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Palimpsestul Lui Arhimede

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Discutii TED.com Codexul pierdut al lui Arhimede 2 Septembrie 2012 AvocatNet.ro http://www.avocatne t.ro/content/articles/id_302 29/Codexul-pierdut-al-lui- Arhimede.html#axzz2BtJQ3LNj Cum citesti un manuscris vechi de 2000 de ani care a fost sters, taiat, scris si pictat pe el? Cu un accelerator de particule, desigur! Restauratorul de carti antice William Noel ne spune fascinanta poveste a palimpsestului lui Arhimede, o carte de rugaciuni bizantine care continea scrieri originale necunoscute pana acum ale matematicianului Arhimede din Grecia antica  precum si ale alto ra. William Noel vorbeste la Ted2xSummit 2012 despre "Codexul pierdut al lui Archimede". Materialul este preluat de pe Ted.com, in conformitate cu prevederile licentei Creative Commons "Attribution -- NonCommercial -- NonDerivative ". Vestitele texte din lumea antica nu ne parvin in forma originala. Au supravietuit pentru ca scribii medievali le-au copiat si le-au copiat si le-au copiat. La fel este cu Arhimede, marele matematician grec.  Tot ce stim despre Arhimede ca matematician stim numai datorita a trei carti, numite A, B si C. Cartea A a fost pierduta de un umanist italian in 1564. B a fost vazuta ultima data in Biblioteca Papei la 100 de mile nord de Roma in Viterbo in 1311. Codex C a fost descoperita doar in 1906 si a aterizat pe biroul meu in Baltimore in 19 ianuarie 1999. Acesta este Codex C.  De fapt, Codex C e ingropat in aceasta carte. E comoara ascunsa. Pentru ca aceasta carte e de fapt o carte de rugaciuni. A fost scrisa de un tip pe nume Johannes Myrones in 14 aprilie 1229. Pentru a-si face cartea de rugaciuni a folosit pergament. Dar nu a folosit pergament nou, a folosit pergament reciclat din manuscrise scrise anterior, sapte in total. Iar Codex C al lui Arhimede era doar unul din cele sapte. A desfacut manuscrisul lui Arhimede ca pe toate cele sapte. A sters toate textele, apoi a taiat foile la mijloc, le-a amestecat, le-a rotit cu 90 de grade, si a scris rugaciuni peste ele. Asadar, aceste sapte manuscrise au disparut pentru 700 de ani lasand o carte de rugaciuni.  Cartea de rugaciuni a fost descoperita de acest tip, Johan Ludvig Heiberg, in 1906. Doar cu o lupa, a transcris din text cat a putut. Dar a gasit doua texte in manuscris care erau unicate. Nu erau deloc in A si B; erau texte complet noi ale lui Arhimede, si se numeau "Metoda" si "Stomachion". A devenit un mauscris faimos.  Evident ca aceasta carte e intr-o stare proasta. S-a deteriorat si mai mult in secolul 20 dupa ce Heilberg a vazut-o. Au fost pictate falsuri pe ea, si a suferit mult din cauza mucegaiului. Cartea aceasta e o pierdere tipica. Este tipul de carte care ai crede ca s-ar afla intr-o institutie. Dar nu se afla intr-o institutie, ci a fost cumparata de un proprietar particular in 1998.  
Transcript
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Discutii TED.com Codexul pierdut al lui Arhimede2 Septembrie 2012 AvocatNet.ro http://www.avocatnet.ro/content/articles/id_30229/Codexul-pierdut-al-lui-Arhimede.html#axzz2BtJQ3LNj

Cum citesti un manuscris vechi de 2000 de ani care a fost sters, taiat, scris si pictat pe el? Cuun accelerator de particule, desigur! Restauratorul de carti antice William Noel ne spunefascinanta poveste a palimpsestului lui Arhimede, o carte de rugaciuni bizantine care contineascrieri originale necunoscute pana acum ale matematicianului Arhimede din Grecia antica

 precum si ale altora.William Noel vorbeste la Ted2xSummit 2012 despre "Codexul pierdut al lui Archimede".Materialul este preluat de pe Ted.com, in conformitate cu prevederile licentei Creative 

Commons "Attribution -- NonCommercial -- NonDerivative".Vestitele texte din lumea antica nu ne parvin in forma originala. Au supravietuit pentru cascribii medievali le-au copiat si le-au copiat si le-au copiat. La fel este cu Arhimede, marelematematician grec. Tot ce stim despre Arhimede ca matematician stim numai datorita a trei carti, numite A, B siC. Cartea A a fost pierduta de un umanist italian in 1564. B a fost vazuta ultima data inBiblioteca Papei la 100 de mile nord de Roma in Viterbo in 1311. Codex C a fost descoperitadoar in 1906 si a aterizat pe biroul meu in Baltimore in 19 ianuarie 1999. Acesta este CodexC. De fapt, Codex C e ingropat in aceasta carte. E comoara ascunsa. Pentru ca aceasta carte e defapt o carte de rugaciuni. A fost scrisa de un tip pe nume Johannes Myrones in 14 aprilie1229. Pentru a-si face cartea de rugaciuni a folosit pergament. Dar nu a folosit pergamentnou, a folosit pergament reciclat din manuscrise scrise anterior, sapte in total. Iar Codex C allui Arhimede era doar unul din cele sapte. A desfacut manuscrisul lui Arhimede ca pe toatecele sapte. A sters toate textele, apoi a taiat foile la mijloc, le-a amestecat, le-a rotit cu 90 degrade, si a scris rugaciuni peste ele. Asadar, aceste sapte manuscrise au disparut pentru 700

de ani lasand o carte de rugaciuni. Cartea de rugaciuni a fost descoperita de acest tip, Johan Ludvig Heiberg, in 1906. Doar cu olupa, a transcris din text cat a putut. Dar a gasit doua texte in manuscris care erau unicate. Nuerau deloc in A si B; erau texte complet noi ale lui Arhimede, si se numeau "Metoda" si"Stomachion". A devenit un mauscris faimos. Evident ca aceasta carte e intr-o stare proasta. S-a deteriorat si mai mult in secolul 20 dupa ceHeilberg a vazut-o. Au fost pictate falsuri pe ea, si a suferit mult din cauza mucegaiului.Cartea aceasta e o pierdere tipica. Este tipul de carte care ai crede ca s-ar afla intr-o institutie.Dar nu se afla intr-o institutie, ci a fost cumparata de un proprietar particular in 1998.

 

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De ce a cumparat aceasta carte? Pentru ca a dorit ca ce era fragil sa fie in siguranta. A dorit cace era unic sa devina larg raspandit. A dorit ca ce era costisitor sa fie gratuit. A dorit sa facaasta din principiu. Nu multi urmau sa-l citeasca pe Arhimede in greaca veche, dar ar trebui saaiba posibilitatea s-o faca. 

Asa ca i-a strans pe prietenii lui Arhimede si a promis ca va plati pentru toata munca. Si era otreaba costisitoare, dar nu atat cat ati crede pentru ca acesti oameni nu venisera pentru bani, ci

 pentru Arhimede. Aveau diferife pregatiri. Erau din fizica particulelor, din filologia clasica,din conservarea cartilor, din matematica antica, din managementul datelor, din imagisticastiintifica si management de proiect. S-au adunat sa lucreze la acest manuscris. Prima problema era o problema de conservare. Iata cam ce am avut de facut: Era lipici pecotorul cartii. Daca priviti cu atentie fotografia, partea de jos e cam maronie. Acest clei e unadeziv organic. Daca esti restaurator, poti inlatura acest clei foarte usor. Partea de sus e cleiElmer pentru lemn. E o emulsie de acetat de polivinil care nu se dizolva in apa odata ce s-auscat. Si e mult mai dur decat pergamentul pe care s-a scris. Inainte sa incepem sa-l scanam

 pe Arhimede, a trebuit sa desfacem cartea. A durat patru ani s-o desfacem. Asta e o rara pozade actiune dinamica in forta, doamnelor si domnilor.

Apoi a trebuit sa indepartam ceara, pentru ca asta era folosita in ceremoniile religioase din biserica ortodoxa greceasca, foloseau ceara de lumanare. Iar ceara era murdara, si nu puteamscana prin ea. Cu mare grija a trebuit sa indepartam mecanic toata ceara. Mi-e greu sa va explic exact cat de proasta e starea aceastei carti, dar se desfacea adesea in

 bucatele mici. Normal, la o carte nu ti-ar pasa de bucatele, dar aceste bucatele puteau continetexte unice ale lui Arhimede. Fragmente marunte am reusit sa le punem inapoi la locul

 potrivit. Apoi am inceput sa scanam manuscrisul. Am scanat manuscrisul in 14 benzi diferite delumina. Daca privesti ceva in lungimi de unda diferite vezi lucruri diferite. Iata o imagine aunei pagini scanata in 14 lungimi de unda diferite. Dar niciuna n-a mers. Asa ca am procesat imaginile la un loc si am pus cate doua pe un ecrangol. Iata doua imagini diferite ale manuscrisului lui Arhimede. Imaginea din stanga eimaginea rosie normala. Iar cea din dreapta e imaginea in ultraviolet. in imaginea din dreapta

 poti vedea un pic din scrisul lui Arhimede. Daca le suprapui intr-o singura coala digitala, pergamentul e deschis la culoare in ambele imagini si apare luminos. Cartea de rugaciuni e

inchisa la culoare si apare intunecata. Textul lui Arhimede e intunecat intr-una si deschis incealalta iar rezultatul va fi intunecat dar rosu, dupa care se poate citi chiar bine. Iata cumarata. Asta-i o imagine "inainte" si "dupa", dar nu asa citesti imaginea de pe ecran. Apropii si iar apropii si maresti si iar maresti, iar acum poti citi. Daca procesezi ambele imagini diferit chiar poti sa scapi de cartea de rugaciuni. Iar asta eextrem de important, pentru ca diagramele din manuscris sunt o sursa unica de diagrame pecare Arhimede le desena in nisip in secolul IV i.e.n. ti iata, vi le pot arata. 

Cu acest tip de imagistica - cu acest spectru invizibil de infrarosu, ultraviolet, n-am fi reusitniciodata sa obtinem imaginea prin stratul auriu de falsuri. Cum urma sa procedam? Ei bine,

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am luat manuscrisul, si am decis sa-l scanam utilizand XRF, fluoresceta cu raze X. O raza Xeste directionata in diagrama din stanga si elibereaza un electron din stratul intern al unuiatom. Acel electron dispare. Cum acesta dispare, un electron dintr-un strat mai indepartat,sare si ii ia locul. Si cand acesta ii ia locul, emite radiatie electromagnetica. Emite o raza X.Aceasta raza X are o lungime de unda specifica atomului pe care il loveste.

 Ce am vrut sa obtinem era fierul. Pentru ca cerneala continea fier. Daca putem identifica deunde provine raza X cand apare, putem cartografia tot fierul de pe pagina, apoi, teoretic

 putem citi imaginea. E nevoie de o sursa de lumina foarte puternica pentru asta. Am dus manuscrisul la StanfordSynchrotron Radiation Laboratory in California, care e un accelerator de particule. Electronii

 pleaca intr-o directie, pozitronii in cealalta. Se intalnesc la mijloc, si creaza particulesubatomice cum sunt quarcii charm si leptonii tau. Nu aveam de gand sa-l punem peArhimede in acel fascicul. Dar cum electronii se rotesc cu viteza luminii, emit raze X. E ceamai puternica sursa de lumina din sistemul solar. Se numeste radiatia sincrotron si e curent

folosita pentru a studia proteine si altele asemenea. Dar noi am vrut sa tinteasca atomii,atomii de fier, ca sa putem citi pagina "inainte" si "dupa". Si ce sa vezi, am aflat ca se poate.A durat cam 17 minute sa faca o singura pagina. Ce am descoperit? Unul dintre textele unice ale lui Arhimede se numeste "Stomachion".Acesta nu exista in Codexurile A si B. Si stiam ca era legat de acest patrat. Acesta e un patrat

 perfect, divizat in 14 parti. Nimeni nu stia ce facea Arhimede cu aceste 14 parti. Acumcredem ca stim. Incerca sa-si dea seama in cate feluri poti rearanja acele 14 parti si totusi saai un patrat perfect. Vrea cineva sa ghiceasca raspunsul? Este 17152 impartit la 536 defamilii. E important pentru ca e cel mai timpuriu studiu de combinari matematice. Iar combinarile matematice sunt o ramura minunata si interesanta a matematicii. Lucrul cel mai uimitor la acest manuscris e ca ne-am uitat si la celelalte manuscrise pe care ledetinea palimpsestul, din care scribul isi facuse cartea, iar unul din ele continea un text al luiHyperides. Hyperides a fost un orator atenian din secolul IV i.e.n, contemporan cuDemostene. in 338 i.e.n., impreuna cu Demostene au decis ca vor sa se opuna puterii militarea lui Filip al Macedoniei. Drept urmare, Atena si Teba au iesit sa lupte cu Filip alMacedoniei. A fost o idee proasta, pentru ca Filip al Macedoniei avea un fiu numit Alexandrucel Mare si au pierdut batalia de la Chaeronea. Alexandru cel Mare a continuat sa cucereasca lumea larga; Hzperides s-a trezit judecat pentru

tradare. Iar acesta e discursul lui din timpul judecatii - e un discurs maret: "Cel mai bine -spune el - e sa castigi. Dar daca nu poti castiga, atunci trebuie sa lupti pentru o cauzanobila, pentru ca atunci nu vei fi uitat. Uitati-va la spartani. Au castigat nenumarate victorii,dar nimeni nu-si aminteste ce sunt pentru ca toate au avut scopuri egoiste. Singura bataliedusa de spartani de care toti iti amintesc e batalia de la Termopile, unde au fost macelariti

 pana la ultimul, dar au luptat pentru libertatea Greciei." A fost un discurs atat de maret incatcurtea ateniana l-a iertat. A mai trait inca 10 ani, cand fractiunea macedoneana s-a luat de el.I-au taiat limba ca batjocura pentru oratoria sa, si nu se stie ce au facut cu corpul sau. Asta edescoperirea unei marete voci din antichitate, vorbindu-ne, nu din mormant, pentru camormantul lui nu exista, ci din curtile de judecata ateniene. 

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Ar trebui sa spun ca atunci cand privesti manuscrise medievale care au fost sterse, nu gasestitexte unice. Sa gasesti doua intr-un manuscris e ceva deosebit. Sa gasesti trei e foarte ciudat.

 Noi am gasit trei. "Categoriile" lui Aristotel, unul din textele de baza ale filosofiei vestice. Am gasit un

comentariu din secolul III e.n. posibil al lui Galen si probabil al lui Porfiriu. Toate aceste date colectate de noi, toate imaginile si imaginile neprelucrate, toate transcrierile

 pe care le-am facut si tot restul le-am pus online sub licenta Creative Commons ca oricine sale poata folosi in orice scop comercial. [...]http://www.avocatnet.ro/content/articles/id_30229/Codexul-pierdut-al-lui-Arhimede.html#axzz2BtJQ3LNj

Vestitele texte din lumea antică nu ne parvin în forma originală. Au supraviețuit pentru căscribii medievali le-au copiat și le-au copiat și le-au copiat.  La fel este cu Arhimede, marelematematician grec. 

Tot ce știm despre Arhimede ca matematician știm numai datorită a trei căr ți,  numite A, B șiC. Cartea A a fost pierdută de un umanist italian în 1564. B a fost văzută ultima dată înBiblioteca Papei la 100 de mile nord de Roma în Viterbo în 1311. Codex C a fost descoperită

doar în 1906 și a aterizat pe biroul meu în Baltimore  în 19 ianuarie 1999. Acesta este CodexC. 

De fapt, Codex C e îngropat în această carte. E comoară ascunsă. Pentru că această carte e defapt o carte de rugăciuni. A fost scrisă de un tip pe nume Johannes Myrones în 14 aprilie1229. Pentru a-şi face cartea de rugăciuni a folosit pergament. Dar nu a folosit pergamentnou, a folosit pergament reciclat din manuscrise scrise anterior, şapte în total. Iar Codex C allui Arhimede era doar unul din cele şapte. A desfăcut manuscrisul lui Arhimede ca pe toatecele şapte. A şters toate textele, apoi a tăiat foile la mijloc, le-a amestecat, le-a rotit cu 90 degrade, şi a scris rugăciuni peste ele. Aşadar, aceste şapte manuscrise au dispărut pentru 700de ani lăsând o carte de rugăciuni. 

Cartea de rugăciuni a fost descoperită de acest tip, Johan Ludvig Heiberg, în 1906. Doar cu olupă, a transcris din text cât a putut. Dar a găsit două texte în manuscris care erau unicate.  Nuerau deloc în A şi B; erau texte complet noi ale lui Arhimede, şi se numeau "Metoda" şi"Stomachion". A devenit un mauscris faimos. 

Evident că această carte e într-o stare proastă. S-a deteriorat şi mai mult în secolul 20 după ceHeilberg a văzut-o. Au fost pictate falsuri pe ea, şi a suferit mult din cauza mucegaiului. Cartea aceasta e o pierdere tipică. Este tipul de carte care ai crede că s-ar afla într-o instituţie. Dar nu se află într-o instituţie, ci a fost cumpărată de un proprietar particular în 1998. 

De ce a cumpărat această carte? Pentru că a dorit ce era fragil să fie în siguranţă A dorit ceera unic să devină larg răspândit. A dorit ce era costisitor să fie gratuit. A dorit să facă asta

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din principiu.  Nu mulţi urmau să-l citească pe Arhimede în greaca veche, dar ar trebui să aibă posibilitatea s-o facă. 

Aşa că i-a strâns pe prietenii lui Arhimede, şi a promis că va plăti pentru toată munca. Şi erao treabă costisitoare, dar nu atât cât aţi crede  pentru că aceşti oameni nu veniseră pentru bani, 

ci pentru Arhimede. Aveau diferife pregătiri. Erau din fizica particulelor, din filologia clasică, din conservarea cărţilor, din matematica antică, din managementul datelor, din imagisticăştiinţifică şi management de proiect. S-au adunat să lucreze la acest manuscris. 

Prima problemă era o problemă de conservare. Iată cam ce am avut de făcut: Era lipici pecotorul cărţii. Dacă priviţi cu atenţie fotografia,  partea de jos e cam maronie. Acest clei e unadeziv organic. Dacă eşti restaurator,  poţi înlătura acest clei foarte uşor. Partea de sus e cleiElmer pentru lemn. E o emulsie de acetat de polivinil care nu se dizolvă în apă odată ce s-auscat. Şi e mult mai dur decât pergamentul pe care s-a scris. Înainte să începem să-l scanăm

 pe Arhimede, a trebuit să desfacem cartea. A durat patru ani s-o desfacem. Asta e o rară pozăde acțiune dinamică în for ță, doamnelor și domnilor. 

(Râsete) 

Apoi a trebuit să îndepărtăm ceara,  pentru că asta era folosită în ceremoniile religioase din biserica ortodoxă grecească, foloseau ceară de lumânare. Iar ceara era murdară, şi nu puteamscana prin ea. Cu mare grijă a trebuit să îndepărtăm mecanic toată ceara. 

Mi-e greu să vă explic exact cât de proastă e starea aceastei căr ți,  dar se desfăcea adesea în bucăţele mici.  Normal, la o carte nu ţi-ar păsa de bucăţele, dar aceste bucățele puteau conţinetexte unice ale lui Arhimede. Fragmente mărunte am reuşit să le punem înapoi la locul

 potrivit. Apoi am început să scanăm manuscrisul. Am scanat manuscrisul în 14 benzi diferite delumină. Dacă priveşti ceva în lungimi de undă diferite vezi lucruri diferite. Iată o imagine aunei pagini scanată în 14 lungimi de undă diferite. 

