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. ststsoE€n atrd Dllttary DoB J udg€al tbr eiza of r orllll by E6a- Tbro16 reaulc polyron surlng the ctrouDf,crcno€. Ptoolua DrntloDa fll oertatn n6!- ber! of oo@unlotio gocletlc! f,ho ohcst€d thrlr fellof, !tgnb6!! Tha Dro( by glrtn8 thcto llots of lend of lalger perlroster tut ltoallc! L.6r I. area than tbr !lot! rhlch tbry took for thaEseLtea. lb says rDil ih! arEr ! aleo that thr thooren thr,t all trlangl,ea fol|lleal on tbo !s,nG ln araa. or equal baaas, rnd alvaya bBiresD tha gelno tro prr|Ucl I.@r II
11n68 are equaL ln BrsE ia! regald€d by thc oro€k! at D3ta. lrr to crsb o lb! doxlcel beqauso thq psrlloeter sould bc esde e! lBrge gt ona trlr,aglsr tbrt r
!16c!€d, lrr of th! leBl! Zenodoru6 rbo lllled probeUy durbg tha paltod lron e00 tbat of tb. t|o 3.c. to 90 A.D. rroto r tlostlss on llguroa of aqurl pcrlnatcr. of tb. rtnlt.r t lhl! rork ba! bacn lost but baa b€€n D$tly lrlloreed tn tht of th€ aou-tt[tl F-1 th. Fork! LI5,19J ot Theon of Alcxandrla rnd Pappu! rbo tarc ooa- Droo t€Dporarles 3|rd Drob.bly llysal durtng the fourth oantury A.D. f olloEr tluDDo! chrlstopber cl!,vlu! (I55?-16L2) referg pf to tlc rerlE of u.lequrl. Conrt! qt th€on and lrplrus ra tbo lourooa froE rhloh bi iool tho lroofr aod lucb tbrt thlqb ha glvc! rsg.lallng trolorl4etrlo flguros. Reocrtly dl!- 8sd Cf lr rqtrrl, cusalon! of tbo thooroEr of Zetlodorus have ieen glvelr by AB Lad !C. !ba! qrtd trlr'ngl. Uertn fr] Chlltnl fA] Ut uotl ro!L! follor tbat of Prppus. A3C l. Thc thgore4s of Zqnodoru! that rrc of latarelt lor our DroblcE tr18,ag1. A8C. t ar6 r of tb. ncr fl.gur on€, rhicb lr oo ( 484) (4S5) rRoot ot 4fomngS
pqlvEona olaar uldsr!ta[d- &gg.re!oJ. leonE all of eq.ual llabar of porlBatorr. polyaon glartGrt !r.. Polyblu! lldsr end aqual thr rcEular lr rry tqon but rllo tn lrta. f r oaal) by Doa- lhloretE f f . lbr otrola 1! sr!3t6! ln rrcr tbrn !I|v reBul$ polyRon rhloh bar r[ cqual pcllnatar. ftf, oertato rer- rlr frllor EcEber! Tba proof ot Thco rAB I rl6pend! oa tro l@la. rtrr lut aDAlIer Lcnl L. t&ong rll trlrtrgle. barlng tbr !a|!6 be!. !alro!. tb aryr end tb! atlr !u|! ot lldsr, tb! l!o!oc1a! trlsrgla t! grortclt lrd oD tDr laeg ln Errr. tro parrllcl ].a@a II. Wbra tro laoroolqt tllrnglcr ara lot rtud,- otcak! aryB.- Lrl to o4ob olhqr, lt ra oolltnrot on tbr arDc ba!.r tr! rt lrrgs cr oat trllrg16r tbrt tr! ll,Er.Lfr to caqb othrr, |'!il luob thrl th. !u! ot tha portro6ter! of tbc sl,.Ellrr trluglc! 1! cqual to tbat of tro tb. !.!tot fro! 800 tb3 orlgl!!1 trtraglct, tbaa tha !!rD ol thr lrler ol rq.ur1 plrlnatar. of tba !1811rr trlin8l€s 1! grcatcr tbra tha llD ol tba rrcr! of tbo non-qiEllrr trlrDglaa. Dtarorved ln ths pua rho rrrc oo!- lbc Droof of Tbcoro! I roqordln8 to Zcnoioru3 1! e! guppocr urtb ocntury A.D. follof,!! Al aad 3C rrr to tbr m*r of ulequal. Colatnrct lI equrl to ! too& th3 proof! CE aad Buqh tbat tba suD of It A !!. R6o.nt1y dta- aod CI' ls rqu&] to the llE of Ban glYo! by A3 a.ad BC. Tbrn bJr L@[. I, th. or tbrt of PaIDU!. trlugls Atr'C 1! grartcr tbaa tb! t It for our problln trlangls l!C. tharofora tbc *so of tb! nor flgura l,! grcateB tb||! tlra lEe. of tbr ortgtnrl o!re, rhtcb l! coatruy to th. blDothsrt! that th! glTrn tlgurs OBICIT A}TDEARIY II{EOBIXS (486) (4e?) be a !ca.xlm!!!. therefore the [axlDur[ polygon nust be equl- notlce tbat 4r
Iaterel. the clrqlc. gl
NoE suppogs tho an8le Ble thllar re
! ls greEter tban tha 8nglc D, is equal to lrill lrhile the polygon la equllat- gleater tbaD tl e!e1. Con6truct i.soaqelea trl- of tha clrcle,
Blgles Atr'C and IOC sl,rollar to ia greater tha! E eash oth€r and guch ths,t the of C 1r, by th. aum of thelr perlmeterr lE equal to one-half tbr to tlre 6u! of the peli-oetera of the leri.Eeter 8 iBC and XDC. then by the uEe of LcEma II re knor that th€ the &rea ol P I sum of the EreaE of iho trla{rglea A}'C a,trd EGC 1! greeter tbaD tbaJr th6 area, o the awl of the 3rea! ol tbc trlangles A-BC a,Dd !OC. therefore The ttt we bave a [eE polygon of equal perlBeter but gleater ar€a ccnclusiou that than the origlnaL onc rhlctt ls contrery to the brpotheels larger alea tha th8t the origlnel polygon is a ua.rlllun poLygon. By repeatlttg But in fheoreD the argument for othe! pailg of 4ngles v€ flnalLy conclude glven peri8etor that the nariEirm polygon Duat be sqularrgular. Therefore thg stetsEe.t rsquL ua.xlmuD polygon nuat be reguLar. plete ploof of
Zenodolus used the follovlng denonstretj.on as 4 proof c4gs rhen the f of theoren II. Let C be a circle {E not a lolygo of perll'eter p, aD'd P be €, reguler Kepler polygon of equal lerimote!. Let tlre readBr to t
P' be a polygo circurnecEibiog C en 8,rgu4eEt Ed s,!d al,dilar to P. Let a aJld a. J@eE ! be th.e apolhens of f and Pr, and (4e6) (487) KTODORUSTO STEItrER
ur6t b€ equl- not1s6 tb8t a,r ls the radius of the clrcle. glnce the poLygona
ere E1E11Er re lmow th,at a/ai
ls equ81 lo p/p'. But p! is
greater tban tlt€ perlEeter I
of ths clrcls, therefors el
LE Ireoter tbao a, The area,
of C ls, by thc ua€ of a theoleE proyed by Arqhlrnedea, equ&l
to one-balf tbs 4res of a recta,ngle the length of rhich i€
the perinetar 8&d the width tbe rsdius of C, or ^tp/z, axld,
I knor that the the 4r€B of P la 4p/2. Ther€fore the area of C 1! grseter
C 13 Erset€r thaD than tb,e are4 of P. 'when d XDC, lhErefore The tvo theo!E!6 conpletely proyed JuEttfy the i gteaEer crgB conclusio! that a, circle of sJSr giveD perineter inclogeB e ib3 \4roth€ela larger area than ery polygon rbi ch ba,E s! equal pgri.Beter, ton. ly repoatlBg 8ut ln TltooreE L,t is essr&gd tbat seong all polygons of a ,aelly soncludo given periDetgr there i6 one that is a Barimutd Bnd thl8 r. lherolore tba atqtotoent r€quireE s, proof. In edditi,on re ne6d for a coo- plet€ proof of the IEoperlmetric TheoreE a diBqusalon of th6
;rail,on ae 4 proof ce6e then the flguFe whlcb ae are coliDaring with tlE circle 16 not a poLygon.
Kepler (15?1-1630) states the theor€m [a]l ana rerere the reader to the !!oof of P4ppua, ealileo (1564-1642) giyes
an argument Egl rhich is egaentially the proof of lenodorue. traneE !ernoulli (L654-l?OS) )neiltiona the probien p! OEICIT AfD IARLY T$ORI3S (4es) (48e)
aDd sry! tb3! tb3 solutlott 1! a qlrslc but sdal! tbr statenut qulta of gl.tan reeds to bs proved. 0n tbe BaDe pegs bs propoEsd to nalhsEr- tbrt tDa coErt tlclarr a loro geEereL problen. EtE probls ls ar foLlorrs . dls:rrtrtlo! .lEoag r]I curves nEN tJr of tbo otral of rqurL lrnagh and bavlnE a rbloh br3 Ll! gllllon (u: oonpB belr ff. to f tnd ona 5 f. rtrcb thrt for a corr€apold: llDtlr! io tlrl
lnr curls B8[ rhoEa ordlErt€ ltrrtlo! L29J ! lE 1! ray funciloB of thr gl,rrn by ZaDod( ordl,nato Pl o! of tba erc IlP. ib6 traa BZllS ahall be s [arl- r.r. |! t'brg ol
EII!. 1895) |!d P. D If ra choolc !Z rqual to Pl' tb6n ths solutio! l! tro Provrd&fl tu equr'l alcs of a clrslc. Tbo solutlon rlll b€ s, corlplrtc o1!- r lad'tlD llcr sla lf rc further chooaa the su!. of tho lcngths of tba tro Dt oonltut. ourtsr to ba tl tlEcs thc lengtb of l[. e..@ Ths nl&€roua Ltteopti ro rolva th. plobtcE of Bqmoutll l!g-s!&. n aad others lllll,l,er to lt led to tbs develofont of s lor D6 tbrd th6orcN! by lorr for aolTlng luch probleEa. thlE nelr lethod Xulor oelLsd th6 polygona rlra ri qalculua of vartetlona. All tbe early rrrlterg on the oaloulu! of tb! ltlataar of variailon! rsrc oonoerBod oul.y lvlth provlng tlut condltlonr grlrrrl tlrl|. ot nsqoEgary fgr a aolutlq!. la e leault tbo rarly lroofa of tbc bcgl,.Dl.[g r the nairlnu! property of th€ clrcLs by th€ ner eethoal as rell rougbly dlrlalaa a! th! older gco ctrle tle tbod rers not conpLste. llrDthctlc dd t lul.6r (t?0?-I?8q) devet opeata tb.eory vhi.ch ls catl.d (r?96-r86s). I thc rulo of 3u1.! s|!d rppltod provlng rl th prorlrg rt lt FA b tlrat lf r (4e8) (48e) I t{otB ctrrtrRl! PnooD 1t lr tba rtat€tlort gulra of gl.ran langtlt l,nclolc! r rltrlEl.a rrcr lt 1a aeoalslry 0!od to Drtbgtu,- tbrt tb. curf. b6 . oirols, Bloc.tt (1?0?-1??6) ba. vrlttrn
[a l| lollorrs a dlBacltrtlo! LeOl f.r nhicb b dllcnamr tba Errln.|! propsr- tt of tbs olrslr t.n. r Eannor lLEl1rr to tDri of Zrnodorir!
