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The cyclic homology of the group rings.

Clearlyail abelian groups hâve property "h.". PROPOSITION III. If G with the cyclic set a géométrie tool not really essential for Theorem I.

MATHEMATICS

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On a simplicial complex associated with tilting modules.

PROPOSITION The géométrie reahzatwn of@N is an n-ball for ail N. Proof The resuit ""* (4 + 1 ». Aw » for m >. 0. They ail correspond to almost ...

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. 43""". Introduction in geometry of the method of analysis x.". Brahmagupta gave the correct answer namely x = mh/(m + 2). In the ...

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Analytic geometry of space

R. CRATHORNE Associate in Mathematics in the University of. Illinois. x " s

The cyclic homology of the group rings.

"right one" for the case of discrète groups since it explains the with the cyclic set

MATHEMATICS

Mathematics" in Edinburgh Jfflncy- dopcedia. and calculation and geometry and astronomy and draughts ... xn + nQ x"".

On a simplicial complex associated with tilting modules.

THEOREM Ifê is fimte the géométrie reahzatwn of(éA is an n-dimensional Hence T détermines a multiplicity-free tilting module f ®"=0 Tn.

THE SCIENCE ABSOLUTE OF SPACE

which geometry had attained.". But Euclid stated his assumptions with the of the founder of the mathematical school of ... /2~i (X".

THESCIENCEABSOLUTEOFSPACE

IndependentoftheTruthorFalsityofEuclids

AxiomXI(whichcanneverbe

decidedapriori). BY

JOHNBOLYAI

TRANSLATEDFROMTHELATIN

DR.GEORGEBRUCEHALSTED

PRESIDENTOFTHETEXASACADEMYOFSCIENCE

FOURTHEDITION.

VOLUMET1IREKOKTHENEOMONIOSERIES

PUBLISHEDATTHENKOMON

2407GuadalupeStreet

AUSTIN.TEXAS,U.S.A.

1896
154

TRANSLATORSINTRODUCTION.

TheimmortalElementsofEuclidwasal

readyindimantiquityaclassic,regardedas

Elementarygeometrywasfortwothousand

yearsasstationary,asfixed,aspeculiarly

Greek,astheParthenon.Onthisfoundation

purescienceroseinArchimedes,inApollon- ius,inPappus;struggledinTheon,inHypa- tia;declinedinProclus;fellintothelong longerunderstandwasthentaughtinArabic bySaracenandMoorintheUniversitiesof

BagdadandCordova.

Tobringthelight,afterweary,stupidcen

turies,,towesternChristendom,anEnglish man,AdelhardofBath,journeys,tolearn

Arabic,throughAsiaMinor,throughEgypt,

backtoSpain.DisguisedasaMohammedan student,hegotintoCordovaabout1120,ob tainedaMoorishcopyofEuclidsElements, andmadeatranslationfromtheArabicinto

Latin.

ivTRANSLATORSINTRODUCTION. lishedinVenicein14SJ.wasaLatinversion fromtheArabic.ThetranslationintoLatin fromtheGreek,madebvZambertifroma

MS.ofTheonsrevision,wasfirstpublished

atVenicein1505.

Twenty-eightyearslaterappearedthe

Ci/itioprincepsinGreek,publishedatBash-

in153^bvJohnHervagius,editedbySimon

Grvnactis.Thiswasforacenturyandthree-

books,andfromitthefirstKnglishtransla

MayorofLondonin15

()1.

Andevento-dav.1S

>5,inthevastsystemof examinationscarriedoutbytheBritishGov ernment,by (>\ford.andbyCambridge,no proofofatheoremingeometrywillbeac- ceptedwhichinfringesKuclidssejiienceof dinaryimmortality. nty-twocenturiestheencourage mentandguideofthatscientificthought whichUonethingwiththe-progressofman fromaworsetoabetterstate.

TRANSLATORSINTRODUCTION.v

"Theencouragement;foritcontaineda bodyofknowledgethatwasreallyknownand couldbereliedon. "Theguide;fortheaimofeverystudent ofeverysubjectwastobringhisknowledge ofthatsubjectintoaformasperfectasthat whichgeometryhadattained."

