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 [1mega/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]1, 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 [1be/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]1, pro\s de\ tau/tais h( tw=n i)di/ws pa/lin kaloume/nwn mhxanopoiw=n [1e)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]1. 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| [1tou=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]1, 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| [1dia\ 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]1, 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 [1tou=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]1. 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