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Kako su Euklidovi elementi vjerojatno napisani?

Kako su Euklidovi elementi vjerojatno napisani?


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Kako su napisani Euklidovi elementi? Mislim, koju vrstu papira, mastila i olovke je vjerovatno koristio Euclid?

Pretpostavljam da na ovo pitanje nema definitivnog odgovora, možete odgovoriti i:

Kako su ljudi 300. godine prije nove ere pisali u staroj Grčkoj?

Je li se pisanje u to vrijeme moglo usporediti s pisanjem na modernom papiru modernom olovkom (posebno u pogledu brzine / tačnosti)?


Stari Grci pisali su isto kao i Egipćani: na svitcima od papirusa. Najstariji evropski rukopis za koji znam je Dervenski papirus, grčki svitak iz 362. godine prije nove ere ... Pisanje je napisano uskom četkom, a ne olovkom. Grci su pisali i o glini, poput fragmenata slomljenih lonaca (ostraka). Učenici bi pisali na tlu ili na voštanim tablicama koje su bile sloj pčelinjeg voska udubljene u dvije ploče koje su imale rezane šarke, tako da su se mogle zatvoriti zajedno:

Na vosku je bio ugraviran šiljati drveni štap zvan a olovka. Zadnji kraj štapa imao je lopaticu koja se mogla koristiti za brisanje, kao što se može vidjeti na gornjoj slici.

Pisanje na papirusu bilo je relativno sporo u usporedbi s olovkom, jer se njime četkalo. To je ipak bilo nevažno, jer kada se nešto pisalo na papirus (vrlo skupo) to je obično bilo važno djelo, pa količina vremena koja je bila potrebna za njegovo pisanje nije bila važna. Da su htjeli pisati bilješke ili nešto brzo, upotrijebio bi se vosak, što je bilo relativno brzo. U nekim slučajevima ugljen se koristio za bilježenje na prikladnoj površini poput gline ili kamena.


Euklid bi, poput drugih drevnih autora, vjerovatno diktirao svoje knjige pismenom robu, koji ih je zapisao na voštane stolove, olovkom, kako je opisao Tyler. Tablete bi tada bile proslijeđene profesionalnom pisaru (također robovu), koji ih je kopirao na svitke napravljene od papirusa.


Euklidska geometrija Prva velika nauka

Evo uvodne zagonetke. U cjelini našeg intelektualnog naslijeđa, koja se knjiga najviše proučava i najviše uređuje? Odgovor je očit: Biblija. Ali koje je djelo nakon njega najviše proučavano i uređivano? To je malo teže reći. Odgovor dolazi iz grane znanosti koju sada uzimamo zdravo za gotovo, geometrije. Djelo su Euklidovi elementi. Ovo je djelo koje je kodificiralo geometriju u antici. Napisao ga je Euklid, koji je živio u grčkom gradu Aleksandriji u Egiptu oko 300. godine prije nove ere, gdje je osnovao matematičku školu. Od 1482. godine štampano je više od hiljadu izdanja Euklidovih elemenata. To je bio standardni izvor geometrije milenijumima. Tek smo posljednjih desetljeća počeli odvajati geometriju od Euklida. U živom sjećanju-mojem sjećanju na srednju školu-geometrija se još uvijek učila koristeći razvoj Euklida: njegove definicije, aksiome i postulate i njihovu numeraciju.


Oxyrhynchus papirus koji prikazuje fragment Euklida Elementi, AD 75-125 (procijenjeno)


Naslovna stranica prve engleske verzije Euklida ser Henryja Billingsleyja Elementi, 1570


Oliverovo Birneovo izdanje prvih 6 Euklidovih knjiga iz 1847. godine Elementi upotrijebio što je moguće manje teksta i zamijenio oznake bojama.


Nedavno izdanje iz Dovera.

Ova duga istorija jedne knjige odražava ogromnu važnost geometrije u nauci. Sada često razmišljamo o fizici kao znanosti koja nas vodi. U sedamnaestom stoljeću Newton je pronašao jedan jednostavan sistem fizike koji je radio i za nebo i za zemlju. To je postavilo standard postignuća na koje su se druge nauke trudile oponašati. Newton je, međutim, učio od druge znanosti koja je već postavila trajni standard postignuća: geometrije.

Možemo identificirati dva razloga za važnost Euklidovih elemenata u našem razumijevanju temelja nauke: njenu strukturu i pouzdanost njenih rezultata.


Kako su Euklidovi elementi vjerojatno napisani? - Istorija

Euklidovi elementi - istorija od 2.500 godina
Bob Gardner
Državni univerzitet Istočni Tennessee
Odsjek za matematiku i statistiku
Johnson City, TN 37614


Slika iz arhive istorije matematike MacTutor -a.

Geometrija nikako nije započela s Euklidom. U stoljećima koja su prethodila Euklidu bilo je mnogo društava na Istoku i Zapadu koja su bila upoznata s određenim geometrijskim idejama, uključujući Pitagorinu teoremu. Na Mediteranu je bilo mnogo geometara koji su prethodili Euklidovom vremenu 300. p. N. E.

Drevni grad Vavilon nalazio se u južnom dijelu Mezopotamije, oko 50 milja južno od današnjeg Bagdada u Iraku. Glinene ploče koje sadrže vrstu pisma koja se naziva "klinasto pismo" preživjele su iz babilonskih vremena, a neke od njih odražavaju da su Vavilonci imali sofisticirano znanje o nekim matematičkim idejama, nekim geometrijskim, a nekim aritmetičkim [Bardi, stranica 28].

Egipćani su takođe dobro poznavali geometriju. Godišnje poplave rijeke Nil izbrisale su granice između različitih parcela na riječnom frontu. To je zahtijevalo od Egipćana da primijene geometriju za mjerenje zemljišta u poplavnoj ravnici kako bi utvrdili ko posjeduje koje zemljište. Ova nekretnina ispred rijeke koristila se za poljoprivredu, a ljudi su se oporezivali na osnovu površine zemljišta koje su obrađivali. Stoga je za društvo bilo od vitalne važnosti da precizno izvrši ovo mjerenje.


Slika sa Wikipedije. Korištene tehnike otkrivene su u papirusu pod nazivom "Upute za stjecanje znanja o svim mračnim stvarima". Sadrži brojne matematičke probleme, napisao ju je pisac po imenu Ahmes, a datira iz oko 1650. godine prije nove ere [Bardi, stranice 26 i 27]. Ponekad se naziva i Iza matematičkog papirusa. Međutim, Egipćani su bili pragmatični sa svojom geometrijom i nisu napredovali u teorijsko područje.


Tales iz Mileta (624. pne., 524. p. N. E. (Oko))
Slika iz arhive istorije matematike MacTutor -a. Tradicija kaže da je Tales prva osoba koja se uključila u dokazivanje teorema. On je iz primorskog grada Mileta u današnjoj Turskoj.


Slika Thalesa iz arhive istorije matematike MacTutor -a. Posjetio je Egipat, tamo naučio geometriju i za njega se kaže da je mjerio visinu velike piramide u Gizi mjerenjem dužina sjena. Pitagora je kao mladić posjetio Tales i na osnovu ove posjete odlučio sam otići u Egipat. Međutim, većina informacija o Thalesu je spekulativna i treba ih uzeti s rezervom [Bardi, stranice 29, 30 i 31]!


