TANLB OF CO}iTBN1S PBge INTRODUCTION ]. gectlon 1, ORIOIF OI THI PROBI,EI{ AND }ARI,Y IROO'S g. CBOI(BfRICIEOO}s O' THE IgOPERIUBTRIC TROTAFtY O' TI{E CIRCAE 11 3. PROOFS BY IIEA}IA OT }OURIEA, SRISS 2A 4. PFOOFS 3Y UE"ANSOF TBE CIICULUS OI VABIATIONS sIILIOCNIPHT \ 4',t7 ) 111 A HISTORY OI' THE CI,ASSICAI ISOIERIIE1BIC PROBI,EII Intloductlon. The falous oltl Isoperimetrlq Probl@ of the a,ncientg res tha't of findi.ng e Elrlply closed eurve of glven Length [hl ch lnclog€a thg lergest a,reE. Aaothcr problen cloBely releted to this problen is tbat of flading qmoug ell curve5 sl-loh lnclose B glvon area tlBt on€ rhloh haE thc shortgat perlnet€r. It ls eesy to proy6 that th6 solutlon to elther of ths tm problels leadr toglcr,lly to the solutlon of thr othcr. tho firat proble! lay be statsd enalytlcelly as that of flndl,ng au arc vlth cquatlonr ln parseetrlc forto r=x(r), y=y(t), r,isf 1^ thlcb, 6atlafloa thg coodl tlou! x(t,) = , y(t,) ' v(t:) , "1111 but doeq not otber'1se lnteraeot i,tsslf, rhlch glvee thr leBgth lntegral (' I /t, d* ( 4'.19) -l- INTRODUqrlOT (480) ( 48L) e flr€d v8luo I, and roaxtEl r6s ths lres lntcgral tbr dloteartl a /t--(x,t' v!lop!r!!i ot tbt t/^ l, - Vx)/*. nlltg of !b! dla thg lolutlo! 1r a otrclo. lbr geBerel lrqbfqm of thc ceJ'cu- psrt of grotlo lus of yarlatloB! for tblch ona lDtogral 1! tg bc glvou a tlEorJr ot gtaUt ftxed taluc thltc aaothar ls to bo Bade a DqrlEuD or nlulluo rar ftrrt aDl[tr lE osLlsd aftrr thlr oBc an lloPerloetrlq problcD. Any prob- lcE by llhtort&l 1e@ rhsre a ftred length lE lDYotved thllc an latsgraL of a.ny ploof! of tb! Ir klnd l! to bo Etdc i lsirLDu! or tltnlloun !! oa]led a! laopoll' b€oD glvel bt !a lostrlo probl6B. I! thl.! pap6r only tha ftrst tto probl€Da tLon! rhlol. iara rhLoh elr fonrulstsd above 1111 be dLscuaaod. irtlonr ars illlo Thc 6ar11o6t attoopted solutlon of tho problso thlob di,lcusliott of th ha! be€n prerorvsd for ua ras that of Z€uodoru!. Hla golu- to tbc lroDortnc tlon rar thc ono geuelally glven for tbo proll€n untll the Tho Blbl tlE€ of gtclDcr. gtolDer gay6 a verJ el.egsJrt aud Blropls proof tldsd lnto fou! of a condltr,on necesgely for a aolutlon but dld llot glve a d8Ecrl,baal lD tht sufflclgncy proof. Tho ftr6t coBplete Droof that ths aolu- Eentr,o! tb Iloll tlon of ths probleE i,B a clrcle tas gl,veD by [eleratraag. fhe Tls !l.qb earlLgEt rrltEra rho atterdlted to aolve thc proble! uaed a follorlag an rut geoEetrl,c method, Late! coBplet€ thaorle€ bar'e beon 81Yea by deatb of thc aut thg rogthod of thc ctlculu. of varlatlonar !y tlr€ geonstrlo rofer to tba El,b nethod, and tlorr reosntly by Eeana of ]'ourler Sorle!. Ths dlscuBalon of the hirtoly of the Ilopcrilttrlc ProbLen 1,9dlvtded belor Lnto four Eect1on6. In geotion I the orlglD of th€ problen and the proof! to tho bsgunlng