ButEuclidstatedhisassumptionswiththe

mostpainstakingcandor,andwouldhave smiledatthesuggestionthatheclaimedfor hisconclusionsanyothertruththanperfect deductionfromassumedhypotheses.Infavor oftheexternalrealityortruthofthoseas sumptionshesaidnoword.

AmongEuclidsassumptionsisonediffering

fromtheothersinprolixity,whoseplacefluc tuatesinthemanuscripts.

Peyrard,ontheauthorityoftheVaticanMS.,

putsitamongthepostulates,anditisoften editionofthetextisthelatestandbest(Leip

JamesWilliamson,whopublishedtheclosest

translationofEuclidwehaveinEnglish,in dicating,bytheuseofitalics,thewordsnot intheoriginal,givesthisassumptionaselev enthamongtheCommonNotions. viTRANSLATORSINTRODUCTION.

BolyaispeaksofitasEuclidsAxiomXI.

TodhunterhasitastwelfthoftheAxioms.

Clavitis(1374)givesitasAxiom13.

TheHarpurEuclidseparatesitbyforty-

eightpagesfromtheotheraxioms.

Itisnotusedintin-firsttwenty-eightpro

positionsofEuclid.Moreover,whenatlength used,itappearsastheinverseofaproposition alreadydemonstrated,theseventeenth,andis onlyneededtoprovetheinverseofanother seventh.

NowthegreatLambertexpresslysaysthat

Proklusdemandedaproofofthisassumption

becausewheninverteditisdemonstrable.

Allthissuggested,atEuropesrenaissance,

notadoubtoftheaccessaryexternalreality andexactapplicability <>ftheassumption,but thepossibilityofdeducingitfromtheother aumptionsandthetwenty-eightpropositions alreadyprovedbvKuclidwithoutit.

Kucliddemonstratedthingsmoreaxiomatic

byfar.Hepnve<whateverydogknows, thatanytwosidesofatrianglearetogether greaterthanthethird.

Yetafterhehasfinishedhisdemonstration,

TRANSLATORSINTRODUCTION.vii

provetheinverse,thatparallelscutbyatrans versalmakeequalalternateangles,hebrings intheunwieldyassumptionthustranslatedby

Williamson(Oxford,1781):

"11.Andifastraightlinemeetingtwo vStraightlinesmakethoseangleswhicharein wardanduponthesamesideofitlessthan tworightangles,thetwostraightlinesbeing producedindefinitelywillmeeteachotheron thesidewheretheanglesarelessthantwo couragetodeclaresucharequirement,along sidetheotherexceedinglysimpleassumptions failthemanwho,askedbyKingPtolemyif therewerenoshorterroadinthingsgeometric thanthroughhisElements?answered,"To

InthebrilliantnewlightgivenbyBolyai

andLobachevskiwenowseethatEuclidun derstoodthecrucialcharacterofthequestion ofparallels.

Therearenowforusnobetterproofsofthe

depthandsystematiccoherenceofEuclids masterpiecethantheverythingswhich,their causeunappreciated,seemedthemostnotice ableblotsonhiswork. viiiTRANSLATORSINTRODUCTION.

SirHenrySavile,inhisPraelectioneson

cherrimoGeometriaecorporeduosuntnaevi, arethetheoryofparallelsandthedoctrineof proportion;theverypointsintheElements whichnowarouseourwonderingadmiration.

Butdowntoourverynineteenthcenturyan

everrenewingstreamofmathematicianstried towashawaythefirstofthesesupposedstains

AttheendofthatyearGaussfromBraun

schweigwritestoBolyaiParkasinKhiusen- burg(Kolozsvar)asfollows:[Abhandlungen schaftenzuGoettingen,Bd.22,1877.]

4iIverymuchregret,thatIdidnotmakeuse

ofourformerproximity,tofindoutmore havesparedm\selfmuchvainlabor,andwould haveheroinemorerestfulthananvone,such

TRANSLATORSINTRODUCTION.ix

asI,canbe,solongasonsuchasubjectthere yetremainssomuchtobewishedfor.