Pitagora sa Samosa (569. pne. Do 475. pne. (Oko))
Slika iz arhive istorije matematike MacTutor -a. Nakon Euklida, najpoznatiji matematičar antičkog svijeta je Pitagora. Rođen je oko 570. godine prije nove ere, a odrastao je u naprednoj kulturi na početku procvata klasične grčke civilizacije.

Rođen je na ostrvu Samos, na zapadnoj obali današnje Turske. On je također živio u Babilonu, proveo neko vrijeme u Egiptu i nastanio se u Crotonu u južnoj Italiji. U Crotonu je osnovao svoju poznatu pitagorejsku školu oko 530. pne.


Image from http://thespiritguides.ning.com/profile/GoldenFlower (novo doba) Pitagorina škola je više ličila na vjersku sektu nego na savremeni akademski institut. Pitagorejci su se zarekli da će svoja otkrića držati u tajnosti i nema pisanih zapisa o njihovom radu. [Moar, stranica 17] Pitagorejci su izbjegavali vino i meso i, što je najpoznatije, zabranili su konzumiranje pasulja ili bilo koje hrane koja izaziva nadutost [Bardi, stranica 36].


Pitagora kako je prikazan u Rafaelovoj knjizi Atinska škola
Image from http://students.roanoke.edu/groups/relg211/johnson/Pythagoras.htm. Uz njegovu dobro poznatu teoremu (koja je vjerojatno bila rezultat pitagorejske škole, ali vjerovatno nije bila posljedica samog Pitagore), Pitagora igra vrlo temeljnu ulogu u povijesti matematike. On je bio prva osoba koja je zaista apstraktno razmišljanje stavila u matematiku. Postavio je geometriju na put s kojeg se više neće vratiti [Bardi, stranica 37]. Nakon njegove smrti oko 475. godine prije nove ere, član njegove škole se razišao, a intelektualni centar matematike preselio se iz Italije u Grčku [Bardi, stranica 41].


Uvod

Euclid & rsquos Elementi čine jedno od najljepših i najutjecajnijih naučnih djela u istoriji čovječanstva. Njegova ljepota leži u logičkom razvoju geometrije i drugih grana matematike. Utjecao je na sve grane znanosti, ali ne toliko na matematiku i egzaktne znanosti. The Elementi proučavali su se 24 stoljeća na mnogim jezicima počevši, naravno, na izvornom grčkom, zatim na arapskom, latinskom i mnogim modernim jezicima.

Ja stvaram ovu verziju Euclida & rsquos Elementi iz nekoliko razloga. Glavni je ponovno pobuditi interes za Elementi, a web je odličan način za to. Drugi razlog je pokazati kako se Java apleti mogu koristiti za ilustraciju geometrije. To takođe pomaže u donošenju Elementi živ.

Tekst svih 13 knjiga je potpun, a sve figure ilustrirane su pomoću Geometrije Applet, čak i one u posljednje tri knjige o čvrstoj geometriji koje su trodimenzionalne. Imam još puno toga za napisati u odjeljcima vodiča i to će me zaokupiti još neko vrijeme.

Ovo izdanje Euclid & rsquos Elementi koristi Java applet pod nazivom Geometry Applet za ilustraciju dijagrama. Ako omogućite Javu u svom pregledniku, tada ćete & rsquoll moći dinamički mijenjati dijagrame. Da biste vidjeli kako, pročitajte Korištenje apleta geometrije prije nego pređete na Sadržaj.

Često čujem da se geometrija više ne uči dobro ovdje u srednjim školama u Sjedinjenim Državama. (Također razumijem da se u nekim srednjim školama uopće ne uči.) To je veliki problem jer se deduktivna logika uči gotovo isključivo u geometriji. Bez razumijevanja logike, studenti će imati poteškoća u svakodnevnom životu i poteškoće na fakultetu ako krenu na fakultet.

Savremena matematika i nauka koriste deduktivnu logiku kao primarno oruđe razumijevanja. Posebno se u matematici ništa ne zna dok se ne dokaže.

Jedan faktor koji doprinosi, možda i najveći faktor, padu geometrijskog obrazovanja u Sjedinjenim Državama je način na koji je predstavljen u udžbenicima. Ako logika nije predstavljena u udžbenicima, nastavniku će biti jako teško da je ubaci u razred.

Nedavni udžbenik, Prentice-Hall's Geometrija: alati za mijenjanje svijeta pokazuje koliko je obrazovanje geometrije danas loše. Za detalje pogledajte moju recenziju knjige.

Za širu kritiku matematičkog obrazovanja u Sjedinjenim Državama, pogledajte web stranicu Matematički ispravna. ->


Kako su Euklidovi elementi vjerojatno napisani? - Istorija

Euklida, starogrčkog matematičara, često nazivaju „ocem geometrije“. Ovo je ime koje se nadaleko nalazi u istoriji matematike i na čitavom polju nauke. Ali postoje neke spekulacije o tome da li je ovaj čovjek zaista postojao. Neki znanstvenici tvrdili su da je Euklid "#8212 ako ne i mitska figura", vjerojatnije ime dato grupi, školi ili tradiciji drevnih matematičara koji su razvili osnove geometrije.

Euklidov potencijalni početak

Bilo je donekle uobičajeno da su vrhunski matematičari u staroj Grčkoj ostavljali biografije svojih života i djela, ali za čovjeka po imenu Euklid ne postoji. Međutim, postoji nekoliko raštrkanih referenci o čovjeku po imenu Euklid u drugim tekstovima. To sugerira da je on možda zaista živio i radio u Egiptu u vrijeme vladavine Ptolomeja I.

Ptolomej je bio Grk koji je vladao Egiptom od 367. godine p.n.e. do otprilike 283. pne., pa se to često navodi kao Euklidov vremenski period. Ne postoje zapisi o rođenju ili smrti Euklida. Neki navode datum Euklidovog rođenja 325. godine p.n.e.

Mnoge reference na Euklida vrlo su problematične. Na primjer, jedan od naših primarnih izvora informacija o Euklidu dolazi iz djela Prokla#8212, ali on je živio 500 do 700 godina nakon Euklida.

Euklidovo djelo

Glavno djelo pripisano Euklidu je Elementi. Ovo je serija od 13 knjiga koje sadrže osnovu za euklidsku geometriju. Elementi opisuje aksiome, teoreme i konstrukcije — šta su oni, kako rade i kako ih koristiti. On također pruža matematičke dokaze za sve gore navedene definicije.

Također je izložena teorija brojeva starih Grka, kao i geometrijska algebra. Ovaj posljednji sistem je dovoljno sveobuhvatan za rješavanje osnovnih problema algebre, uključujući kako pronaći kvadratne korijene brojeva.

Dostignuća koja proizlaze iz Elementi su zapanjujuće. Na primjer, u ovim radovima nalazimo prvu prezentaciju i objašnjenje aksiomatskog sistema. Aksiomatski sistem je svaki skup aksioma (postulata ili premisa) iz kojih se svi aksiomi mogu koristiti za izvođenje teorema pomoću logike. Sve matematičke teorije koriste aksiomatski sistem.