Dar niciuna n-a mers. Aşa că am procesat imaginile la un loc şi am pus câte două pe un ecrangol. Iată două imagini diferite ale manuscrisului lui Arhimede. Imaginea din stânga eimaginea roşie normală. Iar cea din dreapta e imaginea în ultraviolet. În imaginea din dreapta 

 poţi vedea un pic din scrisul lui Arhimede. Dacă le suprapui într-o singură coală digitală,  pergamentul e deschis la culoare în ambele imagini şi apare luminos. Cartea de rugăciuni e

închisă la culoare şi apare întunecată. Textul lui Arhimede e întunecat într-una şi deschis încealaltă iar rezultatul va fi întunecat dar roşu, după care se poate citi chiar bine. Iată cumarată. 

Asta-i o imagine "înainte" şi "după", dar nu aşa citeşti imaginea de pe ecran. Apropii şi iar apropii şi măreşti şi iar mărești,  iar acum poți citi. 

(Aplauze) 

Dacă procesezi ambele imgini diferit chiar poţi să scapi de cartea de rugăciuni. Iar asta eextrem de important,  pentru că diagramele din manuscris sunt o sursă unică de diagrame  pe

care Arhimede le desena în nisip în secolul IV î.e.n. Și iată, vi le pot arăta. 

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Cu acest tip de imagistică - cu acest spectru invizibil de infraroşu, ultraviolet, n-am fi reușitniciodată să obţinem imaginea prin stratul auriu de falsuri. Cum urma să procedăm? Ei bine,am luat manuscrisul, şi am decis să-l scanăm utilizând XRF, fluoresceţă cu raze X. O rază Xeste direcționată în diagrama din stânga şi eliberează un electron din stratul intern al unuiatom. Acel electron dispare. Cum acesta dispare, un electron dintr-un strat mai îndepărtat, 

sare şi îi ia locul. Și când acesta îi ia locul ,  emite radiaţie electromagnetică. Emite o rază X. Această rază X are o lungime de undă specifică atomului pe care îl loveşte. 

Ce am vrut să obţinem era fierul. Pentru că cerneala conținea fier.  Dacă putem identifica deunde  provine raza X când apare,  putem cartografia tot fierul de pe pagină, apoi, teoretic

 putem citi imaginea. 

E nevoie de o sursă de lumină foarte puternică pentru asta. Am dus manuscrisul la StanfordSynchrotron Radiation Laboratory în California, care e un accelerator de particule. Electronii

 pleacă într-o direcţie,  pozitronii în cealaltă. Se întâlnesc la mijloc, şi crează particulesubatomice cum sunt quarcii charm şi leptonii tau.  Nu aveam de gând să-l punem peArhimede în acel fascicul. Dar cum electronii se rotesc cu viteza luminii, emit raze X. E ceamai puternică sursă de lumină din sistemul solar. Se numeşte radiaţia sincrotron, şi e curentfolosită pentru a studia  proteine şi altele asemenea. Dar noi am vrut să țintească atomii,atomii de fier, ca să putem citi pagina "înainte" şi "după". Și ce să vezi, am aflat că se poate. A durat cam 17 minute să facă o singură pagină. 

Ce am descoperit? Unul dintre textele unice ale lui Arhimede se numeşte "Stomachion". Acesta nu există în Codexurile A şi B. Şi ştiam că era legat de acest pătrat. Acesta e un pătrat

 perfect, divizat în 14 părţi.  Nimeni nu ştia ce făcea Arhimede cu aceste 14 părţi. Acumcredem că ştim. Încerca să-şi dea seama în câte feluri poţi rearanja acele 14 părţi și totuşi să

ai un pătrat perfect. Vrea cineva să ghicească răspunsul? Este 17152 împărţit la 536 defamilii. E important pentru că e cel mai timpuriu studiu de combinări matematice. Iar combinările matematice sunt o ramură minunată şi interesantă a matematicii. 

Lucrul cel mai uimitor la acest manuscris e că ne-am uitat şi la celelalte manuscrise  pe care ledeţinea palimpsestul, din care scribul îşi făcuse cartea, iar unul din ele conţinea un text al luiHyperides. Hyperides a fost un orator atenian din secolul IV î.e.n, contemporan cuDemostene. În 338 î.e.n., împreună cu Demostene au decis că vor să se opună  puterii militarea lui Filip al Macedoniei. Drept urmare Atena şi Teba au ieșit să lupte cu Filip al Macedoniei. A fost o idee proastă, pentru că Filip al Macedoniei avea un fiu numit Alexandru cel Mare șiau pierdut bătălia de la Chaeronea. 

Alexandru cel Mare a continuat să cucerească lumea largă; Hzperides s-a trezit judecat pentrutrădare. Iar acesta e discursul lui din timpul judecăţii - e un discurs măreţ: "Cel mai bine -spune el - e să câştigi. Dar dacă nu poţi câştiga, atunci trebuie să lupţi pentru o cauză nobilă, 

 pentru că atunci nu vei fi uitat. Uitaţi-vă la spartani. Au câştigat nenumărate victorii, dar nimeni nu-şi aminteşte ce sunt  pentru că toate au avut scopuri egoiste. Singura bătălie dusă despartani de care toți își amintesc e bătălia de la Termopile unde au fost măcelăriţi până laultimul, dar au luptat pentru libertatea Greciei." A fost un discurs atât de măreţ încât curteaateniană l-a iertat. A mai trăit încă 10 ani, când fracţiunea macedoneană s-a luat de el. I-autăiat limba ca batjocură pentru oratoria sa, şi nu se ştie ce au făcut cu corpul său. Asta edescoperirea unei măreţe voci din antichitate, vorbindu-ne, nu din mormânt,  pentru că

mormântul lui nu există, ci din curţile de judecată ateniene. 

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Ar trebui să spun că atunci când priveşti manuscrise medievale care au fost şterse, nu găseştitexte unice. Să găseşti două într-un manuscris e ceva deosebit. Să găseşti trei e foarte ciudat. 

 Noi am găsit trei. 

"Categoriile" lui Aristotel, unul din textele de bază ale filosofiei vestice. Am găsit un

comentariu din secolul III e.n.  posibil al lui Galen şi probabil al lui Porfiriu. 

Toate aceste date colectate de noi, toate imaginile şi imaginile neprelucrate, toate transcrierile pe care le-am făcut şi tot restul le-am pus online sub licenţa Creative Commons ca oricine săle poată folosi în orice scop comercial. 

(Aplauze) 

De ce proprietarul manuscrisului a făcut asta? Pentru că înţelege atât informaţiile cât şicărţile. Ce trebuie să faci cu cărţile, dacă vrei să fie folosite timp îndelungat, e să le ascunzi

 prin dulapuri şi să laşi pe foarte puţini să se uite la ele. Cu informaţiile însă, dacă vrei săsupravieţuiască, trebuie să le răspândeşti tuturor  cu cât mai mic control posibil asuprainformaţiei. Şi el asta a făcut. 

Instituţiile pot învăţa din asta. Pentru că la momentul actual îşi ascund informaţiile prinrestricţii de copyright ş.a.m.d. Dacă vrei să te uiţi la inscripţii medievale pe web, trebuie săintri pe site-ul Bibliotecii Nationale Y, sau al Bibliotecii Universităţii X, cam cel mai

 plictisitor mod în care poți avea de-a face cu informaţii digitale.  Mai eficient ar fi să le pui laun loc. 

Pentru că Web-ul manuscriselor antice ale viitorului nu va fi construit de instituţii. Va fi

construit de către utilizatori, de către cei ce obţin aceste date împreună, de oameni care vor săadune la un loc tot felul de hărţi  provenite de pretutindeni, tot felul de poveşti medievale dedragoste  provenite de pretutindeni, oameni care doresc să-şi administreze selecţia lor propriede lucruri minunate. Acesta e viitorul Internetului. E un viitor atrăgător şi frumos, doar dacăam reuşi să-l transpunem în realitate. 

 Noi, la Walters Art Museum am urmat acest exemplu, şi ne-am pus manuscrisele pe Internet  pentru ca oamenii să se bucure de ele - de datele neprelucrate, de toate descrierile, de toatemetadatele, sub licenţa Creative Commons. Walters Art Museum e un muzeu mic ce aremanuscrise frumoase, dar informaţiile sunt fantastice. Rezultatul este că dacă cauţi imagini peGoogle chiar acum, şi scrii "Illuminated manuscript Koran" de exemplu, 24 din 28 de imagini

 provin din instituţia mea. 

(Aplauze) 

Să ne gândim la asta puţin. Ce folos trage instituţia din asta? Sunt tot felul de foloase pentruinstituţie. Putem vorbi despre umanitate şi treburi din astea, dar să vorbim despre lucruriegoiste. Ce folos trage instituţia cu adevărat din asta? De ce merg oamenii la Luvru? Se ducsă vadă Mona Lisa. De ce se duc s-o vadă pe Mona Lisa? Pentru că ştiu deja cum arată. Eiştiu cum arată pentru că au văzut poze cu ea peste tot. 

 Nu e nevoie deloc de acest fel de restricţii. Cred că instituţiile ar trebui să ia atitudine, să-şi

facă publice toate datele fără nicio restricţie, şi ar fi un beneficiu enorm pentru toată lumea. De ce nu lăsăm acces liber tuturor la aceste informaţii ca să-şi alcătuiască propriile colecţii de

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cunoştiinţe antice şi de lucruri minunate, să crescă frumuseţea şi importanţa culturală aInternetului. 

Vă mulţumesc foarte mult. 

(Aplauze) 

Manuscris cu teoremele lui Arhimede

descoperit sub un text liturgic bizantin

http://www.doxologia.ro/societate/cultura/manuscris-cu-teoremele-lui-arhimede-descoperit-sub-un-text-liturgic-bizantin

Cercetătoriii şi restauratorii au muncit din greu pentru a aduce la lumină un manuscris ascuns pentru 1000 de ani, care arată din nou geniul lui Arhimede, cel mai mare matematician alantichităţii.

Manuscrisul cunoscut ca "Palimpsestul lui Arhimede", este o carte bizantină de rugăciune dinsecolul al 13-lea, care a fost scrisă folosind manuscrise mai vechi, unul dintre ele conţinândcâteva teorii ale marelui matematician grec Arhimede, copiate în Constantinopolul secoluluial 10-lea. Ele au fost descoperite în 1906 de către savantul danez Johan Ludwig Heiberg,cercetător al lui Arhimede, însă de sub rugăciunile bizantine nu se putea citi decât parţialteoriile geniului grec.

Cartea a dispărut pentru o vreme apoi a fost scoasă la licitaţie într-o stare mult mai degradată

la Casa de Licitaţii "Christies" din New York în 1998. Manuscrisul a fost cumpărat de uncolecţionar anonim american pentru 2 milioane de dolari şi depus la Muzeul de Artă Walter 

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din Baltimore, unde oameni de ştiinţă, conservatori, restauratori, istorici şi matematicieni aulucrat din greu să descopere secretele celei mai vechi copii în care supravieţuieşte opera luiArhimede.

Folosind imagini multi-spectrale şi tehnici de raze X, care au găsit moleculele de fier din

cerneala care a dispărut, cercetătorii au descoperit că Arhimede, care a trăit în secolul 3 B.C.,a luat în considerare conceptul de “infinitate actuală”, pe care umanitatea credea că l-adescoperit în secolul al 19-lea, şi calculele anticipate. De asemenea, 7 tratate scrise de marelematematician şi singura copie a “Metodei de Teoreme Mecanice şi Stomachion (Cutia luiArhimede)”, noi cuvântări ale oratorului atenian Hyperides şi un comentariu pierdut din“Categoriile” lui Aristotel, au fost găsite sub rugăciunile bizantine.

Manuscrisul este acum expus la o expoziţie la Muzeul Walter, iar editura CambridgeUniversity Press a publicat o carte în 2 volume intituată The Archimedes Palimpsest Project,care arată imagini din acest maniscris fascinant, transcrierile textelor şi interpretărileteoremelor marelui geniu grec.

Editorul cărţii, Michael Sharp, a spus că în tratatul său “Metoda Teoremelor Mecanice”,Arhimede a numit conceptul de “infinitate actuală”, “foarte important pentru istoriamatematicii şi ştiinţei”. Arhimede afirma că două seturi diferite de linii sunt egale înmultitudine, deşi este înţeles că ele sunt infinite, o afirmaţie similară în mod remarcabil culucrările din secolele 16 şi 17 care au dus la invenţia “calculului”.

Acest pasaj nu a putut fi citit de marele matematician Heiberg de la începutul sec. 20.

Palimpsestul conţie şi un tratat intituat "Stomachon", în care Arhimede încearcă să descopere

în câte feluri pot fi recombinate 14 piese fixe pentru a face un pătrat perfect. Răspunsul corecteste 17,152 combinaţii. Stomachion înseamnă “durere de burtă”. Această lucrare este un pasînainte în ştiinţa combinatoricii, o ştiinţă matematică care stă la baza matematicii

 probabilistice. Înainte de acest manuscris se credea că ştiinţa combinatoricii a fost descoperităîn sec. 17 sau 18.

Traducerea şi adaptarea: Pr. Ioan Valentin Istrati

 Sursa: http://www.guardian.co.uk 

http://www.doxologia.ro/societate/cultura/manuscris-cu-teoremele-lui-arhimede-descoperit-

sub-un-text-liturgic-bizantin

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Palimpsestul lui Arhimede, descifrat -

scoate la iveala secrete vechi de 2.200 de anihttp://www.ziare.com/magazin/descoperire/palimpsestul-lui-arhimede-descifrat-scoate-la-

iveala-secrete-vechi-de-2-200-de-ani-1130156

Foto: The GuardianManuscrisul numit Palimpsestul lui Arhimede a fost descifrat dupa o munca de mai

multi ani de catre oamenii de stiinta. Acesta contine mai multe tratate ale celui mai

mare matematician antic, scrise in secolul al III-lea i.Hr. si vor fi publicate de catreUniversitatea Cambridge.

Palimpsestul lui Arhimede, manuscris caligrafiat peste o carte de rugaciuni, a fost descifratdupa o munca minutioasa, care a durat mai multi ani, informeaza The Guardian.

Dupa ani de cercetari, oamenii de stiinta au expus manuscrisul ascuns timp de aproape o miede ani, care pune intr-o noua lumina geniul lui Arhimede, cel mai mare matematician alantichitatii.

Cunoscut ca Palimpsestul lui Arhimede, manuscrisul este o carte de rugaciuni bizantine dinsecolul al XIII-lea si a fost asamblat folosind pagini ale unor manuscrise anterioare, dintrecare unul continea mai multe tratate ale matematicianului grec Arhimede, ce au fost copiatein secolul al X-lea la Constantinopol, actualul Istanbul.

Acesta a fost descoperit in 1906 de catre savantul danez Johan Ludwig Heiberg, iar in 1998 afost vandut la Christie din New York unui colectionar american pentru suma de 2 milioane dedolari.

El a fost apoi depus la Muzeul de Arta Walters din Baltimore, unde oamenii de stiinta aulucrat la descoperirea secretelor lui Arhimede.

Folosind tehnica imaginii multispectrale si a razelor X, ei au descoperit ceea ce a conceputArhimede in secolul al III-lea i.Hr., un concept al infinitului, ceva ce s-a crezut ca a fostrealizat doar in secolul al XIX-lea.

De asemenea, au fost gasite si alte sapte tratate ale matematicianului antic, inclusiv singurelecopii care au supravietuit din "Metoda teoremelor mecanice" si "Ostomachion", dar sidiscursuri ale oratorului grec Hyperides si un comentariu asupra lucrarii "Categorii" a luiAristotel, toate descoperite sub textul cartii de rugaciuni.

Manuscrisul este acum expus intr-o expozitie la Muzeul Walters si va fi publicat in aceasta

luna de Cambridge University Press. Editorul cartii, Michael Sharp numeste descoperireaMetodei teoremelor mecanice ca fiind un concept "foarte important pentru istoria matematicii

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si a stiintei".

George Titus Albulescu

Duminica, 30 Octombrie 2011, ora 10:48

Sursa: Ziare.comhttp://www.ziare.com/magazin/descoperire/palimpsestul-lui-arhimede-descifrat-scoate-la-iveala-secrete-vechi-de-2-200-de-ani-1130156

Comoara din pergamentul lui Arhimede - CulturaCultura - O echipa de experti internationali a descoperit ca pergamentele unei carti de

rugaciuni medievale ascund un text foarte important.

http://clujnapocainfo.com/cultura/comoara-din-pergamentul-lui-arhimede-cultura_62_311354.html

O echipa de experti internationali a descoperit ca pergamentele unei carti de rugaciunimedievale ascund un text foarte important.

In aceeasi carte au mai fost deja descoperite lucrari ale lui Arhimede si ale politicianuluiHyperides.

Insa acum, tehnologia de ultima ora a relevat faptul ca pergamentele mai ascund inca un text,cel de-al treilea - care este un comentariu despre Aristotel.

Seful echipei de cercetatori, William Noel, a numit descoperirea ca fiind "senzationala".

Cartea de rugaciuni a fost scrisa in secolul al XIII-lea de catre un scrib numit John Myronas.Insa acesta, in loc sa utilizeze pergamente noi pentru lucrarea sa, a ales sa foloseasca paginidin cinci carti pe care le avea la indemana.

Dr Noel, curatorul manuscriselor de la Muzeul american Walters si co-autor al unei viitoarecarti despre palimpsestul lui Arhimede, a declarat ca "scribul a folosit o metoda brutala dereutilizare a pergamentelor, insa probabil ca nu avea ce face, nu putea cumpara altele, asa caa reutilizat ce avea. A luat niste carti din raft, a curatat textul, si a scris peste, pentru a obtineo noua carte".

In 1906, s-a descoperit ca una dintre aceste carti "reciclate" continea lucrari unice ale luiArhimede, iar in 2002, tehnologiile moderne nu numai ca au oferit o imagine mai clara acuvintelor faimosului fizician, dar au scos la iveala un alt text, si anume singurul manuscris allui Hyperides, un politician atenian care a trait in secolul al IV-lea d.H.

"Ne-am gandit atunci ca acest palimpsest este o mina de aur, insa iata ca s-a intamplat ceva si

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mai senzational" a declarat Noel pentru BBC.

Una dintre cartile reciclate s-a dovedit a fi foarte greu de citit, a explicat Roger Easton, profesor al Institutului de Tehnologie de la Rochester SUA, care a declarat ca se utilizeaza pentru cercetari "o tehnologie multispectrala"

In prezent, o echipa internationala de experti studiaza inscrisurile si o serie de indicii, cum ar fi numele din coltul paginilor, i-a determinat pe acestia sa traga o concluzie: "pasajul filosoficdin palimpsestul Arhimede este identificat acum ca un comentariu al lucrarilor lui Aristotel" adeclarat profesorul Netz.

El a declarat ca aceste Categorii ale lui Aristotel au servit ca fundatie a studiului logicii de-alungul secolelor.

