ ta\s sku­ta/las u(poba/llontes u(po\ to\ forti/on e)n toi=s me/resin, e)n oi(=s h( sfendo/nh e)n tw=| li/qw| ou)k e)pei/lhtai, xalw=si ta\ a)go/­mena tw=n poluspa/stwn a)/xri a)\n e)pikaqi/sh| to\ forti/on tai=s skuta/lais, ei)=t' e)klu/santes th\n sfendo/nhn moxleu/ousi to\ forti/on a)/xri ou(= ei)s o(\n bou/lontai to/pon para/cwsin. ei)=ta pa/lin to\ u(pokei/menon tw=| kw/lw| cu/lon o(/plw| e)pispasa/menoi a)po\ xeiro\s peria/gousin e)pi\ e(/teron me/ros tou= oi)kodomh/ma­tos a(/ma a)nie/ntes tou\s a)poto/mous, kai\ pa/lin e)kdh/santes xrw=ntai, w(s proei/rhtai.