Ostala Euklidova djela

Osim 13 knjiga iz Elementi, postoji još pet djela koja su preživjela do modernog doba i pripisuju se Euklidu kao autoru. Među njima je i knjiga pod nazivom Optika. Ovaj rad se bavio načinom na koji ljudsko oko percipira vanjski svijet. Sugerira da je vid moguć zahvaljujući "zrakama" koje izbijaju ili "izviru" iz ljudskih očiju.

Još važnije je jedna od definicija koje su proizašle Optika. Kaže da kada vidimo objekte iz većeg kuta, ti objekti izgledaju veći, a da kada vidimo objekt iz manjeg kuta, oni se čine manjim. I na kraju, kada stvari vidimo iz jednakih uglova, one izgledaju jednake.

Ovi temeljni aspekti ispitivanja prirode vida i načina na koji opažamo kutove danas se uzimaju zdravo za gotovo, ali u davna vremena malo se toga razumjelo. Da bi Euklid uhvatio te osnove značilo je da ljudi ne samo da mogu početi s preciznim mjerenjima, već su mogli i manipulirati tim "podacima" s dobro razvijenim matematičkim formulama koje su univerzalne i pouzdane.

Čak ni sam predmet „podataka“ u jednom trenutku nije imao odgovarajuću ili univerzalnu definiciju, što nam je danas teško zamisliti. Međutim, to je bila Euklidova knjiga, Podaci, koji je definirao značenje "davanja informacija" koje su pouzdane i koje se mogu primijeniti na probleme matematike i praktične primjene.

Euklidska kontroverza

13 knjiga iz Elementi i pet drugih postojećih knjiga koje se pripisuju Euklidu značile bi da bi, da ih je napisao jedan pojedinac, Euklid bio jedan od najbriljantnijih ljudi u čitavoj istoriji.

Stoga je primamljivo pretpostaviti da je njegovo izvanredno djelo zapravo kombinovani rad mnogih briljantnih matematičara. Možda je slučaj da je Euklid bio jedan od mnogih, ili možda primarni pojedinac iza razvoja moderne geometrije.


IV. Sadržaj elemenata

Prije rasprave o tome šta je u Elementima, mora se zabilježiti šta je ne sadržane u radu. Prvo, iako Euclid opsežno koristi koncept pravolinijskog područja, nigdje ne daje formulu za izračunavanje površine figure. Grci su napravili jasnu razliku između logistike, koju Platon identificira kao "umjetnost računanja", i aritmetika, koja je danas poznata kao teorija brojeva. [48] ​​Elementi su potpuno lišeni logistike. Drugo, budući da Euklid cijelu svoju geometriju temelji na točkama, ravnim linijama i krugovima (i na taj način konstruira samo pomoću ravne ivice i kompasa), takozvana tri poznata problema grčke matematike-kvadrat kruga, udvostručavanje kocke i trisektirajući kut - ne nalaze se u radu. Konačno, konusni presjeci, poznati otprilike 50 godina otkad ih je otkrio Menaechmus, još su se smatrali da su u domeni više matematike u Euklidovo doba i da se stoga ne pojavljuju u Elementima. [49] Kao što je gore spomenuto, Euklid je obrađivao konike u zasebnom radu.

Knjiga I

Definicije I knjige

U knjizi I, Euclid definira osnovne pojmove geometrije ravnine, uključujući tačka, linija, površine, ugao, figura, i tako dalje. Većina definicija u ovoj i kasnijim knjigama je neupadljiva, međutim, neke zaslužuju raspravu ili zbog njihove originalnosti ili zbog njihovog historijskog značaja. Na primjer, način na koji Euclid definira točku, liniju i površinu očito se razlikuje od definicija danih u ranijim udžbenicima. Aristotel je, pišući prije vremena Elemenata, primijetio da su u standardnim definicijama ovih objekata prioriteti definirani u smislu stražnjeg dijela, odnosno da je točka definirana kao kraj linije, linija površine i površina čvrstog tijela. Prema njegovom mišljenju, to je definicije učinilo nenaučnim. Euklid, možda kao odgovor na takve kritike, pokušava definirati svaki pojam neovisno o drugima. Tako je točka "ono što nema dijela" (I.Def.1), linija je "bezdužna dužina" (I.Def.2), a površina je "ono što ima dužinu i samo širina '' (I.Def.5). Nakon što je definirao svaki pojam, tada se vraća na starije definicije kako bi koncepte povezao zajedno. Smatra se da je kompromis prikazan u ovim definicijama Euklidova vlastita ideja. [50]

Druga važna definicija je posljednja u Knjizi I: ``Paralelno ravne linije su prave linije koje se proizvode beskonačno u oba smjera i ne susreću se jedna u drugu u oba smjera '' (I.Def.23). To je u osnovi ista definicija koju je dao Aristotel. [51] Euclid je odlučio da ne koristi drugu popularnu definiciju paralelnih linija kao ravnih linija koje su svuda jednako udaljene jedna od druge. Ova definicija je ključna za takozvani `` paralelni postulat '' (I.Post.5), koji je uživao u živoj istoriji. O tome će biti više riječi u sljedećem odjeljku.

Na kraju, uslovi duguljast, romb, i romboid su definirane, ali, što je zanimljivo, nikada se ne pojavljuju nigdje drugdje u Elementima. Vjeruje se da su preuzeti iz ranijih radova na tu temu. [52]

Postulati i aksiomi prve knjige

Evo postulata koje je dao Euklid:

  1. Da biste povukli ravnu liniju iz bilo koje tačke u bilo koju tačku.
  2. Proizvoditi konačnu ravnu liniju kontinuirano u ravnoj liniji.
  3. Opisati krug s bilo kojim centrom i udaljenošću.
  4. Da su svi pravi uglovi jednaki.
  5. Da, ako ravna linija koja pada na dvije ravne linije čini unutarnje kutove na istoj strani manjim od dva prava ugla, dvije ravne linije, ako se definitivno proizvode, susreću se na onoj strani na kojoj su kutovi manji od dva prava uglovi.

U prva tri postulata Euklid pretpostavlja postojanje točaka, linija i krugova. To je potrebno jer je, kao što je ranije spomenuto, definicija od ovih objekata nije implicirano njihovo postojanje. Postojanje svih ostalih geometrijskih objekata dokazano je u kasnijim postavkama. Prvi postulat se može tumačiti i kao potvrđivanje jedinstvenost prave linije između dvije date tačke. [53] Slično, treće se tumači tako da potvrđuje kontinuitet i beskonačan opseg prostora na ovaj način: polumjer kruga može biti neograničeno mali, što implicira da ne postoji minimalna udaljenost između dvije točke u prostoru - stoga je prostor S druge strane, radijus može biti neograničeno velik, pa ne postoji maksimalna udaljenost između dvije točke u prostoru. Sasvim je moguće da je Euclid uvidio "jedinstvenost" interpretacije prvog postulata, ali je sumnjivo da je treći protumačio na gornji način. [54]

Četvrti i peti postulat dugo su se smatrali teoremama koje se mogu dokazati. Četvrti tvrdi da je pravi ugao determinanta veličine, prema kojoj se mogu mjeriti svi drugi uglovi. [55] Vjeruje se da je peti postulat originalan kod Euklida. [56] Nazvana je "jednom rečenicom u istoriji nauke koja je dovela do objavljivanja više od bilo koje druge." [57] Ideja da se to može dokazati temelji se na njenoj dužini i složenosti, a činjenica da je njegov obrnuti teorem (I.17) koji dokazuje Euklid. [58]

Ovdje se neće pokušavati sa istorijom paralelnog postulata. Neke od značajnijih pokušaja da to dokaže dao je Heath u svom izdanju Elements. [59] Nemogućnost dokazivanja paralelnog postulata uvjerila je neke, naime Carla Friedricha Gaussa (1777-1855) i Nicolaia Ivanoviča Lobačevskog (1793-1856), da neeuklidski geometrije su bile moguće. 1829. Lobačevski je prvi objavio geometriju izgrađenu na postulatu koji je u direktnoj suprotnosti s paralelnim postulatom. [60]

Pet gore navedenih izjava jedine su u Elementima koje Euklid identifikuje kao postulate. Međutim, četvrta i peta definicija u Knjizi V imaju oblik postulata. [61] O njima će biti riječi dolje u odjeljku knjige V.