Ulterior, echipa de experti a stabilit ca cel mai probabil, autorul acestor comentarii de pe pergament este Alexandru al Aphrodisias. Profesorul Robert Sharples de la o universitate din

Londra este de parere ca "avem acum ocazia sa studiem comentarii foarte importante alecontemporanilor lui Aristotel"

Dr. Noel a mai declarat ca "nu exista filosof mai important decat Aristotel. Si a avea texte sicomentarii asupra lucrarilor sale care dateaza din secolul al-II-lea sau al III-lea este fantastic.Aveam pana acum o carte care continea trei texte antice care ne ajutau sa intelegemmatematica si politica, iar acum avem si despre filosofie. Nu am cuvinte sa-mi exprimentuziasmul ca avem ocazia, dupa atata vreme, sa ne aflam in fata unei comori nepretuite".

http://clujnapocainfo.com/cultura/comoara-din-pergamentul-lui-arhimede-cultura_62_311354.html

The Archimedes Palimpsest

Palimpsest Exhibition Digital Community Links 

http://archimedespalimpsest.org/about/

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About the Archimedes Palimpsest

The Archimedes Palimpsest is a medieval parchment manuscript, now consisting of 174

 parchment folios. While it contains no less than seven treatises by Archimedes, calling it theArchimedes Palimpsest is a little confusing. As it is now, the manuscript is a Byzantine

 prayerbook, written in Greek, and technically called a euchologion. This euchologion wascompleted by April 1229, and was probably made in Jerusalem.

The prayer book, or Euchologion, is itself of some interest, and further information on itscontents can be discovered in this website. However, to make their prayer book, the scribesused parchment that had already been used for the writings of other books. The books theytook parchment from were as follows.

The Book's Contents

Firstly, and most importantly, they used a book containing at least seven treatises byArchimedes. These treatises are The Equilibrium of Planes, Spiral Lines, The Measurement of the Circle, Sphere and Cylinder , On Floating Bodies, The Method of Mechanical Theorems, and the Stomachion. Of these treatises, the last three are of the greatestsignificance to our understanding of Archimedes. While the other treatises had survivedthrough other manuscripts, there is no other surviving copy of On Floating Bodies in Greek – the language in which Archimedes wrote, and there is no version in any language of The

 Method of Mechanical Theorems and of the Stomachion. The Archimedes manuscript wasused for the majority of the pages of the prayer book. The Archimedes manuscript was

written in the second half of the tenth century, almost certainly in Constantinople.

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In this short video clip Abigail Quandt, Senior Conservator of Manuscripts and Rare Books atThe Walters Art Museum, turns some pages of the book. It might surprise some people thatAbigail is not wearing gloves. Actually it makes conservation sense. Her clean hands do nodamage to the parchment and she can be more sensitive to the fragile folios if she is not

wearing anything on them.

Another book they used, we now know, contained works by the 4th century B.C. Attic Orator Hyperides. Prior to the discovery of the Hyperides text in the manuscript, this orator was onlyknown from papyrus fragments and from quotations of his work by other authors. The

 Archimedes Palimpsest , however, contains 10 folios of Hyperides text.

Yet further books were used to make up the Palimpsest. Fourteen folios come from anotherwise unknown third century commentary on Aristotle's Categories, and 20 further folioscome from four other books, two of which have yet to be identified.

How Was it Made?

How does one make one book out of seven? The answer is by palimpsesting them. ThewordPalimpsest comes from the Greek Palimpsestos, meaning "scraped again". Medievalmanuscripts were made of parchment, especially prepared and scraped animal skin. Unlike

 paper, parchment is sufficiently durable that you can take a knife to it, and scrape off the text,and over write it with a new text. In this case, seven books were taken apart, the text wasscraped off the leaves, which were then stacked in a pile, ready for reuse.

Medieval manuscripts are constructed like a whole series of newspaper. So each leaf has itsconjoint, just as the front page of a news paper is attached to the back page. One shouldtherefore imagine the leaves of the seven manuscripts as double-sheets, lying in a pile,without any text. To make the prayer book, these leaves were split down the middle, rotatedninety degrees and then refolded to make further double sheets that were half the size. The

scribes then added their prayer book text, which is at ninety degrees to the now almostindecipherable erased writings.

What you see when you open the Archimedes palimpsest therefore, is not a mathematicaltext, nor even a piece of Greek oratory, but a prayer book. Only occasionally can one justdiscern, at right angles to the prayer book text, the erased writings that the current project isattempting to recover.http://archimedespalimpsest.org/about/

The History of the Archimedes Manuscript

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http://archimedespalimpsest.org/about/history/index.php

At 2pm on October 29th, 1998, at Christie's auction house in New York, a very special old book was sold to an anonymous collector for $2,000,000. This collector deposited themanuscript at The Walters Art Museum in Baltimore in order to conserve it, image it, and

study it. The book is special because it contains seven treatises by the ancient Greek mathematician Archimedes. Two of these treatises, The Stomachion and The Method existnowhere else in the world. This book is also the unique source for Archimedes' treatise OnFloating Bodies in the original Greek. The Archimedes Palimpsest, as this book is called, hastrue claims to greatness: it is the earliest surviving Archimedes manuscript by about 400years; it is the most important source for the diagrams that Archimedes drew in the sand inSyracuse, in the third century B.C. It is by far the most important evidence we have for thegreatness of Archimedes. And Archimedes was a very great man.

Paleography, or the study of ancient texts, can allow us to approximately date whenmanuscripts were written. The Archimedes manuscript was probably written in the second

half of the tenth century. It was almost certainly written at Constantinople, for the simplereason that there is no other place that we know of where ancient mathematics wassystematically studied and copied. Constantinople was the one place with a continuedtradition of copying and preserving ancient texts from antiquity through the Middle Ages.

Specifically, the study of Archimedes texts can be associated with the work of Leo theGeometer. Leo the Geometer was the cousin of John VII Morocharzianus, who was Patriarchin Constantinople between 837 and 843. In the 820's, Leo was giving private instruction inConstantinople. Evidently he was successful at inspiring his students: one of them, who hadread Euclid under his supervision, was captured by the Arabs in 830. His report of Leo'slearning was sufficient to cause the Caliph to invite Leo to Baghdad. He did not go. Insteadhe took up the charge of the Byzantine Emperor Theophilus (829-842) to educate the publicin the church of the Forty Martyrs in Constantinople.

Leo was clearly something of a polymath, and a practical one at that. While in Theophilus's service, he built fire stations between the City andthe border of the Empire. Should there be an emergency on the border north of Tarsus, amessage could reach the Capital in less than an hour. In the Late 850's the assistant Emperor,Bardas, founded a school in the Imperial Palace, under Leo's direction. Other professors wereappointed too: Cometas, a literary scholar, Theodegius, an astronomer, and, perhaps mostsignificantly for us Theodore, a geometer. We know few of the details of Leo's school, but wecan assume that it was a center of learning. Two surviving manuscripts containing texts byArchimedes contain inscriptions praising Leo the Geometer. It seems highly likely that it wasas a result of his work that manuscripts of Archimedes were copied in this period.

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The ninth and tenth centuries were glorious centuries for the Byzantine Empire.Constantinople was immensely wealthy, and physically secure. The imperial palace was acenter of culture, and its monasteries flourished.

This is the climate in which it is easiest to see the Archimedes manuscript being copied.

However, the long period of prosperity ended abruptly in 1204. In this year, the FourthCrusade, sanctioned by Pope Innocent III, set out for the Holy Land. However, they stoppedshort of their goal, and sacked Constantinople. Constantinople was the richest City in Europe,and for over 700 years it had been a safe haven for ancient texts.

the sack of Constantinople

But the years after the sack of Constantinople were not years in which there was a great need

for the advanced mathematical treatises of Archimedes, or the Ancient speeches of Hyperides, or a Commentary on Aristotle’s Categories.

While it may well have been in the aftermath of these events that the seven manuscripts were palimpsested to make the prayerbook, the prayerbook was not made in Constantinople. Infact it was almost certainly made in Jerusalem. We know this because many of the prayersare specific to the rites of the Church in Jerusalem and its immediate environs. Quite whatthese rare manuscripts were doing in Jerusalem, or how they got there, is not at all clear.There was a lot of travel between the Holy Land and Europe at this time, not least because of the crusades.

In 2002, Professor John Lowden of the Courtauld Institute, using Ultra-violet light, managedto decipher a colophon, on the bottom of folio 1 verso of the manuscript, which contains thedate of April 13, 1229. This is almost certainly the day upon which the prayerbook wasfinished. It was also just a few weeks after Frederick II, Holy Roman Emperor, Stupor Mundi, released Jerusalem from Muslim control. Clearly the political climate was volatile.Under such circumstances parchment might have been in very short supply, and this would

 be one reason why the scribe of the prayerbook might have recycled the parchment of earlier manuscripts.

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The manuscript survived as a prayer book from that dayuntil it was catalogued by Papadopoulos-Kerameus.

In 1899, this scholar produced a catalogue of the manuscripts which belonged to the Greek  patriarch in Jerusalem, but which were housed at the Metochion – or daughter house – of theHoly Sepulcher, in Constantinople. The book is Ms. 355 in this catalogue. One detail thatPapadopoulos records, and which no longer survives, is that the book contained a sixteenthcentury inscription saying that it belonged to the monastery of St Sabbas.

Traditionally founded in 483 by St. Sabbas, this monastery wasan intellectual and spiritual center in the Holy Land at an early date. It is situated a few milesSouth of Jerusalem and directly east of Bethlehem on the West Bank. The community at Mar Saba had a well organized scriptorium for writing books, some of them lavishly illuminated,at least into the twelfth century, and in 1834 there were more than 1000 manuscripts in theLibrary. The monastery is spectacular, and looks as much like a fortress as a house of God, anecessity in the troubled times that the community has faced through the centuries. A moststriking account of the monastery is given by the Rev. George Croly, who, accompanied by

the artist David Roberts of the Royal Academy, arrived at the Monastery of St. Saba on April4, 1839. Croly records, "The immediate approach to the convent is striking....It was nightwhen after having descended into the bed of a ravine, where the Kidron passes to the DeadSea, and arriving at the foot of the Mountain of St. Saba, we saw the convent above us, by theuncertain light of the moon. It looked a lofty and colossal structure, rising in stories andterraces, one above another, against the sides of the mountain to its summit, and therecrowned with clouds. An old white-bearded monk, leaning on his staff, was toiling up theside of the hill leading a long procession of devotees. Below, apparently growing out of therock, was a large palm tree said to have been planted by the hands of the Saint in the fourthcentury. History, and probably legend, contributed its share to the effect. In a chapel behindan iron grating in one of the grottos was a pile of skulls. The tradition of the convent said theywere those of hermits who, to the amount of several thousand, had been slaughtered by theOsmanlis. We ascended the flight of steps, climbed up a ladder, crept through a small door 

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only large enough to admit one at a time, and found ourselves in an antechamber, surrounded by above a hundred Greek pilgrims….It was Passion Week. The monks receive strangerswith courtesy, and they not merely permitted the artist to sketch their chapel, but as their service was beginning before he had finished his design, they would not suffer him to layaside his pencil."

We do not know when the Palimpsest got to St Sabbas, but it is clear that it was there in thesixteenth century. It is also clear that it had moved again by about 1840, and was in theMetochion by that time. The Biblical scholar Constantine Tischendorf visited the Metochionin the early 1840's and wrote an account of his travels entitled "Travels in the East" in 1846.He says that he visited the Metochion, but found nothing of particular interest except for a

 palimpsest containing some mathematics. Clearly Tischendorf found this book veryinteresting, as one leaf from the Archimedes Palimpsest was sold to Cambridge UniversityLibrary in 1876 from his estate.

(left) It is now C.U.L. Ms. Add. 1879.23. It was onlyidentified as coming from the Archimedes Palimpsest by Nigel Wilson in 1968.

Tischendorf, of course, did not know that the palimpsest contained the writings of Archimedes, and neither did Papadopoulos-Kerameus in 1899. However, Papadopoulos-Kerameus did transcribe a few lines of the under text. These were called to the attention of 

John Ludwig Heiberg, who was the world's authority on Archimedes. Heiberg recognizedthat the under text was from Archimedes’ treatise Sphere and Cylinder, and that this was a previously unreported manuscript of Archimedes that he needed to see. Heiberg visited theMetochion in 1906, and discovered the truth, that this book contained seven treatises byArchimedes, including the unique source for The Method , The Stomachion, and On Floating 

 Bodies in Greek.

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Heiberg took photographs of the manuscript (right), and usedthese extensively for his work on the book. Heiberg incorporated his findings into an entirelynew edition of the complete works of Archimedes, which he published between 1910 and1915.

It is not known how the Palimpsest left the Metochion after Heiberg last studied it in 1908. Itwas auctioned at Christie's in New York on the 28th October 1998, and advertised as from a

 private French collection. The day before the sale the Greek Government and the Greek  patriarch issued an injunction against Christie's in an attempt to stop the sale. They arguedthat the book was stolen. The injunction failed, and the sale went ahead. The court records of the injunction and subsequent proceedings make it clear that the manuscript had been in theFrench collection at least since the 1960's, and the family claimed that it had in fact, belongedto them since the 1920's. Be that as it may, the book has suffered greatly since the time whenHeiberg saw it.

The photographs of the Palimpsest in 1906 are compared with the current state of themanuscript

The damage comes in three main forms

Firstly, some pages are missing. The most important are three missing pages that oncecontained Archimedes text. We know that they were there in 1908 as Heiberg transcribedthem, and even took a photograph of one of them. They are simply not there now.

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Secondly, the book has suffered very severely from mold. Medieval manuscripts tend to bestrong. They are made of the same raw material as leather shoes. Fire and damp can damagethem however, and this book has been very severely attacked by mold, as comparison

 between images that Heiberg took and pictures of the same pages now reveal. It is very oftenthe case that whole areas of text are now missing.

Finally, and most extraordinarily, four paintings of the Evangelists have been added to the

 book, over the top of the prayer book text, and therefore over the top of the under texts beneath that. These images were clearly made after 1929, as John Lowden has shown thatthey were copied from a publication of that year entitled Manuscrits Grecs de la Bibliotheque

 Nationale.

The manuscript was bought at auction by an anonymous American collector who depositedthe book at The Walters Art Museum, Baltimore, for conservation, imaging, and scholarlystudy, in January 1999. Work on the Palimpsest, funded by the owner, has been ongoing ever since.

http://archimedespalimpsest.org/about/history/index.php

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The Conservation of the Archimedes

Palimpsest

http://archimedespalimpsest.org/about/conservation.php

The conservation of the Archimedes Palimpsest is undertaken by Abigail Quandt, Senior Conservator of Rare Books and Manuscripts at the Walters Art Museum in Baltimore. Her first priority was to ensure the continued safety of a very fragile historical document. Her second priority was to prepare the manuscript for imaging. This entailed disbinding themanuscript, as the under texts run through the gutter of the book. Work on disbinding themanuscript started in February 2000, and finished in November 2004.

The Disbinding ProcessWhen the manuscript was deposited at Christie's for sale, it lay loose in its binding. For the

 purposes of safe handling the book was given a conservation binding, that kept the leavessecure between its boards, but which did nothing to alter the original construction.

Having disassembled the modern construction, and having revealed the structure of the spineas it was before the sale at Christie's, Quandt could begin to separate the quires in themanuscript from each other.

To start with all went according to plan. But then Quandt hit an unexpected problem. Part of 

the spine had been glued up in the second half of the twentieth century with a modernadhesive somewhat similar to Elmer's Wood Glue. As Quandt explains, this is actuallytougher than the parchment to which it is applied, and extremely difficult to remove.

This has been the principal challenge, and the main reason why the disbinding of themanuscript has taken such a long time.

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Once separated, the leaves had to be prepared for imaging. One of the first tasks was todiscover the chemical in the parchment, and to discover how stable, or unstable, themanuscript actually was. This work was undertaken by the Canadian Conservation Institute,which has provided expert advice throughout the project. Here you see a core sample, the sizeof a pinhead, taken through the manuscript.

You can see the Archimedes text isnothing more than a stain at the top of the parchment. The parchment itself is made up of 

collagen. The collagen here is reasonably sound. However in moldy areas, it is disintegrating.Abigail Quandt's work on the Archimedes Palimpsest is ongoing. Here you can see hereworking on one of the separated leaves.

Here are some of the procedures that Quandt undertook on the leaves, illustrated withsnapshots.

Documenting the condition of the individual leaves.Quandt draws maps of each leaf, noting where there are problems, be they mold, wax, glue,or modern residues, such as blue tack and paper.

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Mechanically removing the dirty waxfrom the parchment. The manuscript had been used for several hundred years as a prayer 

 book, and it was used by candlelight. Large areas were covered in wax, and it was necessary

to remove the wax that obscured the under text.

 

Unfolding damaged areas to reveal text. Pages of the manuscript were particularly fragile atthe margins an in the gutter of the prayer book. Sometimes the parchment needed to becarefully unfolded to reveal areas that were subsequently imaged. In the illustration below,Ms. Quandt works on a small damaged area in the gutter. You see the area before treatment,during treatment, after treatment, and under ultraviolet light. The last image makes clear theArchimedes text, running vertically down the page.

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Some areas were so fragile that they needed to be carefully supported using Japanese paper,to stabilize them before they were imaged.

 The pages with the forgeries were particularly problematic, as they had been deliberatelydistressed, and in some cases paper guards had been glued onto them. These guards needed to

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 be removed before the text could be successfully imaged.

Some stubs of once complete pages are particularly problematic. Here is a processed image of a stub. Quandt has removed the paper covering the left of this stub, but the paper on the right has yet to be taken off.

All the detritus from the Archimedes Palimpsest is meticulously logged and stored.

Finally the leaves are very carefullyand locally humidified. Since the book is made of animal skin, the individual leaves do not lieflat like paper. To image the manuscript successfully, it was necessary for Quandt slightly torelax the parchment.

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After each leaf was imaged, Abigail undertook further work to stabilize them and preservethem for the future. Finally each leaf was encased in a double sided frame. It is in these

frames that the leaves are shown in the exhibition, and the leaves will stay in them when themanuscript is finally returned to its owner.http://archimedespalimpsest.org/about/conservation.php

Multi-Spectral Imaging of the Archimedes

Palimpsest

http://archimedespalimpsest.org/about/imaging/

The Multispectral Imaging of the Archimedes palimpsest was undertaken by Keith Knox, of the Boeing Corporation based in Maui, William A. Christens-Barry of Equipoise ImagingLLC, and Roger Easton, Professor of Imaging Science at RIT. Here, Roger Easton gives anoverview of the imaging program.

Electromagnetic Radiation

To explain multispectral imaging, we must make a short digression into electromagnetic

radiation. Radiation can be understood as waves of energy traveling through space. Radiationcan vary in its intensity (that is how many waves are received or emitted with a certain

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 period) and in their wavelength (that is, the distance between their peaks as they travel). Theshorter the wavelength, the more energy the radiation has. At one extreme of theElectromagnetic spectrum are long wave radio waves that have the wavelength of football

 pitches. At another extreme are Gamma waves, which have a wavelength much shorter thanthe size of a single atom. For a short interval somewhere between these extremes is visible

light. The only reason that it is visible light (often called "white light") is because humanshave evolved to recognize it that way via their eyes, through the process of Natural Selection.Isaac Newton demonstrated that white light is actually made up of different spectra; red,green and blue. Thus, white light is often referred to as RGB light. At slightly longer wavelengths is infra-red light, invisible to humans but received by your skin, and recognizedas heat by your body.

At slightly shorter wavelengths is the more energetic ultraviolet (UV) light, which is alsoreceived by your skin, and is energetic enough to damage it. That is how you get sunburn.Electromagnetic radiation interacts with matter in different ways, depending upon itswavelength. It is quite possible that something which is invisible in RGB light will be visible,

to a camera anyway, in Infra-red light.