rlrt oh ir! baar glva! llt r Draordl8g pl!!,g!rD! of tbl! s€otlo!. gtlp.on !. (t?rO-tto) gara r dl.lsut.lol [{ of t!. probta llDl,lrr to tbrt of lutcr. f,. allo grr! r Baoctrlo dauo!- )" ltrrtlon L29 ot tb! plob].ca rhlqh l! r8.Gatlrllt th. olc glrca by Zcnodorur. ptoota Othlr BcoDGtRlq @S,S{ Euoh tb hall bc a !|rl- lEa r| tbrl ot SLBDroa brta brra glra! by LgrDdr! (U68- ISSJ) |rd r. Ic'ytur (1?10-1?a9). s. L.8rr11ar (I?so-f8{O) lolutt on i. tm lrovca pfl tbai l.! r snrr6 rbloh brr r glroa l6lgth lnclolsr a cotaplGto olr- r Etd,luD arcr it ta ncoatrlg/ tbat tb€ ridlur of suryetur€ h| ol th, tro bc oolrtrlt. e. GGoErrtElc ploofE 0l tbr lloporlr!€iRl"q plopartv of obl6 of Bortroull,l -!&9.-S!S.IC. Th. $r1y.tteDt! to prore th. IloperLEetrlc ot of r ncr nc tbod lhroroD by !or&a ot gcole try lsoked getcrallty ll!'se oDly
Ilc! oaucd thr Iroly8ou! rlro ulad.r oorFerl.on culte8. Durhg thG bogllltllrg r on the orloulu! o! th. lhctacntb, qoDtury 3n rttGel)! ra! D.da to glTc a Eora t tbl oondttton! garrlal tJrpa of Dloof. lh6 gcoEctrlo proof! attcDpted t!@ lLy lroof! of tbr bc8lE LBg of tbG Dinct€anlh ocntury to tha DleaeBt cra be ao thod e! rsU loughly dltlded lnto tro Btoup!. thr flrlt glouD 1r lEgely it. ljEth€tlc aail la clolcly rsroclrtcd rlth tba rork of gtrt!,G! rbloh ls oaltcd (1796-1863). !h6 srcond group lr rlr.lytlc lnd 1! conoera€d ,ng ihrt lf r tl tb, proilllg alr rurl].lryy theor€!. tmB rhlch thG IEoperlretrlo GEoUBTBTCrfRoCrI'S ( 4so) 12 ( 4s1)
Thooree follos. at onqg. fhis auxiliary theoreu Baya tbst gous proloE1tl( aII]r cLosed the squere of the perineter Dlnu! 4 n for curre clrcle har the ti..!€! the lncloaed slea is greater tba! or equal to zero and tlteae prgloBLtj the equallty sl8n ls valid onLy {hen the curve la e clrcl-g. therd in the lal An atteEpt to prove ihat the area of a cllcle i.B larg- one of t'beB l[ than that of aBy cloaed equal periDete! 4ppe4!ed er curye of geleral enough an artl.cle by an urknovD author. Tbe wrj.te! 1l1 1413 L! L32l perl&eter, but aesuDeo tbat seoDg e1l. ll-€Jxe figur€8 1llcloaln8 equal areaa !ostulatea wltlr th€re ls oDe tbat has the Erla,lleot perheter. He then provea (1) aE' tl8t thiE flgure of alcalleat perimeter nuet be a olrcle. Ee there la at L6l argues tl8t lf one denieg that lt is a circl€ theB Bo&e other the alea of an: huat this property. In that fr.gure have caae he ahors that a (z) l! nev figure can be coDstructed hayi.ng equaL areB but sltraller least one thog pellnete! than the origiual one. lut Buch a constluctlon lreter of any o: poBae6aes contradlctB the h$DotheElE thet the orlglnal figurc Stei!Ie the po9a1ble. fhelefore, sna11e6t ]rerlmeter a&ong all cur.see rft i.E clear t of equal aree the clrcle haa the sll'olteat periEeter. trow equal perlBe to auppose tt8t there ls a figure S having an equal periBeter observea that aad a larger area than e given clrcle Cr. ltrake a clrcle C but aot aE Ia! equal in erea to S. The periDeter ylll of C be les8 tbaD be inclo8ed in the perineter precediDg of S by the theoreli. The al:ea of Ct perlne ber of t will be Ls,rger than the area of C and therefole larger ttl'Jr of the qllqulf the area of S Fhich iE contrary Lo h$rpotheBiB. snootrgtheee tb Steine! gave flve proofs pfl d.ifferent of the llaxL- flgureB, that ttue propelty of the cLrcle. lUE theorema include the ena.1o- but E larger a (490) ( 4e1) IOBK O' SEIXER rr.Et! Eeys tbat goue proposltion th.Bt a,Eong 411 flguleE havlDg equal 4!eaa e rter Dlnu! 4 n clrclo bar the shortest perlraeter. Ee statea ihat e4ch of lurl tO zrFo ard theae propo6itlons lmpliea the other as lndicated for one of 'a 1! a strclq. theB in the l"srt paragraph sboyg' and therefore proYes oDly t cllcL€ la larg- ole of tbeD lu alry partlcular dlacua6ion' Tha progfs are 'lDster s!pes,!ed geleral enough to lnclude el1 closed curves baYin8 a Elrc! lor, Th€ vri ter pelloet€r! but are ln sach ceas baaed on one of the follovlbg .g cqual areEa poatulatea rblch 8re not provedl Iie then proveB (f) e&ong B1l cloaed curtea having e glven periDetor € a olrcIe, He thera LE at tcast aue vhoBe area la equel to o! 8leater tbr$ theB Eoee o ther the area of ary of the othera. Lhe lhows that a (2) tnong a1l cloaed curvea of glven area there !! at ,rea but aualler leeBt one rhoge periDeter LB IeeE tban or equal to the perl- congtructl on Eete! of any of the otlrera. tlgurc poescerea Stelner coisenta o! tbe firEt postulate as follorat alolg EI1 etrver ,'It ts clear that there 1r aD iDfinlty of flgureB xhlch bave ,t!3ts!. trot equal perlrtretera but rhich have differeBt forB and areB. One lual periDeter obaerres that the alea c4n be Dade aE Bnal1 a8 ong pleBlea ks e clrcle C but not e6 large a6 one pleasea, sinco e1l the flgules Eey bc lcs! tbaD be inclosed in a circle rho6e ladlua 1r equel to one Dalf the fhs erea ot gl pelltce te! of the glven flgures and vho6e cente! i.E one poiDt ts larger that of the ctlcruference of one of tbe figuleB. It DuBt be that
srdong these there Ia e llaJrlnu! flgure o! aev€ra1 Darimlr! i3] of the Daxt- figule8, that ls several figules lJhlch bAve equal peliEete! luds the anato- but a lalger area tb8u eny othe! figute not in ths group.' l{ OIOIITRIC PBOO'8 t4e2) (4e3) E
thropco. ^lonB 311 qurrc! ot cqurl parLpctor tbs gllcr tl olro1! lns].olcr tht lsrlurr! aror. only a brllt !u lct IFCII bc a Dar.lEE er! tollovinB I ' llgur!. (b. qen llld r lltl'. fifth Droof!. A3 thrt divlal.r tbr DGrlD.t r illEra ihroug[ ] A l-!to tro lq,ur]. Drrt!. ?bcra- th€ taot that t. fo!.. tb. rrsa t! allrld.d lnto tbat tr tusorlb tro ceual Dartr bac]r!! It Bot flgur. othrr tb oor oa,B Dalllecc tb Drll,c! DErLBotcr 1r L brlf tr't oDr aqu.l to tba lgger nr! flgurc nuat rJrd. thur t.!cr€r!a tD.! 3ref of thr orlgtnal flguEc rlthout fore Euri ba I ch|lgf.ug th! pcrLDtct rlIqh 1! ooatrary to l\rpothoair, It ldlc! r ibr flgBlc t! !o! rlllDtrLe v!tD. ra.past to 1.B, raDlaoo ona a Be@etrlo o0 balf d tb3 tlgrra by r f lgur. rj@,.ttl,o to tbc othc! b|'lf, noai 2B'1 rldsl raa ll.Boa th! fraa ard DadDatat rra Eabrtt8rd, tba ftgurc tr ooDBtruotcd DoI' ltlll o|ta of tba -{.- fl,gursr. tror oboo!. ut por,lt D oD polygou rEd r 3 a ral,Drrbctst. troE lt alret r DlBDaoallsulrr to l3 eaA cr- of tho glv.n Do trlat tbl! D!4)Gndtoul,rt to BoGl tla DarLDcta! rgr1! !t C, larg€1: aurflca rd d,r.r tb! quailrl.lrtcrrl ACID| fht 8r,g1a! of, tbo qurdrt- squel lorlDGto! l,atarrl rt C l|rd D .rc rl8h.t r|lg]'ar, tccaula It trot ona cr! lroof of tha Da tr|lrfoE thr ltgurc !o l! to DrIr tb@ Elgbt tDg1r. rhllc lolygon8. lo t LcaDhg thr D.rt. bctrr.a thr qurilllrt lsl CAD rad thc pcrl- a lolygoD tbl ! lrtcr flxcd, tbn! borcrabg tha frqr rlthout obangllg thc rhtle thc !r!l! pa .Dqtt! rblch Lr oon tnry to \rpoth68lr. 3lnac D l,! art tootor 1! rqErl rrbtbary pobt tha tlgulc Erlt bc r otrolc. rblch ba! al,!o (4e2) (4e3) trlEllsrof,s or gltsnlEBr 3 xElEoD 16
@-_!!e glttcs ths othar proofs ale qui, tc slDllEr to tb6 eboye, only a brtsf EusrDsrjr of the tro proofs nost illEeuEsed by writ- ers follo',lng gtelDer v'lll bs glrsn. fhaae rre hla aeoond End fifth proof8. Ths EecoEd.Dcthod lntroducer a quadlLlateral alratD ihleugh r&y four potEir of tlrc p€riDetgr aBd tbo us6 of
ths faot tba,t th€ nrxllu.@ IroLygoE of glveD perLu€ ter r,r one that lr Luscllbed ln a clrols 1|l order to prore tbrl fo! any flgurc otber than a ol-lslc tlr aloa oara b€ lncrs.aed rhen th6 peliEster 1B kopt flted, ?hc flftb [ethod shor8 tbai a Dqrl, - DUD flgure Dult bs lyDetrta to ra crbl,tlary exl.a ard thore- ,gurt rtthout, fgle [ual ]a a qlro16. sll)othssl.. If 3d1er .tt€lrpted to llal(c a proof f34] by descrtbbg 3, raDlrca ona e ge@etrlc colatFuctlor! for neldDg a rsgul,a! polygon of et
,b. otbc! brlf, nolt 2B-1 lidrs frolo an llr€guIlr potygotr of n sld€a. Tho d' tD. ftaurr t! oo!€truotcal polygoD has B lDellGr p6rllosts! tbaD the glreB I l|Nr poilt D oD polygoB sad i rurfae€ ar€a at l€aal Eqnal to th6 surfec€ l!€e rt to .l3 rra ar- of tho gllrer polygon. Ea alao pror€a tbat r c,-rcle b!.a r ^grrn !t c, larg€r: aurfeca s!€a thrr| a.ny r,Egula! polygo! rbtoh h8r aD qurdrt- of th. equ41 psriDat€r. theEo tro tbeoreEa 9,16 oooblngd 1!rto r It Dot o!! c.J! proof of tho Eaxbrum prop€rty of the qlrqle as coElarcd to . .[g:..r rbll. polygona. To tal.s qErs of tbe caae rhen tbo ftgure ls not C@ rD(t thG p.rl- a polygot! th€ Bethod of gtoh€r la useal to lEoreBse thg BroB , ob.n8lttg thr vhlle tha porltlats! roealns cotatant. A polygon rhoro pe!l- lnoa D l! ra Deter la tqual to ttr6 psllDeter of thr orlginal flgule and tblch bas alEo !J| equal trea lr lnBcELbeil t-u th6 lncreaaeal 16 ctot{rT?rc lRooFg ( 4e4) (4e5) tal llgule. Ths theory regardlng polySon8 ls then applisd to so that lts E[d qoBple te tbs proof. Ee thoil of 8t€1na sturl.l (1,84L-I919) gave a allecuEeioD pe]l or ttre naxr - euxlllery cuFal Du[ property of tb€ circle. llls proof of the condltloD nlng poht A' t tteceaaary for a laf,lDu.E ls the sarae aa that of gteiler. He on g. glnos tb! bBs elso siEpllfled the conatruction of Xdler desclibed ebove. A, lt is Do8llbl {tttins (1861-) prorea ps] tbat arly fLgure, such ore linlt DolDt that tf a lltr€ dll.lde! 1t8 peri.tdeter into tro equal palte, &d tbq llDa 3 | lta sr€s l,B also dlrided into tro equeL peetB, lr s flgure r,, rith a certer polnt. Ee coopletea the lroof of the Iroleri- lDd th6 nrtbod o Detric lheoreD by shoflng that a maxlmuD figule !4rst have Euch arsa tba|! tha 0D properties and thersfore a qento! poiut a,nd flnally that all bave aD uppgr bo dlgl0eterg Bre squaL. llla Droof lB opsn to the s6ee objectlon liroltr and Celat aa thet of Stsller. llBlting ourro t (1s73-) gr.v6u tbe len8th! of t Pados b!,! sll excetteut revfer ffl of the first tro proofs of Eteher. hBve s radlu! cq
Ceratheodory (1S?3-) aod Study (1862-1950) havc rrit- lnclosed brtfsatl g, ten a Jotdt p"p"" fa! la rhich each glvea s aeperate proof. the tbra They aoillfy tlte Dethod of gteine! ao eB to ttra,k€s dllect ln- stesd of sr lndllcst proof. lhe chaDge of Eethod j.s bade l!1 8,ad thr oqu!,Ittt orde! tg 4void the !.eqoasity of nakilg tlrg ealuDptlon that a rhlcb provar tbr rog^ri.Dud erlatE. arer vlth g tbrr
At! outllne of the proof of Caratheodory ls ao fol.lowar gtudy !!