Euklidov aksiom, koji on naziva uobičajeni pojmovi, su sljedeće:

  1. Stvari koje su jednake istoj stvari jednake su i jedna drugoj.
  2. Ako se jednakima doda jednako, cijele su jednake.
  3. Ako se jednaki oduzmu od jednakih, ostaci su jednaki.
  4. Stvari koje se međusobno podudaraju jednake su jedna drugoj.
  5. Cjelina je veća od dijela.

Od ovih, samo četvrti zahtijeva objašnjenje Euklidove namjere. Vjeruje se da u ovom aksiomu on tvrdi da je superpozicija prihvatljiva metoda dokazivanja jednakosti dviju figura. [62] Dakle, u I.4, kako bi dokazao takozvanu teoremu kongruencije "Side-Angle-Side" za trokute, on zamišlja jedan trokut koji treba premjestiti i postaviti jedan na drugi, a zatim pokazuje da su svi strane se međusobno podudaraju.

Kasniji su pisci dodali mnoge druge aksiome, uključujući:

  • (6) Dvije linije ne zatvaraju razmak.
  • (7) Ako se nejednakim doda jednako, cijele su nejednake.
  • (8) Ako se jednaki oduzmu od nejednakih, ostaci su nejednaki.
  • (9) Dvojice iste stvari jednake su jedna drugoj.
  • (10) Polovine iste stvari jednake su jedna drugoj.

Međutim, sve je to izvedeno iz Euklidovih postulata i aksioma. Imajte na umu i da bi "aksiom" šest zapravo trebao biti postulat, jer se bavi geometrijskim objektima. [63]

Propozicije knjige I

U prijedlozima knjige I, Euclid predstavlja poznatu geometriju linija i kutova u ravnini, uključujući rezultate o trokutima, linijama koje se sijeku, paralelnim linijama i paralelogramima. Prve tri propozicije daju tri temeljne 'operacije' Euklidove geometrije:

  1. Na konačnoj pravoj liniji konstruirati jednakostranični trokut.
  2. Postaviti u bilo koju datu točku (kao kraj) pravu liniju jednaku datoj pravoj.
  3. S obzirom na dvije nejednake ravne linije, odrezati od veće ravnu liniju jednaku manjoj.

Većina preostalih prijedloga Elemenata ovisi o tim konstrukcijama.

U prijedlozima Knjige I sadržane su i teoreme o kongruentnim trokutima, poznatim današnjim geometrima u srednjoj školi kao "Side-Angle-Side" (I.4), "Side-Side-Side" (I. 8), `` Angle-Side-Angle '' (I.26) i `` Angle-Angle-Side '' (takođe I.26). Jedinstvenost trokuta sa zadanim stranicama dokazana je u I.7. U I.22, Euclid pokazuje kako konstruirati trokut iz bilo koje tri zadane linije (pod uvjetom da je to moguće). Da se uglovi trokuta zbrajaju u dva prava ugla dokazano je u I.32.

Odredbe 9 i 10 pozivaju na prerezivanje datog ugla, odnosno na segment duži. Da su suprotni uglovi formirani presecanjem linija jednaki dokazuje se u I.15. Odredbe 27-31 se bave paralelnim pravcima u I.31, Euklid konstruiše liniju paralelnu datoj liniji koja prolazi kroz datu tačku koja nije na pravoj.

Postojanje paralelograma je dokazano u I.33. Nakon ovog stava postoje mnoge teoreme o paralelogrammičkim figurama i područjima koja oni obuhvataju. Propozicija 42 zahtijeva konstrukciju paralelograma pod određenim uglom jednakog po površini datog trougla. Ova ideja je proširena u I.45 na konstrukciju paralelograma pod određenim kutom jednake površine datoj pravocrtnoj figuri. Konstrukcija kvadrata na datoj liniji data je u I.46. Konačno, propozicije 47 i 48 predstavljaju tzv Pitagorina teorema i obrnuto. Vjeruje se da je dokaz I.47, zasnovan na liku poznatom kao "mladenkina stolica", originalan s Euklidom. [64]

Knjiga II

Sadržaj Knjige II tradicionalno je shvaćen u smislu geometrijska algebra. Prema ovom gledištu, stari Grci su uzimali algebarske rezultate i iznosili ih u geometrijskim terminima, uglavnom radi strogosti. Ovo je bilo dominantno gledište prema starogrčkoj matematici otkad su ga 1880 -ih razvili Hijeronimus Georg Zeuthen i Paul Tannery. U posljednje dvije decenije, međutim, ovo gledište doživjelo je oštru kritiku onih koji smatraju da nije historijski valjano. Rasprava o geometrijskoj algebri još je daleko od kraja. [65]

Jedna od atrakcija pojma geometrijske algebre je ta što daje udoban okvir za raspravu o rezultatima nekih knjiga u Elementima. Imajući u vidu prethodno upozorenje, algebarski oblik rezultata u Knjizi II bit će predstavljen ovdje radi praktičnosti. Prve četiri postavke su geometrijski ekvivalenti sljedećih algebarskih identiteta: [66]

  • II.1: a (b + c + d +.) = Ab + ac + ad +.
  • II.2: (a + b) a + (a + b) b = (a + b) 2
  • II.3: (a + b) a = ab + a 2
  • II.4: (a + b) 2 = a 2 + b 2 + 2ab

Propozicije 5-10 predstavljaju slične, iako složenije identitete. Važan prijedlog, tipičan za ovu grupu, je II.5:

Primjer ove tvrdnje je sljedeći, parafraziran iz Euklidovog dokaza:

Ako se AB izreže na jednake segmente u C i na nejednake segmente u D, tada je površina pravokutnika koju sadrže AD i DB (pravokutnik ADHK), plus površina kvadrata na CD -u (jednaka kvadratu LHGE), jednaka je površina kvadrata na CB (kvadratni CBFE).

U ovom slučaju postoji nekoliko mogućih dodjeljivanja simbola dužinama na slici, od kojih svaki dovodi do različitog algebarskog identiteta. Ovo pokazuje jedan problem s idejom da su Grci iz Euklida imali pojam algebre. Geometrijski iskaz i dokaz ove teoreme su nedvosmisleni. Međutim, postoji mnogo različitih načina algebarskog predstavljanja ove teoreme. Može se posmatrati kao geometrijski ekvivalent bilo kojeg od identiteta

ili kao osnovu geometrijskog rješenja jednadžbe

Sabetai Unguru, u svom kontroverznom članku "O potrebi prepravljanja historije grčke matematike", osuđuje takva tumačenja i snažno tvrdi da Grci nisu koristili ekvivalent naše algebre, već da je "[to] mi koji koriste algebru. kao ekvivalent od njihova geometrija!'' [68] Unguru je jedan od otvorenijih kritičara tradicionalnog gledišta.