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Multispectral Imaging

Multispectral imaging is a digital imaging technique. Numerous photographs of an area aretaken at different wavelengths of light, resulting in a digital "stack" of images. Algorithms(recipes if you like) are then written in order to enhance particular characteristics of theimaged area. In the case of the Palimpsest, obviously, we wanted to bring out the under text.

The immediate results appear to be spectacular. Above is an image of folio f.70v. Theimagers succeeded in separating the spectral signature of the Archimedes ink from the

 parchment underneath it, and that of the prayer book on top of it. To bring out theArchimedes text even more, they then made the prayer book ink look like the parchment.Below, one can clearly see areas of text and diagrams that are invisible, or at least extremely

hard to discern, under RGB light.

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Despite the extraordinary appearance of these pictures, and the potential that they show for these advanced imaging techniques, the scholars were dissatisfied with the results.

1. The images appeared blurry. They were not out of focus, but rather it proved difficultto register the images accurately. To take images in different wavelengths of light,different filters have to be placed in front of the lens of the camera. Unfortunately, thelens affected the path of the light, and altered the size of the image. When one imagewas combined with another, they did not line up correctly. This is not a problem whenone is imaging large areas from space, but it is a problem when one is imaging thedetails of the Byzantine minuscule script in which the Palimpsest was written.

2. There were lots of "artifacts" in the image. The images that the scientists produced

were highly processed. When one manipulates images, one necessarily createsartifacts or "noise." Again, when trying to read the tenth century script this was agreat hindrance.

3. The imagers were imaging at about 300 dpi, which the scholars found insufficient totheir needs. The scholars liked to “blow-up” individual characters to the size of their computer screens when they try to decipher the text, and to do this successfully, theimages needed to be of a higher resolution.

4. The main problem was the most unexpected. The imagers were working on theassumption that they were supposed to “strip away” the prayer book text. However,

the scholars pointed out that there was no point in stripping away the payer book textunless one could literally read the Archimedes text underneath the individual prayer  book characters. It was merely confusing to them to make the prayer book text look like parchment.

The Initial Experiments on the Archimedes Palimpsest were extremely useful in many ways,as Bill Christens-Barry explains, and a fruitful dialogue developed between the imagers andthe scholars as to how best to use the available technology. However, they were of little usein deciphering the palimpsested text in themselves, and, when imaging on the Palimpsest

 began in earnest, completely different wavelengths were used, and completely differentalgorithms developed.

http://archimedespalimpsest.org/about/imaging/

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http://archimedespalimpsest.org/slideshow/#!prettyPhoto

Archimedes of Syracusehttp://archimedespalimpsest.org/about/history/archimedes.php

Archimedes was born in the city of Syracuse on the island of Sicily in 287 BC. He was the son of an astronomer and mathematician named Phidias. Asidefrom that, very little is known about the early life of Archimedes or his family. Somemaintain that he belonged to the nobility of Syracuse, and that his family was in some wayrelated to that of Hiero II, King of Syracuse.

In the third century BC, Syracuse was a hub of commerce, art and science. As a youth inSyracuse Archimedes developed his natural curiosity and penchant for problem solving.When he had learned as much as he could from his teachers, Archimedes traveled to Egypt inorder to study in Alexandria. Founded by Alexander the Great in 331 BC, Alexandria had, byArchimedes' time, earned a reputation for great learning and scholarship.

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Euclid was one of the most well-knownscholars who lived in Alexandria prior to Archimedes' arrival in the city. Euclid was arenowned mathematician, perhaps best remembered for collecting all of the existent Greek geometrical treatises and assembling them in a logical and systematic order in his book, “TheElements.” This compilation was fundamental to the study of geometry for over 2,000 years,and undoubtedly influenced the work of Archimedes.

After his studies in Alexandria, Archimedes returned to Syracuse and pursued a life of thought and invention. Many apocryphal legends record how Archimedes endeared himself toKing Hiero II, discovering solutions to problems that vexed the king.

 Archimedes' Screw

One such story recounts how a perplexed King Hiero was unable toempty rainwater from the hull of one of his ships. The King called upon Archimedes for assistance. Archimedes' solution was to create a machine consisting of a hollow tubecontaining a spiral that could be turned by a handle at one end. When the lower end of thetube was placed into the hull and the handle turned, water was carried up the tube and out of the boat. The Archimedes Screw is still used as a method of irrigation in developingcountries.

The Puzzle of King Hiero's Crown

The "Eureka" Story

illustrated by Kevin Kallaugher.

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Read the Story » 

King Hiero had commissioned a new royal crown for which he provided solid gold to thegoldsmith. When the crown arrived, King Hiero was suspicious that the goldsmith only usedsome of the gold, kept the rest for himself and added silver to make the crown the correctweight. Archimedes was asked to determine whether or not the crown was pure gold withoutharming it in the process. Archimedes was perplexed but found inspiration while taking a

 bath. He noticed that the full bath overflowed when he lowered himself into it, and suddenlyrealized that he could measure the crown's volume by the amount of water it displaced. Heknew that since he could measure the crown's volume, all he had to do was discover itsweight in order to calculate its density and hence its purity. Archimedes was so exuberantabout his discovery that he ran down the streets of Syracuse naked shouting, “Eureka!” whichmeant “I've found it!” in Greek.

 Archimedes and the Defense of Syracuse

During Archimedes' lifetime Sicily was a hotspot for both geological and political events.The volcanic Mount Etna loomed threateningly over the island, while on all sides the titanicPunic Wars raged between Rome and Carthage. Situated strategically between the two great

 powers, Sicily naturally became an object of contention. Self preservation demanded that thekings of Syracuse negotiate with the great powers, and as a result the small city-state oftenfound itself allied with one against the other. Such was the case in 214 BC, when pro-Carthaginian factions within the city chose to side with Carthage against Rome. Shortlythereafter, legions of the Roman army sailed to Syracuse and laid siege to the city walls.

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King Hiero II had anticipated such an eventuality. Before his death in 216 BC, Hiero setArchimedes to work, strengthening the walls of Syracuse and modifying its great stronghold,the Euryelos fortress. Archimedes also constructed war machines to defend Syracuse.

When the Romans arrived under the command of the famed general Marcellus, Archimedeswas prepared. The Roman historian Polybius relates that Archimedes now made suchextensive preparations, both within the city and also to guard against an attack from the sea,that there would be no chance of the defenders being employed in meeting emergencies butthat every move of the enemy could be replied to instantly by a counter move.…huge beamswere suddenly projected at the [Roman] ships from the walls, which sank some of them with

great weights plunging down from on high; others were seized at the prow by ironclaws….drawn straight up into the air, and then plunged stern foremost into the depths….with great destruction of the fighting men on board, who perished in the wrecks….in realityall the rest of the Syracusans were but a body for the designs of Archimedes, and his the onesoul moving and managing everything; for all other weapons laid idle, and his alone werethen employed by the city both in offense and defense.

The Death of Archimedes

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For two years the genius of Archimedes repelled the Romans, enabling the city to survive thelengthy siege. Nevertheless, in 212 BC the forces of Marcellus prevailed and took the city.Marcellus had great respect for Archimedes, and immediately dispatched soldiers to retrievehis foe. Apparently, the great mathematician was unaware that his enemy had stormed thecity, so deeply were his attentions focused on a mathematical problem. When a soldier demanded Archimedes accompany him to the quarters of Marcellus he simply refused, andcontinued his ruminations. The enraged soldier flew upon Archimedes, striking the 75 year-

old eccentric dead. Marcellus was greatly distressed upon hearing the news of Archimedes'death, and ordered that he be buried with honors. Archimedes' tombstone was, as he hadwished, engraved with an image of a sphere within a cylinder, after one of his geometricaltreatises.

 Archimedes' Legacy 

Despite the many fantastic tales surrounding the life of Archimedes, we are most indebted tohim for his mathematical treatises and the contributions he made to the understanding of fundamental physical phenomena. Through the medium of geometry, he was able to elucidate

the principles for such basic devices as the pulley, the fulcrum and the lever – devices stillutilized today. Archimedes is also credited with the discovery of the principle of buoyancy, or the power of a fluid to exert an upward force on a body placed in it. His further research intovolume and density was fundamental to the development of theories of hydrostatics-the

 branch of physics dealing with liquids at rest.

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The story of the survival of Archimedes' treatises down to our own time is intricate andcomplicated, and has been traced in extraordinary detail. But an essential point is this: it isthrough three manuscripts that we know the texts of Archimedes treatises in Greek. One waslast heard of in 1311, a second was last heard of in the 1550s, and the third is TheArchimedes Palimpsest, now at The Walters Art Museum in Baltimore, and the subject of this website. Because this is just the start of a fascinating story.http://archimedespalimpsest.org/about/history/archimedes.php

Scholarship

http://archimedespalimpsest.org/about/scholarship/

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Scholarly study of the Archimedes Palimpsest began in earnest at the same time that AbigailQuandt began to take the book apart, in April of 2000. The principle scholars on the project atthe stage were Nigel Wilson, Reviel Netz, and Natalie Tchernetska. It was clear, even fromthis early stage, that the book had many secrets to reveal. And here Reviel talks about hisearly impressions, having studied the partially disbound manuscript. The big question that

everybody wanted to know was whether there was more to discover in the book than hadalready been discovered by the great Johan Ludvig Heiberg, 100 years earlier. Reviel Netzdiscusses some of the findings concerning the Archimedes manuscript in other pages of thewebsite.

The truly stunning discovery while working on the Archimedes Palimpsest however, was thatthe manuscript contained other erased texts from the ancient world, texts that were not byArchimedes at all. In 2002 Natalie Tchernetska identified lost speeches by an Attic orator called Hyperides.Hyperides was a contemporary of Demosthenes, and was one of the tencanonical orators of antiquity. However, it was thought that, unlike Demosthenes, the scrollsthat contained Hyperides speeches were never copied into manuscript books, and that

therefore his speeches were unknown in the Middle Ages. In fact the only direct speeches wehad of Hyperides until the year 2000 were papyrus scrolls discovered in thebes in Egypt inthe nineteenth century. So it was extraordinary that Natalie found a Hyperides manuscript inthe Archimedes palimpsest. It took several years to transcribe the text, and scholars from allover the world helped in its decipherment.

But the Palimpsest had yet to reveal all its secrets. In 2005 Nigel Wilson and Reviel Netz began to decipher yet another text in the manuscript. This turned out to be a second or thirdcentury AD Commentary on Aristotle’s Categories.

In conjunction with the exhibition of the Archimedes Palimpsest, which opened at TheWalters Art Museum on October 16, 2011, two of an anticipated 5 volumes have been

 published on the Archimedes Palimpsest by Cambridge University Press. The ArchimedesPalimpsest Volume I catalogues the various manuscripts in the Palimpsest, and provides aHistory of the project to retrieve its texts, a history of the manuscript, its conservation,digitization, and historical importance. The Archimedes Palimpsest Volume two providesimages and transcriptions of the unique texts in the manuscript.

Images of the Archimedes Palimpsest, and transcriptions of the unique texts were first published on the internet at archimedespalimpsest.net on October 28th, 2008. Many additionswere made to this set with a second release of the data, schedule for October 28th, 2011.

http://archimedespalimpsest.org/about/scholarship/

Overview: The Importance

of the Palimpsest to the Study of 

Archimedes

By Reviel Netz of Stanford University

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http://archimedespalimpsest.org/about/scholarship/archimedes-manuscript.php

In discussing the importance of the Palimpsest for our knowledge of Archimedes, care should be taken to distinguish the significance of what Heiberg read in 1906 (call this The 1906Discovery) from that of the further readings we make now (call this The 1998 Discovery).

For both discoveries, one is tempted to say that it is difficult to exaggerate their importance – only that their importance is often exaggerated. Some reports on the web suggest that nothingwas known of Archimedes prior to 1906, or that entire treatises came to light in 1998. Thereality is no less spectacular, but much more complicated. Here are the major facts.

First, we did possess the bulk of Archimedes' writings even prior to the 1906 Discovery, based mostly on a group of Renaissance manuscripts, all of them copied from a singleByzantine manuscript known as The Valla Codex, or Codex A. (The Valla Codex itself disappeared in the 16th century). Another important source was a Latin translation made fromanother Byzantine manuscript known as the Codex Mechanicorum, or Codex B. (This was

lost probably in the 14th century).

Still, The 1906 Discovery was a major event, for several reasons. First, note that Codices Aand B overlapped for a single work at most. The Palimpsest, on the other hand, overlaps withcodices A and B for several works: together with Codex A, it has a text of Spiral Lines,Sphere and Cylinder and Measurement of the Circle; together with Codex B, it has a text of Floating Bodies. While it is of course especially exciting to discover totally fresh works thatare unattested elsewhere, there is great value in having more than a single source for a givenwork: by the comparison between the two copies, our reconstruction of the original text can

 be made more precise. Thus The 1906 Discovery substantially furthered the preciseestablishment of the original text of Archimedes. Further, The Palimpsest did contain fresh,unattested works: the Method, as well as (in fragmentary form) the Stomachion. Indeed,many think of the Method as Archimedes' most interesting work. A special romantic aspectof this discovery lies in Archimedes' statement (which however should be taken with a grainof salt) that the Method contains what was, in some sense, his method of discovery. In other words, The 1906 Discovery appeared to provide a glimpse into the hidden process leading toArchimedes' Eureka moments. Finally, while the extant Latin translation of Floating Bodiesis very precise, we could never know that this was indeed the case prior to the discovery of The Archimedes Palimpsest . The Archimedes Palimpsest thus settled the text of FloatingBodies, and it is only thanks to The 1906 Discovery that we can have the originalformulation, in the Greek, of this treatise by Archimedes.

The 1998 Discovery is important for at least five

reasons.

• First, because it is there. To do any work on an ancient author, one needs to know thatall the evidence is available. The state of affairs following the disappearance of thePalimpsest in the 20th century was especially uncomfortable: we knew that a major source of evidence was (proably) extant, we knew that it was edited, partly – and weknew that it was no longer accessible. And so, for every statement one madeconcerning Archimedes, this worry returned: who knows, perhaps a close reading of the relevant passage of the Palimpsest could refute us? This state of affairs impeded

all study of Archimedes and so of ancient science in general.

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• Second, we now find that there are many corrections – numbering in the hundreds, if not thousands - to the text as printed by Heiberg. Perhaps the majority of these aresmall errors, undetected by Heiberg, made by the Byzantine scribe. These, then, donot change the text of Archimedes himself. In a good number of cases, however, suchcorrections add detail to the text of Archimedes. Sometimes such details are of great

historical value (To take a single example: in Heiberg's text Eudoxus was the first to"discover" a certain proof; we know now the text has Eudoxus as the first to "publish"it). All of those details are important for our understanding of the mind of Archimedesat work.

• Third, while Heiberg had full access to the diagrams in the Palimpsest, he did notcomment on them or publish them. Had it not been for The 1998 Discovery, then, thediagrams would have been simply lost. This is of special significance, as recentresearch has shown the importance of diagrams to science in general and to Greek mathematics in particular. The rediscovery of the diagrams in The Archimedes

 Palimpsest is probably the most important single contribution The 1998 Discovery

makes to our knowledge of Archimedes. (See: The Diagrams as Floating Bodies).

• Fourth, the text of the Method was to a large extent established by Heiberg. Thisachievement, given what we now know of his task, is truly incredible. Even so,certain gaps were left in his text, especially where Archimedes' own thought wasespecially original and therefore unpredictable. One of these gaps was found tocontain a use, by Archimedes, of actual infinity. This passage is unique in ancientmathematics, and surely counts as one of the most original and important passages inthe entire corpus of Archimedes. (See: Methods of Infinity).

• Fifth, Heiberg's reading of the fragment of the Stomachion was so fragmentary that no

conclusions could be made concerning its content. Recent and better readings,together with more recent developments in the scholarship of Greek mathematics,allow us now to provide a guess concerning the nature of the Stomachion. Possibly,this was a treatise in combinatorics – the first written, ever. (See: How Many Wayscan you have a Stomach Ache?)

All in all, The 1998 Discovery is the most important event in the study of ancient sciencesince – well, since The 1906 Discovery.http://archimedespalimpsest.org/about/scholarship/archimedes-manuscript.php

The Diagrams as Floating Bodiesby Reviel Netz of Stanford University

http://archimedespalimpsest.org/about/scholarship/floating-bodies.php

The Archimedes Palimpsest is the closest we get to the hand of Archimedes. It is not hishandwriting, indeed he would be unable to read it. Greek letters have changed so dramaticallyat around the Eighth century A.D. that an ancient author, such as Archimedes, would nolonger be able to recognize his own words. As far as the text is concerned, then, the

Palimpsest provides us with the content, not the form: we know what Archimedes has writtendown, but not how he written it down. The case may be different with the diagrams. It is

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completely feasible that the diagrams we possess in the Palimpsest are essentially the same asthose drawn by Archimedes in the Third century BC. This would have more than a mereantiquarian interest, i.e. this is more than just satisfying our curiousity concerningArchimedes' writing. Instead, knowledge of Archimedes' diagrams would be crucial to our understanding of his scientific personality. This is because in Greek mathematics, unlike

modern mathematics, the diagrams are much more than mere illustrations. Rather, Greek diagrams serve as a crucial part of the logic of the argument. Ancient mathematicians weremostly geometers, studying drawn forms. They thought in diagrams. The Archimedes

 Palimpsest , by bringing us closest to Archimedes' hand, also brings us closest to his mind.

How do we tell that the diagrams in the Palimpsest are, as it were, "originals"? The shortanswer is, we cannot tell for sure. It would be much easier to argue that diagrams are notoriginals. We could do this by comparing the Palimpsest to other surviving manuscripts. If the diagrams at the various manuscripts are radically different, this would raise the suspicionthat medieval scribes have ignored the diagrams present in their originals, and have insteadsimply drawn their own diagrams. In fact, this is – somewhat shockingly – how modern

editors and translators have systematically proceeded. That is, Heiberg not only did notattempt to draw his diagrams based on manuscript authority, but also he did not report howthe diagrams in his manuscripts look like. While his edition from 1915 furnishes us withimportant information concerning the text contained in the Palimpsest, we had no informationat all concerning the diagrams contained in the Palimpsest, prior to the resurfacing of themanuscript in 1998.

However, medieval scribes were not like that. For all the treatises preserved both in thePalimpsest as well as in the remaining manuscript traditions, we can see that the diagrams arein fact very closely allied.

They "share DNA". The medieval scribes did not re-invent their diagrams, but instead copiedthem, as best as they could, from their sources. The reason for this is in fact not hard to come

 by: medieval scribes, unlike modern editors such as Heiberg, did not understand anymathematics and certainly did not follow the contents of Archimedes' text (for which the bestevidence are the great many errors concerning labeling of figures in both text and diagrams).To the ignorant, there is safety in faithfulness.

The medieval scribe could not re-work Archimedes: instead, he has copied him. We should be grateful for that.

The one major exception appears to have been William of Moerbeke. His Latin translation of the works of Archimedes, dating from 1269, is the most significant effort by any MedievalEuropean author to have come to terms with the science of Archimedes. Moerbeke did notalways succeed (some of his translation betrays elementary errors), but he tried hard andusually creditably. Not surprisingly, his diagrams tend to differ from those preservedelsewhere: as part of his translation and his general effort to re-work his material, he has alsore-drawn his diagrams in relative freedom from the sources he had.