of (4eo) (48r) rrrRowcfror !gral, tbr dnoteanth qqatnry Bre diasuEBed. ID S€ctlo! 2 tll6 de- t€loDlari of tb! gcoEctrlc proof! ,,! traceal froE th! bcgln- nlng ot !h! Dlnat€antD. qentury to tha plecent. The grcetor la! of thc cal,cu- It rt of Saotlon g t! coBcerned rr,tb tbo dcyelopaslt of th€ to bc glvca e tbaory ot gtalBrr, .!d of tha thcory of par|.l1sl curres rbich r,rllur or lllllllE ra! fir.t spDlisd to tD! lolutiob ot ttre Iloplrlnetrtc prob- roblca. Any prob- 1@ by lllllonkl,. gaotloa 5 con!l!t! of an outllDa of th€ rr lrtcgral o! a[y Droof! of tbr Ilolorbrtrtc Propclty of a clrcl! rhloh bave orllcd rn L!qps!l- bga! glve! by Beans of I'ourler gcrle8. I! Seqtlo! 4 soLu- rt tro probl8D! tlgn! rhlch hare glyen n. bean by locatls of th6 cA].qulua of var- lit1on! arr dl,Ecu88sd.. Th€ grc.tlr part of S6ctlo! 4 1! a tha !loble[[ rblob dllqusllou of the alpllort1on of ths th.ory of rJyelclstrela oru!. Ula solu- to tb. IroD?rhrtrlc probleB. oblen urtll ths !h6 BlbLtograpt\r rhich fol]orr geotlon 4 ls atso dl- Itt 8[al slapls proof lldsd lnto four gloupr to qorrealond to tho four a6cfi.on! dld not glvo a dsscrlbqal 1tr the lrrcedtlg lq,ragrrplL ReferqnoB! ,blch Eerely I tbat tb€ rolu- Eentlon tba IlolerLErtrlo probl,cE ale not lncl,uded. y Icter8trasE. The Tb€ nr&lcrs ln tha tgrt lnctoBEal ln prrcDthesg! ard BrobloE ulod a follovlag an a,utDor ra ne,Ioe, lafrr to the d.tr! of blrth atld brt6 bern glveu by death of thc author. fhc |luEber! lncloEed ,.n squere braqk€t3 t tb3 gcoDetrLo rqfer to ths flbll,ograptry. tt 36r1q!. I I!opcrlnctrtc , In gsotlon I tb. bogLnntng of ( 485) ORICIX Af,D EIRLY IHEORIXS (4e2) tr'lgurea eppears [email protected] (r.630-16??) t! I ori8tn of the probloE ha! been attlibuted to tbe CreekE but the follorrlng ec It iB Eot knowu who aEong tbso waa the first to atate the iTheae be! attri problsB or att@pt a ro1utlon. lhe roathenatlcel hiBtorlans Isoperlmetry of lIoltucla [a] aoa cantor [s]l quote a atatement of Dlogenea rorkr.rl Laertlus (tblrd century) regardi.lg Pytbs,gorss (580?-5Oo? B.C.) It ls no rbich lraa bee! intsrpreted to Eean that Pythagoras ]drer the q,r€ to lrExinuE property of tbe circle. fhe ate,tement of Pytbagorue lalagraph to the later t!' horever, not at all convlnciag aa lt aelo1y aEselt8 that cot to ./rhi ch th6 nae of elI lIall€ figuleg the cLrcle is the rdost beeutlful. to bBve beed grr The follorllg rena,rk f9l appeara in tbe De Caelo of rtutta IllEtotle (3a4-322 3.C. )! ntrov of llnes whlch return upoE lroorr fto he Bays Fas a th themBel.vea tho line Fhich bounds tb,e clrcle lB the EholteEt. n conclu8lo! that Arr.stotlo docs not givc eny further expleEatlon of the sen- ArshlnedeB, awl tence and hi6 subJ€ct ls e philoaophical rather thaD & tsath- Archieedea €datlcal, one. ltad a: ?rob1eI!. Arqhl-dedes (287-212 !.C. ) Eay