U II.12 i II.13, Euklid predstavlja geometrijski ekvivalent današnjeg `zakona kosinusa`:

gdje je A kut suprotne stranice a. Na donjoj slici, koju je dao Euclid, "kvadrat na AC manji je od kvadrata na CB, BS za dva puta veći od pravokutnika koji sadrži CB, BD" (iz II.13).

Konačno, u II.14, Euclid poziva na izgradnju kvadrata jednake površine datoj pravocrtnoj figuri. Općenito, ovo je problem kvadratura. Kao što je ranije spomenuto, Euklid ne primjenjuje ovu metodu na krivolinijske figure, iako je Hipokrat iz Hiosa "kvadratio" lune -figura u obliku polumjeseca omeđena s dva kružna luka-prije više od stotinu godina. [69]

Knjiga III

Knjiga III bavi se krugovima, segmentima krugova i sektorima krugova (vidi slike ispod). Danas se smatra da nekoliko prijedloga nije sasvim zadovoljavajuće. Čini se da dokazi nekih nisu jasni, ako ne i strogi, neki se oslanjaju na nedokazane pretpostavke, a drugi pate od Euklidove tendencije da izbjegava više slučajeva u svojim dokazima. U svakom slučaju, savremeni udžbenici teže prikazivanju materijala o krugovima drugačije od Euklida. [70]

U III.1 Euklid pronalazi središte datog kruga. Odredbe 5 i 6 dokazuju da dva kruga neće imati isti centar ako se međusobno iseku (III.5) ili dodirnu (III.6). Nekoliko teorema bavi se načinima na koje se krugovi mogu ili ne moraju rezati ili dodirivati. Na primjer, da se krugovi međusobno ne sijeku na više od dvije točke, pokazano je u III.10. Propozicija 16, o linijama tangentnim na krug, historijski je zanimljiva. U njemu stoji: [71]

Na gornjoj slici tangenta EF nacrtana je na kraju BE, ugao polukruga je kut DEC, a preostali ugao je ugao DEF. Posljednji dio ove teoreme uvodi problem prirode kutova koje tvore zakrivljene i ravne linije, posebno tangente. Bilo je velikih kontroverzi oko ove teme u 13. do 17. stoljeću, prije nego što je razvoj računa dao rigorozan način rješavanja tangenti, trenutnih nagiba krivulja i slično. [72]

Odredba 17 pokazuje kako povući tangentu na krug iz bilo koje date tačke izvan kruga. U III. Da su svi uglovi upisani u isti kružni segment jednaki dokazuje se u III.21 (vidi sliku). Kružni luk je podijeljen u III.30. Pretposljednja teorema knjige III (37.) kaže da ako se iz jedne točke izvan kruga povuku dvije ravne linije, jedna dodiruje, a druga siječe, tada pravokutnik sadrži cijela linija rezanja i segment van kruga jednak je kvadratu na tangentnoj liniji (vidi sliku).

Knjiga IV

Četvrta knjiga bavi se figurama okruženim ili upisanim u krugove. Sve propozicije su problemi koji specificiraju konstrukcije koje treba izvesti. Za svaku figuru postoje četiri moguće konstrukcije:

  1. Dati krug za upisivanje date figure.
  2. Dati krug za opis određene figure.
  3. Data je figura za upisivanje kruga.
  4. Data je figura za opisivanje kruga.

Tako obrađene figure uključuju trokut bilo kojeg oblika (IV.2-5), kvadrat (IV.6-9), pravilan peterokut (IV.11-14), pravilan šesterokut (IV.15) i pravilna 15 -strana figura (IV.16).

Knjiga V

U knjizi V Euklid predstavlja teoriju proporcija općenito pripisanu Evdoksu Knidskom (umro oko 355. pne.). Ova teorija ne zahtijeva mjerljivost - odnosno upotrebu brojeva koji imaju zajednički djelitelj - i stoga je superiornija od pitagorejske teorije zasnovane na cijelim brojevima. Euklid predstavlja Pitagorinu teoriju u VII knjizi. [73] Knjige I, V i VII jedine su knjige u Elementima koje su potpuno samostalne i ne zavise od drugih knjiga. Bilo koji od njih mogao koji su poslužili kao polazna tačka za raspravu, stoga je značajno što je Euklid izabrao čisto geometrijski Knjiga I kao temelj cijelog djela. [74] Ova se činjenica može koristiti kao još jedan argument protiv gledišta geometrijske algebre. Euklid očito odgađa raspravu o proporciji što je duže moguće, iako se prirodno može posvetiti algebarski metode.

Dve definicije u ovoj knjizi su od posebnog značaja. Prema četvrtoj definiciji, `kaže se da se za veličine imati omjer to one another which are capable, when multiplied, of exceeding one another.'' That is, A and B have a ratio if nA > B for some integer n and mB > A for some integer m thus A and B are finite and non-zero. This is essentially the so-called ``Axiom of Archimedes,'' which Archimedes himself attributes to Eudoxus. [75] The fifth definition -- of magnitudes being in the same ratio -- is also very important, saying essentially that A : B :: C : D (read `` A is to B as C is to D '') if given any two integers m and n , mC < nD when mA < nB , mC = nD when mA = nB , and mC > nD when mA > nB . [76]

Other definitions present various terms used in the transformation of ratios, including alternate ratio, inverse ratio, composition of a ratio, separation of a ratio, conversion of a ratio, ratio ex aequali, i perturbed proportion. Na primjer, composition of a ratio means the transformation of the ratio A : B into the ratio A + B : B . These definitions simply give names to various manipulations of ratios.

All of the propositions in Book V are theorems. They address multiples of magnitudes, ratios of magnitudes, and magnitudes in given proportions. Most are obvious when expressed in modern symbology. In this case, however, the modern notation is not misleading, since Euclid himself often uses letters to stand for magnitudes. Three examples of the types of propositions contained in this book are the following:

  • V.4: If A : B :: C : D , then mA : nB :: mC : nD , for any integers m and n .
  • V.17: If A : B :: C : D , then (A - B) : B :: (C - D) : D .
    (This is an example of separation of ratios.)
  • V.21: If A : B :: E : F and B : C :: D : E , then D < F if A < C , D = F if A = C , and D > F if A > C .
    (The conclusion of this proposition may also be stated A : C :: D : F .)

Book VI

Book VI deals with similar rectilinear figures and extends the technique of transformation of areas developed in Book I into the method of application of areas, a major constituent of the geometrical algebra view of Greek mathematics. The first definition of this book states: ``Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional'' (VI.Def.1). The dependence of the theory of similar figures on proportions is the reason why the theorems of similarity do not appear in Book I.

In VI.1 it is proved that triangles of the same height ``are to one another as their bases'' -- i.e., that their bases and areas are proportional. Propositions 4-7 concern similar triangles. In VI.11, Euclid finds a third proportional to two given straight lines -- i.e., given A and B , find C such that A : B :: B : C . In VI.12, he finds a fourth proportional to three given lines -- given A , B , and C , find D such that A : B :: C : D . Finally, in VI.13, he finds a mean proportional between two given lines -- given A and B , find C such that A : C :: C : B .