This is most important for the case of Floating Bodies. For this work, two sources survive:the Latin translation by Moerbeke – as well as the Palimpsest. To the extent that the twodiffer, we should assume that the Palimpsest provides us with the more accurate source to the

original form of the ancient diagrams. In point of fact, the manuscript diagrams often differ  between the two traditions. While scholars have yet to study the detail of such discrepancies

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so as to discover their precise meaning, the fundamental issue seems to be clear. In thistreatise, Archimedes considers several hydrostatic configurations, with solids of differentshapes and weights immersed into water in different arrangements.

The question is throughout: given the precise configuration, will the figure stay put, or will it

collapse through the combined action of gravity and buoyancy? Archimedes specifies,textually, the key components of the configuration, but some remain to be decided by thediagram (e.g., is the object immersed face-down, or face-up?). Whenever the Palimpsest andthe Moerbeke manuscript differ on such points, then, we have a substantial piece of thescience of hydrostatics, riding on our interpretation of the manuscript tradition. Without thePalimpsest, we could not even address such questions. Thanks to the Palimpsest, suchquestions are likely to be resolved over the next few years, as the diagrams are edited andstudied.http://archimedespalimpsest.org/about/scholarship/floating-bodies.php

Methods of Infinity

By Reviel Netz of Stanford University

http://archimedespalimpsest.org/about/scholarship/method-infinity.php 

The most fundamental intellectual question in all of mathematics is the nature of infinity. Inmany ways, this is what mathematics is about. Take the most ordinary objects, look at themthrough the infinitely powered magnifying glass that mathematics is – and you come to seeinfinity in one of its many (indeed, infinite) forms. Draw a circle – what is the ratio of itscircumference to its diameter? Right: in order to express this, you need a certain kind of infinity. (To simplify things a bit, this is because the number Pi has infinitely many, endlesslyvarying digits). Draw now a square, presumably a simpler object. But then again: draw thediagonal to that square. What is the ratio of the side to the diagonal? Once again, one needsinfinity, of a somewhat different kind, in order to express this. (To simply again: this involvesthe surprisingly complex nature of the square root of the number two). And now draw just aline, any line. How many points does it contain? Right again – infinitely many. Which kindof infinity? Curiously enough, mathematicians do not know the answer to this. The simplest

line, with the simplest question, already brings us to an enigma of infinity that mathematicsstill cannot solve.

The enigma of infinity has fed into all mathematical progress. Given that there are infinitelymany prime numbers, are there ways in which we can characterize how many precisely theyare? Much of number theory emerges out of this question. And even more fundamentally:given that curved figures involve infinitely complex ratios (akin to the nature of the number Pi), are there ways by which we can characterize, in general, curved lines and measure themas simply as we do straight lines? The attempt to answer this question animates most of Archimedes' works. He measures spiral lines, spheroids, conoids, spheres and parabolas,again and again extending the ways by which measurement may be affected on curves. Out of 

this endeavour would emerge the central line of development in the history of mathematics – from Archimedes' measurements, through early modern attempts to generalize his approach

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into a more systematic tool of calculation, giving rise, in the hands of Newton and Leibniz, tothe Differential and Integral and Calculus, the foundation of modern mathematics.

We may divide all of infinity into two major types, one to be called “potential infinity”, theother to be called “actual infinity”. By potential infinity something such as the following is

meant. Suppose you go into an auction, representing a buyer who mysteriously guaranteesyou that he has limitless funds, that is, he guarantees you that, however much the opponentsare offering, you are allowed to beat their offers. (According to one rumor, such was theauthorization given by the current owner of the Archimedes Palimpsest, going into theChristie's auction). They offer one million – and you offer two. They offer a billion – and youa trillion. However much they put up, you may put up more. You are the representative, then,of potential infinity. Why is this infinity merely potential? Because you are always making afinite commitment. In other words, you know that, if the auction is ever to end, then it willend with a finite commitment – however large that finite commitment may be. It may be amillion, a trillion, or it may be the number 1 followed by a trillion zeroes, but it will always

 be some finite number.

Potential infinity, then, may be viewed as an endlessly extendible, and yet forever finite,magnitude. Not so actual infinity. By actual infinity we mean something such as, say, thenumber of points on a line. If your buyer would have authorized you to offer as your price, inthat auction, a dollar for each point there is on a line, then he would have authorized you tomake an actually infinite bid. This is a very high bid indeed. After all, even the number 1,followed by a trillion zeroes, is still dwarfed by actual infinity. It is a curious bid, too. If ittakes you, say, a minute to pass a suitcase with a million dollars, than it will take you all of eternity to pay your bid actually infinite bid. Actual infinity actually never ends. And for thisreason many philosophers, throughout the ages, have doubted its very existence.

The established historical wisdom on infinity and the history of mathematics, then, used to gosomething like this. At first came the Greeks, who preferred above all to have precise,rigorous proofs. For this reason they have completely avoided the concept of actual infinity,concentrating instead on potential infinity (whose rigorous treatment was perfected byArchimedes in his measurement of curves). Next came the early modern mathematicians,who wanted to get results come what may. To obtain general result concerning curves, theyhave brought in the notion of actual infinity – paying the price that their mathematics was lessrigorous and precise compared to that of the Greeks. Along came the mathematicians of the19th and 20th centuries, slowly and laboriously building up a new kind of mathematics,where the concept of actual infinity is meticulously built up so as to have the same kind of 

rigorous foundations that the Greeks have provided for potential infinity.

The most important discovery made through the Palimpsest over the last few years involvedsome 12 lines of Greek, in an otherwise neglected proposition of Archimedes' Method. Theyhave changed no less than our understanding of how western mathematicians came to handleinfinity. To be sure, the Method was always considered, ever since the discovery of thePalimpsest in 1906, as the most significant contribution made by the Palimpsest to our knowledge of the history of mathematics. It is generally considered to be Archimedes' mostinteresting treatise. And it survives on the Palimpsest alone.

In it, Archimedes sets out to solve many separate problems using a variety of techniques,

most often one involving a striking combination of physics and of mathematics. For instance,two geometrical objects are considered simultaneously, say a triangle and a parabolic

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segment. Then Archimedes shows how each line in the triangle balances a line in the parabolic segment around a given fulcrum, so that it follows that the triangle as a whole, andthe parabolic segment as a whole, balance as well around that fulcrum. Since the property of “balancing” has a precise geometrical correlate (objects balance if and only if their magnitudes are reciprocally proportional to their distances), the discovery of the fulcrum

allows us to deduce the magnitudes of the triangle as well as the parabolic segment – in other words, we have gained, in a surprising fashion, the measurement of a curve – the parabolicsegment.

So much was known since 1906. This was striking indeed – but even more so was thediscovery made in 2001. In proposition 14 of the Method, it turns out, as Archimedes ismeasuring the volume of a cylindrical segment, he makes systematic reference – which

 previous readers could not decipher, in part because this was so unexpected – to actualinfinity itself. This pushes the mathematical use of actual infinity nearly some 2,000 years

 back in time.

Archimedes crucially needs to argue that the number of triangles inside a prism, as well asthe number of lines inside a rectangle, are equal to each other – making a statementconcerning an infinity which is not just potential but is, precisely, that kind of infinity of “thenumber of points in a line”. Archimedes even speaks of the “number” (plethos) of suchobjects, making him the first person ever to consider an infinite numerosity.

The curious twist is this: while we know today that Archimedes' statement is correct – thetwo sets of objects, both infinite, are indeed of the same kind of infinity – we are still no

 better than Archimedes in saying which kind of infinity this is. This – the number of pointson a line – is still the big unsolved riddle of mathematics. The enigma of infinity still hauntsus, staring us, as it were, at the face even from the pages of the Palimpsest.http://archimedespalimpsest.org/about/scholarship/method-infinity.php

How many ways can you have a stomach

ache?

By Reviel Netz of Stanford University

http://archimedespalimpsest.org/about/scholarship/combinatorics.php

A scholarly article published in early 2004 notes that "The Stomachion is something of a poor relation". This has not been the case recently. Indeed, besides the word "Palimpsest" itself,study of the Archimedes Palimpsest has now made this funny Greek word, too -"Stomachion" - into something of a household name. It probably never was so even inantiquity itself. The evidence suggests that antiquity knew a game called "Stomachion",always considered a tough challenge. The very name is not very well understood today, butapparently it refers to the difficulty of the game: our word "stomach" comes from the Greek,so that "Stomachion" means, literally, "that which relates to the stomach". All occurrences of 

the word in classical texts are either references to the game or (much more often) medicalreferences to a belly-ache. Most likely, then, the game literally meant "belly-ache". It appears

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that while modern gamers feel a "headache" with a difficult problem, their ancientcounterparts felt a "bellyache". Anatomy is a matter of history.

The headache was about this: take a square, divided into 14 pieces according to a pre-defined pattern. Now shuffle well the 14 pieces. And then try and put them together into a square.

This is surprisingly difficult.

What turned out to have been even more difficult – what has caused, as it were, even moreheadache – was making sense of Archimedes' foray into the game. We always knew he hassomething to do with this game, because some ancient authorities refer to the game simply as"Archimedes' Box". In 1899, an Arabic text was published based on a manuscript in Berlin,containing a brief passage claiming to derive from a work by Archimedes on the"Stomachion". And then, in 1906, the Palimpsest was discovered.

Here is where the Stomachion is in really bad luck. Archimedes' treatise on the subject wasapparently the last treatise in the original Archimedes manuscript, from which the Palimpsest

was made. Most of its pages were in bad shape already in the 13th century, and so werediscarded by the makers of the Palimpsest. Only a single page survived - the first one. Thus,we have the introduction to the Stomachion, together with a brief introductory theorem. Thisis in principle very useful – the introduction, after all, should state what the treatise is aboutand so should furnish us with crucial information – but worse is to come: the maker of thePalimpsest inserted this page right towards the end of the book. At around the 16th century,apparently, a few pages were lost from the end of the book (perhaps to a fire) and the pagecontaining the introduction to the Stomachion now became the very last page of the book.The result is that this is now among the most difficult pages in the book, indeed was verydifficult to read already when Heiberg came to look at it. Since the Arabic fragment is alsovery tantalizing (and is, anyway, no more than a very late compilation), the end result is that,up until recently, we could not put together this particular jigsaw puzzle. No one knew whatArchimedes' treatise was about. The standard view was, we shall never know.

The interpretation put forward in 2004, based on the new readings made possible thanks todigital technologies, is still tentative. Nothing better is possible, seeing that our evidence is sofragmentary. But this interpretation is widely considered to be very likely. This is quite aresult, too: it makes Archimedes into the first author of an entire field of mathematics, onethat until recently we thought no ancient author has ever worked in. This is the field of combinatorics.

Combinatorics, as its name suggests, is the study of thenumber of possible combinations.

Suppose you have three blue socks: how many pairs can you form?

Answer: three. Each choice of pair leaves exactly one sock "unmatched", so that thenumber of pairs is the same as the number of individual socks.

Suppose you have three green socks, too, and you wish to form bicolored pairs, a green and a blue in each. How many combinations?

Answer: nine. Each choice of blue is allowed, and so is each choice of green, allowingthe entire "multiplication table" of three-by-three, i.e. nine.

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This kind of study, we can immediately see, is a far cry from the world of geometrical curvesstudied so extensively by Archimedes and by his contemporaries. For this reason, no onethought of combinatorics in connection with the Stomachion, looking instead, in vain, for some interesting, strictly geometrical problem associated with the game.

More recently, Fabio Acerbi, an Italian historian of mathematics, has definitely proved theexistence of an interest in combinatorics in antiquity (even if later than Archimedes himself).This has allowed us finally to interpret the treatise. A fuller reading of the introduction hasshown that Archimedes was addressing the fact that there are many ways of solving the

 puzzle, given the possible ways of interchanging one solution for another. The evidencesuggests, therefore, that the treatise was a study in the possible substitutions of one solution

 by another, aimed at calculating the total number of such solutions.

Prodded by the Palimpsest team, modern combinatorists (Profs. Diaconis and Holmes of Stanford, together with Profs. Chung and Graham of UCSD) have tackled this question – howmany ways to solve the Stomachion puzzle – and have come up with the number 17,152 (or 

536 geometrically distinct solutions, each multiplied by 32 symmetries). We cannot tell, of course, whether Archimedes came up with the right answer. But from all we know of him,more likely than not, he did. An ancient game came back to life – and with it, an entireancient field of science.http://archimedespalimpsest.org/about/scholarship/combinatorics.php

Scholarship on Hyperides

http://archimedespalimpsest.org/about/scholarship/hyperides-intro.php

Although the Archimedes Palimpsest is famous for containing the text of Archimedes, manyfolios in the book do not come from the Archimedes manuscript at all. At the time of the saleof the manuscript at Christie's, the erased texts on these texts could not be read at all. But wedecided to treat these folios just like the Archimedes ones, and so we imaged them anyway.Slowly other texts started to be revealed. In 2002, Natalie Tchernetska discovered speeches

 by the Athenian Orator Hyperides. Five bifolia have now been discovered. Deciphering these pages has been a collaborative endeavor, involving very many classicists. It was anextraordinarily difficult enterprise, as you can see from these three "Fly-on-The Wall" videos,

of meetings held in London.

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However good the images are, the text remained extremely difficult to read. An important

 part of the process was interaction and feedback between the scholars, the imagers, and theconservators. Laszlo Horvath from Budapest came to study the manuscript in person, and heand Abigail Quandt discussed the arrangement of the leaves in the original Hyperidesmanuscript. Judson Herrman came also, during the imaging of the manuscript, and advisedthe imagers on capturing raking the Hyperides text with raking light.

 Now, in 2008, the Hyperides texts in the Archimedes Palimpsest have all been transcribedand published in three articles published in the Zeitschrift fur Papyrologie und Epigraphik,

listed in the Bibliography. The authors of these articles are, in sum, and in alphabetical order Colin Austin, Chris Carey, Mike Edwards, Zoltan Farkas, Eric Handley. Judson Herrman,Laszlo Horvath, Gyula Mayer, Tamas Meszaros, Peter Rhodes, and Natalie Tchernetska. Thislist by no means exhausts the many who lent a hand in what was an extraordinarilydemanding task done in a remarkably short anount of time.http://archimedespalimpsest.org/about/scholarship/hyperides-intro.php

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The New Hyperides in the Archimedes

Palimpsest

by Judson Herrmanhttp://archimedespalimpsest.org/about/scholarship/hyperides-texts.php

The Archimedes Palimpsest preserves five leaves from a Byzantine manuscript of thespeeches of the classical Athenian speech-maker and politician Hyperides. This material is

 previously unknown, and its startling discovery by Natalie Tchernetska (see below for detailsof her initial publication) holds great promise, both for our knowledge of Hyperides as anorator and also for our understanding of the history, politics and law of fourth-centuryAthens.

Hyperides lived from 390/389 to 322, and was thus nearly a contemporary with better knownfigures such as Demosthenes and Aristotle (both lived 384 to 322). He was a rhetor , or orator. Athenian rhetores were the most prominent politicians in the fourth-centurydemocracy. They made speeches at public meetings of the citizen assembly and they servedas prosecutors and defendants in the courts. Some of these court-room speeches werecomposed for trials concerning private criminal or civil issues, while other cases had morefar-reaching political implications. As was typical for rhetores at the time, Hyperides wrotespeeches both for others to deliver in minor private cases and he also spoke in person atimportant political trials. Speeches of both types (the distinction can become blurry;sometimes private cases have political overtones) by Hyperides survive. Especially at the

 beginning of his career, he was active as a logographos, or hired speech-writer, who wouldserve almost as a private lawyer and write material for clients to deliver in court. As a

 politician, Hyperides was long allied with Demosthenes in opposition to the expansion of theMacedonian empire under Philip II and his son Alexander the Great. Hyperides came to

 prominence in 343 as the prosecutor of Philocrates for negotiating too conciliatory a peacewith the Macedonian king Philip in 346. Twenty years later, when Alexander the Great diedin 323, Hyperides was an important advocate for rebellion against the Macedonians. Other Greek allies joined the Athenians in taking up arms against Alexander's generals. The so-called Lamian War ended badly for the Greeks in 322, and Hyperides was given the greathonor of delivering the public funeral speech for the Athenian war-dead in Spring of 322 (a

 papyrus copy of the speech survives, see below). Later that year, after the Macedonians had

finally defeated the Greeks and put an end to the political independence of Athens, Hyperideswas rounded up and executed by the Macedonians for his part in the failed rebellion.

After his death, Hyperides' speeches were widely read and admired. He was one of the tencanonical orators recognized by later rhetorical critics. The great literary critic Longinus,active in the first century AD, favorably compared Hyperides with Demosthenes, theacknowledged master of Attic oratory. Longinus describes (On the Sublime, chapter 34)Hyperides as a pentathlete, who displays an extraordinarily well-rounded versatility and

 performs well in many different areas. He praises Hyperides' simple and charming style,seasoned with sarcasm, irony, and wit. Longinus singles out a number of speeches, includingthe famous funeral oration described above and a defense speech for the courtesan Phryne, in

which Hyperides, at least according to later tradition, notoriously bared the breasts of hisclient and overwhelmed the jurors with her beauty. Longinus wasn't the only reader of 

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Hyperides during the first centuries of the Roman Empire. A second-century AD biographyof Hyperides reports that seventy-seven extant speeches circulated under his name (ps.-Plutarch, Lives of the Ten Orators 849D).

These copies of Hyperides' speeches circulated as papyrus rolls. By the fourth century AD a

newer format had become common. These codices, made out of papyrus or parchment andmore or less equivalent in design to modern books, supplanted the older format of rolls.These new books were easier to use and rolls soon fell out of general use. Moreover, the

 papyrus rolls were less durable than codices and authors who were not recopied in the newformat were at risk of disappearing forever. Such was the fate of Hyperides, it was assumeduntil the discovery of the five Byzantine leaves in the Archimedes Palimpsest. We have atantalizing report of a Byzantine edition of Hyperides (the sixteenth-century humanist JohannAlexander Brassicanus claimed that he saw a "complete Hyperides with rich scholia" in thelibrary of king Matthias Corvinus of Hungary), but Nigel Wilson has persuasively argued thatno complete edition of Hyperides existed in the Byzantine period (Greek, Roman and 

 Byzantine Studies 16 (1975) 99-100), and his pronouncement has created "a new orthodoxy

of scepticism" (D. Whitehead, Hypereides, p. 2).

Until these new leaves turned up in the Archimedes Palimpsest, it appeared that only shortexcerpts of Hyperides survived beyond the classical period. Byzantine reference works, suchas the Suda and Photius' Bibliotheca and Lexicon, made frequent reference to Hyperides.They preserved dozens of short quotations by the orator, but it was assumed that their sourcefor this material was not a complete Byzantine edition, but rather earlier compilations anddictionaries, such as Athenaeus and Harpocration's second-century AD Lexicon of the TenOrators. Now we have little reason to doubt Photius' claim to have read several speeches of Hyperides (Photius, Bibliotheca codex 266; cf. Wilson, Scholars of Byzantium, p. 95). Beforethe discovery of more extensive material on papyri, and now in the Archimedes Palimpsest,the surviving corpus of Hyperides consisted of a few hundred short quotations, manyconsisting of only a single word, and none longer than a few lines.

