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Euclidean Geometry

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Bnoh.v\|.ne||eur\ t fe.m.I rr "rut ar ur BooksVlt_IX rnettr,rr ol nunrherr Book X rhe rheor) of trario at lumbcrs Books XI XIII: solid geomerrl ontetry. despilehrs fla\ s The nerl sedjon of rhrs '.u||.1 h: Jre\ ,r.<r|,rIir..ro..r,Jie Jno ner urll lh, I p,n,,f..Jnd tt,<.tJtcrnrlt.i,t.r.ujr proposriions lro tarcf books oI l_.uclidrre rtso uc'u chaprersof $is book. The readcfrs fejerftd xtcnsrveconnnenrary on, Euclid,s Elzcrb lhe nrad/ahon of Enctrdcangeomerrt il can be onnL .(.tu(l Tl. h.t . .ur..trJe. u,rnJd..1,..,.r enLronnuhtons Exercisest.l Bned)de.cnbe.he li\e, _iJ J.....r t,rJ,nnr,r. ., rtc In,,,.qrf j ( J.,. ,,.t r,,J a. rhrle\ b. Pl rnr3urri tl.€JJ. E;;;_i;" j d Ana\agoms 2. ::,1i:,t",., iilll::',,,,::,,,..,". ,,,a,..,,.tpri.n,.,1n,1,rh, r.{r,.q,. .r,. :r:ll,!e\.fbe'her,.ea. crchlnede. e..n,. rrLrr... h ADn unr r. c. pr,,t(l1 ' "., L t.,.,,' f,fp, " FromBook 0ne DEFINITIONS L Apri,/ is thar \rhich llas ro ta.r Srnceone cannotdeine \onrerlnr)Sb! snntl\,tisrnu a| rh. tropcnies ir doesnr{ have lhis cunnoL be takenas a genurnedefinrrion. tucljd In!sr ha\e rea|zed Lhxre!er) d.fillirnnr lnusrrctr on tre!roust] definedre,rns. and so it jmf,,ssibte is logjcaly lo dehnc all ot urc.s rerms.tr)ncad, ths:hould be vreu,edas an attempton Euctid s pai to tellhjs readersrl)rr his poi is sorncrhlrgtjke a doLone rnarks with a pcncil, but ar dre jr jJe, santctinre is an Jartrerthrn r |hlsical enLiri r).n;iriorr\ I and5 belos shoDldbe understoodin rhesaDre \LrI 2. A llr. is breadrl esslengrh Euclid s l,e is our .,/rt 3. The enr..r,iri.r lr1a line de porDh Sinceboth pojn ts and|ncs hale . aLead) beendc h n ed andrhe so d .rrr?,rn ! is subsequently used 10denoictheboDdary(fan)h8ure,!hats)c!er.itishardft)seeexacttywhxtrstcingdeJinedhere It.is possible thar Euclid fett rxt haljrg detinedborh tonrs ard lrncs ]re nou Deertcjto ctarirr rne telJljonshp berucPnrlfn Ir is rmplr!rtin rll|sJehnrU, rh,rre\ef\ rnehas crdpoinrs. erscquenlly. Eucljds lire rs xr ract a nnrre^ ffc, and hr\ rrr:lghr linr\, r,, bc dehDcLlnexr. are l]r ticr lne scgnrerts MDch has beenmade of the Greets iDsisrence(nr tbe finrrencs\of drc ohjecLs.f rh€ir in\esljgrrron\ Neverrhctcss,ELrctid ts qultepragmatic abou r ftis rssuc when e nec.rafis. rs rr rrocsin rhc statemcnt ot propositronl 2 below,Euclid is wiUiDgro bendhi\ o\!n rulesand meDrion r|iiniLe sLraighrtrnes 4. A sffaisht line i! at;ne \,,hichlies ere|ly s,ith rhcpoi s on itsell This too is a soDre\\,har obs.ur€ scnrencc lr rs possiblerhat rhe poi|rs () s.hrch Lhisdeiinrrolr refersne theexireniries ofthc pre\rousore Thus rhisdefinirrcn \lroul<t te understood* sa1:ngrlat ofill the rLxnedr lrnes iurnrfex frir rn tornrs.the nrdrgtr. hnc is ilc ones,bi.h.ousists ol alt rhe pornL,hal lredIe.,) r,.ber$e,.,nerr, c!e1t,.,r.,... .,dd., 5. A r. rh., qni. t, hd .er!,r .,ra I r( o,r J,r , 6, The exrremiriesof a r,'/.r.. aretines TlisInfl..tin r,.denn _ i.r.,fi -!