have beeD the fir8t to etteBpt a eathelaa,tl,cBl aolutlon of the problem. gi.nplicius A rtateEl B. C. ) suggeeto t) eho 1lved sbout the 6ixth century A.D. rdentions a proof ff{ the of Archl.ledes aDd Jenodorua. Horever, ploclus (4I0-4a5) ci rc1e. IIe I slnlfy frol! thei: oaye [f{ tb8t ]-ate! nathenaticiaDE alrlved a,t a aolutlon that L ci.ty partly fron tlre vorks of l.r,rl!d (450-3?4 B,C.) arld partly o! cl be tvlce a6 largl frol]l thoae of Archiuedes. The nathenatlca,t historisr Libri stade8 the atater [e] givee a list of booke fourd in the CosEogr4p]\y of tho trouble triBurolytus (1494-15?5) aod one c&lled the IsoperiBettlc ia tl -- (4e5) EARI! OREF< SRIENS ( 4e2) tr'igures eppeals under tbe ua'Be of Arqhlned,e!. Isa,ac Ba,troy L!b99!199. rhe (1650-16??) tE bta prefaca to tur Book of L€Enaa na.keI . to the Oreeksbut fzo] the fo1lo?lng connrgLt h. e note on the Eargtn of hLE bookt .rat to Etste the 'Tbess nen attribute to lriu (Alchl,tn€des) a book rhloh is thc rEatlcal hl8 toriano Iaoperlnetry of ZenoaloruE, frBgoents of stetlcs, 6.nd otber |!!ert of Dl ogenes roaka. I' prsr (580?-500?3.C. ) It ia lot clesr rha ttler the leference. of the laBt rythagp!46 loet the paregraph are to a rolk of ArcblEedes rlLch haa bee! lost o! ier ent of PYthBgorus to the later collection of altlcle! cElted tbe look of I€Dbae uerely araert8 thBt to vhich tbe nEJoe of Arcb.inedeg lraa by so&e crltlca thought )at beautlful. to have been erroneously appended. Zenodorus in one of hLs l! ihe De Ceelo of prooro itultBch ed., vol. 3, p. 1f9! ueed a th€olen rhlch rhlch leturn upou [o he Beyg waB a theoreB of ArchlrDedeg. Thta lE evl,denco fo! s rle tr the Ehorte6t. n concluaioo thst ZeDodorua f,as acquainted rith the rork ol ustlon of th€ 8en- Archlmedes, and tha,t he rould ha,ve Bentloned the lact lf ratber than a Bath- ArchiBedeB l|Ad alrlved at a solutlon of thg Iroperlnotric Probl em. re beeD the first to A Ets,tedent of tho hlEtorla! Polyblu! (ZO1-120 robl€E. SillpliciuB l10l 3.C,) suggeoto that he n8y havg hlo'n Lhe [a.xlmula ploperty of lntionE B lroof flt the circle. He aayB thBt looBt people Judge the aize of c1tleo rocluE ( 4lO-485) slr0ply frotd their circueferenc6 aud that sben one tella thern red at E aolutiou thet 8, city or ca&p with a. clrcuBference cf forty Etades lnay B.C. ) and p8ltly be twlce aE large ae one irith a cirqu.Efeience of one hundled ral hiBtollan Ltbrr. 8tade8 the stateloent Besna agtounding to theb. Ue adds that )snogrcp\y of the Lrouble iR tlBt Fe heve forgotten ou! lessona ln geouetry. t lEolerimetric OBIOIT AID XARI.T Tf,!ORI3E (484) ( 485) fha orrat! 1r gsasrel dlal lot tr8re r olcsr ulderttrnd- lltgrts tn8 legaldlng tho r€18tton of Derltlatar to 3148. Polybtu! lldoa and aqurl add! to tlrs abov6 lsts rk tbat aot only ordlnrry Bcn but rllo 1n atgt.