Several theorems follow concerning figures whose sides are in various proportions. Proposition 18 calls for the construction on a given straight line of a figure ``similar and similarly situated to a given rectilineal figure.'' Another powerful technique in the transformation of areas is given in VI.25: ``To construct one and the same figure similar to a given rectilineal figure and equal [in area] to another given rectilineal figure.'' An example of this would be to construct a quadrilateral similar to a given quadrilateral, but equal in area to a given triangle.

Propositions 28 and 29 are examples of the method of application of areas. This is a controversial subject, since the traditional interpretation of this technique is that the Greeks used it to solve quadratic equations. (Transformation of areas, by the way, is not controversial in the least, since it simply means constructing one figure equal in area to another figure.) An example of application of areas is VI.29: ``To a given straight line to apply a parallelogram equal to a given figure and exceeding by a parallelogrammic figure similar to a given one.''

In terms of the figure above, given straight line AB, figure C of given area, and figure D of given shape, the problem is to apply to AB a figure (AEFH) equal in area to C and exceeding by a figure (BEFG) similar to D. According to those who hold the geometrical algebra view, the Greeks used this construction to solve quadratics of the form

where the parallelograms are taken to be rectangles, and where a = AB , x = BG , b/c is the ratio formed by the sides of D , and S is the area of C . [77]

Personally, I find it hard to believe that Euclid had any notion of the algebraic equivalent to his geometrical problem. This is supported by the statement of Proclus that ``the application of areas, their exceeding and falling short [are] drevni discoveries of the Pythagorean muse,'' implying a date of discovery long before Euclid's time. [78] Thus their origin is surely not algebraic in nature.

In VI.30, a line is cut in extreme and mean ratio. This means cutting a straight line into unequal segments so that the smaller segment is to the larger as the larger is to the whole. This construction, known today as the zlatni presjek, appears again in Book XIII.

Proposition 31 is a generalization of the Pythagorean Theorem of Book I: ``In right-angled triangles the figure on the side subtending the right angle is equal to [the sum of the areas of] the similar and similarly described figures on the sides containing the right angle.'' Note that the figures in this proposition may be of bilo koji shape. According to Proclus, this theorem is original with Euclid. [79]

Book VII

In Book VII, Euclid presents Pythagorean number theory. This is the last book of the Elements that is entirely self-contained. The first definition is of the unit, being ``that by virtue of which each of the things that exist is called one.'' Definition 2 states that ``a number is a multitude composed of units.'' Thus the Pythagorean concept of number includes only integers greater than one. Subsequent definitions subdivide the set of numbers into smaller categories of numbers: čak i odd even-times even (e.g., 4 = 2 × 2 ), even-times odd ( 6 = 2 × 3 ), and odd-times odd ( 15 = 3 × 5 ) composite i prime plane i solid (the products of two and three numbers, respectively) square i cube i perfect (number equal to the sum of its divisors, as 28 = 1 + 2 + 4 + 7 + 14 ). Euclid also defines what it means for numbers to be prime to one another -- relatively prime numbers -- and composite to one another -- numbers which have a common divisor. These definitions serve as the basis for Books VII-IX, the so-called ``arithmetical'' books, which are almost entirely independent of the first six books. [80]

Propositions 2 and 3 give a method for finding the greatest common divisor of two and three numbers, respectively. Propositions 4-20 set out the Pythagorean theory of proportion based on commensurables, numbers which have a common divisor. Most of these theorems are direct analogs of those in Book V, which deals with general magnitudes. However, Euclid does not treat them as special cases of the earlier theorems, but instead proves all of them from the principles of commensurability. Primes and relative primes are treated in VII.21-32. For example, in VI.31 it is proved that any composite number is ``measured by'' (has as a divisor) some prime number. Least common multiples are discussed in VII.34-39. Here appear problems on finding the least common multiple of two (VII.34) or three (VII.36) numbers. The method employed may be extended to find the least common multiple of as many numbers as desired. [81]

Book VIII

The propositions of this book deal with numbers in continued proportion that is, geometrical progressions of the form

A typical proposition is VIII.1, which states: ``If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.'' That is, if A, B, C, . N be a geometrical sequence such that A and N are relatively prime, then there exists no other sequence of integers A', B', C', . N' with the same common ratio such that A' < A, B' < B, . N' < N . [82] The remaining propositions are along similar lines, dealing with square, cube, plane, and solid numbers in continued proportion.

Book IX

The last of the so-called ``arithmetical'' books deals mainly with multiplication and the classification of numbers in geometric progressions from the unit -- that is, sequences of the form

Proposition 14 is the fundamental theorem in number theory that a number may be resolved into prime factors in only one way. [83] In IX.20, Euclid proves that ``prime numbers are more than any assigned multitude of prime numbers'' -- that is, that the number of primes is (countably) infinite. Propositions 21-29 deal with the sums, differences, and products of odd and even numbers taken in different combinations. In proposition 36, Euclid proves that if

be prime, then S × 2 n will be a perfect number. This is still the only known method for finding perfect numbers. [84]

Book X

Most of this book is devoted to the classification of irrationals. According to a commentary by Pappus on this book, much of the theory is due to Theaetetus (425-369 B.C.). [85] It is considered the most difficult book of the Elements and is by far the longest at 115 propositions. Euclid's notions of rational and irrational are slightly different from those of today. According to the third definition, a rational line is any which is commensurable in length or in square to a given reference line, which is a priori agreed to be rational. Hence, if P is assumed to be rational, then given any proper fraction M/N that is not the square of another proper fraction (as is 1/4 = 1/2 × 1/2 ), Euclid would call both

rational, whereas today's mathematician would identify the latter quantity as irrational. According to Heath, Euclid differed from his predecessors in this extension of the concept of rational quantities. [86]

In the first Proposition of Book X, Euclid gives the theorem that serves as the basis of the method of exhaustion credited to Eudoxus. The theorem states:

This result depends on definition 4 of Book V, the so-called Axiom of Archimedes. It is this theorem which allows Euclid to compare the areas of curvilinear figures and volumes of solids in Book XII.

In X.21, Euclid begins classifying irrationals. The first category is arrived at in this way: if the sides of a rectangle be commensurable in square only (e.g., 3 and sqrt <2>), then the side of the square equal to the rectangle ( 3 × sqrt <2>, in this case) is an irrational called medial. The rest of Book X is devoted to naming and proving theorems about medials and 24 other types of irrationals. In modern algebraic symbology, all of these are of the form sqrt < sqrt± sqrt > .

Irrationals of this form with a plus sign are called binomials (discussed in X.36-72) and those with the minus, apotomes (X.73-110). [87] In X.111, it is proved that a magnitude connot be both a binomial and an apotome.

Book XI

This book contains the final group of definitions, all of which concern solid geometry. Euclid departs from tradition in many of his definitions of solid figures. For example, Aristotle defines a sphere as a solid whose ``extremity is equally distant from its center.'' [88] Euclid, however, defines a sphere as a figure described by the revolution of a semicircle about its fixed diameter (XI.Def.14). Similar ``motion-based'' definitions are given for the cone (XI.Def.18) and cylinder (XI.Def.21). Other definitions concern parallel planes, similar i equal solid figures, solid angles, prisms, piramide, and the other four regular solids -- the tetrahedron is not distinguished from the pyramid.