InmyownworkthereonImyselfhavead

vancedfar(thoughmyotherwhollyhetero geneousemploymentsleavemelittletime therefor)buttheway,whichIhavehitupon, leadsnotsomuchtothegoal,whichone wishes,asmuchmoretomakingdoubtfulthe truthofgeometry.

IndeedIhavecomeuponmuch,whichwith

mostnodoubtwouldpassforaproof,but whichinmyeyesprovesasgoodasnothing.

Forexample,ifonecouldprove,thatarec

begreater,thananygivensurface,thenIam incondition,toprovewithperfectrigorall geometry.

Mostwouldindeedltetthatpassasanaxiom;

Inot;itmightwellbepossible,that,howfar

apartsoeveronetookthethreeverticesofthe triangleinspace,yetthecontentwasalways underagivenlimit.

Ihavemoresuchtheorems,butinnonedoI

GausswasstilltryingtoprovethatEuclids

istheonlynon-contradictorysystemofgeome- xTRANSLATORSINTROIHVTION. try.andthatitistin-systemregnantinthe externalspaceofourphysicalexperience.

Thefirstisfalse;thesecondranneverhe

proven.

Janus,thenunhorn,hadcreatedanotherpos

sibleuniverse;and,strangelyenough,though ourphysicalexperiencemay,thisveryyear, be-atisfactorilvshowntobelongtoBolyai

Janus,yetthesameisnottrueforEuclid.

TodecideourspaceisHolyais,oneneed

onlyshowasinglerectilinealtrianglewhose angle-summeasureslessthanastraightangle.

Andthiscouldheshowntoexistbvimperfect

mustalwa\she.Forexample,ifuurinstru- mrntsfurangularmeastiivmentcouldhe broughttomeasureanangletowithinone millionthofa-rcond,thenifthelackwereas givat a>twomillionthsofasecond,wecould makecertainitsrxisten-

ButtoproveKuclidssystem,wemustshow

angle,whichnothinghumancaneverdo.

Howeverthisisanticipating,forin17

()()it

Karkas.wasinpreciselytilesamestateas

TRANSLATORSINTRODUCTION.xi

thatofhisfriendGauss.Bothwereintensely tryingtoprovewhatnowweknowisinde monstrable.AndperhapsBolyaigotnearer thanGausstotheunattainable.Inhis *Kurzer

GrundrisseinesVersuchs,etc.,p.46,weread:

Geradensind,ineineSphaerefallen,sowaere

AutobiographywritteninMagyar,ofwhich

myLifeofBolyaicontainsthefirsttransla tionevermade,BolyaiFarkassays:"YetI couldnotbecomesatisfiedwithmydifferent treatmentsofthequestionofparallels,which wasascribabletothelongdiscontinuanceof mystudies,ormoreprobablyitwasdueto myselfthatIdrovethisproblemtothepoint whichrobbedmyrest,deprivedmeoftran quillity."

ItiswellnighcertainthatEuclidtriedhis

owncalm,immortalgenius,andthegeniusof question.Ifso,thebenignintellectualpride ofthefounderofthemathematicalschoolof notletthequestioncloakitselfintheobscuri tiesoftheinfinitelygreatortheinfinitely xiiTRANSLATORSINTRODUCTION. [ThisistheformwhichoccursintheGreek ofKu.i.:

Letusnotunderestimatethesubtlepower

ofthatoldGreekmind.Wecanproduceno

VenusofMilo.Euclidsowntreatmentof

whichStolxdevotestoitin1885aswhen continuousnumber-system;

Butwhatfortunehadthisgeniusiuthelight

withitsself-chosensimpletheorem?Wasit foundtohededuciblefromallthedefinitions, otlu-rPostulatesoftheimmortalElements?

Notso.ButmeantimeEuclidwentahead

camethepracticalpinch,thenasnowthetri 44A

Straightfallingupontwoparallelstraights

Butfortheproofofthi-heneedsthatre

calcitrantpropositionwhichhashowlong beenkeepinghimawakenightsandwaking

TRANSLATORSINTRODUCTION.xiii

himupmornings?Nowatlast,truemanof science,heacknowledgesitindemonstrableby spreadingitinallitsuglylengthamonghis postulates.