The p181e is dj.Tided Lnto he1ve6 by e Line g, 9rld from a' poilt ge ltsrtr rl tb I
A on thig Line En arbltrary curre of length r| la conatlucted and !hor! lo[ to (4e4) ( 49s) EXEIISIOTE OT SBlS8R I S IBTHOD t7
applied to so tl|at lte end point lB aLgo olt g. A lodlflcatLon of thc nethod of Steiner is uasd to coostluct a,D lnfr.ulte rellcs ot 5]l of the uexi- aurlllBry qurveg all of leDgth tT sral baylng ! colonoD beglD- oondltloa nlng polnt A, but f,hosc enil pohts !,rg s !61169 of pointr
gteiler. ile on g. Slnce these gnd polnts are at !!6t a dLBtance rT fro! lescribed ebove. A, lt ls posslble to aeleci I subaet vhich lrar one ead ool,y lgure, euch otte lirlt pglnt at , fhe areea bclosed bctrsen tba curvc! :quel parts, ilrd th€ llno g ere dgnoted by
l! s, flgure I,' Ir' Ijr ------' In' --- f the Isoperi- |!d th6 ne thod of construqtlon rhor! tbat no one l! 8Daller 1! t lttBt have aucb er6s tha,u th€ olle pleqedfug. gllss tbE areaa of tboaa curvg! lally that 41L haye aD upper bound and sro lncrcaalag tbsy Erst oonvcrg! to r aa&e obJectlon IlEit, alrd Caretheodory proveg thr,t the6g currs! ltlve ar a lltliting ourre the 6eliiclrol,6 alrer! through A a,ad o, . glnqq rrler [{ of tbe Lengtlrs of tbo qurreE re&ailt €que1 tofl, tbs slrcle lrult beve a rediua equBl to ons. Flrlslly lf Io alctrot€r tb8 rrcr )50) havc yrlt- lnclo8ed betreen tho flrat arbltratlly seleoted curre sld tb6 rperat€ proof. llne g, than t e dlrect Ln- ro.h=ft/z, tod ie nEde 1n aIId tho equellty Blg! c8.E bold oBbr rba! Io lt . EcDLclrolc,
Dption that a thl,ch ploves thst a asEl,qlrcl,c of Lrngt! T' l.uqloac! a largor
a!e4 rlth g thar ally otber curia of cqual langth.
1r as follow6 t Study B.ke! ulc of tbs ay@ctEla De thod of gtelacr. nd f!o!r a poiat Ile ltart! rttb a polygo!' rblqh ha! r pellDcter zltt.D lcrgtb s con8truqted end shoE b,or to conEtruct a lror conver Dglygob tbrt ha! ! CXOIETRTC PBOCIDE ( 496) (497) exes of !y@c !rt. Fo! tbd trer polygoD tb! parllctc! rcEaiDr tho Beqond Stoup conrte.Dt a|!d thc sroa ls Dot lesa ibr|t tho orlglDrl ona. lr ot tbis lcstlu. n Lncrgrrsa th.rc i.r produq€d e ssrle! of polygoa! thrt rD- E(u,v) rhloh rpp proech a! a ll,al t E clFcle rhoae redlu! r 1! to bc dotcrli,nsd.
SlDcc the perlratclr of the polygo!! r@41"a rqu31 to ztT.ad lb t trlttt d Brc eltho! aqurl to or gleater tha! tb! perlDctcr of tba r, ara r ilcf L!a3 llD[ttng clrele, lt fouors tba,t r E 1. turtherrore thc arrl,. lb ar.. ar€8! of the polygono nuat approech tbc arra tlr', aad henca ths Brtr of thc originrlly glve" polygon ro.rot bc 3t. f rbqla E ta dat!,n ibs equaU ty .19! hold!, fU lbo polygou! of th. lcrl€r nrlt qqntcr of grrrtq be cqqal La rr!a. &rt t-a tlEt care lt lE poltrted out, tbat gravlty rr tL o thc letbod of oonstrEctl,o! Isad! to 8 oontrqdlcilon. It lo1- lolat o( ' p o! U Iorr thrt r olEcla of DarLEctcr 27r hrE s largcr sror th|'rr |ny P= stD€. Cbool Dolygon of 6qua1 DdrLlctar. tbr outcr lortfl
8. DG I.cbor bas g{veD tbtrG lroof! [q] D|rca .€sftr E qtr tb" Dolblrta thrt tbcrc 1r a EarlDr! flgulo $[.t!g all gup!ora tbolc tlgura! brrLDg r glya! porl)D€ t6r. El aDd aolatanat 3l.borbacb. (1e86-) ploof bst Blven ., ol th. thoor€t8 E4l ilcnotc tha .rca ' tbrt allolg r,IL d@alllr thrt .rr Dl|'!s, fllltc, .Dd bayc r glvon tr dlacte! tba cl,rcla hBr tb! lugert rt!a, rb6r! br !6rtt! by rlrcre t ard tr. r dleretor tha naxltouD dtttalco bstr€cn |Iry tro poLEt! of tbe glvoa W tba 6qul bo rd cr. r.v2, al xlnkorlkl (1864-1909) drycloped a tboory pfl rcgera- tbfr qu|ntllt I i bg palrs of trrc'l,lcl oyrls tbBt raducer to thr Iropsrbotrto tre!!lrtlonr. I thlotcD fo! a lpeclat crsc. Thls rorl 1! tba b6glnrlng of ourtc!. ft tr D: ( 4e6) (4s7) IIjIID IREA O' TM OVALS 19
Iratc! rrDAlDa th9 second groqp of proof! ngntloncd t! ths flrrt parrgrrpl
Bhrl oB.. lr of tbia lectloD. Ee dcflDe! .|! orrl Br !aaD! ot a futrotlo! Bolr tbrt rF E(u,v) rhlcb rl)Dc.rs tD tb tllqrrltttr o bo datc:rEred. urtE'fg(t't1' [r1 to ?n ald fbr tatllliy ot ruo! llcquall tla. tor rl1. pq!.1b14 Yrlqc! of tor ol tb! n rla ! drfb.. a d6rLB ol poLEt! x,y raleb tr sallqd |! lrnorc tha atrl. !b rrra ol r! ovrl h r)r'" r', tld haaoc aG+ Vz./. | +llloJO. 5q 9n. tt rhar! E l.r datt!6d f| ?ollonr Drrr r urllt olral,r a,bolrt tDa ia acrl6! nrlt crDtcr ol grrrlty qf tha otal rDd conlldo! thl! c.tltrr of t.d out tbat grarl tJ' u lhr orlgtB ot tb lJrrt.L of soorall-D,.tc.. to! aoD! ctlon. It fot- polnt .(, p o! th. prtlDtcr of, tb u!1,t clroh vrlta.<.Coa4.
I frtr tba[ ||ry p = Sl|r 9. Choo!. tlr Dobi oD tb. bquad.ry ot th. ovat rbarq tbc otlts! t!ot|1!d b. tb. dt iDd vrltc rlctl,oE < , P I Dfrcc rgFrn E("(,p). E(oor€, rrbp ) = s a Ellg rll EupDolc ibrt rG brvs tro ruoh ovela doflErd by E, |!d E1 l|od eoEstruct . lhlra ltrqb thrt g. (l-t)H, + tHs., If r! i tac tmoro pl]l dcnotc tha .r.a of thlr tlltrd oval by t ihc! |Id bara r glYoB a. 1r-tfr, + e(t-t)ut + tlr! r ba !3r!! by rLsre t, and tr am thc a!ea! of tbr fl,lat tvo ovrl! |!d I l!