In the nineteenth and twentieth centuries numerous important classical works werediscovered on papyrus manuscripts, including four sizeable manuscripts of Hyperides(discovered between 1847 and 1891). These papyri were written in Egypt, and the date of composition ranges from the second century BC to the second century AD. They confirmHyperides' popularity in the ancient world and they have reconstituted a corpus for the orator that now fills more than fifty pages in a modern printed edition. We now have extensivesections from five court-room speeches (one survives complete) and also Hyperides' famous

 Funeral Oration. These speeches evince the lively style described by Longinus andcontribute much to our understanding of fourth century literature and history: the Funeral Oration, with its focus on the individual general Leosthenes and the historical events of theLamian War, stands in sharp contrast to Thucydides' Periclean Oration; the speech Against 

 Athenogenes offers an unusual legal argument to invalidate an unfair contract and also presents a vivid characterization of a duplicitous perfumier and his prostitute/pimp partner;the defense speeches For Lycophron and For Euxenippus are vital sources for thedevelopment of treason law in Athens; the prosecution Against Demosthenes is a key sourcefor the politics of the 320s.

The five leaves in the Archimedes Palimpsest are the most significant discovery of new

Hyperides text since the last big papyrus discovery in 1891 (several highly fragmentary papyri published in the twentieth century may also add a little to the corpus, but none can be

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attributed to Hyperides with much certainty; see Whitehead, Hypereides, pp. 473-476). Thefive bifolia were probably originally written in the eleventh century and they preserve ten

 pages of text, with thirty-two wide lines on each page. These 320 lines of new text willincrease the size of Hyperides' corpus by some 20%.

The palimpsest appears to preserve extensive sections of at least two previously lostspeeches. Natalie Tchernetska has edited the recto of the first bifolium (135/138) and alsooffers provisional readings of parts of the verso and the other Hyperides pages ("NewFragments of Hyperides from the Archimedes Palimpsest," Zeitschrift fur Papyrologie und 

 Epigraphik 154 (2005) 1-6). The recto of 135/138 includes a fragment of Hyperides previously known from a Byzantine encyclopedia (fr. 165 Jensen, attributed to the speech Against Timandra), which allows us to attribute the new material to Hyperides. The newlydiscovered context of the fragment calls for a reinterpretation of the case. It previouslyappeared to be a prosecution of a notorious courtesan, but it is now clear that we have aspeech in prosecution of a man, Timandros (previously unknown), in a dispute over aninheritance. The remaining pages are currently being studied: the folio 136/137 appears to

come from the political speech Against Diondas, in which Hyperides defends his proposal tohonor the rhetor Demosthenes prior to the battle of Chaeronea; 144/145 also refers to Philipand may come from the same political speech.

Study of the remaining material is underway. The pages are extremely fragile and are verydifficult to read. They were well erased by the second scribe and they have suffered extensivedamage. The imaging team of the Archimedes Palimpsest project is experimenting with newtechniques to make the material more readable, and an international team of scholars has

 been assembled to work in collaboration on the project. The scholarly team features expertson Hyperides and Greek oratory, Greek palaeography and textual criticism, and Athenianlegal procedure, and we are meeting regularly in 2006 and 2007 with the goal of interpretingand publishing the new material as soon as possible. Just the small amount of material so far deciphered reveals that we have much to learn about the career and style of Hyperides, aboutAthenian legal procedure, and about fourth-century politics and history.http://archimedespalimpsest.org/about/scholarship/hyperides-texts.php

Scholarship on Other Texts

http://archimedespalimpsest.org/about/scholarship/other-texts.php

In 2004, Reviel Netz and Nigel Wilson started to pay some attention to yet another set of leaves in the palimpsest. These were particularly difficult to read, but they seemed to be

 philosophical in nature. Many people became more interested in this text after Nigel Wilsondiscovered the word "Aristotle" on one of the leaves in June 2005. During meetings held atThe British Academy, one group of scholars would work on the Hyperides text, while another group would work on the philosophical text. In this fly-on-the-wall video Robert Sharples,Richard Sorabji, and Nigel Wilson tackle a very difficult problem.

Progress is being made, but this is was difficult text to read and progress was slow. A reportwas written by Robert Sharples in 2007 that makes clear just how challenging this task was.

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This manuscript was the hardest to image and the hardest to transcribe. But by the end of 2010, Roger Easton and his students had processed the images in new ways, and scholarsincluding Natalie Tchernetska, David Sedley, Bob Sharples, Marwan Rashed and NigelWilson had recovered most of the text.

 Now, in 2011, we know that The Archimedes Palimpsest consists of eight manuscripts: the prayerbook and seven erased texts. The erased texts are Archimedes Treatises, HyperidesSpeeches, the Commentary on Aristotles Categories, two Byzantine liturgical manuscripts,and two manuscripts that we have not yet been able to identify.http://archimedespalimpsest.org/about/scholarship/other-texts.php

The New Commentary on Aristotle's

Categories

in the Archimedes Palimpsest

by R.W. Sharples

http://archimedespalimpsest.org/about/scholarship/commentary-aristotle.php

We have a more or less speculative substantially continuous text for about 55% of the seventwo-sided leaves. A rather lesser amount has so far been worked over by several members of the group in such a way that the decipherment can be regarded as reasonably certain. Havinga more or less continuous text is not the same as fully understanding the structure of theargument at every point, which is made more difficult by the way in which a commentarywill cite a range of interpretations and arguments for them; distinguishing between what theauthor of the commentary is reporting, what represents his objections to a particular view,and what is an expression of his own view is not always easy, and this is an area where work is on-going.

The text is part of a formal commentary on Aristotle's Categories, covering 1a20-1b24. Thiswork was a focus of philosophical debate from the first century BC onwards; the argumentswhich it prompted are most accessible to us now in the massive commentary by Simplicius,

 but even that is necessarily selective, and Simplicius does not always name the participants in

the debates he echoes. Not surprisingly, there are numerous parallels between the new textand the debates in Simplicius; it contains named references, not all in Simplicius, toAndronicus and Boethus, the leading Peripatetics of the first century BC, and to Herminus,the second-century AD teacher of Alexander of Aphrodisias. It also includes a reference toStrato, the third head of Aristotle's school, parallel to others in the doxographical tradition.

Reviel Netz has suggested that the structure of the commentary indicates that the author took 1b10-15 with what preceded it rather than with what follows, i.e. that he did not regard thestart of the modern chapter 3 as a significant division in the argument.

A terminus post quem is given by the reference to Herminus. Marwan Rashed has argued,

and I am convinced, that the way in which Plato's view on the origin of the world is referredto shows that the author of the commentary is not himself a Platonist, as he does not himself 

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take sides on the question of Plato's view as a Platonist surely would. That the text is later than Herminus and not by a Platonist suggests strongly that it is a part of Alexander of Aphrodisias' lost commentary on the Categories, though we don't as yet have anything thatwould show this decisively. Another theoretical possibility is Galen's lost commentary, butthere's nothing particular to suggest this. On the issues in 1b15-24 there are, as Marwan has

 pointed out, links with a quaestio by Alexander extant in Arabic (Dietrich no.10).

R.W. SharplesSeptember 2007

 Members of the group working on this text: Sophia Kapetanaki, Stephen Menn, Reviel Netz,Marwan Rashed, David Sedley, Bob Sharples, Richard Sorabji, Natalie Tchernetska, NigelWilson.http://archimedespalimpsest.org/about/scholarship/commentary-aristotle.php

Capturing Images of the Archimedes

Palimpsest

http://archimedespalimpsest.org/about/imaging/capture.php

Path to this page: Home » About » Imaging 

In the initial imaging experiments on the Archimedes Palimpsest, the camera used by KeithKnox and Roger Easton was a monochrome camera with a liquid crystal tunable filter. It wasset up on a conventional copy stand, as you see below. The imagers used the tunable filter tocapture images in narrow wavebands. With this equipment the imagers could neither achievesufficient resolution, nor could they create satisfactory "data cubes."

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Standard Imaging

From 2001 through 2006, approximately 15 leaves of the Archimedes Palimpsest wereimaged every six months, in a period of 10 days. In order to create images of sufficientresolution (about 600dpi), each leaf was imaged in ten sections.

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For the purpose, Roger Easton built an X-Y stage on which the leaf traveled underneath the camera.

Onlymembers of the conservation staff were permitted to handle the leaves.

The various images of sections of each leaf were then stitched together to form images of 

entire pages. Instead of using filters, the imagers simply used different light sources:ultraviolet, Tungsten, and Strobe.

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Thereare nearly 90 leaves of the Archimedes Palimpsest, which need to be imaged back and front,in three different wavebands of light. That makes about 5,400 raw images. The specifics of each image had to be recorded for metadata.

In practice Roger Easton was at the computer in charge of the camera and the movements of the

XY stage; Keith Knox managed the lights, and Bill Christens-Barry kept the log. It was a pretty basic system, but it worked.

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Experiments

While the imagers created images that scholars deemed adequate, they also experimented inorder to improve performance. Bill Christens-Barry developed a system of collecting imagestaken in narrow bands of light using light emitting diodes. In an early version, this systemworked in conjunction with the XY stage built by Roger Easton. This necessitated capturing avery much larger number of images. The imagers also tried imaging in transmissive light andraking light. Here are some pictures of this system at work.

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Final Imaging Session

The final Imaging Session took place in August 2007. Bill, Keith, and Roger collaboratedwith Stokes Imaging of Austin Texas. The Stokes capture station was used in conjunctionwith a Sinar camera.

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It wasadapted to our purposes, and Bill Christens-Barry designed LED panels for the illumination.

Each leaf was imaged in twelve different wavelengths of light, from ultraviolet throughinfrared. Such was the resolution of the camera, that we could image entire leaves, and stillmaintain a resolution of 800 dpi, which was deemed sufficient by the scholars. Stokessoftware was used to manage the prodigious amount of metadata we were capturing.

Using this system we were able to image the entire manuscript in just under three weeks. Sixyears of work was redone in three weeks.

However: the magic has not happened yet. None of these images produce easily legible text.

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This isa montage of 12 shots of the leaf 105-110 of the Archimedes palimpsest, containingArchimedes method Proposition 14.

In none of these images is the text legible. Legible text is only produced in the post processing, and this is a whole different stage in the process, and one of which Keith Knoxwas the master.http://archimedespalimpsest.org/about/imaging/capture.php

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Project Management

http://archimedespalimpsest.org/about/management/

The Archimedes Palimpsest Project is managed by Michael B. Toth. Mike is the Presidentand CTO of R.B.Toth Associates, which provides a range of strategic services for organizations seeking to structure appropriate and practical responses to complex issues. He

 provides systems integration, program management and strategic planning for the study, preservation and display of cultural objects for museums and libraries. This includes planningand managing the hyperspectral imaging of the Waldseemuller 1507 Map at the Library of Congress and other technical studies.

Mr. Toth brings extensive experience in program management, strategic planning andsystems integration with his work on advanced information and space systems and national

 policy issues. During his 28 years of US Government service, Mike managed the

development, integration and operation of imagery and geospatial information collection, processing, dissemination and storage systems around the globe. He retired from governmentservice in September 2007, and continues to work as a consultant on a range of nationalsecurity issues.

Mike received his Bachelor's Degree in History from Wake Forest University, Winston-Salem, North Carolina, and completed graduate courses at George Mason University, Fairfax,Virginia. He also participated in Executive Management Seminars at the Kennedy School of Government at Harvard University and the Massachusetts Institute of Technology, and

 National Policy Seminars at the Brookings Institute, Washington D.C.

Report by Michael B. Toth

While government and industrial programs have integrated complex technical systems and processes for decades, the integration of advanced technologies and techniques to support thenew and unique set of requirements of the Archimedes Palimpsest Program required effective

 program management and systems planning in the museum and conservation community. Byadapting integrated program management techniques from industry to produce uniqueadvanced digital images of Archimedes original work, the Archimedes Palimpsest Programhas linked the range of skills, techniques and disciplines available in the conservation,scientific and academic communities to use advanced imaging techniques developed for the

study of earth and space. Advanced Imaging Magazine awarded the Archimedes PalimpsestProgram the "2003 Imaging Solution of the Year" award for its innovation and technicalexpertise with electronic imaging technology solutions.

When he first started to plan the Archimedes Palimpsest Program, Dr. William Noel, theArchimedes Palimpsest Curator, found he had to rapidly form a management team to assessthe best path forward for the efficient digital collection of Archimedes work. As the ProgramDirector, supported by Program Manager Michael B. Toth and Conservator Abigail Quandt,he developed a plan with performance goals that could be accomplished within cost and onschedule. They based their plan on the input from virtual and real teams of scholars and

museum professionals, who defined their needs, working closely with the imaging scientistsunder the leadership of the project management team. This process resulted in the users

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clearly defining the needed system capabilities and technological needs, as opposed to havingthe system defined by the available technology and the professionals who work with thetechnology.

The user needs were defined simultaneously with the development and integration of theappropriate imaging technology through a multi-phased program. These included phases for:

• Scientific Study and Experimental research• Development of Prototype Systems

• Testing and Review

• Systems Integration

• Full Operational Capability

During these phases the imaging team worked on the research, development and integrationof a range of advanced technical capabilities to reveal the underlying original writings of Archimedes. Research into a variety of promising imaging technologies identified thetechnical capabilities most appropriate for this project and how they could be best used.Based on research and study supported by the Archimedes Program, integrated technicalcapabilities were developed to collect all the required imagery, store the digital information,and display the available information for academic researchers.

The Archimedes team rapidly realized the following key capabilities needed to be developedto support the Archimedes Palimpsest Program:

• Data and Image Collection• Data and Image Processing

• Data Validation and Storage

• Information Access

• Academic Analysis and Study

With so much data to be collected, processed, stored and retrieved, the success of theArchimedes Program was dependent on the integration of ever growing amounts of data from

multiple sources. Early in the program, Mike Toth, with the assistance of Bob Toth and BillChristens-Berry, developed the Archimedes Palimpsest Metadata Standard to facilitate onlineaccess and further study of Archimedes data in library and museum digital collections. Thisstandard utilized existing remote sensing, multispectral and geospatial information standards,in particular Dublin Core and the Federal Geographic Data Committee Content Standard for Digital Geospatial Metadata (FGDC-STD-001-1998) [link]. As with any technical researchand development effort, this program required taking risks in a number of areas. To ensure

 project success, the program management team also constantly identified and managedvarious risk areas.

The challenge of retrieving all available information from the digital data and preserving it

for future generations has required effective management and teamwork among conservationand other technical and scientific disciplines to reach common goals for the preservation and

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study of digital information about the Palimpsest. Each of the various disciplines contributedto a multidisciplined team in the integration of diverse work processes and technical skills:

1. Conserving parchment in poor condition,2. Imaging text in appropriate spectral bands,

3. Processing the data to yield useful information,

4. Storing the digital data and associated information about the data, and

5. Making the information available for researchers and the public.

Research into a variety of promising imaging technologies identified the technicalcapabilities most appropriate for this project and their best application. As new technologieswere required to meet new technical challenges, the program management team opened upthe effort to a broader scientific community to ensure the best available technologies wereconsidered for this program. An example of this was the ADITUP Conference (Conferenceon Applied Digital Imaging Techniques for Understanding the Palimpsest) in April 2004,which resulted in two major paths of study: X Ray Fluorescence and Character Recognition.Based on research and study supported by the Archimedes Program, a full suite of integratedtechnical capabilities was developed to collect all the required imagery, store the digitalinformation, and retrieve and display the available information for academic researchers.

This project is a multiyear effort requiring long-term support from a range of individuals andorganizations with other responsibilities. Management of this program with a far-flung teamwith varied technical and academic knowledge proved challenging. That the team was up tothe challenges is evident from the huge amount of useful data, volumes of new information,

and the significant contributions to our knowledge of Archimedes, as well as imaging scienceand new discovery techniques.April 2004http://archimedespalimpsest.org/about/management/

Image Processing

http://archimedespalimpsest.org/about/imaging/processing.php

If you look at things with different lights you see different things. It has long been known, for example, that we can retrieve under-drawings below paintings by looking at them in infraredlight. This is because the carbon used by artists to make their under-drawing absorbs light atthat range of the spectrum. On the whole, infrared light was not very useful in imaging theArchimedes Palimpsest (although there are one or two interesting exceptions in the non-Archimedes leaves), because the ink used by by the scribes did not have much carbon in it.Rather, the ink contained a lot of iron, and iron absorbs ultraviolet light. Ultraviolet light hasoften been used to look at faint text in medieval manuscripts, and it is extremely helpful inthe case of the Archimedes Palimpsest.

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These days, however, it is possible to post-process images to bring out more of what youwant than is possible simply through image capture. And, of course, it is possible to useinformation from more than one image to do this. Keith Knox explains:

This pseudo-color technique that Keith developed is extremely useful. It is clever but it is not

complicated, and it works well over the entire palimpsest. It is extremely powerful, and has been used to great effect. Here are a few examples.

 before

after 

These are before and after images of a detail of the leaf 105-110 of the ArchimedesPalimpsest. The pseudo-color allowed Reviel Netz, Nigel, Wilson, Ken Saito, and Natalie

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Tchernetska to read Archimedes' Method Proposition 14 as never before, and come to a newunderstanding of Archimedes conception of infinity.

 before

after 

These are before and after images of a detail of the leaf 172-177 of the ArchimedesPalimpsest. The pseudo-color allowed Reviel Netz, Nigel Wilson, and Fabio Acerbi to readArchimedes Stomachion as never before, and come to a new understanding of th role of Archimedes in the development of combinatorics in the mathematical tradition.

 before

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after 

These are before and after images of a detail of the leaf 135-138 of the Archimedes

Palimpsest. Pseudo-color images like this allowed Natalie Tchernetska to discover five leavesin the Palimpsest that do not contain writing by Archimedes at all, but rather otherwiseunknown speeches by Hyperides, one of the ten canonical orators of antiquity.

Keith Knox's pseudo-color post-processing technique is but one of the ways in which theimages can be processed. The imaging team also processed the same set of images to peel of the prayerbook text completely, and reveal the undertext alone. On the whole, the scholarsseem to prefer the Pseudo-Color images to these pictures. But they do give very satisfactoryimages of the diagrams in the Palimpsest. This is important because the ArchimedesPalimpsest is a unique source for the diagrams that Archimedes himself drew in the sand, inSyracuse, in the third century BC.

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 before

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The post-processing possibilities on the images that the imagers have taken of theArchimedes Palimpsest have by no means been exhausted. Now that the images are online,anyone will be able to try their hand at improving on the legibility of the texts.http://archimedespalimpsest.org/about/imaging/processing.php

Optical Character Recognition

http://archimedespalimpsest.org/about/imaging/optical-character-recognition.php

Even the most advanced imaging techniques will not recover all of the undertexts in theArchimedes Palimpsest, because sometimes these texts are simply no longer present.Individual characters have been partly erased, and sometimes completely obliterated. It is

 proposed that computers could serve as valuable tools in reconstructing partial characterswhen ink is missing, or when characters have been obscured by the prayerbook text. In recentyears much progress has been made in making computers compare images of faces with thosein a database, and finding a match. If this is possible with images of the human face, thethought is that it should be possible to achieve a similar result with the limited range and very

 particular shapes of the Byzantine script that was employed by the scribe of the Archimedestext. It also significantly helps that only one scribe wrote the Archimedes text, and that he hada very regular hand — that is, his letter shapes are consistent.

The limits of optical character recognition by computer are well understood: the best that acomputer can come up with are probabilities that a certain shape is a particular Greek character. These probabilities can be narrowed down both by looking at the spatialcharacteristics of a character, and then by considering the likelihood that this character would

 be preceded or would be followed by a particular identified letter. The results will facilitatescholars by presenting them with a range of possibilities from which they might choose.