SinceSchiaparellihasrestoredtheastron

omicalsystemofEudoxus,andHultschhas publishedthewritingsofAutolycus,wesee thatEuclidknewsurface-spherics,wasfamil iarwithtriangleswhoseangle-sumismore thanastraightangle.Didheeverthinkto carryoutforhimselfthebeautifulsystemof geometrywhichcomesfromthecontradiction ofhisindemonstrablepostulate;whichexists lessthantworightanglesyetnowheremeet ing;whichisrealifthetrianglesangle-sum islessthanastraightangle?

Ofhownaturallythethreesystemsofgeom

etryflowfromjustexactlytheattemptwe supposeEuclidtohavemade,theattemptto demonstratehispostulatefifth,wehaveamost romanticexampleintheworkoftheItalian priest,Saccheri,whodiedthetwenty-fifthof

October,1733.HestudiedEuclidintheedi

tionofClavius,wherethefifthpostulateis givenasAxiom13.Saccherisaysitshould notbecalledanaxiom,butoughttobedem onstrated.Hetriesthisseeminglysimple .\i\-TRANSLATORSINTRODUCTION. oftheabsolutenecessityofEuclidssystem, hemighthaveanticipatedBolyaiJanos,who ninetyyearslaternotonlydiscoveredthenew worldofmathematicsbutappreciatedthe transcendentimportofhisdiscovery.

HithertowhatwasknownoftheBolyais

camewhollyfromthepublishedworksofthe fatherBolyaiFarkas,andfromabriefarticle byArchitectFr.SchmidtofBudapest"Aus clemLi-benxweierungarischerMathematiker,

JohannundWolfgangBolyaivonBolya."

GrunertaArchiv,Bd.48,1868,p.217.

kindlyandgraciouslyputatmydisposalthe resultsofhissubsequentresearches,which1 willhen-reproduce.ButmeantimeIhave fromentirelyanothersourcecomemostunex pectedlyintopossession <foriginaldocuments [tensive,sopreciousthatIhavedetermined whollytothelifeoftheIJolyais;buttheseare notusedinthe>ketchheregiven. liolyaiFarkaswasbornFebruary (th,1775, atBolya,inthatpartofTransylvania(Er-

TRANSLATORSINTRODUCTION.xv

dely)calledSzekelyfold.Hestudiedfirstat

Enyed,afterwardatKlausenburg(Kolozsvar),

thenwentwithBaronSimonKemenytoJena andafterwardtoGoettingen.Herehemet

Gauss,theninhis19thyear,andthetwo

formedafriendshipwhichlastedforlife.

ThelettersofGausstohisfriendweresent

byBolyaiin1855toProfessorSartoriusvon

Walterhausen,thenworkingonhisbiography

ofGauss.EveryonewhometBolyaifeltthat hewasaprofoundthinkerandabeautiful character.

Benzenbergsaidinaletterwrittenin1801

thatBolyaiwasoneofthemostextraordinary menhehadeverknown.

HereturnedhomeinT]3%andinJanuary,

1804,wasmadeprofessorofmathematicsin

theReformedCollegeofMaros-Vasarhely.

Herefor47yearsofeictiveteachinghehad

forscholarsnearlyalltheprofessorsandno bilityofthenextgenerationinErdely.

Sylvesterhassaidthatmathematicsispoesy.

Bolyaisfirstpublishedworksweredramas.

Hisfirstpublishedbookonmathematicswas

anarithmetic: legeisenrichedwithnotesbyBolyaiJanos. xviTRANSLATORSINTRODUCTION.

Nextfollowedhischiefwork,towhichhe

inLatin,twovolumes,8vo,withtitleasfol low-:

TKXTAMEN

|JUVENTUTEMSTUDIOSAM

INICLEMENTAMATHESEOSPURAE,ELEMKN-

TARISAC

|SUBLIMIORIS,METHODOINTUI-

TIVA,KVIDKNTIA

|QUEHUICPROPRIA,IN-

TRODUCENDI.

|CUMAPPENDICETKiPUci. |AuctorePro- fessoreMatheseosetPhysicesChemiaeque|

Publ.Ordinario.

|TomusPrimus.. |Maros

Vasarhelyini.1832.

|TypisCollegiiRe- forinutoruinperJosKPHUM,et |SlMEONKM

KALIdefelsr.Vist.

|Atthebarkofthetitle:

Imprimatur.|M.VasarhelyiniDie

|12Octo- bris,1829.PaulusIlorvathm.p.|Abbas

ParocliuselCensor

|Librorum.