9ot!t! ol tbr glvcD bJr thc 6qurt1otr, . ('n r t/z I E,(E! + 4+) de. yzf y,9.,' * {:J1r,lde . " /o d0- /o Je' Ft Eg !c8rrd- tbt! quanu tJr I l! r! lnvarl.at rlth rclpoot to al,L prrrllcl I I!oparbotrio tralrlatlon!. ll1.ntof,lkl c.LIE X the mlred arce of th! tro baglnlllng o! ouF c!. ft lr prored tbat i 20 CEOIE1BIC PROOTS ( 4s8) (4es) (r) u" - r',r'. i e F(t) car t a|ad tbat tha oquali ty slgD 18 va].ld only rhe! the tvo oy41B then the dlBcrtlllnan are aLnller (h@othEttc ). If tfig Eoconaloyel ls ohoeen aa t a unit ctrcle, Hr becorea ths lnteger one €Jrd the expreaalon The BotlYc ts not a olrcla r for l{ !hov! that An l! the perl-E€ ter of the ftrst ov€,1. If ara tharrlorr t tbo pgrlmster anal ares of tlr€ flrst ovg'l are denoteal by L and greeter prrt I the! (1) gtves of lt -lnnlo tht 6 lalt ctate of tho theory o 8.nd tha equrllty rlgn lB valld only f,hen th€ ftret oval iE a publlBhed tr 1.9 clrcle. It !,! eeslly verlflsd thst thts laat aolrtenco ia a perlrstrla flso rteteDent gf the lEgporiEetrlc propsrty of tho clrqle. l3 !t4114r to t The thcory of pera'llrl qurves hAs be€B alt8ouascd by 1t6 stlpllolty. .everr,l rltterc. crotle (fezf-) has ao ertlct€ $! on thrr Theor@ lubJest thet eDpeBled about one yeBr a,fter tha rorL of ulnlorskl. (reaSJ) pelllteter L of llaEohke gave e dtscusaron ftil or trrc theory of ltlnlorlkl lB att artlcl6 that apDearod ln 1914. trs anal fo! etery o dlacuaBeal flrlt tba seae rh€! tro qurdrr,Let6ral,! t€,ks tho place of th6 oveLc o! IlDkor.kl. The theory lr then exteDd- l,rt C I ed to polygone eltd flrlally qloEsd to ary currea. .1,paper [O] scrlbed on lt, trrltts! by tr'robcdtus (1849-191?) lea publlEheat tn 1915 a.Dal illsiq.trco? f,"tth auo ther p9J bJr Llcbnarrrr {t8t4-) rn L919. the lart tro papers lo th,s targsntr uEg th€ theoly of quadretlc foErs to proye tho lsoperlDetriq uDlt cl1016. I property. If the eles I'(t) of ariy curvg of r fallt,ly ls ex- D Yhose eles co presged in tern8 of thg area 3 and the perldgt€r L of BD arbltrary culvs, for erardlle by a foBula THEORYO' PITII.IXL CURIEE 3l (4ee) ( 4se)
r(t)=F + tL +trt!' Y&Iue. of t ollly rhatt th! tro ovalg thett x(t) ceB be roads Degetlre for real quadratlc gr€ater ,l! chosen aa the dlEcrlnlnant of the fo!! 1! tbanr zcrg. tha exprsssloD lhe Eotlve la then to shor tbat for aly elo6ed ourve lhat lt t ed.6ts for thtcb t(t) 1! neBatlvr rlt ova"I. If not s clrclq a value of greater than tha enot€d by L ard srd therefore the dlscrlrdlnant ls zrro. greeter pErt of l,lebtrannrE p4per iE ooncemsd rlth prgvllg th'.s Laat atetement, Bonneosn (18?3-) hBs glyen en account boot lltt oYaI lr . of th. tbsoly of tbe Dlxeal eree of tro ovala 1! hf8 fsil publt ploof labtoncc l! e rheat 1Il 1929. Bleacli
trluglc. of parallel tangeltB celted by tbest tor Ct ald Cr. tr'€t t bs the flgures t! Iength of r 1lne neasured h a posltivE EenEe along t t.a- end thsrltorc 1 gslt frou c poiut of cogtact lP- qn C to tbt l''tersootlq! oi tror p ' 6'1, 1, thia tragGnt vlth s sid€ of leaE tbr,! or a( D. Let tp a8al tt be tha sor- lootr of tr'f . ( rccpondl.ng lcngth! for pare- l,le! ta,Dgent! to Cl' aod C). Let If th.
U. tL. Englo b€ trson tb. t8n- It l! !o.t tlvr gont anal r fl$d dlrsotlon. Let thG lDtcgrrl n d, dp, ran dr b. the ar6s,a of D' ltaat. th6rs!( Dp' arl'dDr. tret f' lpl aad I'r tangetrt8 roil I be the ]reas of C, CD, 8rd Cr. sirculllcrlbed I Lct ! be thr radlu! of th€ lnlorlbcd olrcla to thc trlEgl6 suqb, that tbr d D. fh.n w. bNe the follor1lg rclBtlons, suoh thal d.I I /n tl(f) = t({) . !t'(f) t FpE dp - la), ti a(1 flnal roBark oI ..qfn (r) Fr. (p + r)-o' - L/2 (t + D tr)r il the porlaete! ( f_n 0. Thq aRea of th! I'$rllc1 curro llay be oxprca.al by th! lquatlo! lhes. (s) tp=r+l! + D't. Lsbs!g! rbe r€ L dcnotsr thc p6rh6tor of C. lhl! oaa te leen er fol- bs thor! tbat t Io'.rr r Consldo! thc EnBII trapcaold lnoLoBod by tha s,rc 16[gth 'ti.y. Hla ltat€ da of Cr tb! tdo lengthE Boaaulgtl along tho norEBL! to C donallr !o! rhlo thlougb ths end6 ol dE, 4Dd the erc of ths outgr curre lntor- to tht Bar, !l (5oo) ( 501) ll4l'CE(I 23
cepiqd by ths96 l6ngihs. Thc Erge ot caoh of theoc rDall flgures lg p(ar+rrzEpad) a!!d thorslolc by trklnA th€ aur of thoE ooc !€e! tb.l I !p - r. lp(a. + lz e af). Dt * rFlT l'or p - 6,1', la posltly€ by equ.tloa (S) .ad for P + 1.61" LeaE tha! or equ&I to zsro by squrtlon (1), lhereforc tbo rootB of xf . 0 ara r€a1 aad lt tollorn tsrt r,- - lnrlo lf thr equr,llty slgn hold8 tr'p Eur t Dot cba,[Be 1-o !1g!.
It 1! positlvo for !. o, thersfors i.f oEa !!t! n.gfn (f) thc lntogrrl Euat vasish altd tJr. rstlo t/t' cqual to r 1! co!- ltaDt. fheroforo one cen rrely orle aldc of tha trlrn8lr of tengeBta alid r rl11 reEa,ln ulctralged. It follorB thrt rll
clrcuBscrlbed trlang1e5 to th€ oyBl C brre atr lnscrlbEa clrqla I tb! t!r.ugl6 Euoh thrl tbc dtceetor3 of all thr or,rcles 8rs equ.l. l flBurc sucb thrt d.l !t8 dtse€tcrs rr! cqull Eult bc a o1ro1c. !h€ fi a0,, flnaL roBark of ths altbor t. thst thc proof is rElld onLy rhen rl the perllieter of tbc flgurg ogntol,ua nc cornerB o. llrol8ht ra ty th! cqurtlo! llue a. (18?5-) Lebesgue vrot€ a paper [s]f in 19L4 in vhtch bc lcan ar !o1- bs rhora tbat th6 IrqpellEatrlc ProblsE Day be ateted lD 8 net
)y th! rro lrrgth riay. Hl! rtrtenont of tll! I'robleo ls 8! follorEr llld a lErlr to C doEa1n for rhloh tho ra,tlo LrA, the Equar. ol the perheter lor curta lntar- to tha rrqa, sbal,l be tJ1g leaLleEt po8rtbls. 21 GsotdBtRrc PR00'g ( 5o2) (503)
3le8chke lraa glven e proof of th€ lBoperinetrlc th€o- Theor€n fot oonr re,r by Eeans of E nethod ehloh ls dlffereBt froE that descllb- ourv€ Bhioh i! ! sil ebovg. fls h8s vrltten tro arti.cleB ln vhlch th6 Ee,ns the erpres8ion t eetbod is uaed. Ute laper f4t of l9t5 i.! a brlea accourt curve. H€ rtat6 of I ploof rbich, 1! glysu lrore ln dctail fn frfa toot fafl L, . : rhlch ras publlshsd j,n L916. It ls proved that lf one bas 3 cso be oaally d€ oloaad, contlnuour, reatlftabl€, pLans curr€ of lsngtb ! a,!d aad for tho gstrer lroa t' one c.! oon.truct e polygon Euoh tbat lf A ,,8 Ltr He proreo tbat lellnotsr alra d ft,r srea then 1t te tEue tha,t . lL-Alze,lF-61 € tor a oonvex oun rbeEo € 1r !r albltDrry quentlty, ard it sL6o foltorir that Ecri.bed alrd inror
A'- un$ 7 6 t! 1905 rolkod ;u Nor lupDosc th8t sphere. Olla o! t ft _ 'tnC __ t o L1 +rrt of e !1ane 1! tbc then 006 can qoDEtruct !, polygon Euqh that te! eboyq €xoept , X-_tn$'o r/8(t + zn)L. e rhicb 18 qontrary to tbe raaultE of e prevlouo proof a.nd thele- Eorrr8t€l! and laln fola one Eust bave Eype to b9 dcrtTad Tl '-sr > a L -.1 rrL iv A brl6! ot lhe uetbod qf Stelncr nay nor be uaed to ahor that the equatlty Constd6! the qurdr rlgn hold! onLy fo! a clrcls. Sonneae[ ba! ElttoD 8 nulbe r of arllc].er [OrSfrSfl Its dtscltntnrat i o! tha IEoDerlEetrtc Problca but bc ha8 glveu tba csssntlsl! Alrlde! out thc fl of r11 thls Datrrlat tn a Uoof $S]l . Ee begln! th! dtgqusllol equal to aet,o rbol rlth tbG r€ttrark tl|at lt l! sufflclent to provc tha Isoperltletrlo Thc f ora tr !o!tt, (5o2) (503 ) roNxEttEt 26 a Ilopsrlnetric th€o- qurrreg Theoreq for oonvex becauao lt 1! o3!y to !€c that r 6Et lro! that d98cr1b- oulve ahioh ia llot ooav€x oa! not furarigb g natl&u&. IIr callE n rhl.ch tho aalre ,'Iaop€rtDetrlc the erpreBalon tlTetr - F tbc Deftolti o! e 1! e brlel acooutl! cureE. He Etete8 that th6 !ol?ula! book tn h1! L4U L, =l+8nt r r,.I+!tt rrt{- lf one haa . 8d thst ca,Db€ earlly derlvoal for the oeeo ot tro paral1ol, pol.ygons, [trs of lengtb L aDd end foE gsneral the oaac lt tE rufficLsnt to DaEE to tha ttuo1t. tbat 1f Atrttr lle proveE that I tbat r?/qr - r'= ( n/a)(n_"i qonvex lor e curt€, rhela R a,Dd r are thr rsdtt of tlra qlrourb that t a18o follot8 scrlbed and lnscrtboat clrolor of thr ourtc. Bomrtaln (18?g_) i! 1905 rolked out a at&b€E of Lnequa,lltl6! fo! ourro! oB a spherg- one of tbe rslttlon! ,blch h€ obttl]oca for thr orlc of e llatre ts the lnequallty of lonneEon rhlch ba! bacn rrlt_ It teD aboie ercept tbat the conetaot laalor Tl/4 l! roDlsqcd tttr t/AG + znlL. Don!6sen rsfrr! to thi! iaequrtlty of :6vtous Proof grid there- 8€rnst6tn and !ay! thet lt ,at thr flrlt lnaquallly ol tblr type to bs d6rlved. 0 A brlef outlln€ of tbr proof of Bounosen l,! ar fol!.orar r lhor thrt the oqualltY Conslder the quadlatlo foE F+r,t+fto. i 3rttcrc! er,s! It8 dlsqllnl,ndrt [0, l! prcolacly thc dstlolt o! tha oure bolt onB th. lrasntlal! al?lde! Elvao out th6 faato! 417. Ihc d€fr,ott l! grsrter thrD o! t bogln! tbt dlaousslott equel to z€ro llhen tbs lootE of tho qutdrttlc foft arg !ral. I plova tba lloporlee trt o The for? polttlvr l! for t . O. If thslo srlrt va,luc! of t ta (lor.lttro DnootB ( 5o4) (5o5) tor rhloh tb! tollr 1r l!t. tbr! or .qurl to l.ro tD.! tba eld r tht rra, loota aFa rrll. It tbrla .d!gr r ral,u! of t for rhlob Lr-I- n*lo, zLu^ tb6 tba aaSttlya of thlr nlu. of t rlU nrl. Bnt ttrrtt ntn 6 o. fb Droof l. coElrl.t. rha !t l. DroyrA tb.l e|t(l thrlrfort Lt-r- ttt' t o fc! t .qual to r o! R, tb. rrdll, of th. brortb.C rad olrol!- ald thc 6qudl rqllbaC olrolaa to tba oott"ar flgura. clr. But (1) ( lontlaran Dlovaa tba lut atrtalant tn tba Draoarlbg lrlrSrrDb rhaa tba flgura lr I oonr6: poly3on rr follorrt fhe! a abllrr
Itao!1la . olrola 1! tba Doltton louohl,ng tro .l.d!a, rrd lf ol 1016 rnotbcr t$. ald.a rra Drrrllal ihtr l. tb. 1$8.rt otrol. thrt o|! bc ( lnaotlDad, but !.! aot ona o|rt rlrryr dlar tb olrola to touoh lbc luoqud.ltt thra. atd... Drr' t||r3!!t. to tha otrala at tbr thrra Dototr ( rb|l. tt gduob.a ti. DoLySon. Tbr|r t&B.nt. rlll. for! ., trl- 1481.. Lova arob .taa ot tha Dolygon D$rl,lal, to r fl:.d lha Isorcrllat dlraotloar .ry D.r.ll.l to iha bl..otor. of tb. |Jrgl.r ot !u1t. tua trlrn8la cr Durll.l to rot tro pa U.t lldar of tb. Sonnc! Doly8on untll arob .td. of oonvor ourvc b tba DclySo! l. trag.nt to ths portnrtot ti. otrolc. It C.lr tb ths polygon. l.n3tb, ot I rlrla of tba slr@e t rl o Es tl l|olySon, I ltr dt.t.[o. of Stolncr. I fro! t)l. o.ntar ol tla olrola, ln th! prcaadl ( 504) (505) aoxf,tgrr 2l rato tDa! tb. eDal ! tha ndlua of tbc qlrolc thrtt onc bas t lo! rbtab lL,r^Ha- (H- = (tra -2r1ruo):nr" ")A ta hrt F' tr/a{at L=Zct eral thcruf ora (t) rL - r 3 rr' flDaa uA olroul- rld tbc 6qu lltlr elg! t! vrlld olly rhon ths fl8urc tE r stD- cle. 3ut (t) oaa uc Bittsn,,8 tba fo'r .D tb Drac.rlbS (z) t/en -l"z fto./zn -t)L o ra lollotrr Vhe! a ll.ElLrr ploccsr 1! c8rrleal out for r, olrcutrloribad D alo.a, rld tt clrolc srothcr lnoquallty 1r pFoduc6d, !.D!ly, ,trola tbrt o|l tr (s) t/"n - rl rr(n -Ven)L a olrota to touob Thc lnequelltlcs (2) rld (3) Ery bc ooEbtasdtnto tbs lncqurl- , tha tbrar trolntr ltv a rul fot! r trl.. (4) I) /q, - v I n7a6. -r1L al !o I ltr.d lha llopcrh6t!1s thlorrlq oa'n bo reEily detluqed froD thl,! 16- tha Egl.. ot lult.