The optical character recognition has two distinct advantages. Firstly, compared to theadvanced imaging techniques that are being employed, it is inexpensive. Secondly, the

character recognition is not based upon the manuscript itself, but upon the best available

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images of the manuscript. Therefore this investigation does not rely on access to a fragile,unique, and ancient document.

The Archimedes project asked for submissions. The guidelines for the

submission were very specific: to provide a tool that a scholar could use to give practical helpin the decipherment of the Archimedes text.

The submission that the owner chose to fund was provided by Derek Walvoord (left). Theinterface that he devised is depicted below.

Scholar s deemed that the machine had potential for further development, and the owner has agreed tofund the ongoing character recognition effort.http://archimedespalimpsest.org/about/imaging/optical-character-recognition.php

X-ray Fluorescence Imaging

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http://archimedespalimpsest.org/about/imaging/xray-flourescence.php

There is a well known equation in program management circles that for 20% of the effort onecan get 80% of the result, but that one can spend 80% of the effort achieving that final 20%of the result. The imaging procedure that was employed to image the majority of the

Archimedes Palimpsest was an extremely effective way of recovering most of the text.However, it was of very little use in reading the text underneath the forged pages, and someareas of text still proved recalcitrant.

Reviel Netz sent in this example on September 29, 2003, circling the area he could not read,and writing “this example is from the Method, middle of 158v. col. 1. Here Archimedesdraws some lines to do something, but I am not sure which something it is (I have a guess,

 but it's just a guess). If I get the letters referring to the diagram describing where the lines gothrough, and get a couple of words before that, I think I'll be able to settle with great certainty

the mathematical action of this proposition.”

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A small conference was held n April 1, 2004 to determine how best to retrieve the remainingtext from the Palimpsest. There were three main approaches: X-Ray Florescence Imaging,Synchrotron Radiation.

 X-ray Fluorescence Imaging

For the April 2004 workshop three different scientists suggested independently that we

employ a technique called X-ray Fluorescence Imaging. This technique would involveexciting the Palimpsest with relatively high energy, short wavelength photons (X-rays).

 Normal X-ray images are taken in transmission: that is, a reactive plate is placed behind asample, and a negative image of an object is created on the plate. This is what happens whenyou go to the dentist. However, this transmission technique would not work for thePalimpsest because the image contrast from the traces of ink would be much too small. Sothey proposed to use a different way to enhance the contrast. Some of the X-rays that do not

 pass through the leaves excite individual atoms in the palimpsest, which would then generatetheir own secondary X-rays. The crucial thing about these so called fluorescence X-rays is

the fact that different elements generate X-rays with different wavelengths. A fluorescenceX-ray sent out from an iron atom is different from an X-ray sent out by a calcium atom, or anatom of gold. These fluorescence X-rays could then be detected with a device that canrecognize these different X-rays. The idea is simple: by mapping these X-rays, we couldcreate “element maps” of individual pages of the Palimpsest.

The imaging scientists who worked on this aspect of the project were Gene Hall, Professor of Chemistry at Rutgers University, Uwe Bergmann, Staff Scientists at the StanfordSynchrotron Radiation Laboratory, and Bob Morton of the Children of the Middle Waters.

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Gene Hall Uwe Bergmann Bob Morton

Initial experiments were conducted by Gene Hall at Rutgers University, using an X-raymicrofluorescence system provided by the EDAX Corporation – an EDAX Eagle Probe – inhis lab. He presented his initial results in April 2004.

Following the conference it was decided to do further imaging with an EDAX Eagle probe.The EDAX firm, a subsidiary of Ametek, were extremely generous in providing their facilities in New Jersey for five days of intensive research by Bob Morton and Gene Hall.

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They were greatly helped in

their efforts by Tara Nylese and Bruce Scruggs of EDAX inc.

Tara Nylese

Notable Results

A regular image of a small section of the Archimedes Palimpsest. You can seethe Archimedes text clearly but faintly, underneath the prayerbook text.

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A pseudocolor image of the same area, in which the Archimedes text appears in

red

An X-ray fluorescence image, which shows all the iron in this area. This is whatwe had hoped to see because it brings out the texts on the page, and we knew that the ink hada large concentration of iron.

This is a “gray-scale” version of the figure above, because it is easier to seedifferences in intensity in a gray image. Note that the X-ray image measures not the surfaceof the page, but the intensity of the iron signal on a particular part of the parchment. Thus,when the Archimedes text overlaps the prayerbook text, the iron signal becomes particularlyintense. This is the first time that scholars could make out the shape of the Archimedes letters

underneath the letters of the prayerbook text.

An X-ray fluorescence image, which shows all the potassium in this area. As youcan see, there is a much greater concentration of potassium in the prayerbook text than thereis in the Archimedes text. It is still hoped that through post-processing of these images, and

 by combining the information provided by the iron and the potassium, we might be able to

reveal the entire Archimedes text, stripped of the prayerbook text, for certain difficultsections.

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Imaging Through the Forgery 

This image is of one of the pages in the Archimedes Palimpsest containing a forgery. A smallsection was imaged, and the resulting iron map produced a section of the Archimedes“Equilibrium of Planes”, which our scholars could decipher.

These images were exciting, but the small section of Archimedes text that we recovered fromthe forged page took 10 hours to generate! It was clear that we would need more intense X-rays, and for this we needed a more powerful instrument. It was for this reason that we turnedto Uwe Bergmann, and to the Stanford Synchrotron Radiation Laboratory (SSRL).

The important point about synchrotron

radiation is that it is much more powerful than the radiation that a normal X-ray tube can provide, and it is also much more tunable to the specific wavelengths that we wanted. It

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therefore had the potential to produce similar results to those of the EDAX Eagle Probe, butmuch more quickly. The potential danger of using synchrotron radiation is that these highenergy X-rays have the potential to damage the parchment. Therefore careful tests werecarried out by Gregory Young of the Canadian Conservation Institute, to see how parchmentreacted to exposure to X-rays, and we were guided by these results at Stanford.

For this experiment we usedBeamline 6 at the SSRL, which is encased within a lead lined hutch.

A leaf of the ArchimedesPalimpsest was mounted vertically on an XY stage, and placed before the beam of highenergy photons. A detector, tuned to pick up X-rays of the wavelength on Iron was placed at90 degrees to the beam.

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Mike Toth, Will Noel,Abigail Quandt, Uwe Bergmann, Martin George and others worked for six days on theseexperiments.

The results can be seen below. We retrieved a whole column of Archimedes text from thesame forged page of the Palimpsest in about 24 hours.

http://archimedespalimpsest.org/about/imaging/xray-flourescence.php

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Manuscrisul lui Arhimede

De la Wikipedia, enciclopedia liberă

http://ro.wikipedia.org/wiki/Manuscrisul_lui_Arhimede

Stomachion este un joc logic din Manuscrisul lui Arhimede (prezentat după Suter dintr-osursă diferită; această versiune necesită o alungire laterală cu factorul 2 pentru a fi în acord cuManuscrisul)

Manuscrisul lui Arhimede este un manuscris pe pergament sub forma de codex. Acesta afost inițial o copie a unei alte lucrări necunoscute ale lui Arhimede din Siracuza și a altor 

autori, fiind rescrisă cu un text religios.

Arhimede a trăit în secolul al 3-lea î.Hr, dar copia este o lucrare făcută în secolul al 10-lead.Hr. de un scrib anonim. În secolul al 12-lea codexul original al lui Arhimede a fostdezlegat, răzuit și spălat, împreună cu alte șase manuscrise pe pergament, incluzând și unadin lucrările lui Hypereides. Apoi aceste pergamente au fost îndote pe jumătate și refolosite

 pentru un text liturgic creștin de 177 de pagini, astfel încât o foaie veche a fost împăr țită îndouă foi noi de carte liturgică. Răzuirea nu a fost completă, iar manuscrisul lui Arhimede

 poate fi citit acum după lucrările științifice și academice din 1998-2008, obținute prin procesarea imaginilor manuscrisului cu raze ultraviolete, infraroșii, vizibile și raze-X.[1][2]

În 1906 manuscrisul a fost sumar inspectat de filologul danez Johan Ludvig Heiberg. Cuajutorul unor fotografii alb-negru pe care le-a luat, el apublicat o transcriere a textului luiArhimede. La scurt timp textul din greacă a fost tradus în engleză de T.L. Heath. Înainte dedescoperirea lui, manuscrisul nu era cunoscut printre matematicieni, fizicieni sau istorici.

Manuscrisul conține:

• "Echilibrul Planelor"• "Linii Spirale"

• "Măsurarea Cercului"

• "Despre Sferă și Cilindru"

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• "Despre Plutirea Corpurilor" (singura copie cunoscută, în greacă)

• "Metoda Teoremelor Mecanice" (singura copie cunoscută)

• "Stomachion" (singura copie cunoscută)

Manuscrisul mai conține cuvântările politicianului Hypereides, care a trăit în secolul al 4-lead.Hr, comentariile lui Alexandru din Aphrodisias aspura Categoriilor lui Aristotel, precum șialte lucrări..[3]

Cuprins

• 1 Conținutul matematic• 2 Istoria modernă

• 3 Note

• 4 Referințe

• 5 Legături externe

Conținutul matematic

Cea mai remarcabilă lucrare din cele de mai sus este  Metoda Teoremelor Mecanicii, iar manuscrisul conține singura copie cunoscută.

În operele sale, Arhimede demonstrează de multe ori egalitatea a două arii sau a două volumefolosind metoda epuizării a lui Eudoxus, metodă folosită în Grecia antică, similară cu metodamodernă de trecere la limită. Deoarece grecii erau conștienți că unele numere erau iraționale,notația lor pentru numere reale era cantitatea Q aproximată prin două secvențe, una dândlimita superioară, iar cealaltă limita inferioară. Dacă se găseau două secvențe S și I, cu S > Qși I < Q, și dacă cele două secvențe se apropiau mai mult decât orice valoarea specificatăanterior, atunci Q se găsea, sau epuiza, între S și I.

Arhimede a folosit de multe ori metoda epuizării pentri a-și demonstra teoremele. Acestlucru implica aproximarea figurilor a căror arie trebuia calculată în secțiuni a căror arie eracunoscută, furnizând astfel limita superioară și inferioară a figurii. Astfel el dovedea că cele

două limite deveneau egale când subdiviziunile deveneau arbitrar de mici. Aceste dovezi,considerate încă riguroase și corecte, rareori foloseau geometria cu rezultate precise. Maitârziu, scriitorii l-au criticat adesea pe Arhimede pentru că nu a explicat cum a ajuns la acesterezultate. Aceste explicații sunt conținute în lucrarea Metoda Teoremelor Mecanicii.

Metoda pe care o descrie Arhimede se baza pe investigațiile lui din fizică în ceea ce priveștecentrul maselor  și legea pârghiilor . El compara aria sau volumul unei figuri, căreia îicunoștea masa și centrul de greutate, cu aria sau volumul unei figuri despre care nu știanimic. Împăr țea cele două figuri în foarte multe păr ți mici, apoi cântărea pe o pârghie fiecare

 parte a unei figuri cu cea corespunzătoare celei de a doua. Punctul esențial este acela că celedouă figuri sunt orientate diferit, astfel încât păr țile corespunzătoare se află la distanțe

diferite de punctul de sprijin, iar condiția de echilbru a păr ților nu este aceeași cu condiția deegalitate a lor.

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Odată ce arăta că fiecare parte a unei figuri echilibra fiecare parte a celeilalte figuri,concluziona că cele două figuri se echilibrau una pe alta. Dar centrul de masă al unei figurifiind cunoscut, întreaga masă putea fi plasată în centrul ei și rămânea în echilibru. A douafigură avea masa necunoscută, dar poziția centrului ei de greutate putea fi aflată prinobținerea echilibrului față de punctul de sprijin, ceea ce permitea calculul masei totale a celei

de a doua figuri. Arhimede considera metoda ca folositoare euristic, dar întotdeauna a făcut-oca să dovedească rezultatele obținute prin metoda epuizării, deoarece metoda nu furniza nicilimita inferioară și nici pe cea superioară.

Folosind această metodă, Arhimede a fost capabil să rezolve multe probleme care la oraactuală sunt rezolvate prin calcul integral, dat în forma sa modernă în secolul al 17-lea deIsaac Newton și Gottfried Leibniz. Printre problemele pe care Arhimede le-a rezolvat a fostcea a calculului centrului de greutate al unei emisfere solide, centrul de greutate al unuitrunchi al unui paraboloid circular și aria unei zone a parabolei limitată de parabolă și odreaptă secantă a ei (vezi tratatul Metoda Teoremelor Mecanicii).

Când a demonstrat riguros teoremele Arhimede a folosit ceea ce azi numim suma luiRiemman. În tratatul Despre Sferă și Cilindru el a dat limita superioară și inferioară pentrusuprafața sferei prin tăierea sferei în secțiuni de lungimi egale. Asfel a limitat aria fiecăreisecțiuni prin aria unui con înscris și unul circumscris, dovedind că au arie mai mică șirespectiv mai mare. Apoi a făcut suma ariile conurilor, care sunt sume de tip Riemman pentruzona din sferă considerată ca suprafață de revoluție.

Există două diferențe esențiale între metoda folosită de Arhimede și ce din secolul XIX:

1. Arhimede nu știa nimic despre diferențiabilitate, deci nu putea calcula nici o

integrală, decât acelea date prin considerarea centrului de greutate, adică prin simetrie.Deși avea noțiunea de liniaritate, pentru a găsi volumul unei sfere el a echilibrat douăfiguri în același timp; dar nu a înțeles nici schimbarea de variabile, nici integrarea

 prin păr ți.2. Când calcula sumele aproximative, el impunea anumite constrângeri pentru ca sumele

să stabilească cu precizie limita inferioară și superioară. Acest lucru a fost necesar deoarece grecii antici nu aveau metode algebrice cu care puteau stabili dacă eroare eramică sau nu.

O problemă rezolvată exclusiv în lucrarea Metoda Teoremelor Mecanicii este calcululvolumului unei pene cilindrice, rezultat care reapare ca teorema XVII (schema XIX), în

lucrarea lui Kepler  Stereometria.

Câteva pagini din tratatul Metoda Teoremelor Mecanicii au rămas nefolosite de autorulmanuscrisului și astfel ele sunt pierdute. Între ele, există un rezultat care dă volumulintersecției a doi cilindri, o figură pe care Tom M. Apostol și Mamikon Mnatsakanian de laInstitutul de Tehnologia din California au redenumit-o globul lui Arhimede cu n = 4 (iar 

 jumătatea lui cupola lui Arhimede cu n = 4), al cărui volum se referă la piramida n- poligonală.

În timpul lui Johan Heiberg, a fost acordată o mai mare atenție folosirii strălucite a calcululuiinfinitezimal de către Arhimede, pentru a soluționa problemele referitoare la arii, volume șicentre de greutate, și mai puțină atenție a fost acordată jocului logic Stomachion, o problemătratată în manuscris care pare a fi un joc de copii. Reviel Netz de la Universitatea Stanford a

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argumentat că Arhimede discută despre numărul de moduri în care se poate rezolva problema, adică de a pune piesele la locul lor în pătrat. Nu au fost identificate piese avândaceastă formă; nu s-au găsit regulile de plasament al lor; dacă este permisă sau nu întoarcerea

 pieselor cu fața în jos, existând dubii asupra figurii. Figura prezentată aici de Netz, este una propusă de Suter dintr-o traducere a unui text arab în care egalul  și de două ori sunt ușor de

confundat. De asemenea Suter a făcut cel puțin o greșală topologică într-un punct crucial,egalând lungimea unei laturi cu diagonala, caz în care figura nu mai poate fi pătrat. Dar,deoarece diagonalele unui pătrat se intersectează în unghi drept, prezența triunghiurilor dreptunghice face ca prima propoziție din Stomachion să rezulte imediat. Mai exact, prima

 propoziție asamblează o figură constând din două pătrate alăturate (ca într-un Tangram). Oreconsiderare a figurii lui Suter cu figura din Codex a fost publicată de Richard DixonOldham, în revista Nature din martie 1926, ceea ce a creat o manie Stomachion în acel an.Combinatorica modernă a dezvăluit că numărul de moduri în care piesele lui Suter pot fiasamblate pentru a se obține un pătrat este de 17152. Numărul este mult mai mic – 64 – dacănu este permică întoarcerea pieselor cu fața în jos. Unghiurile ascuție ale figurii lui Suter facdificilă asamblarea, în timp ce jocul poate fi incomod dacă piesele cu puncte ascuțite suntîntoarse cu fața în jos. Pentru figura din Codex există trei moduri de a grupa piesele; ca două

 pătrate alăturate lateral; ca două pătrate unul deasupra celuilalt; sau ca un singur pătrat culatura radical din doi. Dar cheia acestor grupări este formarea de triunghiuri isoscele drepte,așa cum, luându-l în considerație Meno al lui Plato, Socrate a obținut copilul sclav,susținând cunoașterea prin amintire, și aici recunoașterea modelului din memorie pare a fimult mai pertinent decât numărul de soluții. Figura din Codex poate fi considerată ca oextensie a argumentului lui Socrate într-o grilă pătrată de șapte pe șapte, sugerând oconstrucție cu un număr de diametre care să dea o aproximare rațională a numărului radicaldin doi, iar starea fragmentată a manuscrisului lasă multe dubii. Dar cu siguranță se adaugămisterului dacă Arhimede a folosit prioritar figura lui Suter față de figura din Codex. Totuși,

dacă Netz are dreptate, acest lucru este cea mai sofisticată lucrare de combinatorică dinGrecia antică.

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Istoria modernă

O pagină tipică din manuscrisul lui Arhimede. Textul preotului apare de sus în jos. Textuloriginal apare dedesupt, mai slab,de la stânga la dreapta.

După imaginea unei pagini din manuscris, textul original al lui Arhimede apare mai clar.

Savantul Biblic Constantine Tischendorf a vizitat Constantinopolul în 1840, și intrigat detextul matematic grec vizibil din manuscris, a adus o pagină din el acasă. (Această pagină se

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află acum la Librăria Universității din Cambridge). Cel care și-a dat seama că este un text allui Arhimede a fost Johan Heiberg, atunci când a studiat manuscrisul la Constantinopole în1906, manuscris care conținea lucrări considerate pierdute. Heiberg a făcut fotografii dupăcare a realizat traducerea publicată între 1910 și 1915, cu operele complete ale lui Arhimede.

 Nu se știe cum manuscrisul a fost adus ulterior în Franța.[4]

Din 1920, manuscrisul a stat la Paris într-un apartament necunoscut al unui colecționar demanuscrise și la moștenitorul acestuia. În 1998 dreptul de proprietate al manuscrisului a fostîn disput la curtea federală din New York între Patriarhia Ortodoxă Greacă din Ierusalim șiCasa de Licitaț ie Christie Inc. Cu mult timp în urmă, manuscrisul lui Arhimede a stat înlibrăria mânăstirii Mar Saba, de lângă Ierusalim, mânăstire înapoiată Patriarhiei în 1625, iar reclamantul a susținut că manuscrisul a fost furat din mânăstire în anul 1920. JudecătorulKimba Wood a decis în favoarea Casei de Licitație Christie din nepăsare, iar manuscrisul afost cumpărat de un anonim pentru suma de 2 milioane de dolari. Simon Finch, cel care l-areprezentat pe cumpărător, a spus că a fost un cumpărător privat american, care lucrează înindustria high-tech, dar nu este Bill Gate.[5] (Revista germană Der Spiegel a raportat că celcare l-a cumpărat a fost probalil Jeff Bezos.)[5]

La Muzeul de Artă Walters din Baltimore, Maryland, manuscrisul a fost subiectul uneicercetări extensive din 1999 până în 2008, precum și de conservare, datorită mucegaiului.Cercetările au fost conduse de Dr. Will Noel, curator al manuscriselor de la Muzeul de ArtăWalters și coordonate de Michael B. Toth de la R.B. Toth Associates, împreună cu Dr.Abigail Quandt care a realizat conservarea manuscrisului.