Tomu>Serundus.

|MarosVasarhelyini.

Thetirsl\-oluniecontains:

Indexrerum(IXXXII).l.rrata

XXXIIIXXXVII).

Profvrotnhusfinmavicelegentibusno-

tandasequentia(XXXVIIIL1I). f-rrores(LIULXVI).

TRANSLATORSINTRODUCTION,xvii

Scholion(LXVIILXXIV).

Pluriumerrorumhaudanimadversorum

numerusminuitur(LXXVLXXVI).

Recensioperauctoremipsumfacta

(LXXVIILXXVI1I).

Erroresrecentiusdetecti(LXXV-

XCVIII).NowcomevSthebodyofthetext(pages

1502).

Then,withspecialpaging,andanewtitle

page,comestheimmortalAppendix,here giveninEnglish.

ProfessorsStaeckelandEngelmakeamis

takeintheir "Parallellinien "insupposing thatthisAppendixisreferredtointhetitle of title,includingthewordsCumappendice dernur28Seitenumfasst,hatJohannBolyai seineneueGeometrieentwickelt."

ItisnotathirdAppendix,norisitrefer

redtoatallinthewords"Cumappendice triplici."

Thesewords,asexplainedinaprospectus

intheMagyarlanguage,issuedbyBolyai

Farkas,askingforsubscribers,referredtoa

realtripleAppendix,whichappears,asit xviiiTRANSLATORSINTRODUCTION. should,atllu-endofthehookTomnsSecun- dus,pp.J(o^J.?.

ThenowworldrenownedAppendixby

HolvaiJanoixwasanafterthoughtofthe

Staeckelsays,buttotranslatefromtheGer

manintoLatinacondensationofhistreatise, ofwhichtheprincipleswerediscoveredand properlyappreciatedin1SJ.\andwhichwas i^iveninwritingtoJohannWaltervonKck- \vehrin1SJ5.

Thefather,withoutwaitingfor-Vol.II,

rtedthisLatintranslation,withseparate pa^in.u(1Jo),asanAp])endixtohisVol.I, where,countingapa.j^eforthetitleanda pa-v sixnumberedpai^vs,followedbytwounnum beredpa^esofErrata. twenty-fourpa^* >themoste\traordinar\ twodo/i-nia.i:esinthewholehistoryof thought!

Miltonreceivedbutapaltry/,"5forhis

l of

TRANSLATORSINTRODUCTION.xix

printingofhiseternaltwenty-sixpages,104 florins50kreuzers.

ThatthisAppendixwasfinishedconsider

ablybeforetheVol.I,whichitfollows,is seenfromthereferencesinthetext,breath ingajustadmirationfortheAppendixand thegeniusofitsauthor.

Thusthefathersays,p.452:Elegansest

ometriamproomnicasuabsoluteveramposuit; quamvisemagnamole,tantumsummeneces- saria,inAppendicehujustomiexhibuerit, brevitatisstudioomissis.

Andthevolumeendsasfollows,p.502:Nee

operaepretiumestplurareferre;quumres totaexaltioricontemplationispuncto,inima penetrantioculo,tracteturinAppendicese- quente,aquovisfideliveritatispuraealumno sultsofhisowndetermined,life-long,desper ateeffortstodothatatwhichSaccheri,J.H.

Lambert,Gaussalsohadfailed,toestablish

Euclidstheoryofparallelsapriori.

xxTRANSLATORSINTRODUCTION.

Hesays,p.4

)<):"Tentaminaidcircoquae olimfeceram,breviterexpom-ndaveniunt;ne osophiquesdelageometricetsolutiondes additionthatitisnotthesolution,thatthe finalsolutionhascrownednothisownintense efforts,butthegeniusofhisson.quotesdbs_dbs47.pdfusesText_47

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