ald.. ot tb, BonnoEea ray! tb8t thc lle tbod csr bs epplled to r|ry gonvor ourve by rdovlng rll trngents and all ltral,Bht llnea of th€ p6rLmotcr parallel to e flred dlleqtlon a! ln the car6 of
tbe polygon. 116 bas gr.ven a seconal proof fs{ by Eeanr of a lJr@etrla nethqd. Hr has slso lodlfted th€ lJr@ctrlc Eothod of Stel.ne!. In eBoh cals he arrlve! at ths leBult (a) gtren 1n tlra proordlng prragrrlb. (507) 2A !BOO33 lr IOUnIn SRIIS (506)
glven 3. Proof. bv Eranr of lourlcr Ecrlra. Thr Dloot of by tb. .ql tha laoDcrhatrtc ThaoraE by E!.a! of tourlar Srrl!! dcpanalr fol,lorEt on r throrrE rhlcb lirrri tz (1859-1919) o.l.l.d tb. funds'@ntrl x.ao/2. thlor.u. lhia thlorcE lr ar 3ollorrt If r=bo/z, Blur.-../2 t:(.i glTen l. Thr Droof ot by tbe squetlo! u = ZttS/L. r aod y eay be erprcclcd a! lr 8at1!! drpand. follor! r oo tb! fundaolntrl t= to/z + !(a*Cor ku+r'KriEku), r = b./2 + ;(brco! k u { brr ar.D k u). r) fb6 derlvatlv€s rltb. reopsct to u er6t dy'clu ^ coE - Tt(".x t u rsrla I u ), 1l aylau-!t(tr - Kcos t u b I rla k u). vhcn tb€ rarlrbte ! lB tho €quatlo! layaalr + (af/d,.lL - I t( rtxb ) i! r€placsd !0r u tb€ Equatlon beooEg! ;or ttnl !6rt! tbrt (az,raula | (dy/do,lL . (V2 tr \L. ,laoat laDltaalt lhc lutegretton of thl! equrtton Blylrr 'utctlona [Y, I o a-t !ro!Dt )o [ur/u"f t (avlau)ilau- ztr (Uztr lL . f/zn. a of tba Tbo a,ppLlcation !c!la! of th€ fuada,esntEt thaorrE delqrtb€d abovs Eo t. lU. .qulyrl6nt thlE lalt equatlon glves pn thr fulotlo! (1) n * * . ,1=.r,"/z7 i ,*, "i ti 16i* The areq, lncloaed D!opartlc!. by ths cury6 Eey bc reprGsented by th. equa_ L br th. .qurtlonr tloD /'tf gurvo tbtrr ! lr = (:'dr/dulau. ! Jo It But egaltt FrlD.tr!. 1! on Bqcount of the funda&rntEl thsoE@ tbi. leduces to rrlo(ltc, srd ht! r (2) I . )'7) {a,,r;, - e'* b71). [!l ta nulber ot It fotlows frora (1) alrd (zt tb..t I Dourlrr Srllas t/zn -* . i"7 (r'-r) f [u.-u1 )'r(rei + u*)rr (t,.0.""t, It ua auDpola and stlcs tfrc rfgit brad !ld,c of thl! cquatlon lr Dltcr uega- lbc ourve Do!!r!! tlY. lt Eult be tnr! that a tb! grl|llatlr u r." - jDr 3 o TBOOTS BY }OURIN, EIRIEA so (so8) ( s09) O|tc oblorve! thrt th€ equrllty ltgn val,ld l! only rhen lDterEoct thollc a', I . - r (f - U, O, r, D', 0r rr' rrK= bh:brR8 0 a,3r4----). tuuLng trlgalt But thcl y tla ssrtc! for r rltd roduoc to rppllc! gDa fEod x= e"/2 + rtCot I u, rr,.lD u, prrqgraDb of tbl y=bo/z-lrCo!u+3r!l!u argS f,hlcb Err tha par&06tr1o oquttlon! of e ol.roLc. lhatafols for rll llnply olosed, reotlflrbLc, plaac currc! rhosa equrtlon! 3!d urar tL Llt x = x(r) , y.ds) barc tho proportles desortbed tt thc bogla- ntn8 of thls pare€reph lt tE truq that to coq)utc tbo a f - {ny-o thc pcrlrGtat of ard th6 aqual,l ty llgo LE valld only rbsn tha ourrc l! r otrcla. atanta of r(r) r Tbo tloDarilctrlo fhroroD' follorc e6!t1y f ron thls lnoqurll,ty. tha arlr ol r ol, Thr ftREt proof of ths IlopcrlEotrto throret! fS@ vhloh lrr curr€ rhlob Eadc ulc of th6 lourr.er gerlea rag glven by Hurvlta. Lcbclgue ullcr! ti! ragul' lr8! also glvln e Droof ablfrr to tbrt gtvon by lturrtts. Sfl rgtl! Drlscdcd t Iaob of tha6c proofs 1r pleoeded by c, dotalled dlloullton ol outtl.!. Et oC gerles, thc thlory of tha i'ourler ?hc appllc.tloq oi tbr !1. Vor1rruDSu thco proof ry to thc of tbe IsoDerilletrlo Thaorrt! i! b sach a. Proot oElt tb,6 sauc crqaDt for I fer dettlls, fha tJrpe of currs! atrtcocai nt r to rhr,oh the lroqf! apply lE dotemtned by the dcyclopDcnt of llotcrlD tRlc lL tbo throry of tho tr'oullor Scrles. Elraqhlqr b.r ',ri,ttsn r rtl! ooucalilrd q theata b rhloh th6 .!rr of a logular oloeod Fd ourvs t! !bl! ltrt.Dtt o cordlarsd rlth thst of a olrcLc by mean! of tba lourlcr g6rl€!. !!r. Dublirb.A D He dsflnes a regular curyg aa fotloEt A rogulsr curv€ t! knor!. frrv ol contlnuous, coallsts of f flnltc al4bar ol aro! rhteh do !o! pRob1a rb1ob, |! (5o8) (soe) HISTOBTCAI REII&(6 sl I only rheu I'.ateraect theDBel-ve!, cDd eroh r.ro polsclro! a contlnuoully t0 (1 . a,5,4----). tuEing ta|tgert.t cysry bterlor lotnt a4d 6nd pobt. Nr rppllo! tba ftlrdtlloDtrl thaor€E rbloh lr rtttcd t!' tD! ttrlt lrlagraDb of tbtr ssot!,on to thr !yr!a!r!c fora .t fora ttrc ar9l rla. tbrrcfore for lrn t. t/z / (ry. _ Jr') d s, t, r tholr squati on! r,Dd uror t&! htagnl of thr lrlatlon llbqd at tbo bsgln- lar,/as)z+(aylar)a=r to couputc thc 3rt.' of r olrota rhlch b8r a perLBotcr cqurl to tbc lerlDct€r of tb! !sEuJ.B! currc l-o telar of tbs loullc! CoD- rourlcl!rolrqlc. ltant! of r(c) .-a y(s). tr! proye! tblt th. dlfferclor b.trssD )r lhl! lnoqusllty. thc lrcr of r qllclc of glre|r pcrl-E€ tcr rlld ihc srar q? a rcgu- irtdo lhoor€E whloh hE ourre rhi.ob brr !a squal porllsct r lr a loritlvs q.u|!u ty f,ulvltz. LcbrrBue ur16!! tb! rcgulr! corrc 1! ltlclt r otrclG. lb6 proof la by lturrlts. Blvcn agrttr preocdeal by r CcyclolEnt ot tba tourla! g€rtGr. l, good Larl dlloulrlon ot outltnr L59l of th4roal ot Errt,ltr l' glv€! bJr Elarotikc t-B Lo.tlou of tbc bl! Vorlrrulgu ebr! DLffar.Bgld Oc@ctrlc. torao l! lr! €Eoh ,1. Proot! b]' laraa of tha orlattur o! rrrtrtl,on!. l 1 tl4)o ot qurea rtattacrt rat trdc l.! Ecotlo! 1 tbt th! rrrly lroofr o! thG iba rlaYelopDent of Iloporlllatrlo Thcor@ rhloh rpDltoal tb! oll,culu! of yallrtlonr br rrlttsn r i ror€ ooEccrrcd only r{tb tha lsoqlrtry conalltlo! for r tlarlEr!. ilorcat orrrrc l! lblr !t.trt!c!t eotttLtluc! to bc tr,o for all luoh Dloof! Thlch ba louricr gcrler. rcrs lublllhqd blfor. ibr rorl o!'falc!.trur (tatS-IAel) ru gulrr curv€ 1! knor!. lfrny ol tb! vdtarr dtloqltcd r Eodlfloatlo! of our ro! rhloh do not problcu ihr,ob' ra rt[l oru, !I:ob16! I, eld cf rhtoh our Drobla (51r) 32 PROOI'EBT CAICIILI'E Otr VARI./ITIONS ( 510) tbai tbe lrthod i la B ap€clal cgae. cotnsLdc. @Ig!.!. to flnal aaonr all ourrer vhlqh have a sLYet! Pmots a lensth arrd,loln tro flx€d pointa o[ 3 stlalEht llBs that one besn gl,rsD by !o trhlch i.nclo8eE vlth the lln€ 3 nlrrllru arca. Jellrtt (181?-t8r I flrst leoesssry oqndltlott that a tllEp1y qlosed' atroa reglrIar cur"e of glvelr longtb 1no1os6 a !&Elnu! alear 1f aEcb @, a,adr grdresld (f865-) a culr.e exlsts, Ea,y be deElvaal froD tbc thsory of Problen I by Vclarrt! alrualng tbat th6 tro polnts on th€ t!n6 qolnclde. A fl!!t tbD lrtbod tbl Deooaaary condttlon on e solutlon of lrobleE I 18 ibat tho ol eA by E uaxlmiztDg qurre Blr811 sattlfy thi f!!at Bece€rsry oondltlonr slralG lr lr! thet the lntegral clorod curtr f i,F ^ . F-a------othrr pl (r) J--Utzts' - y:<')+ A V-'' + r''Ja t c@plrta Dot hl! r have t EaxlDuD T3]uc rhqE6 A la a lul tably Bolected oonrtsnt. !ub11!b. by Dgrrrs of tba If Te rrlte I{ for the htegra,ad of th6 lntegrel, these condl- l rolula of hls soJ tlons are that ths delLvativea ltx, , Il?,, bc conthuou! on of thslr notsa, th6 Darirlzlng aro and tbrt th€ dlffcroDtlel BquatlonB of trule! t gohrars Ioult b6 sstilfled. Tha qonAitlona th.t EX,, an tbat tbe lsthod l! the Ealle Ehen th6 tro pollta oE tho llne colnol.dc. rhloh baYe a qlven Proot! alyivhg e€Bontla11y at tbe reault abova baye rt, llna that on€ besn gtv.n by Doldonl (1?89-1860) fsfl , ateog"" (tera-rese)S! , JeUctt (lBl?