The propositions of Book XI begin with the elementary theorems of three-dimensional geometry: that two planes but each other in a line (XI.3), that two lines at right angles to a given plane are parallel to each other (XI.6), and so on. Two typical problems are to drop a perpendicular to a given plane from a point not on that plane (XI.11) and to construct a solid angle from three given plane angles, if possible (XI.23). The latter third of the book is devoted mainly to parallelepipedal solids, which are solids contained by three pairs of parallel planes. [89]

Book XII

Book XII concentrates on pyramids, cones, and cylinders. It is in this books that Euclid employs the celebrated method of exhaustion. Since this method is attributed to Eudoxus, it is assumed that most of this book is due to him. [90] After proving that ``similar polygons inscribed in circles are to one another as the squares on the diameters [of the circles]'' (XII.1), Euclid uses the result and the method of exhaustion to prove that circles themselves are to one another as the squares on the diameters (XII.2). This is accomplished by assuming that the circles do not obey this property and getting a contradiction. By inscribing a square in each circle, then an octagon, then a 16-sided figure, and so on, it is shown that eventually one will obtain polygons that are not to one another as the squares on the diameters. Thus the assumption is wrong and the proposition is proved correct.

The method of exhaustion is also used to prove that pyramids, cones, and cylinders of the same height are to one another -- with respect to their volumes -- as their bases (XII.5, XII.11) and that a cone is one third of a cylinder with the same base and height (XII.10). Proposition 17 foreshadows the construction of the five regular solids in Book XIII: ``Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.'' Euclid uses this in the final proposition (XII.18) to prove that spheres are to one another as the cubes on their diameters.

Book XIII

The final genuine book of the Elements is devoted to the construction of the five regular solids. Much of this book is assumed to be based on a work by Aristaeus entitled Comparison of the Five Figures . [91] The construction of these so-called ``Platonic figures'' was thought by Proclus to be the objective of the entire work -- a view which supported his contention that Euclid was a Platonist. [92] However, there is no real significance to the placement of the regular solids in the final book. The propositions of this book had to come last because they depend on most of the preceding books. On the other hand, Euclid could have easily skipped the ``arithmetical'' books (VII-IX) if his purpose was merely the construction of the regular solids.

Book XIII begins humbly enough with six propositions on lines being cut in extreme and mean ratio -- that is, in the golden ratio. This ratio arises in regular pentagons and is therefore necessary for the discussion of the dodecahedron. Various propositions follow concerning pentagons, hexagons, and equilateral triangles inscribed in circles.

Finally, Euclid constructs in spheres the five regular solids: the tetrahedron, composed of four equilateral triangles and called by Euclid a pyramid (XIII.13) the octahedron, of eight equilateral triangles (XIII.14) the cube (XIII.15) the icosahedron, of twenty equilateral triangles (XIII.16) and the dodecahedron, of twelve regular pentagons (XIII.17). Finally, in XIII.18, Euclid sets out in the same semicircle the sides of the five figures and compares them to one another. Also in this proposition he proves that no other regular polyhedra are possible.

Apocryphal Books

A fourteenth book was added to Euclid's original thirteen by Hypsicles (fl. c. 170 B.C.). Working from treatises by Aristaeus and Apollonius, he compares the five regular solids with respect to their faces, surface areas, and volumes. [93]

A fifteenth book is due to Isadorus of Miletus (c. 530 A.D.), perhaps written down by one or more of his students. It deals with inscribing certain of the regular solids in others, determining the number of edges and vertices on each solid, and finding the angle of inclination between adjacent faces in each solid. This so-called ``Book XV'' is quite inferior to the previous one, being imprecise and even inaccurate in some passages. [94]


Euclid of Alexandria

There is other information about Euclid given by certain authors but it is not thought to be reliable. Two different types of this extra information exists. The first type of extra information is that given by Arabian authors who state that Euclid was the son of Naucrates and that he was born in Tyre. It is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors.

The second type of information is that Euclid was born at Megara. This is due to an error on the part of the authors who first gave this information. In fact there was a Euclid of Megara, who was a philosopher who lived about 100 years before the mathematician Euclid of Alexandria. It is not quite the coincidence that it might seem that there were two learned men called Euclid. In fact Euclid was a very common name around this period and this is one further complication that makes it difficult to discover information concerning Euclid of Alexandria since there are references to numerous men called Euclid in the literature of this period.

Returning to the quotation from Proclus given above, the first point to make is that there is nothing inconsistent in the dating given. However, although we do not know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in On the sphere and the cylinder. The obvious conclusion, therefore, is that all is well with the argument of Proclus and this was assumed until challenged by Hjelmslev in [ 48 ] . He argued that the reference to Euclid was added to Archimedes' book at a later stage, and indeed it is a rather surprising reference. It was not the tradition of the time to give such references, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference. Despite Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes in [ 1 ] :-

( i ) Euclid was an historical character who wrote the Elements and the other works attributed to him.

( ii ) Euclid was the leader of a team of mathematicians working at Alexandria. They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death.

( iii ) Euclid was not an historical character. The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier.

It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage about Euclid was added to Archimedes, favours the second of the three possibilities that we listed above. We should, however, make some comments on the three possibilities which, it is fair to say, sum up pretty well all possible current theories.

There is some strong evidence to accept ( i ) . It was accepted without question by everyone for over 2000 years and there is little evidence which is inconsistent with this hypothesis. It is true that there are differences in style between some of the books of the Elements yet many authors vary their style. Again the fact that Euclid undoubtedly based the Elements on previous works means that it would be rather remarkable if no trace of the style of the original author remained.

Even if we accept ( i ) then there is little doubt that Euclid built up a vigorous school of mathematics at Alexandria. He therefore would have had some able pupils who may have helped out in writing the books. However hypothesis ( ii ) goes much further than this and would suggest that different books were written by different mathematicians. Other than the differences in style referred to above, there is little direct evidence of this.

Although on the face of it ( iii ) might seem the most fanciful of the three suggestions, nevertheless the 20 th century example of Bourbaki shows that it is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name of Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Of course if ( iii ) were the correct hypothesis then Apollonius, who studied with the pupils of Euclid in Alexandria, must have known there was no person 'Euclid' but the fact that he wrote:-

certainly does not prove that Euclid was an historical character since there are many similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki team are all well known in their own right and this may be the greatest argument against hypothesis ( iii ) in that the 'Euclid team' would have to have consisted of outstanding mathematicians. So who were they?

We shall assume in this article that hypothesis ( i ) is true but, having no knowledge of Euclid, we must concentrate on his works after making a few comments on possible historical events. Euclid must have studied in Plato's Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.

None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces. Pappus writes ( see for example [ 1 ] ) that Euclid was:-

Euclid's most famous work is his treatise on mathematics Elementi. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in Elementi were first proved by Euclid but the organisation of the material and its exposition are certainly due to him. In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong, a rhombus, and a rhomboid.

The Elements begins with definitions and five postulates. The first three postulates are postulates of construction, for example the first postulate states that it is possible to draw a straight line between any two points. These postulates also implicitly assume the existence of points, lines and circles and then the existence of other geometric objects are deduced from the fact that these exist. There are other assumptions in the postulates which are not explicit. For example it is assumed that there is a unique line joining any two points. Similarly postulates two and three, on producing straight lines and drawing circles, respectively, assume the uniqueness of the objects the possibility of whose construction is being postulated.