( ,..)4t rrrc.,r,l L Aptdr, td^ci1,.r J.e\ti., rr\c\-,tr vr,t .t. .!t, Possiblythh mc.ns thrr rhee\ltmiiies ol a trane nura.e .ltc srrarghLlrnes and rhal rhc surace lrese\,cnly $ith ihcnr. 8 Apl.'r. dngls i\ the m.rrnarior ro onea'odrcr of rwo ri'es in r frane whrch meerone anorher and do not lie in a st.aightline 9. And Dten the lines colrrajninsthe rngte aresraight rherngle L\ca|ed fdr r/rrd It is clearfronr Lheabo\e rwo deh irioDsrhrr ELrclidh,s so r. interc\Lin flritjneal rngles. Ie, anglesformed b) cunes r trer rlr.rnby srraighLtine\ Ne\,erthetes:.he n,bscquenllyrefer ro suchangles only ju once. P()fosthon t6 of BooI ll1. $Icr. he sr),\ that the anglcbctween rle Iangenrro a circle and irs c;cujnfercncejs smrller rhrn an) rc.r|rnert x|glc Ir h,s b;cn fosrted thar geomebiclans of therime wcrt dxllyns wirlrsuch urrlle\ wirhour crrnjnr ro dnyserrcu\ .onclusons and Ihat Enclid fell rhe necd to l.kno$tcd8e thcjr elt Ls ru.L\ no i sm,ghl trnc setnf ur l slLrtshrlme nrakosrIc rdla.enr orglcs eqnnl or oe rnsres.rs Loone another,each fiCr..lrd th. surrltrt t,rc sLrndir!!D rtrcolhci rs ca ct ar,tpdttrLutdtta thard1 j l hls inlocno[s took g dcnniuonhrs L\rodrt oi angles Docshe hjw equr|L) in nrasuk I rr )t\,loreo!er: consnterce lhe rsjtreofthc c\breLL. ( by equaliL).is clearty sidestcl)f.d Th. trlrcr ,ss cqL',,r1).rhe rudrorbelicves that l_tLrclidi\ r(ifenins Loequatit).ot.mea{It rarhcr rtrrn co,e,uerce A-,noredetaitcd di*rssio,) 01rh..s. of equ.rt,r) n t.hr L:t.t.rt\ l. st,..,,1",i.,;",,,;; i,,,,,.,," tl. Ar r/r4. angtcis u rnllc lfe er rhrn r Jrshtrnile o' 11 ""'"' 1 'eh'' 13. Ahr1,,, n r..n.U\ tr., .,,,..,, ., ,,. ,., li)'-'r'r'" ,,',"",.' ,1i,"' r ptJn(lielrr !lnturmd hJ I oneiire $ch rharx lhc slrrighltincs laltins nrcpoll-Id;r1,r-,:,- \ ..r..,i, up.n rl lrcL L ur.,.rrt rr..r.t,, | .r.. Jr .r1"1 ro. qnr rtrL . p.rnr (dti-.t rt.e,, ., , ,,t ,, . ,t- 17.A,.liahvet al rhc.rcte rs an! soatlh Ln. d.awr thrrrh rhece|lcr anil ler|nlnrrediD both djfectionsby the cncuirfererce olrtrc urcie. ur sch a srlr'hr hoc at$ brse.rsthe cjfctc. 18. A rerln n.i c LsLbe ltgur. conraineitbl r}r rd t}" .s,r? crr off b) ,1 the ol rhesemrcrr.le is ,,," *." *" ,,."llileLci 'rNmferu.e -^.Dd ^ 15. R.tilihcdt fisu"s Ne rhosc{,hj.lr " ry e,J,',;;;;,;,,;,;:,ilJ::lre co l:',:'l:i5:l,j;;;.liiij::1j:ili:;:; Ihan"o,',"',"alour srfatshr'r,.i]ines. li::::: .l ,:1,,j.'1,. an L'lttttdt dt rlLutlL .11 lr:^, )\.txar lvrlch rras|ls rirrccsjdes equat, an ,rnql, that\\hi!h hasr$,, ,rde\ i,.,.cr.r oi i^ Jlof. ellrAt tttr) L tt'tlr" tttdiill' qhich jr' srdesunequat. rhal has th.ee of t tateral tiEstes. x titht arytt.t jtrdnrl..is rhar utich argl..i_21-,Furrhef, has ! righL angtc.an obtujr r?rr8l. rharwhi.h h.rsa. ohrusean!te .,,S/./ r/rr?qL,rbrr$,hich has irs Lhrec i\.both cqurlaLeral.nd fishr angled.rn,r/dr8 rh!( ther jr uhjch equitarfatbul not.ighrmgted:aDd r1 | 1,,.,n..Jn.l \ .qr.it..,Jr .eq.
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