-leae) uolgno-&lndelof end !urd- lEply olo6ed, f6d, F , !t!@ and Dole reqeDtly by rhonj (1841-1910) .na !t! r!6t, 1f lnch @ , [d, Il|drlr,rd ( eo- pdJ ry of ?robl.s! I by ra ) . VclcrltraEa rar th! flrlt to DAJte a coEplctc proof by rcld6. A fl!!t tba l!€ tboal o! ths eaJ.qulu! of varlatlon! that the ar€B lncloa- I ls tbat thr eal by e etrclq lr lerger tbaa thst trcl,osed by eny other !!rarf oondlttola regu- Ifr clored ourrc of squel, length. III! proof alEo plecedea the FJu' othc! c@plate proofa by othcr !ethod!. Unfortuletely be dld not publlcD. b,l! ForL. l'he reoold of lt res pleaerreal ho{€yer, roleetod oonatsnt. by DerJra of th. lecture note! of b.l! ltudent!. The aerrenth ral, those qond!- toltac of hts oollqqtsd basod on a€yerBl collectlon! ! oonthuou! on "o"t" [{ ot tholc noteE, raa publllhcd b 192?. rquatlon! of sule! Schrars (1843-192r) urcal a Eethod etulf$ to tfre aod I y'r arc oon- fol] Elthod of Yslclltris|a qornar!. lbr dlf- to shor tbat rooug ell olosed ourres rhlcb lnoloEc a glven srea tbo elrolc ba8 tbe lhort€st perl- rf' ) . o' B€ter. ft rsa DolDted out ln Seqtlon 2 that tbe Iaoperllatrlc -;7 lhoorero fotlon froB tb1! r€luIt. ) " o. ItrrEer (L862-1950) ha! I dlaq1rr6loa of tbc lloperl- Detrlc Throrell pll V4iE ' la hta bool on ttc oaLculue of rarlattoo!. E! dctal,op! both thr BcoGsnr? a!d. tbr luftlotoat condltlon! for a Eartnuro lrsla. It l! ol,cer for ProbIcE I. I! ttrr lasoud oalltlou of hl! bool 34 PROOFSSr CItgttLUE Ot VlBrlrlor8 (5r2) (515) be deduc6! tbe fsoDcri&ctrlc lhegro4 froD tha rolqu,oB fot EBy bs erlrrssreA PEobleD I. g1! argreent !s ao follorer Conlldet a oloscal , ' o/2(.!/rl'- curse conBlEtlng of a flnlto Duroborof rogula! part!, rnal laJ(s th€ dsrtyatt l-et tbe lelgth of tht qurr€ be L. tr'tnal tro potnt! B , Pr- tlvo oquel to s!) rhlqh. dlrlaie tho perbater i.nio tro squal p&!ta ard JolrL Iht! gl!s! the6e points vith e strelght llne. Booauac of tbc lolulton -04- o"cgg o- I 2 O{ to Probl,olr I, the potlrt6 Ea,y bs Jolned, by arcE of 61!ol,s! . 1/e coE eech of lcngth L/2 L\ a Belllor ruqh thr,t, tbc iErl llrqlolaA o/2(.1 lhls har by thB tro arq. le greate! thrn tbr rrea lncl,oled by tba onty onl tlon glv6s e Dexl orlglnEl curre. It rollal,la to be lrorcd thBt a qlrqlc rhls!. to TT tbo ftgur. haa a qlr"craferenco equal to I lnqloreE a larEpr rrrt tb!,! qlo8e! rlth lts ( tbe area lnclosed by thr tro iro! of clrolet th! ru! of equ4l Lslgth. li r!o8e lolgth! l! I, uad rhlcb Jol! tb. tro polatr Er, !r. ouch Eonlcltol!a fncaer does thtB by provbg tbrt a lcDlelrolG rbloh ba! r tongths ot thr tr Lenglh, !/2 ldoloacr a l.rgar rrca culgo gad a'|l arta rlth, tt! chold tbar llry otboE rls tt! arqa. of lenBtb. I/2 lncl,oBer rlth thc E8ncoat la|oe chord. To provc tbl! oon6l- I Vslorstra8! der aE albltrarjr 3rc of r olrclc ln a pubuchod. tn 190{ rht oh b8! lcJlgth V2 aDd JolD tb6 lary .,oil lufllolr end! of th1! arq by r 1116, Drar a Dartntu arsl. ths redlt ol thc arq at tb. trq Sotzt (1€ end8. CeLl tb. crltr.l arglc bc- of Problelr I. Ht tGcD tlelc tro radll I slrd dcDotr ls dl,ffer€nt fro! tho aros b€trlr! tho arc rtd, Lt! cbord by t. !han tba ercr ' la (512) (513) B0T,ZA !l lollrlt,o|r for Eay be e)rpre.BEalby tb! equ.,tlon - [!!de! a oloEod t - e/z(-!/al/ - (.l/c)L arn s/z coa a/Z ' (0 sllt 0). g- lB !rrt!, rnd fake the derivetlyE of I' rlth respeot to O, aet this derlya- polnt! P/ , P?_ tlvs equel to zero, aud allylds out thr ooaBtart fector l!,/8. ]!tr rsd Joih thl,r glvar of tbr lolullo! -0"- o'c-eg-q-!:a-o--g!E -g, . -2 o' coaz eh _+_-.19_.12'.atn-efu coq e/z ro! ot clrclo! 0Y 64 o 4/8 coa e/z(al\ e/2 - e/2 coe o/tl . g. ! r!6r lnqlolsd fhtr ha. only one solutlon betreetr ololed by th. r7 0 end 2rI enal thls solu- tlon gtveB a loaxlrau.n. But rhen rt a clrqlc rhich. lhc centret angle ls equEl -ft to tho fLgu!€ tB a E€atcLrqle. lh€reforo trgar rrcr tb!,! a E€|!'lclrcle i.a- qloges tr1th lt6 chold a larger area thaa any r tha au[ of otlrar are of equal lsEgth, It follora that a olrolc polatr t PL. conrtructod fro! Ero such aenlcirqloE pgae€EaeEa perlneter equat to th€ l€ rbloh ba! l 8un of tho Iengths of the t\io arca Jotnlng the tro potrlt! on tbs ortglnBL - Eancock (186?-) gave a dl,eculllon of the tbcory of 7 llgigrstlala 1n a book on the oalqulus of vsrletlons rhlch rr,s ln gave lub1lEhed 1904. E6 a proof fe! of both thr neqec- lrry ard Eufflolcni condlfi,onB fgr a glolod ctrrra to Lncl-oss a llallnutr areg. Bolzs (1e5?-) giyen h8'r an erceLtent afeaueeton @{ of Probten L HI! dell,vstLon of the filBt necesBary condltlon 16 dlfferent fro!0 tllat glven th€ paraglaph '. Tbdr th rrcf tr tn thlrd of thlE s6 PROOTERT CAJ,SI'LI'g OF VARI.ArIONTI (514) (515) Sectto!, Ee leducea the dlfferentlal equatlon€ of. luler to Thlg qougrugnq! ihs equiYalent f orn lnequ.l i ty Ex?, - n?X + Er(rr y" - yr x't);o, B, Fhere E, - ^lg7=-;7)3 flll! up e dell ftold. Ths l-t a,!d therofore the dlfferential equation !6duces to (:r, :.,/1. (1r y', - y,'x,')/((;u-7jr1s . -\/A. f rrr r, rhcre )3tr ac thiE Ehors tnat ) ig diJferelt from zelo eld thst the curve rhole curve Cr, €ought Eult be a,a 8!c of a circle. lha polnts aoC lhs Euftlclsney proof of Bolze for lrobleto I LB as of a c6rtein dc f,olLovB! Drer B ctrcular alc Co of leugth 2L througb the naJZ bs reducci I)91!t8 Pt r Pr. trEer thLr drav aa albitlgry, adlrlsslble curve A(t3 , K.1tt Cr of tbs 8€I!e loagth tbrough the Ese.€ tvo lotnts. Let !o be a It 16 ql6ar tba poiat o!' the extensLoa of Ce aod Euppose tbBt it !B not a point of Cr. Cboose a! arbltrEry poi,Dt !9 o! Cr. lhea p3 i! dlffer- Th€lefo!€ lt fo cttt f!@ Po 4nd ths ata Zt sf tJlo lenAth of the erqa Poi/ plug iutegrat (I) Et !, P3 is larger tha,n the dlEtsnqe ?op3. Tbere can be drertl only the Eaoe tut€g! ons aro of a clrele Ca throuAh !o elrd tt rhich has a lelrgti 8od diffors[t f equ4l to ZS eual shlch ls d.e8cribeal i! lsenarcofa, B poaltlve seno€. Bolza b8r ahonr ln Bolzr d e lreyloua p4ragraph tb,Bt a coEgtlrence e qooplete ot!c. of a!ec6 [email protected] through !o nay bs After o deflned by the equatlon€ cutve Lncloag g r-5 . -2 ACog(T+K)e1.nfry-yor -2\ei.B (r+x)Elr r., z = -e\r1 Length the proo vhe!6 T, ,l ,K ere Lbited to tbe dolaLnl conditiona, oD I o Lla.fl , 0iKz21T , Aro. (514) ( 5r.5) IOXE',II ASD IONXETIIf, t? of !ule! to Ttrlo congrugnce Eal flsld. The B-Sunctlon of VcLerstralr for thla Droblee l! io (xr, . - - -vA. f r3 t D3, e3t n" , te r )3| )tfr cor(o'" su\ rtrcre )3lr.ogativc aad ooa(e'3 - e3 ) la not zeEo atorg tha hat the curYe rholo curye Cr, rbcs m ruppoEe tbst Cr t! dlffereat fro! Co. tho polnta colJrlgats to P6 nay be egrresacd by tbo acr! taluss leE I la es of B csrta,in deteD[ina,ttt. lbl! d6t6r:oheat for thl! probla@ hrougb the Eay bc reduecd to olEElble curYe Aej , Kj ',\3) = a 13 rln Tr(ab t? - t3 cos t3 ). iB. tret Po be a It lE qlear that tb€ detoralnartt t8 not equ.a1 to zero rho[ t ie Dot e poi.nt O / tszfl , )r. O. e[ P3 l! dl,ffer- tho!