The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem "obvious" but it actually assumes that space in homogeneous - by this we mean that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this a postulate led to Euclidean geometry. It was not until the 19 th century that this postulate was dropped and non-euclidean geometries were studied.

There are also axioms which Euclid calls 'common notions'. These are not specific geometrical properties but rather general assumptions which allow mathematics to proceed as a deductive science. For example:-

Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have been the first to show that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions.

The Elements is divided into 13 books. Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Heath says [ 9 ] :-

Book six looks at applications of the results of book five to plane geometry.

Books seven to nine deal with number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers. Book eight looks at numbers in geometrical progression but van der Waerden writes in [ 2 ] that it contains:-

Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.

Books eleven to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly similar pattern to the two-dimensional analogues previously given in books one and four. The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five. This book appears to be based largely on an earlier treatise by Theaetetus.

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigour was to become a goal for the inventors of the calculus centuries later. As Heath writes in [ 9 ] :-

The next fragment that we have dates from 75 - 125 AD and again appears to be notes by someone trying to understand the material of the Elements.

More than one thousand editions of Elementi have been published since it was first printed in 1482 . Heath [ 9 ] discusses many of the editions and describes the likely changes to the text over the years.

B L van der Waerden assesses the importance of the Elements in [ 2 ] :-

Elements of Music is a work which is attributed to Euclid by Proclus. We have two treatises on music which have survived, and have by some authors attributed to Euclid, but it is now thought that they are not the work on music referred to by Proclus.

Euclid may not have been a first class mathematician but the long lasting nature of Elementi must make him the leading mathematics teacher of antiquity or perhaps of all time. As a final personal note let me add that my [ EFR ] own introduction to mathematics at school in the 1950 s was from an edition of part of Euclid's Elements and the work provided a logical basis for mathematics and the concept of proof which seem to be lacking in school mathematics today.


Evidence for Inclusion in Wythe's Library

Listed in the Jefferson Inventory of Wythe's Library as "Euclid by Simpson. 4to." This was one of the titles kept by Thomas Jefferson and later sold to the Library of Congress in 1815. Based on Millicent Sowerby's entry in Catalogue of the Library of Thomas Jefferson, Ε] both George Wythe's Library Ζ] on LibraryThing and the Brown Bibliography Η] list the 1756 English edition published by Foulis and translated by Robert Simson. Jefferson's copy no longer exists to conclusively verify the edition. It is possible that Wythe instead owned Simson's Latin translation, also published by Foulis in 1756. As a noted Greek and Latin scholar, Wythe often collected Greek classics in the original language as well as in multiple translations. He owned a second version of Euclid which Jefferson listed in his inventory Euclid. Eng. 8vo. was Jefferson making a language distinction between that version of Euclid and the "Euclid by Simpson"? Since we cannot definitively say which of the Simson editions Wythe owned, the Wolf Law Library chose to purchase both versions to illustrate the frequent problems in precise edition identification.


Sources and contents of the Elements

Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (flourished c. 440 bce ), not to be confused with the physician Hippocrates of Cos (c. 460–375 bce ). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle (384–322 bce ). The older elements were at once superseded by Euclid’s and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own, culminating in the construction of the five regular solids, now known as the Platonic solids.

A brief survey of the Elements belies a common belief that it concerns only geometry. This misconception may be caused by reading no further than Books I through IV, which cover elementary plane geometry. Euclid understood that building a logical and rigorous geometry (and mathematics) depends on the foundation—a foundation that Euclid began in Book I with 23 definitions (such as “a point is that which has no part” and “a line is a length without breadth”), five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions. (Vidi the table of Euclid’s 10 initial assumptions.) Book I then proves elementary theorems about triangles and parallelograms and ends with the Pythagorean theorem. (For Euclid’s proof of the theorem, vidi Sidebar: Euclid’s Windmill Proof.)

Euclid's axioms
1 Given two points there is one straight line that joins them.
2 A straight line segment can be prolonged indefinitely.
3 A circle can be constructed when a point for its centre and a distance for its radius are given.
4 All right angles are equal.
5 If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
Euclid's common notions
6 Things equal to the same thing are equal.
7 If equals are added to equals, the wholes are equal.
8 If equals are subtracted from equals, the remainders are equal.
9 Things that coincide with one another are equal.
10 The whole is greater than a part.

The subject of Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book II contains a construction of “the section,” the division of a line into two parts such that the ratio of the larger to the smaller segment is equal to the ratio of the original line to the larger segment. (This division was renamed the golden section in the Renaissance after artists and architects rediscovered its pleasing proportions.) Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines (vidi plane trigonometry). Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon.

Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus (along with Book XII) to Eudoxus of Cnidus (c. 395/390–342/337 bce ). While Book V can be read independently of the rest of the Elements, its solution to the problem of incommensurables (irrational numbers) is essential to later books. In addition, it formed the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century. Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means.

Books VII–IX contain elements of number theory, where number (arithmos) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers Book VIII examines numbers in continued proportions, now known as geometric sequences (such as ax, ax 2 , ax 3 , ax 4 …) and Book IX proves that there are an infinite number of primes.

According to Proclus, Books X and XIII incorporate the work of the Pythagorean Theaetetus (c. 417–369 bce ). Book X, which comprises roughly one-fourth of the Elements, seems disproportionate to the importance of its classification of incommensurable lines and areas (although study of this book would inspire Johannes Kepler [1571–1630] in his search for a cosmological model).

Books XI–XIII examine three-dimensional figures, in Greek stereometria. Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’s method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters. Book XIII culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere, as displayed in the animation .

The unevenness of the several books and the varied mathematical levels may give the impression that Euclid was but an editor of treatises written by other mathematicians. To some extent this is certainly true, although it is probably impossible to figure out which parts are his own and which were adaptations from his predecessors. Euclid’s contemporaries considered his work final and authoritative if more was to be said, it had to be as commentaries to the Elements.


Sadržaj

Iako Elements is a geometric work, it also includes results that today would be classified as number theory. The contents of the work are as follows:

Books 1 through 4 deal with plane geometry:

  • Book 1 contains the basic properties of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
  • Book 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted as algebra.
  • Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.
  • Book 4 is concerned with inscribing and circumscribing triangles and regular polygons.

Books 5 through 10 introduce ratios and proportions:

  • Book 5 is a treatise on proportions of magnitudes.
  • Book 6 applies proportions to geometry: Thales' theorem, similar figures.
  • Book 7 deals strictly with number theory: divisibility, prime numbers, greatest common divisor, least common multiple.
  • Book 8 deals with proportions in number theory and geometric sequences.
  • Book 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a geometric series, perfect numbers.
  • Book 10 attempts to classify incommensurable (in modern language, irrational) magnitudes by using the method of exhaustion, a precursor to integration.

Books 11 through 13 deal with spatial geometry:

  • Book 11 generalizes the results of Books 1&ndash6 to space: perpendicularity, parallelism, volumes of parallelepipeds.
  • Book 12 calculates areas and volumes by using the method of exhaustion: cones, pyramids, cylinders, and the sphere.
  • Book 13 generalizes Book 4 to space: golden section, the five regular (or Platonic) solids inscribed in a sphere.


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