O echipă de specialiști în prelucrarea imaginilor, care a inclus pe Dr. Roger Easton de la[[Institutul de Tehnologie Rochester, Dr. Bill Christens-Barry de la Equipoise Imaging și Dr.

Keith Knox de la Boeing LTS, au folosit calculatorul pentru procesarea digitală a imaginilor în benzi spectrale diferite, inclusiv lumina vizibilă și cea ultravioletă, pentru a descoperitextele care stau la baza manuscrisului. După ce au scanat și au procesat digital întregulmanuscris în trei benzi spectrale până în 2006, în 2007 au reprocesat imaginile manuscrisuluiîn 12 benzi spectrale plus scanarea cu: UV: 365 nanometri; Lumină vizibila cu lungimea deundă de: 445, 470, 505, 530, 570, 617 și 625 nm; Infraroșu: 700, 735 și 870 nm; și luminăînclinată de: 910 and 470 nm.[6] De asemenea au procesat imaginile digitale ale originaluluilui Heiberg. Dr. Reviel Netz[7] de la Universitatea Stanford și Nigel Wilson au produs o copiea textului, completând lacunele descrierii lui Heiberg cu aceste imagini. Toate imaginile suntținute pe website-ul http://archimedespalimpsest.net/

După 1938, un proprietar al manuscrisului a falsificat patru imagini religioase  bizantine înefortul de a-i crește valoarea. Se pare că acest lucru a făcut ca textul de bază să fie ilizibil

 pentru totdeauna. Totuși în mai 2005, Drs. Uwe Bergman și Bob Morton au folosit raze X  pentru descifrarea păr ților textului de 174 de pagini care nu fuseseră încă dezvăluite. Luminasincrotronă se creează atunci când electronii călătoresc cu viteză apropiată de cea a luminiimișcându-se pe o curbă în jurul inelului de stocare și emițând raze X pe lungimi infraroșii.Fascicolul de lumină rezultat are caracteristici care îl fac ideal pentru descoperirea arhitecturiicomplicate a obiectelor, în acest caz, lucrarea ascunsă a unuia din părinții tuturor științelor.[8]

În aprilie 2007, s-a anunțat că un nou text a fost găsit în manuscris, text care este uncomentariu al lucrării lui Aristotel atribuit lui Alexander din Aphrodisias. Dr. Will Noel aspus într-un interviu: Începi să te gândești că manuscrisul este aur și apoi că este absolut uimitor. Dar apoi ceva și mai grozav se întâmplă. S-a referit, desigur, la descoperirea

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anterioară a texului lui Hypereides, un politician atenian din secolul al 4-lea î.Hr, aflat tot înmanuscris.[3] Este vorba de discursul Împotriva lui Diondas, publicat în 2008 în revistaștiințifică Zeitschrift für Papyrologie und Epigraphik , vol. 165, devenind primul text nou dinmanuscris publicat într-o revistă științifică.[9]

Transcrierea căr ții a fost codificată digital cu ajutorul liniilor directoare din Text Encoding  Initiative, precum și catalogarea imaginilor bazată pe elementele Dublin Core Metadata.Catalogarea datelor a fost coordonată de Dr. Doug Emery de la Emery IT.

Pe 29 octombrie 2008 (aniversând 10 ani de la cumpărarea manuscrisului la licitație), toatedatele, inclusiv imaginile și transcrierile au fost stocate pe Digital Palimpsest Web Page,

 pentru liberă folosință sub licența Creative Commons, iar imaginile procesate alemanuscrisului au fost stocate pe Google Book .[10]

Note

1. ^ „"Reading Between the Lines", Smithsonian Magazine”. Accesat la 31martie 2009.

2. ^ „The Archimedes Palimpsest Project”. Accesat la 31 martie 2009.

3. ^ a b „"Text Reveals More Ancient Secrets", BBC News”. 26 aprilie 2007.Accesat la 31 martie 2009.

4. ^ „History of the Archimedes Manuscript”. Accesat la 31 martie 2009.

5. ^ a b Hisrhfield, Alan. Eureka Man, Walker & Co, NY, 2009; p. 187.

6. ^ „File Naming Conventions”. Accesat la 31 martie 2009.

7. ^ „The Scholarship of the Palimpsest”. Accesat la 31 martie 2009.

8. ^ „Placed under X-ray gaze, Archimedes manuscript yields secrets lost totime”. Accesat la 31 martie 2009.

9. ^ Carey, C. et al., "Fragments of Hyperides’ Against Diondas from theArchimedes Palimpsest", "Inhaltsverzeichnis", Zeitschrift für Papyrologie und 

 Epigraphik , vol. 165, pp. 1-19. Retrieved 2009-10-11.

10. ^ Archimedes Palimpsest . Accesat la 31 martie 2009

Referințe

• Reviel Netz and William Noel, The Archimedes Codex, Weidenfeld & Nicolson, 2007• Dijksterhuis, E.J.,"Archimedes", Princeton U. Press, 1987, pages 129–133. copyright

1938, ISBN 0-691-08421, 0-691-02400-6

Legături externe

• The Archimedes Palimpsest Project Web Page•

Digital Palimpsest on the Web

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• The Archimedes Palimpsest web pages at the Walters Art Museum

• The Nova Program outlined

• The Nova Program teacher's version

• The Method : English translation (Heiberg's 1909 transcription)

• Did Isaac Barrow read it?

• May 2005 Stanford Report: Heather Rock Woods, "Archimedes manuscript yieldssecrets under X-ray gaze" May 19, 2005

• Will Noel: Restoring The Archimedes Palimpsest (YouTube), Ignite (O'Reilly),August 2009

• The Greek Orthodox Patriarchate of Jerusalem v. Christies’s Inc., 1999 U.S. Dist.LEXIS 13257 (S.D. N.Y. 1999) (via Archive.org)

• Ultima modificare efectuată la 15:08, 9 august 2012.

http://ro.wikipedia.org/wiki/Manuscrisul_lui_Arhimede

Metoda Teoremelor Mecanicii

De la Wikipedia, enciclopedia liberă

http://ro.wikipedia.org/wiki/Metoda_Teoremelor_Mecanicii

 Metoda Teoremelor Mecanicii este o lucrare a lui Arhimede, care pentru prima dată atestăexplicit folosirea calculului infinitezimal. Lucrarea originală s-a crezut a fi pierdută, dar a fostredescoperită în celebrul Manuscris al lui Arhimede din 1902. Manuscrisul include șidescrierea lui Arhimede despre metoda mecanică, numită așa deoarece s-a folosit de legea

 pârghiilor (demonstrată pentru prima dată de el însuși) și de centrul de greutate al obiectelor.

Arhimede nu a admis infinitezimalul ca parte a rigorii matematice și de aceea nu și-a publicat metoda în nici un tratat formal, care să conțină acest rezultat. În tratatul MetodaTeoremelor Mecanicii, el a demonstrat câteva teoreme prin metoda epuizării, găsind în modriguros limita inferioară și superioară, limite care conduc spre răspunsul cerut. Cu toateacestea, metoda mecanică a fost folosită pentru a descoperi relații pentru care, mai târziu, s-au găsit demonstrații riguroase.

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Cuprins

• 1 Aria unei parabole• 2 Prima propoziție din manuscris

• 3 Volumul unei sfere

• 4 Aria sferei

• 5 Forme curbilinii cu volume raționale

• 6 Alte propoziții din manuscris

• 7 Vezi și

• 8 Referințe

 Aria unei parabole

Pentru a explica azi metoda lui Arhimede, este mai convenabil să facem uz de geometriecarteziană, care evident, nu era disponibilă în antichitate. Ideea lui Arhimede a fost aceea de afolosi legea pârghiilor pentru a determina aria unei figuri cunoscând centrul de greutate alaltei figuri. Cel mai simplu exemplu în limbaj modern este aria parabolei. Arhimede foloseșteo metodă mult mai elegantă, dar în limbaj cartezian, metoda lui este aceea de a calculaintegrala:

care are ca rezultat valoarea 1/3.

Pentru a găsi rezultatul integralei, considerăm un triunghi în echilibru cu parabola. Triunghiuleste o regiune din planul x- y aflat între axa x și dreapta y = x, cu x variind de la 0 la 1.Parabola este o regiune aflată în planul x- y între axa x și curba y = x2, cu x variind deasemenea de la 0 la 1.

Descompunem triunghiul și parabola în fâșii verticale subțiri, pentru fiecare valoare a lui x.Să ne imaginăm că axa x este o pârghie cu punctul de sprijin în x = 0. Legea pârghiilor spunecă produsul dintre masa și distanța la punctul de sprijin trebuie să fie egal pentru cele douăobiecte în echilibru. Masa fâșiei verticale a triunghiului la distanța x de punctul de sprijineste egală cu înălțimea ei, astfel că va echilibra fâșia de parabolă, având înălțimea x2, dacă vafi amplasată la o distanța egală cu 1 de cealaltă parte a punctului de sprijin.

Deoarece fiecare fâșie este în echilibru, întreaga parabolă va fi în echilibru cu întregultriunghi. Acest lucru însemnă că, dacă parabola este atârnată de un cârlig în punctul x = -1, ea

va echilibra triunghiul aflat între x = 0 și x = 1.

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Centrul de greutate al triunghiului poate fi ușor aflat prin următoarea metodă, datorată tot luiArhimede. Dacă o linie mediană este desenată din oricare vârf pe latura opusă în E , triunghiulva fi în echilibru pe mediană considerată ca punct de sprijin. Motivul este acela că dacătriunghiul va fi împăr țit în segmente paralele cu latura pe care se află E, fiecare segment arelungimi egale față de mediană, iar echilibrul se stabilește datorită simetriei. Acest argument

 poate fi ușor făcut riguros prin folosirea de dreptughiuri foarte mici în loc de linii, iar acestlucru l-a făcut Arhimede în lucrarea Despre Echilibrul Planelor .

Deci centrul maselor unui triunghi trebuie să fie la intersecția medianelor. Pentru triunghiulîn chestiune, o mediană este linia y = 1/2, în timp ce a doua mediană este linia y = 1 -x.Intersecția celor două mediane se află în punctul x = 2/3, deci întreaga masă a triunghiului seaflă în acest punct. Aria întregului triunghi este 1/2, deci momentul total al triunghiului fațăde punctul se sprijin este egal cu 1/3. Rezultă că masa parabolei, precum și aria ei, trebuie săfie 1/3.

Această metodă poate fi folosită pentru a a calcula aria oricărei secțiuni arbitrare a unei parabole. Similar argumentele pot fi folosite pentru a găsi integrala oricărei puteri a lui x,deși pentru puterile de ordin superior calculul devine complicat fără algebră. Arhimede amers în măsura posibilului până la integrala x3, pe care a folosit-o pentru a găsi centru demasă al unei emisfere.

Prima propoziție din manuscris

Curba din figură este o parabolă.

Punctele A și B se află pe curba. Dreapta AC este paralelă cu axa parabolei. Dreapta BC estetangentă la parabolă.

Prima propoziție afirmă că:

 Aria triunghiului ABC este de trei ori mai decât aria limitată de parabolă și dreapta secantă AB.

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 Demonstraț ie: Fie D mijlocul segmentului AC . Punctul D este punctul de sprijin al pârghiei, care este segmentul JB. Punctele J  și B se află la distanță egală față de punctul de sprijin. După cum a demonstrat Arhimede, centrul de greutate altriunghiului se află în punctul I pe pârghie, astfel încât DI : DB = 1:3. Este deci

suficient să arătăm că dacă întreaga masă a triunghiului se află în I  și întreaga masă a parabolei în J , pârghia este în echilibru. Dacă întreaga masă a triunghiului se află în I ,ea exercită același moment pe pârghie ca și fâșia infinit mică a oricărei secțiunitransversale EH paralelă cu axa parabolei și care acționează în punctul G careapar ține pârghiei. De aceea este suficient să demonstrăm că dacă greutatea fâșiei EH  se află în G și greutatea fâșiei EF din parabolă se află în J , atunci pârgia este înechilibru.

Cu alte cuvinte, este suficient să demonstrăm că EF :GD = EH : JD. Dar acest lucru esteechivalent cu EF : DG = EH : DB, care este echivalent cu EF : EH = AE : AB. Dar acesta

este tocmai ecuația parabolei.

Volumul unei sfere

Din nou, pentru a clarifica metoda mecanică, este convenabil să folosim coordonategeometrice. Dacă o sferă de rază 1 este plasată în punctul x = 1, secțiunea transversală înorice punct x aflat între 0 și 2 este dată de formula:

Masa secțiunii transversale, în scopul echilibrării pârghiei, este propor țională cu aria:

Arhimede a considerat regiunea dintre y = 0 și y = x din planul x- y rotindu-se în jurul axei x, pentru a forma un con. Secțiunea transversală a acestui con este un cerc cu raza egală cu

iar aria acestei secțiuni este

Deci, dacă fâșiile conului și al sferei sunt luate împreună, aria secțiunii transversalecombinate este:

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Dacă cele două fâșii sunt plasate împreună la distanța 1 de punctul de sprijin, greutatea lor vafi balansată de un cerc cu aria egală cu aflat la distanța x de cealaltă parte a punctului desprijin. Acest lucru însemnă că sfera și conul luate împreună vor balansa un cilindru de pe

 partea opusă a pârghiei.

Pentru a echilibra fâșiile pe axa x, fiecare fâșie a sferei și a conului trebuie atârnate ladistanța 1 de punctul de sprijin, astfel încât momentul va fi propor țional cu aria. Dar fâșiacorespunzătoare cilindrului trebuie atârnată la distanța x pe partea opusă, Cum x variază între0 și 2, cilindrul va avea centrul de greutate la distanța 1 de punctul de sprijin, astfel încâttoată greutatea cilindrului poate fi considerată la distanța x = 1. Condiția de echilibru asigurăfaptul că volumul conului plus volumul sferei este egal cu volumul cilindrului.

Volumul cilindrului este egal cu aria secțiunii transversale înmulțită cu înălțimea careeste egală cu 2, adică . Arhimede a putut să afle volumul conului folosind metodamecanică, deoarece, în termeni moderni, integrala implicată este aceeași cu cea folosită

 pentru calculul ariei parabolei. Volumul conului este 1/3 din aria bazei înmulțită cu înățimea.Baza conului este cercul cu raza 2, având aria și înălțimea 2, iar volumul conului este

. Scăzând volumul conului din cel al cilindrului obținem volunul sferei:

Dependența volumului sferei vine evident de la suprafața ei. Metoda ne dă formula familarăa volumului sferei și înmulțind liniar dimensiunile, Arhimede a putut ușor extinde volumulrezultat la sferoizi.

Argumentele lui Arhimede sunt aprope identice cu argumentele de mai sus, dar cilindrul lui aavut o rază mai mare, deci conul și cilindrul atârnă la o distanță mai mare de punctul desprijin. El consideră acest argument a fi marea lui realizare, cerând ca figura cu echilibrulsferei, a conului și a cilindrului să fie gravate pe piatra de mormânt.

 Aria sferei 

Pentru a găsi aria sferei Arhimede argumentează că, așa cum aria cerului poate fi împăr țită

într-o infinitate de triunghiuri mici în jurul circumferinței (vezi Măsurarea cercului), tot așavolumul sferi poate fi divizat în multe conuri cu înălțimea egală cu raza, iar baza să fie pesferă. Toate conurile vor avea aceeași înălțime, deci volumul lor va fi 1/3 multiplicat cu aria

 bazei și înălțimea.

Arhimede stabilește că volumul sferei este egal cu volumul conului a cărei bază are aceeașiarie ca a sferei, iar înălțimea egală cu raza. Nu există detalii pentru acest argument, dar evident, conul poate fi divizat într-o infinitate de conuri, fiecare con aducându-și contribuțiaconform cu aria bazei, la fel ca la sferă.

Fie S suprafața sferei. Volumul conului cu aria S  și înălțimea r este egal cu: , caretrebuie să egaleze volumul sferei, egal cu: . De aceea suprafața sferei trebuie să fie

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egală cu: , sau de patru ori aria cercului mare. Arhimede demonstrează riguros acestlucru în lucrarea Despre Sferă și Cilindru.

Forme curbilinii cu volume raționale

Unul din lucrurile remarcabile din Metoda mecanică este acela că Arhimede a găsit douăforme definite prin secționarea cilindrului și al căror volum nu implică valoarea π , deșiforrmele au margini curbilinii. Acesta este punctul principal al cercetării dacă anumite formecurbilinii pot fi trasate cu rigla și compasul, astfel încât să existe relații raționale netrivialeîntre volume definite de intersecții geometrice prin solide.

Arhimede a accentuat acest lucru la începutul tratatului și i-a invitat pe cititori să reproducărezultatul prin alte metode. Spre deosebite de alte exemple, volumele acestor forme nu sunt

riguros calculate în nici o altă lucrare. Printre fragmentele manuscrisului ar apare căArhimede a înscris și circumscris formele pentru a demonstra riguros limitele volumului,deși detalii despre acest lucru nu există.

Arhimede a considerat două forme, una este intersecția a doi cilindrii sub un unghi drept,aflată în regiunea ( x, y, z) care satisfac condițiile:

(2Cyl)

și prisma circulară, care satisfac condițiile:

(CirP)

Ambele probleme au o por țiune care produce o integrală simplă pentru metoda mecanică.Pentru prisma circulară, tăiem axa x în felii. Regiunea din planul y- z la orice x este interioarăunui triunghi dreptunghic de lungime a cărui arie este , astfel că volumultotal este:

(CirP)

Care poate fi ușor rectificat folosind metoda mecanică, adăugând fiecărei secțiunitrunghiulare o secțiune a unei piramide triunghiulare cu aria echilibrând o prismă a căreisecțiune este constantă.

Pentru intersecția celor doi cilindrii, por țiunea din manuscris s-a pierdut, dar poate fi evidentreconstituită prin comparație cu restul documentului: dacă planul x-z are direcția feliilor,ecuația pentru cilindru ne arată că în timp ce , definind o regiune care esteun pătrat în planul x- z având lungimea laturii egală cu math>\scriptstyle 2\sqrt{1-y^2}</math>, astfel că volumul total este:

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(2Cyl)

Iar aceasta este aceeași integrală ca cea din exemplul precedent.

 Alte propoziții din manuscris

O serie de propoziții de geometrie sunt demonstrate în manuscris cu argumente similare. Oteoremă afirmă că locul centrului de greutate al unei emisfere este la 5/8 din distanța dintre

 pol și centru sferei. Această problemă este remarcabilă, deoarece trebuie evaluată o integralăcubică.

Vezi și 

• Metoda epuizării• Calculul infinitezimal

• Integrală

Referințe

• Archimedes (translated by Thomas Little Heath), The method of Archimedes recentlydiscovered by Heiberg; a supplement to the Works of Archimedes, CambridgeUniversity Press, 1912.

Ultima modificare efectuată la 12:53, 1 noiembrie 2011.

http://ro.wikipedia.org/wiki/Metoda_Teoremelor_Mecanicii


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