€for€ lt follova flom the tlEory of VclsrEtEaaa that th€ arcs !ol/ plus i.lrte8ral (I) atong tlr6 a,rc of the cLrole Co le gr€etar tba.[ 9r bs d!av! only the sE&e lutegral aloug any other cllrre Cr rhtch lr ad|lllrlbl€ ba! a lelgtll eDd difforent fro8 Co thlB rays tba,t the Eotu ,on to lrobleD I ia an arq of a clrcle. SoLza doe6 not extend the ns thod !o tbAt t.t tlpllar to a coEplete clrcle but Etates tha,t gucb a! extelsion ca! be Eede- Afte! one haa found the condi,tlonE necegeary that e curve inclose B larger sreB tlran any other curre of cquBl !1rrrz= -2h, length the proof toay be made cotlplete by prorLtrg that thaBc conditiong, or e oodlficatlo! of them, are EufftclEnt to insurs (51?) 3A IRooFE BY CI].gUlUs 0F VARTIIIIONS (516) curr€ at rlgbt a oaxtEr.@, or ono Eay ua6 t EeconalEethod rhich oonolEts ln llnr tho Daxlrlr provtqg tbat thsre exlst! a curve of glYeu lengtb thtcb t!t- ths lrotr€rlEotr oloees a Eariltul! erea. toncl-U (1885-) bas uE€d ths Eecond an exteatlYe ltr E€thod EA ln trovl'ng the laoporl-Eetric fhqoreD. oble end potnta, Bonnaaen detotoE a ch8ptor lD his book psJ to tuc oB the rerleblo proof of tbs lEoporlmetrlc lllgorGs. ge qeer poLgr tenBsntiel r"yr ta rtu ooordhrler e!.d th€ ac thod of Vclerrtra€E' Ilo ltateE hlr re- F9 problco of th, I rult! a! folloi!! Lst 3 b€ s olrc1e of radlu6 r' (l) tlr€ le@ ol Dtdo. oomllote onger0ble of trlingl,e! ald paLra of parallel llneg olroun€gllbeal to 3. Lel C be the ans€qobleof convex qurYeE rblch EAy b6 lnlcrtbeal l,! oae af th! fi8qres l. !e tf,oelr ths. perlEster I, 8nd thg arer t of any onc ot th9!6 curto! C one baB tho lnoqualt ty rr-I? nrz ald iho cquaLlty !lg! ts valid oBly then C l! a cLrcle. f}rle sglec! rlth hl! reBult rhlcb bAs beca degcrlbed ln S€ctlon e. lDothgr probl€@ clo!61y lelateal to the IsoperlDetllo theorr[ 1.6 tbc proble[ rblch ll8! beeE cellod th€ problsE of Dl,do. It D.y bs stst€d B! follorlt lo flld ellong 811 qurrer of gty€n langtb rhl cb Joln tro polEt! oD rB a'rbl,trary cureo on€ rhl cb, lncloses rlth tb! Brbltraly curvg & lorxitoum are4. If tha srbltrary qurrs l! a otralBht ll,ne ard tho polnt8 ale flxeal tt reduees to ProblsD I, but lf tho snd polntE ere varlBble lt reduco3 to a nsr proble!. It ls bcorn that lf the end polnts are yerlabla the llaxlDuq curre Bugt lDtsraect the elbltrary (s16) (51?) A REIABD PRO3IETI 39 .ch oonslatE !n curv€ at rlght EngIeB. If the Erbltrary curre tB a Btralght rngth rhlch ilr- lins tho roaxlBuo cureq lB B EeBlcLrcle. It lE ea8y to deduce lBsd the secotrd thg lroperlEetrlo theorelq froE thls leeuLt. lc!!e6er hEE Dade r16. ar! extenslve etuay fofl of the probletd for the ceBe of vBrl- Bble end polnts. ltrelrlIl (18s?-) haB alBo yrritten pap"t ,k Et tq the e fn] oB the point lo1ar talrgontlsl rarl.eble end ceae. lh6re i,a elEo a pBper by C, l€ ltates his re- feyf pf, 1| rhlqh eufflctent condltlong for a' raore genelal r! r, (T) the proble! of tho calcuLu8 of verlati.on! aro applled to the prot- rrtallel llnes f@ of Dld.o. gonl€x curTea t. 3etveen thc. g curv€6 c one a clrcle. fhts ed ln gectlon e. |9 IEolerlne tllc th€ pBoblgta of aaon8 all curvoa rbitrary curye , IqtlEullul crea. If € polnts are flred a ale yerlabLe lt f ths end lolrlt8 tbe albi trery (51e) BIALIOCA.trET 1,[, sl4)llotut Ialltton ot tlo! ln 0r 31, D. 6. OnlR.|'J. NXIEREfGS 15. fhcon of ,l t. Lecota Eacyclopidtc d€6 scleacer ltath;oat1qu6!, II {uT.(uno ).913), pp. 252-243. 16. ?r!!u! of (168t 2. Ptrc trnea€r, ,Tloykf ogLlc der l4cthoms.tl6chrn la6enEchaf tsn, of ftr1t.ol II A 8 !19901, pp: 608-609, 611-516r Z€rrqlo uld !rrb6, 1189-1194, tbld.. Il l' 8r (1904), pp, 535-638. 1?,ohlt.toDbl 3. Canto!, iibe! GeEhtchte der )trEtbeDatlk, (1892),pp.,I9!l?.u"e"ft loloo 0@ vo1. I 16?, 341-542, ,{I8, 695i rol.. Z, !. to5. tdLtlor tll 4. Ioltucle, Nlstolr€ dEs ltretb.iDatlqu6!, yo1. 1 (1?99), 18. KlDlcri J, p. 1l?. (16L5)' Dl (1863)' P, 5. AIfDsJr, cloek Geo&etelE froE Thsles to 3ucltd (IgB9), D.46. 19. c.llllo Ol by Elnry ( 6, ltbEi, Ul!totre des Sci,ence. ltrathdnatlques En It&l,tc, rol. g (t86o pp. ), 2SZ-A3|4. e0. Bqrrarr I, (1676), D, ?, Hesth,,A_El.BtorJr of Greek tlathEDEtics, vot. II (f9p1), pp.205-a15. 21. lcrnouul 8. Cblrlat, Eulls frorla XteEentere Degll Iooperlnetrl, ln lnr1quo6, qucEtlont ?2. 3ulor, Lr RlgualdaDtl ie Uatttiaoaiicfre- IEtlo !l a^l1l EJ'cnsntqrl, vol,. g, pp.201_910. 1?58, Dp. 1. OBIOItr OI TIII PROBIS AXD XARIY PBOOFE 25. llYl,u!, P, (r?{1), & 9, Arirtotl6, Dc Caclo, p. 28?a; !b€ rforkr of A!1Btotl. bv A. O. I EraDlletcd lnto.3ngl16h by^{t.^, goitb aDd V. D. Rosa, (izar),rr vol, 2, D.rt Z (190e), p. 28?4. 25. ghp.oDr : I0. Th. HtErort€s, book lr, p. Xngltlb Rr!olutlot lgrytlul|:trEnlt.ttoa pBton, .264, by r. R. ro1, ft 1:.sZ!_reIf), r. Of. PhlIo.opb 3d1tlo! I Il. Dlog€ner laerttus, I,lveE of fuinent phtlo!oDher!. D!.623-6 trng1t!tr tranrlerlon by youge (Igot), ;.-Jsg;----' 26. Rloortlr I 1,2. Proelu8, Lyolu!, luEztarled,Diadoclu!, lha Colorlgnterles lc r!l!!L! or r/roctu! o! thl TLbaeu! ol plato, p. 984, &lallEh Drolnrsool trrnalatlo! by Trylor (18A0), !. 450: pp. 2-1{. 13. ProoLuE, _Lycius, Eur!6E€d DlBdochuE, Co!@6nt8ry on--- look-f of luctld, p. 403; Sec atao ;Ef6r;;;; pp.206-20?. i: 40 (sr8) (5re) BIILIOCRAPEI al 10. gl!!Du,olu8, CoEoantarl€s oD tbt Dc Ciclo of Arlltotlr, Edttlon of HrlberB', ro1. Il (1.894), p. 4lar A quot.- tlolt ln ero6k vItL r tralllrtr,o! ln CeErD l-! rcfara[or 31, D. 5. 15. fhlon of Alsxr'Ddrl,r, @€'uvoI A\ef *v/Pe'ul eit ra Lqu6!, II < u t< uh oPYtip.('?orv pipli4 a't , gela t f;? t, I6. Pappuq of AlBxaDdltr,_ Psppl Alexr,ndrt-ul Colleotlone!, Pt6c (1588)r v.nlla (15S9), lool 5, pp. 83-I15r Edttt,oa l!EetrBchaf ten, of nultloh [18?8), ro1. 1, DD. 325-53?, yol. 5, DIr. alo uld ILb!, 1r89-119{. l?. ChrlltoDhar Clavlu!, In SDbrors!- Joa.Dtrl! Dc gaoro. thceattk, Bosoo Co@antarly., Vcntra (fSg:.), pf. Yol..2, 9?-9at 3tcl !. 105. Edltlqn Sslnt-o€ryrl! (1608), pD. U0-113. . (1?ee), r 18. KcDlcf, J., troya gtrraoEctrlr DollonE Yo!.arlorun, Llna (1615)' Irart IIr thlor@ IVt O!rr., Idltlon of C. trrtloh crtd (rss9), (1863), p. 607. 19. GallLro Orlll€1, Tro lllr golcnoc!, trngl,llh tranllatlon by Ecnry gr€r aEd Alfonso De sclrlo (1914), pp. .ln I tallc, 58-60. B$rcrr I.. ArshlEod€! OpcEr l{cthodo Uovr ll1ultr.tr (16?0); 1. rr (re21), p. 26s. 2r. lrnroulll, J8,De8, opsla, vql. 2 (1?{4), D. ??0. o!e!l&etrl,, trulor, haetlch€ Loorhart, Co@sntart,l Acadenr,tq Eohntl.ruD InlI8ltall! Prtlopolltana., !ot. 6 (t?SA-1?39), llftto! 1?98, !p. 142-145. tot9 25. lfvtui' P., K. Vct.!rL. AIrd. 8rtrd1. lttoothob, vol. 2 (UCl), ldlttotr l?45, D!. 136-I3?t Ccurr tEenll.tlo! : A!1! totlr PV rr..g. l(artncr, gohrad. lhd. ll!!. Abh. I rol. 3 I Y. D. Ross, (1?41), Xdltlon 1?50, pD. 160-161. 25. glaploD, 1., An llvostlgrtlon ot r 0€!613I Rulc for tha trnBll!h Relolutlon of l8opcrlDctrlc.t '192?), Probl6D! of rll Ordrrr, !. 6r. !bl,l,o.ophloal frs.aleqtlo!!, Loldon, ro1. {9 (l?55). trdltlon 1756, pp. 4-]5t tol. 50 (l?sS), Edltton U59. roDbsr!, pp.623-63I. 'll. 26, Rlcortlr 0., Dcltc tlgulr plr,n6 llopcrtletra ooatcDentt lo@antBrl6E Lc nalsllla lupcrflclc, dr.lar r tazl,onG I Xuorr Raoosl tr ,! &tgu.h DrOpuaqoll lolrntlflot q fttologlol, vo1. 19 (t??e), pp.2-14. ntary on cr ?. (s18) (521) t2 BIEIJOCBJI}SY (520) 2?. 9., De problcEBtlbu! ques vooatltor tloDcrl- 38, Bor|rBtsln' Lrlirflcr, dor K!el!€s ErtFlol!, PrinclplorrE CaLcult DLff srentlall! !t p!. I!?aarlll! ( 1?95), pp. 3er-32a. U?-I56 flttlng' A. 98. foodhou8c, Roberg,A lrcetlss o! Iloperlllsirlo Probltmr 39, (18f0), pp, lE de! trbct! rld th! Caloulu! ol VrrlBtloEr 4f-al. 5, Yol. 12 29. sLrqraon, t., ll@c!t! of CcoDetry (fe21), pp. 195-199. 40. lrdoa' 4.' Dc tlrthdnst 10. Lrgrttdr, EIcDeDts of Ga6rtrt, IDgllsh trsrrlrtloB by (1885), pp. Jobr I'lrrrr 99-10a. 41, C3rathaodor gl. golDldt, da! il6r llre vllheLr!, Sur Oelshlcht€ dcr fsopsrLErlrlo La (rorrtgn fr llt€rtu.D3, Blbllotbca. llcttr Iq tlor, yol. 22 (1901), pp. pD. 5-8. Irgro), 42. Dc Lsb€r' I 2. OIOTtrIRIC PROOTgO' TE ISOPEBIIIIIRIC lrEnBolg!@ PROINIT O' TBT CIRSII 384. 51. A.aonl|loua, Annale! De U!,th;Eatlquc! Purc! El Appllque!, 45. Lcb€Eguor S rol. 13 (1822-1825), pp. 132-139i yo1. l{ (1823-1824), Ic! dolrtnc gd8tta6! (19 t D. 516-3U. gtaller, J., Ilnfaob€ Bcretsc d6r l.ope rLue trllch€n l{. 31cb€rbecb' lbuptrat3e, Clqllers JouEnaI, rol. lS (1838), pp. lcrcl8t!, Ja 2A6-257 j gur 10 Errlluu ct EtulE'E. de! ll,gurqr daa. Vrr o1!l gultg Ic pIaD,.6ur Le spbsre, et l.c.paoc ea geuerel, !!!!., voI. 2{ (18i12), pD. 105-106, 19?, 205, 809. {5. PIBlchkG' I irbsr dlc llt E4. Idl6r, F., Vertoll! talrdlgurg drr 9trlnerilobatl D6r D€utlctl clgEeDtar-geonetrlscbrn Botrlsc far dcn gatz, dasg der ,!. 210:2314 Kr€la groaaere! tr'Iachsahal t belltz i1-! Jedc a|!d€rr 195-I99. EboBs tlgu! Bl€lslgroEss! UIlf arge!, c6tttngoJr trachllohtsa (f882), pp. 73-80; lranrtatcA tnto trenoh 46. trobeulu!, Bad prlnt€d ln th. Bullstl!. Dca gol6Dce! lfsthiloatlque 6, 0Ta1e, Sltr aocond serl€r, vol. ? (1885), !p. 19S-204. 4?. Blaacbkq, Il Stun!., R. , -3€oerk'.rngoa rrld Zurrt5r zu gt61D€rrs pa!tloul,at AufEetzon libor UErlnuI! und ltrlntDu!, Cr€llsrE Journal, vol. 96 (r8S4), pp. 50-53t l(ad,-88 urd Ilnltos, ln d6r 48. trorsy, Y., clenentarsn OeoD€trlc (19f0), pp. 31-51. \urd lu d6r traturql!ast Ulnlorakl, H., Vo1rDetr uld Obsrf],ac]ro, UathoEatlsohc 281-293, Ann8len, yot. 5? (1903), pp. 1159-46?. 49. Ltotuarul, l 3?. Cfone, C. I Oa PrlsnatotdeaE Rurfetlg I Nyt. lld8lkrtft dls lEop€rl fur !,lathsmatlk, 15 ! (f904), pp, ?3-?5. tI!ehe ZEtl ! srt.ts€ t rl, ! ( ftath. Ph. I ( (520) s21) BISIJOCRA?T t5 38. SoEnsteln, Fe1lx' UbeF dLe laopellletrtEchc Elgonsqhaft cl|rtu! lloperl.- (1905)1 atlrll! rt ds! Krelaes. Matheloatisch€ Antlelen' voI. 60 pp. U?-136. Irutrlo ProbllE! 39. trlttlng, A., lln Beltrag zuxalE operlDc tFl schsD Probl.@ lbyoll, aerler ItIr.4l-49- ln de! Ebele, Archlv der Uatba'tlatik urd 5, Yol. t2 (1907), 91. 288-290. I), pp. re5-r99. 40, Pedoa, 4,, Uno qu6atlon de Ba,i,Eum, ]iouvsll,e! lnarle ! yol. b tt|lralitlo! by D6 lbthdlatlquea, scrlea 4, I (1908)' p!, 529-535. 4L. Ciretheodory, C. asd Study, 3., Zrel Bs'e166 de! grtao!' loDtrllltrlo l|! de! der Krels ulter B1loE Rtguron 8lelohe! UEfa!8e! de! r.-22(lsor), gross!,en Inhalt lrat, ldathoEatlrohs Atualen, tol. 68 (1910), pp. 133-140. 42. De leber, Ed)-er, I,e _oontclu du oerclc et Ie-lphergl IIITBIC LrEnselgncment NBthd,Datlque!' voI. 15 (1915)' pp. 569- 384. tca Il ApllLqu?r, 45. Lsb€Egue, sur 1ea probl€@eo aea lcopddn)tma et aur I' tr825-1S24), Ic! doB8Lne! de IBrBqur-?e-? conatslta, Co&pt6! Roudu8 DcE sdancee (tsta), pp. 6. rlnotrl!qhe! a'[. 31€borbacb., I,udr{g, Ub6r eLao Ertreealclgonloha.f t d6! 1858)' pp. lcrets6s, Jahr€rberloht Der D6utachatt ltrathetlatlkcr- ltgurlr drls VoretBlgu.lg, vo1. 24 (L9I5), pp. 247-250. !38:*'stc.' 45. nlBlcl rs, w. , SeretE€ zu Satz€s'ron Brunn uDd ltrlntotrkl iber dl€ UlElEalolgeBraha.ft dss tr!€l!€s, Jahrosborloht )rr lohtlr DeE reutachon ltathostlksr.Yere lElgutrg, ro1, 25 (f914)' I 8ata, delg d€r pp. 2IO:234i Klcls uld lcrgql, &!q:, iol. 2a (1915)' DD. J sda rodorr 195-I99. itllgen l.nl tnto trrench 46. trobenluE, c., ubcr den g6!1scht9a tlFohenlrr^lt Ev8ler tr ltrthdtratlques, OvaLe, gltzun8sberlqhte, 3erltn (1915), p!. 38?-404. t0{. 47. Bl.aschke, lf., Klei.s und Kug6I (1916), pp. 1-52, ta Itcla€r. g partloula! pp. 50-31. tllal! JournEl, ItDbr, l|l drr 48. Lorey, w., Uber lEoperlEetrlEch€ Prgbl€EE ln drr Scbul€ und ln der lorEchung, Zeltsohrllt fu! llatb@atllohen u.ttd NaturqlEEenocbrf tli aheu Untorrlcht, To1. 49 (1918), pp. llrtbrDrtl Eqhr 281-293, 49. Llqhralur, Hctnrlch, Das I'lobenluqachs l(appendlcLeol uttal 4. tldsrkrttt dl€ lEoperi&e trlsche Elgen5cba,ft - des Krol seo, uatheror- tlrche Zeltrohrilt, vo1, 4 (19f9)' pp. 288-294t Dle 180- Derlnetrllcbe Elqenscbaft d€s Krelaea, .gl t zulRlborl cbto Aeth. Ph. KI. Beter Aladatnle, Uunqhen-(1919)' pp. 1t1-114. BIBLT@RATE (s22) (Ess) 50, Bonno.san,T., Sur unc .aeuoratlon do lrlnequallty tlopirtldtDlque du cerolc ct dqoonstratlo! alruna tn- 65. Dl.rr8.r, t, cqusLlte ds ltllr*orlkl, ColDtsa RenduE, vo]. 1?2 (19S1), 9D. 52-61. l0a?-1089. Dp. 6,[. gohrrrrr & 51. Borlele!, 1., Uobcr cln€ V€rlqbarfuBg d6t lloDsElactrllchcn C.a rrDlt. Itn6l.tqbhslt de! Kr.lrs. 1|r dlr trbeBG uDd url der &€sl- 99. 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Bolra, (1909). ritrtqu.! dr. 0., Yorl..utrgo ibcr Vrrtrtlonrr.obar[c lluDarlcura, Dtr. a6!-alL, a8g-r8., l9i[-{96r t98-t99. 502. 6r0-6tt. ?0. Edlrfil, LogoBr rur Ia odoul d.a vutrttoar (1910), nbtrlquer DD. 408-30{.' 71. I.rrtll, l. 8.,, ln lloDlltlrtrlo iroDt6' rttb vrrl.bla lnd l-oltrir, Alartoa! Journd ot htb[rtlor, vol. at |llar gallrr (1919 tbrlt., ), DD: ao-?8. ,! ?e. lon.lt!, !.r tuDd||! ntl dl a.lo.lo Aatla rrltrlloll, !o1. ft, DD. 542-5{1. I,ltloIS rf.yl, ?E, O., lltnralqlr.nd. !callngr.ra8.! dcr lxtrr|ar b.l rlrtcntl ru alnrE nauaD .lrt ro! l roDlltlrtriraba! problelan. Journtl ot c tlrlot tU!. Xrtho|'tlk, vol, l5a (fges), pD. ?6-?0. ?{. Yatarr t!r|r, -I(a!l, Vorlarun8aa ibar valtttlonaraohnung 154-I60. (192?), pp. t5?-e6{, l0I-502. r.ntlrl .t ?t. 8o!nc.an, t., Ea. R.tar.ao. 5! rbovc, Dtt. I06-16C. DIr. S03- r Problcltcr! Itn{Y.r!l ty